Quantum Chemistry

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QUANTUM
CHEMISTRY

HENRY EYRING

DEAN OF GRADUATE SCHOOL

University of I r ta/j

THE LATE JOHN WALTER

PALMER PHYSICAL LABORATORY
Princeton University

GEORGE E. KIMBALL

PROFESSOR OF CHEMISTRY

Columbia I University

JOHN WILEY & SONS, INC.
NEW YORK LONDON

TOPPAN COMPANY, LTD., TOKYO, JAPAN

Authorized reprint of the edition pub-
lished hy John Wiley & Sons, Inc.
New York and London

COPYRIGHT, 1944, BY
HENRY EYKINQ
JOHN WALTER

AND
'GEOKGE E. KIMBALL,

Alt Right* Reserved.
No part of this book may be reproduced in
any form without the written permission
of John Wiley & Sons, Inc.

Wiley International Edition

This book is not to be sold outside the country
to which it is consigne-d hy the publisher

Printed in Singapore by Toppan Printing Co. (S) Pte. Ltd.

PREFACE

In so far as quantum mechanics is correct, chemical questions are
problems in applied mathematics. In spite of this, chemistry, because
of its complexity, will not cease to be in large measure an experimental
science, even as for the last three hundred years the laws governing the
motions of celestial bodies have been understood without eliminating
the need for direct observation. No chemist, however, can afford to
be uninformed of a theory which systematizes all of chemistry even
though mathematical complexity often puts exact numerical results
beyond his immediate reach. In this book we have attempted to put
into a systematic, condensed form the tools which have been found
useful in efforts to understand and develop the concepts of chemistry
and physics.

We have included a wider range of subject matter than is to be found
in other introductory textbooks in quantum mechanics and have indi-
cated this in the title. The effort has been to build a reasonably com-
plete and unified logical system on which the serious student can con-
tinue building. This has necessitated a somewhat formal mathematical
style. Much interesting illustrative material might have been added.
Including it would have lengthened a book already long enough.
Instead we have tried to present group theory, statistical mechanics,
and rate theory in a usable, if condensed, form.

The book has been written at the level of the graduate student in
chemistry. It contains somewhat more material than has been pre-
sented in the year course given at Princeton since 1931. The good
student who has mastered calculus will be able to follow the arguments.
The way will be made easier by whatever he has learned of differential
equations, vector analysis, group theory, and physical optics. Often
unfamiliarity is mistaken for inherent difficulty. The unavoidable for-
mality of quantum mechanics looks much worse on first reading than
it is. The important fact emerges from the experience of the last
twenty years that mastery of the subject is worth what it costs in
effort.

Our general indebtedness to others is great and is acknowledged in
part throughout the text and in Appendix IX. We want, in addition,

iii

iv PREFACE

to thank Dr. Hugh Hulburt for much help in the writing of the chapter
on rate processes. We also thank our many friends who, over a period
of years, have decreased the number of errors and obscurities in the
material finally assembled here and hope thatf they and others will feel
the unfinished task to be worth continuing.

HENRY EYRING
JOHN WALTER
GEORGE E. KIMBALL

CONTENTS

CHAPTER

I. INTRODUCTION: THE OLD QUANTUM THEORY

The Composition of Matter, 1. Black-Body Radiation, 1; The Photoelectric
Effect, 2. Bohr's Theory of the Hydrogen Spectrum, 3. The Old Quantum
Theory, 5. The Dual Nature of Light, 6. The Dual Nature of Electrons, 7.

II. THE PRINCIPLES OF CLASSICAL MECHANICS

Generalized Coordinates, 8. Lagrange's Equations, 9. Generalized Momenta and
Hamilton's Equations, 14. Vibration Theory and Normal Coordinates, 16.

III. THE PRINCIPLES OP QUANTUM MECHANICS

The Uncertainty Principle, 21. Wave Mechanics, 23. Functions and Operators,
25. The General Formulation of Quantum Mechanics, 27. Expansion Theo-
rems, 31. Eigenf unctions of Commuting Operators, 34. The Hamiltonian Oper-
ator^ 37. Angular Momenta, 39. ~~~

IV. THE^ DIFFJJRENJIAL EQUATIONS JC
The Linear Differential Equation of the Second Order, 48. TheJLegendre Poly-
nomiajs, 52. The Associated Legendre Polynomials, 52. The General Solution
c>f the^Associated Legendre Equation, 53. The Functions Qi, m (0) and F/, m (0, <p),
57. Reciiyr^nlFoj^n.ulae^for the Legendre Polynomials, 59. The Hermite Poly-
nomials, 60. The Laguerre Polynomials, 63.

V. THE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS
The JFree Particle, 68. The Particlejn aJkix. 70. The Rigid Rotator, 72. The
Rigid Rotator in a Plane, 75. The Harmonic Oscillator, 75.

VI. THE HYDROGEN ATOM

The Hydrogen Atom, 80. Hydrogenlike Atoms, 84. Some Properties of the Wave
Functions of Hydrogen, 85. The Continuous Spectrum of Hydrogen, 90.
VII. APPROXIMATE METHODS

Perturbation Theory, 92. Perturbation Theory for Degenerate Systems, 96.
The Variation Method, 99. The Ground State of the Helium Atom, 101.
VIIL TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY
Time-Dependent Perturbations, 107. The Wave Equation for a System of
Charged Particles under the Influence^T^iT^EHernal Electric or Magnetic
Field, 108. Induced Emission and Absorption of Radiation, 110. The Einstein
Transition Probabilities, 114. Selection Rules for the Hydrogen Atom, 116.
Selection Rules for the Harmonic Oscillator, 117. Polarizability; Rayleigh and
R,ainan Scattering, 118.

IX. ATOMIC STRUCTURE

The Hypothesis of Electron Spin, 124. Electronic States of Complex Atoms, 128.
The Pauli Exclusion Principle, 129. The Calculation of Energy Levels, 132.
Angular Momenta, 133. Multiplet Structure, 135. Calculation of the Energy
Matrix, 143. Fine Structure, 151. The Vector Model of the Atom, 155. Selec.
tion Rules for Complex Atoms, 159. The Radial Portion of the Atomic Orbitals,
162. The Hartree Method, 163. The Periodic System of the Elements, 167.

X. GROUP THEORY

Matrices, 172. The General Principles of Group Theory, 175. Group Theory and
Quantum Mechanics, 184. The Direct Product, 187.

vi CONTENTS

CHAPTER

XI. ELECTRONIC STATES OF DIATOMIC MOLECULES

Separation of Electronic and Nuclear Motions, 190. Molecular Orbitals; The
H2* Ion, 192. The Electronic States of the H^" 1 " Ion, 201. Homonuclear Diatomic
Molecules, 203. Heteronuclear Diatomic Molecules, 208i
XII. THE COVALENT BOND

The Hydrogen Molecule, 212. The Covalent or Electron-Pair Bond, 218. The
Quantitative Treatment of H 2 O, 225. The General Theory of Directed Valence,
227.

XIII. RESONANCE AND THE STRUCTURE OP COMPLEX MOLECULES

Spin Theory and Bond Eigenfunctions, 232. Evaluation of the Integrals, 240.
The Two-Electron Problem, 244. The Four-Electron Problem, 245. The
Concept of Resonance, 248. The Resonance Energy of Benzene, 249. The
Resonance Energy of Benzene by the Molecular Orbitals Method, 264.

XIV. THE PRINCIPLES OP MOLECULAR SPECTROSCOPY

Diatomic Molecules (Spin Neglected), 258. Symmetry Properties of the Wave
Functions, 261. Selection Rules for Optical Transitions in Diatomic Molecules,
262. The Influence of Nuclear Spin, 265. The Vibrational and Rotational
Energy Levels of Diatomic Molecules, 268. The Vibrational Spectra of Poly-
atomic Molecules, 273.

XV. ELEMENTS OP QUANTUM STATISTICAL MECHANICS

The Maxwell-Boltzmann Statistics, 282. The Fermi-Dirac Statistics, 285. The
Bose-Einstein Statistics, 287. Relation of Statistical Mechanics to Thermo-
dynamics, 289. Approximate Molecular Partition Functions, 292. An Alterna-
tive Formulation of the Distribution Law, 296.
XVI. THE QUANTUM MECHANICAL THEORY OP REACTION RATES
Formulation of the General Theory, 299. General Behavior of the Transmission
Coefficient, 311. Transition Probabilities in Non-Adiabatic Reactions, 326.
Thermodynamics of Reaction Rates and the Effect of Applied External Forces, 330.
XVII. ELECTRIC AND MAGNETIC PHENOMENA

Moments Induced by an Electromagnetic Field, 332. Dipole Moments and Dielec-
tric Constant, 337. The Theory of Optical Rotatory Power, 342. Diamagnetism
and Paramagnetism, 347.

XVIII. SPECIAL TOPICS

Van der Waals' Forces, 351. The Quantum-Mechanical Virial Theorem, 365.
The Restricted Rotator, 358.
APPENDIX

I. Physical Constants, 361.
I. Vector Notation, 361.

III. The Operator V 2 in Generalized Coordinates, 363.

IV. Determinants and the Solution of Simultaneous Linear Equations, 368.

V. The Expansion of , 369.

Tij

VI. Proof of the Orthogonality Relations, 371.

VII. Symmetry Groups and their Character Tables, 376.

VIII, Some Special Integrals, 388.
IX. General References, 389.

INDEX, 391.

CHAPTER I
INTRODUCTION: THE OLD QUANTUM THEORY

la. The Composition of Matter. It is now well established that
all matter is composed of a small number of kinds of particles. We
cannot discuss here the long chain of experimental evidence on which
our belief in the existence of these particles is based, but at present the
following types are known:

Electrons, with a mass of 9.107 X 1(T" 28 gram and a negative charge
of 4.8025 X 10~ 10 electrostatic unit.

Protons, with a mass of 1.6725 X 10~~ 24 gram and a positive charge
equal numerically to that of the electron.

Neutrons, with a mass very nearly equal to that of the proton but
carrying no electric charge.

In addition, the following particles have been observed in nuclear
phenomena but are not of chemical importance:

Positrons, with the mass of the electron and the charge of the proton.

Mesotrons or mesons, with the charge of the electron but with a mass
intermediate between that of the electron and that of the proton.

Neutrinos, with a mass of the same order of magnitude as that of the
electron but without electric charge.

The evidence for the last two particles is less conclusive than that
for the first four.

From the viewpoint of the chemist, matter may be regarded as being
composed of atomic nuclei, of charge +Ze and mass M , where Z is the
atomic number and e the magnitude of the electronic charge, and elec-
trons of charge e and mass m. If matter is composed only of these
particles, then it must follow that all the properties of matter are
properties of assemblies of these particles. A knowledge of the forces
between these particles and their laws of motion is at least in principle
sufficient to determine the behavior of all matter under any conditions.

It was only natural that the first attempts in this direction should
have been made with the laws of classical mechanics as laid down by
Newton. It was soon found, however, that these laws were inadequate
to cope with problems of atomic mechanics.

Ib. jJtockjBpdy Radiation. The failure of classical mechanics in
this realm was first recognized clearly by Planck 1 in 1901. By applying

Planck, Ann. Physik, 4, 553 (1901).

1

2 INTRODUCTION: THE OLD QUANTUM THEORY

classical mechanics to the problem of the equilibrium between a perfect
absorber and emitter of radiation, a so-called black body, and its radia-
tion field, Rayleigh and Jeans had found that the amount of energy
per unit volume, E v dv y in the frequency range between v and v + dv,
in equilibrium with a black body at the absolute temperature T should
be

,

E v dv = 3 dv M

where k is the Boltzmann constant and c is the velocity of light. This
law is in hopeless disagreement with the experimental facts, since it
states that E y becomes infinite as v approaches infinity, whereas experi-
mentally E v approaches zero for large values of v.

Planck overcame this difficulty only by making a violent departure
from the concepts of classical mechanics. In order to simplify the
problem he assumed that the black body was composed of harmonic
oscillators, that is, of small charged particles bound by Hooke's law
forces to their equilibrium positions. According to classical theory
such oscillators will absorb or emit radiation of their natural frequency,
the absorption or emission taking place continuously. According to
Planck's "quantum theory/' however, this absorption or emission is
not continuous. Instead, the energy of each oscillator must always
be an integral multiple of a certain " quantum " of energy, the size
of the quantum depending on the natural frequency j> of the oscillator
according to the law

c = hvQ 1-2

where e is the quantum of energy and h is a factor of proportionality,
the now famous Planck's constant, the value of which is 6.6242 X 1Q~ 27
erg seconds. Application of the same statistical methods as those used
by Rayleigh and Jeans now led Planck to the black-body radiation
formula

,
dv 1-3

which is in agreement with experiment. It will be noted that, when the
condition hv^kT is satisfied, the two expressions for the radiation
density are identical.

Ic. The Photoelectric Effect. The next successful application of
Planck's quantum hypothesis was made by Einstein. 2 It was known
that, when light falls on a clean metal surface, electrons are emitted.

2 A. Einstein, Ann. Physik, 17, 133.(1905).

BOHR'S THEORY OF THE HYDROGEN SPECTRUM

If the kinetic energy of the electrons is plotted against the frequency
of the incident light, we obtain a graph of the type shown in Figure 1-1.
Varying the intensity of the incident light at constant frequency does
not affect the kinetic energy of the emitted electrons but merely changes
the number which are emitted in unit time. The equation of the curve
in Figure 1-1 is

Kinetic energy = h(v *Q) 14

where VQ is a minimum frequency below which no electrons are emitted
and h is numerically identical with the value required by Planck to
make equation 1-3 reproduce the experimental data on the density of
radiation in equilibrium with a black body. This photoelectric effect
was completely explained by Einstein by the hypothesis that the energy

Kinetic

Energy

"0 V-*-

Fio. 14. Energy of photoelectric electrons.

of light is not spread out through the wave, as classical electrodynamics
would have it, but is concentrated into corpuscles, or photons, of energy
hv. It is further assumed that the emission of an electron from the
surface takes place only when the electron is struck by and receives
all the energy of the photon. The kinetic energy with which the elec-
tron is emitted will be less than hv by an amount hv^, the energy with
which the electron is bound to the surface. The intensity of the light
therefore determines only the number of electrons emitted; the fre-
quency of the light determines their energ

Id. Bohr's Theory of the Hydrogen SpecttliS^^erhaps the great-
est of these early successes of the quantum theory w11|fche theory of the
hydrogen spectrum as developed by Bohr. 3 Empirically, it has been
found that all the lines in the hydrogen spectrum could be represented

by the formula

/i 1\

1-5

where n and m are integers with n < m t and where n has the values 1,
*N. Bohr, Phil. Mag., 26, 1 (1913).

4 INTRODUCTION: THE OLD QUANTUM THEORY

2, 3, 4, 5 for the Lyman, Balmer, Paschen, Brackett, and Pfund series
of lines, respectively. The Rydberg constant for hydrogen, /2 H , has
the value 109,677.581 if the frequencies are measured in wave-
numbers V. (p is the reciprocal of the wavelength in centimeters.)
Bohr postulated, following Rutherford, that a hydrogen atom was
composed of a positive nucleus around which moved one electron.
Classically, it can be shown that an electron of mass m moving about

a proton of mass M is equivalent to a particle of mass u =

m + M

moving about a fixed center. Considering only the case in which the
particle moves in a circular orbit of radius r under the influence of the
coulomb forces between the two charges, we have, since the centrif-
ugal and attractive forces must balance,

where <p is the azimuthal angle. The kinetic energy T, the potential
energy F, and the total energy E = T + V are

V = - - 1-7

r

1 e 2

E = T+V = --- = -5P

2 r

Bohr now made the assumption that only those orbits are allowed

for which the angular momentum is an integral multiple of ; that is,

2?r

only those values of r are allowed for which

9 d(p nh , . , N
pr* = (n an integer) 1-8

dt 2iTc

Using this relation, the allowed vahies for the energy are
_ 11 n 2 A 2 le 2

or, eliminating r from these equations,

Classically, such a system would radiate light of a frequency equal to

THE OLD QUANTUM THEORY 5

the frequency of revolution of the electron about the nucleus. Bohr
therefore made the additional assumption that the atom could exist in
the states characterized by the .above energy values without radiating
energy, and that a transition from the state with the " quantum
number " m to that with the quantum number n could take place with
emission of light of frequency

E m E n 4 44

7 Ml

ch

Substituting the values for the allowed energy levels into this expression
gives the allowed frequencies

/I 1\

; 2) 1*12

k n" mrj

with B H = ~" /T . We see that (1-12) is.identical in form with (1-5).
ch

The calculated value of R H agrees with the experimental value within
the limits of error involved in the determination of the values of the
fundamental constants e, h, and m.

le. The Old Quantum Theory. The work of Planck, Einstein, and
Bohr formed the basis of what is now known as the old quantum
theory. This method of attacking atomic problems consisted of two
definite parts: first, the problem was solved by the methods of classi-
cal mechanics; and then, from all the possible motions of the system,
only those which fulfilled certain " quantum conditions " were kept.
These quantum conditions, for a system of / degrees of freedom, were
of the form

nh (i = l, 2-.-/) M3

where qi is any of the coordinates representing a degree of freedom
of. the system and pi is the corresponding momentum. The symbol

V means that the integral was taken over a complete cycle if the motion

was periodic; non-periodic motions were not quantized. The anal-
ogous quantum condition in the above treatment of the hydrogen
atom is given by equation 1-8. The number n on the "ight side of
equation 1-13 was called the quantum number corresponding to that
particular degree of freedom. In most cases n was taken to be an
integer, but in some problems it proved better to use aalf quantum
numbers, that is, to give n a value from the series 1/2, 3/2, 5/2 . In
this way a number (generally an infinite number) of quantized motions

6 INTRODUCTION: THE OLD QUANTUM THEORY

of the system were found. The energies of these motions were called
the energy levels of the system. These energies substituted in equa-
tion 1-11 determined the spectrum of the system. By the absorption
of light of the proper frequency the system could be raised from a low
energy level to a higher one, and by the emission of light the system
could pass from a high energy level, or excited state, to a lower one.
Although it did not prove possible to calculate the actual energy levels
of any atom except hydrogen (and other one-electron atoms such as
He 4 " and Li" 1 " 1 "), the scheme of energies could be found from the spec-
trum, and in this way a large amount of knowledge concerning atomic
structure was gained.

In spite of the success of the old quantum theory in simple problems,
it finally became evident that it could not produce correct quantitative
results in the more complicated ones. For this reason the old quantum
theory was finally abandoned in favor of what is known as quantum
mechanics.

If. The Dual Nature of Light. The essential feature of quantum
mechanics is the dualism of all the fundamental particles. At times
these particles behave like forms of wave motion; at other times they
exhibit the ordinary properties of particles. Consider, for example,
the photon. Although Einstein was forced to assume that the energy
of a light wave was concentrated into corpuscles, his success in explain-
ing the photoelectric effect in no way invalidated the old well-tested
evidence that light is a form of wave motion. It is just as inconceivable
that a stream of particles should show the phenomena of diffraction as
that a wave should suddenly concentrate its energy at one point to
knock an electron out of a surface.

Let us examine the conditions under which light behaves as a wave.
In a typical diffraction experiment, light from a point or line source
passes through a slit system and the diffraction pattern is recorded on
a photographic plate. Now it has been found experimentally that the
sensitization of a photographic plate is a quantum process as much as
is the photoelectric effect. Regarded from the photon theory, there-
fore, the experiment may be considered as the passage of a stream of
photons from the source to the plate. If it were possible to perform
the experiment with a single photon we could not possibly obtain the
complete diffraction pattern; at most one grain of the emulsion on the
plate would be sensitized. The experiment with a large number of
photons can be regarded as the experiment with a single photon re-
peated a large number of times. The diffraction pattern is then an
expression of the probability that a photon emitted from the source
will strike a given part of the plate. The waves themselves are not

THE DUAL NATURE OF ELECTRONS 7

observed in this or any other optical experiment: the actual observa-
tions of the light are always quantized, whether we detect the light with
a photographic plate, a photoelectric cell, or the human eye.

Since we always observe photons, and not light waves, we must
logically conclude that light is " really " a stream of photons. The
waves are the mathematical expressions of the way in which the photons
move. The photons of a beam of light do not obey Newton's laws of
motion but the laws of wave motion.

Ig. The Dual Nature of Electrons. It is now known, as a result
of the experiments of Davisson and Germer 4 and of G. P. Thomson, 5
that diffraction experiments very similar to those on light may be per-
formed with a beam of electrons. These experiments form a brilliant
confirmation of the hypothesis, first advanced by de Broglie, 6 and put
into mathematical expression by Schrodinger, 7 that electrons, instead
of having laws of motion similar to the classical laws, actually obeyed
the laws of wave motion in the same way that photons do.

Instead of starting with the classical motion and applying quantum
conditions to it, as in the old quantum theory, the new quantum me-
chanics abandons classical concepts almost entirely. In the new
quantum mechanics all our information concerning an electron is con-
tained in the mathematical expression of a function ty. The square
of the absolute value of this function expresses the probability of
finding the electron at a given point. The law according to which ^
changes with time is known, so that a statistical knowledge, but nothing
more, can be obtained of the position of the electron at any future time.

In the chapters to come we shall see how this new mechanics gives
all the results of the old quantum theory and goes on to solve problems
in which the old theory failed. Before starting a discussion of the
principles of quantum mechanics and its application to problems of
chemical interest, it will be of value to review, in the next chapter, the
principles of classical mechanics, since the new mechanics is expressed
largely in the terminology of the old.

4 C. Davisson and L. Germer, Phys. Rev., 30, 705 (1927).

6 G. P. Thomson, Proc. Roy. Soc. London, 117, 600 (1928).

6 L. de Broglie, Ann. phys., 3, 22 (1925).

7 E. Schrftdinger, Ann. physik, 79, 361, 478; 80, 437; 81, 109 (1926);

CHAPTER II
THE PRINCIPLES OF CLASSICAL MECHANICS

2a. Generalized Coordinates. In classical mechanics, the motion
of a particle is determined by Newton's law of motion, which may be
written in the form

d?x , d 2 y d'z

where x, y, and z are the Cartesian coordinates of the particle, m is its
mass, and f x , f yy and f z are the three components of the total force
acting on the particle.

In many problems it is not convenient to use rectangular coordinates.
In the problem of planetary motion, for example, the forces are simple
expressions in polar coordinates but are quite complicated in rectan-
gular coordinates. Again, if the motion is not free but is subjected to
constraints, such as the requirement that the particle move on a given
surface, it is usually preferable to use a coordinate system such that
the condition of constraint takes the form of a requirement that one
or more of the coordinates remains constant.

The most frequently met example of a motion subject to constraints
is the motion of a rigid body. We may regard a rigid body as a system
of particles moving in such a way that the distance between any two
of them remains constant. If there are N particles, their positions
are specified by 3N rectangular coordinates xi, y\ y z\ XN, y^ y ZN-
These variables are not all independent, since we must allow them to
vary only in such ways that the distances between particles remain
constant. If we fix the coordinates of three particles which do not lie
in a straight line, the coordinates of all the other particles are then
determined. Between the nine coordinates of the three chosen par-
ticles there are three relations corresponding to the three distances
between the particles, so that the positions of the three particles, and
hence of the whole body, are fixed by only six variables. The only
exception to this rule occurs when all the particles lie on a straight
line; then the position of the whole system is seen to be determined
by five variables. The number of variables necessary to specify the

8

LAGRANGE'S EQUATIONS 9

position of a system is known as the number of degrees of freedom of
the system; thus, a rigid body has six degrees of freedom.

In general the position of a system of N particles with F degrees of
freedom is determined by F coordinates, which we shall denote by
gi, q F . The conditions of constraint may then be expressed by the
constancy of 3N F other variables q F+ i qw- In terms of the
q's the rectangular coordinates of the particles are given by the 3N
functions

2-2

XN = ^3^-2^1 ' ' <?3Ar UN = <f>3N-

2b. Lagrange's Equations. We shall now find the equations of
motion in terms of the generalized coordinates <?,-. In the derivations
which follow, it is convenient to make a slight change of notation.
Instead of representing the Cartesian coordinates of the first particle
by xi, 2/1, zi, we shall use xi, x 2 , x%. For the second particle we use
^4) x 5) XQ, and so on. We number the forces and masses in the same
way. Newton's laws may then be expressed by the single equation

Likewise, equations 2-2 become

24

and a variation in the x's may be expressed in terms of a variation in
the q'a by the relation

dxi = 53 * dq$ 2-5

j 30;

Suppose that the system is subjected to an infinitesimal displacement
of the coordinates. In this displacement, the forces do an amount
of work

dW - /, dxi = L E/i ~dqj = L Qj dqj 2-6

i i 3 d & 3

where

* 2-7

10 THE PRINCIPLES OF CLASSICAL MECHANICS

is the generalized force associated with the generalized coordinate
qj. Using equation 2-3, the work done may also be written as

i ii

dW = Em, -g dxi = EEm, w - dg, 2-8

so that we have the relation

lq 2-9

i j ar oqj j

Since the q y s are independent variables there is no relation between
the dq's'y equation 2-9 will therefore be true only if the coefficients of
each dqj on both sides of the equation are equal. This condition gives
us the set of j equations:

w * -77 T- 1 = Qj 2-10

The left side of this equation can be simplified. By the ordinary
rule for the differentiation of an implicit function, we have

jr _ y; -^ 2-11

dt j dqj dt

so that, upon differentiating with respect to = q kj we obtain

at

= 2-12

dfa dq k

We can obtain the additional relation

^ = 2-13

dt dq k dq k

by noting that

dXi _ d dXi dqj 914.

and

= Z -r 2 * 15

at dq k j dqj dq k dt
We now write the left side of equation 2-10 as

_ d Xi dxi _, d /dxi dx\ _ dxi d ~~ %

LAGRANGE'S EQUATIONS H

which becomes, after using the relations expressed by equations 2-12
and 2-13,

d (dxi di

The kinetic energy of the system is T = iw( -) ; the set of

i \dt/

equations 2-10 may therefore finally be written

d dT dT ^

---- = Qi 2-18

dt dfa dqj 3

If the/i's are derivable from a potential, that is, if there exists a function
Y(XI XZN} such that

*"

then

.__! *!!_! 2.20

7 f - ax<dfly dqj

In this particular case, which is the most important one, equation
2-18 reduces to

^_A (r _ F)==0 2.21

dt dfa dqj

Since V is a function of the coordinates only, this equation can be
written more simply in terms of the function L = T V; it then
becomes

^-^ = 2-22

dt d<ij dqj

The function L is known as the Lagrangian function for the system,
and the set of equations represented by 2-22 as the Lagrangian equa-
tions of motion.

Example 1. Two particles of masses mi and m% are connected by
a massless rod of length R. The system moves in a vertical plane
under the influence of gravity. Discuss the motion.

Let the coordinates of the center of gravity of the system be x and
y t the y axis being vertical, and let the angle made by the line from the

12 THE PRINCIPLES OF CLASSICAL MECHANICS

center of gravity to the first particle with the vertical be <p. In terms
of these coordinates, the rectangular coordinates of the two particles are

Rcos<p

w ,
mi +

2-23

mi . mi _

x 2 = x -- - xc sin p y% = y -- - R cos <p
mi + m,2 mi + m%

The kinetic energy

is easily found to be, in the new coordinate system,

where M = mi + w 2 and I = - - ^ R 2 . The forces acting on the

mi -j- m>2

particles are, first, the forces due to gravity, which may be derived
from the potential V = Mgy, where g is the gravitational constant;
and the forces exerted by the connecting rod. Suppose that the x
component of this force on the first particle is F. In order that the
force may be directed along the rod, the y component must be F cot <p.
The law of action and reaction then requires that the components of
the force on the second particle be F and F cot <p. It is easily
verified that the generalized forces due to the connecting rod vanish.
This result is perfectly general: the generalized forces due to the con-
straints of any system always vanish and hence never need be con-
sidered. The Lagrangian function for the system is therefore

and Lagrange's equations of motion become

2-26

LAGRANGE'S EQUATIONS 13

The first two of these equations show that the center of gravity of the
system moves as a particle of mass M under the influence of the grav-
itational field. The last equation shows that the angular velocity of
the system is constant.

Example 2. A particle of mass m moves in a plane under the in-
fluence of a potential which is a function only of the distance of the
particle from a fixed point in the plane. Discuss the motion.

Let the origin of coordinates be at the fixed point; let r be the dis-
tance of the particle from the fixed point, and let <p be the angle between
the x axis and the line from the fixed point to the particle. The rela^-
tion between the two coordinate systems is therefore

x = r cos (p y = r sin <p 2-27

so that the kinetic energy is

If we represent the potential energy by V(r), the Lagrangian function is

-fKD'-'d)']-*'

and the Lagrangian equations of motion are

d( 2 dv\
^( mr Jt) =

* w (3?y + FW .o 2-30

at \fl*7 vf

The first of these may be integrated to give the result

= Pv 2-31

at

that is, the angular momentum of the particle about the fixed point is
a constant. By i
equation, giving

a constant. By use of this result may be eliminated from the second

at

V(r}ss Q 2 . 32

dr '

which may be solved when an explicit form is taken for V(r). The
second term in this equation represents the centrifugal force.

14 THE PRINCIPLES OF CLASSICAL MECHANICS

2c. Generalized Momenta and Hamilton's Equations. In many
problems it is convenient to express the kinetic energy in terms of
momenta instead of velocities. For generalized coordinates we define
the momentum pi associated with the coordinate g t - as

dL
Pi = 2-33

%

In order to find the laws of motion in terms of the Pi's and g/s instead
of the qSs and &'s, consider the differential

r . \

2-34

The coefficient of d$i is by definition p^ while from Lagrange's equations

dL d dL

T~ - T ~T = Pi 2-35

dqi dt dqi

Hence

2-36

i
If we subtract equation 2-36 from the identity

2-37

i i

we obtain

2-38

The quantity (Sp$i -~ ^) is known as the Hamiltonian of the system

and is represented by the symbol H. If H is regarded as a function
of the p's and g's, then, from 2-38,

dH = E (fcdpi ~ pidqfi 2-39

240

and by the definition of a partial derivative
dH dH

This set of equations is known as Hamilton's equations of motion.

Since the kinetic energy T is a homogeneous quadratic function of
the x'a, and the x's are homogeneous linear functions of the #'s, it
follows that T is a homogeneous quadratic function of the #'s; that is,

T = Z)ZXv#t<k> where o# may be a function of the q's but not of the

i J

HAMILTON'S EQUATIONS 15

q's. If the forces are derivable from a potential, then

dL dT
Pk = ^I" ^ ^T " Eajy& + Ea&

d#& d$k j i

2r 241

k j i k

that is, the Hamiltonian function H is equal to the total energy of the
system.

Example 3. Solve example 1 by the use of Hamilton's equations.

The generalized momenta are:

dL ^<h

PX= = M

dx dt

--*

so that

H = (A + Pi) + 7TrPl + Mgy 243

and Hamilton's equations of motion become

dx^px dpx^

dt M dt

The equations are readily seen to be equivalent to those obtained from
Lagrange's equation.

Any pi and the corresponding qi are said to be conjugate variables.
By extending this definition to include any pair of variables which
satisfy equations of the form 2-39 the concept becomes more general.
For example, if a system of F degrees of freedom is described by the
coordinates q\ qp and the time t, with t being formally treated on
the same basis as the g's, the analysis shows that for systems in which

16 THE PRINCIPLES OF CLASSICAL MECHANICS

the total energy is constant the variable conjugate to t is the negative
of the Hamiltonian function.

2d. Vibration Theory and Normal Coordinates. Many problems
of mechanics are concerned with the vibrations of a system of particles.
By a vibration we mean the oscillations of a system when it is slightly
disturbed from a position of stable equilibrium. In such a motion no
coordinate ever departs by a large amount from the value it would have
if the system were in the equilibrium position. It is convenient in
these problems to choose a system of coordinates such that all the g/s
vanish at the point of equilibrium; then all the g/s will remain small
throughout the motion.

If we were to express the coordinates of each particle by a set of
rectangular coordinates with the origin at the equilibrium position of
the particle, the kinetic energy of the system would be

Usually a more general coordinate system would be used; the kinetic
energy of the system in generalized coordinates is of the form

/f^ 246

at at

where the a t -/s are functions of the g/s, but for small vibrations it will
be a sufficiently good approximation to regard the a t -/s as constants,
with the value they have at the equilibrium position.

The potential energy may be expanded in a Taylor's series in the
q's about the point of equilibrium:

V = 7o + Z ff, + LL 55- M/ + 247

i \d<li/o i j ^ V%d<?//o

where the derivatives are evaluated at # = 0, the position of equi-
librium. The constant term F is arbitrary, and so for the sake of sim-
plicity we take it to be zero. Since g t - = is a point of equilibrium,

V must be a minimum at this point, so that ( 1=0. If we denote

\<Wo

/ 3 2 F \
the constant ( - J by &#, we may therefore represent V approxi-

\dqi dqj/Q

mately by

V = jEL&fc& 248

VIBRATION THEORY 17

Lagrange's equations for the system are then

?; = 249

If the system has F degrees of freedom, there are F of these differential
equations corresponding to i = 1, 2, F. In order to solve these
equations let us try to find a set of constants c such that, if each equa-
tion of the set is multiplied by c; and the results added together, the
new equations will be of the form

^-f + XQ = 2-50

dr

where Q is an expression of the form

Q = EMy 2-51

3

The equations which must be satisfied in order to obtain this result are

1

2-52
X i

The equations given by the equality sign on the left are just sufficient
to determine the c/s; the remaining equations will then give the hj's.
If we write the left-hand equation in the form

ZCXfltf- by)* = 2-53

it is seen that one solution is c = 0. Now by an algebraic theorem
(Appendix IV) this is the only solution unless the determinant of the
coefficients vanishes, that is, unless

|Xa tV - bij\ = 2-54

This equation may be satisfied if we choose X properly. Let Xi be one
of the F roots of this equation. Then the equations

E(Xity - &</)* = 2-55

i

have a set of non-vanishing solutions for the c/s, which are unique
except for one arbitrary constant factor.

When this set of c/s has been determined, the Vs are fixed by equa-
tion 2-52. Let these be denoted by /^ and the corresponding Q by Qi.
In the same way each of the other roots of 2-54 gives a set of hi& which
in turn determines a possible Q, so that we arrive at a set of F Q's, each

18 THE PRINCIPLES OF CLASSICAL MECHANICS

of which satisfies the equation

-~jjT + *&i = <> (* = 1, 2, F) 2-56

If we regard the Q's as a new set of coordinates, these equations are just
Lagrange's equations in the new coordinates. Because of the simple
form of these equations the Q's are known as the normal coordinates of
the system. In terms of the normal coordinates, the kinetic and po-
tential energies have the simple form:

r = - v = texdJ 2.57

(
X t - = -^ 1 are real and
dQt/

positive. But for positive X's it is easily seen that the solution of
equation 2-56 is __

Qi = Ai cos (V\it + i ) 2-58

where Ai and e^ are arbitrary constants.

If the solution is desired in terms of the original coordinates, the g t -'s
must be expressed in terms of the Q's by solving simultaneously the
equations defining the Qi's. Suppose that the result is

qi = srQy 2-59

Then the equations of motion are

qi = I^gijAj cos (V\jt + ey) 2-60

3

If all the A/s are zero except one, then each qi varies sinusoidally with
time, each with the same phase. Such, a motion is called a normal
mode of vibration of the system. Corresponding to such a mode of vi-

bration there is a definite frequency given by vj = -^ . The most

2ir

general vibration of the system can be regarded as a superposition of
the various normal modes, with arbitrary amplitudes and phases.

If the equilibrium is unstable, the above formal treatment can still
be carried out. In this case, however, at least one of the X's is real and
negative, so that the corresponding frequency is imaginary. The
motion then is not sinusoidal, since the Q corresponding to the negative
X will be an exponentially increasing function of the time.

Example 4. Three springs, whose force constants are fc, 2k, and
k, are joined in a straight line, and the ends of the system are fixed.

VIBRATION THEORY

19

The joints between the springs are balls of mass m (Figure 24). Deter-
mine the motion if the balls are set vibrating in the line of the springs.

Fio. 24.

Let the displacement of the first ball from its equilibrium position
be xi and that of the second ball x 2 . The changes in length of the three
springs are then #1, x 2 x\, and x 2 , so that the potential energy of
the system is

V == "2/uJ/i ~T" ^%\~K)\X 2 "~"~ X\) |~ "2 A/ ^ "~~~ X 2 )

= f kx 2 2kxiX 2 + %kx 2 2-61

If we consider the springs to have zero mass, the kinetic energy is

2-62

Lagrange's equations of motion are therefore

m

m

dt 2

d 2 x 2
dt 2

2kx 2 =

3kx 2 =

2-63

If we multiply the first of these equations by ci and the second by c a
and add, we obtain

2-64

In order that this be of the form 2-50 we must have

_ 1
= X

2kc 2 ) = hi

2-65

- (2/cci + 3/cc 2 )

A

There will be a non-trivial solution for Cj and c 2 only if
Xwi - 3fc 2k
2k \m - 3

2-66

20 THE PRINCIPLES OF CLASSICAL MECHANICS

5k k

The roots of this determinant are Xi = ; X 2 = . Substituting

m m

\i in equation 2-65 gives ci = c 2 . Hence if we take ci, which is arbi-
trary, to be 1, c% must be 1, and

Qi = h^xi + /4 1} ^2 = mxi mxz 2-67

In the same way we find for Q 2 the value

Q 2 = mxi + mxz 2-68

The equations of motion in the new coordinates are

The first normal mode is that in which Q 2 = 0> that is, x\ + x% = 0.
In this normal mode the balls move in opposite directions with a fre-

quency x / . In the second normal mode x\ = # 2 ; the balls move
2?r \ w

1 fk

in the same direction with a frequency A /

2x \?n

CHAPTER III
THE PRINCIPLES OF QUANTUM MECHANICS

3a. The Uncertainty Principle. A typical problem in classical
mechanics involves the finding of the values of the various dynamical
variables of a system, given the values of the p's and q's at one instant.
In general the motion of a system of / degrees of freedom requires the
knowledge of 2/ variables at some time in order to be completely speci-
fied. It is assumed in classical mechanics that it is possible to deter-
mine these 2/ variables to any desired degree of accuracy. Within the
last two decades it has been found that this specification cannot be
carried out beyond a certain limit.

Microscope
Objective

Light

Suppose that we set out to measure the position arid momentum of
an electron in order to determine its motion. The natural instrument
to use in the determination of its position is a microscope, illustrated
diagrammatically in Figure 3-1. The accuracy with which a microscope
can measure distance along the x direction is limited by the wavelength

of the light used, this limit being . At first it might seem that

2 sin c

this difficulty could be overcome by using light of a very short wave-
length. However, a new difficulty then arises: the Compton effect.

If a photon of energy hv and momentum strikes an electron at rest,

c

the photon after the collision will have an energy hv' and momentum

r /

, while the electron will have a kinetic energy %mv 2 and momentum
c

mv, where m is the relativistio mass of the electron and v is its velocity.

21

22 THE PRINCIPLES OF QUANTUM MECHANICS

The motions of the photon and electron are illustrated in Figure 3-2,
The law of conservation of energy gives the relation

hv = hv' + \m^ 3-1

FIG. 3-2. The Compton effect;

while the law of conservation of momentum gives the relations

hv hv'

= cos a + mv cos 3-2

c c

hv'

= sin a mv sin /3 3*3

c

The x component of the momentum of the electron is therefore

h

Px = - (v v cos a) 34

c

As may be seen from equation 3-1, v is less than v] that is, the scattered
light is of longer wavelength than the incident light. For our purposes,
however, we will obtain a sufficiently accurate value for the momentum
of the electron if we put / = v in equation 34, giving

p x = r (1 ~ cos a) 3-5

A

If we are to see the light in the microscope it must be scattered by the
electron into the objective, so that a must lie between the limits 90 e
and 90 + . Since there is no way of telling through which part of the
objective the light from the electron has passed, we know only that the
x component of the momentum of the electron lies between the limits

- (1 - sin ) < p x < - (1 + sin c) 3'5

A A

so that there is an uncertainty in the momentum of the electron of an
amount

&Px ~ - sin 3*7

A

WAVE MECHANICS 23

Owing to the finite resolving power of the microscope, there will be an
uncertainty in the position of the electron, as mentioned above, of an
amount

Ax ~ - 3-8

sin e

The product of these uncertainties is

Ap x Ax ~ h 3-9

which is independent of the wavelength of the light used. The attempt
to gain accuracy in position by using light of short wavelength is there-
fore defeated by the loss of accuracy in the measurement of momentum.

This difficulty is quite general. Whatever experiments are devised
to measure at the same time two conjugate variables, the limit of accu-
racy always appears to be given by a relation similar to 39, This result
has been assumed by Heisenberg 1 to be a fundamental law of nature,
and is generally known as the uncertainty principle.

3b. Wave Mechanics. In order to explain this principle a new
mechanics is necessary. This mechanics must not, like the old me-
chanics, assign a definite position and momentum to each particle, but
must allow an uncertainty in these variables. This is accomplished
by introducing functions which express, not the fact that a particle is
at a given point, but the probability that the particle is at that point.
Such functions are used in the theory of electromagnetic waves. As
stated in Chapter I, light is corpuscular in nature, at least when it inter-
acts with matter. The motion of these light corpuscles, or photons,
is governed by the electromagnetic field, which according to Maxwell's
equations moves in the form of waves which obey the usual equation of
wave motion

where c is the velocity of light and W is the amplitude of the wave.
The photon is not definitely located in any part of the wave, but the
probability of finding a photon at any point is given by the square of the
amplitude at that point. The energy E of the photon is connected with
the frequency v of the wave motion by the Einstein relation

E = hv 3-11

and the momentum of the photon is

,-J a

*W. Heisenberg, Z. Phy&ik, 43, 172 (1927).

24 THE PRINCIPLES OF QUANTUM MECHANICS

As stated in Chapter I, the de Broglie assumption that electrons were
accompanied by waves controlling their motions in the same manner
that photons are controlled by electromagnetic waves has been verified
by the electron diffraction experiments of Davisson and Germer, G. P,
Thomson, and others. By using crystal lattices as gratings and com-
paring the wavelength of the electrons as calculated from the diffraction
pattern with that calculated from the momentum of the electron, the
validity of equation 3- 12 for electron waves has been verified.

If we assume that the Einstein relation (3-11) is valid for electron
waves, the velocity of the waves is given by

v = \ v = - 343

P

The differential equation of the wave motion is then

^ a^ ^_!f^_P!^

dx 2 + dy 2 + dz 2 " v 2 dt 2 "" E 2 d# 3

where ^ is the amplitude of the wave; the square of this amplitude
gives the probability of finding the electron at a given point. If we
wish a solution that will represent standing waves (such as those on a
string fastened at both ends), we may write S in the form

y = ^ 6 -a* 3-15

where ^ is a function of x, y, and z, but not of the time t. For the
probability of finding the electron at a point we must take the square
of the absolute value of ^ in order that the probability be real and
positive. Substituting 3-15 into 3-14 gives

aV.aV.aV W. , lfi

^ + ^ + ^ = ~ir* 346

as the differential equation for ^. The kinetic energy of the electron
is T = E V, where V is the potential energy and is connected with

P 2

the momentum by the relation T = - . We may therefore write

2m

equation 3-16 in the form

This is the first of Schrodinger's equations, 2 by means of which most of
the applications of quantum mechanics are made. From the form of

2 See Chapter I, reference 7.

FUNCTIONS AND OPERATORS 25

equation 3*15, it may be noted that this equation may also be written
as

. .. _ 3-18

dx* dy 2 dz 2 - 318

which is the second of Schrodinger's equations. Although the greater
part of this text will be devoted to the solution of equations 347 and
348 for particular systems, the above " derivation " of these equations
does not furnish a very satisfactory foundation for an exposition of
quantum mechanics. We shall therefore proceed to formulate the
principles of quantum mechanics in more general terms.

3c. Functions and Operators. Because the mathematics used in
the general formulation of quantum mechanics is rather unusual, we
shall first develop the elements of the theory of operators. A function
is nothing but a rule by which, given any number, we can find another
number. Thus the function x 2 is merely the rule: take the number
x and multiply it by itself. Similarly, we define an operator as a rule
by which, given any function, we can find another function. Thus we
may define an operator | as follows : multiply the function by the inde-
pendent variable. This rule is written symbolically

|/(z) = xf(x} 349

Another operator, 8, is differentiation with respect to the independent
variable:

/(*) = /'(*) 3-20

We may develop an algebra of operators, just as we can develop an
algebra of numbers. The sum of two operators a and p is defined by
the equation

(a + p)/(z) = a/(z) + p/(x) 3-21

The product of two operators is defined by the equation

3-22

The resemblance of operator algebra to ordinary algebra is only super-
ficial. Although the operator a + p is the same as the operator p + a
by definition, the operators ct,p and pa may be quite different. If
o,p and pa are the same, a and p are said to commute. An example of
two operators which do not commute is given by the operators | and
S defined above, since

= I/O*) - "if (*) 3-23

3-24

26 THE PRINCIPLES OF QUANTUM MECHANICS

Operators are not limited to functions of one variable. We may have
such operators as 8*, defined by the equation

8*/(z, y) = r-/(s* 2/) 3-25

An important group of operators are the vector operators (Appendix II),
of which the one most frequently used in quantum mechanics is the
Laplacian operator, defined by the equation

We define a class of functions as all those functions which 6bey cer-
tain specified conditions. Thus there is the class of continuous, single-
valued functions of one variable x which vanish at x = 1 and x = 1.
It may happen that in a given class there are functions which, when
operated on by an operator a, are merely multiplied by some constant
a, or in symbols

a /(z) = a f(x) 3-27

Those members of the class which obey such a relation are known as
the characteristic functions or eigenfunctions of the operator a. The
various possible values of a are called the characteristic values or
eigenvalues of the operator. For example, if our class is the class of
functions of x which are finite, continuous, and single-valued in the

range w< x <ir, and if our operator is 8 = , the eigenfunctions

ax

are of the form e kx , where k may be real or complex, since

Jcx l*Jcx Q OQ

fj K,G d'4o

dx

and every number is an eigenvalue. The class of the functions is very
important, for, if in this example we had restricted our class by adding
the condition that the value of the function must be the same at x IT
and x = TT, the eigenfunctions would have been only those functions
of the form e imx t where m is an integer, and the eigenvalues would
have been the imaginary integers im.

In quantum mechanics the eigenfunctions which are allowed are
always chosen from the class of functions which are single-valued and
continuous (except at a finite number of points where the function
may become infinite) in the complete range of the variables, and which
give a finite result when the squares of their absolute values are inte-
grated over the complete range of the variables. If $ is such a f unc-

GENERAL FORMULATION 27

tion and ^* its complex conjugate, the last condition requires that
I \W* dr be finite, where the element of volume dr = dxi dx% dx n , the

x's being the cartesian coordinates of the particles. By appropriate
transformations dr is expressible in other coordinate systems. We shall
refer to this class of functions as the class Q.

If \l/ and <p are any two functions belonging to the class Q, and a is an
operator operating on ^ and <p, the operator a is said to be Hermitian if

J

*>*(at/0 dr = I iKaV) dr 3-29

Hermitian operators have the property that their eigenvalues for func-
tions in class Q are always real, for, if a is an Hermitian operator and
\l/ is an eigenfunction of a from class Q with the eigenvalue a, then

a\f/ = a\l/ 3-30

Taking the complex conjugate of every quantity in this equation,

o*\l/* = a*\[/* 3*31

Then

/^*(a^) dr = a I \l/*$dr 3-32

J

and

JV(aV) dr = a* fw* dr 3-33

But if a is Hermitian these quantities must be equal, so that a = a*,
which is true only if a is real.

Operators do not generally have the property expressed by the
equation

3-34

For example, if a is the operator which squares the function, that is, if

</(*) = Lf(*)] 2
then

] 2 3-35
and

a/ 2 (x) = [/i (a:)] 2 + [f 2 (x)} 2 * a[/i (*) + / 2 ()l 3-36

If equation 3-34 holds for a given operator, the operator is said to be
linear.

3d. The General Formulation of Quantum Mechanics. We are
now ready to put forward the general principles of quantum mechanics.

28 THE PRINCIPLES OF QUANTUM MECHANICS

We shall do this by making certain postulates, which, like the axiomn
of geometry, are not proved. From these postulates the whole theory
will follow, and finally we shall be led to conclusions which can be
checked experimentally. The theory will stand or fall on the strength
of these experimental checks. There is no unique set of postulates in
quantum mechanics, but the following formulation seems most con-
venient for our purposes.

Let us consider a system of particles of / degrees of freedom which
could be described classically by the values at a given time t of the /
coordinates q\ <?/ and the / conjugate momenta p\ pf. We
then state

POSTULATE I. Any state of the system is described as fully as pos-
sible by a function ^(#1 <?/, t) of the class Q. This function
^(<7i ' * * #/> i g called the state function of the system, and has the
property that ty*^ dr is the probability that the variables lie in the
volume element dr at time t] that is, qi has a value between q and
qi + dqi 9 etc. Since each variable must have some value, the total
probability must be unity, so that

g/, t)dr = l 3-37

where the integral is taken over all possible values of the q's.

POSTULATE II. To every dynamical variable M there can be
assigned a linear operator p. Since we are interested only in observ-
able quantities, which are real, we may restrict ourselves to the case
where these linear operators are also Hermitian. The rules for finding
these operators are the following:

(a) If M is one of the q's or t, the operator is multiplication by the
variable itself.

(6) If M is one of the p'& 9 the operator is where qi is con-

jugate to p^

(c) If M is any dynamical variable expressible in terms of the q's,
the p's, and t, the operator is found by substituting the operators for
the g's, the p's, and t as defined above in the algebraic expression for
M , and replacing the processes of ordinary algebra by those of operator
algebra. If there is any ambiguity in the order of the factors, they
must be arranged so that the resulting operator is Hermitian.

POSTULATE III. The state functions ty(q y t) satisfy the equation

;f; )*(.)- ~S< 5 *(.) 3-38

GENERAL FORMULATION 29

(h d \
#, : t] is the Hamiltonian operator for the system.
2iri dq /

This is Schrodinger's equation including the time.

POSTULATE IV. If the state function *&(q,t) is an eigenfunction
of the operator |x corresponding to a dynamical variable M, that is, if

= m*(q, 3-39

then in this state the variable M has the constant value m precisely;
and conversely. Such a state is known as an eigenstate of M.

COROLLARY I. If the state function is an eigenfunction of the
energy operator, the Hamiltonian operator H, with the eigenvalue E,
then ^(g, ) satisfies the equation

2^ Tq '

If equations 340 and 3-38 are to be consistent, ^(q, t) must be of
the form

~T 341

where \l/(q) is a solution of the equation

^ ) 342

Equation 342 is Schrodinger's equation for a stationary state, that is,
for a state which is an eigenstate of the energy operator.

As an example of the manner in which these postulates work let us
consider the system consisting of a particle of mass m moving in a
potential field V(x, y, z). The kinetic energy of the particle is

T = ^( P 2 x + pl + p 2 ,) 343

and the classical Hamiltonian function is

H = b (p * + p2y + p * } + V(x> y> s)

According to the above postulates, the Hamiltonian operator is

l

i dq

+ V(x,V,*> 345

, A _ [( IV , (JL AY 4. (JL 1Y1

) 2m \\2iri dx) \2iri dy) \2;rt dz) J

30 THE PRINCIPLES OF QUANTUM MECHANICS

The Schrodinger equation including the time is therefore
-h 2 '* 2

(d d d \

dx 2 dy 2 dz*/

r 2/> *> J

-h d ,

,y,*> 346

If we look for a solution of this equation in the form

we obtain the equation

5 \ T

This is identical with equation 342, so that, if ^(x, t/, z) is a solution
of equation 348 and is a function of class Q, it is an eigenf unction of
the energy operator and describes a state with the precise value E for
the energy. Equation 348 may also be written as

so that in this special case the general theory gives the same result as
the theory of section 3b.

In most of our discussions we shall be dealing with state functions
which are eigenf unctions of the energy operator for a given system.
For such state functions equations 3-37 and 3-39 are equally valid if
SF (g, /) is replaced by ^ (q), as may be seen from the relation 341 between
these functions.

From the postulated form of the operators for position and momen-
tum we can obtain an important rule for their product. Since

~ q 3-50

Zin dq ~ ' ZTTI Zin dq

and

qp^ = q I - ^ J = - q 3*51

\2iTri dq f 2iTri dq

we have the result that

h

3-52

2in
a rule first discovered by Heisenberg, 1

EXPANSION THEOREMS 31

3e. Expansion Theorems. If two functions <p\(x) and <f>z(x) have
the property that

-&
vl(x)<pi(x) dx = 3-53

for a certain interval (a, 6), the functions are said to be orthogonal in
this interval. A set of functions

such that any two functions in the set are orthogonal in the interval
(a, 6) is called an orthogonal set for the interval (a, 6). If in addition

f.

3-54

for all values of i, the set is said to be normalized.

The great importance of orthogonal functions lies in the possibility
of expanding arbitrary functions in a series of these orthogonal func-
tions. Suppose that/(rc) is any function and that it is possible to expand
in the interval (a, 6) in a series

f(x) = CI<PI(X) + C2<? 2 (x) H ----- h CfPi(x) -I ---- 3-55

where the c's are constants. If we now multiply both sides of equa-
tion 3-55 by <p%(x), where <p n (x) is a member of the set of normalized
and orthogonal functions given above, we have

f v* n (x)<Pi(x) dx-\ ----- h c n C <p*(x)<pn

*/a Ja

(x) dx -\ ---- 3-56

A1J, the integrals on the right side of 3-56 vanish because of the orthog-
onality of the functions except

hence

/ 6

>)f(x)dx 3-57

f

J a

so that the coefficients are easily found if the expansion is valid. The
question as to when such an expansion is possible is beyond the scope
of this text, but it is possible for all functions of the class Q.

We shall now show that the eigenf unctions of any Hermitian operator

32 THE PRINCIPLES OF QUANTUM MECHANICS

are orthogonal functions in the interval corresponding to the complete
range of the variables. Let the operator be a, the eigenfunctions ^i
and \f/2j with the eigenvalues ai and a 2 , that is

a\l/i = ai^i Ufa ^2^2 3*58

Now consider the integral

. dr = ai I ifoti dr 3-59

Since a is Hermitian, we have

/* r**j * r * r *
\f/ 2 Q,\t/i dr = I y/iO, \l/2 dr 0,% I yn/^! dr = a 2 I ^1^2 dr 3-60
J J J

the last equality arising from the fact that a 2 , being an eigenvalue of
an Hermitian operator, is real. Hence

/* j C * ,

Y2\yi dr = CL 2 I Yi^2 dr
J

(GI a 2 ) I ^2^1 dr =
If <*i 9* a 2 , this requires that

r = 3-62

3-61

so that eigenfunctions corresponding to different eigenvalues are orthog-
onal. When two or more eigenfunctions have the same eigenvalues
this argument breaks down. In this case, however, a set of orthog-
onal eigenfunctions can always be found. Suppose that

I i

dr == 6 3-63

If we replace ^ 2 by ^2 = ^2 ?>^i,then

f^2*^i dr = f^i dr - 6 JVfyi dr = 6 - 6 = 3-64
and

== a (^/ 2 - 6^0 = a^ 3-65

80 that ^2 is also an eigenfunction of a and is orthogonal to fo, and if
we use ^2 instead of ^ 2 our eigenfunctions are all orthogonal. The
process by which i/4 was found is known as orthogonalization.

EXPANSION THEOREMS 33

If fa and fa are two eigenfunctions of the operator |t corresponding
to the dynamical variable M, with eigenvalues mi and m% (m\ j& m^),
the state represented by \p c\^\ + c 2 ^2 is not an eigenstate of |i, since

c 2 fa) 3-66

I \l/*\l/ d

In order that I \l/*\l/ dr = 1 we must have the relation

(cf tf + c^f) (dfc + c 2 fa) dr = c? Cl + c|c 2 = 1 3-67

between the coefficients. We may give the following interpretation
to the state ^ by a postulate, known as the principle of superposition.

POSTULATE V. If $\ and fa are eigenfunctions of the operator \L
corresponding to the variable M with the eigenvalues m\ and m 2 , then
the state represented by ^ = Ci\[/i + c 2 ^2 is that state in which the
probability of observing the value of M to be mi is c* ci and the proba-
bility of observing the value ra 2 is c|c 2 . Since the set of eigenfunctions
^b ^2, * fa> of an operator JJL is a normalized, orthogonal set, we
may expand any state function <p in terms of these ^/s:

where

Ci = / i/'iV dr

Consider now the integral / ^*|i^ dr. Expanding <f> in terms of the
eigenfunctions of \i we have

/<p*W> dr = / (E4^*)Ht(Z^i) dr
/ *

3-68
t

But Smc*c t * is merely the average value of M in the state <p, so that

we have the important theorem:
THEOREM I. The average value of any dynamical variable M in

any state <p is given by / >*ji#> dr, where [i is the operator corresponding

toM.

An important example of an expansion of this type occurs when <p is
formed by operating on one of the eigenfunctions \l/j of an operator p

34

THE PRINCIPLES OF QUANTUM MECHANICS

by the operator a of some other variable, so that <p$ = a^y. In this
special case we have

where

j dr

3-70

The set of quantities oy found by expanding all the functions a^s is
called the matrix of a. The quantities a t -y are usually written in the
form of a table:

an

^32

023 "

v

3-71

If a is Hermitian

j dr = /VjttV? dr = a*
Suppose that we also have an operator p such that

3-72

3-73

and we wish to find the matrix c# of the operator y o,p. We have

so that

3-74

3-75

3f. Eigenfunctions of Commuting Operators. In most of the
problems in atomic and molecular structure with which we shall be
concerned we will be interested in several operators at the same time.
The eigenfunctions of one operator are usually different from those of
another operator, but there is a very important exception in which one
set of functions is a set of eigenfunctions of two operators at the same
time, namely, when the operators commute. In this case we have the
theorem:

EIGENFQNCTIONS OF COMMUTING OPERATORS 35

THEOREM II. If two operators a and p commute, there exists a set
of functions which are simultaneously eigenf unctions of both operators.

Let a and p be two commuting operators and let $* and $ be the
eigenfunctions of these operators, so that

o# = cktf; p$ = W$ 3-76

If we expand \l/* in terms of the ^/'s, $ = Sc^t^y we have

or E<V;(a - ;)$ = 3-77

y

The function (a a)^y is an eigenfunction of p, or is identically zero,
for, since a and p commute

= (a - ad&yitf = b f { (a - <*<)$} 3-78

and the eigenvalue of this function is the same as that of $ itself. Let
us suppose that there are no two of the &/s which are equal, that is,
the \l/j's form a set of non-degenerate eigenfunctions. Then it follows
that \(/j and its multiples are the only functions which satisfy the equation

M = btf 3-79

The function (a a)^y must therefore be some multiple of \f/j ; that ia

(a - Oi)$ = 0tf# 3-80

where 0# is some numerical constant. Substituting this relation in
3-77, we have 0^7^ == 0. If we now multiply by ^* and integrate

over the variables we have

Z<V^ /V* Vy * = c^ft = 3-81

y J

so that either c& or g^ is zero. If g^ is zero, then

(a - ad$ = 0; a^ = a$ 3-82

and ^/ is an eigenfunction of a with the eigenvalue a t -, as well as an
eigenfunction of p. If cu = 0, then

dr = Ec;i * Vy ^ = c fci = 3-83

and we see that ^? is orthogonal to ^|. Since the set of functions $
form a complete set, no function can be orthogonal to all of them;

36 THE PRINCIPLES OF QUANTUM MECHANICS

there must be at least one value of k for which g k i = 0. We thus see
that when there is non-degeneracy every eigenfunction of (3 is also an
eigenfunction of a.

If, as usually happens, there are several ^/s with the same eigen-
value, the above argument does not hold. Suppose, for example, that
^i and \l/2 have the same eigenvalue. Let us then make the following
change in the $'&. We write

+
and construct another function

with the constants d\ and d 2 chosen so that $ is orthogonal to $.
If there are other degeneracies we adopt an analogous scheme. We now
carry out the analysis using the t/i''s. Owing to the degeneracy of
$i and $, we have in place of 3-80 the two equations

(a - a,-)*i' = gut" + 0*2$

v v 3 ' 80a

(a a-)^ 2 = huti

where gn = / ^i*(a a^^'dr, etc. It is now possible to form

linear combinations of $ and \l/\ which are eigenfunctions of a, that
is, we can find constants kn and ki2 such that

(a - a,) {fcutf + k&ft} = Ai{kid% + k&$} 3-84

By multiplying the first of equations 3-80a by kn and the second by
ki2 and adding, we find that kn and fc; 2 must satisfy the relations

AJCH = kngn +

3-85

When these relations are satisfied, the functions (kn\l/\ + ^2^2') are
eigenfunctions of a with the eigenvalues A f * + a*. They are of course
eigenfunctions of |3 with the eigenvalue 61.

Theorem II has thus been proved. The analogous theorem holds
for the case of more than two operators; we shall use this theorem
without proof.

We may also prove the converse of this theorem:

THEOREM III. If there exists a complete set of orthogonal functions
\l/i which are eigenfunctions of two operators a and p, then a and p
commute.

Let us expand any function <p in terms of the &'s, and then operate

THE HAMILTONIAN OPERATOR 37

on <p with (o,p pa). We have

(ap - paV = (ap - pa)Lc^ = I>(a& - 6*a)^ = 3-86

Since <p is an arbitrary function we see that a and p commute.

The physical interpretation of these theorems is that, if two physical
quantities have operators which commute (as do the operators for the
energy and the total angular momentum of a system), then it is pos-
sible to have states of the system in which both variables have definite
values. Conversely, if it is possible for two variables (physical quan-
tities) to have definite values for a complete set of states at the same
time, then the corresponding operators commute.

Another theorem which we shall find useful is:

THEOREM IV. If p is an operator which commutes with an operator
a (where both are Hermitian), and \l/\ and t 2 are eigenfunctions of a,

then the matrix element / \l/* p^ 2 dr vanishes unless a\ = a 2 , where

ai and a 2 are the eigenvalues of fa and fa.
To prove this theorem consider the integral

dr = a 2 $\ p^ 2 dT 3-87

Using the fact that a is Hermitian,

dr = Via(fc) dr = J* (p^ 2 )(aVi) dr

dr 3-88

Therefore (ai - a 2 ) *P*2 dr = 3-89

so that / ^* p^ 2 dr = if a x ^ a 2 .

3g. The Hamiltonian Operator. We have seen that the Hamil-
tonian operator for a single particle is, in rectangular coordinates,

m

3 ' 90

but we have not as yet proved that this operator is actually Hermitian.
Such a proof is necessary, for if we had written the classical Hamil-
tonian in the form

38 THE PRINCIPLES OF QUANTUM MECHANICS

which is algebraically equivalent to equation 344, and then substituted
the operators for the p's, we would have the quite different result

Since H = H*, the condition that H be Hermitian is

fff <p*H*dxdydz= f f f tH<p*dxdydz
provided that <p and \f/ belong to class Q. This condition requires that

r) 2

<p and \l/ vanish at infinity. Consider the operator . We have

** * dy dz =

r r r VcM
_ i i i

J J J_oo a^ to

3^

Since <p* vanishes at infinity, and is finite or zero, the first integral on

dx

the right is zero, hence

In the same way, we find

d 2 d 2 d 2

so that the operator %, and analogously the operators % and %, are
dx dy oz

Hermitian. The Hamiltonian operator 3-90, being the sum of Her-
mitian operators multiplied by real constants, is therefore Hermitian.

n

For the operator , we find
dx

r r r 00 a* r r 00 T"" 1 "*

/ I / v*-rdxdydz= I I <p*tdydz\

/// -oo OfX t/ /_oo Ja;=~QO

-fff t dxdgdzytfff 4> dxdydz 3-93

J J J -<t> dX J J J -a> OX

ANGULAR MOMENTA 39

\

so that the operator is not Hermitian. (From the above equations,
dx

\

however, it is apparent that the operator i is Hermitian.) The

ox

1 1

operator -- will likewise be non-Hermitian, so that the Hamiltonian
^ xdx

written as 3-92 is not valid from the quantum-mechanical viewpoint.
Written in vector notation, the Hamiltonian operator 3-90 is

V 2 (x, y, z) + V(x, y, z) 3-94

m

Rectangular coordinates are not always the most convenient coordi-
nates to use. Since the Laplacian operator is invariant under a trans-
formation of coordinates (Appendix III), the Hamiltonian operator
in an arbitrary coordinate system will be

H = - V 2 & ij, f ) + F& 77, f) 3-95

O7T M,

where , r;, f are the new coordinates. The transformations from rec-
tangular to various other important coordinate systems are given in
Appendix III, as are the expressions for the Laplacian operator V 2 and
the volume element dr in these coordinate systems.

3h. Angular Momenta. Of almost as great importance as the
Hamiltonian operator are the operators connected with angular momenta.
For a single particle, the angular momentum about the origin is
M = r X p, where r is the distance from the origin and p is the linear
momentum, or, in terms of its components M x , M yy and M z in rectan-
gular coordinates,

M x = yp s - zp y

My zp x xp z ' 3-96

M z = xp y - yp x

Replacing the p's by the corresponding quantum-mechanical operators,
we obtain the operators for the components of angular momenta:

h d d

3-97

40 THE PRINCIPLES OF QUANTUM MECHANICS

The total angular momentum is, of course,

M = \M X + jM y + kM,

We shall never have occasion to use M itself, but only its scalar product
with some other vector, or its square:

M 2 = M* + M* + Mf 3-98

The angular momentum operators are usually expressed in spherical
coordinates. By means of the transformations given in Appendix III
the operators in spherical coordinates are readily found to be: 3

8 We may illustrate this calculation for M x . From the transformations
x = r sin 6 cos <?
y = r sin sin <f>
z r cos

we obtain the reverse transformations

2 - 2 2 2

cos =

! + * 2

tan v? = -
x

from which we calculate the partial derivatives:

dr & * -

~- = cos r- = sin sin ?>

dz dy

^ ~ 8m ? dQ _ cos sin <?

dz r dy r

d<p d<p cos <p

dz dy rein0
Therefore

h r . . /dr d , do a , dv d\

I r Bm Biu<f>[ - --- ---- --- I

27ri L \dzdr dz do dz d<f>/

dr d

I (r sin sin y> cos r cos sin sin <p)
27rt L or

+ (~ sin 2 sin <p - cos 2 sin *>) ~ -f- (~ cot cos *) ~-

00 dy>J

A r . a ^ ai

-. I sm ^> - -- cot cos <p I
2iri L de * d<pj

ANGULAR MOMENTA

* - d
cot * sm ^ "

( d

~- V cos <? ^ ""
rt \ d0

3-99
M = ~

9 27TI d<f>

a

The commutation rules for M x , M v , and M^ are most readily found
from the expressions in rectangular coordinates

3 ' 100

By

^2 \

3 ' 101

dx dz dx dy dz* dy dy dz

so that

~ . i ~ ~ ^ ~ i ~ 3*102

^&7T

Similarly, we find

*^M,

3-103

If we now consider the commutation of M 2 and M z , we see immediately
from the expressions for the operators in spherical coordinates that

M 2 M, - M,M 2 = 3-104

so that M 5 * commutes with M z . Because of the equivalence of M x ,
M y , and M a , M 2 will also commute with M, and M y , as may be proved
explicitly from the expressions for the operators. These commutation
rules will be shown later to hold lor systems of particles as well as for a
single particle.

From the commutation rules derived above, it is possible to deduce
the most important properties of angular momenta. Since M 2 and

42 THE PRINCIPLES OF QUANTUM MECHANICS

M 2 commute, it is possible to find a set of functions which are eigen-
fimctions of both operators simultaneously. We shall denote these
eigenfunctions by YI, m , where Fj, m satisfies the equations

3-105
3406

where the subscripts I and m identify the eigenvalues ki and k m asso-
ciated with Yi t m . Writing M 2 in terms of its components, 3-105 be-
comes

(M* + M 2 + M 2 ) Y lt m = ktY lt m 3-107

and applying M to equation 3-106 yields

M*F,, w = AF,, TO 3-108

Subtracting 3-108 from 3-107 gives the result

(M 2 + M 2 )F Z , m = (ki- k*JY lt m 3-109

so that Yi t m is an eigenfunction of (M 2 + M 2 ) as well as of M 2 and M z .

tet us now write Fj, m = X) c^-, where the &'s are eigenfunctions of

t

M x with eigenvalues m x i. Then

M 2 F Z , m = Zcim'i^ = EE an, P F n>p 3-110

1 n p

where

= f (

t/

Mdr = Zc<X 3-111

i

Similarly, we can find for M 2 F^ m the expression

f pF.p; where 6n, P = EMi|X, 3-112

n p t

Comparing 3-109 with the sum of 3-110 and 3-112, we see that

ki - kl = E (\ Ci \ 2 m 2 xi + |<fc | 2 mi) 3-113

Since M^ and M y are Hermitian, m 2 z - and w 2 % - are positive; the right
side of 3-113 is therefore positive, so that

ki > k 2 m 3-114

We now make use of the commutation rules for M x , M y , and M z . It
is easily verified that

3415
= (M. - <M,)M, - 3-11S

ANGULAR MOMENTA 43

Operating on Fj, m with both sides of 3-115 gives

(M, + iULy)yL z + j^Y lt m

+ iM,) F,, m } 3417

so that (Ms + iM y )Yi, m is seen to be an eigenfunction of M 2 with the

eigenvalue k m + or zero. Since M 2 commutes with all the operators

2?r

in 3-115, (M x + iM y ) F/ t m is still an eigenfunction of M 2 with the eigen-
value ki. In an analogous manner we see that (M^ iMj,)Fz >m is

an eigenfunction of M z with the eigenvalue k m or zero, and of M 2

2?r

with the eigenvalue k\. We may therefore obtain a whole series of
eigenfunctions of M 2 and M 2 , all having the eigenvalue ki for M 2 but
having the eigenvalues

'y k m , k m , k m , k m + , fc m + , 3-118

for MZ. This series must terminate in both directions, since 3-114
states that the square of the eigenvalue for M z must be less than ki.
Let us denote by k m t the lowest and by fc m // the highest allowed eigen-
value of M z , and the corresponding eigenfunctions by Yi t m r and Fj, m >/.
(M x + iM^Fj, m ff should be an eigenfunction of M 2 with the eigen-

h

value k m " + or zero, and since by hypothesis k m u is the highest
2iir

eigenvalue we must conclude that

(M x + iM y )r. ro // = 3-119

For similar reasons, we must have

(M x ~iM y )F z , m r = 3-120

Operating on 3-119 with the operator (M x iM y ) gives

{M 2 + M 2

+ M 2 - M.r, iW ^ = n 2 - M 2 -

f k i - k 2 m >< - k m ,,) Yi, m n

3-121

44 THE PRINCIPLES OF QUANTUM MECHANICS

so that

2 h

Jf. - Tf* .. JL, Tf mt Q.I 99

n> l "- ivffi'' i^ rt A/JH// Q-J.^/^

By operating on 3-120 with (Ma- + iM y ), we find

1U_ l2 Z . Q 1OQ

A/i *~~ A/jyj/ '* / tU f O A^5O

In order that equations 3-122 and 3-123 will be consistent, with k m n
> k m ') we must have k m == k m t. According to 3-118, k m n must be

greater than k m t by an integral multiple of so that k m n must be of the

form - , where n is a number in one of the series 0, 1, 2, 3, or |, f , f , .
Since we have so far put no particular significance on Z, and since k m rt
depends on I only, we may put I = n, so that k m *r = . Then

2T

by 3-122, or

3-124

The possible values of k m are then

a, _,) ... s.125

27r' 27r' 2ir

and we may specify m by

*,-?, Z>m> ~Z 3-126

ZTT

If Z is an integer, m is also an integer; if I has half-integral values, w
likewise has half-integral values.

We may now write equations 3-105 and 3-106 as

M 2 Y ltm = mYi. m 3-127

ZTT

As these results were derived entirely from the commutation rules, they

ANGULAR MOMENTA 45

will be valid for any operators whose commutation rules are similar to
those for angular momentum.

With the aid of the operators (M* iM y ) we may obtain the form
of Yi t m for the case of a single particle where the explicit form of the
operators is known. We have seen that (M x iM v ) Fj , m is an eigen-
function of M 2 with the eigenvalue I (I + 1)^2 an( ^ f ^* with the eigen-
value = (m 1) . The eigenfunction (M x iM y )Fj, m

2?r 2?r 2?r

is therefore related to the eigenfunction Yi t m-i by the relation

(M, - <M,,)y| f m = NY lt ^ 3-128

where N is a constant numerical factor. In order to determine N we
use the requirement that YI, m be normalized. We therefore multiply
3-128 by its conjugate and integrate over the coordinates. This gives

- NN*J*Yf. n^iYi. m _i dr = N 2 3-129

since we may choose our normalizing factor to be real. M x and M y are
Hermitian, so that the first integral may be written as

N 2 = f{ (M x - OILJYi. m }*M x Y t , dr

- if{ (M, - iM,) Y,, m ] *M,r lf dr

i. m M:{ (M. - iM v ) Y t . m ] * dr

+ M* 2 + i(M,*M* - MX*)} Yt m dr 3-130

Taking the complex conjugate of this equation, and making the obvious
substitutions, we have

2 - Mj + M. Y l>m dr
ra { Z ( Z + 1) - m (" - 1)} 3-131

46 THE PRINCIPLES OF QUANTUM MECHANICS

so that

(M, - m v )Yt, n = Vi(Z + 1) - m( m - 1) F,, ,_,

-m + l)Yi.^i 3-132

61T

In a similar way, we find

^ _&_

= V(Z + m + 1)(Z - m) YI m+l 3-133
2?r

Let us now assume that Fz, m (0, <p) may be written as the product
QZ, m(^) ^m(^). Using the operator for M 2 in spherical coordinates,
we then have

h d <

3-134

AlTl O(p Z>TT

or

<Mv) - iw* ro (^) 3-135

d<^3

The solution of this equation is

*(?) - JVc i7n ^ 3-136

In order that $ m (<p) be a single- valued function of position, we must
have

or = 6 ^+2*> 3-137

This requires e 2?rmi to be unity, which is true only if m is an integer.
The normalization condition

pi*

= N 2 / ^ =
/ o

3-138

gives the value . for N

V27T

In order to calculate 9z, m (8) we apply 3-119 in the form

(M, + tM If )F|, i = 3-139

In spherical coordinates the operator (M x + tMj/) is

(M, + tM v ) - A f <?* ~ + te r cot 4-} 3-140

^7T \ OV u<P/

ANGULAR MOMENTA 47

Equation 3-139 is therefore

A JL 6 " 1(1 + i cot 1) 8,, ,(0)6*4 = 3441

2jr V27T IV0 <w j

which reduces to

d

Ta e *. *(0) ~ Z cot Qi, i(0) = 3-142

dO

or

^l = Z^ 3-143

/, j(0) sin 6

Integrating both sides of this equation with respect to 0, we readily find
the solution

6j. j(0) = Nusm l e 3-144

The normalization factor NU is given by

1 /* T / lT

r-2 = I sin 2Z (9 sin d0 = -sin 22 cos 0]j + 2i I cos 2 sin 21 " 1 d0
^VH /o *^o

sin 21 - 1 d0 3445

26 + 1

If the integration is repeated I times, we obtain

- 2) 2

. 3-146

The normalized solution of 3-142 is therefore

8,, ,0) = (-1)' Jifi^-iil-ij sin'fl 3-147

where the factor (I) 1 has been introduced for our convenience later.
From the expression thus obtained for Yi t i, Yi, m may be found by
repeated application of 3432. The result obtained in this way is
equivalent to the following compact expression, as will appear in the
next chapter (section 4e).

-i)' R

2*irV-

M

where \m\ is the absolute value of m.

CHAPTER IV
THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS 1

4a. The Linear Differential Equation of the Second Order. We

have seen that the Hamiltonian operator is a second-order differential
operator, so that the equation H^ = Ety is a second-order differential
equation. In order to find the eigenfunctions of H we must therefore
develop the technique of solving differential equations of this type.
For the present we shall confine ourselves to the case of one inde-
pendent variable. The general equation to be solved is then of the form

d 2 y dy

2 "f" P( X ) l~~ Q\%jy ^ 0' 4*1

where p(x) and q(x) are given functions of x.

Suppose that, in the neighborhood of the point x = # > y can be
expanded in the form of a power series

y = O>Q + Q>\ (x XQ) + #2 (X XQ) -)- 4'2

Taylor's theorem then states that

1 //72,\

: 4-3

d 2 y d 2 y dy

o, we find an expression for -7-5 in terms of
dx 21 F dx 2 dx

and y. If we differentiate 4-1, we then find an expression for -7-3 in

d 11 di/
terms of % , , and y. By repeated differentiations we may therefore

(tX CLX

find any derivative of y in terms of the lower derivatives. Now suppose

that at the point x = we are given the values of y and -7- . The

dx

differential equation then gives all the higher derivatives, and from
these the constants a , ai, a 2 in the series 4-2 are determined by

4-3. Thus from the values of y and at one point we may determine

dx

1 Some readers may prefer to delay the study of this chapter until they require
the specific results for the solution of the equations in later chapters of the text.

48

LINEAR DIFFERENTIAL EQUATIONS 49

the whole function throughout the interval in which the series
converges.

In practice this method usually turns out to be rather clumsy. If,
however, the differential equation can be put in the form

Q(x) ^ + R(x)y = 44

where P(x\ Q(x\ and R(x) are polynomials in x, the solution is much
simpler. We expand y as a power series in x

y = a + aix H + a v x v -\ 4-5

and differentiate the series term by term, obtaining

fy

dx

dv

f- = ax + 2a 2 x + ... + ( + l)a v+l x' + 4-6

uX

2 -2a 2 + 3-2a 3 a; H ----- h (v + 2)(i> + l)a v+2 x v H ---- 4-7

If we substitute these series in the differential equation, the coefficient
of every power of x must vanish. Putting the coefficient of x v equal
to zero then gives a relation of the form

a v = cia v ^i + c 2 a v -2 + + c k a v _ k 4-8

where the c's are constants. The number of c's occurring in this
Expression will depend on the form of the differential equation. Such
a relation is known as a recursion formula; by means of it the whole
series can be found when the first few terms are known.
As a simple example of this process consider the equation

d 2 y

where P(x) = 1, Q(x) = 0, R(x) = -1. If we put

y == a + aix + a 2 x 2 -\ ---- 4-10

and substitute in the equation, we find

(2a 2 - OQ) + (3-2a 3 - ajx H ----

+ l(v + 2) (p + lK+ 2 - aftf + 4-11

Equating the coefficient of x v to zero gives

or

O x. rr^; " ^ "? TT

2)(v + 1) v(v - 1)

50 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS
If we take ao = ai = 1, we obtain the particular solution

= e* 443

*L + . . . +

3-2 v\

which is seen to satisfy the differential equation 4-9.
If we attempt to use this method on the equation

4 ' 14

x dx
we find

= -

+ [("* - iK + 0^2]*' + 445

so that the recursion formula is

In order that the constant term and the coefficient of x vanish, we must
have a = &i = 0. The recursion formula then requires that all the
remaining a's vanish, so that we get no solution at all. A little con-
sideration shows why this has happened. If we write equation 4-14
in the form of 4-1, we have

=

When x = 0, p(x) and q(x) become infinite, so that % cannot be

found from this equation. The point x = is called a singular point
of the equation. If it is possible to write a differential equation in
the form

~2 >

* 2 Tl + P'(*) -T + JWV = 4 ' 18

ax ax

where p f (x) and q'(x) are finite at x = 0, the point x = is called
a regular point of the equation. A singular point which is not regular
is called an essential singularity.

In the neighborhood of a regular point a differential equation can
usually be solved by the following method. Instead of starting the
power series with a constant term, we use a series of the form

y = aQ x + a& l + azX H ---- 4-19

LINEAR DIFFERENTIAL EQUATIONS 51

where ao j& and L may have any value. Then

^ = La^ 1 + (L + l)a^ L H ---- 4-20

ax

^-f = L(L - l)^* 1 "" 2 + (L + IJLais*- 1 + 4-21

dx

If we substitute these series in equation 4*14, we find
= (L 2 - ^]a Q x L + { (L + I) 2 - l

+ {[(L + v) 2 - ila, + a,_ 2 }z L +" + 4-22

which leads to the series of equations

(L 2 - i)oo =

[(L + I) 2 - J]ai =

{[(L + -) 2 - JK + a^ 2 } - 4-23

The first of these is called the indicial equation. Since a ?* we
must have L 2 = j, or L == ^. The other equations then determine
the other o's in terms of OQ. There are thus two solutions :

4-24

4-25

If we are interested only in the solutions of a differential equation
which belong to the class Q, it is essential to investigate the behavior
of the solutions at all the singular points of the equation, for if the
exponent L is negative or fractional the function y is either infinite or
multiple-valued and therefore cannot be of class Q.

Infinite series are at best awkward things with which to work. For
example, it is very difficult to derive the properties of the function sin x
from the series

x 3 x 5 x 7
8** = *--+--- + ... 4-26

whereas the properties are easily derived from the ordinary trigono-
metric definition. For this reason the various functions which are met
in quantum mechanics are defined by some other definition than the
power series derived from the differential equation, and are then iden-
tified with this series.

52 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS
4b The Legendre Polynomials. Consider the equation

(1 - x 2 ) ~- + 2nxy = 4-27

ax

If we write this in the form

dy ^ _ 2nxdx 4 2g

y (1 x 2 )

it may immediately be integrated to give

y = c(l - x 2 ) n 4-29

where c is any constant. If we differentiate equation 4-27 (n + 1)
times, the result is

which may be written as

(1 x 2 ) g 2x + n(n + l)z == 431

where

z ._ -. c (1 _ x ^n 4.32

rfa; n do: n

Equation 431 is known as Legendre's equation. The particular
solution

is called the Legendre polynomial of degree n. [Po(x) is defined to
be unity.] It is possible to show that these functions form a system
of orthogonal functions in the interval 1 < x < 1, and also that these
polynomials are the only functions of class Q which satisfy equation
4-31 in this interval. As these functions are special cases of a more
general set of functions which we shall now discuss, we shall prove
these statements only for the general case.

4c. The Associated Legendre Polynomials. If equation 4-27 is
differentiated (m + n + 1) times, we obtain the equation

a -

n+l)(n-m) = 4-34

THE ASSOCIATED LEGENDRE POLYNOMIALS 53

which may also be written as

n d z . N dz

dx ax

where

tjm+n^ tfn jm+n

L - x 2 ) n 4-36

Let us now put

z = u(l - z 2 )""^ 4-37

Then
dz

dx '1 -
d 2 u

2mx du / m m(m + 2)x 2 \ 1 2 -2

r^^^ + Vi^ 1 ^ (i-x*) 2 ) u \ (l ~ x >

so that the differential equation for u is

- 4.38

This equation is known as the associated Legendre equation, and the
function u, denoted by u = P%(x), is called the associated Legendre
polynomial of degree n and order m. From equations 4-36 and 4*37,
we see that

P(x) - (1 - x^z = (1 - x*)*~ P n (x) 4.39

or, using 4-33,

m
*\% n+m

;2 - 1)n 44

It is apparent that P2C*0 = P n (x). Also, since P n (x) is a polynomial
of degree n, PnO*0 is zer if w > n.

We shall now show that the functions P(x) and Pf(x) are or-
thogonal in the interval l<x<lifZ^n. The equations arising
satisfied by these functions may be put in the form

54 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS
If we multiply 441 by Pf and 442 by P% and subtract, we obtain

dx n dx

+ {n(n + 1) - 1(1 + 1)}P%PT = 443

Upon integrating between 1 and +1, the first term vanishes because
of the factor (1 x 2 ), so that, if n 7^ I, we have

r +i

I F%(x)PT(x) dx = 4-44

'-i

In order to normalize these functions, let us now consider the integral

/-t-l
iKwl ^

If 4-27 is differentiated m + n times, and the result multiplied by
(1 - a 2 )" 1 "" 1 , we obtain

dx

jm Ip

+ (n + m)(n - m + 1)(1 - a; 2 )- 1 -^ = 4-46

which is equivalent to

' di

= - (n + m)(n - m + 1)(1 - a; 2 )"- 1 ~? 4-47
Substituting this result in 4-45, we find

/

^ (i-a-*) i

- (n + m) (n - m + 1) ( [FT 1 (*)1 2 *c 4-48

J-i

THE ASSOCIATED LEGENDRE POLYNOMIALS 55

If this process is continued, we finally arrive at the result

f [P(x)] 2 dx = ( n + m j; ( [p n ( x )] 2 dx 449

/_! (n m)! J_!

This last integral can be evaluated by means of the explicit expression
4-33 for P n (x). We have

/~HI i s*^~^ d n f% n '
\P (x\^ dx ~ I (x^ l^ n (x^ l^ w dx 4*50

Integrating by parts n times, this reduces to

t+l (-. 1\n r +l j2n

-l) n dx

/.

-1 I- 6 '-!.

(x 2 - ir(2n)\dx

[2"n !]
(2n)!

Since

x) n dx

n(H ^j * I /^ , __\2n v / - ,1 KO

^t'O^

;/:

(w + 1) (w + 2) 2w
equation 449 may finally be written as

(n - m) ! 2n + 1

, /2n + 1 (n - m) !

tso that the normalized functions are

4.53

x /
\

2 (n + m) !
4d. The General Solution of the Associated Legendre Equation.

Up to this point we have been concerned with particular solutions of
Legendre's equations 4-31 and 4-38. We shall now show that these
are the only solutions for the interval 1 < x < 1 which belong to
class Q. Since 4-31 is a special case of 4-38, we shall discuss 4'38 for
the sake of generality; the results, of course, will hold for 4-31 as well.
Equation 4-38 has two singular points, x = 1 and x = 1. If we
make the transformation x = + 1, the resulting equation is

56 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS
which is of the form 4-18, with

- -( + 1) - - 4-55

Since these functions are finite for = 0, the point = (x = 1) is a
regular point. If we now clear of fractions we obtain our equation in
the form of 44

+ [-n(n + 1) 2 2n(n + 1) m 2 ]u = 4-56
Substituting for u the power series

u = ao + c&i -h 4'57

we obtain the indicial equation

4L(L - 1) + 4L - m 2 = 0; L = =b ~ 4-58

If we take m real and positive the series beginning with 2 becomes
infinite at = in such a way that the square of the function is not
integrable, and so the resulting function is not of class Q in a range
including the point = (a; = 1). The solution which remains is,
returning to re as the independent variable,

u = ( x - I) 2 " (ao + a(x -\ ) 4-59

A similar analysis of the other singular point shows that u must also
be of the form

/* { *** I 1 \ ~9 f t*f ? I ^.^^x*. I \ A C(\

u = (x ~r i)* (Q,Q -f- #1 x -j- / 4'OU

Now the function

) 4-61

is of the form 4-59 if we expand the factor (1 + #) 2 and also of the

m

form 4-60 if we expand the factor (1 - z) 2 , so that the requirements
at both singular points are satisfied if we put

m

22

u = (1 z 2 ) 2 z; z = & + M + " 4-62

The differential equation satisfied by z is

T - 2(m + l)a; + [n(n + 1) - m(m + l)]z - 4-63
ax ax

SPHERICAL HARMONICS 57

giving the recursion formula for the coefficients in 4-62

(p + l)(p + 2)6,4.2 = K" - 1) + 2(m + 1>

- n(n + 1) + m(m + l)]b, 4-64

If n m is an integer, we find that, for the particular case in which
v = n m, &n-m+2 is zero, and thus the alternate coefficients &nm-f-4,
fen-m+e, are also zero. Therefore if the solution is written in
the form

/ m(m + 1) ~ n(n + 1) 2 \

2 = &0 I 1 H X* H 1

. -4- ( +!)("* + 2 > ~"(" + l)s , \ 4fi .
+ x + 1 4-65

either the even series or the odd series degenerates to a polynomial
of degree n m. The complete solution of 4-38 is therefore

u = (1 - z 2 ) 2 z = APJTCc) + 5Q^(o;) 4-66

where Q(x) represents the series which does not terminate. The
terminating series must, of course, be the polynomial F(x) which we
have already studied. Since

^Hh? _ yfr - 1) + 2(m + 1> - n(n + 1) + m(m + 1) m
b, "" fr + l)fr + 2) J

lim 5tH - l 4-67

the series for QJ* converges for 1 < x < 1, but at the points x = 1
the series becomes divergent. The function [Q^(x)] 2 is therefore not
integrable over the range 1 < x < 1 and hence is not of class Q in
this range. We therefore see that, aside from a numerical factor,
F(x) is the only function of class Q which satisfies equation 4-38 in
the range (1,1). If n m is not an integer neither series ter-
minates. The same considerations which were applied to the function
Q? apply here to both series, so that there is no solution of 4-38 which
belongs to the class Q.

4e. The Functions 0/, m (8) and Fj, m (8, $>). We will now establish
the connection between the functions 8j, m (0) introduced in the pre-
vious chapter and the functions P(x) discussed above. We there
derived the results:

J , m 4-68

4-69

58 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS
YJ m was found to be equal to p=9/ OT (0)e* m *, where m is an integer.

V27T

Using for M 2 the expression in spherical coordinates (3-99), we obtain
as the differential equation for 9j f m (0)

- 4 - 70

sin e
or

Q 4-71

dO z sin d0 I sin" 5

In order to be an acceptable solution, 9j, m (0) must be of the class Q
over the range < 6 < ir. Let us now make the change of variable
x = cos 0. Then

d - d <* 2 2 , <** * <*

= -sin ~2 = sin 2 2 - cos 4-72

dB dx dQ 2 dx 2 dx

and equation 4-71 becomes

.-'

Expressed as a function of re, 9j t m must be of the class Q over the range
1 <J # ^ ! Comparing 4-73 and 4-38 we see that GZ, m must be
identified with w = P. Since m enters equation 4-73 only as m 2 ,
we must have 6j, m = 9*, ^^j. The exact identification is, therefore,
for the normalized functions:

where |m| indicates the absolute value of m. Further, since m is an
integer, and since equation 4-38 or 4-73 has an acceptable solution only
if I m is an integer, I must be integral valued. From 4-40, we see
that the explicit expression for 9j, m (0) is

The explicit expressions for the normalized associated Legendre poly-
nomials are given in Table 4-1 for I = 0, 1, 2, 3. The normalized
spherical harmonics YI, m (0, <p) are obtained by multiplying 9j, m (0)

RECURSION FORMULAS 59

TABLE 4-1
THE NORMALIZED ASSOCIATED LEQENDEE POLYNOMIALS 0j, ra (0)

I 0, m = 60, o = -71

V2

I 1, m = 81, o = "^ cos 9

Z = 1, w = dbl Oi, 1 = V| s i n

Z 2, m = 2f = V| (3 C os 2 0-1)

I = 2, m = 1 6 2 , 1 = v/ ^? sin cos

Z = 2, m - 2 6 2 , 2 = V^f| sin 2 6

I = 3, m = 9 3 , o = ^f- (f cos 3 -

Z3, m = 1 e 3 , i = v^T(5 CC) s 2 -

Z = 3, m = 2 G 3 , 2 = **/*- sin 2 cos

Z - 3, m = 3 3 , 3 = VH sin 3

4f. Recursion Formulas for the Legendre Polynomials. In our

later work, we shall have occasion to evaluate integrals of the form

/ QJ, m cos 6j/, m / dr and / 0^, m sin 0j/ t m / dr

The evaluation will be greatly simplified if we have available explicit
expressions for the quantities cos 6 Pj m| (cos 6) and sin Pi 1 (cos 6)
in terms of a series of Legendre polynomials. From the formula

W=^]|i(* 2 -l)' 4-76

it can readily be shown that the equations

4-77

and (I + 1)P, = -j- PI+! - x P t 4-78

ax ax

are valid. If 4-77 is differentiated m times and the resulting equation

m+l

multiplied by (1 a; 2 ) 2 we immediately have, using the definition
4-39 for the associated Legendre polynomials,

(21 + 1)(1 - x 2 ^P? = PJS 1 ~ PT-5 1 4-79

Differentiating 4*78 (m 1) times, we have

60 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS

Multiplying this expression by (21 + 1)(1 # 2 ) 2 and rearranging

gives

(21 + 1)*PP = (21 + 1)PJ5. 1 - (Z + m){ (21 + D(l - s^PT 1 }

- (21 + i)pr+! - (i + ){p?+i - PT~I\

= (Z - m + l)Pft! + (Z + m)PH 1 4-81

If equation 4*27 is differentiated (m + I) times, the resulting equation
for PI is

4-82

We now multiply by (21 + !)(! a; 2 ) 2 and rearrange, obtaining

- 2m (2J + l)xP? - (I + m) (I - m + 1) (21 + 1) (1 - z^Pf" 1 4-83

Substituting for the terms on the right their values as given by 4-79
and 4-81, we obtain

- x 2 )*PT +l = PT +1 { (m-lHl-m + 1)}

or, replacing m by (TO 1),

(a + 1)(1 - x*)HPT = -(I - m + l)(l -m

+ (I + m)(l + m - 1)PE? 4-84

Expressing equations 4-79, 4-81, and 4-84 in terms of cos 6 = x gives us
the desired relations

cos 0Pl m| (cos 0) = -y^ { (I - \m\ + l)Ptft(oos 6)

(cose)\ 4-85

sin 6 P\ ml (cos 0) = - { P^'i +1 (cos 6) - P^ 1 (cos ) 4-86
M + 1

^j^ { (Z + H)C + M - DPft-Vooe*)

- (Z - |m| + l)(i - |m| + 2)Plff 1 (cos0)} 4-87
4g. The Hermite Polynomials. Consider the equation

d ^- + 2xy = 4-88

ax

THE HERMITE POLYNOMIALS 61

The solution of this equation is readily seen to be

y = ce~** 4-89

If we differentiate equation 4-88 (n + 1) times, we get

d 2 z dz rt/

2 + 2x h 2(n + l)z = 4-90

where

d n y d n _ a . 2

Z ^ "T T == C "T T ^6 y Tt*yX

ax ax

z is a function of the form u(x)e~ x *, where u(x) is a polynomial of
degree n. Substituting this expression for z in equation 4-90 we find
that u (x) satisfies the equation

d 2 u ^ du

z 2x + 2nu = 4-92

ax ax

Equation 4-92 is known as Hermite's equation, and the particular
solution obtained by putting c = ( l) n is known as the Hermite
polynomial of degree n. The usual symbol for these polynomials is
H n (x); explicitly

4-93

The first five Hermite polynomials are:
H (x)

In general

1

fjf ( /v 1 O/Y

n i \x ) AX

4x 2 -

2

H 3 (x) =&c 3 - 12x

16s 4 -

- 48a; 2 + 12

4-94

n(n

l)(2z)"- 2

1!

JL. . JL , . A.QF\

2!

Differentiating this series term by term, we find
dH n (x) a

dx T ' 1!

= 2nH n _i(x) 4-96

62 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS

Differentiating once again, we find

d 2 H n (x) dH^(x)

-~2 = 2n = 4n(n l)# n -2 4-97

dx 2 dx

If we now substitute 4-96 and 4-97 in the differential equation 4-92, we
obtain the useful recursion formulas

4n(n - l)ffn-2 - 4nxH n -i + 2nH n =

4-98

= (U -

or, upon replacing n by n + 1, we obtain the relation analogous to

equation 4-85 '

xH n = nHn-i + i^n+i / 4-99

We shall now show that the functions H n (x)<T~* form an orthogonal
set in the interval ( , ). Let m be less than n] then

/ r d n (e~ x *}

H m (x)H n (x}e~ x2 dx = / H m (x) \ n ) dx 4- 100
^.00 . . t/-00 " X L.

Integrating by parts, we have

(-!) C H m (x)H n (x)e-** dx ^^fj^ ^1

v oo (* J oo

J-oo dx dx n ~~ l . ^

The first term on the right is zero, since e~ x * and all its derivatives
vanish at x = Q . Replacing - by 2mH m ^i, we have

/ / J-l

ff m ( a; )H n (a ; )e- a:! dx = (- l) +1 2m / H m ^ 1 ~, (e~ x *) dx 4-102
.00 */-.< GW? /

Repeating this process, we finally obtain
f ff m (*)/U*X- 2 cte= (-ir+^-w! T H (x) f^ (e-**) dx

%/.00 */_oo ^ X

[jn~m-l -|QQ

-^(e- 8 ) = 4-103
"^ J 00

(

lim <= n, the same process leads to

f [H A (x)] 2 dx = (-I) 2n 2 n n! f e-*'dc = 2 n w!V^ 4-104

*/ 00 / 00

THE LAGUERRE POLYNOMIALS 63

1 -?!

We have thus proved that the functions - 7= H n (x)e 2 form a

__
normalized, orthogonal set. " -===__.

We must now show that these polynomials are the only solutions of
equation 4-92 belonging to the class Q for the range (~ 00,00). As
there are no singular points in this equation (except x 00), we
may expand our solution about any point. Expanding about the
origin, we find the recursion formula for the coefficients in the series

u = a + aix + a 2 x 2 + 4-105

to be (v + 2) (v + lK +2 + (2n - 2*)a, = 4-106

so that the solution is

4-107

y 2

Since lim ^ = - , the series always converges. In the series
> a v v

<? = 1 + x 2 + + - + 4408

a v ~ , ,^'. f r v even; a v for v odd, so that lim -^ = - We
(v/2)l v->oo a v v

therefore see that the series 4-107 behaves like e xZ near x = dbco.

But neither e 2x * nor e x * is integrable from oo to oo , so that neither u
_&

nor ue 2 is a function of class Q. If n is a positive integer, one of the
two series in 4-107 reduces to a polynomial, which will be a constant

~ x -
times the particular solution H n (x). Then, as we have seen, H n (x)e 2

is quadratically integrable and hence is a function of class Q.

4h. The Laguerre Polynomials. The polynomials L a (x), of the
ath degree in x, are defined by the equation

L a (x) = e*^(x a e-*) 4-109

ox

and are known as the Laguerre polynomials. The |3th derivative of
L a (x), called an associated Laguerre polynomial, is denoted by L%(x).
The general formula for the associated Laguerre polynomials is

64 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS
The Laguerre polynomials are not orthogonal functions, but the func-

_5

tions e 2 L a (x) form an orthogonal set for the interval (0, oo). In
order to prove this, consider the integral

f e- x x*L a (x)dx~ C X* ^- (x a e~ x ) dx 4-111

/o JQ d%

If we integrate by parts 7 times, we find

/ ,*> fla-y

I e-*x*L a (x) dx = ( - 1)^7 ! / T-= Grtf) dx
t/o t/o dx T

For 7 < a, this gives

r r d a ~^~ l T

I e-*x*L a (x) dx = (-1)^1 ^=^i (x a e~*) = 4412
*/o Lax ^ J

while for 7 = a we have

y^ 00

^e" 35 ^ = (~l) a (a!) 2 4-113

Since L y is a polynomial of the 7th degree, 4-112 gives

r" -.

t/O

or, exchanging the roles of a and 7,

f e-*L y (x)L a (x) dx = y^a 4-115

t/o

The highest power of x in L a (x) is ( 1) V, so that
/ /

I e~ x [L a (x}] 2 dx = ( l) a I e~ x x a L a (x) dx = [a!] 2 4-116

t/o ^0

1 ~-

The functions e 2 L a (x) therefore form a normalized, orthogonal set.
a!

To show the orthogonality properties of the associated Laguerre
polynomials, we consider the integral

r r rf 5

I e-*x*li dx = I 6""*^ 3-5 1/ a cte 4-117

J Jo ds*

Integrating this by parts j8 times gives

r / d!' 3

J e-^Lf dx - (- 1)* J L a ^ (e-*z Y ) cte 4418

as the integrated part always vanishes. The first term in (e~*x y )

THE LAGUERRE POLYNOMIALS 65

is (--l)^""*^ 7 . For 7 < <*, we therefore see irom 4412 that

Now the function x^L^ is a polynomial of degree 7, so that we have
the result

^

/ e-*xfti*li dx = 7 < ct 4-119

Jo

or, since a and 7 may be exchanged

/*

/ e^rftffJi <fo = y^a 4-120

Jo

For 7 = a, the first term in x^L^ is 7 - r; x a . We therefore have, by

(a - 0) I

4-118 and 4-113,

/ _ I^O^vf / 00

L^(fo = f^. / e-*x"L' a dx

(a /3)!t/o

f 00 (!) 3

e-^L dx = ^ ^ 4-121

The functions A / , 3 e 2 x 2 L% therefore form a normalized orthog-

/(-/! -1 I

(!) 3
onal set in the interval (0, <*> ).

In the theory of the hydrogen atom we will need the value of the in-
tegral I e~~ x x^ l L%La dx. By expanding x^ l L? a by means of 4-110,

and retaining only the terms in x a+l and x a , since lower powers of x
will integrate to zero, we have

(

J

dx = 7 '

(a-ftllt/o

4-122

Using 4-118, and retaining the first two terms in the expansion of
(e~*x a+l ), this reduces to

4-123

/ /^-l^a/vf f /*

/ e^^LgLgdx^ ) Aj " / e~*x" +1 L a dx
Jo ( j8)'l/o

66 THE DIFFERENTIAL EQUATIONS OF QUANTUM MECHANICS

The last integral has the value ( l)"[a!] 2 , according to 4413. For
the first integral, we have

/* r d a

m > S/r**"^ 7" /7l --< I /y a "4"l (v* lt 0~"^\ /7l

JQ a JQ dx a

or, after integrating by parts a times,

/ r

I e~ x x a+l L a dx = (-l) a (o: + 1)! / x a+l e~* dx
J o ^o

= (-!)[( + I)!] 2 4424

Substituting these values in 4423 gives us the final result

; dx = . ONI { (a + I) 2 - [0(a + 1) + a (a ~|8)]}
(a p)!

. (2 - g+i) . ( " 3

In order to find the differential equations satisfied by the Laguerre
polynomials and their derivatives, consider the equation

x d ^+(x-a)y = Q 4426

ax

which is satisfied by y = x a e~ x . If we differentiate this equation
(a + 1) times, we find

x \ + (x + 1) Y + (a + l)z = 4427

where

Substituting this value of 2 in equation 4427 gives us the differential
equation for the Laguerre polynomials

+ (1 - *)^r +L = 4-128

and differentiating this equation 13 times gives

^ + (ft + 1 - 3) ^ + (a - j8)ti - 4429

7j9r

where u = j- = Z/f as the differential equation for the associated
Laguerre polynomials.

THE LAGUERRE POLYNOMIALS 67

We shall now consider the general solution of this equation. The
point x is a singular point, but it is regular, so that we take as our
solution the series

u = a Q x L + aix 1 * 1 H ---- 4-130

Substituting this series in 4- 129 gives the indicial equation
L(L + ft) = L = 0, L = -0

The series beginning with x~& cannot be of class Q in any region in-
cluding the point x = 0; therefore there remains only the series

u = a + aix + 2# 2 + - 4-131

The recursion formula for the coefficients is readily found to be

( + + !)( + 1K +1 = ( + - )a; 4-132

so that

lim

a v v

The series always converges, but, as v > oo , the limiting ratio of the
coefficients is the same as that in the series expansion of e x . Hence

__x

the function u is not of class Q, nor is e 2 x 2 u unless the series terminates.
This is possible only if a ft is an integer greater than zero, in which
event we have a constant multiple of Lf . Where we shall use this
function, /3 will be a positive integer, so that a. must be an integer
greater than if there is to be a solution of class Q.

CHAPTER V
THE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS

In this chapter we propose to treat by the methods of quantum
mechanics certain of the simple systems for which the Schrodinger
equation can be solved exactly. These systems are an idealization
of naturally occurring systems, but the consideration of them is not
without value, as they furnish an insight into the methods of quantum
mechanics and give results which are useful in the discussion of many
problems of physical and chemical interest.

5a. The Free Particle. The simplest imaginable system would be
a particle of mass m moving in the x direction under the influence of
no forces. The classical Hamiltonian function for this system is

*--Lrf 5-1

where the value zero has been chosen for the constant potential V.
The eigenvalues E x for the energy are given by the solution of the
equation

6-2

A possible solution of this equation is

V 5-3

In order that the wave function remain finite at x = > , the quantity
V2mE x must be real, and so E x must be positive. As this is the only
restriction on E x , we conclude that all possible values of E x from to
+ are permissible; that is, we have a continuous spectrum of energy
values. Classically, the energy is related to the momentum by the

relation E x = pj*, so that in terms of the momentum we may write

2771

?I*V"T
t = N( Px )e *** 54

Let us now calculate the average value of the momentum associated

68

THE FREE PARTICLE 69

with the wave function 53. According to equation 3-68, this is

5 . 5

An equally valid solution of equation 5-2 is

27r *v^TF-~

f TiT/TTT \ -- r- v *USrx# j^f f \ -- r- x p.

^ = N(E x )e h = N(p x )e h 5-6

The average value of the momentum associated with this wave function

is p 2 .. The wave function 54 therefore represents a state in which
the particle is moving in the +# direction with the definite momentum
Vpf ; the wave function 5*6 represents a state in which the particle
is moving in the x direction with the same absolute value of the
momentum. Let us now consider the probability that the particle,
in the state represented by 5-3, will be in the region between x and
x + dx. According to Postulate I, this is \l/*\[/ dx = N*N dx. We
thus see that all regions of space are equally probable, so that the un-
certainty in the position of the particle is infinite. This is, of course,
required by the uncertainty principle, for, if Ap^ is zero, as we have
found it to be, Ax will be infinite in order that the relation Ap x Ax ~ h
will be true. Any linear combination of the solutions in 5-3 and 5-6
involving the time

5-3a
5-6a

= L ,

2 ~

is aiso a solution of the wave equation

_ h 2

87r 2 m dx

If we take the combination representing the sum of all possible wave
functions 5-3a and 5-6a, the waves will reinforce one another at some
particular point x = XQ and will interfere everywhere else, so that in
this event the particle can be located exactly. However, the uncer-
tainty in the momentum is infinite. A more complete analysis 1 shows
that the smallest possible simultaneous uncertainties in position and
momentum are governed by the relation Ap x Ax~h; in this way
the uncertainty principle is derivable from our fundamental postu-
lates.

1 R. C. Tolman, Principles of Statistical Mechanics, Oxford, 1938, p. 231.

70 THE QUANTUM MECHANICS OP SOME SIMPLE SYSTEMS

6b. The Particle in a Box. We now consider a particle of mass m
constrained to move in a fixed region of space, which for simplicity we
take to be a rectangular box with edges of length a, 6, and c and volume
v = abc. The potential V may be put equal to zero within the box;
at the boundaries of the box and in the remainder of space we put
V = oo . We take our coordinate system to be the cartesian coordinate
system with the origin at one corner of the box and the x, y y and z
axes along the edges of length a, 6, and c, respectively. The potential
energy is then

V x 0, < x < a; V x = co otherwise

V y = 0, < y < bj V y = oo otherwise

Vg = 0, < z < c; V g = oo otherwise
Schrodinger's equation for the system is

We have a differential equation in three variables. In order to solve
this equation we seek a solution of the form

* = X(x)Y(y}Z(z) 5-8

where X(x) is a function of x alone, etc. If we now put V = V x + V y
+ V z , and substitute 5-8 in 5-7, we obtain

. 9

On the left we have a function of x and y only; on the right, a function
of z only. If we vary x and t/, keeping z constant, the right side re-
mains constant; therefore the left side must be a constant. Let us

call this constant -TJ- &* We then have the equations

ffl 1 */ (T\ Rir^m

= 5-10

w

THE PARTICLE IN A BOX 71

By the same reasoning followed above, both sides of this last equation

8?r 2 m
jnust be equal to a constant, which we call 2 E y . This gives us the

additional equations

f (ii\ fc7T 2 m

= 542

;*) = o 5-13

dy 2

d?X(x)
dx 2

where, in 5-13, we have put E = E x + E y + E z . We thus have three
differential equations, each involving one variable only, and which are
thus readily solvable. Let us consider first the equation in x. Since
V x = oo for x > a, x < 0, 543 will hold in this region only if we put

X (x) = 0, r-o = 0. For the region inside the box the equation is
dx

rfY(^ S-rrV

5-14

for which the general solution is

. . ~^m x x D -~mc*

X(x) = Ae h + Be h 545

or

X(x) = A f cos -^ \/2mE x x + B' sin -^ \/2mE x x 5-16

h h

In order that the solution for the region inside the box join smoothly
with the solution for the region outside the box, we must have X(x) =0
at x = 0, x = a. The first of these considerations requires that

A 1 = 0; the second requires that \/2mE x a = n x ir, where n x is an

h

integer (not including zero, as this value of n x would make X(x) equal
to zero everywhere). This limitation on the nature of the wave
function at the edge of the box requires that the energy be quantized
that is, only those values of the energy given oy

E x = -^ n x = 1, 2, 3, - 547

will give an acceptable wave function. This wave function may now
be written as

X(x) = B'sin x 5-18

a

72 THE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS
The normalization factor is given by the integral

f a [X(x)] 2 dx = B' 2 r sin 2 ( x}dx = 1 5-19

t/o ^0 \ O /

/2
which gives ' = + I-. The equations in y and 2 are solved in the

same manner. The final results are therefore

/ \ -Vf \\7t \rjf \ . x . y . Z

\f/(x, y, z) = X (x)Y(y)Z(z) = * / sin x sin y sm

aoc a o c

/8

./-
\v

= + /- sin -^- x sin -7- ?/ sin -^- 2 6-20

A f; a 6 c

/n 2 /n2\

5-21

where n x , n v , n z = 1, 2, 3, 4, . These results will be of importance
in connection with the theory of a perfect gas, and also in the theory of
absolute reaction rates.

6c. The Rigid Rotator. The theory of the rigid rotator in space will
be of value in the discussion of the spectra of diatomic molecules. As
an idealization of a diatomic molecule, we consider that the molecule
consists of two atoms rigidly connected so that the distance between
them is a constant, R. As we are not interested here in the transla-
tional motion of the molecule in space, we may regard the center of
gravity as fixed at the origin of our coordinate system. Suppose that

the polar coordinates of one atom are a, 6, <p, where a == - - JK,

mi + w,2

and mi and ra 2 are the masses of the two atoms. The polar coordinates
of the other atom are then 6,
energy of the first particle is

of the other atom are then 6, 0, <p, where b = -- ~ - R. The kinetic

mi + 77^2

Similarly, the kinetic energy of the second particle is

2
BO that the total kinetic energy is

THE RIGID ROTATOR 73

If we put m\a 2 + w 2 6 2 = I (the moment of inertia), then the kinetic
energy may be written as

which is the same as that of a single particle of mass I confined to the
surface of a sphere of unit radius. From the expression for the La-
placian operator in spherical coordinates (Appendix III) we see that the
Hamiltonian operator is

-h 2 Pi a / 2 a\ 1

H ~ 8A Lr 2 Or V dr) + r* si

a .

r* sin Sm

2
r 2 sin 2 d<p

+ V 5-25

If no forces are acting on the rotator we may put 7 = 0, and, putting
r = 1, mr 2 = m = I, we find that Schrodinger's equation is

0V

We have once again a differential equation with more than one inde-
pendent variable, so that we look for a solution of the form

$ = 6(0)$ (<p) 5-27

After making this substitution we may write equation 526 as

sine d / n ae\ , ST^IE . 2 n i a 2 $

f sin e J -\ -j- sin 2 e = 2 5 ' 28

By the same line of reasoning as was employed in the previous section
both sides of this equation must be equal to a constant, which we take
equal to m 2 . We thus obtain two differential equations

- = -m 2 $ 5-29

JL^/ in ^\_^_ 8^ 6 = 5>3Q

sin e de \ a0/ sin 2 h 2

Equation 5*29 is seen to have the solution

5-31

We have previously shown (section 3h) that this is an acceptable wave
function if m is an integer; the normalized solution of 5-29 is therefore

m = 0, 1, 2, 3 5-32

74 THE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS

o 2rijj7

In equation 5-30, let us replace 75 by 1(1 + 1). This equation then

becomes

si

-sin - -~ 6 + 1(1 + 1) 6 = 5-33

sin0d0\ 36 ) sin 2 v

which is identical with equation 4-70, and therefore has acceptable
solutions only for integral values of Z; Z > |m|. The normalized
solutions of 5-33 are therefore

e(0) 55 BZ, m (0) =

The restriction on Z requires that only those values of the energy given
by the relation

E = ~^- 1(1 + 1) I = 0, 1, 2, 3 - 5-35

O & T ^ ' ' 777

are allowed. The total wave function is, of course,

As we have seen in sections 3h and 4e, these wave functions satisfy
simultaneously the equations

5-36
M 2 7 Z , m = dbm Fj, w

In addition to being eigenfunctions of the Hamiltonian, with the eigen-
values for the energy E = 1(1 + 1), these functions are also

eigenfunctions of the total angular momentum, with the eigenvalues

h 2
M 2 = Z(Z + 1) g, and of the z component of the angular momentum,

with the eigenvalues M z = dzra . All these variables thus have a

2?r

constant value simultaneously. In general, when only two coordinates
are needed to describe a system, we would expect only two dynamical
variables to be simultaneous eigenvalues. The reason that we have
three here is that classically there is a relation between the energy and

THE HARMONIC OSCILLATOR 75

M 2

the square of the total angular momentum, this relation being E = - >

2tL

which is also true for the eigenvalues given above.

5d. The Rigid Rotator in a Plane. As a special case of the above
problem, let us restrict the motion to rotation in a given plane, which
we may without loss of generality take to be the xy plane. In this
event, 6 has the constant value of 90, so that equation 5-26 becomes

5-37

Comparing this with 5-29 we see that the eigenf unctions are ^ = $ m (<p)

h 2
with the energy eigenvalues E = 27 m 2 . These eigenf unctions for

O7T 1

the energy are also eigenfunctions for M g , with the eigenvalues m

2iTT

5e. The Harmonic Oscillator. Many systems of interest can be
approximated by harmonic oscillators; for example, the vibrations of a
diatomic molecule and the motions of atoms in a crystal lattice can be
treated to a first approximation as motions of a particle in a harmonic
field. A harmonic oscillator is a particle of mass m moving in a straight
line (along, say, the x axis) subject to a potential V = ^kx 2 , so that
the force on the particle is kx. Classically, the equation of motion is

m 9 = kx 5-38

dr

with the general solution

x a cos 2irv(t IQ) 5*39

1 Ik
where a and IQ are constants and v = ^ / . The kinetic energy is

2?r \m

\m ( J = 2w7r 2 j> 2 a 2 sin 2 2vv(t < ); the potential energy is %kx 2
\dt/

2ra7rVa 2 cos 2 2irv(t J ), so that the total energy is E = 2w7rVa 2

k

- a 2 ; all positive values of E are allowed.

&

The classical Hamiltonian for the system is

so that the Hamiltonian operator is

TT , 2

H = - o- 5 + ~ x 2 541

87r 2 m dx 2 2

76 THE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS
The wave equation for the system is therefore

dx 2 h 2 \ 2 /

and our problem is to find those functions of class Q which satisfy this
equation. This equation may be rewritten as

87r 2 m _ 2irv mk ___ , ... . .

where a = j- E, = . We can further simplify this

fi fl

_ * 2

equation by making the change of variable == Vpx. Then

J o * an d the equation in becomes
d

%

5-44

1..2 I 1 - * I T -" ** **

Let us first see what form we must take for iK) for very large values
of in order that we may have an acceptable wave function. For

sufficiently large values of , - may be neglected in comparison with

2 , so that in this region ^() must approximately satisfy the equation

T-J ^ fV 5*45

The solutions of this equation are approximately \l/ = ce 2 , since
_ (e^) = e 2 ~ ( 2 db 1), and the factor dl may be neglected in

comparison to 2 in the region of large . We cannot use the solution

+1! -L 2

e 2 , as this is certainly not of class Q; the solution e 2 will, however,

behave satisfactorily at large values of . These considerations sug-

_
gest that a solution of 5-44 of the form ^() = w( )e~" 2 might be found.

Making this substitution, we find that w() must satisfy the differen-
tial equation

Comparing this with equation 4-92, we see that the two equations are

THE HARMONIC OSCILLATOR 77

fa \

identical if we replace I - 1 ) by 2n. u() is therefore H n (), and
\P /

-L 2
the wave function \p() is cH n ()e 2 , which, as we have seen, is a

function of class Q for all positive integral values of n, including zero.
This restriction on n gives us a corresponding restriction on E. We have

~ = 2n + 1 547

P

or, substituting the full expressions for a and /3 and simplifying

+ ^ = (n + i)/i 5-48

m

The allowed energy values are thus ^, -f , -| , times the energy hv
associated with the classical frequency of oscillation.
We now need to find the constant c such that

f tfa dx = 4= f lH(M 2 e~*<% = 1 549

'-oo V^-oo

^

From equation 4-104, we see that we must take c = , =. The

n

normalized wave functions for the harmonic oscillator are therefore

5-50

The first few energy levels and the corresponding wave functions are
shown graphically in Figure 5*1. We note that the wave functions
are alternately symmetrical and antisymmetrical about the origin. Of
particular interest is the fact that, on the basis of quantum mechanics,
the harmonic oscillator is not allowed to have zero energy, the smallest
allowed energy value being the " zero-point " energy \hv. This is in
accord with the uncertainty principle; if the oscillator had zero energy
it would have zero momentum and would also be located exactly at the
position of minimum potential energy. The necessary uncertainties
in position and momentum give rise to the zero-point energy. A sim-
ilar situation exists for the particle in a box, which also has a zero-point
energy. In the rotator, the lowest state corresponds to the situation
in which all orientations in space are equally probable; the uncer-
tainty in position is therefore infinite, hence the momentum, and
therefore the energy, may have the precise value zero.

78 THE QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS

Let us now calculate the average value of the displacement of the
particle from its equilibrium position. We have

wszlO^gm. T=1186 cm. 1

6.00--

6.00-

=4.00--

3.00--

2.00- -

1.00- -

0.00

FIG. 64. Energy levels and eigenf unctions for the harmonic oscillator.
The recursion formula 4-99 for the Hermite polynomials states that

-20 -16 -.12 -.08 -.04 0.0 .04 .08 .12 .16 .:

The integral therefore vanishes because of the orthogonality of the

THE HARMONIC OSCILLATOR 79

wave functions, so that x = 0. This was, of course, to be expected,
since the squares of the functions are symmetrical about the origin,
tor the average value of # 2 , we have

^tt)^tt)^ 5-52

From the recursion formula, we have
2 H n = ntH^ + &H n+1

= n[(n - l)ff n _ 2 + %H n \ + |[(n + l)H n + lff n+2 ] 5-53
Since only the terms in H n contribute, the integral 5-52 becomes

? = W" ^ (n + tifL ^-tt)!*^ 1 * = ^ 5-54

7?

Using the relations derived above, this may be written as x 2 = or

rC

- 2

E = kx 2 . Now, classically, a: 2 = 7-, so that in terms of the mean

2

square displacement from the equilibrium position we see that the
energy is given by the same relation in both cases.

CHAPTER VI
THE HYDROGEN ATOM

6a. The Hydrogen Atom. In this chapter we will treat by the
methods of quantum mechanics the simplest atomic system, the hy-
drogen atom. The hydrogen atom consists of a proton, of charge +e
and mass M y and an electron, of charge e and mass m. These two
particles attract each other according to the Coulomb law of electro-
static interaction. If we denote the coordinates of the proton by xi,
?/i, and 21, and the coordinates of the electron by #2, 2/2, and z 2 , the
potential energy is

e 2

The classical Hamiltonian function, in this coordinate system, will
then be

Because of the form of the potential energy, the Hamiltonian function
is not at all simple in this coordinate system. We therefore introduce
the following change of variables. Let #, y, and z be the coordinates
of the center of gravity. Referred to the center of gravity of the
system as origin of a spherical coordinate system, let the spherical
coordinates of the electron be a, 0, <p and those of the proton 6, 0, <p,

M m

where a = - r. b = -- r and where r is the distance between
M + m M + m

the proton and the electron. In this coordinate system, the potential

e 2

energy now has the simple form V -- . In terms of the new coor-
dinates, the rectangular coordinates of the proton and electron are

M A*

x\ x -- r sin B cos <p x% = x ~\ -- r sin coe <p
M m

jj, jj,

2/1 = y TT T sin B sin <p y 2 = y H -- r sin 6 sin <p 6-2

M m

M M

Zi = z r cos B z 2 == z H -- cos B

M m

80

THE HYDROGEN ATOM 81

where u = is the " reduced mass " of the system. With the

M + m

introduction of this change of variables, the classical Hamiltonian
function becomes

]tf-|_ m r/^A2 /^,\2

H =

(~dt) + (&) \

+

The first term is the kinetic energy, in rectangular coordinates, of a
particle of mass M + m; the second term is the kinetic energy, in
spherical coordinates, of a particle of mass ju. The Hamiltonian oper-
ator for the system is therefore

H = _r* 2 ' ' '

_ _
m) dx 2 dy 2 d

fi d / 2 a\ l d / . d

V 2 dr \ dr)

r 2 sin 8 d6

+ 2 " 3 2 - - 6-4
r 2 sin 2 d d<<?\ r

so that Schrodinger's equation for the hydrogen atom is
h 2 id 2 . d 2 . d 2

m) [dx* dy* dz-
. h 2 11 d

87T 2

+

r 2 sin 2

It is apparent that we can separate this wave equation into two equa-
tions, one containing x, y, and z only, the other containing r, 0, <p only.
Carrying out this separation in the manner employed in the previous
chapter, we set \l/' = x(#? 2/> z)^( r j 0> <p) ^ n( l obtain the two equations

3 2 X , d 2 x 9 2 X S* 2 (M + m) ,
" 2 + + + - 2 - (E -

^_
r 2 sin0

6-7

82 THE HYDROGEN ATOM

Equation 6-7 contains only the relative coordinates of the two particles,
so that E is the internal energy of the hydrogen atom. Equation 6-6
is just the wave equation for a free particle of mass M + m, with
translational energy E f E. This equation has already been dis-
cussed, and we will not consider it further in this chapter. It is ap-
parent that in the treatment of any atomic or molecular problem the
translational degrees of freedom of the system may be separated from
the internal degrees of freedom in the same manner and thus need not
be considered in general.

To separate the variables in 6-7, we make the substitution

obtaining
1 d / 2 dR
ft dr\ dr

-|r 2

i d t . m i d 2 Y n

_ . ( sm ) - -T~O o- 6-8

F sin 6 3d \ dO/ Y sin 2 6 d<p 2

By our usual argument, both sides of this equation must be equal to a
constant, which we call X. We then have the two equations

sin
d

M

Equation 6-9 is by now quite familiar. The allowed solutions are
^ = Yi.m(0 9 ^)? where X = 1(1 + 1), with I and m integers, and
I > |m|. It should be mentioned at this point that, in all problems in
which, in spherical coordinates, the potential energy can be written
as a function of r only, the separation of the wave equation will proceed
in the same manner as above, giving YI, m as the angular part of the
wave function. Introducing the required value X = 1(1 + 1) into
6-10, and expanding the first term, we have

*

We must now consider separately the two cases where E is positive
and where E is negative. With negative E, equation 6-11 may be
simplified by the introduction of a new parameter n, defined by the
relation

THE HYDROGEN ATOM 83

and a new variable x defined by

nh 2
r = o 2 2 s 6-13

Equation 6-11 is reduced by these substitutions to

dor x dx \ 4
If we look for a solution of the form

R = w(a?)^"5 645

we find that w(x) must satisfy the differential equation

This is the same as equation 4-129 if we put = 21 + 1 and a = n + I.
We have seen that equation 4-129 possesses satisfactory solutions only
if a /3 is a positive integer. Now a is equal to n Z 1, and,
since Z may have the values 0, 1, 2, 3, , n may have the values
1, 2, 3, , with the restriction that n > Z + 1. This gives the
allowed negative values of the energy

n = 1, 2, 3, - - - 647

which are identical with the values obtained in Chapter I by means of
the Bohr theory.

The radial wave functions for the hydrogen atom are therefore

p( r \ - vr l t> *JrTf(r}' y r 648

L\> \i ) t/it/ O jL/7j_l_J \Jif J j Jls 72' ^ *&

nil
To determine c, we take

f [R(r)] 2 r 2 dr = 1 649

But

' [fl (r)] 2 r 2 dr = c 2 f * 2 ^[L*ff (x)] 2 r 2 dr 6-20

t/n

1,2

x, where ao = o
2 4?r

Bohr orbit, as is easily verified from the equations in Chapter I. Then,

We may write r = x, where ao = o o is the radius of the first

84 THE HYDROGEN ATOM

by means of equation 4-125,

= c 2 (^-] r . V'. 6.21

so that

where the minus sign has been introduced to make the functions
positive.

6b. Hydrogenlike Atoms. The problem of the ionized atoms He + ,
Li" 1 "*", etc., is identical in principle with that of the hydrogen atom, the
only distinction being a slight difference in the reduced mass n, and
a numerical factor Ze 1 in place of e 2 in the potential energy. We can
therefore write down immediately the solution for the general hydrogen-
like atom of nuclear charge Z and nuclear mass M z . The solutions
are

t = R n ,i(r)Yi, m (0, v -) 6-23

6-24

b24

1 - Zr /*2Zr\

r^lSi'f 1

/

&

where p = r. Certain of the normalized radial wave functions for
a

hydrogenlike atoms are given in Table 6-1. The energy levels for these
atoms are

+m

Since is very close to unity, the energy levels of hydrogenlike atoms

are given to an excellent approximation by the relation E z = Z 2 E H9
where E& represents the energy levels of hydrogen.

WAVE FUNCTIONS OF HYDROGEN 85

The energy levels as given by 6-26 are in essentially perfect agree-
ment with the experimental results. The above treatment of the
hydrogen atom neglects certain small energy terms, the first being
that due to " electron spin/' which, although very small for hydrogen,
is important in the general theory of atomic structure and will be con-
sidered in a later chapter, the second being a small relativity correction
which we shall entirely disregard, as these relativistic effects are neg-
ligible for small energies and hence can be ignored in problems of chemi-
cal interest.

6c. Some Properties of the Wave Functions of Hydrogen. From
the form of the wave functions, we see that they satisfy simultaneously
the equations

H \l/ nt i t m = E n \l/ n , I, m

M 2 * n . ,, m = 1(1 + 1) j~ t n9 ,. m 6-27

^ n , I, m

so that the energy, square of the total angular momentum, and z com-
ponent of the total angular momentum are simultaneous eigenvalues.
Each wave function of hydrogen is specified by the three quantum

TABLE 6-1

NORMALIZED RADIAL WAVE FUNCTIONS R (r) FOR HYDROGENLIKB ATOMS

nr

p = r; n ^ I + 1

--
n = 2, I = R 2 , o = T= ( ) (2 - p)e

2V 2 \ a o

n = 2,

2 /zy*

jRa, o ~p \ ) (27 18p +
81 V \o/

n3,

86

THE HYDROGEN ATOM

2.00
1.60
LOO
0.50
0,00

\

0.50
0.40
0.30
0.20
0.10
0.00

#20

0.30

>0.20
*
* 0.10

*0,00

0,30
0.20
0.10
0.00

0.10
0.05
0.00

0.04
0.03
0.02
0.01
0.00

\

3.

4.

Pia. 6*1. Radial eigenf unctions of hydrogen.

WAVE FUNCTIONS OF HYDROGEN 87

numbers n, Z, and m. The quantum number n determines the energy
of the atom; the quantum number I determines the total angular
momentum; and the quantum number m determines the z component
of the angular momentum. In Figure 6-1 we have plotted the radial
wave function R (r) for several of the lowest energy levels of hydrogen.
It is noted that the radial wave function has a non-zero value at r =
only for those states for which I = 0, that is, only for those states which
have no angular momentum. The radial wave functions become
zero n I 1 times between r and r = oo .

According to Postulate I, the probability of finding the electron at
a distance between r and r + dr from the nucleus, with its angular
coordinates having values between and 6 + dd, <p and </? + d<p, is

W dr = [R n , j(r)] 2 [7j, w (0, *>)]V sin dr de d^> 6-28

An alternative way of viewing this situation is to consider the electron
as having a spatial distribution, the density of the " electron cloud "
at any point in space being given by the square of the wave function
at that point. The angular distribution of the electron density is given
by the square of the spherical harmonics F/ t m (0, <?). Referring to
Table 4-1, we see that the state with I = 0, m = 0, is spherically sym-
metrical about the origin. The state with I = 1, m = 0, has the maxi-
mum value for the electron density along the z axis; the states with
I = 1, m = 1, have the maximum value for the electron density in the
xy plane.

To determine the probability that the electron be between the dis-
tances r and r + dr, regardless of angle, we must integrate over 6 and
<p. Since the spherical harmonics are normalized to unity, this inte-
gration gives us simply

P(r)dr = [R n ,i(r)] 2 r 2 dr 6-29

In Figure 6-2 we have plotted this probability distribution for several
of the lower states of the hydrogen atom. The average distance of
the electron from the nucleus is given by the integral

r =

[R n , ,(r)]V d r

For the hydrogen atom in the lowest state (n = 1, Z = 0), the average
value of ris

'

88

THE HYDROGEN ATOM

that is, the average distance of the electron from the nucleus is three-
halves the radius of the first Bohr orbit. The most probable value of r
is found from the equation

6-32

dr

so that the most probable distance of the electron from the nucleus is
exactly a. The average value of r 2 is

i a

oJo

6-33

u.ou
0.40
0.20

o.oo

1 0.20

"iuoo

0.20
0.10
0.00

f

fx

V

I

^

1 s

/

"^,

"* ^

^K

2s

^x.

^^*

'

'

**

-*^.

x**

-^.

^

^

- *~^

- ^

2p

^^K^

^^

^**""^

"" '.

^

"^**.

-^

_^

^

*->.^.

*** ^

- .

i

I 2 3 4^5 6 7 8 9

FIG. 6-2. Probability distribution in hydrogen.

States of the hydrogen atom with Z = 0, 1, 2, 3, 4, 5, are known
as s, p, d, /, g, h, states, respectively. A state with n = 3, Z = 1, is
called a 3p state; a state with n = 1, I = 0, is called a Is state; etc.
It will be noted that there are three p states, with m = 1,0, 1. These
may be designated as p+i, po> and p_i states, respectively. The angu-
lar factors associated with these states are, aside from a numerical
factor,

' sin

' COS0

6-34

I ~ sn

WAVE FUNCTIONS OF HYDROGEN 89

For many purposes it is more convenient to replace these functions
by the following linear combinations:

P-i

-_ _ - -

V2
Po ^ cos ^ 2! 6-35

p = i ~ 1 /^/ sin sin y> ~ y

V2

the designations p xi p y , and p z indicating that the angular part of these
wave functions have their maximum values in the x, y, and z directions
respectively. Similarly, for the d functions, we take the linear combina-
tions

d Z 2 = d ~ (3 cos 2 6 - 1) ~ 3z 2 - 1

_ d+i + ^i . n
d xz = - 7= ~ sin ^ cos 6 cos <p ~ xz

V2

rf i ^ d__j_

dj, z = i 7= - '^ sin 6 cos &in<p ~ yz 6-36

V2

, ^M-2 I* ^* 2 9 rt 9^/9 9\ 9 O

" 2 6 cos 2<0 ^ sin 0(cos <p sm z <p) ~ ^ y 2

-7=

i ^ ,- ~
V2

sin 2 ^ sin 2<p ~ sui 2 ^ cos <p sin ^

In Table 6*2 we give the complete normalized hydrogenlike wave func-
tions for n = 1, 2, 3, using the above form for the angular part of the
wave functions.

TABLE 6-2
NORMALIZED HYDROGENLIKE WAVE FUNCTIONS

Z

p r
a

i

n = 1, i 0, m *i = -7= -

V 7T

n 2, Z0, w=0 02

cos

90 THE HYDROGEN ATOM

TABLE 6-2 (Continued)

i /zy* ~

n = 2, 1 = 1, m = 1 ^20 = 7=- [ ) pe 2 sin cos ?>

* 4V27r\ao/

i __f

pe 2 sin sin v>

n = 3, Z = 0, m = ^ == - 7=^ I ) (27 - 18 P

/

(6 P - p 2 )<f cos

2 /zy* --

n 3, 2 = 1, m = 1 ^3 P = 7=( ) (6p p 2 )e 3 sin cos v>

* 81 V TT \ a v/

1 sin ^ sin <p

-
n3, Z - 2, m=0 fe 2 = - 7= ( "" ) P 2 e 3 (3 cos 2 (? - 1)

* 81V GTT V*0/

V2 /Z\^ ~^
n = 3, i = 2, m = 1 ^sd = - T=[ ) pe 3 sin ^ cos 6 cos ^>

\/2 9 - .

) P e sm ^ cos sm >

--

n - 3, Z - 2, m = 2 ^i^i = ^_ /-- ( ~ ) P 2 e 3 sin 2 (? cos 2y>

-

7= ) P 2 e 3 sin 2 sin 2^

81 V 27T \o

6d. The Continuous Spectrum of Hydrogen. We must now con-
sider those states of the hydrogen atoms for which E is positive. In
this case we make the substitutions

E - w- 6<37

M 2

6 ' 38

Equation 6-11 then becomes

'*_

For very large values of x this is approximately

i*-0 640

CONTINUOUS SPECTRUM OF HYDROGEN 91

x

for which the solutions are R = ce *. The solutions of 6-39 are there-
fore finite at infinity. The only possible difficulty is at the origin,
where there is a singular point. If we let

R = a^c L + aix L+1 + a 2 x L+2 H 641

the indicial equation is found to be

L(L - 1) + 2L - 1(1 + 1) = 642

so that L = Z or L = (Z + 1). The solution beginning with L = I
is finite at the origin and at infinity. There is therefore a solution of
Schrodinger's equation of class Q for all positive values of E. We thus
have a continuous range of positive eigenvalues, corresponding to
ionization of the hydrogen atom, as well as the discrete set of negative
eigenvalues for the un-ionized atom.

CHAPTER VII
APPROXIMATE METHODS

7a. Perturbation Theory. The number of problems which can
be solved exactly by the methods of quantum mechanics is not very
large. This is not surprising if \ve recall that even in classical mechanics
such problems as the three-body problem have resisted solution in a
closed form. The great majority of the problems of quantum mechanics,
including the problem of the structure of all atomic systems more com-
plicated than the hydrogen atom, must therefore be treated by ap-
proximate methods. The most important of these approximate
methods, at least for our purposes, is the quantum-mechanical per-
turbation theory.

Suppose that we wish to solve the problem of the motion of a system
whose Hamiltonian operator H is only slightly different from the
Hamiltonian operator H of some problem which has already been
solved. Associated with H we have a set of eigenvalues E ( \, E% - -
E^ , and the corresponding eigenfunctions ^i, ^2 * ' ^ ' ' ' satis-
fying the equation

H ^ = Eftfi 7-1

Since by assumption H is only slightly different from H , we write

H - HO + XH (1) 7-2

where X is some parameter, and the term XH (1) , which is called a " per-
turbation," is small in comparison to HO. The equation which we wish
to solve is therefore

(H +XH (1) )^ n - E n t n 7-3

If X is placed equal to zero, equation 7-3 reduces to 7-1, so that it is
natural to assume that for small values of X the solutions of 7-3 will lie
close to those of 74; that is, the effect of the perturbation XH (1) will
be to change slightly the " unperturbed " eigenvalues E^ and eigen-
functions \l/n- Now suppose that \f/ n and E n are the eigenfunction and
eigenvalue which approach ^ and E^ as X > ; and for the present
we shall assume that no two of the E^s are equal. Since \l/ n and E n
will be functions of X, we may expand them in the form of a power

92

PERTURBATION THEORY 93

series as

+ 74

7-5

where t%\ ^, ; E, E, are independent of X. If we sub-
stitute these series in 7-3 we find

+^f) + --- 7-6

In order that this equation may be satisfied for all values of X the co-
efficients of the various powers of X on the two sides of the equation
must be equal. Equating the coefficients of the various powers of A
gives the series of equations

7-7

(Ho - Sftt = E>& - H<* 7-8

(H - EM> = < 2 V + EW - H<*i 7-9

The first of these, by assumption, is already solved. If the second
can be solved, we can find # n and E% } . The solution of the third
equation then gives ^ 2) and E\ and so on.

In order to solve equation 7-8, let us assume that the expansion of
the function \[/^ in terms of the normalized and orthogonal set of
functions ^J, ^ ' * ' ^n> is

^> = Arfl + A 2 $ + + A m j& + 7-10

where the A m 's are to be determined. The function H (1 Vn can also
be expanded into the series

H<tf = tf g/tf + Hgfi + + H<W m + 7-11

where

Substituting these series into 7-8 gives

(H - ) (Adi + Ad$ + ) = 'j?V2 ~ ^HV? - Hgfi . . 742
which can be reduced by means of equation 7-7 to

7-13

94 APPROXIMATE METHODS

The coefficient of each $J must be equal on both sides of the equation.
On the left the coefficient of $J is zero; on the right it is (E ( ^ H)*,
so that

- H%> = 7-14

The first-order perturbation energy has thus been determined to be

dr MB

By equating the coefficients of ^ (wi ** w), we obtain

7-16

This relation gives us the values of all the A's except A n . The coeffi-
cient A n may be determined by the requirement that ^ n be normalized.
We may express $ n as

X 2 (- )

m

where / means that we are to sum over all values of m except n. Then

m

fo dr = ^VU dr + X E' ^ m JVS 1 *2 ^r

VS dr + X 2 (- ) 7-17

Since the functions ^ are normalized and orthogonal, equation 747
reduces to

n dr = 1 + 2\A n + X 2 (- ) 7-18

If the functions \l/ n are to be normalized, the right side of this equation
must be equal to unity for all values of X, so that we must put A n
equal to zero. The results to the first order in X are therefore

E n = E n + Atf> + X 2 (- ) 7-19

* + X 2 (- ) 7-20

We obtain ^ 2) an d ^n 2) by a similar process. If we assume that

5a^ + . . + B^ + 7-21

PERTURBATION THEORY 95

where the B 9 a are to be determined, then

(H - E n )t = (B? - EjDBrf? + (E 2 - E)Ba$ + - - - 7-22
From equations 7-10, 7-11, 745, and 7-16, we have

7.23

77(1)17(1)

/ -^km^mn ,Q

-p sr ^*

Jfc m ^n ~ ^m

so that, with the use of these results, equation 7*9 becomes

v~" v/ ^ L km l - L mn /Q *y oe

- EE ffQ _ go tk 7-25

If we now equate the coefficients of $[ on either side of the equation,
we find

7-26
or

Equating coefficients of ^2 (k 9* n) gives

77(1) rr(l) 17(1)17(1)

-"nn^fen ^/ -"fem^mn

- -

- OQ

7-28

^n ^* m &n -^m

so that

17(1)17(1) 77(1)17(1)

O . ' ^fan^tnn -"nn^fen

The normalization of ^ n requires that B n vanish. The results, correct
to the second order in X, for the energy levels and wave functions are
therefore

1(1)(1)

x'o- ) + 7.30

96 APPROXIMATE METHODS

^n = ^n "f" X S "-,

rT(l)rr(l)
-fcmwn n nn n kn

[/rCDr/d)
V _ -"fcm^wn
tr (Js2-*2)(#-

7-31

In most applications it proves convenient to absorb the parameter X
into the function H (1) , in other words, to place X equal to unity in the
above equations.

7b. Perturbation Theory for Degenerate Systems. In the last
section we made the assumption that the unperturbed energy levels
E^ were all different. Problems frequently arise in which two or more
orthogonal eigenfunctions have the same eigenvalue. Such eigen-
values are called degenerate. Suppose, for example, that

= Eft 7-32

and

Efa 7-33

Then, if c\ and c% are any constants such that c*ci + c*C2 = 1,

c 2 fa) 7-34

so that GI\!/I + ^2^2 is also an eigenfunction. Hence, if there are two
eigenfunctions having the same eigenvalue, there are an infinite num-
ber of eigenfunctions having that eigenvalue.

A set of n eigenfunctions is said to be linearly independent if there
is no relation of the type

cnh + c 2 fa + + Cntn = 7-35

connecting them. In the above example there are no three eigen-
functions which are linearly independent since all the eigenfunctions
are expressible in the form c^i + 02^2. If the number of linearly
independent eigenfunctions corresponding to a given eigenvalue is n,
the eigenvalue is said to be n-fold degenerate. Once a set of n linearly
independent eigenfunctions has been chosen, any other eigenfunction
with the given eigenvalue can be expressed in the form

+ C2$2 H ----- h C n \l/ n 7-36

where the c's are constants.
Now suppose that we wish to find the solution of

(Ho + XH (1) )* - B* 7-37

DEGENERATE SYSTEMS 97

where the eigenvalue approaches an m-fold degenerate eigenvalue of

7-38

as X approaches zero. With no loss of generality we may assume that
the m linearly independent eigenfunctions corresponding to this eigen-
value are tf, $ tf&, so that S? = E% = = E m . We may also
assume that ^?, ^2 * * * tm have been made orthogonal. As X approaches
zero, ^ must approach some solution of 7-38 whose eigenvalue equals
/?, that is, some linear combination

+ c m ^ m 7.39

where the c's are constants. This linear combination may be spoken
of as the " zeroth-order " approximation to \t/. The expansion of E
and \t/ in powers of X must therefore be of the form

E = #? + XB (1) + X 2 7 (2) 740

741

Substituting these expansions in 737 and equating coefficients of like
powers of X gives

m m

HO E c $j -Ei E Qffrj 742

y=i y-i

(H - ^)^ (1) = E cj(E - H< 1 >)^ 743

.7=1

Since JS? = El = = E? m , equation 742 is already satisfied. As
before, let us put

744

where ^ = JV*H (1 Vy dr, and the A's are constants to be deter-

mined. Then

745

i y

Substituting these expressions in 743 gives

ffVctf - E( E c*H5')*? 746
y-i i k-i

98 APPROXIMATE METHODS

If j > m, equating the coefficients of $ on both sides gives

(E - 01) Aj = ECJ - H$c k 747

**!

But, for j < m, E$ = 1%, so that

748

We thus have the system of m simultaneous equations for the c/s

+ ffgca + + Hc* =
+ (H - E)c 2 + + fffik. =

= 749

A possible solution of this set of equations is c\ = c% = = c m 0.
According to a theorem of algebra (Appendix IV) this is the only solu-
tion unless the determinant of the coefficients of the c's vanishes, that
is, unless

as? flffi

= 7-60

flffl

This is known as the secular equation. Since the flyj^s are known
constants, it is an equation of the rath degree in E (l \ and therefore
has m roots. Let these roots be E^\ E%\ E$. Unless some of
these roots happen to be equal, there are therefore m different perturbed
states whose energies approach ^ as ^ approaches zero.

In order to find the eigenfunction corresponding to the root
we substitute this value for E (l) in the set of equations 749 and solve

for the ratios -i *.... A knowledge of these ratios, plus the nor-
malizing condition

+ C*C2 + + CmCm = 1 7-51

is sufficient to determine completely the c's and hence the zeroth-order
eigenfunctions v?/, the subscript indicating the root of 7'50 to which the
eigenfunction corresponds.

THE VARIATION METHOD 99

The first-order eigenfunction may now be found by equating the
coefficients of the remaining $ (j > m) in equation 746. The result is

- E c*flg>; Ay = SK 7-52

In order that ^ be normalized we must put Aj = (j < m). Hence,
if fa is the eigenfunction whose zeroth-order approximation is <pi, the
first-order perturbation theory gives

+ X a (--0 7-53

7-54

for the perturbed eigenfunctions and eigenvalues.

7c. The Variation Method. Another, completely different, method
of finding approximate solutions of the wave equation is based upon
the following theorem: If <p is any function of class Q such that

I <p*<p dr = 1, and if the lowest eigenvalue of the operator H is J5? , then

/V"!!?? dr > E 7-55

The proof of this theorem is very simple. Consider the integral
(H - EQ)<P dr = /Vlfy dr - E Q f<p*<p dr

- E Q 7-56

J

If we expand the function <p in a series of the eigenfunctions ft,
ft of H, we will have

/
J

(H - E )<f> dr = (LcftT)(H - E )(E*fc) dr 7 ' 57

Since the ft's are eigenfunctions of H, H^- = JStV'i, and so equation
7-57 becomes

/V(H - E*b dr - f (cjtf ) ( (< -
/ t/ f - <

7-58

100 APPROXIMATE METHODS

Now c*c t - is a positive number, and by definition Ei > E Q , hence

> 7-59

JV*(H -

and

r > E 7-60

The equality sign can hold only when <p = ^ > where \l/ Q is the eigen-
f unction with the eigenvalue EQ.

The method of applying this theorem is equally simple in principle.
A trial eigenf unction #>(Ai, \2, ), normalized to unity, is chosen, this
trial eigenf unction being a function of a number of parameters AI,

X 2 , . The integral J = I <p*H<p dr is then found. The result will,

of course, be a function of the parameters AI, A 2 , . The integral J
is then minimized with respect to the parameters. The result is an
approximation to the lowest eigenvalue, and the corresponding <p is
an approximation to the corresponding eigenfunction. By taking a
sufficiently large number of parameters in a function of a well-chosen
form, a very close approximation to the correct eigenvalue and eigen-
function can be found.

In simple cases, and with a judiciously chosen trial eigenfunction, the
results of the variation method are identical with the results obtained
by the solution of the Schrodinger equation. As an example, let us
consider the harmonic oscillator, for which the Hamiltonian operator is

H- **+^ 7-61

8w 2 m dx 2 2

If we try <p = ce~ Xa; ', the condition that <p be normalized is fulfilled if

/2A
we put c = + / . Then

and

/+ r ^2^2 .2)L -I

J = / - rT- ( 4x2a;2 - 2X)e- 2X ^ + C - x 2 e- dx 7-62
^_oo L OTT in 2 J

Upon evaluating these integrals (Appendix VIII), we obtain

GROUND STATE OF HELIUM ATOM 101

The condition that J be a minimum is

_ Q or x 7.64

_

d\ Sir 2 m 8X 2 h

so that the lowest eigenvalue is

= > 7-65

and the corresponding eigenfunction is

-v&*
h 7-66

The fact that these results are identical with those of section 5e is, of
course, due to the " well-chosen " form which we took for our trial
eigenfunction.

It is also possible to use the variation method for the calculation of
energy levels and eigenfunctions for excited states. After we have
obtained an approximation to the ground state by the above method,
we choose a second trial eigenfunction which is orthogonal to the one
that we obtained for the ground state. A repetition of the above pro-
cedure will then give us an approximation to the first excited state.
This process may be continued indefinitely, although the errors involved
will be cumulative.

7d. The Ground State of the Helium Atom. As a further example
of the use of these approximate methods we shall calculate the energy
of the ground state of the helium atom both by the application of the
first-order perturbation theory and by the variation method. The
Hamiltonian operator for the helium atom (or for other two-electron
atoms such as Li + , etc.), is, if we neglect the terms arising from the
motion of the nucleus,

h 2 ,_, . _, Ze 2 Ze 2

e

,2

H = - rV (Vf +V|) -- + - 7-67

Sir m 7*1 r 2 ri 2

where Vi and V 2 are the Laplacian operators for electrons 1 and 2;
ri and r 2 are the distances of these electrons from the nucleus; r*i 2 is
the distance between the two electrons; and Z is the nuclear charge.
For the purpose of simplifying the calculation of the integrals involved
in problems of this type it is usually more convenient to use atomic
units. The transformation to atomic units is obtained by expressing

h 2

distances in terms of the Bohr radius a = 7-5 ., . We therefore

4ir we"

102 APPROXIMATE METHODS

a 2 i a 2

write ri = Oo-Ri, r 2 = flo#2, 7*12 = o#i2, 7-5 1 ^2 > etc - With
this transformation, the Hamiltonian operator becomes

1 2 2 Ze 2 Z^ e 2

" ""

e 2

or, in units of
a

7-69

In applying the methods of the perturbation theory to this problem
we set H = H + H (1) , where

H = -iCV? + VI)-|--J- 7-70

K\ 1%

7 ' 71

The zeroth-order eigenfunctions are the solutions of the equation

ff ^o = 0^o 7 . 72

If we now set * = * (1)^ (2), E = JB(1) + S(2), equation 7-72 is
immediately separable into the two equations

7-73

(2) =

These equations are just those for a hydrogenlike atom with nuclear
charge Z. For the ground state of the helium atom, we therefore have
I i

^ (1)>== v^ Z e ' * (2) = 7; Z e
z 3

7.74

7T

where #i fl (H) is the energy of the ground state of hydrogen, and, in

02

ordinary units, is \ . The first-order correction to the energy is,

GROUND STATE OF HELIUM ATOM 103

according to equation 745

e 2 Z^ C re

OeMT 2 J J

^ TI ^ T2 7-75

where

dri ^i sin 61 dfi

dr 2 == fil sin G 2 dE
Written in the above form, the integral cannot be evaluated because
of the presence of the term . According to the theorem proved in

Appendix V, this quantity can be expanded in terms of the associated
Legendre polynomials as

7 ' 76

where ,B< is the smaller and R > is the larger of the quantities RI and
R 2 . The wave functions themselves do not involve the angles ex-
plicitly; in other words, only the constant functions PQ (cos GI) and
PO (cos 62) are involved in the wave functions. Since the associated
Legendre polynomials are orthogonal, all the terms in the summation
will vanish except those for I = 0, m = 0. For these terms P (cos 6)
= 1, so that equation 7-75 reduces to

rrfii e * z * C re~ 2ZR *e- 2ZR * J J

E = ~ "T / / - 5 - d 'i d '* 7-77

OQ 7T J J /t>

The integration over the angles gives a factor (4*-) 2 , so that we have
only the integral over RI and J?2, which may be written as

\dR l 7-78

which may be evaluated in a straightforward manner to give

2
#ci) 6 z L. 7 . 79

OQ

To the first order, the energy of the lowest state of helium (or helium-

104 APPROXIMATE METHODS

like atoms) is therefore

E = E Q + E = (2Z 2 - fZ/- -

\ 2 o

The energy of the ground state of He + is Z 2 E ls (Il). The first ioni-
zation potential of helium, that is, the energy necessary to remove one
electron from the atom, is thus calculated to be

(Z 2 - |Z)# l8 (H) - f Si. (H) = |(13.60) = 20.40 electron volts

The observed value is 24.58 e.v., so that our calculated value is in error
by 4.18 e.v. or about 16 per cent. Our results will look better if we
compare the calculated and observed total binding energy. These
values are: calculated -^-(13.60) = 74.80 e.v., observed 78.98 e.v.
an error of 4.18 e.v. or about .5 per cent. The actual error is the same
in both cases; the percentage error is, of course, decreased in the
latter case.

For the two-electron atoms Li + , etc., the percentage error is consid-
erably less than for helium, since the interactions between the electrons
and the nucleus become relatively more important than the interaction
between the electrons. In 7-80 we see that for helium the perturba-
tion energy is -fg of the zeroth-order energy. As this is certainly not
a " small " perturbation, we should not be greatly disappointed at the
failure of the first-order perturbation theory to give more accurate
results. We shall now show how the binding energy of the helium
atom can be calculated to any desired degree of accuracy by means of
the variation method.

In order that we may choose a reasonable trial eigenfunction for use
in t,!u> variation method, let us consider that one of the electrons of the
helium ai.mn 5s in an excited state and the other in the ground state.
The electron in thr ground state is subjected to the full attractive force
of the nucleus, so that ^ u (l) should be essentially the same as in 7-74.
The electron in the excited state, however, moves essentially in the
field of a nucleus of ch'irgo c, as the (Is) electron screens the nucleus
more or less completely, and so ^(2) for an excited state should ap-
proximate more closely a hydrogen wave function with Z = 1. These
considerations suggest that a good trial eigenfunction would be

9 - 6 -3 f <*ri-ai> 7.8!

7T

where Z f is between 1 and 2, the best value being determined by mm-

GROUND STATE OF HELIUM ATOM 105

imizing the energy with respect to Z'. The functions written above
are the solutions of the equation

7-82
where, in units of ,

e 2

E' = 2Z' 2 E ls (H)
We must evaluate the integral

E = v*H<p dri dra 7-83

where H is given by 7-G9. We can write H^ as

= -i <VS + VI) -

+ p 7-84

^12

BO that the integral 7-83 is reduced to
B=E> -(Z- Z>)

The two integrals in the brackets are equal, as they differ only in the
interchange of the subscripts. The first of these is

dft 2 = Z r 7-86

The last integral is identical with 7-75 if 2 is replaced by Z'. The
result for the energy is thus

2 2

= i' - 2Z'(Z -Z')- + Z'-

a (i

= [2Z' 2 +4Z'(Z - Z') - Z']tfi.(H)

7-87

According to the theorem of the variation method, we obtain the best
approximation to the true energy by giving Z r the value which will

106 APPROXIMATE METHODS

make the energy a minimum. This condition requires that

ftp 1

, = (-4Z' + 4Z - f )tfi.(H) -

or

z ' ~ z - A 7>88

Substituting this result in 7-87, we obtain for the energy the value

E = 2Z /a ffi.(H) - 2(Z - A) 2 #i.(H) 7-89

For the first ionization potential of helium we therefore obtain
[2(fi) 2 - 4]# U (H) = 1.695(13.60) = 23.05 e.v.

The discrepancy between calculated and observed values is thus re-
duced to 1.53 e.v., or about 6 per cent. The total binding energy is
calculated to be 77.45 e.v. as compared to the experimental value
78.98 e.v., or an error of about 2 per cent.

By introducing more parameters into the trial eigenfunction <p, we
can approach more and more closely to the experimental result. Par-
ticularly good results are obtained, even with quite simple trial eigen-
functions, if the variable Ri2 is explicitly introduced into the trial
eigenfunction. For example, Hylleras, 1 using the trial eigenfunction

9 = A { -*'<*+*> (1 + cfii a ) } 7-90

where A is the normalization factor and Z r and c are adjustable con-
stants, obtained a value for the ionization potential which was in error
by only 0.34 e.v. By using the more general function

<p = A { 6 -s'cai+*i> (polynomial in R l9 R 2 , R 12 ) } 7-91

fie was able to reduce the error still further. The function involving
a polynomial of fourteen terms gave a value for the energy which agreed
with the experimental value within 0.002 e.v. It is thus apparent that
the accuracy of the results obtainable by the variation method are lim-
ited only by the patience of the calculator, a very important limitation,
however. In our later work, we shall therefore be satisfied with a smaller
degree of accuracy in most of our numerical calculations.

1 E. Hylleras, Z. Physik, 65, 209 (1930).

CHAPTER VIII
TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY

8a. Time-Dependent Perturbations. In the previous chapter we
discussed the problem of determining the new energy levels and wave
functions for a system subjected to a perturbation which depended only
on the space coordinates of the system. For certain problems, particu-
larly those dealing with the emission and absorption of radiation, we
need to calculate the effects produced by a perturbation which is a func-
tion of the time; therefore, we shall now develop the time-dependent
perturbation theory.

The wave equation, in the form which expresses the manner in which
the complete wave function ^ changes with time, is

-

2irt dt dt

where h = . Let us now' write the Hamiltonian operator as

ZiTC

H = H + H', where H is independent of time and H' is a time-
dependent perturbation. The unperturbed eigenfunctions ^ satisfy
the equation

Ho*-tfc - 8-2

ot

- t

and are of tne form ^(<Z> t) = ^2(<z)e * . In order to obtain a solu-
tion of 8-1 we expand the function <& in terms of the unperturbed eigen-
functions ^nt with coefficients which are functions of the time; that is,
we write

, = 2X(0*2(, 8-3

Substituting this expression for V in equation 8-1 gives

dr

84

Since the unperturbed eigenfunctions ^2 satisfy equation 8-2, the above

107

108 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY
equation immediately reduces to

r*2 8-5

t

Let us now multiply both sides of this equation by ^^T and integrate over
coordinate space. Then

XX /

*2TH'*2 dr = HE

In any particular problem we will thus have a set of simultaneous differ-
ential equations which can be solved to give explicit expressions for the

8b. The Wave Equation for a System of Charged Particles under
the Influence of an External Electric or Magnetic Field. The most
important problem to which the time-dependent perturbation theory
will be applied is that of radiation. In order to discuss radiation theory
we need the Ilarniltonian operator for a charged particle in an electro-
magnetic field. In deriving the classical Hamiitonian function it is
more convenient to use the vector potential A and the scalar potential <p
rather than the electric and magnetic field strengths E and H. The
relations between these quantities are given by the equations

H = V*A: E = A - V<? 8-8

c dt

where c is the velocity of light.

A particle of mass m and charge e moving with a velocity v in an elec-
tromagnetic field is subjected to a force

F = e (E + - [v*H])
c

The equations of motion are therefore

m~= -e^r-- + '{- H z --//) 8-9

dt 2 dx c dt

c

d 2 y d 2 z

with similar expressions for z and -$. Using the relation H = V X A,

at at

these equations become

d 2 x d<p e dA x eTdy /dAy dA

m$ = e r ~ T I

at ox c dt c[_ m dt\dx dy

THE WAVE EQUATION 109

etc. It is not difficult to demonstrate that these equations of motion
are derivable from the Lagrangian function

nr

From the definition of generalized momentum, pi = , we see that

%

sjy p

p x = m + - A x , with analogous values for p y and p z . The Hamil-
dt c

tonian function is therefore
dx dy dz

i\-/dx\ 2 /dy

LU) + (*

or, in terms of coordinates and momenta,

m\
2

The procedure for constructing the Hamiltonian operator is identical
with that followed before; in the classical Hamiltonian function the

h d d

momentum p x is replaced by . = ih , etc Operating on a

2irl dx dx

wave function ^, the first term in the Hamiltonian gives

x

c dx c dx

+ ifi-A x + ih-A x + -z\A x r ] 8-14

C dx C dx C /

After collecting terms, and expressing our results in vector notation, we
see that the Hamiltonian operator is

p p p \

H = ~[ ft 2 V 2 + ih- V A + 2t'ft~A V + - I A 2 J + &p 8-15
c c c I

110 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY

For an electromagnetic field such as that associated with a light wave,
V A = and <p = 0. Since the perturbation of a system by a light

o

wave will be a small perturbation, the term -$ |A| 2 may be neglected in

discussing radiation, although it must be retained when discussing per-
turbations due to strong magnetic fields. For a system of charged
particles, with an internal potential energy F, we will therefore have
the Hamiltonian operator H = H + H', where

Ho = -L U? + V-, H' = Z--iMj V; 8-16

The operator H is just the operator for the system in the absence of an
electromagnetic field; the perturbation H 7 may equally well be written

aaH' = -E^"A,-.p,,

y ntjC

8c. Induced Emission and Absorption of Radiation. Let us con-
sider an atomic or molecular system subjected to the perturbation H 7 of
an electromagnetic field. For simplicity, we first will assume that the
field is that of a plane-polarized light wave with A y and A z equal to
zero.* We shall need to calculate the values of such matrix elements as

8-17

Since molecular dimensions are of the order of 1/1000 the wavelengths of
visible light, a sufficiently good approximation for our present purposes
will be to take A as constant over the molecule. The matrix elements

* That this situation represents a plane-polarized wave may be seen as follows:
We take A "L

A x

0-0

This represents a wave moving in the z direction with a velocity equal to c. The
associated electric and magnetic fields are then

c dt

The electric and magnetic fields therefore have equal amplitudes and are at right
angles to each, other as well as to the direction of propagation, and hence represent
a plane-polarized light wave moving along the z axis with a velocity c.

iA '2 ( -*^
c * 8m *A c/

H - VxA J Ajsin 2*v(t - -J

INDUCED EMISSION AND ABSORPTION
can then be written as

Ill

A

. A x e

c^

8-18

In order to obtain usable results, we wish to express the matrix elements
\

of in terms of those of Xj. Let us for the moment consider that the

3

^'s are functions only of one coordinate x. Then
the equations

and $J satisfy

8-18a

8-18b

If we multiply the first of these equations by x$ and the second by
a^^T an( i subtract, we obtain, upon integrating the resulting equation,

- f (B - -

The first two terms may now be integrated by parts; since the wave
functions vanish at infinity, we have

dr 8-18d

or

(f - * f) * - f < - - ->

of we integrate the first term by parts, we see that it is equal to the
second. For this one-dimensional example, we have thus obtained
the result

- c*. -

8-18f

This result can be generalized. The matrix element (^JJf |H'|*JJ) may
therefore be written as

(ST|H'|JD - - -A x (E m - EnlX^e*^ 1 8-18g

112 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY
where X mn = (^kZX'l^n) is the matrix element for the x compo-

3

nent of the dipole moment.

If we now assume that the system was originally in the state n,
so that c n = 1 and all the other c's are zero at the time t = 0, we have,
for times sufficiently small so that all the c j s are negligible except c n :

dr 1

2 *m(E m - E n )e * 8-19

dt cfl z

If the light has the frequency j>, the time dependence of A x may be
expressed as

A x = Al cos 2irrf = |A2(e 2 "" + e~ 2 "' 1 )
Then

,E m -E n -hy]

n OT e ~~*~ 8-20

etc

Integrating, and choosing the constant of integration so that c m equal
zero when t = 0, we have

vm 2ch ~ x """" '" {(E m - E n + hv) ' (E m -E n - A,)J

Let us consider that E m > E n , so that the transition corresponds to
absorption. c m will be large only when the denominator of the second
term in brackets is nearly zero, that is, when E m E n ~ hv. The
probability that the system will be in, the state m at time t will be the
value of the product c^c m . The first term in brackets being neg-
lected, this product is

. s \E m - E n -h

sm^ 1\

\2

E m - E n

8

So far we have considered only a single frequency *>. To obtain the
correct value of c^c m we will have to integrate over a range of fre-
quencies. Since c m c m is very small except for frequencies such that
E m E n ^ hv, it will be satisfactory to integrate from o to + oo
and regard A as constant and equal to A x (v mn ). This integration gives

2 2

^m ~ c2ft2 |-,.

where E m E n has been replaced by

INDUCED EMISSION AND ABSORPTION 113

Equation 8*23 has been derived upon the assumption that the light
wias plane polarized. In the general case in which A y and A z are not
zero, equation 8-23 will, of course, contain additional terms in A v and
A z . In calculating c^c m the cross terms will vanish because of the
randomness of the phase differences between the various components
of A; the final expression for c^c m will therefore be

&m = ^ {\A(* mn )\ 2 \X mn \ 2 + \A v ( Vmn )\ 2 \Y mn \ 2

+ \A t ( Vmn ~)\ 2 \Z mn \ 2 }t 8-24
If the radiation is iso tropic, then

Vmn ) 2 8-25

We may express |A(v mn )| 2 in terms of the radiation density p(v mn ).
From Equation 8-8,

/ \

u Omn) sin

Then

since the average value of sin 2 2wvt is ^. In electromagnetic theory, it

j _ 2 c 2

is shown that p(v mn ) = E*(v mn \ so that |^0mn)| 2 = - y- p(^ mn ).

47T 3 7T^ mn

The product c^c m may now be written in terms of the radiation den-
sity as

c* m c m = ^ { |X mn | 2 + | Y mn \ 2 + \Z mn \ 2 } p(v mn )t 8-26

Since the probability that the system will be in the state m is zero at
time t = 0, and is the value given by 8-26 at time , the probability
that a transition from the state n to the state m will take place in
unit time, resulting in absorption of energy from the electromagnetic
field, is

2 p(Vmn) 8-27

where

\R mn \ 2 = \X mn \ 2 + \Y mn \ z + \Z mn \* 8-28

If the system is originally in the state m, then the same treatment
shows that the probability of transition to the state n, resulting in the
emission of energy, due to the perturbing effect of the electromagnetic

114 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY
field, is

8-29

8d. The Einstein Transition Probabilities. The coefficients B m ^ n
and Bn-*m are known as the Einstein transition probability coefficients
for induced emission and absorption, respectively. Since a system in
an excited state can emit radiation even in the absence of an electro-
magnetic field, the completion of the theory of radiation requires the
calculation of the transition probability coefficient A m ^ n for spon-
taneous emission. The direct quantum-mechanical calculation of
this quantity is a problem of great difficulty, but its value has been
determined by Einstein 1 by a consideration of the equilibrium between
two states of different energy. If the number of systems in the state
with energy E m is N m , and the number in the state with energy E n is
JV n , then the Boltzmann distribution law states that at equilibrium

_5

Nm e kT -^

T^ = ^ = e w 8-30

where T is the absolute temperature and k is the Boltzmann constant.
Expressed in terms of the transition probability coefficients discussed
above, the number of systems making the transition from m to n in
unit time is

Similarly, the number of systems making the reverse transition is

Since at equilibrium these two numbers are equal, we have, after elim-

N
inating the ratio ~~ by means of equation 8-30,

N n

i

p<*) - A ~ 8-31

B m -^ n e kT + B n ^ m
Using relation 8-29, the energy density may therefore be wmten as

8-32

-I

1 A. Einstein, Physik. Z., 18, 121 (1917).

THE EINSTEIN TRANSITION PROBABILITIES 115

The energy density in a radiation field in equilibrium with a black body
at a temperature T may be calculated by the methods of quantum
statistical mechanics (Chapter XV) and, as mentioned in Chapter I,
is given by Planck's radiation distribution law as

n(v ^

PWmn)

c 3
Comparing 8-32 and 8-33, we see that

mn

m >n '

c 3
or

q o

4m-*n = 3 \Rmn\ 8 '34

To the degree of approximation used above, the coefficient for spon-
taneous emission depends only on the matrix element for the electric
dipole moment between the two states. If the variation of the field
over the molecule is not neglected, there will be additional terms in the
expression for A m _ n , the first two additional terms being those corre-
sponding to magnetic dipole and electric quadripole radiation. In-
cluding these terms gives 2

8-86

We may estimate the relative orders of magnitude of these terms as
follows. Disregarding the constant term, we have

|(mk|n)| 2 ~ (ea ) 2 ~ 6-5 X 1(T 36 c.g.s.

2

l ^f |(m|err|n)| 2 ~ 6-8 X KT 43 c.g.s. (X = 5000 A)

We thus see that the probability of transition due to magnetic dipole
or electric quadripole radiation will be negligible in comparison to the
probability of transition due to electric dipole radiation. The higher
terms in 8-35 will therefore be of importance only in those cases in which

2 E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, p. 96, Cam-
bridge University Press, 1935.

116 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY

the electric dipole matrix element (ra]er|ra) vanishes because of the
symmetry properties of the states m and n.

The actual intensity of radiation of frequency v mn due to sponta-
neous emission will, of course, be

8-36

8e. Selection Rules for the Hydrogen Atom. According to the
results derived above, the only transitions of importance in the hydro-
gen atom will be between those states a and 6 for which

(a\ei\b) = e{i(a\x\b) + j(a\y\b) + k(a\z\b)}

is different from zero. The eigenfunctions for the hydrogen atom may
be written as

so that we must investigate the values of such integrals as

/ tn, I, ro^n'.l . m' dr = J t^ n , tt m (r COS 0)tfV f r.m' dr

This integral may be written as the product of integrals in r, 0, and <p.
The integrals in r will be non-vanishing, so that we may concentrate
our attention on the integrals over the angular coordinates. For the
z component of the electric dipole moment we must therefore investigate
the integral

fp\

cos e Pf l sin 6 dd e i(m ~ m ^ d<p 8-37

From the recursion formulas for the associated Legendre polynomials
(equation 4-85), we have

cos OPfr' 1 = ~ 7 , ' , P\?+( + . \ P\^_{ 8-38

21+1 21+1

The integral 8-37 is therefore non-vanishing only if

m = m'; l=l'l 8-39

These relations are the selection rules for the emission of light polarized
in the z direction. Rather than calculate the matrix components for
x and y separately it is more convenient to calculate those for the com-
binations (x + iy) = r sin e l * and (x iy) = r sin 6 e~~ % *. For the
combination. (# iy) we have the integrals

fp| m| sin 9 P\T'\ sin dO J*

840

SELECTION RULES FOR THE HARMONIC OSCILLATOR 117

The integral over <p is non- vanishing only if w = w' + 1. From the
recursion formula 4*86

sin P\r'l = ^~ \P\^ 1 - P^.[ +l } 841

we see that the integral over 6 is non- vanishing only if I = I' 1.
Similarly for the combination (x + iy) we have the selection rules
m=m / 1, Z = Z' 1. The selection rules for the hydrogen atom
may therefore be written as

AZ = dbl; Am = 0, 1 842

The selection rules for the different types of polarization are, of course,
significant only when there is a unique z direction, due, for example,
to the presence of a uniform magnetic field. This subject will be dis-
cussed more fully in the following chapter, where the Zeeman effect is
considered. It is apparent from the derivation of the above selection
rules that they are not limited to the hydrogen atom but are valid
for any central field problem where the angular portion of the wave
function is identical with that of the hydrogen atom.

8f. Selection Rules for the Harmonic Oscillator. Let us now con-
sider the system consisting of a particle moving along the x axis with a
harmonic motion. Let the charge on the particle be +e, and let the
charge at the position of equilibrium be ~e. The instantaneous value
of the electric dipole moment will then be ex. The wave functions for
the system will be the eigenfunctions of the harmonic oscillator (sec-
tion 5e). A transition between the states n and m will be possible
only if the integral

= e I tn

843

is different from zero. From the recursion formula (equation 4-99)

+ f tf^H-i 844

we see immediately that the integral 843 will be zero unless m = n rb 1.
This is the selection rule for the harmonic oscillator; only the funda-
mental frequency v can be emitted or absorbed by this system. We
shall later require the exact values of the integrals in 843. The wave
functions are

*(*) = N n e~*H n ($ 845

where

118 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY

so that

(n\ex\n + 1) = N n N^~ Je^Hr,(^H n+l (^ dx 846

By the use of the recursion formula 844, this reduces to
(n\ex\n + 1) = N n N n+ i -^ fe~^H n (^(n + 1 )#() dx

(n + 1) )] 2 <fe = - (n + 1) 847

a

Introducing the explicit expressions for the normalizing factors, we
obtain

| i

848
Similarly,

(n\ex\n - 1) = N n N n -.i -7= fe^H n (^H n ^(^ dx

849

8g. Polarizability; Rayleigh and Raman Scattering. If a is the
polarizability of an atomic system, then an electric field E induces a
dipole moment R = aE in the system. Let us consider classically a
system in which an electron of charge 6 is bound elastically to an
equilibrium position at which there is a charge +e. If the system is
subjected to an alternating electric field of strength E = E cos otf,
the classical equation of motion is

m -rx + kr = eE Q cos wt 8*50

at

where r is the displacement of the electron from the origin and k is the
force constant of the forces binding the electron to the equilibrium
position. The steady-state solution of this equation is

e

E cos at

eE cos wt m

* * k-mu 2 " a? 2 ~a> 2

/*
where = \/ The dipole moment of the system is R = er;

POLARIZ ABILITY 119

the classical polarizability of the system is therefore -s - o . If we

co w

now write r = A cos otf, then the acceleration of the electron is

i2-

r-j = Aco 2 cosori. The time average of the square of the accelera-
tion is

(^2 r \ 2 _

^2 } = \A\ V cos 2 co* = JU| V

8-52

According to classical electromagnetic theory, an electron moving with

t\ 2 2

acceleration a radiates in one second the amount of energy 3--

oc

Classically, the system described above would therefore radiate light

CO

of frequency v = , the amount of energy radiated per second being

*Wv

3c 3

According to quantum mechanics, if a system is in a state a, then
the probability of a transition to a state 6 with the emission of light
of frequency v j is, as we have seen earlier in this chapter,

3k ak\

12

3c 3 A
The amount of energy radiated per second is

8-54

Let us now calculate the dipole moment associated with the transi-
tion between the states a and 5, which we define as

2(a|erj&) cos - t = 2(a|er|6) cos w^* 8-55

where w a & = -r = ^-r - . By analogy with the classical argu-
n n

ment, we therefore calculate the rate of emission of energy of frequency

120 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY

v a b to be

V ^7y2 / ^,A AJ.^.4,,4

- \R ab 2 8-56

which is identical with the quantum-mechanical result in equation
8-54. This is a justification for the definition given in 8-55 for the
dipole moment associated with the transition between two states a
and 6. R a & may also be written as R ab = 2/fe{(^*|er|^){, where
Re means that we are to take the real part of the quantity in brackets.
For the dipole moment associated with a given state

A system which in the absence of an electromagnetic field is in the
state represented by ^ has, in the presence of the field, the perturbed
wave function

*a = *a + Ec 6 *g 8-57

b

where c&, according to equation 8-21, is given by the relation

f i*< t <5p,l

Cb ~ Vr *8 < 6 M a ) |-z- + i - + constant 8 ' 58

ACTl [&ba + &ba J

where E ba = Eb -~ E a ; e = Jiv. The dipole moment associated with
the state ^ a is

8-59

= (o|R|o)+Be'

i * > \^J-J V / V J "| ^/ U I 77, j 7-7

6 CH- L^ba "T" JC/fea

where R = er. In writing this last expression the constant in equa-
tion 858 has been placed equal to zero. This can be done in this case
without loss of generality, as the inclusion of this constant would give
us no additional terms which would be functions of the perturbing
fields. Equation 8-59 may be simplified to

(a|R|a) + E ~- ~ -r- X

|a).Agsin~< 8-60

Since EO = --- r- = AQ sin - , the dipole moment as a function
c at en n

POLARIZABILITY 121

of the electric field becomes

R = (o|R|a) + (~ -- F~V) (|R|&)(MR[a) E

6 \tiba tiba ~T /

= (a|R|a) + E y-2 ("**0 (& R <0 ' E o

b &ba *

= (a R|a) + 7 E T^-* (a|R|6)(b|R a) - E 8-61

A 6 "6a ~ ^

The first term in this expression is the permanent dipole moment p,
of the system. In order to determine the polarizability, we must
write the remaining terms in the form aE , which can be done by
averaging the vector quantities over all orientations of the system with
respect to the field, assuming all orientations to be equally probable.
The quantity which we must average is of the form RR E. If 6 is the
angle between R and E, this is equal to Rl/i'J \E\ cos 6, which is a vector
in the direction of R. The component of this vector along E is
\R\ \K\ \E\ cos 2 9. Averaging over all values of 0, we obtain for the
magnitude of the vector along E the value ^\K\ |/?| fi|, so that the
average value of the vector quantity RR E is ^R RE. Making this
transformation in 8-61, we obtain

Vba V

8.G2

The second term is now of the form E () , so that we have obtained the
quantum-mechanical expression for the polarizability

8>63

The dipole moment R a contains a term which varies with the frequency
v of the incident light. According to the above discussion, the system
will therefore radiate light of frequency v, which will be in phase with
the incident light and which is therefore called coherent. This co-
herent scattering of light by an atomic system is known as Rayleigh
scattering.

Of more importance for the elucidation of problems in molecular
structure is the study of the phenomenon known as incoherent or
Raman scattering. In order to understand the origin of this phenom-
enon, let us calculate the dipole moment for the transition a * b when
the system is subjected to radiation of frequency v. The wave funo

122 TIME-DEPENDENT PERTURBATIONS: RADIATION THEORY
tions are ^o = ^2 + C A^&

8-64

I

The dipole moment for the transition is therefore

Eab.

i Ekb.

where the coefficients cj and ci are

i \ A o.
.\a) A

+

Cl =

2ch

AS

.jxip-rm -fio *.

+ 6

+

e

8-65

8-66

8-67

Substituting these expressions for the coefficients in equation 8-65, we
obtain for the dipole moment of the transition between the two states
a and 6 the result

2Re

(a|R|6)

.ab M

'

* -

(*|R|6)(fe|R|a) Ag X

f ,*=!, .-5ftJ] .

f4- + | - + Z ^ (

L^fca + ^?4a - CJ 2cft

X

.Sob-i,

Since

'K" 1 "')---^)'

= sin f

8-68

8-69

the expression for the dipole moment R& may be simplified to
= 2(o|R|6) cos27r Vo 5< - Z (ft|R|6)(fc|R|o) A X
sin 2^-^)* Bi

{si

I

+

_ ( a

I Cft

fsin 2v(v + Vab)t sin 2r(i> Vofr)'!

AoX
8-70

POLARIZABILITY 123

Since the summations over k and I are entirely equivalent we may re-
place both of them by a summation over j and write 8-70 as

X

= 2(a|R|6) cos 2ir Vab t + ^ --- ^L X

Vaj + V Vjb +

j V b j V

A
(a|R|j)0'lR|&) * sin27r ^ - *<*)* 8 ' 71

where v a j = -~, etc. The dipole moment for the transition thus con-
h

tains three terms varying with the time. If the matrix element (a]R|&)
is not zero, light of frequency v ab will be emitted. This, of course, is
just the term corresponding to ordinary emission, as stated earlier in
this chapter. We see, however, that if there exists some state j for
which (a|R|j) and (j|R|6) are simultaneously different from zero, then
light of frequency v + v ab and v v ab are also emitted by the system.
The first frequency arises when the system goes from an excited state
to a lower state, thus adding energy to the field; the second arises
when a system absorbs energy from the field.

For a harmonic oscillator, the selection rule for the Raman effect
is the following. Let us assume that the harmonic oscillator is origi-
nally in the state a with quantum number n. Then the matrix ele-
ment (a|R|j) will be different from zero only if the state j has the
quantum number n 1. Similarly, if state b has the quantum num-
ber m, 0|R|k) will t> e different from zero only if state j has the quantum
number m =fc 1. Both matrix elements will be simultaneously different
from zero only if m = n or m = n 2, so that we may conclude that
the selection rule for Raman scattering by a harmoriic oscillator is
An = 0, 2. The first possibility corresponds to scattering of light
*of the incident frequency v] the second corresponds to scattering of
light of frequency v 2*> e , where v e is the fundamental frequency of
the harmonic oscillator.

If we assume that v v a j we may write

V j

(a|RR|6) 8-72

v

According to 8'72, the frequency v + v ab will appear in the Raman
effect if any matrix element of the type (ala^a^), where o^, x% equal
x, y, or z, is different from zero. We shall use this formulation of the
selection rules for Raman scattering.

CHAPTER IX
ATOMIC STRUCTURE

The spectrum of a many-electron atom will in general consist of
hundreds of lines with little apparent regularity. Before the develop-
ment of the quantum theory it was found empirically that when the
observed wavelengths were expressed as frequencies, a set of numbers
could be assigned to a given atom such that all observed frequencies
would be given by the difference of two numbers in this set, although
not all differences would appear in the observed spectrum. In modern
terms these experimental facts are expressed as follows. For a given
atomic system, there exists a set of discrete energy levels, or " stationary
states/' of energy EI, E 2 E n , . Transitions between certain of
these states are allowed, these transitions resulting in the emission of

T^F TyT

radiation of frequency v = - m ~ if the transition is from the state

h

designated by the letter m to that designated by the letter n. In this
chapter we shall determine the number and characteristics of the states
present in a given atom, derive the selection rules regulating the transi-
tions between these states, and calculate theoretically, so far as these
calculations are feasible, the eigenvalues and eigenfunctions of these
states. This program will necessitate the inclusion of several additional
features in our general theory, the first of these being the hypothesis of
" electron spin."

9a. The Hypothesis of Electron Spin. If the energy level diagram
of an alkali metal, which has one valence electron, is drawn from the
spectroscopic data, it is found that most of the levels appear in closely
spaced groups of two. In the alkali-earth metals, with two valence
electrons, the levels may be divided into two series; in one series the
levels appear separately, in the other in groups of three. This multi-
plicity of levels does not appear if the electron is assumed to have no
properties other than mass and charge, as was done in our previous
treatment of the hydrogen and helium atoms. This difficulty was
resolved by Qoudsmit and Uhlenbeck, 1 who introduced into the old
quantum theory the hypothesis that the electron possesses an intrinsic

1 G. Goudsmit and S. Uhlenbeck, Naturwissenschaften, 13, 953 (1925).

124

THE HYPOTHESIS OF ELECTRON SPIN 125

angular momentum , with which there is associated a magnetic

eh
moment of magnitude - . The interaction of the magnetic moment

of the electron with the magnetic fields produced by the motion of the
electrons in their orbits was then found to give rise to the splitting of the
energy levels into the observed multiplets.

The spin of the electron was measured experimentally by Stern and
Gerlach. 2 A beam of silver atoms was passed through a strong inho-
mogeneous magnetic field. It was found that the beam divided sharply
into two beams. One beam was deflected as though each atom were a

eh
magnetic dipole of magnitude - - oriented along the field direction;

the other was deflected like a magnetic dipole of the same strength
but oriented in the opposite direction. There are good grounds for
believing that the total angular momentum due to the electrons mov-
ing in their orbits vanishes for the normal silver atom, and that the
observed magnetic moment is due solely to the single valence electron.
The Stern-Gerlach experiment is therefore a direct measurement of the
electron's intrinsic magnetic moment.

It is impossible to explain by classical mechanics why the beam of
silver atoms is split into two distinct beams; classical mechanics would
predict only a broadening of the beam. If, however, we assume that
the spin angular momentum obeys the quantum-mechanical laws of
angular momentum, and assume that Z = ^, then we would expect that
the beam would be split into two beams corresponding to the states with
m = +i and m = ^.

In view of this fact we make the assumption that the electron has spin
angular momentum, represented by a set of operators S 2 , S^, S 2 , and S 2
which are analogous to the operators M x , M^, M 2 , and M 2 for orbital
angular momentum and obey the same commutation rules. According
to the discussion in the previous paragraph, we assume that there is

h 2
2

"rTT

only one eigenvalue ^(1 + iO 2 ^ or ^ 2 , so that there are only two

eigenvalues of S z , namely, ^ and \ . In referring to a general

2?r 2iir

spin eigenfunction we shall use the quantum number s, writing s for

27T

the eigenvalues of S z . s may therefore have the values ^.
Let us denote, by a the spin eigenfunction corresponding to s =

2 O. Stern and W. Gerlach, Z. Physik, 8, 110; 9, 349 (1922).

126 ATOMIC STRUCTURE

and by the spin eigenf unction corresponding to s = f . These
eigenfunctions then satisfy the equations

3r 2 o z2

C2 " o n 1

1 k 1 % 9-2

In order to find the effect of S x and S y on these eigenfunctions we use the
equations analogous to 3-132 and 3-133. These are, making the proper
substitutions,

rq ^ - h R

(S.-tb,)*-^

( s * ~ is y^ = Q o

(S,+fS> =

- A

From these equations we find by addition and subtraction that

h h

18 S x & = a 94

4?r 4?r

/O Q O _, Q C

~ p &yP . & y O

47T 4?T

The effect of such operators as S^Sj/, etc., can be easily found by the
successive application of the above rules. For example,

i.li 2 ih 2

9-6

16**'

so that

ill h ih

(S X S V - S v S x )a = a = S z a 9-7

2-jr 4?r 2?r

as, of course, it must be if the commutation rules are to be satisfied.
This formal method of treating electron spin was developed by Pauli. 3
In order to fix in our minds more clearly the relation between angular
momentum and magnetic moment, let us consider a particle of mass m
and algebraic charge e moving in a circular orbit of radius a with a
velocity v. The motion of the charged particle is equivalent to a current
flowing along the orbit, the magnitude of the current being equal to the

8 W. Pauli, Z. Physik, 43, 601 (1925).

THE HYPOTHESIS OF ELECTRON SPIN 127

charge divided by the time required for a complete cycle, or

e __ ev
2wa/v 2?ra

According to electromagnetic theory, a current / moving about a loop

I A

of area A is equivalent to a magnetic moment . The magnetic

c

moment associated with the motion of the charged particle is therefore

ev TO? e e

p, = - -- = av = - p 98

2ira c 2c

where p is the angular momentum of the particle. The ratio of mag-

netic moment to angular momentum, - , is called the gyromagnetic

2mc

ratio. The classical result above gives the correct relation between
orbital angular momentum and associated magnetic moment in quantum
mechanics as well. For spin angular momentum, however, we obtain
correct results only if we take the gyromagnetic ratio to be twice as
large. The magnetic moment of the spinning electron is therefore
represented by the operators

V* = $x
me

Ik, = S y 9-9

me

|i. = -S,

me

where e and m are the charge and mass of the electron. In a magnetic
field with the components H x , H y , and H z , the classical energy of a
magnetic dipole is

The corresponding operator is obtained by replacing the /i's by the
operators in 99, so that in a magnetic field the Hamiltonian operator
contains the operator

- (H 9 S, + H y S y + H,S 2 ) 940

me

If the z axis is taken to be the field direction, so that H x and H y vanish,
this reduces to

- ffS, 9-11

me

128 ATOMIC STRUCTURE

The spin eigenfunctions a and ft are therefore the correct eigenfunctions
of the Hainiltonian operator as well as of the operators S z and S 2 . If
the field strength is allowed to become zero this argument still holds,
so that even if there is no magnetic field we still use a and as our spin
eigenfunctions of the Hamiltonian operator.

9b. The Electronic States of Complex Atoms. For an atom of
atomic number Z, the approximate Hamiltonian is

9-12

where V? is the Laplacian operator for the zth electron, r^ is the distance
of the z'th electron from the nucleus (motion of the nucleus can be
neglected in heavy atoms), and r# is the distance between the zth andyth
electrons. The exact Hamiltonian contains additional terms represent-
ing the magnetic interactions of the electronic orbits and spins, which
we will disregard for the present.

e 2

Because of the presence of the terms in the Hamiltonian operator

?Vy

it is not possible to separate the variables in the wave equation. The
problem must therefore be treated by approximate methods, but it is
gratifying to find that all the qualitative features of atomic structure can
be obtained with little actual computation. As the starting point of
the perturbation theory we use the set of one-electron eigenfunctions
which are the solutions of the wave equation with the Hamiltonian

H = --Zv? + ZF(r f ) 943

where F(r t -) is an effective potential for the ith electron. The differ-
ence between 9-12 and 9-13 is treated as the perturbation

H' = - + E" - F(r,) 944

In order to obtain a good set of zero-order functions, F(r t -) should be
taken to be such a function of r z - that U' is as small as possible. If r f -
is much greater than any of the other r/s, the field due to the remaining
electrons will approximate that of a charge (Z l)e at the origin, so

e 2
that V(ri) = -- , in which case we say that the nucleus has been

Ti

screened by the other electrons. If r t - is much smaller than any of the
other r/s, the field due to the other electrons is approximately the field
inside of a spherical shell of charge, which is a constant, so that

THE PAULI EXCLUSION PRINCIPLE 129

V(ri) = --- h constant. As the general features of the electronic

Ti

states are not dependent upon the explicit form chosen for F(r), a
further discussion of the radial part of the wave equation will be reserved
for later sections of this chapter.

The eigenfunctions of the approximate Hamiltonian 9-13 are known
as atomic orbitals. Since we have chosen a potential function which
is a function of r only, they will be similar in many ways to the hydrogen
eigenfunctions. The factors of the orbital depending on 6 and <p are
of the form Pj ml (cos d)e im<f> , and the radial factor is similar to the radial
function of hydrogen. Each orbital may therefore be labeled by quan-
tum numbers n, Z, ra, analogous to those in hydrogen, and we may speak
of 2s orbitals, 3p orbitals, etc. One important difference between these
atomic orbitals and the hydrogen eigenfunctions is that the energy of a
hydrogen eigenfunction is a function of n only, whereas the energy of an
atomic orbital depends on I as well, so that, for example, a 2s and 2p
orbital have different energies. Since the energy of an atomic orbital
is not a function of m, the value of this quantum number is not specified
when we describe an orbital.

9c. The Pauli Exclusion Principle. Having once chosen an appropri-
ate set of atomic orbitals we must combine these orbitals into approxi-
mate eigenfunctions for the entire atom. This can, of course, be done
in a variety of ways, depending on the choice of orbitals from the chosen
set. Let us suppose, however, that we have decided to construct an
eigenfunction corresponding to a state in which the numbers of Is, 2s, 2p,
etc., electrons are all fixed. To be concrete, suppose that the state is
one with two Is electrons and one 2p electron. Let us denote our orbital
by <p(n, lj m\i), giving the first part of the parentheses the three quantum
numbers, and in the other part the number of the electron occupying
the orbital. Then we might take the product

9-15

as an approximation to the eigenfunction for the desired state. We
might equally well have chosen

9-16

ss <p(2, 1, l|l)} or any similar product. Up to the present
we have not considered the spin of the electrons. Since the Hamiltonian
942 does not involve the spin in any way as yet, we may take the
functions a and (8 of section 9a as representing the two possible spin
states of each 1 electron. If we insert the s value after the three quantum
numbers n, Z, m in our symbols for the orbitals the eigenfunctions for our

130 ATOMIC STRUCTURE

chosen state might take on the appearance

9-17

In this way we can write an eigenfunction, in fact a number of eigen-
functions, for any state. But we could by this procedure write eigen-
functions corresponding to states which are never observed. For
example

9-18

would represent a state in which the atom had three Is electrons, all
with positive spin. If we were to calculate the energy of this state in
lithium by the variation method we would find a lower energy than that
of any known level, and yet the theory of the variation method tells us
that our calculated energy must be too high.

The way out of this difficulty was discovered by Pauli. 4 Before
quantum mechanics took on its present form Pauli put forth the follow-
ing exclusion principle: no two electrons can have all their quantum
numbers, including spin quantum number, the same. With the new
quantum mechanics this has been changed to the following statement,
from which the previous one may be derived: every eigenfunction must
be antisymmetrical in every pair of electrons (or more generally in every
pair of identical elementary particles); that is, if any two electrons are
interchanged, the eigenfunction must remain unchanged except for the
factor 1. The electron density distribution for a given state is given
by the square of the corresponding eigenfunction. If two electrons are
interchanged, the electron density distribution must, of course, remain
unchanged. This means that the eigenfunction must either be unaf-
fected by this interchange of electrons or must change only in sign. It is
found empirically that we obtain a correct description of nature only if
we always adopt the second alternative.

It is not immediately apparent that the new form of the exclusion
principle includes the old. Let us, however, consider the determinant

V3!

9-19

If we interchange any two electrons in this eigenfunction, say electrons 1
and 2, then 'two columns of the determinant are interchanged. This
causes a change of sign of the determinant, so that this eigenfunction
satisfies the exclusion principle in the new form. Any antisymmetric
eigenf uaction can be written as such a determinant or as a linear combi-

4 W. Pauli, Z. Physik, 31, 765 (1926).

THE PAULI EXCLUSION PRINCIPLE 131

nation of such determinants. Now if we attempt to set up an eigen-
f unction in which two electrons have the same set of quantum numbers,
and write the eigenfunction in its determinantal form, it is seen that two
rows of the determinant are identical. But such a determinant, one
with two identical rows, vanishes, and the eigenfunction vanishes.
Therefore the old form of the exclusion principle is included in the new.

Accepting the exclusion principle in its new form as a new postulate
to our theory, we are no longer permitted to use the simple products
such as 9*15 in our future discussion, but only linear combinations of
determinants such as 9*19. In other words, though we can construct
eigenfunctions corresponding to states in which a definite set of orbitals
is occupied, we cannot specify which electron occupies a particular
orbital.

In most atoms the energies of the various orbitals lie in the order
Is, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4/, 5d, 6p, 7s, 6d, the Is orbitals
having the lowest energy. When the atom is in its normal state, its
Z electrons occupy the lowest orbitals possible without violating the
exclusion principle. Thus hydrogen, with one electron, normally has
this electron in the Is orbital, but in lithium one electron must go into
the 2s orbital, since there is no way in which three electrons can occupy
the Is orbital without violating the Pauli exclusion principle. The
group of two Is electrons is presen in all the elements of higher atomic
number than 1. Such a complette' group is known as a closed shell.
With beryllium, the 2s shell is completed, and the fifth electron in boron
must occupy a 2p orbital. Since there are three different 2p orbitals,
corresponding to the values 1, 0, 1, for the quantum number w, and
since each of these may have either of two values for the spin quantum
number associated with it, the 2p shell can hold six electrons. In the
elements from boron to neon the 2p orbitals are being filled. When we
come to sodium, however, there is no more room for 2p electrons, and so
the eleventh electron goes into a 3s orbital. Magnesium completes the
3s shell, and from aluminum to argon the 3p orbitals are filled. In this
way we build up all the atoms in the periodic table. There are a few
irregularities which will be discussed in section 9m, where we shall con-
sider the properties of the atoms in the periodic table from the view-
point of their electronic structure.

A state of an atom in which the number of electrons in each shell is
specified is called a configuration. The usual notation for a configura-
tion is the following. We write the designations Is, 2p, etc., for the
various shells, with exponents indicating the number of electrons in the
shell. Thus the normal configuration of lithium is (Is) 2 (2s), and the
normal configuration of phosphorus is (ls) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 3 .

132

ATOMIC STRUCTURE

9d. The Calculation of Energy Levels. 8 For a closed shell of
electrons only one eigenfunction may be written. Thus for a shell of
six 2p electrons we may write

V6!

9-20

sible. ('

No other determinant is possible. [ The factor ;= is a normalizing

\ V6!

factor; the determinant as written will be normalized if the functions v

are normalized. For N electrons the factor is , ) Hence for a

VAT! /

closed shell the eigenfunction is completely determined by the configura-
tion. For an incomplete shell, however, there are always a number of
possible eigenfunctions. For example, the configuration (ls)(2p) has
twelve possibilities. Abbreviating the determinants by writing just
the principal diagonal between bars, these are

*- I /^x>^v1l^\ Sr*. -* j 1 I ^ \ I r\

Z> 2 =

V2T

Dz = V2!

*-4-'

V2!

D 5 = ^
De =

D 8 =

D 9 =

Dlo = "7^

V2!

~

D 12 =

5 The methods used in this and several subsequent sections are essentially due to
J. C. Slater, Phys. Rev., 34, 1293 (1929).

ANGULAR MOMENTA 133

Let us suppose that in the general case there are n determinants
DI, Z>2, D n corresponding to a given configuration. The first-order
perturbation theory for degenerate systems then tells us that the energy
levels will be given by the roots of the equation

H-21

9-21

where Hy = / D*HDj dr] Sij = fD*Dj dr. Equation 9-21 is analo-

gous to 7-50, except that the D functions have not been assumed to
be orthogonal.

9e. Angular Momenta. The large number of states arising from
most configurations makes the direct solution of equation 9-21 a very
difficult task. Equation 9-21 can be simplified by the use of the opera-
tors representing the various angular momenta of the atom. Represent-
ing the operators for the components of orbital angular momentum of the
various electrons by M x i, Mj/i, M z i; M^, Mj, 2 , M Z 2, etc., we define the
operators for the components of the total angular momentum by

M* = M xl + M* 2 + -

M y = M^ + Mj,2 H ---- 9-22

The operator for the square of the orbital angular momentum of the
atom is

M 2 = Mf + M + Mf 9-23

At the same time we define the components of the total spin angular
momentum by

S v2 H ---- 9-24

For the square of the total spin angular momentum the operator is

S 2 = S 2 + S* + Sf 9-25

Finally there are the operators for the components of total angular

134 ATOMIC STRUCTURE

momentum and its square

J TV/T -L Q
x J "*aj i &x

J Z = M Z + S z

T 2 = T 2 4- T 2 4- T 2

J Jx i Jy i^ Jz

Classically all these angular momenta are constant. The quantum
analogue of this is that all these operators commute with the Hamilton-
ian operator H. That S x , S^, S z , and S 2 commute with H is obvious,
since H does not contain the spin coordinates. To show that the others
commute with H we first consider M z . In polar coordinates this
operator is

* = 2^
The only terms in H which involve the ^?'s are the parts of the Laplacian

containing o > which commutes with , and the terms in . Now
d<p dtp Tij

involves <p\ and <p2 only in the combination (<p\ <p2\ so that

7*12

+ (eju2, +

=

Hence M commutes with H. Since H is spherically symmetrical,
MS and M y must therefore also commute with H, as must M 2 . The J's
must likewise commute with H, for they are combinations of operators
all of which commute with H.

These operators do not, however, commute among themselves.
Making use of equations 3-103, and the fact that the various angular
momenta of one electron commute with those of any other electron, we
find that

-^M.

ZTT

= M s 9-27

MULTIPLET STRUCTURE

135

Similar equations hold between S x , S y , and S z , and between J x , J y , and
J z . Just as M 2 commutes with M x , M y , and M 2 for one electron,
M 2 , S 2 , and J 2 commute with their components. Of course M 2 and its
components commute with S 2 and its components. M 2 and S 2 also
commute with J 2 and its components, but the components of M 2 and S 2
do not commute with either J 2 or its components (except like components
such as MS, S x , and J^).

We can therefore select from among the angular momentum operators
various sets which commute among themselves and also with H. Thus
we might choose M 2 , M 2 , S 2 , $ z , and J z or M 2 , S 2 , J 2 , and J z . Now
according to Theorem II of Chapter III, the eigenfunctions of H can be
chosen so that they are simultaneously eigenfunctions of all the operators
in any one of these sets. Moreover, if this is done the matrix compo-
nents of H between eigenfunctions which have different eigenvalues for
any one of these operators will vanish. Therefore, if we take a set of
linear combinations of the D functions described above such that each
combination is an eigenfunction of all the operators of a set of commut-
ing angular momenta, then in equation 9-21 the Hij's and /S t -/s will
vanish except between those combinations which have the same eigen-
values for all operators of the set. The appearance of 921 will then be

000

000

9-21a

if we arrange the combinations of D's in such a way that the combina-
tions with the same eigenvalues for the angular momenta are together.
The shaded portion of the diagram represents non-vanishing H's and S's;
the remainder of the determinant is occupied by zeros. Now a determi-
nant such as the one in 9-21a is equal to the product of the shaded
determinants along the diagonal, so that the roots of 9-21a are just the
roots obtained by putting each of the shaded determinants successively
equal to zero. The solution of 9'21 is thus reduced to the solution of a
number of equations similar in form but of smaller degree.

9f. Multiplet Structure. As long as we use the Hamiltonian 9-12,
which does not contain the spin terms, it does not matter what set of
angular momenta we use to reduce the secular equation 9-21 to the form
9-21a. If the complete Hamiltonian were used, the only angular
momenta which would commute with it would be J 2 and its components.

136 ATOMIC STRUCTURE

Although the other angular momenta do not commute with the exact
Hamiltonian, it is found that usually M 2 and S 2 " almost commute "
with H; that is, HM 2 - M 2 H is small. On the other hand, M*, S z , etc.,
do not " almost commute " with the exact Hamiltonian. It would
therefore seem logical to pick the set M 2 , S 2 , J 2 , and J e as the set of
angular momenta to be used to reduce the secular equation. We shall
presently see, however, that the eigenfunctions of the type described
above are already eigenfunctions of M z and S. For this reason it is
more convenient to choose M 2 , S 2 , M, and S 2 as our set of angular
momenta as long as we are using the approximate Hamiltonian 9'13.
If it is desired to find the eigenfunctions of J 2 and J z , these may be found
later in terms of this set.

Our next problem is then to combine the determinantal eigenfunc-
tions, the D functions, into new trial eigenfunctions which are eigen-
functions of M 2 , S 2 , MS, and S z . We shall first show that these trial
eigenfunctions are already eigenfunctions of M and $ z . Let us take a
general determinantal eigenfunction

D = . \(f>(nilimiSi\l)<f>(n 2 l2m2S2\2) * *l 9-28

vNl

A typical term in the expansion of this determinant is

Operating on this term with M, we find that the term Mi gives the

same product multiplied by mi , the term M z2 the same product

2?r

multiplied by w 2 , and so on. The total result of operating with M

27T

is the sum of these, or (mi + ra 2 + ) times the original product.

2ir

Since this holds for every term in the expansion of the determinant, it
holds for the whole determinant, which is therefore an eigenfunction of

JA Z with the eigenvalue ML , where ML = mi + w 2 + By an

2x

exactly similar argument the D functions are eigenfunctions of S z with
the eigenvalue M 8 , where M s = s\ + 82 + For the list of

27T

possible D functions for the configuration (ls)(2p) given above, we

MULTIPLEX STRUCTURE 137

have, for example,

M,D! = ~ D l M J) 6 = - - D 6 9-29

ZiTf &1T

S z Di = ^-Di S 2 I> 6 = 9-30

2>TC

Let us now consider the effect of M 2 on the D's. A typical term in the
expansion of a D function is

<p(ttiZiWiSi|lV(n 2 Z 2 ra 2 s 2 |2) 9-31

Now

M 2 = (M xl + M* 2 + -) 2 + (M yi + M^ + -) 2

+ (M f ! + M, 2 + -) 2
= M? + Mi + Mi + -

+ 2M 21 M, 2 + 2M,iM, 3 + 2M Z2 M 23 +

Using equations 3-127, 3432, 3-133, we find

(n 1
h 2

+ 1) - m l (m l
[<p(nili, m

-f-

mi - 1, i|l)p(n 2 Z 2 , ^2 + 1, S2|2V(n 3 ; 3 m 3 s 3 J3) ]
+ -} 9-32

If we represent by D{(niZiWi$i)(n 2 Z 2 w 2 s 2 ) } the D function whose
principal diagonal is 9-31, we see that M 2 affects each term in its expan-
sion in the same way, and hence that

h 2
= 71 {[Lli(k + 1) +

47T

+ E [(li(k + 1) - m

Vj

D{ (niZiWii) (nili, mi + 1, ,) (n^y, m y - 1, sy) } } 9-33

138 ATOMIC STRUCTURE

Analogously

h 2
= 7-2 HEk|(kl + 1) + 2 E s i s j ]D{(n l l l m l s l )(n 2 l2m2S2) }

47T i <>y

+ E [(f + *)( - *)(f - *)(i + /)]*/>{ (*iZimii)
<*y

(n^-m;, Si + 1) (rijljmj, Sj - !)}} 9-34

In the same way we find the effect of (M^ d= iM y ) and (S x =t iS y ) on
the D functions to be

db l)]

(mltm* db 1, s^ } 9-35

db ^-)(l =F *)]*

(n^m t -, s t - db 1) } 9-36

We thus see that the D's are not themselves eigenfunctions of M 2 or S 2 .
We can, however, form linear combinations of D's which are such eigen-
functions. First, it is obvious from 9-33 that M 2 does not change the
value of M L or M s . The combinations that we desire will therefore
involve only D functions with the same ML and M s . We therefore put

A = Ec/Z); 9-37

i

where Di is one of a set of D functions arising from a given configuration
and having the same M L and M s . We wish A to satisfy

= \A 9-38

h 2
and we know in advance from 3-127 that A is of the form L(L + 1) 5

4r

Since M L has an integral value, L must also be integral. Substituting
9-37 into 9-38 gives

i i

Multiplying through by each of the D* in turn gives a set of equations
analogous to 749

-X,-,)c< = 9-39

where M = / D*M 2 Z)* dr. These equations are solved in the same
way as 749.

MULTIPLET STRUCTURE 139

Having found a set of A's which are eigenfunctions of M 2 , M*, and
Sz, we repeat this same process with S 2 taking the place of M 2 . The
resulting linear combinations of A's are then eigenfunctions of M 2 , S 2 ,
MS, and S 2 . We thus obtain a set of trial eigenfunctions Bi which
satisfy the relations

h 2

M 2 J3; = Li(Li + 1) 2 Bi 940

4?r

M Z B { = M Li Bi 942

S Z B, = M gj B< 943

.<&7r

The final step is to use the B's (taking only a set with the same I/, S, ML,
and M g ) in equation 9-21 to determine the approximate energy levels
and eigenfunctions of H. Before going into this last step in detail, we
shall note some of the properties of the B functions.

The L and S values of a S function (or an energy level whose eigen-
f unction of H is a combination of B functions) is denoted by a capital
letter showing the L value, the letter being chosen from the same code
which is used in denoting the I value of an atomic orbital. The value
of 2S + 1, the multiplicity of the term or group of levels with the same L
value, is placed as a left-hand superscript. Thus a B function with
L = 1, S = f is denoted by 4 P (quartet P); a B function with L = 0,
S = is denoted by 1 S (singlet S).

If any one of the B's is picked out, a set of associated B's can be found
by applying the operators (M^ db iM y ) and (S x db iS y ) to the original B
function until no new functions arise. All the members of such a set
will have the same L and S values but will have all the possible ML
and MS values consistent with the L and S values. Such a set is known
as a " term." If there is more than one term of a given L and S value
arising from a given configuration, the B's arising from one of the B
functions with a given ML and M s upon multiplication by (M r zM v )
and (S x dz iS y ) may not be the same as those found directly from per-
turbation theory. These may, however, always be combined in such a
way that the terms do not get mixed. In future work we shall assume
that this has been done.

Now let us consider a matrix component of H between any two B

functions, I 5*KLB 2 dr, where E\ and B% have the same value of L, S,

140 ATOMIC STRUCTURE

, and M s . Let B( be the member of the term of B\ derived from B\
by the application of the operator (M x + iM y ), thus having an ML
value greater by 1 than BI has. Let B% be the corresponding function
derived from B%. Then from equations 3 '133

B{ = ~ (L(L + 1) - M L (M L + l)r w (M. + i

/I

B't = ~ (L(L + 1) -
The matrix element of H between B{ and B' 2 is

[L(L + 1) - M L (M L + I)]" 1

iM y ) (M + iM. y )B 2 dr

- M2 -

= [L(L + 1) - M L (M L + l)]

/^?H5 2 dr
= J*B*HB 2 dr

This same relation holds for all the members of the term. Hence if we
use any set of B's belonging to a term, regardless of the ML and M s
values, the matrix of H is the same, and therefore the approximate energy
levels are the same. This is the reason that the ML and M s values of a
B function are not specified.

As an illustration of the construction of B functions let us consider
the configuration (2p) 2 . The possible D functions are:

ML MS
D l -Z>|(21lJ)(21li)J 2

Z> 3 -Z){(211|)(210)( 1

D 4 -D{(21lJ)(210i)J 1

D 6 Z>f(21lJ)(2l6J)| 1 ~1

MULTIPLEX STRUCTURE

141

Dt -Z>|(211i)(2lIJ)J
Di = D{ (2114) (2114)}
Ds = Z> { (211 J) (2114)}
Dg = D{(210j)(210j)t
D 10 = >{(21lJ)(2lIJ)}
D n -DK210J-) (21T4)|
Du-DK2104)(2lTI))
JDis = D{(2l6J)(21l4)}

Z> 15 =

M L

M,

1

-1

-1

1

-1

-1

-2

Since Z>i is the only D function with M L = 2, M^ = 0, it must be a
# function. Applying M 2 and S 2 to it, and using 9-33 and 9-34, we find

s 2 ^ = f^

4?r

= 2 (Di -Z>i) = 0(0

we conclude, therefore, that, having L = 2, 8 = 0, DI is a J5 function
belonging to a 1 D term.

As an example of the procedure when there is more than one D func-
tion with given M L and M s values, consider Z> 7 , D 8 , and Z> 9 . Equa-
tion 9-33 gives

**-!*+*

M 2 Z> 8 =

M 2 Z) 9 = 2f,Z) 7 -

The secular equation for the eigenvalues is therefore
2 - L(L + 1) 2

2 - L(L + 1) -2

2 -2 4 - L(L + 1)

142 ATOMIC STRUCTURE

which is satisfied by L = 0, 1, and 2. For L = 0, equations 9-39 become

2c x + 2c 3 =

2c 2 - 2c 3 =
2ci - 2c 2 + 4c 3 =
which gives Ci : c% : c 3 = 1 : 1 : 1. Hence we put

choosing the magnitudes of the c's so that A? is normalized. It is easily
verified that A 7 is an eigenf unction of M 2 with the eigenvalue zero. In
a similar way we find for L = 1

and f or L = 2

AQ = 4=

V6

We now calculate S 2 D 7 , S 2 D 8 > and S 2 D 9 by the use of equation 9'34.

h 2 h 2

S2D7 = ^ D7+ 4? D *

= tf_ D J^.n

S 2 D 9 =
Hence

7,2

=

It follows that A 7 , A 8 , and AQ are eigenfunctions of S 2 and may be used
immediately as B functions. B 7 = A 7 belongs to a 1 S term; B 8 = AS
belongs to a 3 P term; and B g = Ag belongs to a X Z) term. The complete
list of B functions is found to be

CD) B l = D l CD) B 9 = -L (Z ) 7 _ D 8 + 2D 9 )

( 3 P) B 2 = D 2 ( 3 P) B 10 =

'10

( 3 P) B 3 - 4- (D 8 + D 4 ) ( 3 P) fin - D u

V2

CALCULATION OF ENERGY MATRIX 143

= (D 8 -

( 3 P) B 6 = D 6 (*D) B 18 = | (D 12 -

( 3 P) B 6 = D 6 ( 3 P) J5 14 = D 14

(ig) 7 = -1, (-D 7 + D 8 + D 9 ) ( X

V3

( 3 P) B S =^=(D 7

By operating with (M^ iM^) and (S x it iSj,) on these functions we
find that all the 1 D B functions belonging to a single term. In the same
way the 3 P B functions form a term, and the single 1 S B function repre-
sents a complete term.

A configuration representing a closed shell has been seen to have only a
single D function. This D function is always a B function of the type
1 S ) having L = and S = 0. If we compare the D functions arising
from a configuration which includes one or more closed shells with the
D functions arising from the configuration obtained by omitting the
closed shells, we find that their properties are the same for operations
with the operators M 2 , S 2 , M z , etc. The terms arising from these con-
figurations are therefore the same. Hence, in the treatment of any
atom, the closed shells may be omitted during the processes described
in this section. Thus our example of the configuration (2p) 2 serves as
well for the configuration (Is) 2 (2s) 2 (2p) 2 of the carbon atom.

9g. Calculation of the Energy Matrix. The matrix Hij of equa-
tions 9-21 is formed from integrals of the type

Since the B's are linear combinations of the D functions, these in-
tegrals may be expressed in terms of the integrals

= f

, dr
Let us therefore consider this integral, taking as D t

.. 9-44

144 ATOMIC STRUCTURE

and as Dj

) 945

where we have abbreviated the sets of quantum numbers n, Z, w, s to
the single symbol a, and N is the number of electrons in the configuration.
If the determinants are expanded, 944 and 945 become

946

D, - ; (-iyP,{*(aniMai'|2) -1 947

where P v is an operator which permutes the variables 1, 2, 3, among
themselves, and v is the ordinal number of the permutations. Hy may
then be written

) }] dr 948

Since all the variables are integrated out in each of the terms in this
expression, we may interchange them in any way that we wish without
changing the values of the integrals, provided that all the functions in
any one integral have their variables changed in the same way. Sup-
pose, therefore, that in each of the terms in 948 we change the variables
by the permutation P" 1 which restores the order of the variables in the
first product to the normal order. Then

) ...}]dr 949

since H is not changed by permuting the variables. If we now sum
over M we find that all the terms in the sum are equal, for the application
of P" 1 to the whole set of permutations P, gives the same set in a differ-
ent order. Since there are Nl permutations PJ 1 , we find

)...}dr 9-50
The last product in this expression is of the form

where the aj are the same as the a", or at most in some new order.

CALCULATION OF ENERGY MATRIX 145

If we put

equation 9-50 becomes

H'<j = EIX-D" f {**(aJ|lV*(aS|2) }U*{^(ai|lV(a* 2 |2) } dr
k v **

f{

9-51

Since U^ is a function only of the coordinates of the fcth electron, we
may now integrate over all the variables in the first of these integrals
except those of the /feth electron. The result is

(-iyU(ai, al) = (-1)" JV*(aJ|fc)TMaj|fc) dr 9-52

if aj = a for all i except i = k. Otherwise the integral will vanish
because of the orthogonality of the orbitals. But if one of the P^ makes
all the aj except one equal to the corresponding a{, then any other permu-
tation P M will destroy this equality, since no two aj are equal. Hence
only one term in the summation over v will remain. If we now sum
over ft, we find the total contribution of U& to Hy. This contribution is

(a)

(6) [/(a-, a") if Dj differs from Di only in that a< ^ a?.

(c) If, by permuting the order of the orbitals in Dj by a permutation
P, Dj may be made to satisfy the conditions of (a) or of (6), then the
contribution is that of (a) or (6) times +1 if the permutation is even,
or times 1 if the permutation is odd.

(d) If two or more orbitals of Dj are different from those of D^ there
is no contribution.

Rule (a) is a consequence of the orthogonality of the orbitals and the
fact that no term in H involves the coordinates of more than two elec-
trons. In the second integral of 9-51, we integrate first over all the
coordinates except those of the fcth and Ith electrons. This integration
gives zero unless every a{ = aj except for i = k and i = Z, when it
becomes

(-l)'F(oJ, al; a' k , of)

(-1)" v*(cJ|*V*(of |l)Vj((iS|fc^(aF|0 dr 9-53

146 ATOMIC STRUCTURE

In considering the various permutations ?, we see that, if any P v gives
a set of aj such that aj = a[ except for i = k, Z, then any other permuta-
tion will destroy this property except the permutation which differs
from this P,, only in the interchange of I and fc. . The sum over v there-
fore gives only two terms. Summation over k and I now leads to the
following rules for determining the contribution of V&z to //. This con-
tribution is

() {V(a' t , a/; a' k , a/) - V(a(, a/; a,', )} if Dy = A.

K*

(6) {F(a, (; a' k , aJ) - F(a, <; <C a)j if D,- differs from >,-

ft^m

only in that a f m 7^ a^.

(c) F(c4 a' n ; a'J, <0 - V(a' m , <; < 7 , a^) if Dy differs from D* in
that a f m T* a f m and a^ T a" but is otherwise the same.

(d) That of (a), (6), or (c), if, by permuting the a^, />/ can be made
to obey the conditions of (a), (6), or (c), times +1 if the permutation is
even, or times 1 if the permutation is odd.

(e) Zero if Dj contains three <p's different from those of !).

Since neither H k nor V k i depends on the spin coordinates of the orbi-
tals, these coordinates may be integrated out of the expressions for 11^
and V&z. Using the orthogonality properties of a, and /3 we see at once
that C/(ai, a 2 ) = if si 7^ s 2 and that V(ai, a 2 ; a 3 , a 4 ) = unless
si = 83 and 2 = 4.

As the operators U^ do not depend on the 6 or $ coordinates of the
electrons, these may also be integrated out of 9-52. If we put

?(nfoiwli) = R(nl\i)Y(lm\i) j^jl 9-54

then

r

dr
, s 2 ) 9-55

where

*r 9-56

and where 5(sj, s 2 ) = 1 if s\ = s 2 ; = otherwise. Because of the
orthogonality of {he Y's, the integral

f r*(Zimi|l)r(Z 2 m 2 |l) si

CALCULATION OF ENERGY MATRIX 147

vanishes unless l\ = Z 2 and mi = ra 2 , in which case it is unity. Hence

; n 2 Z 2 ) 5(Zi, Z 2 ) (wi, m 2 ) 5(i, s 2 ) 9-57
Since V^ involves , which depends on the variables 6 and <p, we

cannot integrate out these variables in V(aia 2 , a^a^) directly. If,
however, we use the expansion
after integrating out the spins,

however, we use the expansion of as given in Appendix V, we find,

Z T^TTl | f r

i,m 21 + I (J J r >

B(n 4 Z 4 |l)rJr|dri dr 2 X
T rF % (Zim 1 |^ 1 )r(Z 3 m 3 |Vi)I rHc (^|^i) sin ^ d6i d<pi X

/ / ^(Z 2 m 2 |^ 2 )F(Z 4 m 4 |^ 2 )F(Zm|^2)sin^ 2 ^ 2 ^ 2 } X

9-58

The integrals vanish unless m = m^ mi = w 2 m 4 , so that the
summation over m is reduced to a single term. The remaining integral
over the angles is of the form of a product of two integrals which we
denote by

4-7T

... sm6d6d(p

where m = mi w 2 . Now unless mi + ra 2 = m 3 + m 4 in equa-
tion 9-58 it is impossible to find an m such that m = ra 3 mi =
m 2 w 4 . Hence F(ni ) = unless raj + ra 2 = ra 3 + 7n 4 , and of
course si = 83; s 2 = s 4 . When these conditions are fulfilled

; 713/3771353;

9*59

where

n 4 Z 4 ) =

dr a 9-60

148 ATOMIC STRUCTURE

The integrals I(nil\\ n 2 Z 2 ) and Ri(nil\, n 2 Z 2 ; ^3, nl] cannot be
evaluated unless the forms of the <p's are known, so that we have reduced
the calculation of 77^- to its simplest form.

To summarize the procedure for calculating fl#, there are four steps :

1. We are given twoD functions, D(a( J a^a f ^ OandZ^a^a^a" ).
We first permute the sets of quantum numbers a", a", until as
many sets a" as possible are identical with the corresponding sets a.
If the permutation which effects this arrangement is an odd permutation,
we multiply the result by 1.

2. We now express //# in terms of the J7(a, ay) and V (a, ay; a&, a/).
There are four cases :

(a) After the permutation, the two D functions are identical. Then
k , a k ) + { V(a fc , a z ; a fc , a z ) - V(a kt a z ; ai, a k )} 9-61

(&) After the permutation the two D functions are identical except
for one set of quantum numbers, say the /cth, so that a f k 5^ a k '. Then

H*= U(a' k , 40 + Z{F(a, a,; a! k , a,) - 7(aJ, a z ; a l9 a f k ')} 9-62

(c) After the permutation the two D functions differ in two sets of
quantum numbers, say a k = a k and a[ ^ a" '. Then

H'ti = V(a' k , ai; a' k ', a'/) - V(a' k , a,'; a,' 7 , a(') 9-63

(d) After the permutation, the two D functions differ in more than two
sets of quantum numbers. The contribution to II ^ is then zero.

3. The Ufa, ay) and F(a z -ay; a k ai) are expressed in terms of the
integrals /(n t -Z t -, Ujlj) and Ri(riili, Ujlj] n k l k , nili). We have

j) = I(nili\ n,lj) d(li, lj) 5(m-, my) 5fe, sy)
a 3 a 4 ) ==

i

nzk', n^ n 4 ^ 4 ) <5(si, s 3 )

6 (mi + m 2 , m 3 + m 4 ) 9-64

4. The integrals J(n t -Z z -; nyZy) and R(n-ili ) are evaluated and the
results for HH are substituted in 9-21. The integrals Sij are equal to
6y, since the D's and B's are orthogonal.

In most of the calculations of this type it will be necessary to calculate
only diagonal elements for the energy matrix. This will be the matrix
element between one D function and itself, so that we have case 2a.

CALCULATION OF ENERGY MATRIX
The energy matrix in this case may be written quite simply as

M) -

fc<J

where the " coulombic " integral J(fc, Z) is defined as

J(k, I) = V(a k a l ] a k ari = E<* z (a*, a t )F l (a k , aj)

149

9.65

9-66

9-67

and the " exchange " integral K(k, I) is defined as

K(k, 1) = 7(a A , an a,, a*)
where

, a k )

F l (a k , ai) = Ri(a k ai] a k a{)
G l (a k , ai) = Ri(a k aij a^a fc )

In Tables 9-1 and 9-2 are listed the values of the a's and 6's for states
involving s and p electrons. Values of the c's, as well as more complete
tables for the a's and b's, may be found in Condon and Shortley. 6

As an example of these rules, let us calculate the energies of the terms
arising from the configuration (2p) 3 . Referring to the list of B func-
tions for this configuration, we see that the secular equation has the form

9-68

including eight determinants of the first order, two of the second, and
one of the third. Now Z>i is a B function corresponding to the X Z> term,
so that the energy of this term is immediately given by the root of the
determinant in the upper left-hand corner as

E( 1 D) = H' n 9-69

Similarly, the energy of the 3 P term is immediately given as

E( 3 P) = H'v 9-70

8 E. U. Condon and G. H. Shortley, Theory oj Atomic Spectra, Cambridge Uni-
versity Press, p. 1V8, 1935.

50 ATOMIC STRUCTURE

TABLE 9-1

o'(Z, m; I', m')

ELECTRON

CONFIGURATION

I m I'

_' _o
m o>

a 2

ss

00

1

sp

1

1 1

01

1

PP

1 1 1

1 1

A

1 1 1

1

-A

1 1

1

A

The remaining a' a are zero for s and p electrons.

TABLE 9-2

b l (l, m\ l' t m')

ELECTRON

CONFIGURATION

I m I'

m' 6

6 1 6 2

88

00

1

sp

1

dbl

* o

01

i

PP

1 +1 1

+1 1

o A

1-11

-1 1

n x

u "SIT

1 1 1

o A

1 +1 1

-1

o A

1 t) 1

1

o A

The energy of the 1 S term will be given by one of the roots of the third-
order determinant. The other two roots of this determinant correspond
to the 1 D and 3 P terms. Now the sum of the roots of a determinant is
equal to the sum of the terms on the principal diagonal, so that the energy
of the 1 S term is given as

E( 1 S) = H' 77 + H 8S + Hgg HH #2

22

9-71

Referring to the list of D functions, we readily find, by means of equa-
tions 9-65-9-67 and Tables 9-1 and 9-2 that the pertinent matrix com-
ponents of H are

, 1; 2, 1)

, 1; 2, 1)

H' u = 27(2, 1; 2, 1) + F(2, 1; 2, 1) +
H^ = 27(2, 1; 2, 1) + F(2, 1; 2, 1) -

'r, = 27(2, 1; 2, 1) + F(2, 1; 2, 1) + ^F 2 (2, 1; 2, 1)
' K - 27(2, 1; 2, 1) + F(2, 1; 2, 1) + ^ 2 (2, 1; 2, 1)
9 = 27(2, 1; 2, 1) + F(2, 1; 2, 1) + ^F 2 (2, 1; 2, 1)

FINE STRUCTURE 151

from which we calculate the energies for the various terms to be
E( 1 D) = 21(2, 1; 2, 1) + F(2, 1; 2, 1) + ^F 2 (2, 1; 2, 1) 9-72

= 27(2, 1; 2, 1) + F(2, 1; 2, 1) - ^F 2 (2, 1; 2, 1)

2,1; 2,1) 9-73

E( 1 S) = 27(2, 1; 2, 1) + P(2, 1; 2, 1) + ^F 2 (2, 1; 2, 1)

2, 1; 2,1) 9-74

It is, of course, impossible to make an exact comparison of theory with
experiment without evaluating the integrals for a particular choice of
trial eigenfunctions. It is, however, possible to make some qualitative
calculations. As the integrals F and G are positive quantities, we see
that the orders of the levels will be 3 P, 1 D, 1 S, with the 3 P the lowest.
This is in accord with Hund's rule in atomic spectroscopy : Of the terms
arising from a given electron configuration the term with the highest
multiplicity will lie lowest; of terms with the same multiplicity, the one
with the highest L value will lie lowest. For equivalent electrons,
F l (nl, nl) = G l (nl, nl). For the electron configuration (np) 2 , our
results state that

3

y ''

E( 1 D) - #( 3 P) TJ^F 2 + v\G 2 2

For the atoms C, Si, Ge, Sn, all having this electron configuration,
Condon and Shortley give the values for this ratio as 1.13, 1.48, 1.50, 1.39,
respectively, although for certain other atoms having this same con-
figuration the agreement is less satisfactory. Although this method of
approach by no means gives exact quantitative results, it does give the
number and type of terms correctly, and usually the relative positions
of these terms are qualitatively correct.

9h. Fine Structure. To the approximation we have been consider-
ing, we have seen that the various B functions of a term all have the
same energy. If, however, we add to our Hamiltonian operator the
terms representing the interaction between the magnetic dipoles associ-
ated with the motions of the electrons in their orbits and the magnetic
dipoles associated with the spin of the electron, this is no longer true.

We can determine the general form of this " spin-orbit " interaction
by the following argument. Consider an electron moving about a
nucleus of charge Z\e\ at a distance r. The electric field at the electron

7\p\
will be E = -4p r. If the electron is moving with a velocity v, it will

152 ATOMIC STRUCTURE

be subjected to the action of a magnetic field of strength

mcr

where M is the angular momentum of the electron. The classical
energy of a dipole in a magnetic field is jx H, so that the energy cor-
rection should be

A more careful analysis, 7 including a relativity correction, shows that this

Lf

expression should be reduced by a factor \ t Since a x = S^, etc.,

me

the spin-orbit interaction thus contributes a term to the Hamiltonian

i 2m c Ti
For the more general case of a non-coulomb field, the term -y- will be

replaced by ~ , so that we obtain for the spin-orbit interaction

operator the result

1 fl dV(r-)}

Ht v""* J ___L_. I /"iv/r Q i -. iv/r .Q . i TV/T Q ^ Q TQ
Z^ ~ a o 1 ~ ( (Ml-xi&xi ~T~ IVlj/tOitt -f- jyi zl & zl ) V'JO

This operator commutes with J 2 and J z , but not with M 2 , S 2 or their
components. The eigenfunctions of the complete Hamiitonian will
not therefore be eigenfunctions of M 2 , S 2 , M z , or S z . If, however, the
term H 7 is small compared to the other terms in the Hamiltonian, the
energy levels of the complete Hamiltonian will approximate those of the
approximate Hamiltonian which we have previously considered. There-
fore each term will break up into a number of energy levels whose ener-
gies will be close to that of the unperturbed term. The eigenfunctions
corresponding to these energy levels will be approximately represented
by linear combinations of the jB functions corresponding to the unper-
turbed terms.

Since J 2 and J 2 commute with the complete Hamiltonian, the final
eigenfunctions of H must also be eigenfunctions of those operators with

eigenvalues of the form J (J + 1) 5 f r J 2 an( ^ M for J z . If we

47T 2ir

7 E. Kemble, Fundamental Principles of Quantum Mechanics, p. 502, McGraw-
Hill Book Company, 1937.

FINE STRUCTURE 153

represent the final eigenf unctions of H by C(7, M ), we must therefore
have

J 2 C(/, M) = J(J + 1) A_ C (J, M) 9-79

J 2 C(J, M) = M ~ C(J, M) 9-80

^7T

Since J z = M 2 + S 2 , the 5 functions involved in C(J, M) must have
ML + MS = M , since there is but one B function with a given ML
and MS value. We may label the B functions by their ML and M 3
value as B(M^ M$), so that the C's must have the form

C(J, M) = fc(M L , M S )B(M L , Ms) 9-81

where k(ML, MS) is a constant.

The values of the constants k are most easily found by means of the
operators

] x =fc ij v = M x =fc iMj, + S* =fc iS y
We have

(J* iJy~)C(J, M) = fc(Jlf L , M S )(M X tM,)B(M

+ Zk(M L> M S )(S X t
from which

- Af (M 1)] H C(J, M db 1)

- M L (M L 1)]* 5(M L 1, MS)

1) - M 8 (M a l)] w J?(M L , M 5 zfc 1) 9-82

and therefore, from a knowledge of C(J, M), equation 9-82 gives
C(7, M zb 1) and hence all C's with the same J value.

The highest value of M is M = L + /S, and there is only one B func-
tion with this value of ML + M s . Moreover, there is no B function
with a greater value of M. Hence if J = L + S

C(J, J) - B(L, S)

and by 9-82 all the C(/, M) with / = L + S can be found. In particu-
lar, (7(J, J 1) is found as a linear combination of B(L 1, /S) and
B(L, S - 1). Now C(J', J'), where J r - L + S - 1, must also be a
linear combination of the same two B functions; in addition it must be
orthogonal to C(J, J 1). We therefore put

C(J', J 7 ) = k(B(L - 1, S) + k' 2 B(L, S - 1)
If

C(J, J - 1) = kiB(L - 1, S) + & 2 B(L, /S - 1)

154 ATOMIC STRUCTURE

the orthogonality condition is, since the B's are normalized and
orthogonal,

kik[ + k 2 k' 2 = 9-83

which, with the condition k{ 2 + k 2 2 1, determines k{ and k 2 . Having
determined (?(/', </') we now determine all the C(J f , M} by 9-82.
We may now determine C(J", J"), where J n = L + S 2, by finding
that combination of B(L - 2, S), 5(L - 1, S - 1), and B(L, S - 2)
which is orthogonal to <7(/, J 2) and (7(/ / , J' 1). This in turn
gives all the C(J", M). This process is continued until 2S + 1 or
2L + 1, whichever is smaller, sets of functions have been found, with the
/ values \L + S\, \L + S\ 1 [L S\. If we attempt to carry
the process beyond this point, we find that the number of B functions
corresponding to M = J (n+1) is only 2x + 1, where x is the smaller of
L and S, and we already have (2x + 1) C(J (n) , M)'s. It is therefore
impossible to find a new linear combination of B's which is orthogonal
to the linear combinations already found.

To illustrate this process, let us consider a 2 P term, so that L = 1,
S = i The B functions are J3(l, J), 5(1, -|), 5(0, |), 5(0, -),
B(-l, i), B(-l, -4). The process therefore starts with C(f, -f) =
5(1, ^). Equation 9-82 (with the lower signs) then gives

C(f, f ) =

c(t, -i) =

C(|, -f ) = B(-l, -
We now put

and 9-83 gives for orthogonality to C(f , J)

+ V%k 2 = 0; A; 2 = -

The values fc x = vj and k 2 = V f satisfy this as well as the relation
fcf + fcl = 1. . Hence

If we apply 9'82 again, we obtain

, -i) =

THE VECTOR MODEL OF THE ATOM 155

We have now formed six C functions out of the six B functions, and no
more linear combinations can be made. The process is therefore
complete.

The energy levels derived from the C functions are dependent only
on J and are the same for C functions with the same J but different M.
Thus the four C functions C(f , f ), C(f , |), C(f , -|), <?(f , | ) have
one energy, and the two C(^, ^-) and C(J, ^) have a different energy.
The energy levels are denoted by adding the value of / as a subscript to
the term symbol. Thus the two energy levels arising from the 2 P term
are denoted by 2 P^ and 2 P^.

In retrospect, the energy levels of an atom may be thought of as aris-
ing from the various electron configurations. If the interactions
between the electrons were completely neglected, the various eigen-
functions arising from the configuration would have the same energy.
The electrostatic interactions split the configurations into terms, Denoted
by their L and S values, with different energies. Finally, the spin-orbit
interaction splits each term into 2L + 1 or 2>S + 1 energy levels, dis-
tinguished by their J values, which run from |L + /S| to |L S|.
Each of these energy levels is still degenerate, having 2J + I eigen-
f unctions corresponding to M values which run from J to J. As we
shall see in the next section, this degeneracy can be removed by the
application of an external magnetic field. *

In many atoms, particularly atoms of high atomic number and with
almost complete shells, the spin-orbit interaction may become of more
importance than the electrostatic interaction. In considering these
atoms, it is a better approximation to neglect the electrostatic energy
at first and consider the spin-orbit interaction alone for the first approxi-
mation. We shall not treat this case here. Atoms of this type are said
to have j-j coupling, as contrasted with the type we have been consider-
ing, which are said to have Zr-S, or Russell-Saunders, coupling.

9i. The Vector Model of the Atom. The results obtained above
form a basis for a discussion of the " vector model " of the atom. This
model is of value in visualizing some of the energy relations involved, as
well as in giving a simple method of calculating the terms arising from a
given electron configuration. From the configuration (np) 2 , we saw
that one obtained the terms 1 D, 3 F, 1 S; that is, we had possible L
values of 2, 1, and 0, and possible S values of 1 and 0. Since we had
Zi = 1, 2 = 1, where Zi and Z 2 are the I values of the two electrons, we
see that the above L values are obtained by adding li and Z 2 vectorially,
if we require the resultant vectors to differ in length by steps of unity.
In the same manner the above S values can be obtained by vector addi-
tion of li| = ^ and |s 2 | = ^. The resulting L and S values cannot be

156

ATOMIC STRUCTURE

combined arbitrarily, since the exclusion principle must be taken into
account. The possible terms can be determined by the following
scheme. 8 We write as a table all the possible values of ML which can
be formed by combination of mi and m^

m 1 = 1

-1

X 2 X

1

1

~^

~oT~l

-1

I

-0"j -1 >x2
I I ^

-1

ii

values

of Jl

If L are

2 1

-1 -2

1

-1

which are just the values required to form one D, one P, and one S term.
The spins can be combined to form either S = or S = 1. For 5=1,
both electrons have the same spin quantum number, so that they must
differ in their values of m. We cannot, therefore, combine any of the ML
values from the diagonal of the above table with S = I . Also, we can
use only the ML values from one side of the diagonal, as those on the
other side merely correspond to a different numbering of the electrons.
When S = 1, we are thus limited to the ML values 1, 0, 1, so that we
have a 3 P term. When S = 0, the electrons differ in their spin quantum
numbers and there is no restriction on the values of ML which may be
combined with this value of S. As the set of values 1,0, 1 has
already been used to form the 3 P term, we thus have the sets of ML
values 2, 1, 0, 1, 2, and to combine with S = 0, giving 1 D and 1 S
terms.

For the configuration (ns) 2 , there is, of course, only the one term 1 S.
For the configuration (nd) 2 , our table for the ML values is

-1 -2

2

1

Z 2 =2
-1
-2

8 G. Breit, Phys. Rev., 28, 334 (1926).

THE VECTOR MODEL OF THE ATOM 157

For S = 1, we are again limited to the ML values from one side of the
diagonal, which are sufficient to give 3 F and 3 P terms. For S = 0,
we have the remaining sets of ML values, which are sufficient to give
1 0, 1 D, 1 S terms. This method can be extended to more than two
equivalent electrons, although the calculations are slightly more compli-
cated. The possible terms that can arise from combinations of equiva-
lent s and p electrons are

M 2 s

(ns) 2 X S

(np) 2 P

(np) 2 1 S 3 P 1 D

(np) 3 *S 2 D 2 P

(np) 4 1 S 3 P 1 D

(np) 5 2 P

(np) 6 S

For such electron configurations as (np, mp), the exclusion principle
is already satisfied by the differing values of the principal quantum
number. There is thus no restriction on the combinations of ML and S,
so that this electron configuration gives the terms 3 D, 3 P, 3 S, 1 D, 1 P, 1 S.
Similarly, for the combination (np, nd), the exclusion principle is satis-
fied by the differing Z values. From this configuration we thus obtain
the terms 3 F, 3 D, 3 P, 3 S, 1 F ) 1 D, 1 P, 1 S.

For a given term we have an electronic angular momentum L* >

2?r

where |L*| = Vl/(L + 1); and a spin angular momentum S*

where |S*| = VS(S + 1). As we have seen in the previous section,
these angular momenta combine to give a total angular momentum

J* , where |j*| = Vj(J + 1), and where J has the values from
JTT

|L + S| to |L S\, the J values differing by unity. These are just the
/ values we would obtain if we imagined the orbital and spin angular
momenta to combine vectorially to give the resultant angular momen-
tum, the resultant values being limited by the quantum principle. As
we have seen, a magnetic moment is associated with the orbital angular
momentum as well as with the spin angular momentum. Also, the
motion of the electrons in their orbits produces a magnetic field. Ac-
cording to equation 9*78, we see that the spin-orbit interaction will be of
the form

A# - AL* S* 9-84

158 ATOMIC STRUCTURE

where A will be some function of L and S and the electronic state. Since

|T*|2 |T*|2 |Q*|2
T*. <** - U I - I L I |0 I

L b - 2

this becomes

1)} 9-85

The magnetic field produced by the orbital motion is in the same direc-
tion as the orbital angular momentum. The state of lowest energy will
be that in which the magnetic moment of the electron spin is parallel
to the field. Now the magnetic moments of both orbital and spin
angular momenta are in the direction opposite to that of the associated
angular momenta, owing to the negative charge of the electron. The
state of lowest energy will thus be that in which L and S are in opposite
directions. The lowest state is therefore that with the lowest value of J.
[In the list of terms arising from configurations of equivalent electrons
as given above, it will be noted that (np) 5 gives the same terms as (up) ;
that is, a closed shell minus one electron gives terms similar to those from
the same shell containing only one electron. In terms arising from
these " almost-closed shells " the order of the states is reversed, the
lowest state being that with the highest J value. For example, the
halogens, with the (np) 5 configuration, have 2 P^ as the ground state.]
Since L and S are the same for all members of a multiplet, the sepa-
ration between the state with quantum number J and that with / + 1 is

1) 9-86

that is, it is proportional to the larger J value. For example, the states
3 P , 3 Pi, 3 P2 have relative separations 1:2; the states 4 D^, 4 D^,
4 D^, 4 Z)j have relative separations 3:5:7. This relation, the Lande
interval rule, is closely obeyed in atoms with Russell-Saunders coupling.
As stated above, a state with angular momentum quantum number J
is (2J + l)-fold degenerate, there being this number of C functions
with the same value of J. Returning to our example of the terms
arising from the electron configuration (np) 2 , we have 1 /S , > non-degener-
ate; 1 D 2 , 3 P2, five-fold degenerate; 3 Pi, three-fold degenerate; 3 Po,
non-degenerate. This accounts for the fifteen states corresponding to
the possible, linear combinations of the fifteen D functions for this con-
figuration. If the atom is now placed in a magnetic field, the energy will
depend on M as well as on /, and this degeneracy will be removed. If
|ij* is the magnetic moment associated with a state with quantum

SELECTION RULES FOR COMPLEX ATOMS 159

number J", the energy in a magnetic field will be

E(J, M) = # (/) - I*/* H 9-87

where EQ (J) is the energy in the absence of the field. As we have noted
previously, the magnetic moment associated with the orbital motion is

- T* eh

4irmc

that associated with spin angular momentum is

~* &h

The magnetic moment associated with the total angular momentum is
therefore

4-irmc

= 4.

cos (L*> J*) + k*l cos (s

9-88

is the " Lande g factor." The energy levels in the presence of a mag-
netic field are therefore

eh
E(J, M) = E (J) - g - - J* H

47T771C

M=J,J-1 ---- J 9-89

Each level with quantum number J is thus split into 2J + 1 equally
spaced levels by the magnetic field, the lowest level being that with
M = J. In Figure 9-1 we have plotted the levels arising from the
configuration (np) 2 for the various degrees of approximation. For each
degree of approximation, the relative separations within a given multi-
plet are correct; the other separations are not to scale.

9j. Selection Rules for Complex Atoms. As the field of atomic
spectroscopy lies to a certain extent outside the scope of this book, we
shall not treat the selection rules for the allowed transitions in complex
atoms in any detail but shall merely summarize some of the more impor-
tant results. According to the general theory of Chapter VIII, a transi-
tion between two states a and b may take place with the emission or

absorption of electric dipole radiation only if the integral / ^*

s

160

ATOMIC STRUCTURE

different from zero. Now any atom has spherical symmetry; the
Hamiltonian operator is unchanged if the position coordinates of every
electron are subjected to an inversion through the origin of coordinates.
Because of this spherical symmetry, the product f *&, must be unchanged

-2J

No electronic
interaction

Electrostatic
repulsion
included

Spin-orbit

interaction

included

Magmetia

field
present

FIG. 94. Schematic energy levels from the electron configuration (np) 2 .

if every coordinate is replaced by its inverse. Upon inversion, therefore,
the function \l/ a must either remain unchanged or must change only in
sign, so that the atomic wave functions can be classed as even or odd
with respect to inversion. The vector r is, of course, an odd function
with respect to this operation. Now, if $ a and \f/ b are both even func-
tions or both odd functions, the product ^T\f/ b will be an odd function.
If we carry out an inversion, the integrand will change sign, its magni-
tude, however, remaining unchanged. Thus for every contribution to
the integral from a particular volume element dri there will be a con-
tribution of equal magnitude but opposite sign from the inverse volume
element; the integral will be identically zero. We have thus derived the

SELECTION RULES FOR COMPLEX ATOMS 161

selection rule for dipole radiation: only transitions between an even
state and an odd state are allowed. This selection rule is perfectly
general, depending as it does only on the symmetry properties of an
atom and not on the particular set of wave functions which we use to
give an approximate description of the electronic state of the atom. The
following selection rules are less strict but hold approximately for many
atoms.

If we consider that the states of the atom can be described by wave
functions which are the proper linear combinations of products of one-
electron orbitals which are solutions of a central field problem, we can
make the following statements. Many transitions can be described by
assuming that only one electron is involved. The angular portions of
the orbitals are then identical with those of the hydrogen atom, the
selection rule will therefore be the same, so for a one-electron transition,
to this approximation, we have the rule AZ = dbl. For example, a
transition of the type (up) 2 > (np, md) is allowed; transitions of the
type (up) 2 > (np, mp) or (up) 2 > (np, ra/) are forbidden. As may
readily be seen, either from the expressions for the hydrogenlike wave
functions or from the nature of the spherical harmonics, these functions
are even for I even and odd for I odd. To the approximation we are here
considering, an atomic wave function is therefore even or odd according
as ^i is even or odd. If we have a transition involving two electrons

i

simultaneously, then the general theorem that transitions are allowed
only between even and odd states tells us that AZ must be even for one
electron and odd for the other. Since we have Al = 1 for a one-
electron transition, for a two-electron transition we would have
AZ X = 1; AZ 2 = 0, 2.

Of more general interest are the selection rules for S, L, and J, as
these quantum numbers describe an atomic state with greater accuracy.
To the approximation that we neglect spin in the Hamiltonian operator,
the spin wave functions are independent of the coordinate wave func-
tions. The dipole moment integral will vanish because of the orthogo-
nality of the spin functions unless the spin quantum numbers match in
the initial and final states. To this approximation, we thus have the
selection rule AS = 0; that is, only transitions between terms of the
same multiplicity are allowed. The selection rules for L and J cannot
be derived so simply; the results are 9

AL = 0, 1; A/ = 0, =fcl 9-90

9 E. Kemble, Fundamental Principles of Quantum Mechanics, p. 543, McGraw-
Hill Book Company, 1937.

162 ATOMIC STRUCTURE

In the presence of a magnetic field, the energy depends also on M. The
observed phenomena in the Zeeman effect are explained by the selection
rule for M

AM - 0, 1 9-91

this selection rule being derivable in the same manner as the selection
rule for m in the theory of the hydrogen atom. The selection rule
AM = corresponds to the non-vanishing of the z component of the
dipole moment integral; the light emitted during a transition for which
AM = is polarized along the z axis (the direction of the magnetic field).
Similarly, for transitions for which AM = 1, the emitted light is
circularly polarized in the xy plane. In passing, we might mention that
investigations of the Zeeman effect provide one of the most powerful tools
for the determination of the characteristics of the states involved in
atomic spectra.

9k. The Radial Portion of the Atomic Orbitals. Up to the present
we have not specified the exact nature of our atomic orbitals, aside from
the specification that the angular portion of the orbitals would be the
ordinary spherical harmonics, since they were assumed to arise from the
solution of a central field problem. For any quantitative calculation of
energy levels some form must be chosen for the radial portion of the
orbital. The best one-electron orbitals are found by the method of
Hartree, which is discussed in the next section. In approximate work
it is often desirable to use orbitals which, although less accurate than
those obtained by Hartree's method, are simpler in form and hence
easier to use. For example, Zener 10 and Slater 11 have used orbitals of
the form

^ r (n*~l) e -(^)^T oF ^ m |^ ^) Q.Q2

where n* and s are adjustable constants and TV is a normalizing factor.
These eigenfunctions are solutions of the central field problem where
V(r) is given by the relation

For large values of r this approaches

V(r) ~ - ~ 9-94

corresponding to a screening of the nucleus equivalent to s atomic units;

10 C. Zener, Phys. Rev., 36, 51 (1930).

11 J. C. Slater, Phys. Rev., 36, 57 (1930).

THE HARTREE METHOD 163

in other words, the effective nuclear charge is equal to Z s. From its
resemblance to the quantum number n of the hydrogen atom, n* is
known as the " effective quantum number. " Qualitatively, the eigen-
functions 9-92 differ from the hydrogen eigenf unctions in that there are
no nodes in the radial portion, whereas the hydrogen eigenfunctions
have n I 1 nodes.

By varying n* and s so as to minimize the energy, Slater has been
able to give the following rules for the determination of these constants :

1. n* is assigned according to the following table, according to the
value of the real quantum number n :

n = 1 2 3 4 5 6
n* = 1 2 3 3.7 4.0 4.2

2. For determining s, the electrons are divided into the following
groups: Is; 2s, 2p; 3s, 3p; 3d; 4s, 4p; 4d, 4/;

3. The shielding constant s is formed, for any group of electrons, from
the following contributions:

(a) Nothing from any shell outside the one considered.

(6) An amount 0.35 from each other electron in the group considered
(except in the Is group, where 0.30 is used).

(c) If the shell considered is an s or p shell, an amount 0.85 from each
electron with principal quantum number less by 1, and an amount 1.00
from each electron still farther in; but if the shell is a d or/ shell, an
amount 1.00 from each electron inside it.

For example, carbon has two Is electrons, two 2s electrons, and two 2p
electrons. The approximate radial orbital (unnormalized) for a Is
electron is, according to the above rules:

-5.70

<p(ls) ^ e
while for a 2s or 2p electron the radial function is

3.25 r

<p(2s) = <p(2p) ^re 2a

Since the constants in these functions were determined by the use of
experimental data, the functions will be satisfactory for rough quantita-
tive calculations. Various other sets of screening constants have been
proposed for use in wave functions of the above type, for example, those
of Pauling and Sherman. 12

91. The Hartree Method. According to our discussion of the helium
atom in Chapter VII, the best wave functions for an atomic system

12 L. Pauling and J. Sherman, Z. Krist., 81, 1 (1932).

164 ATOMIC STRUCTURE

should include the distance between the electrons r t -y, explicitly.
Because of the complexity that would result from the use of such wave
functions for many-electron atoms, we are limited in practice to the use
of the one-electron wave functions of the type 9-19. The best possible
wave functions of this type would be obtained by means of the variation
method. In the earlier work of Hartree, the simpler product form of
one-electron functions was used instead of the determinantal form. If
we write the wave function for an n-electron atom as

-<pn(an\n) 9-95

then the best wave function of this type is that which makes the energy
E = / ^*H^ dr a minimum, in other words, that which satisfies the
equation

dE = 6 Vlfy dr = 9-96

The above method of expressing the energy is valid only if ^ is normal-
ized. We can insure that this condition is fulfilled if we require that
each <p be normalized, that is

<pfyt dr = 1 ; i = 1, 2, n 9-97

h 2 Ze 2 } e 2

iff o *~*V I . i . V

The Hamiltonian operator for the system of n electrons is

9-98

e 2
= EH; + iEZ-

i=l i jv^iTij

The solution of equation 9-96, subject to the condition 9-97, shows that
the best possible wave function of the type 9 '95 is obtained by using the
set of <p's which are the solutions of the n simultaneous equations

L dr\ W - en 9-99

Now e 2 |<py| 2 is just the charge distribution of the jth electron, and

I -I 2
e 2_iL j s h e potential energy of the zth electron in the field of the yth

Tij

electron. The term in brackets thus represents the potential energy of
the ith electron in the field of all the remaining electrons. The method
of solution of the set of simultaneous equations is the following. First

THE HARTREE METHOD 165

an arbitrary set of ^/s is chosen (the choice, of course, being guided by
any previous knowledge of approximate wave functions for the atom).
The field arising from all the electrons except the ith electron is calcu-
lated from this set of <p's, and <?i is then calculated from equation 9-99.
This procedure is carried out for each of the n electrons. The calculated
set of <>'s will not in general be identical with the original set; the calcu-
lations are repeated with a new set of <p's which have been altered in a
manner suggested by the results of the first calculation. This process is
continued until the assumed set of <p's and the calculated set are identical,
at which point the solution of the set of simultaneous equations has been
achieved. Since an assumed charge distribution gives a set of <p's
which will reproduce this charge distribution, the above method is fre-
quently called the " method of the self-consistent field. "
The energy associated with the wave function 9-95 is

E = /VW dr

! dr . dr . 9400

2 f f

J J

Comparing this equation with 9-65, we see that it contains terms repre-
senting the energy of the one-electron wave functions and terms repre-
senting the " coulombic " interaction of the electron cloud distributions,
but that the term representing the " exchange " energy is missing. If
the <p's are the solutions of equations 9-99 the energy will be given by

E = Z et - iZ E e 2 l^W- 2 drt drj 9-101

The reason why there are no energy terms in 9400 corresponding to the
exchange energy is that the original wave function $ was written as a
simple product rather than as a determinant, as it should be if the exclu-
sion principle is to be satisfied. By writing ^ as a determinant, and pro-
ceeding as above, with the additional requirement that the <p's be orthog-
onal, one is led to Fock's equations, 13 which are similar in form to the
Hartree equation 9-99, but contain additional terms corresponding to the
potential energies arising from the electron interchange. The results
for the energy levels and wave functions do not differ appreciably from
those obtained by the Hartree method. Since our nrnin interest is in
the problem of molecular structure rather than in that of atomic struc-
ture, we shall postpone a more thorough discussion of exchange energies
to later chapters, where they will be of more significance.

13 V. Fock, Z. Physik, 61, 126 (1930).

166

ATOMIC STRUCTURE

It should be mentioned at this point that, although the above wave
functions, particularly those obtained by the solution of Fock's equa-
tions, are the " best possible " one-electron wave functions, they give
values for the energies of atoms which are incorrect by approximately
0.5 volt per electron. For this reason any results obtained from calcu-
lations based upon one-electron wave functions can be only quali-
2

1

>(r)

10
5

60
40

30
D(r)

20

10

\

V

JL

He

Rb"

1.0

2.0

3.0

FIG. 9-2. Electron distribution in He, Na + , and Rb + by the Hartree method.

tatively correct; it is somewhat unfortunate that these are the oftly
wave functions which can be used in most problems concerning the
energy levels of complex systems.

In Figure 9-2 we show the electron density distribution, as calculated
by the Hartree method, 14 for He, Na + , and Rb + . It will be noted that

14 D. R. Hartree, Proc. Cambridge PhiL Soc., 24, 89, 111 (1928).

THE PERIODIC SYSTEM OF THE ELEMENTS 167

the various shells of electrons can be clearly distinguished, and that
the " radii " of these shells decrease as the nuclear charge Z increases.

The diameter of the first shell, the K shell, is essentially , where OQ is

Zt

the radius of the first Bohr orbit. In Figure 9-2 the. electron density
D (r) is normalized in such a way that the area under each curve is equal
to the number of electrons in the atom.

9m. The Periodic System of the Elements. We will conclude this
chapter on atomic structure with a brief discussion of the periodic system
of the elements from the viewpoint of the electronic states of the various
elements. In Figure 9-3 we have reproduced the periodic chart, the
arrangement being essentially that suggested by White 15 on the basis
of the spectroscopic characteristics of the various elements. In this
table are included the electron configurations of the ground states of
the elements, the designation of the lowest level, and the ionization
potentials 16 both numerically and graphically. As a matter of con-
venience, the members of the rare-earth group between La and Lu have
been omitted. In giving the electron configurations, only those electrons
which have been added to the electrons of the preceding rare-gas con-
figuration have been explicitly noted. For example, the electron con-
figuration of Ne is (ls) 2 (2s) 2 (2p) 6 ;the designation (3s) for the elec-
tron configuration of Na implies (Is) 2 (2s) 2 (2p) 6 (3s).

As mentioned in section 9c, the electron configuration of the ground
state of any atom is obtained by adding the electrons one at a time to the
lowest possible orbitals, taking account, however, of the exclusion
principle. As previously mentioned, the orbitals lie approximately in
the order Is, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4/, 5d, 6p, 7s, Qd.
In the hydrogen atom the energy depends only on n, so that, for example,
the 3s, 3p, and 3d orbitals all have the same energy. The new order
of the orbitals in the heavier atoms may be qualitatively understood in
the following way. The outer electrons move in an effective field which
is the resultant of the nuclear field and the field of the inner electrons, or,
as we have previously said, the inner electrons " screen " the outer elec-
trons from the nucleus. The energy binding an outer electron to the
atom will thus depend on how effectively this electron is screened from
the nucleus; for example, if a 3d electron is more completely screened
than a 3p, and the 3p is more completely screened than a 3s, then the
order of the orbitals will be 3s, 3p, 3d, with the 3s orbital the most stable.
This is exactly the case in heavy atoms. If we look at the hydrogenlike

15 H. E. White, Introduction to Atomic Spectra, p. 85, McGraw-Hill Book Com-
pany, 1934.

16 G. Herzberg, Atomic Spectra and Atomic Structure, p. 287, Prentice-Hall, 1937.

168

ATOMIC STRUCTURE

t- a,

s -

*1S

gf "S

S3 *

g

s

5 ' 1

'

I

I

CO
Oi

THE PERIODIC SYSTEM OF THE ELEMENTS 169

eigenfunctions in Table 64, we see that, disregarding the constant
factor, the angular factor, and the common exponential factor, for small
values of r the 3s eigenfunction is a constant, the 3p eigenf unction varies
as r, and the 3d eigenfunction varies as r 2 . This means that the proba-
bility of finding the electron in the immediate neighborhood of the
nucleus is greatest for the 3s electron and least for the 3d electron, so
that the 3s electron is screened least by the inner electrons and the 3d
is screened most; the order of stability is thus 3s, 3p, 3d. Because of
this difference in energy between the 3p and 3d, it is possible for the 4s
orbital to be even more stable than the 3d, and we observe that this is so.
Similar arguments hold for the relative stability of / orbitals; it is
observed that the 4/ orbitals are so completely screened by the inner
electrons that the 6s orbital is more stable.

By adding electrons into the various possible orbitals, taking proper
account of the exclusion principle, we get the electron configurations
as shown in Figure 9-3. The Is shell is complete at He, the 2s shell at
Be, and the 2p shell at Ne. From the viewpoint of electron configura-
tion alone, He should properly be in the same column as Be; from the
viewpoint of chemical properties, it belongs with Ne. This point will
be discussed later. The 3s shell is completed at Mg, the 3p shell at A,
and the 4s shell at Ca. At this point the 3d shell begins to fill up. Since
the 4s and 3d orbitals are close together, it happens that the 3d shell does
not fill up smoothly; there is a tendency for the 3d shell to fill up at the
expense of the 4s shell. At Cu the 3d shell is complete, and there is one
4s electron present; at Zn both the 3d and the 4s shells are filled. The
4p electrons are now added, this shell being completed at Kr. The
same procedure is repeated with the 5s, 4d, and 5p electrons. The 6s
shell is completed at Ba, then one 5d electron is added, followed by the
fourteen 4/ electrons. The 4/ shell is completed at Lu, which, as far as
electrons outside closed shells are concerned, has the same configuration
as La. The addition of the remaining 5d electrons and the addition of
the 6p electrons brings us finally to Rn. Little is known about the
remaining elements.

We now note that all the elements in a given column have the same
ground state. For example, the elements H, Li, Na, K, Rb, and Cs
have 2 S# as the ground state; the elements Zn, Cd, and Hg have 1 So
as the ground state; the elements 0, Si, Ge, Sn, and Pb have 3 Po as the
ground state. Since the chemical properties of an element are deter-
mined by the electron configuration of the element, we see why the ele-
ments in any given column have similar chemical properties. The
above statement? are not strictly true for the elements in those portions
of the table where the d shells are being filled. Owing to the competition

170 ATOMIC STRUCTURE

between the d and s shells, certain irregularities arise; for example,
while Ti, Zr, and Hf have the same electron configuration, that of Cb
differs from that of V and Ta. In the first four and last eight columns
the correspondence is exact; the intermediate columns show more or less
variation.

We also note that the chemical characteristics of an element are not
completely determined by the electron configuration of the ground state,
since, for example, K and Cu both have 2 S^ as the lowest level, and
Ca, Zn, and Kr all have 1 /S . In order to understand the differences
between such elements several other factors must be considered. The
differences between K and Cu are undoubtedly to be attributed to the
fact that in Cu we have a 4s electron on top of the rather loosely held
3d shell, whereas in K we have a 4s electron on top of the tightly held A
configuration. The result of this is that the 4s electron in Cu moves in a
stronger effective field than the 4s electron of K, hence the ionization
potential of Cu is considerably higher than that of K. The 4s electron
in K is therefore more readily removed than that of Cu, so that, though
there are many similarities between the two elements, K is the more
reactive.

As will be seen in the following chapters, the normal covalence of an
atom is equal to the number of unpaired electron spins in the atom.
F, O, and N have multiplicities of 2, 3, 4, respectively, so that the spin
quantum number S has the values J, 1, |> respectively, for these atoms.
These values of S can arise from 1, 2, 3 unpaired spins, so that F, 0,
and N have normal covalences of 1, 2, and 3, respectively. Be, on the
other hand, should have a normal covalence of zero. However, com-
paring the ionization potentials of Be and B, we see that the 2p orbital
is only slightly less stable than the 2s. One of the 2s electrons in Be
can thus be rather readily moved up to the 2p level, giving two unpaired
spins and hence a covalence of two. Similarly, by moving one electron
from the 2s to the 2p level in B and C, we obtain the normal covalences
of 3 and 4 for these elements. The behavior of Zn is analogous to that
of Be. For the rare-gas elements, a quite different condition exists.
For example, the ionization potential of Ne is 21.45 e.v., while that of
Na is only 5.12 e.v. There is thus a great gap between the 2p and 3s
levels. The promotion of a 2p electron in Ne to the 3s level, which
would give Ne a covalence of 2, requires so much energy that this con-
dition does not occur. Similarly, there is a large gap between the Is
and the 2s levels; for this reason He has the chemical properties of Ne
rather than of Be.

AS we fill in a given shell, say the 2p shell, the ionization potential
increases with increasing atomic number; since the electrons in a given

THE PERIODIC SYSTEM OF THE ELEMENTS 171

shell screen each other very little, the effective nuclear charge is increas-
ing. An exception will be noted in the oxygen group; the ionization
potential of is less than that of N. As long as there are three or less
electrons in a p shell they can all be in different ones of the three available
p orbitals (p x , p yj p z ) y so that the electrostatic repulsions are a minimum.
The fourth p electron must go into an orbital which is already occupied
by one electron; the increased electrostatic repulsion causes the electron
to be bound less firmly than would be expected from considerations of
effective nuclear charge alone.

CHAPTER X
GROUP THEORY

The Schrodinger equation can be solved exactly in only a very few
simple cases; in general, we are limited to the approximate methods
of solution discussed in Chapter VII. There exists, however, a large
class of results which depend only on the symmetry properties of the
system under consideration; these results can be obtained exactly by
use of the branch of mathematics known as group theory. We pre-
sent in this chapter an elementary treatment of group theory, although
we make no claim to completeness. Several of the important theorems
are presented without proof; the interested reader may refer to one of
the complete expositions of the field (see General References). Group
theory, in the form which will be of interest to us, makes considerable
use of matrix notation, so that we shall first present the elements of
matrix algebra, with particular reference to the matrices involved in
linear transformations of coordinates.

FIG. 10-1.

lOa. Matrices. Let us consider a point in the xy plane, this point
being specified by the coordinates fa, y\) (Figure 10-1). These two
numbers may also be thought of as defining the vector ri. A rotation
of this vector through an angle 6 will transform it into the new vector
r 2 , defined by the numbers x% and y 2 . The relation between (& 2 , 2/2)
and (a?i, y\) and the angle of rotation 6 is given by the set of linear
equations

#2 = Xi cos 2/1 sin

10-1

2/2 = #1 sin + y\ cos &
172

MATRICES

173

The reverse transformation is

cos 6 sin 6

+

A
sin

A
cos 6

10-2

A =

= cos 2 6 + sin 2 = 1

where A is the determinant of the coefficients in 10-1 :

cos 6 sin

sin cos 6
The set of equations 10-1 can be written as

;cos 6 sin
sin cos 6

where

(x 2 \ (cos 6 sin 0\ /xA

t/2 / \sin0 cos0/\t/i/

(cos sin 0\

sin cos 0/

10.3

104

is the matrix of the transformation which takes r\ into T 2 . The corre-
sponding matrix for the reverse transformation is

(cos sin 0\
sin cos 0/

10-5

In the n-dimensional case, we have the corresponding set of equations
x( = a n x 1 + 0,12X2 H ---- + ainX n

+ 0,22X2 H ---- + a 2n Xn 10-6

x' 2 =

where the xfa are the new coordinates and where the a's satisfy the
relations

ft = 1, 2, n

k = 1, 2, n

10-7

Jc 7 1 V *n If s

n/j v " * Aj &y Ib j iv r~*

s*

y-i

174 GROUP THEORY

In addition, the determinant of the a's is unity. The matrix formed
by a set of a's which satisfy the relations 10-7 is said to be a unitary
matrix; the matrices representing rotations, reflections, and inversions
are unitary. It will be noted that the matrices 10-3 and 10-4
satisfy the relations 10-7. The transformation which is the reverse
of 10-6 is given by the set of equations

a n x{ + 0,21X2

10-8

X n = ln^l 2n#2 ' * '

The matrix of the reverse transformation is thus obtained from the
matrix for the original transformation merely by changing columns
into rows.
The set of equations in 10-6 can be written in the compact form

2>y*&* *, j = 1, 2, . n 10-9

An even simpler notation which expresses the same thing is obtained
if we write

x' = ax 10-10

where a is the matrix

a =

of the transformation from the unprimed to the primed coordinates.
A second transformation could be written as

x" = bz'; or x? = EM^ 10-11

3

These two successive transformations are equivalent to some one trans-
formation

x" = ex; or x" = ^c ik x k 10-12

k

Combining the above equations, we have

10-13

j y

The components of the product matrix c = ba are thus given by the
relation

Cik = LMy* 10-14

GENERAL PRINCIPLES OP GROUP THEORY 175

which is the rule for matrix multiplication. The product of the ma-
trices of the transformations 10-6 and 10-8 must be a matrix which
represents no transformation of coordinates at all, that is

0\

10-15

'On Oi2 ' ' * Oin^

O21 O22 " ' ' O2n

/On 021

=

r i u u
1

W 2 ' aj

\0ln 02n

\0 1

From the rule for matrix multiplication, and the relations in 10-7, we
see that the matrix equation 10-15 is indeed true.

Before proceeding with the formulation of group theory,- we need
certain general concepts concerning vectors. In three-dimensional
space, any three numbers may be thought of as defining a vector, the
vector from the origin of coordinates to the point specified by the three
numbers. If we have the two vecors A and B, defined by the num-
bers (Ai, A 2 , AS) and (Bi, B 2l 3), the vectors are said to be orthog-
onal if

AA + A 2 B 2 + A 3 -B 3 =

In the three-dimensional case, this means that vectors are perpen-
dicular, or, in the notation outlined in Appendix II, the scalar product
A B is zero. In more general terms, we may consider n numbers
(Ai, A 2 A n ) as defining a vector A n in n-dimensional space. If
B n is another n-dimensional vector, the two vectors are said to be
orthogonal if their scalar product

A n E n = A& + A 2 B 2 + + A n B n

is equal to zero. If the numbers which define the vectors are com-
plex, the vectors are said to be orthogonal if the Hermitian scalar
product

(An B n ) = A?B! + A$B 2 + - . + AjBn

is zero.

In three dimensions any arbitrary vector can be expressed in terms
of a linear combination of three orthogonal vectors, for example, the
three unit vectors along the coordinate axes. In other words, it is
possible to construct only three independent orthogonal vectors in
three-dimensional space. Analogously, in n-dimensional space, it is
possible to construct only n independent vectors. This concept of a
set of numbers defining a vector will prove useful later.

lOb. The General Principles of Group Theory. The set of opera-
tions which send a symmetrical figure into itself are jaid to form a

176

GROUP THEORY

group. Let us consider the symmetrical figure formed by three points
at the corners of an equilateral triangle, as in Figure 10-2. The opera^
tions which send this figure into itself are :

1. The identity operation E, which leaves each point unchanged.

2. Operation A, which is a reflection in the yz plane.

3. B reflection in the plane passing through the point b and per-
pendicular to the line joining a and c.

4. C reflection in the plane passing through c and perpendicular
to the line joining a and b.

5. D clockwise rotation through 120.

6. F counterclockwise rotation through 120.

)ft

FIG. 10-2.

Other symmetry operations are possible, but they are all equivalent
to one of the operations given above. For example, a clockwise rota-
tion through 240 is a symmetry operation, but it is identical with
operation F; a rotation through 180 about the y axis is identical with
operation A.

The successive application of any two of the operations listed above
will be equivalent to some single operation. Rotation in the clock-
wise direction through 240 is obtained by applying operation D twice;
this is equivalent to the single operation F we denote this fact by
the equation DD = F. Operation A interchanges points b and c; if
operation D is applied to the resulting figure, c is returned to its original
position, b goes to the position originally occupied by a, and a goes to
that originally occupied by b. Operation A followed by operation D
is thus equivalent to operation C, or DA = (7. If we work out all
possible products of two operations, we obtain the following multi-

GENERAL PRINCIPLES OF GROUP THEORY 177

plication table, Table 10-1, where the operation which is to be applied
to the figure first is written across the top of the table. The set of
operations E, A, B, C, D, F forms a group, and Table 10-1 is known
as the multiplication table for this group. The number of operations
in the group is called the order h of the group; here the order of the
group is 6.

TABLE 10-1

E

A

B

C

D

F

E

E

A

B

C

D

F

A

A

E

D

F

B

C

B

B

F

E

D

C

A

C

C

D

F

E

A

B

D

D

C

A

B

F

E

F

F

B

C

A

E

D

More generally, any set of elements P, Q, R, S is said to form a
group if the following conditions are satisfied:

1. The product of any two elements in the set is another element in
the set.

2. The set must contain the identity operation E which satisfies the
relation ER = RE = R, where R is any element of the set.

3. The associative law of multiplication, P(QR) = (PQ)R, must
hold; that is, P times the product of Q and R must be equal to the
product of P and Q times R.

4. Every element must have a reciprocal such that, if R is the re-
ciprocal of S, then RS = SR = E.

That all these conditions are satisfied by the group given above is
easily verified. The commutative law of multiplication does not neces-
sarily hold. From Table 10-1 we see that AB = D; BA = F, so that
AB ? BA. If PQ = QP for all elements of the group, the group is
said to be Abelian.

In the group of symmetry operations on three points as given above,
we have three distinct types of operations: the identity operation E\
the reflections A, B, and C; and the rotations D and F. We say that
each of these sets of elements forms a class; that is, E forms a class by
itself, A, B, and C form a class, and D and F form a class. Usually
the geometric considerations will enable us to pick out the classes;
more precisely, two elements P and Q which satisfy the relation
X~ 1 PX = P or .Q, where X is any element of the group and X~~ l is
its reciprocal, are said to belong to the same class. From the multi-

178 GROUP THEORY

plication table, we have

EDE = D EFE = F

ADA = F AFA = D

BDB = F BFB = D

CDC = F CFC = D

FDD = D FFZ) = F

DDF = D DFF = F

The elements D and F therefore form a class; in the same way it is
found that A, B, and C form a class, and that Z? forms a class. If the
group is Abelian, then X~ 1 PX = X^XP = P for all X's and P's. Each
element of the group then forms a class by itself, and the number of
classes is equal to the number of elements. The concept of a class of
operations has the following geometric meaning. If two operations
belong to the same class, it is possible to pick out a new coordinate
system in which one operation is replaced by the other. For example,
in the group given above, we could equally well have taken our y axis
through the point b and perpendicular to the line joining a and c. The
operation A in the new coordinate system is the same as the operation
B in the old coordinate system, since A has been defined to be a re-
flection in the yz plane.

Any set of elements which multiply according to the group multi-
plication table is said to form a representation T of the group. For
the group given above, we immediately see that the sets of numbers
assigned to the various elements in the following way form represents
tions of the group:

E A B C D F

111111
1-1-1-1 1 1

The corresponding matrices will also form a representation of the group
if we replace ordinary multiplication by matrix multiplication. If we
denote by e, a, b, c, d, f the matrices of the transformations of coordi-
nates associated with the corresponding operations, we see that these
matrices form a representation of the group. That is, the product of,
say, A and B is AB = Z); the product of the matrices a and b must

GENERAL PRINCIPLES OF GROUP THEORY

179

therefore be ab = d, so that the matrices multiply according to the
group multiplication table. We have therefore found three matrix
representations:

E

A

B C D F

(D

(D

(D (D (D (D

(D

(-1)

(-D (-D (D (D

/

1 v/ 3\

/ i v^iX/ i ^X

/ 1

vi\

10\

/-, .X

2 """F

2 T

19 9
2 2

2

2

oi/

\ +!/

\

^3 1

T" "i/

v^ 1
\ 2 "2/

V- --J

\ 2 2/

\T

1

In F 3 , the matrices e and a can be written down immediately, d and
f are obtained from 104 by inserting the proper value of 0; b and c
can then be found by means of the group multiplication table and the
rule for matrix multiplication.

It is possible to find other representations of the group. For ex-
ample, if we assign to the points a, fr, and c the coordinates (x a , y a \
etc., the matrices of the transformations would be of dimension 6
(six-row matrices) and would form a representation of the group.
Let us suppose that we have found some such representation, and let
us call the corresponding matrices e 7 , a r , b 7 , c 7 , d 7 , f 7 . The new set
of matrices e 77 = p~Vp; a 77 = P'Vp 7 ; b 77 = (TVp, etc., also form
a representation of the group, as may be seen as follows. Assume that

a /7 b 77

10-16

Then p~ 1 a / pp~ 1 b / p = p"" 1 d 7 p. From the associative law of multi-
plication, we have

If we now multiply from the left by p and from the right by p"" 1 we have

aV = d 7 10-17

Since 10-17 is true, 10.16 must also be true; the transformed matrices
e 77 , etc., therefore also form a representation of the group. The trans-
formations of the type a 77 = p'Vp are called similarity transfor-
mations. Let us now suppose that it is possible to find a similarity
transformation which will transform all the matrices e', a' into

180
the f orm

GROUP THEORY

a

// _

/

ai'

\

tf

J'

1048

where aj is a square matrix which has the same dimension as
bi', c 7 / , and where there are only zeros outside the squares.
Since a /7 b 77 = d 77 , we have from the law of matrix multiplication
the relations

10-19

The sets of matrices a", a", b" ; e", a", b^' ; etc., therefore
form representations of the group. The matrix representation e 7 ,
a 7 , b 7 is said to be reducible and to have been reduced by the simi-
larity transformation with the matrix p. If it is not possible to find a
similarity transformation which will further reduce all the matrices
of a given representation, the representation is said to be irreducible.

The representations PI, F 2 , F 3 given above are all irreducible. Since
matrices representing transformations of interest to us are unitary,
we may restrict ourselves to representations which involve only unitary
matrices and to similarity transformations with unitary matrices.
Two irreducible representations which differ only by a similarity trans-
formation are said to be equivalent. We shall now show that the
non-equivalent irreducible representations FI, F 2 , F 3 given above are
the only non-equivalent irreducible representations of the correspond-
ing group, and we shall then state, without proof, certain general
theorems regarding irreducible representations.

We denote by Tt(R) the matrix corresponding to the operation R
of the Oh irreducible representation, and by F,-(Jf2) mn the mnth com-
ponent of this matrix. For the above representations, we therefore

GENERAL PRINCIPLES OF GROUP THEORY 181

have the relations: 1

=6

R

Lr a (fi)iir a (B)ii = 6

R 10-20

R R

Er 3 (#) 21 r 3 (7i!)2i = Zr 3 (#)2 2 r 3 (#) 23 = 3

In addition, we note that

Eri(R) n r 8 (fl)ii = o

H 10-21

All the relations of the type 10-21 can be expressed by the equations

Er f -(fi)mr/(flW = 0; i^j

R 10*22

Er<(fi) mll r<(/Z) m / n / =0; m ^ ro'; n ^ fl,'

72

while the relations 10-20 can be written as

l I\(fl) wn = 10-23

where A is the order of the group and li is the dimension of the ith
representation. Equations 10-22 and 10-23 can be combined into the
general relation (see Appendix VI)

* a mm , 5 nn , 10-24

where 5*y = 1 if i = j; % = otherwise. Equation 10-84 may be shown
to be true for the non-equivalent irreducible representations of any group.

From equation 10-22 we see that the matrix components Tt(Ri) mn9
Ti(R2)mn Ti(Rh)mn of the h elements of the group can be regarded
as the components of an /^-dimensional vector which is orthogonal to
any one of the vectors obtained by a different choice of the subscripts

1 To be more general, these expressions should be replaced

R
etc., but this form of the equations will be sufficiently general for our purposes.

182 GROUP THEORY

m and n, as well as being. orthogonal to any of the similar vectors ob-
tained from a different irreducible representation. If there are c such
irreducible representations, each of dimension l^ there are l\ + Z| + *
+ 1 2 C such orthogonal vectors. But it is possible to construct only
h orthogonal A-dimensional vectors. Actually,

l\ + l\ + + % = h 10-25

This result is perfectly general and follows directly from 10-24. The
representations TI, T 2 , and T 3 are therefore the only non-equivalent
irreducible representations of the symmetric group of three points.

The sum of the diagonal elements of a matrix is known as the char-
acter of the matrix. We denote by Xi(R) the character of the matrix
of the operation R belonging to the ith irreducible representation of
the group, that is

10-26

The characters of the representations TI, TZ, FS of the group which we
have been discussing are

E A B D F
xi 1 1 1 l 1 1

X2 1-1-1-1 1 1

Xs 2 0-1-1

The character of a matrix is unchanged by a similarity transformation.
The character of a matrix P is xp = EP- The character of

Q = X~*PX is

i j k j k i

- EL SvP = ZP# - XP

3 * 3

If two operations belong to the same class, the corresponding matrices
for a given representation have the same character, as may also be
verified from the character table above.
From 10-24, we have the result

^Ti(R) mm Tj(R) m ' m f = r- dij 3 mm t
R l >

Summing over m from 1 to Z and over m' from 1 to lj gives

Zx<(B)x/(B) = r 5 *' S mm > = *</ E 1 - hfy 10-27

R l>j m-lm'-l lj m'-l

GENERAL PRINCIPLES OF GROUP THEORY 183

We see, therefore, that the characters of the matrices of the irreducible
representations form sets of orthogonal vectors. Since the character
is unchanged by a similarity transformation, we see that two non-
equivalent representations have different character systems and that
two irreducible representations with the same character system are
equivalent.

Since the characters of all matrices of a given representation which.
correspond to operations in the same class are equal, 10-27 can be
written as

or

10-28

p=l

where g p is the number of elements in class p, R p is any. one of the
operations in this class, x(Rp) is the corresponding character, and k is

the number of classes. The normalized characters Xi(Rp) \l~r & Q

> h

therefore the components of a set of orthogonal vectors in fc-dimensional
space. Since there can be k such vectors, we see that the number of
irreducible representations is equal to the number of classes.

From the relations already developed it is possible to obtain further
interesting results. Any matrix representation of a group must be
some one of the irreducible representations or some combination of
them; otherwise it would be an additional irreducible representation,
but the number of irreducible representations is limited to the number
of classes. Any reducible representation can be reduced to its irre-
ducible representations by a similarity transformation which leaves the
character unchanged. Thus we can write for the character of a matrix
R of the reducible representation the expression

k

X (R ) = E a j Xj (R) 10- 29

0=1

where a, is the number of times the jth irreducible representation
occurs in the reducible representation. From 10-27 we have

Ex()x(fi) = EE<WCy(fl)x<(fi) = ^ 10-30

R R 3

so that the number of times the irreducible representation I\ occurs

184 GROUP THEORY

in the reducible representation is

10-31

Since there is a one-to-one correspondence between the character sys-
tems of a group and the irreducible representations of the group, we
will usually find it sufficient to deal with the characters themselves
rather than with the irreducible representations. For any group, the
character table can be built up by means of the relations already de-
rived. We here summarize these rules in a convenient form. -

RULE 1. The number of irreducible representations is equal to
the number of classes of the group.

RULE 2. The sum of the squares of the dimensions of the irre-
ducible representations of a group is equal to the order of the group,
that is,

li + l\ + + ll = h
Since lj = X;C#)> this is equivalent to the relation

= h 10.32

RULE 3. The character systems of non-equivalent irreducible
representations form orthogonal vectors; that is

Zx<(fl)xX) =0; i^j 10-33

R

RULE 4. The sum of the squares of the characters of a given ir-
reducible representation is equal to the order of the group; that is

Etxi()] 3 = h 10-34

R

lOc. Group Theory and Quantum Mechanics. We consider now
the Schrodinger equation

for some atomic or molecular system. Suppose that R is some trans-
formation of coordinates which has the effect of interchanging like
particles in the system. For example, in helium, R could be the trans-
formation which interchanges the two electrons; in H 2 O, R could be
the transformation which interchanges the hydrogen atoms. We sub-
ject both sides of the Schrodinger equation to the transformation R,
obtaining RHfa = RE^. Since R interchanges ohly like particles,
it can have no effect on the Hamiltonian, so that RH = HK. R y of

GROUP THEORY AND QUANTUM MECHANICS 185

course, commutes with the constant J5 t -, so that we have

HBfc = EiRfa 10-35

that is, the function R\l/i is a solution of the Schrodinger equation with
the eigenvalue Ei. If Ei is a non-degenerate eigenvalue, then ^ or
constant multiples of \l/i are the only eigenfunctions satisfying 10-35,
so that for this case we have R^i = c^; in order that Rfa be normalized,
c = 1. If Ei is fc-fold degenerate, then any linear combination of
the functions ^i, ^2, fak will be a solution of 10-35, so that in this
case we have

k

Rtu = EM'tf 10-36

j-l

where the a's must satisfy the relation

L4 = i

y-i

If $ is another operation which interchanges like particles, we also have

k

Stij = E Zwfc* 10-37

m=l

Applying operation <S to 10-36 gives

k k k

SRtu = E ayiSfcy - E E a/&Am 10-38

;1 y-lm-1

Now the product of 5 and R, which we may denote by SR = 7 1 , is
likewise an operation which interchanges like particles, so that

Tt - E Cmrfim 10-39

m=l

Comparing 10-38 and 10-39, we see that

k

Cmi = E bmyty* 1040

y-i

If we now form the matrix a from the coefficients ayj, and the matrix
b from the coefficients fe m y, we see that the product of these two ma-
trices is equal to the matrix c formed from the coefficient Cmi', more-
over, all the matrices are unitary. In other words, the matrices
obtained from the coefficients in the expansion of Rfai, etc., form a
representation of the group of operations which leave the Hamiltonian
unchanged. The set of eigenfunctions ^*i, fat is said to form a
basis for a representation of the group, since the representation is
generated by the application of operations Jf2, S, etc. The dimension

186 GROUP THEORY

of the representation is equal to the degeneracy of the corresponding
eigenvalue. The representations generated by the eigenfunctions
corresponding to a single eigenvalue are irreducible representations, as
otherwise it would be possible to form sets of linear combinations

of the original eigenfunctions such that operations of the group would
send one of the new eigenfunctions into a linear combination involving
only members of the same set. But, if this were possible, the eigen-
values corresponding to the new sets could be different, which would
contradict our original assumption, except for the extremely rare case
of " accidental degeneracy " (where two eigenvalues are the same
even though the corresponding eigenfunctions behave differently under
the operations of the group). We may therefore in general assume
that sets of eigenfunctions with the same eigenvalue form a basis for
an irreducible representation of the group of operations which leave
the Hamiltonian unchanged. Returning to our original notation, if
Py is an irreducible representation of dimension fc, and if \l/{, fy $
is a set of degenerate eigenfunctions which form the basis for the jth
irreducible representation of the group of symmetry operations, these
eigenfunctions transform according to the relation

Bitf = Erxfi)**/ 10-41

1-1

If we are dealing with a symmetrical atomic or molecular system, these
considerations place a severe restriction on the possible eigenfunctions
of the system. All possible eigenfunctions must form bases for some
irreducible representation of the group of symmetry operations. From
a knowledge of the irreducible representations of the group, we there-
fore know immediately what degrees of degeneracy are possible. The
form of the possible eigenfunctions is also determined to a large extent,
since they must transform in a quite definite way under the operations
of the group. For example, if our system had the symmetry of the
group of three points which we have discussed in detail in this chapter,
our eigenfunctions would be of the following types. There would be
a set of eigenfunctions which would form bases for the representation
PI. These eigenfunctions would be non-degenerate and would remain
unchanged if subjected to any of the operations of the group.
There would be another set of non-degenerate eigenfunctions which
form bases for the representation P 2 ; these would remain unchanged
if subjected to operations E, D, and F, but would change sign if sub-
jected to operations A, J5, or C. Finally, there would be a set of doubly

THE DIRECT PRODUCT 187

degenerate eigenvalues; two eigenf unctions with the same eigenvalue
would behave in the manner determined by the matrices for the irre-
ducible representation F 3 and equation 1041. No other types of
eigenf unctions would be possible; for example, there would be no
triply degenerate eigenvalues, nor would there be any non-degenerate
eigenf unctions which changed sign when subjected to operations D
orF.

lOd. The Direct Product. Let us suppose that R is some operation
of a group, and that AI, A 2 A m \ J3i, B 2 - B n are two sets of
functions which form bases for representations of the group. Then

m

RAi = E djiAj

RB k = E
j-i

and

RAiB k =

y-i J-i j i

The set of functions AiB k forms a basis for a representation of the group
of dimension mn. The matrix c of this representation has the character

i - E Ea#&H = x(a)x(6) 1042

j i j-u-i

The set of functions A^Bk is called the direct product of the sets of func-
tions Ai and Bk. Equation 1042 then tells us that the character of
the representation of the direct product is equal to the product of the
characters of the individual representations. The representation of
the direct product of two irreducible representations will in general
be a reducible representation but may be expressed in terms of the
irreducible representations by means of equation 10-31. For example,
for the direct products of the irreducible representations FI, F2, F 3 of
the symmetric group of three points, we have

FiFx = Fi FiFs = F 2

F2F2 = FI FiFa = FS

r 3 r 3 = F! + r 2 + r 3 F 2 r 3 - r 3

The importance of the direct product appears when we wish to evalu-
ate integrals involving functions which are bases for representations of

the group. If we have an integral / <PA<PB dr, this integral will be

different from zero only if the integrand is invariant under all the
operations of the group or may be expressed as a sum of terms of which

188

GROUP THEORY

at least one is invariant. The integrand belongs to the representation
Pint = PAP# where TA^B is the direct product of the representations
of <PA and <PB. In general, TA^B will be reducible, that is, will be ex-
pressible as

where the F/s are irreducible representations of the group. The in-
tegral will be different from zero only if TA^B contains the totally
symmetrical representation PI. It may readily be verified from the
tables in Appendix VII that, if the characters of the representation
are real, as they will be in all cases of interest to us, then TA^B con-
tains TI only if F^i = T#. For our purposes, therefore, we may state

the following corollary to this theorem: The integral / <PA/<PB dr is

different from zero only if TA^B = P/. Moreover, since the Hamil-
tonian operator belongs to the totally symmetrical representation PI,

the integral / <PA^<PB dr is different from zero only if TA = P#.
the secular determinant of the type

In

HH

H\ n

=

the terms HH and Sij will be different from zero only if <pi and <pj be-
long to the same irreducible representation. By classifying the eigen-
functions <p according to the representation to which they belong, it
is often possible to reduce the order of the secular equation.

It may happen that in certain problems we start a perturbation
calculation with zero-order eigenfunctions which do not themselves
form bases for irreducible representations of a group. If we take the
proper linear combinations of these eigenfunctions so that the new
eigenfunctions form bases for ii reducible representations, the secular
equation will be simplified. These linear combinations can be found
by the following procedure. Denote the original eigenfunctions by
<p f and the new by <p, where <pj m is the eigenfunction belonging to the
ith irreducible representation of dimension li with the eigenvalue E^.
Suppose further that there are s eigenvalues corresponding to the
representation I\; that is, the representation F occurs s; times in the
reducible representation to which the <p''a belong. Then any of the

THE DIRECT PRODUCT 189

may be expressed in terms of the <p's by

* = E E E **m*L. 1043

If / is any operation of the group, then by 1041

V = E E E c ikm E IX/ZWL 1044

i k~l m=l n=l
*i

If we now multiply by Xj(R) = E Tj(R)tt and sum over all operations

<=i

of the group, we have

xXK)V = E E E c ffcw E E ErXR)r<(K)nmbi 10-45

R i k=*l m=l n=l *=1 #

From 10-24, we see that this expression reduces to

= E E<v^rE E*Mn0[ 1046

= E Lrcjwrfi 1047

A?=i <=Wy

Equation 1047 has the following meaning. If both Sj and lj are unity,

then ^Xj(K)R<p' will give a constant times <pji, regardless of which
R

<p' we use. If lj is unity, but s^ is, say, Uvo, then ExX^)^' will give

R

expressions of the form cup{i + btp^. There will be two linearly inde-
pendent expressions of this form; the combinations of those which
correspond to the two eigenvalues are determined in the usual way
after the corresponding two-row determinant has been solved. For
Sj equal to unity, lj equal to 2, we obtain two independent linear com-
binations which have the same eigenvalue. For Sj and lj equal to 2,
we obtain four independent linear combinations, the solution of the
corresponding secular determinant will then enable us to form two
sets of two combinations each, one set for each of the two eigenvalues.
The other cases are analogous.

A description of the various symmetry groups of interest in the theory
of molecular structure has been included in Appendix VII. Appen-
dix VII also contains the character tables for these groups, as well as
the transformation properties of certain quantities which will be of
interest in our later work. We shall make considerable use of group
theory in later chapters; the actual applications are much easier to
perform than might be expected from some of the rather complicated
equations which appear above.

CHAPTER XI
ELECTRONIC STATES OF DIATOMIC MOLECULES

lla. Separation of Electronic and Nuclear Motions. A molecule
is usually defined as a stable group of atoms held together by valence
forces. We shall here, however, use the word molecule in a somewhat
wider sense, to denote any system of atomic nuclei and electrons,
whether stable or not. If we regard the problem of the motion of such
a system from the classical viewpoint, we see that, because of the great
masses of the nuclei as compared with the mass of the electron, the
electrons will move with much greater velocities than the nuclei, so
that, to a first approximation at least, the motion of the electrons is the
same as it would be if the nuclei were held fixed in space.

This same approximation is stated quantum mechanically by the
assumption that the eigenfunction ^ for the whole system may be ex-
pressed as the product of the two factors $ n and \f/ ei where \p n involves
only the coordinates of the nuclei, while ^ e is an eigenfunction of the
electronic coordinates found by solving Schrodinger's equation with the
assumption that the nuclei are held fixed in space. The coordinates
of the nuclei would thus enter $ e only as parameters.

In order to test the validity of this assumption we need to see if such
an eigenfunction can satisfy, to a good approximation, the wave equa-
tion for the whole system. The exact Hamiltonian operator may be
written as

h 2 h 2

H = 271,*- ^a ~~ S o o ^* ~l~ Vnn + V ne + V ee ll'l

8w*M a i Sir'm

where the first term represents the kinetic energy of the nuclei, the
second represents the kinetic energy of the electrons, and V nn , V ne ,
and V ee are the contributions to the potential energy arising from
nuclear, nuclear-electronic, and electronic interactions, respectively.
If the nuclei were assumed to be fixed in space, the Hamiltonian for
the electrons would be

H. = - rV V? + Vne + Vee 11-2

i OTT m
190

ELECTRONIC AND NUCLEAR MOTIONS 191

If we now represent the remaining terms in 11-1 by H n we have

and

H = H n + H. 114

We define \[/ e as the function which satisfies the equation

H e ^ - Erf. 11-5

where E e is the electronic energy. Now if we write the wave equation
for the complete system, assuming ^ to be of the form ^ n ^ e , where
^ n is a function of nuclear coordinates only, we have

HM W = EM n 11-8

or

+ (V nn + Vne + V..)M* = E^ n 11-7

Now

so that 11-7 becomes

7-2 7 2 1 7 2

-E ^r v^ - v a ^ n - z -27-

' i '

( V nn + V ne + V ee )M n = E^ n 11-8

OTT m

If we neglect the terms in braces, this reduces to
te^ h 2 o

(Vne + 7 M >
- Et e + Vnnte = 11-9

or, from 11-2 and 11-5,

E -

7T M-a.

which, because of 11-3, is equivalent to

(H n + EM n S* n 1M1

We thus see that, if our approximation is valid, the effective Hamil-
tonian for nuclear motion is just that which would arise if we assumed

192 ELECTRONIC STATES OF DIATOMIC MOLECULES

that the electronic energy E ej which will be a function of the inter-
nuclear distances regarded as parameters, behaved as a part of the
potential energy of the nuclei. For equation 11-11 to be valid, the
terms in braces in 11-8 must be small in comparison to the term

^eZ) 2 ^atn, which represents the kinetic energy of the nuclei.

a STT J\l a

\l/ e is usually only a slowly varying function of the nuclear coordinates,
so that Vate ^ s much smaller than V a ^ n ; hence the approximation will
be valid. Otherwise the neglected terms may be treated as a perturba-
tion and wiH s.;I\e rise to energy terms representing the interactions
of electronic and nuclear motions. We shall postpone further con-
sideration of equation 11-11 to Chapter XIV, devoting the remainder
of this and the following two chapters to the solution of equation 11-5.

lib. Molecular Orbitals; The H 2 + Ion. The problem of the
electronic structure of molecules bears many resemblances to the prob-
lem of atomic structure. Just as the eigenf unctions of atoms are
usually built up as linear combinations of atomic orbital s, so may the
eigenf unctions of molecules be approximated by a series of molecular
orbitals. No method for the construction of molecular orbitals, how-
ever, is comparable in accuracy to the Hartreo method for atoms. It
is true that the principles of the Hartree method apply equally to atoms
and molecules, but the difficulties encountered in the numerical inte-
grations needrd to calculate the Hartree field of even the simplest
molecules have not yet been overcome.

The chief source of trouble in molecular problems is the absence of
the spherical symmetry of the isolated atom. The operators M 2
and its components, which played such an important part in our treat-
ment of atomic structure, no longer commute with the Hamiltonian,
and so they lose their usefulness. It is true that many molecules have
some elements of symmetry, when the problem can be simplified with
the aid of group theory, but these symmetry elements are properties
of individual molecules and cannot be used in the general theory of
molecular structure.

In order to set up a system of molecular orbitals we are almost
forced to use linear combinations of some set of functions in terms of
which an arbitrary function m&y be expressed. Such a set might be
the atomic orbitals of any one of the atoms composing the molecule.
This set, however, would converge very slowly if we tried to expand
in terms of it an orbital belonging to some other atom of the molecule.
If we use as our set the orbitals of all the atoms of the molecule we
should expect rather rapid convergence of our molecular orbitals. The
value of this method can be estimated only by actual trial.

MOLECULAR ORBITALS

193

Let us therefore suppose that we have set up an approximate po-
tential field V for the molecule, in which an electron is to move. This
field might, for example, be that obtained b} r the superposition of the
Hartree fields of the component atoms. Let us also suppose that we
are given a set of functions <pi, <p2, m <pn, which we shall think of as
atomic orbitals of the various atoms of the molecule, although any set
of independent functions could be used. The approximate orbitals
can then be found by the method of trial eigenf unctions, using <pi, <p n
as the zero-order functions. The approximate energies of the molecu-
lar orbitals will therefore be the roots of the secular equation

=

11-12

H nn S nn

where, if H is the one-electron Hamiltonian H =

= J pjHpy
= / <

dr

<P*<PJ dr

11-18
1144

Unless the atoms of the molecule are infinitely far apart, the integral
Sy will not in general vanish, since <pi and <pj are not necessarily eigen-
functions of the same Hamiltonian and are therefore not necessarily
orthogonal.

If, however, the atoms are a large distance apart, all the off-diagonal
terms of 11-12 will vanish; those between orbitals of different atoms be-
cause each orbital vanishes over the region where the other has a finite
value, those on the same atom because then H is just the atomic Hamil-
tonian in the region where the orbitals do not vanish, and the orbitals
are eigenfunctions of the atomic Hamiltonian. If the atomic orbitals
are normalized, equation 11-12 is of the form

HH E

//nn ~ E

= o

11-15

194 ELECTRONIC STATES OF DIATOMIC MOLECULES

and the roots are E = H\\, #22, H nn . In other words, the energies
of the molecular orbitals are equal to the energies of the atomic orbitals
if the atoms are far apart.

If we now bring the atoms together, the roots of equation 11-12 will
change continuously. We may thus correlate each of the n roots of
11-12 for any configuration of the nuclei with one of the energies of
one of the separated atoms. This does not, however, imply that each
molecular orbital will become an atomic orbital on separation of the
nuclei, for, if any two of the atomic orbitals have the same energy,
the molecular orbitals whose energies approach this energy on separa-
tion will usually go over into some linear combination of these orbitals.
To illustrate this effect let us consider two orbitals which have the same
energy at infinite separation. When the separation is large, equation
11-12 has the form

- E

Ei - E

= 11-16

where EI is the energy of each orbital and is the small value of H\^.
The roots of this equation are E = EI =t e. If we take E = EI + c,
the linear combination of <p\ and <?% given by equation 7-49 is

~~7= GPI + ^2) 5 while if we take E = EI e, the linear combination is

V2

=. (^ <p 2 Y If we go to the other extreme and let the distances
V2

between the nuclei become zero, the Hamiltonian of the molecule re-
duces to that of an atom whose nuclear charge is the sum of the nuclear
charges of the atoms composing the molecule. The proper orbitals
are then just the atomic orbitals of this " united atom." Each mo-
lecular orbital may therefore be designated by the united atom orbital
into which it degenerates when the nuclei are brought together, as well
as by the atomic orbital (or combination of such) which it becomes
when the nuclei are separated.

As an example of this procedure let us consider the hydrogen mo-
lecular ion H2 + . In this molecule just one electron is moving in the
potential field of the two nuclei. We may get a rough description of
the lowest orbital of this molecule by considering it as a linear combina-
tion of the Is orbitals of the two hydrogen atoms. Let us designate
the nuclei by the letters a and 6, the Is orbital of an electron in the field
of nucleus a alone by &, and the Is orbital of an electron in the field

THE H 2 + ION
of nucleus b alone by fa. Analytically (in atomic units)

195

If we put

8

vV

afa dr Hbb =

/r
^ a H^ a dr H a b = I ^aHfo dr = #&

J

the determination of the coefficients in the linear combination

leads to the following secular determinant for the energy

-0

H ab - SE Hbb - E

11-17

11-18

11-19

11-20

From the symmetry of the problem it is evident that H aa = Hu Us-
ing this relation, the roots of the determinant are found to be

or

Haa~E~ ^(Hab- SE)
tlaa \ flab -r-, H-aa

i- s

11-21

The first root gives the following set of simultaneous equations for the
coefficients in 11-19

- H ab )c a + (H ab - SH aa )c b ] =

11-22

Equations 11-22 are satisfied only if c = c&. In order that the eigen-
f unction 11-19 be normalized, we must have

so that

2c a c b S
1

196 ELECTRONIC STATES OF DIATOMIC MOLECULES

Similarly, the second root gives

c a = -c b

1

'2- 2S

The wave functions and their associated energies are therefore

$a + tb H aa +

V2 + 2S

ta-tb

'2 -25

1 + 8

HT]
aa "<

1 - 5

11-23

The integrals S, H aa , and H ak may all be evaluated exactly. In atomic
units the Hamiltonian is

11-24

where R is the internucloar distance in units of a . Since

where EH is the energy of the ground state of hydrogen, the matrix
elements for the energy become

-/,'!_.

R aa '

11-25

Hab = I EH + } S db9 Cab = / -~- dr
\ '<

so tnat the cnorgj" levels are

11-26

In order to evaluate the integrals involved in 11-25 and 11-26, it is
convenient to transform to elliptical coordinates (Appendix III)

R

" 2 ) dfj. dv dtp

THE H 2 + ION 197

For the " overlap " integral S we have

S =

r>3 /. /il p2

~ / *-**<* / ( -*)&- /

07T /] /_! /Q

P3 /

^e-^dn- \ e- K d
o t/j

8

33

2i i" ~ | n | ~j-w J> JL *rl

The integrals involved are special cases of the general integral (Appen-
dix VIII).

Y__*i _ It

, u .e

..,. -n = A n (a) 11-28

so that S is readily found to be

/ # 2 \

rr _ R 1-t i r> i * v I 11 Qr\

S = e ^l + fl + -y 11-29

The integral e aa is
1 re~ 2I{
irJ R b

R 2

9 T-

30

~i *^i ^-i J

The integrals in v are special cases of the integral

r l

J x n e~ ax dx - (~l) n+1 ^ n (-a) - A n (a) 11-31

Inserting the proper values for the integrals gives

6 = ~ 1 1 - e~ 2R (l + R)} 11-32

In the same way, we find for c^:

R / 1 I D \ 1 1 O O

*ab & (1 + R) Il-t30

For large values of R we see that S 0, H aa = ^?H H a b = 0> so that
EI = jB 2 = ^H; that is, just the energy of a normal hydrogen atom,

as of course it should be. For R = 0, S = 1, H aa = E& 1 + ~- >

^5 = ffaa., Neglecting the nuclear repulsion term -- for the time

K

being, we see that the electronic energies are E{ = 3Z? H , E' 2 0.

198 ELECTRONIC STATES OF DIATOMIC MOLECULES

When R = 0, the lowest molecular orbital should become the Is atomic
orbital of He, with an energy 4B H . Our approximation is therefore
in error by an amount H for R = 0, although it is correct for large R.
The reason for this is clear. For large R the orbital \l/i is the correct
orbital for a hydrogen atom, but for R = it is again a hydrogen or-
bital, but surrounding a nucleus whose charge is two instead of one.
In order to get a good approximation for small R we should take a
number of orbitals for each atom.

Referring to 11*26, and for simplicity neglecting S as compared with
unity, we see that the difference in energy between the two states, to
this approximation, is just 2e a &. Also, to this approximation, state
^2 is unstable with respect to a hydrogen atom and a proton by an
amount (1 + R)(e~ R + e~ 2R ) while state ^i is stable by an amount
(1 + R)(e~ R e~ 2R ). That this should be so may be seen qualitar
tively in the following manner. For state fa, the electron density is

While for state fa it is

At a point midway between the two nuclei, we have

4

Pi

28

State fa thus has a much greater accumulation of charge between
the two nuclei tnan state ^ 2 ; -the attraction between this accumulation
of charge and the two protons may be considered as producing the
stability of state ^i.

It is of some interest to look at this problem from the following view-
point. The wave functions, including the time-dependent term, are,
if we neglect S as compared with unity,

-A 1 -<*

*i = iM * ==4^ + ^)0 *
v2

-f i l - t

^ 2 = fae * = : (^ a - fa)e *

V2

Any linear combination of these two solutions will represent some
particular distribution of electron density. Let us consider the com-
bination * = 7=. (^i + ^2)- The electron density corresponding to
V2

THE H 2 + ION

199

this state is

Forf = 0, p = 5(^1 +

> so that the electron is on nucleus a.

FIG. 11-1. Binding energy of H2**" as a function of the internuclear distance.

so that the electron is on nucleus 6. From this viewpoint (which should
not be taken too literally), the electron oscillates between a and 6,

V 1?

the frequency of the oscillation being v = ^-7 . We thus have

200

ELECTRONIC STATES OF DIATOMIC MOLECULES

the result AS = hv, where v is the frequency of the oscillation between
the two states and AE is the difference in energy between these two
states.

In Figure 11-1 the energy levels for these two states are plotted as
a function of the intermi clear distance 7?, along with the experimental
curve as determined from spectroscopic data. In Figure 11-2 the dis-

R
FIG. 11-2. Electron density distribution in Ha + .

tribution of charge along the internuclear axis is shown. It is seen
that this approximation gives qualitatively correct results, although
quantitatively the treatment is not very satisfactory. The results
can be somewhat improved by taking more complicated zero-order
functions.

A simple method of improving the agreement would be to introduce
a parameter into $ a and ^; for example, we might take

^flLl -*

and then vary a so that the energy is minimized. We should then get
agreement at R as well as at R = o> . The energy may be further

THE ELECTRONIC STATES OF THE H 2 + ION 201

improved by including in the secular equation the 2p orbitals of the
hydrogen atoms. The inclusion of these terms partly takes into ac-
count the polarization of the hydrogen atom by the other nucleus.

We shall see in Chapter XIV how the depth D e , the internuelear
distance r e , and the curvature of the energy curve at the minimum
may be determined from spectroscopic data. For H^ 4 " the band
spectra indicate the values D e = 2.791 e.v. and r ti = 1.06 A. The
simple theory as described above gives D e = 1.76 e.v. and r e = 1.32 A.
Introduction of the parameter a improves these results to D e = 2.25 e.v.
and r - 1.06 A, 1 and inclusion of the 2p orbitals gives D e 2.71 e.v.,
r e = 1.06 A. 2 The value of D e could, of course, be improved by add-
ing more and more hydrogen orbitals.

The best orbitals which have been obtained were found by a different
method of approach. This method is similar to that used in the varia-
tiorial treatment of the helium atom; hydrogenlike orbitals are given
up completely. For the H^* ion, the natural coordinates to use are
the elliptical coordinates ju, v, <p. James 3 found that a good approxi-
mation to the lowest orbital of this molecule is

where d and c are parameters. For the observed internuclear dis-
tance 1.06 A, the best values of the parameters give D fi = 2.772 e.v.,
which is quite close to the experimental value. Even better results
may be obtained, however, since the wave equation is separable in
these coordinates and may be solved by numerical integration. We
shall discuss the results in detail in the follow ; ng section.

lie. The Electronic States of the Ha" 1 " Ion. In the theory of the
electronic states of molecules, particularly of diatomic molecules, the
simplest example, H^, plays a role of importance equal to that of the
hydrogen atom in the problem of the electronic structure of complex
atoms. We wish at this point, therefore, to present a general dis-
cussion of the possible states of this molecule and its exact energy
levels before proceeding to the description of more complicated dia-
tomic molecules.

The wave equation for the hydrogen molecule ion may be written as

11.34

1 B. Finkelstein and G. Horowitz, Z. Physik, 48, 118 (1928).
2 B. Dickinson, J. Chem. Phys., 1, 317 (1933).
1 H. M. James, J. Chem. Phys., 3, 7 (1936).

202 ELECTRONIC STATES OF DIATOMIC MOLECULES

We now transform to the elliptical coordinates

r a

In this coordinate system, the Laplacian operator V 2 is (Appendix III)
4

M 2 -

Equation 11-34, in the new coordinate system, is then

*\f* n ^1 _L a F/1 2x ^

(M "" } + ( }

Gu 2 - 1)(1 - v 2

+ { / 2 2\ i o D ) i f\ ^ i o c

{ (/A V ) + ZfJ,l\Y ^ ll'OO

where

2ft 2 \" rai >
We now try to find a solution of the form

11-36

Since <p enters equation 11-35 only in the term % , it is at once appar-
ent that this equation can be separated into a part dependent on <p
alone and a part dependent on /z and v. We call the first separational
parameter X 2 , so that $(??) satisfies the equation

11-37
Equation 11-35 is thus reduced to

i2

f 2 ^ 2

T; T- (M - 1) ~ -- -a 7 "" /* +
Af djL*

1

We set both sides of this equation equal to r and thus obtain the
final differential equations for M (/z) and N(v).

\

I - 11-39

HOMONUCLEAR DIATOMIC MOLECULES 203

-Q 1140

The set of equations 11-37, 11-39, 1140 will possess satisfactory solu-
tions only if the parameters X, r, and 6 have certain definite values.
The set of simultaneous equations has been solved by Teller, 4 and, for
the ground state, by Burrau, 5 Hylleraas, 6 and Jaffe, 7 leading to results
in complete agreement with experiment. The solution of the <p equa-
tion leads to the familiar result

where X takes on positive or negative integral values. The energy
depends on X through |x|, since only X 2 occurs in the equations which
determine the energy parameter e. For r a & = 0, the wave equation
is the same as that for He + , except for the difference in nuclear mass.
The quantum number X is a " good " quantum number at all inter-
nuclear distances, since the equation in <p can always be separated
from the remainder of the wave equation. For r a & = 0, that is, for
the united atom, X becomes equivalent to the atomic quantum num-
ber w. In describing the molecular orbital, it is therefore character-
ized by the designation of the state of the united atom to which it
reduces, 2s, 2p, etc., plus the designation of the X value; the symbols
a, TT, d denote |x| = 0, 1, 2 . Since, for the united atom,
m = I, I 1 I, the possible molecular orbitals are Iso-, 2s<r, 2p<r,
2pw y 3s<r, 3po", 3pir, 3d<r, Scfor, 3dd . The cr states are non-degener-
ate, the TT, 6, etc., states are doubly degenerate because of the equiva-
lence of the two values X. In Figure 11-3 the electronic energies of
several of the lowest states of H 2 + are plotted; in Figure 114 the total
energy for the Istr and 2pcr states are plotted as a function of the inter-
nuclear distance. 4 It will be noted that the state Iso- corresponds to
the state {^ a (ls) +^6 (Is)} of our earlier treatment; the state 2pcr
corresponds to {&*(!$) ^6 (!$)}

lid. Homonuclear Diatomic Molecules. At this point it is of
value to consider, on the basis of group theory, the possible states of a
homonuclear diatomic molecule. Such molecules belong to the sym-
metry group DOOA; the characteristics of the possible states, including
their degeneracies, are determined directly from the character table

4 E. Teller, Z. Physik, 61, 458 (1930);

8 O. Burrau, Kgl. Danske Videnskab. Sdskdb., 7, 1 (1927).

6 E. Hylleraas, Z. Physik, 71, 739 (1931).

7 G. Jaffe, Z. Physik, 87, 536 (1934).

204 ELECTRONIC STATES OF DIATOMIC MOLECULES
0.0

4.0

0.0 2.0 4.0 6.0 8.0-

FIG. 11-3. Calculated electronic energy

oo

-0.70

-0,80-

-0.90 -

-1.00-

-L10-

-1.20- -

1.0 2.0 3.0 4.0 6.0 6.0 T.Q 8.0

FIG. 114. Calculated total energy of H2^".

HOMONUCLEAR DIATOMIC MOLECULES 205

of the irreducible representations of this group. This character table
is shown below. The symbols in the first column are those used to
describe the electronic states of homonuclear diatomic molecules.

1 1

J J

1 -1

2-1 1 -1 -1 -1 1

2 cos <?

2 cos ?>

E

2CV

00

iE

1

1

1

1

1

1

1

~1

]

1

-1

1

1

1

-1

-1

2

2 CO8<f>

2

2

2 cos^>

-2

2

2 cos 2<?

2

2

2 cos 2<p

2

States which are invariant under rotation about the symmetry axis
are called 2 states. If only one electron is present, these are the states
for which X = 0; if more than one electron is present, these are the
states for which A = ]C\ t - = 0. n states are those for which A = 1;

i

A states those for which A = 2, etc. If only one electron is piesent,
these are equivalent to the TT and d states discussed above. In addition,
the S states are characterized by the designation + or according
to the manner in which they behave when subjected to the operation
<r v , which is a reflection in a plane in which the symmetry axis lies. All
states are further characterized by the symbols g (" gerade ") and u
(" ungcrode ") These symbols tell whether tho wave function re-
mains invariant or changes sign u]^n inversion at the center of sym-
metry. For the special case of the H2 + ion, we readily see that a
orbitals give S + states, ?r orbitals give H states, etc. The g or u prop-
erty is independent of the internuclear distance. If the molecular
orbital is written as a linear combination of atomic orbitals, then we
immediately see that combinations of the type {^ a + fa] ar ^ g, those
of the type \^ a ^b\ are u. For the united atom, the center of sym-
metry becomes the atomic nucleus, a molecular orbital thus has the
same g or u property as the atomic orbital to which it reduces. For
a one-electron atom, the wave function is g or u according as I is even
or odd; for a many-electron atom, the atomic wave function is g or u

according as ]C? t - is even or odd. We thus see that a 2scr orbital gives

a S/" state, a 2po- orbital gives a Sj state, and a 2pw orbital gives a
n u state. Referring to Figure 114, we note that in H^ 4 " the S/ state
is stable, the 2 J state is unstable.

It is possible to make a unique correlation between the states of the
united atoms and the states of the separated atom by means of the

206 ELECTRONIC STATES OF DIATOMIC MOLECULES

theorem which says that two levels with the same symmetry properties
cannot cross as the internuclear distance is varied. The following
proof of this theorem is that given by Neumann and Wigner 8 as modi-
fied by Teller. 9 Suppose that we know all the electronic wave func-
tions except two. These two may be written as linear combinations
of the functions fa and fa, which have been chosen to be mutually
orthogonal and orthogonal to all the remaining wave functions. The
energy levels are given by the solutions of the equation

HH E Hi2
H 22

In order that the two roots be equal, the conditions HH = #22* #12
must be satisfied simultaneously. If ^i and fa have different symmetry
properties, then Hi 2 is identically zero. Since Hn and #22 are func-
tions of the internuclear distance, it is possible for the two to be equal
at some distance; when this is true, the two energy levels are equal,
and crossing can occur. If fa and fa have the same symmetry, #i 2
will not be zero. It will not, in general, be possible to satisfy the two
conditions by varying one parameter, hence the two energy levels
can never be equal, and crossing is impossible.

Since X is a good quantum number at all times, and since the g and
u properties must be preserved as the internuclear distance is varied,
the correlation between the states of the united atom and those of
the separated atom must be a g > <r g) <r u > <r u , ir g > K g , etc. Further,
two ff g orbitals cannot cross; neither can two <r u orbitals, etc. In
Figure 11-5, we have on the left the various states of the united atom
and the possible molecular orbitals into which they can split; on the
right we have the various states of the separated atoms and the sym-
metry properties of the molecular orbitals which can be formed by
taking linear combinations of the atomic orbitals. The correlation
between the two sets of states is given by the connecting lines which
were drawn in accordance with the above rules. It will be noted that
this schematic representation corresponds exactly (as of course it
must) with the exact energy level diagram for H2 + (Figure 11-3).

In determining the electron configurations of complex atoms, we
added the electrons to hydrogenlike orbitals, placing the electrons into
the lowest orbitals allowed by the exclusion principle. Similarly, for

8 J. v. Neumann and E. Wigner, Physik. Z., 30, 467 (1929).
D E. Teller, /. Chem. Phys., 41, 109 (1936);

HOMONUCLEAR DIATOMIC MOLECULES

207

complex homonuclear diatomic molecules, we add the electrons to
the lowest allowed H2 + -like molecular orbitals. Taking account of
electron spin, the exclusion principle allows us to place two electrons
in <r orbitals, and four electrons in TT, d orbitals, because of the
two possible values of X in the latter. This procedure gives us the
electron configuration of the lowest state, which will then be char-
acterized by the values of A = X t - and S = s t \ States with S = 0,

are called singlet, doublet, triplet, states, just as for atoms.

4p

United Atom Separated Atoms

FIG. 11-5. Correlation diagram for like nuclei.

This procedure will be perhaps clarified by several examples. For the
simplest homonuclear diatomic molecule, H2 + , the electron configura-
tion is (<701s), the notation referring to the states of the separated atoms,
which gives A = 0, S = |, so that the state is 2 S*. (<r orbitals can,
of course, give only + states; TT, d orbitals can lead to either +

208 ELECTRONIC STATES OF DIATOMIC MOLECULE

or states. If the number of electrons in u orbitals is even, the
resulting states will be g\ if the number of electrons in u orbitals is
odd, the resulting states will be u.} For H 2 , the electron configura-
tion is (ffgls) 2 , so that A = 0, S = (because of the exclusion prin-
ciple), and the ground state is r 2^~. For Li 2 , the electron configura-
tion is (ffgls) 2 (<r w Is) 2 (ff g 2s) 2 ; the corresponding state is 1 S^. For
more complex molecules, there is a possible ambiguity in the order of
the orbitals, as this order sometimes changes as the internuclear dis-
tance is varied. For these molecules it is necessary to resort to ex-
perimental evidence to determine the electron configuration of the
ground state. For example, the electron configuration of N 2 is found
to be (v g ls) 2 (<r u ls) 2 (a g 2s) 2 (<r u 2$) 2 (w u 2p)*(<T g 2p) 2 . Only closed shells
of electrons are involved, so that the state represented by this con-
figuration is 1 S^'. In O 2 , the two additional electrons go into the
(ir g 2p) orbital. The quantum number A can be either or 2, depend-
ing on whether or not the X's are directed oppositely or are parallel.
If A is 2 the spins must be opposed; if they may be either parallel
or opposed. The configuration (v a 2p) 2 thus leads to *Z, 3 2, and
*A states. All states arc, of course, g; the detailed theory 10 shows
that they are in fact 1 2^" > 3 2^, and 1 A a . The lowest state of O 2 is
found experimentally to be the 3 2~ state; as a result, O 2 is para-
magnetic (Chapter XVII).

An interesting interpretation of the character of the chemical bond
in diatomic molecules can be given in terms of the above considerations.
If an orbital maintains the same principal quantum number as the
transition is made from the separated atoms to the united atom, it is
said to be a " bonding orbital." If the principal quantum number
increases, it is said to be an " anti-bonding orbtial," and an electron
in an orbital of this type is said to have been "promoted." As may
be seen in Figure 114, occupied bonding orbitals will tend to form
stable states; occupied anti-bonding orbital 9 will tend to form un-
stable states. The difference between the number of pairs of electrons
in bonding orbitals and the number of pairs in anti-bonding orbitals
may be regarded as the effective number of " electron pair " bonds.
For Li 2 , O 2 , and N 2 this difference is seen to be 1, 2, and 3, respectively,
which corresponds to the usual designation of the bonds in these mole-
cules as single, double, and triple.

lie. Heteronuclear Diatomic Molecules. For heteronuclear dia-
tomic molecules we no longer have the center of symmetry which was
present in the homonuclear diatomic molecules. Ileteronucloar dia-
tomic molecules belong to the symmetry group Cx> vt for which the

10 E. Wigner and E. Witiner, Z. Physik, 61, 859 (1928).

HETERONUCLEAR DIATOMIC MOLECULES 209
character table is

"Bf Q/"V ~

Hi &\s(p (TV

S+ 1 1 1

s- i i -i

n 2 2 cos <f>

A 2 2 cos 2^

We note that we have the same types of possible states as for the group
Daoh> except that the g and u property has been lost. In drawing the
correlation diagram analogous to Figure 11-5, we must take into
account the fact that there arc now, for example, two Is states for the
separated atoms, since, if we designate the atoms by a and 6, the atomic
orbitals ^ (ls) an d ^>(ls) will have different energies because of their
different nuclear charges. Let us suppose that we again write the
molecular orbital as a linear combination of atomic orbitals, for ex-
ample

1141

= 1142

The secular equation is

Haa E H a i SE

H ab -SE H bb -E

where the symbols have their usual meaning. We now wish to inves-
tigate the values of the coefficients c a and c b . We imagine the inter-
nuclear distance to be sufficiently large so that the overlap integral S
may be taken equal to zero. We further assume that a has the greater
nuclear charge, so that H aa > Hbb- The energy eigenvalues, given
by the solution of 1142, are

E = J{ (H aa + H bb ) (H aa - H bb ) 2 + 4H 2 ab } 1143

For H aa = H bb , the upper oign ,ivi - c^ -- c ; thv, low^r sign gives
c a = c b . For H aa > H bb we may therefore conclude that the upper
sign corresponds to the case where c a and c b have the same sign; the
lower sign corresponds to the case where c a and c b have opposite signs.
Further, for the limiting case where II a b = 0, the upper sign gives
E = Haa^ cl = 1, c b = 0; the lower sign gives E = HM, cl = 0,
c b = 1. For any internuclear distance we therefore conclude that
the proper linear combinations of atomic orbitals are

where c a > c b ] c" > c a f . Using the correlations a <r, ir>7r, etc.,

210

ELECTRONIC STATES OF DIATOMIC MOLECULES

and noting that two cr or two T orbitals cannot cross, we obtain the
correlation diagram for heteronuclear diatomic molecules given in
Figure 11-6, where atom a is considered to have the greater nuclear
charge. The method of determining the electron configuration for
the ground state is identical with that applied previously. For ex-
ample, LiH should have the electron configuration (ls(r) 2 (2s<r) 2 if we
use the united atom notation, which leads to the ground state 1 S + .

Is Iscr

United Atom Separated Atoms

FIG. 11-6. Correlation diagram for unlike nuclei.

If we consider the molecule (LiH) 4 " 4 ", we see that the electron con-
figuration for the ground state is (ls<r) 2 . Upon separation of the
nuclei, Figure 11-6 tells us that we obtain Li 4 " and H+; that is, both
electrons remain on the Li nucleus, which is in accord with our con-
clusion that Ca > c' b . However, if we separate the nuclei in LiH x the
correlation diagram states that we would obtain Li 4 " and H~~. Actu-
ally we would obtain Li and H. These considerations suggest that
we would obtain better molecular orbitals by the following procedure.
We consider that the Is electrons of Li are unaffected by the formation
of the molecule, and we form molecular orbitals, not between Li 4 " 4 " 4 "
and H 4 ", but between Li 4 " and H 4 ". The problem then becomes quite
similar to that of the hydrogen molecule; we take as the molecular
orbital which leads to a stable molecule the linear combination

1145

HETERONUCLEAR DIATOMIC MOLECULES 211

This orbital can, of course, be occupied by two electrons, provided
that their spins are antiparallel.

The above considerations have enabled us to determine the types
of states that may arise in diatomic molecules and have shown us how
to write approximate orbitals to describe these states. It cannot be
expected that the use of these simple orbitals will give quantitatively
correct results. In the next chapter we shall see what results can be
expected in the simpler cases, and, on the basis of these results and the
above considerations, we shall attempt to develop a satisfactory quali-
tative theory of valence.

CHAPTER XII
THE COVALENT BOND

12a. The Hydrogen Molecule. For the hydrogen molecule, the
Hamiltonian operator, in atomic units, is

where the subscripts 1 and 2 refer to the electrons and the subscripts
a and b refer to the nuclei. According to the results given in the pre-
vious chapter, the lowest molecular orbital, expressed as a linear com-
bination of the atomic orbitals of hydrogen, is

*!., = *a(l) + *&(!*) 12-2

We can, according to the exclusion principle, place two electrons in
this orbital, with their spins opposed. If we designate the electrons
by 1 and 2, the unnormalized wave function for the ground state of
the hydrogen molecule would then be, to this approximation,

12-3

where

etc.

VTT VTT

/Vfy dr

The energy E - of the ground state of hydrogen as given
Cwdr

by this wave function has been calculated approximately by Hell-
mann. 1 He finds the equilibrium distance to be R ~ 1.6a ; the dis-
sociation energy to be ~2.65 e.v. values which are not at all in
agreement with the experimental values R = 1.40a ; D 4.72 e.v.
One reason for this disagreement may be seen immediately from the
following argument. If we multiply out equation 12-3, we obtain

* - *a(l)**(2) + * a (l)ih(2) + ih(l)*(2) + * 6 (l)lfc(2) 124

1 H. Hellmann, Einftihrung in die Quantenchemie, p. 133, Franz Deuticke, 1937;

212

THE HYDROGEN MOLECULE 213

The first and last terms in this expression represent electron density
distributions in which both electrons are on the same hydrogen nucleus;
that is, they represent ionic states such as H+H~. Since it is known
that the electron affinity of hydrogen is very much less than the ioni-
zation potential of hydrogen, we would expect that such states are not
very stable, and hence that we might obtain a better representation
of the ground state of the hydrogen molecule by dropping these terms.
This leads us to the function used by Heitlcr and London 2 in the first
successful attack on the problem of chemical valence. These authors
wrote

2) 12-5

as the function representing the ground state of the hydrogen mole-
cule. The energy corresponding to this function may be written as

-#$

where

dr z

From the form of H and the functions &,(!), etc., it is readily seen
that the energy may be written as

E - 2E lt (H) + + a 12-7

where

0- 1 "" 12 ' 8

J = a(0fc(2) - - _ _ + _ + _ (W6(2 ) dri dT2

J [ tlbi K>a2 K>12 K>al)

J. l 4. r^a(l){ 2 {^(2)i 2 , , ...

= ~ 2faa + + - - dT1 dT2 12 ' 9

r

J

S2 9 a- /Va(l>fe(2(2)fe(l)
= -- 2^ 6 + I - - dri art 12- 10

** W. Heitler and F. London, Z. Physik, 44, 455 (1927).

214 THE COVALENT BOND

The integrals S, e aa , e a & have been evaluated in section lib. The re-
maining integrals have been evaluated by Heitler and London 2 and
by Sugiura. 3 The results gave for the internuclear distance and the
dissociation energy the values R = 1.64a , D = 3.14 e.v. Although
this value of the dissociation energy is only slightly better than that
obtained from the strict molecular orbital treatment, the Heitler-
London method is somewhat easier to handle than the molecular
orbital method, and we shall make considerable use of this method of
writing approximate wave functions in the following pages.

It is of interest to investigate the state of the hydrogen molecule
that arises when one electron is placed in the ls<r orbital and the other
in the 2pv orbital. The wave function is then

* = {*(!) +*&(!)} {* (2) - ^ 6 (2)} 12-11

which, if we drop the ionic terms as in the Heitler-London approxima-
tion, may be written as

l) 12-12

For this state, the energy is

E = 2fi ls (H) + (?'-' 12-13

where

Of--*. K

v - i-s 2 ' a - i - s*

and where the remaining symbols have the same significance as before.
The quantities Q and a are called the " coulombic " and " exchange "
energies, respectively; the integrals J and K are called the coulombic
and exchange integrals. The state represented by 12-12 is unstable.
It is noted that the stability of 12-5 relative to 12-12 is due essentially
to the difference in sign of the exchange energy, the binding energy of
12-5 being Q + a, and of 12-12 being Q 1 - a. In Figure 12-1 the cou-
lombic energy and the total energy, as calculated from the above equa-
tions, are plotted as a function of the internuclear distance. It is
observed that the greater part of the binding energy arises from the
exchange term, the coulombic energy being only about 10-15 per cent
of the total. We 1 further note from equation 12-10 that the exchange
integral K will be at least roughly proportional to the overlap integral
S, so that if two orbitals overlap only slightly the exchange integral will
be small, and hence the binding energy of the two orbitals will be small.
This result will later become of great importance.

8 Y. Sugiura, Z. Physik, 45, 484 (1937). '

THE HYDROGEN MOLECULE

215

We have as yet made no explicit mention of electron spin. To the
approximation that we have been using, the Hamiltonian operator
contains no terms dependent on spin, so that the spin wave functions
and the orbital wave functions are separable. As before, we designate
by a. the spin eigenf unction which has the eigenvalue +^, and by ft

Interatomic distance, A

FIG. 12-1. Calculated coulombic and total energies of H2 as functions of the

interatomic distance.

the spin eigenf unction which has the eigenvalue J. If we have two
electrons, we can form one spin eigenfunction

which is antisymmetric in the electrons, and three spin eigenfunctions

{0(1)18(2)}

which are symmetric in the electrons. Of the orbital wave functions
which we have thus far considered for the hydrogen molecule, 12-5 is

216 THE COVALENT BOND

symmetric in the electrons and 12-12 is antisymmetric in the electrons.
According to the exclusion principle, the complete wave function for
a system must change sign if two electrons are interchanged; in order
to form acceptable wave functions of the above type we must there-
fore combine a symmetrical orbital with an antisymmetrical spin func-
tion, and conversely. We thus obtain the complete wave functions

1244

12-15
{0(1)0(2)}

We note that in the stable state 1 S^" the spins are opposed; in the
unstable state 3 2t they are parallel. The general theory of valence
will be based almost entirely upon the material thus far presented in
this and the preceding chapter. Before beginning this general discus-
sion, we shall see how the above method of approach should be modified
in order to obtain better quantitative agreement with experiment.

Several simple modifications of the function 12-5 have been made.
A considerable improvement in the calculated dissociation energy is
obtained by introducing an effective nuclear charge as a variational
parameter, that is, writing

/ 3 V*

y/d^L) == i i 6 , ei/Cr

This function, with a = 1.17, gives D = 3.76 e.v. 4 A more general
function is obtained by writing the unnormalized function as

where the nuclei lie along the z axis, that is> including the 2p z hydrogen
orbital in the variational function in order to take account of the fact
that one hydrogen atom will polarize the other. With a. = 1.17,
c\ = 0.10, a value D = 4.02 e.v. is obtained for the dissociation energy. 5
if in addition the ionic terms in 12-4 are included, multiplied by a
parameter C2, then, with a = 1.19, c\ = 0.07, c 2 = 0.175, the dissocia-
tion energy is calculated 6 to be D = 4.10 e.v. as compared with the
experimental value D = 4.72 e.v. This small value for c 2 is further
confirmation of the fact that, for the hydrogen molecule at least, the

4 S. Wang, Phys. Rev., 31, 579 (1928).

8 N. Rosen, Phys. Rev., 38, 2099 (1931).

6 8. Weinbaura, J. Chem. Phys., 1, 317 (1933).

THE HYDROGEN MOLECULE 217

Heitler-London or " valence-bond " method is superior to the method
of molecular orbitals.

As for H2* 4 " , it is again found that an accurate value of the binding
energy can be obtained only by a method of approach which is not
based upon the use of one-electron approximate wave functions.
James and Coolidge 7 used a variation function which was written as
a function of elliptic coordinates and which included the interelectron
distances #12 explicitly. These authors investigated the function

* = e- 5( ^ 2) c klmnp {^2^ n 2U p + Mi/&WuP| 12-16

klmnp

where

Hal + Rbl Ra2 + Rb2

-

Ral ~~

p p

Mab n>ab tCab

The form of the function is such that it is symmetric in regard to inter-
change of electrons. In order that it may also be bymmetric in the
coordinates of the nuclei, only those terms which have (ra +- ri) an
e\en interger were included; the indices were taken to be positive
integers or zero. With 5 = 0.75, the calculations, for a thirteen-
term function, gave R a b = 1.40 a and D = 4.698 e.v., in essentially
complete agreement with experiment. The accuracy could no doubt
be further improved by the inclusion of additional terms. We have
had several examples (He, H 2 + , H 2 ) of the exact results which can be
obtained by means of the variational method with a wisely chosen
variation function. However, the labor involved in these calculations
is so great even for these simple systems that it does not appear to be
a profitable method of attack on molecular problems in general. Be-
cause of the mathematical difficulties involved, we are forced to use
much less accurate approximations; usually we are forced to write the
wave function as some linear combination of one-electron wave func-
tions. Although these will not give satisfactory quantitative results,
they should in general be qualitatively correct, and should enable us
to correlate experimental chemical facts. It is to be noted that the
James and Coolidge treatment of H 2 contains nothing corresponding
to the separation of the binding energy into a coulombic part and an
exchange part which appeared in the Heitler-London treatment.
This separation is a mathematical result of the use of one-electron
orbitals in forming the valence-bond function. Thus, though we shall

7 H. James and A. Coolidge, /. Chem. Phys. y J, 825 (1933).

218 THE COVALENT BOND

continually use the terms " coulombic energy " and " exchange en-
ergy, " the reader should remember that these terms have more of a
mathematical than a physical significance. It might be added, how-
ever, that in principle any wave function can be written in the form

* = E(-im {/(},<& at,- <)}

V

where P v is the operator which interchanges the subscripts of the sets
of quantum numbers a}, etc., this form of function being that which
has the proper symmetry in regard to the interchange of like particles.
The associated energy will then be a sum of integrals. Certain of these
integrals will be of the type

'/(a!, ai, . OH/CaJ, ai )
which may be called coulombic integrals; others will be of the type
l, o|, OH/Xai a?, .- <), etc.

which may be called exchange integrals.

12b. The Covalent or Electron-Pair Bond. 8 According to the
Heitler-London theory, which gives a satisfactory qualitative descrip-
tion of the hydrogen molecule, the covalent bond in this molecule is
represented by the orbital wave function

{lk(l)fc(2)+ik(2)ih(l)} 12-17

where the functions ^ a (l), etc., have the significance previously stated.
This orbital wave function must be combined with the spin wave
function

representing oppositely directed spins, in order that the complete
wave function be antisymmetric in the electrons. Parallel spins lead,
as we have seen, to unstable states. The valence bond in LiH, ac-
cording to the discussion in the last chapter, will be represented, in
the molecular orbital method, by the function

12-18
or, in the Heitler-London method, by the function

12-19

8 The methods employed in this section are essentially those employed by Pauling.
See L. Pauling, Nature of the Chemical Bond, Cornell University Press, 1940.

THE COVALENT OR ELECTRON-PAIR BOND 219

where $u is the 2s wave function of Li and ^H is the Is wave function
of H. The mathematical treatment follows the same lines as for H 2 ;
the function 1249, which must be combined with an antisymmetric
spin function representing oppositely directed spins, leads to a stable
molecule.

The Heitler-London method is obviously not limited to the treat-
ment of the formation of the bonds in diatomic molecules. In CEU,
for example, if we let fob ^C2> fos, ^C4 be the orbitals occupied by
four of the electrons of the carbon atom, and ^HI> ^112, ^H3> ^H4 be
the Is orbitals of the four hydrogen atoms, then the set of functions

(5)^ H3 (6) + ifcs (6)^3 (5) }

would represent four covalent bonds formed by eight electrons. With-
out at this time further specifying the nature of the four carbon orbitals,
we can state that they will be orbitals with the principal quantum
number n equal to 2. The remaining two electrons of carbon will be
in the Is orbital and will have their spins paired; as each of the four
valence electrons has its spin paired with an electron from a hydrogen
atom, we see that all electron spins in the molecule are paired, so that
the resultant spin S of the molecule is zero. Similarly in LiH and
in H2 all the electron spins are paired. The experimental fact that
stable molecules (with a very few exceptions) are non-paramagnetic
(see Chapter XVII) is a confirmation of the above-derived results that
the resultant spin should be zero for molecules in their ground states.

An electron which has its spin paired with another electron from
the same atom obviously cannot take part in the formation of a co-
valent bond, so that the covalency of an atom is equal to the number
of electrons with unpaired spins possessed by the atom. The results
of this rule have been mentioned in Chapter IX; we shall not repeat
that discussion at this point but shall consider a number of examples
in greater detail.

The electron configuration of the ground state of the nitrogen atom
is (ls) 2 (2s) 2 (2p) 3 ; the lowest term is 4 /S, indicating that the spins
of the three 2p electrons are parallel. Nitrogen should thus have a
covalence of 3, in accordance with experimental facts. The three
valence orbitals may be written as $2 PXJ ^2p vt fap,', if we were to carry
out actual numerical calculations, we would use, for example, Slater-

220 THE COVALENT BOND

type eigenfunctions with the effective nuclear charge appropriate for
the 2p orbitals of nitrogen. If we wished to describe the valence
bonds in Nils, then, according to the Heitler-London theory, we would
write

with analogous expressions for the two remaining bonds. According
to the results of the similar calculation for H 2 , the energy of the bond
represented by 12-21 is determined largely by the value of the exchange
integral between fa Px and ^ n . This exchange integral is, as we have
seen, proportional to the overlap integral between \l/ 2px and ^ H . Now
\f/2 Px has its maximum value along the x axis. Consequently the over-
lap integral arid hence the exchange integral will have their maximum
values when the hydrogen atom lies along the x axis; that is, this
particular N II bond will be stronger when the hydrogen atom is on
the x axis. The same argument can be carried out for the other bonds;
the stable configuration for the ammonia molecule is thus that in which
the three hydrogen atoms lie along the x, y, and z axes, respectively.
Since the three bond* are equivalent except for orientation in space,
this simple theory of directed valence predicts that the ammonia mole-
cule should be a triangular pyramid, with all H N H angles equal
to 90. The experimental value is about 108; the manner in which
the simple theory must be modified will be discussed later.

The oxygen atom has the electron configuration (ls) 2 (2s) 2 (2p) 4
the lowest term being 3 P. Since there are four electrons to be placed
in the three 2p orbitals, one of the orbitals, which we may take to be the
2p z orbital, must be occupied by two electrons with their spins paired.
The two valence orbitals are then ^2 Px and $2 Pv ', the arguments used
above for NH 3 lead us to expect that the II O H angle in H 2 O would
be 90. The actual angle is 105, indicating that the simple picture
of the oxygen valence also requires some modification.

The description of the covalent bonds formed by the carbon atom
is less simple than the description given above for nitrogen and oxygen,
and it requires a new concept : that of the formation of valence orbitals
by the " hybridization " of the simple atomic orbitals. The lowest
electron configuration of carbon is (ls) 2 (2s) 2 (2p) 2 ; the lowest term
is 3 P. If this electron configuration represented the valence state
of carbon, it would be divalent, with the spatial distribution of the
bonds similar to that of oxygen, in complete contradiction to experi-
mental facts. However, the first excited electron configuration of
carbon is (ls) 2 (2s)(2p) 3 ; the lowest term arising from this electron
configuration is 5 S; carbon in this state had four electrons with un-
paired spins and hence has a valence of 4. This excited state, arising

THE COVALENT OR ELECTRON-PAIR BOND 22]

from the " promotion " of a 2s electron into the 2p level, has not been
definitely located experimentally, but according to calculations by the
Hartree method, as well as certain indirect experimental evidence,
the 5 S term lies about 3-4 e.v. above the ground state. 9 There is
probably no very close correlation between this energy difference in
the carbon atom itself and the energy required to " promote " a 2s
electron when a compound such as CH* is formed, since the perturba-
tion of the energy levels of carbon by the hydrogen atoms will be ex-
ceedingly large. This promotional energy would be more than com-
pensated for by the energy furnished during the formation of the two
additional covalent bonds. It may therefore be regarded as a reason-
able assumption that quadricovalent carbon is represented by the
electronic state (ls) 2 (2s)(2p) 3 , 5 S.

The four valence orbitals of carbon could then be written as fast
fap x > fap y t fap,- However, we need not limit ourselves to this particu-
lar set of four valence orbitals but may use linear combinations of
them. We will therefore take as our valence orbitals the set of linear
combinations ^i, fa, fa, ^ 4 , the combinations being formed in such a
way that they fulfill the necessary requirements of being normalized
and mutually orthogonal. In addition, we will require the four va-
lence orbitals to have the maximum possible bond-forming power.
According to our previous discussion regarding exchange and overlap
integrals, this means that we want fa, for example, to be that linear
combination of \l/ 2s , fa Px , fa Pv , fap, which has the largest possible
magnitude along some arbitrary direction in space, subject, of course,
to the condition that it be normalized. The functions fa a , fap x , fap v >
fap, may be written

fa Px = R* P (r) sin B cos <p ^

fa Pv = K 2p (r) sin 6 sin <p

fap, = #2p(r) COS0

The radial functions /^(r) and R 2p (r) will not differ greatly; if we
use Slater-type eigenf unctions the two are identical. Assuming the
radial functions to be identical, and writing only the angular parts of
the wave functions, we have

fas = 1 t2 Pv = 3 sin 6 sin <p

,- / 12-23

fapx = ^3 sin cos <f> fa Pg = v 3 cos 6

where the angular parts of the wave functions are normalized to 4ir.
9 C. W. Ufford, Phys. Rev., 63, 568 (1938).

222 THE COVALENT BOND

In accordance with the result that the strength of the bond formed
by two valence orbitals will be proportional to the overlap of the two
orbitals, Pauling 10 has defined Jbhe bond-forming strengths of the above
orbitals to be 1 for ^ 2s and V 3 for fap x , fap v , and \f/ 2pg , these being the
maximum values possessed by the angular parts of the wave functions.
We now write

+ Wzp, + c^ 2pv + dfap, 12-24

where the possible values of the coeffi-
cients are restricted by the normalization
condition a 2 + 6 2 + c 2 + d 2 = 1. The di-
rection in space in which this first orbital
has its maximum value is arbitrary. We
choose this direction to be along the (1,1,1)
(diagonal of a cube with the carbon atom
at its center and with the x, y, and z axes
parallel to edges of the cube, as illustrated
in Figure 12-2. This choice of direction
FIG. 12-2. Coordinate system requires the coefficients 6, c, and d to be
for tetrahedral carbon orbitals. equal, SO that ^i may be Written as

^i = at 28 + b(t 2ps + fa Pv + t 2pg ) 12-25

with the normalization condition a 2 + 36 2 = 1. Along the (1, 1, 1)

diagonal, sin.^ = cos <p 7= , cos 6 = 7=- , sin 6 = ^ , so that

V 2 V3 v3

faps == $2p v = fop, = 1 along this diagonal. The bond-forming strength
of ^i is thus (a + 36), or, by use of the normalization condition
(a + V3 V 1 a 2 ). The condition that the bond strength be a maxi-
mum gives us the relation

da

so that a = ^. The condition a 2 + 36 2 = 1 gives b = ^; the valence
orbital ^i is thus

*1 = |(^2s + fa Ps + *2 Pv + *2p,) 12-26

This valence orbital, which has the bond-forming strength 2.00, has the
maximum strength which can be obtained from any linear combina-
tion of 2s and 2p orbitals.

10 See page 78 of reference 8.

THE COVALENT OR ELECTRON-PAIR BOND 223

If we form the set of valence orbitals

1>2p, ~

_

we note that these functions are normalized and mutually orthogonal,
each function having a bond-forming strength of 2.00 (the maximum
possible value for 2s and 2p functions), with the maximum density
along the diagonal of the cube indicated. All members of the set are
equivalent except for orientation in space. Each member of the set
has its maximum value along one of the lines from the carbon atom to
a corner of a regular tetrahedron with the carbon atom at its center.
This set of orbitals, determined by the condition that their bond-
forming power should be a maximum, thus gives a description of the
valence of carbon in complete accord with the experimental facts.

It seems probable that promotion of a 2s electron also takes place
to a certain extent in nitrogen and oxygen. Since the strongest pos-
sible bonds that can be formed from a combination of 2s and 2p orbitals
are the tetrahedral type with the coefficient of ^2* equal to f , we
would expect sufficient promotion in nitrogen and oxygen to make the
bond angles tetrahedral in NHa and H 2 0, provided that bond strength
were the only criterion for the type of bond formed. However, the
necessary promotional energy increases as we go along the feeries,
C, N, 0; in addition, promotion does not increase the valence of nitro-
gen and oxygen. The experimental facts indicate that a compromise
is reached in N and 0; the angles are slightly less than tetrahedral,
showing that promotion takes place to a lesser extent than would be
necessary to give pure tetrahedral bonds.

Actual calculations made with these valence bond functions cannot
be expected to lead to accurate values of bond energies, nor could they
be expected to lead to any results not given by the above qualitative
procedure. The above procedure must be looked upon as a quantum-
mechanical description of covalent bond formation based largely on a
previous knowledge of experimental fact. In the next chapter we will
find that it is possible to treat many problems in a more quantitative
manner; we will here largely limit ourselves to this descriptive method.

The carbon-carbon double bond, such as that in ethylene, may be
described as follows. From ^2, and two of the 2p orbitals, which we
may take to be fap 8 and fap v > we form three equivalent bond orbitals,
which will have their maximum values in the xy plane and will be

224 THE COVALENT BOND

separated by angles of 120. Taking one bond direction to be the x
direction, these bond orbitals are

1 . V2 .

~ fc* 12-28

The remaining carbon valence orbital is ^4 = $2p e - Using the first
three valence orbitals, the carbon atom can form two C H bonds
and a single C C bond. With the fourth orbital, an additional
carbon-carbon bond can be formed with the corresponding orbital of
the second carbon atom. The bond formed by orbitals of the type
^i is called a <r bond; that formed by orbitals of the type 1^4 is called
a TT bond. Let us denote the coordinate system of the first carbon
atom by x, y, z\ that of the second, by x', y', z'. The strongest x
bond will be formed when the relative orientation of the two carbon
atoms is such that fop, and ^ 2p ,, overlap as much as possible, that is,
when z and z' are parallel. The stable configuration should thus be a
planar configuration, with the H C C and H C H angles equal

FIG. 12-3. Stable configuration of C 2 H 4 .

to 120, as illustrated in Figure 12-3 (the z directions are upward, per-
pendicular to the plane of the paper). This is in agreement with the
experimental facts for ethylene. It also gives a reason for the rigidity
of the double bond; if we were to rotate one end of the molecule rela-
tive to the other, we would have to supply energy to compensate for
the weakening of the TT bond caused by this rotation. Since the carbon-
carbon or bond is not equivalent to the C H bonds, we would not

THE QUANTITATIVE TREATMENT OF HgO 225

have been required to make ^2 and ^3 equivalent to ^i ; in other words,
the H C C angle could have been made unequal to the H C H
angle. The angles in ethylene are as indicated; these angles may not
persist in substituted ethylenes.

In acetylene, we may describe the carbon-carbon triple bond as
consisting of one <r bond and two TT bonds cf the type given above.
This valence form would give acetylene the known linear configura-
tion. Since the 2p orbitals in a TT bond overlap less than the orbitals
in a a bond, we expect a TT bond to be weaker than a a bond, and hence
expect a C=C bond to be less than twice as strong as a C C bond.
The experimentally observed bond strengths are C C, 59 kcal.;
C=C, 100 kcal.; C=C, 123 kcal., in agreement with this expectation.

When we come to second-row atoms, there are available for the
formation of valence orbitals not only the 3s and 3p orbitals, but also
the 3d orbitals, five in number. It is thus possible to have a covalence
greater than 4. Rather than discuss the many individual cases by
the methods used above, we shall later present the general theory of
directed valence and show how it is possible to predict, by application
of group theory, the types of bonds formed by any combination of s,
p, and d orbitals. First, however, it is of interest to see to what extent
the above qualitative conclusions can be justified by approximate
calculations; as an example, we consider the H^O molecule.

12c. The Quantitative Treatment of H 2 O. n In the valence bond
method, the valence orbitals of the oxygen atom may be taken to be
^2p x (O) and ^2p y (0), those of the y.
hydrogen atoms to be ^i s (H). }[
We first investigate the energy of TT

the valence structure shown in f

Figure 12-4, where <p is the angle /

between the O H bond and the 0/
v axis. For a particular valence 7 ^H

structure, the energy may be
written as

o

E = Q + a 12-29 FIG. 124.

where Q is the coulombic and a is the exchange energy. According to
equation 13-30 of the next chapter, the exchange energy for this case
will be

a = KM + K 24 - f (# 14 + #23 + K 12 + KM) 12-30

where the K'a are integrals of the type 12-10. The oxygen 2p x orbital
11 J. Van Vleck and A. Sherman, Rev. Modern Phya. t 7, 200 (1935).

226 THE COVALENT BOND

can be expressed as

fap x = fap<r cos <p + \l/2p T sin <p 12-31

where \l/ 2pff is an oxygen 2p orbital with its axis along the H bond
and fapv is an oxygen 2p orbital with its axis perpendicular to the bond
direction. We may therefore write KI% as

n dr 2

= K v<f cos 2 <p + KM sin 2 <t> + 2J T(r sin <p cos <p 12-32
where

rx dr 2

with similar expressions for the other exchange integrals between
oxygen and hydrogen orbitals. The integral K\ 2 is independent of
<p. If we neglect the hydrogen-hydrogen exchange energy, the part of
the exchange energy that depends on the angle <p will be

K ffff (2 cos 2 <p sin 2 <p) + K vv (2 sin 2 <p cos 2 v?)

+ 2K vff cos <p sin <p 12-33
The integrals K vff are of the form

dn dr 2

where the z r axis is along the H bond direction and R\ s is the radial
part of \l/i 8 , etc. The integrals K V9 are therefore zero, since they in-
volve odd powers of x f . The.orbitals \l/ 2p7r and \{/i a are orthogonal;
the integral K vir thus contains only the electron repulsion term, so that
this integral is positive. The orbitals $ 2pff and \f/i 8 are not orthogonal;
the integral K ffff is therefore similar to the analogous integral in the H2
problem and is negative (corresponding to attraction). The value of
<f> which gives a the greatest negative value is therefore 90. On the
basis of the exchange energy alone, and without considering the effect
of hydrogen-hydrogen repulsions, this calculation gives the same
results as our previous qualitative investigation. The coulombic
energy for this structure is

Q = J 13 + J 24 + J 14 + J 23 + J 12 + / 34 12.34

and Js4 being neglected, the coulombic energy becomes
Q - 2 J 99 + 2J*

' If IT

GENERAL THEORY OF DIRECTED VALENCE 227

where

W dn dr 2

The coulombic energy is thus independent of (p, so that only the ex-
change energy is effective in giving directional properties to the va-
lence bonds in H^O. Inclusion of the hydrogen repulsions would, of
course, tend to increase the bond angle, as would the use of s-p hybrid
orbitals for the oxygen atom.

12d. The General Theory of Directed Valence. 12 In discussing the
valence of the carbon atom, we found that it was possible to construct
four equivalent valence orbitals from the four atomic orbitals ^2>
faps, fap y , fap,, these valence orbitals having their maximum values
in the directions of the corners of a regular tetrahedron. Similarly,
from the orbitals ^2, fap x , fa Py we found that it was possible to construct
three equivalent valence orbitals in the xy plane, separated by angles
of 120. We now wish to investigate the possible sets of equivalent
valence orbitals which can be formed from any combination of s, p,
and d atomic orbitals, that is, from any combination of some or all of
the atomic orbitals s, p x , p yy p z , d z ^ d xzj d yz , d xy , d x ^-.y^. These sets of
equivalent valence orbitals can be most easily found by means of group
theory. Any set of such orbitals has a characteristic symmetry group.
The set will form a basis for a representation of the group which will
in general be reducible, but which can be expressed in terms of the
irreducible representations of the group by means of the group char-
acter table. The s, p, and d orbitals of the atom will also form repre-
sentations of the group. By comparing the component irreducible
representations of the set of valence orbitals with those of the atomic
orbitals it will be possible to tell which combination of atomic orbitals
will lead to a set of valence orbitals of the required symmetry. The
.general procedure can be best described by an example; we will discuss
the possibilities of forming three equivalent coplanar bonds, separated
by angles of 120.

This set of valence orbitals has the symmetry D^; the character
table for this group is reproduced in Table 12-1. The table also in-
cludes the transformation properties of the coordinates and the perti-
nent combinations of the coordinates. The atomic orbitals are func-
tions of r multiplied by the function written as subscript. Hence
they transform under the operations of the group in the same way
as their subscripts. The atomic orbitals therefore form bases for
representations of the group as given in the table. It is now necessary

12 G. Kimball, J. Chem. Phys., 8, 188 (1940). J. H. Van Vleck and A. Sherman.
Rev. Modem Phys., 7, 174 (1936).

228

THE COVALENT BOND

to investigate the transformation properties of the set of valence
orbitals. These are represented schematically in Figure 12-5. Apply-
ing the operation E leaves the three orbitals unchanged; the char-
acter of the representation for the operation E, applied to this set of
orbitals which we denote by <r, is consequently 3. The operation <r hj
reflection in the xy plane, likewise leaves the orbitals unchanged; the
character for this operation is thus 3. The operation 3, a rotation

Y

FIG. 12-5. Valence orbitals with symmetry D^.

by 120 about the z axis, changes the position of all orbitals, and thus
has the character zero. The operation 83, a rotation by 120 fol-
lowed by a reflection in the xy plane, likewise has the character zero.
The operation C^ a rotation by 180 about an axis which we may take
to be the x axis, interchanges orbitals 2 and 3 and leaves orbital 1 un-
changed, and hence has the character 1. Similarly, the operation
<r v , which we may take to be a reflection in the xz plane, is seen to
have the character 1. The character of the representation <r is given
in the table. Either by inspection or by applying equation 10-31, we
see that this representation may be broken up into the sum of irre-
ducible representations

<r = A( + E' 12-35

The set of valence orbitals <r can therefore be made up of any of the
combinations

cr = s + p x + p y

a = s
<r = d 9

xy

that is, it is possible to form the set of three equivalent valence or-
bitals from any of the electron configurations sp 2 , sd 2 , dp 2 , d 3 .

GENERAL THEORY OF DIRECTED VALENCE

TABLE 12-1
CHARACTER TABLE FOR TRIGONAL ORBITALS

E <TH 2C 3 2S 3

229

s, d*

1, x 2 + y\ z 2

A(

1

1

1

1

1

1

A' 2

1

1

1

1

-1

-1

A'{

1

-1

1

-1

1

-1

P

z

A' 2 f

1

-1

1

J

J

1

1 PPv I

[d xv , rf x 2_ y 2j

(of 2 - y 2 , xy) (x, y)

E f

2

2

-1

1

d X Z, dy Z

(xz, yz)

E"

2

-2

-1

1

or

3

3

1

1

IT

6

-2

The possibility of double bond formation can be discussed in a
similar fashion. In describing the valence bonds in ethylene, we stated
that a double bond can be considered as a cr bond, with valence orbitals of
the type discussed above, plus a TT bond formed by the interaction of two
p electrons in orbitals which have their axis perpendicular to that of the

FIG. 12-6. Double bond formation for group

<r bond. In a polyatomic molecule, consisting of a central atom with a
number of external atoms bound to it, bonds of this type can also be
formed; as far as the external atoms are concerned, the condition for
the formation of TT bonds is the presence of p orbitals at right angles
to the bond axes. Since there can be two such p orbitals per external
atom, the configuration of these p orbitals can be represented by the
arrows in Figure 12-6, where orbitals 1, 2, 3 are in the xy plane and
orbitals 4, 5, 6 are parallel to the z axis. Designating this set of or-
bitals by the symbol T, we proceed to find the character of t^3 TT repre-

230 THE COVALENT BOND

sentation in the same manner as for the <r representation. We have :

E: all orbitals unchanged; character 6.

cr&: 1, 2, 3 unchanged; 4, 5, 6 changed in sign; character 0.

3: all orbitals changed in position; character 0.

3: all orbitals changed in position; character 0.

2 (considered as rotation about the x axis) : orbitals 2, 3, 5, 6 changed
in position; orbitals 1 and 4 changed in sign; character 2.

a v (considered as reflection in the xz plane) : orbitals 2, 3, 5, 6 changed
in position; orbital 4 unchanged; orbital 1 changed in sign; character 0.

The character of the w representation is given in the table.

Breaking the TT representation down into its irreducible components,
we have

TT = A' 2 + A' 2 ' + E' + E" 12-37

In order that T bonds may be formed, the central ,atom must have
available orbitals which transform in the same manner as the p or-
bitals of the external atoms, so that any orbital of the central atom
which belongs to one of the irreducible representations in 12-37 may
be used in double bond formation. Consequently the orbitals avail-
able for the formation of TT bonds are the p orbital belonging to A f 2 f
(p z ), the d orbitals belonging to E" (d xzy d yz ), and the two orbitals be-
longing to E r which are not used in forming the original a bonds. The
original a orbitals probably contain a mixture of both the p and the
dj E f orbitals, so that the TT orbitals formed from these orbitals are
probably weaker than those formed from the A" and E" orbitals.
We may divide TT bonds into two classes, calling them " strong " if
they belong to representations not used in a bond formation, and
" weak " otherwise.

The results of such calculations for a central atom surrounded by
two to eight external atoms are shown in Table 12*2. The columns
give the coordination number, the configuration of electrons used in
the formation of a orbitals, the arrangement of the bonds obtained
from this electron configuration, the orbitals available for the formation
of the strong and weak TT orbitals, in this order. Where there is a choice
df two or more orbitals when only one can be used, the orbitals are
enclosed in. parentheses.

It should be noted that this method does not predict directly the
type of bond arrangement formed from any given electron configura-
tion; it merely tells whether or not a given arrangement is possible.
If, as often happens, it is found that several arrangements of the bonds
are possible for a single configuration of electrons, the relative sta-
bility of the various arrangements must be decided by other methods,

GENERAL THEORY OF DIRECTED VALENCE 231

such as Pauling's strength criterion discussed earlier in this chapter,
or considerations of the repulsions between non-bonded atoms.

TABLE 12-2
STABLE BOND ARRANGEMENTS AND MULTIPLE BOND POSSIBILITIES

1 II III IV V

2 sp Linear p 2 d 2
dp Linear p 2 d 2
p 2 Angular d(pd) d(sd)
ds Angular d(pd) p(pd)
d 2 Angular d(pd) p(spd)

3 sp 2 Trigonal plane pd 2 d 2
dp 2 Trigonal plane pd 2 d 2
ds 2 Trigonal plane pd 2 p 2
d 3 Trigonal plane pd 2 p 2
dsp Unsymmetrical plane pd 2 (pd)d

p 3 Trigonal pyramid (sd)d*

d 2 p Trigonal pyramid (sd)p 2 d 2

4 sp 3 Tetrahedral d 2 d 3
d 3 s Tetrahedral d 2 p 3

dsp 2 Tetragonal plane d*p

d 2 p 2 Tetragonal plane d 3 p

d 2 sp Irregular tetrahedron d

dp 3 Irregular tetrahedron s

d 3 p Irregular tetrahedron a

d 4 Tetragonal pyramid d (sp)p

5 dsp 3 Bi pyramid d 2 d 2
d 3 sp Bipyramid d 2 p 2
d 2 sp 2 Tetragonal pyramid d pd 2

d 4 s Tetragonal pyramid d p 3

d 2 p 3 Tetragonal pyramid d sd 2

d 4 p Tetragonal pyramid d sp 2

d 3 p 2 Pentagonal plane pd 2

d 5 Pentagonal pyramid (P)P 2

6 d 2 sp* Octahedron d 3
d 4 sp Trigonal prism p 2 d
d 8 p Trigonal prism p 2 s
d 3 p 3 Trigonal antiprism sd!
d 3 sp 2 Mixed

d 6 s Mixed

d*p 2 Mixed

7 d 3 sp 3 ZrF 7 ~ 3 d 2
dhp ZrF 7 ~ 3 P 2
d 4 sp 2 TaF 7 ~ 2 dp
d 4 p 3 TaF 7 ~ 2 da
d*p 2 TaFr 2 ps

8 d 4 sp 3 Dodecahedron d
d B p 3 Antiprism
d 5 sp 2 Face-centered prism p

CHAPTER XIII

RESONANCE AND THE STRUCTURE OF
COMPLEX MOLECULES

13a. Spin Theory and Bond Eigenfunctions. We will now develop
the methods which will enable us to -give an approximate quantitative
treatment of many-electron molecules. Because of the complexity of
the problem, we again use products of one-electron eigenfunctions as the
basis for a perturbation calculation. For a system of n atoms, each
with one valence electron, we denote the eigenfunctions of the valence
orbitals of the atoms by a(x, y, z), b(x, y, z), c(z, y, z) - - n(x, y, z).
Let the coordinates of the ith electron be (x*, y^ zj). Then a possible
eigenfunction for the n electrons, with spin neglected, will be
a(l)6(2)c(3) n(n), where a(x\, y\, z x ) has been abbreviated to
a(l), etc. Any function which may be obtained from this by permu-
tation of the numbers 1, 2 n is an equally good eigenfunction. To
be complete, each of the above orbital eigenfunctions must be multiplied
by a spin function of the type a(l)a(2)/3(3) - a(n). According to the
exclusion principle, only those functions are allowed which are anti-
symmetric with regard to electron interchange As for the atom, our
zero-order eigenfunctions are therefore the linear combinations

(Mi

(6) 2

tp =

(c/3) 2

(oat), (6) n (c/3) n

(na) n

134

where (aa)i = o(l)a(l), etc. There are 2 n such determinants, since
each column may contain either a or ]3. We will denote the above
eigenfunction by

(a b c n\
]
a a a/

The energy, to the first order, is given by the roots of the secular equation

232

SPIN THEORY AND BOND EIGENFUNCTIONS 233

where

Hij I (piHpjdr; Sij = I <Pi<pjdr

For the ?i-electron problem, the order of this equation is 2 n , so that the
problem will be tractable only if we can break the secular determinant
down into a product of determinants of lower order. In the atomic
problem this was done by means of the operators M 2 , M 2 , S 2 , S^ all of
which commuted with the Hamiltonian. Owing to the lack of spherical
symmetry of most molecular systems, the operators M 2 and M z will no
longer commute with the Hamiltonian, and so they lose their usefulness.
To the approximation we are considering, in which spin interactions are
neglected, the operators S 2 and S z commute with the Hamiltonian and
may be used bo reduce the order of the secular determinant. Each of
the eigenf unctions <> is already an eigenf unction of S 2 , since each term
in the expansion of the determinant of y is an eigenf unction of S z with
the same eigenvalue. The eigenvalue of any <p for S z is found from the
relation

, h

&z<P = (n a ~ n ) <p
4ir

where n a is the number of columns of a's and n$ is the number of columns
j3's. For n = 6, we have 2 = 64 eigenfunetions <p. Classifying these
according to their eigenvalues of S z , we have:

Eigenvalue of S z (units of ) 32 1 -U -2 -3

\ 27T /

Number of eigenfunetions 1615 20 15 1

Since H^ = Sij = if <pt arid <pj have different eigenvalues for S z , this
classification results in a considerable simplification of the secular equa-
tion. Further simplification is possible if the <p's are combined into
linear combinations which are eigenfunetions of S 2 as well as of S z .

Let us imagine the n atoms to be divided into pairs (a, b) (c, d), etc.,
with each pair at a great distance from all the other pairs. The system
will be most stable if there is a bond between a and 6, one between
c and d, etc. According to our previous discussion, a and 6 will have a
stable bond between them only if the spins of the corresponding electrons
are paired; the same condition holds for each of the other pairs. Such a
distribution of spins corresponds to the condition S^ = 0. Even if the
atoms are not separated, it would seem reasonable to assume that the
most stable configuration would be that corresponding to the maxi-
mum number of bonds. We will therefore focus our attention on the
determinants containing only the <p's which have the eigenvalue for the

234 COMPLEX MOLECULES

operator S z equal to zero; that is, in the six-electron case for example,
we assume that the ground state of the molecule is given by one of the
roots of the twenty-row secular determinant. The eigenfunctions <p
are not in general eigenfunctions of S 2 , but we can form linear combina-
tions of the ^>'s which are. Rather than treat the general case, let us
consider the system of four electrons. The eigenfunctions <p which have
the eigenvalue zero for S are

a

b

C

d

at

at

ft

ft

at

ft

at

ft

ft

at

ot

ft

at

ft

ft

at

ft

ot

ft

ot

ft

ft

a

ot

We will now form a linear combination of the ^>'s which corresponds to a
bond between a and 6 and one between c and d. This requires a and b
to have opposite spins, so that we are limited to the functions <P2 9 <PZ,
<p> and <f> 5 . The combination will therefore be of the form

&, cd

If we interchange the spins on a and &, the function ^ a &, c d must change
sign, since the spin function associated with a stable bond is antisym-
metric in the electrons. We therefore obtain

tab, cd = 02^3 ~ <*>3<P2 ~ 4p5 ~ 5^4 13-3

Performing the same operation with the spins on c and d, we have

fab, cd = ~" ^2^4 #3^5 #4<P2 ~~ ^5^3 134

The above equations are consistent only if a 5 = a 2 , a 3 = a 4 = a 2 .
We have, therefore, for the (unnormalized) function representing the
two bonds a-6, c-d, the bond eigenfunction

&*&, cd = <P2 <Ps <P4 + <Ps 13-5

In a similar manner, we find the bond eigenfunctions

^ad, be = 91 ~ <f>2 <P5 + tf>6 13-6

&ic, fcrf = 91 <P3 ^4 + <PQ 13-7

We note that, these bond eigenfunctions can be written as

6

13-8

SPIN THEORY AND BOND EIGENFUNCTIONS 235

where 5#(n) is 1 if i has the spin a and j has the spin p; 5i/(n) is 1 if
i has the spin ft and j the spin a; tf(n) = if i and j have the same spin.
This expression can readily be generalized.

If we want the bond eigenfunction corresponding to a bond between
a and 6, but none between c and d r we proceed in an analogous manner.
Only <f> 2 , <f>3j <p4, vs can be used, as they are the only functions in which
a and b have opposite spins. This function must be antisymmetric
with respect to exchange of the spins on a and 6 but symmetric with
respect to the exchange of the spins on c and d. We readily find the
appropriate bond eigenfunction to be

6

tab = <?2 ~ <P3 + <f>4 ~ <P<> = &ab(n)<p n 13'9

nl

The other one-bond eigenfunctions are

tbc = Vl <P2 + <P5 ~ <?6

ted = <P2 + <P3 ~ <P4 <P5

tad = <P1 + <P2 <P5 <f>6 13-10

tac = <P1 <P$ + <P4 ^6

fad = 91 + <f>3 <P4 <P6

The bond eigenfunction corresponding to no bonds is obviously

# = <Pl + <f>2 + <P3 + <P4 + <P5 + >6 1341

This eigenfunction is symmetric with respect to exchange of the spins
on any pair.

Not all the above bond eigenfunctions are independent. As our
independent set we take the bond eigenfunctions $ abt c d, tad, be, tab,
tbc, ted, t- The remaining bond eigenfunctions can be expressed in
terms of these as

tac, bd - tab, cd + tad, be

tad = tab + tbc + ted -

lo'l^

tac = tab + tbc
tbd = tbc + ted

We will now show that the bond eigenfunctions derived above are eigen-
functions of S 2 , and further, that bond eigenfunctions corresponding to
different numbers of bonds have different eigenvalues for S 2 . Accord-
ing to equation 9-3, the spin operators S,- S y and the spin eigenfunctions

236 COMPLEX MOLECULES

<x and obey the relations

(Sxi + iS tfl )a(l) = (S x + iSy) = L(Sxn + tS yn )

(S al - tS Bl )a(l) =

ZTT

(8,1 - 18,0/5(1) = (S*

where S^i is the operator for the x component of the spin of electron (1),
etc. Denoting the <p eigenfunctions by the abbreviated notation
(a/to/3), etc., we have

h

) =

ZTT

(S x + iS s )G3aa0) = - {(aaa/3)

ZTT

(S x + iS v )(a/3j8a) = ^- { (aa/?a) + (o/3aa) }

JTT

(S x + t'S^Gfctfa) = -

(S, - iS s )(aaj8|8) = - { (j9a/3j8) + (aj8/3|8)} 13-14

/7T

(S x - iS y )(a/3a^) = ^- {(,8j3a/3) + (/

^7

(S x - iS v )G9aa/J) =

(S, - iS v )(a|S|8a) = ~ {(j9l3/Ja) + (a

(8, - tS v )(/Si9a) =

(S,

SPIN THEORY AND BOND EIGENFUNCTIONS 237

Operating on the bond eigenfunctions we therefore obtain

(S* + iSJtab. cd =

13*15
(S, + tS,,)^, ic =

Since

and since the eigenvalues of the bond eigenfunctions for the operator S 8
are zero, we see that the bond eigenfunctions representing two bonds are
eigenfunctions of S 2 with the eigenvalue zero.
Operating on \t/ a b, we obtain

ft f(aaa/3) + (afiaa) - (aaaff) - (j3aaa) }

(S. + iSyWab = 7T\ 13 ' 16

^1 + (cuxpa) + (apaa) (ctaftct) -
-2 {(a/Jaa) - (/3aaa)}

h i i h

4?r 4?r 2

ft2

^ a & is therefore an eigenf unction of S 2 with the eigenvalue 1(1 + 1) 5*

In the same way we find that the and \[s c d are eigenfunctions of
S 2 with the eigenvalues 1(1 + 1) ^; and that ^ is an eigenfunction

of S 2 with the eigenvalue 2(2 + 1) - Since matrix elements be-

47T

tween functions with different values of their eigenvalue for S 2 vanish,
the six-row secular determinant for the four-electron problem is thus
reduced to one three-row, one two-row, and one one-row determinant.
The energy of the ground state of the system should be given by one of
the roots of the two-row determinant, since the bond eigenfunctions
involved in this secular determinant correspond to the maximum number
of bonds. This is further confirmed by the observation that the ground
state of most molecules is non-paramagnetic, indicating zero spin.

The above results are readily generalized for more than four electrons.
To summarize the procedure when we are interested in the ground state
of a complex molecule, we first form the zero-order eigenfunctions <p ia

238 COMPLEX MOLECULES

determinantal form. These functions are eigenf unctions of S*; we
consider only those for which S z ^ = 0. From these functions <p we
form the bond eigenf unctions ^ by the rule

*mW ' ' ' ft 13-17

Bond eigenfunctions corresponding to different numbers of bonds have
different eigenvalues for S 2 ; we consider only those corresponding to the
maximum number of bonds, which have the eigenvalue zero for S 2 .

i 1 = V

d c d *c

a b a
d

a 6 a

^

tf

a 6

X

d c - c

*. + ^ = ^

FIG. 134. Bond eigenfunctions for four electrons.

The energy of the ground state is then given by the solution of the
secular equation \Hij StjE\ = 0. (In the above discussion, we
implicitly assumed the number of electrons to be even. For an odd
number of electrons, the problem is first solved for the next higher even
number, and then all integrals in the secular equation which involve the
extra electron are placed equal to zero.) In the general case, no further
reduction of the secular determinant can be made. If the molecule
possesses external symmetry, a further reduction is possible, as will be
shown later.

The possible bond eigenfunctions for the four-electron problem and the
relations between them are illustrated graphically in Figure 13*1 (from

SPIN THEORY AND BOND EIGENFUNCTIONS 239

equations 13-12). We note that a kind of vector addition law holds for
r the bond eigenfunctions. Usually we are interested only in bond eigen-
f unctions corresponding to the maximum number of bonds. Figure 134
tells us how to determine the number of independent bond eigenfunctions
for any number of electrons. We arrange the symbols for the orbitals

x / / /

I d e - d ed

e - e

i ii in rv v
Bond eigenfunctions representing 3 bonds

a/ b a b a b a b a 6

/ f / c / o f c

e d e - d e d e d \ d

VI VII VIII IX X

a b a b a b a

f G f f f

e d

e d e d e d e - d

XI XII XIII XIV

Bond eigenfunctions representing 2 bonds

o 6 a> 6^ a b a b a b

f f e / / / of o

e d e d e d e d e cL

XT XVI XVII XVIII XIX

Bond eiganfanofckms representing 1 bond

a b

f Bond eigenfunctions

representing bonds

e d
xx

FIG. 13 -2. Bond eigenfunctions for six electrons.

in a circle, which, of course, need have no correlation with the actual
structure of the molecule. We then draw the bonds in all possible ways.
Some of these will involve bonds which cross, such as ^ OC| &<* Any two
bonds which cross can be uncrossed by the application of equations of
the type

too. bd = tab, cd + tad, bo 13-18

240 COMPLEX MOLECULES

This procedure can be continued until the original bond eigenfunction
has been expanded into a linear combination of bond eigenfunctions
which do not involve crossed bonds. Conversely, any eigenfunction
can be built up from eigenfunctions which do not involve crossed bonds.
We thus have the theorem due to Rumer: 1 arrange theorbitals in a
circle, draw all the structures which contain the maximum number of
bonds, but draw only those which contain no crossed bonds. The
eigenfunctions corresponding to these structures are linearly inde-
pendent, and, as all other bond eigenfunctions representing the maxi-
mum number of bonds can be expressed in terms of them, they form a
complete set.

In order to form the bond eigenfunctions with the eigenvalue of S z
equal to zero, but representing less than the maximum possible number
of bonds, we proceed in an analogous manner, including only those bond
structures which are not derivable from others by the rule of vector
addition. For six electrons, for example, we have the bond eigenfunc-
tions as given in Figure 13-2. There is a total of twenty, as this is the
number of <p's with the eigenvalue zero for the operator S z . The deter-
mination of the energy of the ground state for a six-electron problem
would thus involve the solution of a five-row determinant.

13b. Evaluation of the Integrals. In order to complete the solution
of the problem, we need the values of the integrals of the type

HAB = I tfftyB dr and SAB = / tirta dr

Since the \l/'s are expressed as linear combinations of the <p's, we will first
focus our attention on the integrals of the type I <p*H<py dr. Return-
ing to the four-electron case, we consider the eigenfunctions

V4!

The integral / v?tH^ 2 dr is therefore
#12 =

]dr 13-19
1 G. Rumer, Goltingen Nachr., 1932, 377.

EVALUATION OF THE INTEGRALS 241

If the operator Pj is multiplied by another permutation operator, the
result will be unchanged, since the integral already contains all possible
terms. As in dealing with the atom, we therefore multiply by the
inverse operator ( l) l '[Pj]~ 1 , obtaining for the first summation

=4! {(

since there are 4! identical terms in this summation. The integral is
thus reduced to

H 12 =

HE(-ir>,{(aaM&/3) 2 M 3 (d0) 4 }] dr 13-20

/'

Because of the orthogonality of the spin functions, all the terms in 13-20
will vanish except those for which the spins match identically. Inte-
grating over the spins, we therefore have

#12 = -(abcd\R\acbd) + (abcd\H\acdb)
+ (abcd\H\cabd) - (abcd\H\cadb)
where

(abcd\B\acbd) = / (aib^c^d^^a^b^d^ dr, etc.

If we now assume that the one-electron functions a, &, etc., are mutually
orthogonal (or are willing to set all integrals involving multiple exchanges
of electrons equal to zero), all the above integrals with the exception of
the first are equal to zero; this integral will be further abbreviated to
(be). We thus have the result

#12 = - (be) 13-22

This result is perfectly general: The matrix element HIJ between two
different <p functions is zero unless the functions differ only in the spins of
two orbitals; then it is the negative of the corresponding exchange integral.
For the integral HU we obtain the result

- (abed\B\abde) - (abcd\-R\bacd)
- Q - (cd) - (a&) 13-23

That is : The matrix element HU between a <p function and itself is the cou-
lombic integral Q minus the sum of all exchange integrals between orbitals
having the same spin.

242 COMPLEX MOLECULES

Continuing with the four-electron problem, and denoting \l/ a b, cd by
\I/A and t a d, be by ^B, we require the matrix elements HAA, HBB, HAS-
In terms of the integrals #12, etc., we obtain from 13-5 and 13-6 the
results

= #22 + #33 + #44 + #55

+ 2(# 25 + #34 - #23 - #24 " #35 - #4 5 )

= #11 "t" #22 + #55 + #66

+ 2(-H 12 - H 15 + H 16 - ff 26 + H 25 - H 56 )

= #12 ~ #13 ~~ #14 + #15 ~~ #22 + #23 + #24 ~~ 2#25

+ #35 4~ #45 ~~ #55 + #26 ~~ #36 "" #46 + #56

The necessary matrix elements #y are readily found by application of
the rules given above; we have

#11 = Q - (aft) - (cd) #12 = - (be) # 25 =

#22 = Q ~ (ac) - (bd) #13 = ~ (ac) # 26 = - (ad)

#33 - Q - (ad) - (be) #14 = ~ (bd) # 34 -

#44 = Q - (ad) - (be) # 15 = - (ad) #35 = - (cd)

#55 = Q ~ (ac) - (bd) # 16 = 7/36 = - (bd)

1 #66 = Q - (ab) - (cd) #23 = - (ab) # 45 = - (ab)

-(cd) # 46 = -(ac)

#56= -

Therefore

HAA = 4Q - 2 { (ac) + (ad) + (be) + (bd) } + 4 { (ab) + (cd) }

HBB = 4Q - 2 { (ab) + (ac) + (cd) + (bd) } + 4 { (ad) + (be) } 13-24

HAB = -2Q + 4{ (ac) + (bd) } - 2{ (ab) + (ad) + (be) + (cd) }

Since / v*<pjdr = 6^-, we have the results SAA SBB = 4; SAB = 2.

In general, SAB will be equal to the coefficient of Q in HAB*

The above procedure, although straightforward, becomes rather
tedious for more than four electrons. By an extension of arguments of
the above type, the following general formulation can be made.

To find the matrix element of H between two bond eigenfunctions,
say (1) tab, cd, ef, O h and (2) \l/ ac , bd, eg, fh> we note that the two electrons
in any bond must have opposite spins. If we let the spin of a be a, that
of 6 must be from (1); according to (2) the spin of c must be and
that of d must be a. The spin of e can now be chosen arbitrarily; for a
given choice of the spin of e the remaining spins are fixed. We say that

EVALUATION OF THE INTEGRALS 243

a, 6, c, d form a cycle and e, f, g, h another. Let us assign the spin a to e.
This situation is denoted by

(ad\ feh\

where the orbitals associated with a's are written aboVe the line and
those associated with P's are written below the line. The matrix ele-
ment of H for the bond eigenfunctions (1) and (2) is then, where x is
the number of cycles, and v is the number of interchanges of orbitals
which are assigned different spins in the diagram in \l/ 2 required to
make it equal to ^.

#12 = ( 1)"2*{Q + f[Z (single exchange integrals between
orbitals in the same cycle with opposite spins) ] (single
exchange integrals between orbitals in the same cycle with
the same spin)] fZ)(all single exchange integrals)} 13-25

Equation 13-25 is perfectly general; the proof goes as follows. Let us
consider the matrix component of H between the two functions

$A = tab, cd, ef, gh &nd $B = tac, be, dh, fg

for which the diagram is ( -r ). This integral will be expressed as a sum
\bfhc /

of integrals involving <p functions, of the type I >*H^y dr. There are

three possibilities:

1. ^ == (pj' } according to 13-17, the sign of the integral is ( 1)".
This gives a contribution to HAB, according to equation 13-22, equal to

(-l)"{3(Q-[(ae)-type])}

the factor 2 arising from the fact that either a or J3 could be assigned to a.

2. The two <p functions differ in the spins assigned to, say, a and 6.
This integral is multiplied by 1, relative to the coefficient of Q, as
can be seen from 13-17, and therefore, according to 13-22, gives a con-
tribution + (ab). Since there are two such integrals, and since the
spins of any pair of this type could have been interchanged, the total
contribution to HAB arising from integrals of this type is

(-!)"{ +2Z[(o&)-type]}

3. The two <p functions differ in the spins assigned to, say, a and d.
This integral has the coefficient +1> relative to the coefficient of Q, so
that the total contribution to HAB arising from integrals of this type is

244 COMPLEX MOLECULES

and the integral HAB is

HAS = (-1)"2{Q + ElW-type] - 2EtM)-type]}

in agreement with equation 13*25.

For the functions $ c = tab, cd, e f. gh and $D = tac, bd, eg, fh we have

/ad\ /eh\
\bc)\fg/'

the diagram ( J ( ) . Again there are three cases.

1. The two <p's are identical. There are then four different ways of
assigning spins to a and e :

a~a,e~a, gives ( 1)"{Q [(od)-type]
a ~a, e~/3, gives (-1)'{Q - E[(0-type] E[(/)-type]}
a~ft e~a, gives (-1)*{Q - E[(ad)-type] - E[fo/)-type]}
a 0, e ft gives (-1)"{Q - E[(*)-type] - EIM-type]}

Cases 2 and 3 are identical with the similar cases above, except that
the numerical factor is 4. Since it is impossible to change the spin on
only one orbital in a given cycle, there are no integrals of this type in
which the two functions differ in the spins of a and e, etc. We therefore
have the result

ype] + E[(<*)-type]
~ and (a/)-types]

which is equivalent to the general expression 13-25. The proof can
readily be extended to more general cases.

Returning to the four-electron case, for HAA we have the diagram

( 7 ) ( 5 ) The matrix component of H is therefore

2 2 {Q + |[(ob) + (cd)}

- i[(o6) + (oc) + (ad) + (be) + (bd) + (cd)}

= 4Q - 2[(oc) + (ad) + (be) + (bd)] + 4((ab) + (cd)]
in agreement with the previous calculation. For HAB we have the
diagram ( 1 , so that

HAB = (-1)2{Q + ft (06) + (ad) + (be) + (cd)} - f[(oc) + (W)]

- *[(o6) + (ac) + (ad) + (be) + (bd) + (cd)]}

= -2Q + 4[(ac) + (bd)] - 2((ab) + (ad) + (be) + (cd)]

13c. The Two-Electron Problem. Although the two-electron prob-
lem has previously been treated in detail, it is of interest to consider at

THE FOUR-ELECTRON PROBLEM 245

this point the results obtained by the above method. Denoting the
orbitals by a and 6, there are four possible <p functions:

a b
<pi a ft

(f>2 ft <X

<f>3 a a

<P4 ft ft

Of these four functions, <pi and <p2 have the eigenvalue zero for S^. The
linear combination having the eigenvalue zero for S 2 is the bond eigen-
function

$A = tab = <Pl - <P2

The energy of the ground state is therefore given by the solution of the
one-row secular determinant

- S AA E =
According to the rules developed above,

HAA = 2[Q + (06)], S^ = 2

so that the energy is

E = Q + (aft) 13-26

Comparing this with the result given by equation 12*7, we see that they
become identical if the overlap integral S I ta&b dr in 12-7 is placed

equal to zero, as it has been assumed to be in the present method.
Although this discrepancy may appear to be quite serious, in actual
practice a compensation is made, as will become evident later.

13d. The Four-Electron Problem. The matrix elements have
already been determined. Inserting these in the secular determinant

HAA SAAE HAB SABE

=
HAB S A sE HBB SB BE

we obtain

-W -|-(ai + 2 +ft +02-71 -7 2 ) + iTF

= 13-26

where the common factor 4 has been removed from each term, (a&),
(cd), (ad), (be), (ac), (bd) have been replaced by i, e* 2 , ft, &, 71, 72,
respectively, and

W~E- Q+(a l +aa+p l +fa + 71+72)

246 COMPLEX MOLECULES

Writing a = i + 2, fr = ft + 2, T = Ti + 72, 13-26 reduces to

TF 2 - (a + + v)W - f (a + - 7) 2 + 3a0 =
so that

w = ?LZJ1ZJ1 IV ( a + ft + y) 2 + 3 (a

^

or

E = Q + 2 + T 2 - a/3 - ay - 187 13-27

Of these two solutions the one with the negative sign represents the more
stable state; this may be written as

E = Q - ^{ (a - 0) 2 + 08 - r) 2 + (7 - ) 2 } 13-28

a result first obtained by London. 2

If we now remove the electrons c and d to infinity, equation 13-28
reduces to

Eab = Qab + l 13-29

(the sign of a\ is taken to be + since i is a negative quantity; equa-
tion 13-29 therefore represents the lowest state of the system), which is
identical with 13-26, as of course it should be. This result forms the
basis for the " semi-empirical " method of treating complex molecules.
We assume that the energy of any electron pair bond a-b can be repre-
sented by an equation of the form E a b = Qab + a6. The energy
of this bond is obtained from the experimental data on some simple
system, usually the data on the corresponding diatomic molecule.

Some assumed ratio for ^- being taken, each of these quantities can be

o6

determined independently. It is then further assumed that the quanti-
ties i, etc., which appear in the energy equation for a many-electron
system, are identical with the corresponding quantities for the two-
electron system, and that Q for the many-electron system is equal to
where Qa is the coulombic energy for the two-electron system

i j. For example, if we had the problem of four atoms A, B, C, D
with the corresponding orbitals a, 6, c, d, the various terms in 13-28 are
obtained as follows. We draw the diagram illustrating the various
energy terms (Figure 13*3). From experimental data on the diatomic
molecules AB, CD, AC, BD, CB, AD we determine the energies E a b>
E d, E ac , E bd , E cb , E ad . These energies may be expressed as functions

2 F. London, Z. Ekktrochem., 36, 552 (1929).

THE FOUR-ELECTRON PROBLEM

247

of the corresponding interatomic distances by means of an empirical
equation such as the Morse function (equation 14-31). We now assume

some value for the ratios , etc., and thus calculate the individual

ai

quantities. These values of <*i, etc,, as obtained from the diatomic
molecules, are now used directly in 13-28. Q in this equation is assumed
to be given by

Q = Qab + Qcd + Qac + Qbd + Qcb + Qad

It will be observed that this procedure is actually an extrapolation of
the binding energy between atoms from the diatomic to the polyatomic
problem. Some such extrapolation is necessary if any treatment of poly-
atomic molecules is to be made, since actual evaluation of the integrals,
even if a satisfactory set of zero-
order functions were known, is
completely out of the question.
Since the above method of treat-
ment involves the same type of
approximation for both the dia- o*
tomic and the polyatomic prob-
lem, we would expect the defects
arising from these approxima-
tions to be at least partially
compensated for by the method
of determining the integrals from
experimental data. Actually it
must be recognized that this is an empirical method, and its validity
must be determined largely by the results obtained.

In determining Q a b and a\ from E a b, it is necessary to assume some
definite ratio for the two integrals. In the Heitler-London treatment
of the hydrogen molecule, where the corresponding integrals can be
evaluated, the total binding energy is about 10-15 per cent coulombic.
In actual practice satisfactory results are usually obtained if the coulom-
bic energy is assumed to be about 14 per cent of the total binding
energy for a given electron pair.

We may summarize the situation with regard to quantum-mechanical
calculations involving polyatomic molecules as follows. The correct
wave function for a system of n electrons would be a complicated func-
tion of the 3n coordinates. Judging from the results on H^, a satis-
factory wave function should contain the interelectron distances r t -/
explicitly. However, with such a wave function, the n-electron prob-
lem becomes completely intractable; we are forced to make drastic

FIG. 13-3. Coulombic and exchange en-
ergies for a four-atom system.

248 COMPLEX MOLECULES

simplifications. The first approximation that is made is to assume that
the eigenfunction for the system can be written as a product of one-
electron functions. Even in the simple case of H^, these wave functions
will not give very accurate results. Using these functions as a basis,
we carry out a first-order perturbation calculation in which the further
approximation that the one-electron functions are orthogonal is made.
The energy of the system is then obtained as a function of certain cou-
lombic and exchange integrals. These particular integrals arise from
the method of writing the zero-order wave function and might not appear
in this particular form in a more accurate formulation of the problem.
These integrals are in general evaluated, not by direct calculation, but
from experimental data on simple systems. In spite of the approxima-
tions, these calculations usually give results of considerable value in the
interpretation of chemical phenomena.

13e. The Concept of Resonance. We have seen that, for a system
involving many electrons, it is possible to write a number of bond eigen-
functions representing various ways of pairing the electrons to form
two-electron bonds; the energy of the system is then given by the
solution of a secular determinant of high degree. In many cases only
one of these bond eigenfunctions will be of importance in determining
the energy of the ground state of the molecule; we say then that the
electrons are localized in particular bonds. If we assume that the ground
state of a given molecule can be represented with sufficient accuracy
by one bond eigenfunction \(/A, the energy of the ground state is given by

E = -~ . From equation 13*25, we have

SAA

E = Q + 2 (exchange integrals between bonded orbitals)

TT (exchange integrals between non-bonded orbitals) 13-30

This equation has been used in section 12c in the discussion of the water
molecule, where it was assumed that the bonds could be drawn in a
unique way. In many molecules there does not appear to be any single
way of drawing the bonds which is a better representation of the actual
state of the molecule than other possible ways of drawing the bonds; in
this event the complete problem must be solved by the above methods.
We then say that the molecule " resonates " among the various states
represented by the individual bond eigenfunctions; the difference in
energy between the actual molecule and the energy of the most stable
bond eigenfunction is called the " resonance energy." This concept of
resonance has been applied, especially by Pauling, 3 to the elucidation of

8 See L. Pauling, The Nature of the Chemical Bond, Cornell University Press, 1940.

THE RESONANCE ENERGY OF BENZENE 249

many problems in molecular structure. As an example of this concept
and of the methods derived in this chapter, we calculate the resonance
energy of benzene.

13f. The Resonance Energy of Benzene. 4 Experimental evidence
shows that benzene is a planar molecule, with the carbon atoms at the
corners of a regular hexagon, and with all C C C and C C H
angles equal to 120. Taking the plane of the molecule to be the xy
plane, we can form three carbon valence bonds with bond angles equal
to 120 from the proper linear combination of the s, p x , and p y carbon
orbitals. This leaves the six p z orbitals, one on each carbon atom. The
valence structure is thus similar to that of ethylene. In benzene, how-
ever, there is no unique way of pairing the spins of the p z electrons to
form TT bonds. We shall not attempt to solve the benzene molecule
completely, but shall assume the s, p x , and p y carbon electrons and the
hydrogen electrons to be localized in a bonds, and shall limit ourselves
to the calculation of the binding energy arising from the interaction of
the p z electrons. This is a six-electron problem; if we denote the p z
orbitals on the various carbon atoms by a, 6, c, d, e,/, the five independent
bond eigenfunctions corresponding to the maximum number of bonds are

b P> 4 a b \ \ a

c f c / \ * /

\
e d e d e d e ^d e

TA \I/ \1/ \b \1/

A *B ~C VD T E

We note that, owing to the symmetry of benzene, the following rela-
tions hold between the matrix elements :

HAA HBB', HOC

HAC = HAD = HAE = HBC =

HCD H DE II CE

We need calculate only the matrix elements HAA, HAB, HAC, HCD, HCC,
which is readily done by use of equation 13-25. For HAA we have the

diagram , so that

= 2 3 {Q + f[(o&) + (cd) + (ef)} - f(all exchange integrals)}

- *[ (06) + (be) + (cd) + (de) + (ef) + (a/)]

- i[(oc) + (fid) + (ce) + (df) + (ae) + (bf)]

- i[(ad) + (be) + (cf)]}

Again, owing to the symmetry of benzene, there are only three different
4 L. Pauling and G. W. Wheland, J. Chem. Phys., 1, 362 (1933).

250

COMPLEX MOLECULES

exchange integrals: the type (a&), the type (ac), and the type (ad}.
We denote these types by a, /?, 7, respectively, so that

Since exchange integrals will decrease rather rapidly as the distance
between the atoms increases, we now make the approximation that &
and 7 can be neglected in comparison to a, and write

HAA =2 3 {Q

In the same way we find for the other necessary matrix elements the
values

Hcc = 2 3 {Q}

H AB = 2{Q + 6a}

H AC - -2 2 {Q + 3}

HCD = 2{Q + 6}

The secular determinant, after the common factor 2 3 is removed,
is therefore

where x = Q E and
minant reduces to

= . After a slight manipulation, this deter-

f*

(f * + v)

-(f + *)

-(H

+*)

(t* - fj/

) o

(1* - 2y)

THE RESONANCE ENERGY OF BENZENE 251

The roots of the secular equation are therefore

* = 0, |0, 40, f(l + Vl3)t/, f(l-Vl3)y

so that the energy levels are

E = Q E = Q - (1 + Vl3)a

E = Q - 2a E = Q - (1 - VT3)a
E = Q - 2a

a is a negative quantity, so that the energy of the ground state is

E = Q - (1 - \/13)a = Q + 2.61
In determining the coefficients in the expression

for the wave function of the ground state of benzene, we note that,
owing to the symmetry of the problem, CA = c#, GC = CD c^. (It is
not necessary to make this assumption; the solution of the five simul-
taneous equations will give the same result.) From the first equation
in the set which determines the coefficients, we thus obtain the ratio

cc 5 V13 -17

CA 24-6 V 13

so that the unnormalized wave function for the ground state of benzene is
* = ifct + *B +

Had we assumed that the ground state of the benzene molecule was
represented by one of the Kekul6 structures, say ^, the calculated
energy would have been E '= Q + l.SOa. According to the above
calculation, the actual benzene molecule is more stable than this hypo-
thetical molecule by the amount 1.1 la. The resonance energy of ben-
zene is therefore 1.1 la. Had we based our calculation on the two
Kekul6 structures alone, we would have obtained the secular determinant

with the roots x = 0, x = ft/, corresponding to the energy values
E = Q, E = Q + 2.40a. The resonance energy here is 2.40a
1.50a = 0.90a. We therefore see that the greater part of the resonance
energy arises from resonance between the two Kekul6 structures.

252

COMPLEX MOLECULES

The factorization of the secular determinant is not accidental. In
the general six-electron problem the determinant would not be further
reducible; however, on account of the presence of symmetry in the
benzene molecule we may reduce the determinant by the aid of group
theory. Benzene belongs to the symmetry group D 6 h', in this particu-
lar problem the operation i gives no new results, so that we may use the
group DQ. In Table 13-1 we present the results of the application of the

a 6

\ /

A

a b

/ \

c f c

ed
B

TABLE 131

a b

f c

e d
C

a b

K

D

a

K

E

b

/

c

E

C 2

ct

<7

ct

C7 <

?i(*)

C 2 (b)

C 2 (c) (

r a '(o&)

C 2 (bc)

C' 2 '(cd)

A

A

B

A

A

B

B

B

B

B

A

A

A

B

B

A

B

B

A

A

A

A

A

B

B

B

C

C

C

E

D

D

E

E

D

C

C

E

D

D

D

D

C

E

E

C

D

C

E

E

D

C

E

E

E

D

C

C

D

C

E

D

D

C

E

A)

5

3

2

2

1

I

1

3

3

3

TABLE 13-2
CHARACTER TABLE FOR

E

C 2

2(76

3C 2 3C 2

ri

Ai

1

1

1

1

1

1

r 2

A 2

1

1

1

1

-1

-1

r s

BI

1

-1

1

-1

1

-1

r 4

B 2

1

-1

1

-1

-1

1

r 6

Ei

2

2

-1

-1

r 6

E 2

2

2

-1

1

t

5

3

2

1

3

operations of this group to the five bond eigenf unctions, which have been
denoted by the letters A, B, C, D, and F rather than by $A, etc. In this
table, C^ means a rotation by +120, etc.; c (a) is a rotation about the
symmetry axis through atom a; c" (ab) is a rotation about the symmetry
axis perpendicular to the bond a 6, etc. The character of the repre-
sentation which has this set of bond eigenf unctions as its basis is given in
Table 13-2, which contains the characters of the irreducible representa-
tions of the group. xM was determined directly from the results given
in Table 13-1. T(^) can now be expressed in terms of the irreducible
representations of the group as

2r x + r 4 + r 6

THE RESONANCE ENERGY OF BENZENE

253

By means of equation 1045 we now find the proper linear combinations
of the original set which are the bases for these irreducible representa-
tions. The various sums x;(^)^> etc., of interest are given in

Table 13-3. From the three combinations denoted by (1), (2), (3), we

TABLE 13-3
= 6 (A + B)

6(A + B)

4(C + D + E)

=

R

R

I

R

R

If.

2E 2D

i 2(7 -- 2E

-2D -2C

- 6(A - B)

- -6(A - B)
=

=

(D
(2)
(3)

can form the independent linear combinations (2) (3) = 6(D E)
and (1) (2) = 6(C D), so that the linear combinations which are
bases for irreducible representations of the group are

Since matrix elements between eigenfunctions belonging to different
irreducible representations of the group vanish identically, our problem
is reduced by this method to the solution of two two-row and one one-row
determinants. These are:

(fa + 4y) - (3x + 6t/)

"12

#44 ~
#45 ~~

"22

(#33 - S 33 E) - f z =

#45 ~
#55

= (1)
(2)
= (3)

254 COMPLEX MOLECULES

where the notation is the same as above. The matrix elements
etc., are easily found from the previously derived matrix elements
HAB, etc. The roots of (1) are

x = f (1

For the coefficients in the expression \f/ = crfi + c^2 we obtain, for the
root x = -1(1 Vl3)2/, the ratio = 0.434, which is identical with the

Cl

previous result. From (2) we have x = 0; from (3) we have x = |^/,
x ^ lh/> so that the results are in accord with our previous findings, as of
course they must be. Determinant (3) can be reduced further, since the
two roots are known to be equal from the fact that the functions ^ 4 and
\[/5 belong to a doubly degenerate representation.

The calculated resonance energy of benzene has been found to be 1.1 la,
where a is the exchange integral between the orbitals on adjacent carbon
atoms. It would be of little value to attempt to calculate this integral
directly; we may estimate its value in the following manner. From
thermochemical data, Pauling finds that the strength of the C C bond
is 59 kcal., that of the C=C bond is 100 kcal. From our previous dis-
cussion of ethylene, we see that the difference between these values,
41 kcal., should be approximately equal to the strength of the TT bond
between the p z orbitals. Since the energy of a bond is largely exchange
energy, we would therefore estimate the value of the integral a to be
about 40 kcal. (Since the carbon-carbon distance in benzene is greater
than that in ethylene, this estimate should be a maximum.) We would
therefore expect the resonance energy in benzene to be of the order of
40 kcal. From thermochemical data, it is found that 1039 kcal. is
required to break all the bonds in benzene, while only 1000 kcal. would
be required to break three C C bonds, three C=C bonds, and six C H
bonds; that is, the experimental resonance energy is 39 kcal. This
agreement is perhaps better than should reasonably be expected,
although we shall see later that this method gives internally consistent
results for benzene and the other condensed ring systems. First, we
wish to treat the resonance energy in benzene by the molecular orbitals
method.

13g. The Resonance Energy of Benzene Molecular Orbitals
Method. 5 In the molecular orbitals method, we do not attempt to
find a set of functions with the significance of the bond eigenfunctions
used above. Rather, we form molecular orbitals which are linear combi-
nations of the atomic orbitals on the six carbon atoms. From the six

6 E. HUckel, Z. Physik, 70, 204 (1931); 72, 310 (1931); 76, 628 (1932).

THE RESONANCE ENERGY OF BENZENE

255

atomic orbitals, we can form six independent molecular orbitals. Two
electrons can then be placed in each of the three most stable molecular
orbitals, giving the state of lowest energy. We denote the six orbitals
on the six carbon atoms by $ a , fa, to td, te, and ^/. These are con-
sidered to be the solutions of a Hartree-type calculation, although we
shall not specify them further. The molecular orbitals are then of the
form

t = C a t a + Cbtb + Cote + C<$ d + C e ^e + Ctff

Carrying out the usual first-order perturbation calculation, we are led
to the secular equation

W ft

W ft

18 W p

ft W

ft W ft

ft ft W

13-31

where W = Q - E, HU = Q, #,-/ = ft if i = j 1, Hy = otherwise;
that is, the t a 's are assumed to be normalized and mutually orthogonal,
and the integrals of H between non-adjacent carbon atoms are placed
equal to zero. The roots of this equation can be shown by an algebraic
method to be

W

2p cos
o

k = 0, 1, 2, 3, 4, 5

or

W - 2ft ft -ft -2ft -ft ft

13-32

13-33

We shall later obtain these roots by a different method. The integral ft
is presumably negative, representing attraction, so that the three
lowest levels are those for which W = 2ft ft ft or

E l = Q + 2ft E 2 = Q + ft E 3 = Q + ft

If we place two electrons in each of these levels, we find the energy of the
six p z electrons in benzene to be E = 6Q + 8ft For one of the Kekul6
structures, we have the corresponding secular equation

W

ft

W

W

18

|8

W

W

ft

W

256 COMPLEX MOLECULES

with the triply degenerate lowest root W = /3. For this structure the
energy is therefore E = 6Q + 6ft so that the resonance energy of
benzene, on the basis of this treatment, is 2.00/3. It does not seem
possible to estimate the value of /3 directly; we may test the theory as
follows. The resonance energy of the series of compounds benzene,
naphthalene, anthracene, and phenanthrene is calculated by this
method. The resonance energies will be functions of the parameter |3.
If the values obtained by comparison with experimental data agree, we
may assume that our description of the structure of benzene is sub-
stantially correct. The same test can be applied to the method of bond
eigenf unctions. The results obtained are tabulated in Table 13-4. 6

TABLE 13-4

ABODE

Benzene 39 1.1 la 35 2.000 20

Naphthalene 75 2.04a 37 3.68/3 20

Anthracene 105 3.09 34 5.32/8 20

Phenaritlirene 110 3.15a 35 5.45/3 20

A = experimental resonance energy:
jB, D = calculated resonance energies.
Cj E = calculated values of a and ft:

It is seen that both theories give consistent values for the parameters;
the molecular orbital method seems slightly better here. In general, it
appears that both methods give about the same results; 7 sometimes the
method of bond eigenf unctions is slightly superior to the molecular orbi-
tals method, as it was for H 2 .

By means of group theory, we can find directly the molecular orbitals
which form bases for irreducible representations of the symmetry group
of the benzene molecule. The procedure is identical with that followed
in finding the proper linear combinations of bond eigenfunctions. In
place of the set of five bond eigenfunctions we use the set of six atomic
orbitals as the basis for the reducible representation. The character of
this representation is

E C 2 2C 3 2<7 6 3CJ 3CJ'
x ty) =600 2

which, in terms of the irreducible representations gives

r(^) = r t + r 3 + r 5 + r 6

Proceeding as above, we find the linear combinations which are the bases

6 E. Htickel, Reference 5. L. Pauling and G. Wheland, /. Chem. Phys., 1, 362
(1933); G. Wheland, ibid., 3, 356 (1935).
T G. W. Wheland, /. Chem. Phys., 2, 474 (1934).

THE RESONANCE ENERGY OP BENZENE 257

of these irreducible representations to be

to = -7= {*. + *5 + *e + * + *+*/} ( r i)
V6

f 2 = -

V6

(r 6 ) 13-34

2to + to + frf - 2to +*/}

V12

+ f 6 - to - % - *. + */}

(r.)

to =

From these linear combinations, we can immediately write down the
pertinent matrix elements of H. These are

Hu = Q + 20 #34 = i(0 - 0)
#22 = Q - 2/3 H 55 = Q + p

= Q -

The solution of the one- and two-row determinants now gives us the
energy levels :

E = Q + 2(3 (r x ) E = Q-p (r 5 ) 10QA

jg = Q-2/3 (r 3 ) = Q + ^ (r e ) 13 ' 36

The six electrons go into the lowest three molecular orbitals ^i, ^5, ^G
The molecular orbital with the lowest energy, ^, represents a charge
distribution in which the electrons are distributed symmetrically about
the ring. The second level, which is degenerate, has associated with it
an unsymmetrical charge distribution. ^ 6 , for example, represents a
charge distribution in which the electron density on atoms a and d is
four times greater than the density on any of the other atoms. This
does not mean that the resultant electron distribution is unsymmetrical.
If we take the normalized linear combinations

- ^6 + 2* c - ^ + ^ -

we see that we have a symmetrical distribution of charge if both
and \l/i a^e occupied by electrons.

CHAPTER XIV
THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

In the preceding chapters we have discussed the energy levels and
electronic wave functions for molecules in which the nuclei were assumed
to be at rest. For a diatomic molecule the electronic energy levels were
given by the solution of the equation

{H --Eo(r)}* = 144

where the electronic energy E (r) was a function of the internuclear
distance r, the stable position for the nuclei being that which made
E Q (r) a minimum. We wish in this chapter to drop the restriction that
the nuclei remain in fixed positions, and to consider the possible wave
functions and energy levels for actual molecules in which the nuclei
may move relative to one another or relative to a set of axes fixed in
space.

14a. Diatomic Molecules (Spin Neglected). For the discussion
of diatomic molecules we will use the following coordinate system.
Let x 1 , y f , and z f be the axes of a rectangular coordinate system fixed in
space, and let x, y, and z be the axes of a similar movable system with
the same origin. The orientation of the movable system relative to the
stationary system will be defined by the Eulerian angles 0, x> <P, as illus-
trated in Figure 14-1. The equations giving the relation between
x, y, z and x', y', z 1 are:

x' = x (cos <p cos x cos sin <p sin x)

y(sm <p cos x + cos 6 cos <p sin x) + # sin 6 sin x

y f = a; (cos <p sin x + cos 6 sin <p cos x) 14'2

2/(sin <p sin x cos 6 cos <p cos x) z sin 6 cos x

z 9 = x sin 6 sin q> + y sin 6 cos <p + z cos

The concept of a molecule possessing electronic, vibrational, and rota-
tional energy levels is quite familiar. Similarly, several investigations
along the lines indicated in section lla have shown that the complete
wave equation for a diatomic molecule can be separated into parts corre-
sponding to electronic, vibrational, and rotational motion, the terms
which must be neglected in order to make this separation being in general

258

DIATOMIC MOLECULES

259

very small. 1 The results of these investigations show that if the com-
plete wave function ^ is written as

* = F(x, y, z, r)R(r}U(e, x) 14-3

then the component wave functions satisfy the equations

{Ho(*, y, z, r) - E Q (r)}F(x, y, z, r) = 14-4

o / v

8?r 2 /zr 2 dr \ dr/ j

(.sin c

U(0, x) = 14-6

In these equations E is the total energy of the system, r is the internu-

clear distance, v is the reduced mass, and X is the angular momentum

2?r

about the z axis associated with the electronic wave function F(x, y, z, r).
R(r) and 7(0, x) are the vibrational and rotational wave functions,
respectively; we see that the rotational and electronic energies enter the
vibrational wave equation as effective potential energies. We shall first
investigate the possible states of the system and the symmetry proper-
ties of these states before we consider the energy levels in detail.

Equation 144 is just the equation for the electronic energy which was
discussed in Chapter XI. The electronic wave functions F(x, r, z, r)
therefore have the symmetry properties of the various irreducible repre-
sentations of the groups DOOA r
C<x> v according as the nuclei are
identical or different. The vibra-
tional wave function R(r) de-
pends only on the distance
between the two nuclei and
therefore belongs to the totally
symmetrical representation. The
complete wave function will thus
have the symmetry properties of
the product F(x, y, z, r)J7(0, x).
In order to discuss the nature of
the solutions of 14-6 it will be con-
venient to consider first the wave
equation for a symmetrical top,

Fia. 14*1. Rotating coordinate system

that is, a rigid body with rotational symmetry about one axis. Using
the coordinate system of Figure 14-l,with the symmetry axis of the rigid

1 M. Born and E. Oppenheimer, Ann. Physik, 84, 457 (1927). R. Kronig, Z.
Phyaik, 46, 814; 60, 347 (1928). J. Van Vleck, Phys. Rev., 33, 467 (1929).

260 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY
body along the z axis, the wave equation 2 for the system is

I sin 6 ~~ ) -p ~ 5~~r T o i \ ~~~. o i* ~7^ ) o
sin d dO \ SB / sin" 5 dx \siir 6 C/ dip 2

2 cos 6 d 2 ^ Sw 2 A

i- -f Ef = 14-7

sin ox <V /i

where C is the moment of inertia about the symmetry axis and A is the
moment of inertia about an axis perpendicular to the symmetry axis.
If we now set

we obtain the equation
1 d ( . ^dO\ ( M*__ /cop 2 6 A

2 n I \ 2 / I /~v

sin ^ ^

2 r,os f) Xir 2 A 1

) = 14-8

Q j.-^*rj. Q

sin 2 ^ A J

For the special case in which K = 0, this equation reduces to

Id/ d6\ f M 2 87r 2 A 1

-T-- T-lsin^- ) - -r-o 7T~ E \ e = 14 ' 9

sin d0 \ d^ / Ism 2 6 h 2 J

which is identical with the equation discussed in section 5c. The
energy levels are therefore

- 1) 14-10

and the wave functions are

QJ, o. M = P I J M] (cos e) 14-11

For K ^ 0, the solutions of 14-8 are quite complex, and we shall give
here only those results which are significant for our discussion. The
energy levels in this case are

h 2

i i\i

C~ A)\

where J > \K\, and the solutions 6/ f Kt M have the property that

e/. K, *(* - fl) = (- i) J ^- M e Jf Xf M (e) 14-13

Returning now to equation 14-6, we see that if we write

U(0, x) = e'(0)e iMx

2 F. Reiche and H. Rademacher, Z. Physik, 39, 444; 41, 453 (1927). R. Kronij? and
L Rabi, Phys. Rev., 29,' 262 (1927). D. Dennison, Rev. Modern Phys., 3, 280 (1931):

SYMMETRY PROPERTIES OF WAVE FUNCTIONS 261

we obtain the equation

sin 6 dQ '\ ( M * 4- C X*

sm + ~ ~

* C S e ****** '

= 14-14
This is identical with equation 14-8, provided that we replace K by X,

A by /zr 2 , and put equal to zero in equation 14-8, so that
C

14b. Symmetry Properties of the Wave Functions. 3 If all the
particles of the system are subjected to an inversion through the origin
of coordinates, the transformation is equivalent to changing from a right-
handed to a left-handed coordinate system. The Hamiltonian operator
is invariant under such a transformation, so that the complete wave
function must either remain unchanged or must change only in sign.
This inversion at the origin is equivalent to a rotation of 180 about an
axis perpendicular to the z axis, which we may take to be the y axis,
followed by a reflection in the xz plane. If the transformed angles are
denoted by the superscript , then 6* = T 6, <p* = 2ir <p t x t = TT + x-
If ^ = $, the wave function is said to be positive; if $* ^, the wave
function is said to be negative (the superscript t denotes the transformed
wave function). The rotation about the y axis cannot change the elec-
tronic wave function since it leaves the relative positions of electrons and
nuclei unaltered; the reflection in the xz plane is equivalent to the
operation <r v . S" 1 " states are unchanged by the operation a v ; for S~
states, the operation a v changes the sign of the wave function. For the
S states we thus have the result that (S+)' = S+, (S~)' = -S~.
For X 5^ 0, we have the doubly degenerate states n, A, etc. For these
states the electronic wave functions F\ will be of the form F \ =
/xe =tlX * > , where / \ is a function independent of <p. Since e lX ( 27r ~* J ) =
e* we conclude that F*+\ = F_ x , f*-\ = ^+\. From 14-13, we see
that eS. x, M = (- l) J -*- M Qj. x, M. Since e {M( ^> = (- l) M e iMy ,
we have, for the transformed value of U(8, x), the result [/!/, x, M =
( l) / ~ x t/'/, _x, M. The complete wave function will transform in the
same way as the product of F and U. For X = 0, we therefore have the
following relations :

1. If the electronic state is S + , then tf - ( l)*fy, so that a rota-
tional state is positive for even J and negative for odd J.

2. If the electronic state is S~~, then \l/ 1 = ( l) J+1 \l/, so that a rota-
tional state is negative for even J and positive for odd J.

3 E. Wigner and E. Witmer, Z. Physik, 61, 859_(1928);

262 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

When ^ 7^ 0, we can form two linear combinations of the functions
such that ^i = ^i, ^4 = fe these combinations are

*1 = Uj. X, Mf+X^ + (- 1) J -*U Jt -X, Mf-K*-**

= Uj, x, Mf+xe** - (-iy-*Uj.

-x, -x

For X 7^ 0, we thus have both a positive and a negative state for each
value of J.

The positive or negative character of the wave functions is independent
of the nature of the nuclei. For identical nuclei we have in addition
the g or u property; the electronic wave function either remains the
same or changes sign upon inversion at the origin. Since an interchange
of the nuclei does not affect the Hamiltonian, the complete wave function
must either remain unchanged or must change sign when the nuclei are
interchanged. Wave functions of the type that remain unchanged are
called symmetric in the nuclei; those that change sign are called anti-
symmetric. Interchange of the nuclei is equivalent to an inversion of all
particles through the origin, followed by an inversion of the electrons
only. Under the first operation the wave functions transform according
to their positive or negative character; under the second they transform
according to the g or u property of the electronic wave function. We
therefore have the following relations :

1. For molecules in g states, positive terms are symmetrical in the
nuclei, negative terms are antisymmetrical in the nuclei.

2. For molecules in u states, positive terms are antisymmetrical in
the nuclei, negative terms are symmetrical in the nuclei. In Tables 14-1
and 14-2 we give the symmetry properties of the various allowed states
for diatomic molecules, 3 where the positive and negative property is
designated by the symbols + and , the symmetric-antisymmetric
property by 5 and a.

14c. Selection Rules for Optical Transitions in Diatomic Molecules.
According to the results derived in Chapter VIII, a transition between
two states ^i and fa, resulting in the emission or absorption of radiation,

is possible only if the integral / tix'fa dr is different from zero (x f

represents x', y* , or z 7 )- Using the symmetry properties of the wave
functions we can immediately derive several selection rules for transi-
tions in diatomic molecules. Since x changes sign upon inversion, the

integral I tf'ix'fa dr will be different from zero only if $\\l/% changes sign

upon inversion. We thus have the selection rule :

1. Positive terms combine only with negative terms.

SELECTION RULES FOR OPTICAL TRANSITIONS 263

Since interchange of the nuclei does not affect the electric moment
integral for molecules with like nuclei, and since x f is unaffected by this

interchange, the integral I ty\x f fa dr will be different from zero only

if the product -fy% does not change sign under this operation. This
gives the additional selection rules for homonuclear molecules :

2. Symir^tric terms combine only with symmetric terms.

3. Antisymmetric terms combine only with antisymmetric terms.
We need in addition the selection rules for changes in J, X, and M .

TABLE 14-1

SYMMETRY OP ROTATIONAL STATES FOB HOMONUCLEAR DIATOMIC MOLECULES
J = 1 2 3 4

a a s a s

n, +-+-+-+-

s a s a s a s a

TJ I I I I . ^

as as as as

A, + - + - + -

a a a a a a

A u +-+-+-

a a as as

TABLE 14-2

SYMMETRY OF ROTATIONAL STATES FOR HETERONUCLEAR DIATOMIC MOLECULES
/ = 1 2 3 4

n +- +- +- +-

A +- +- +-

(Since the energy does not depend on M in the absence of a magnetic
field the selection rule for AM is of little significance.) Using the trans-
formations 14-1 for x, y, and z, we will obtain a series of integrals, each
of which may be written as the product of two integrals, the first involv-
ing the functions F \, B(r), and the coordinates x, y, z; the second
involving the functions Uj, \, M (0, x)e iK<f> and the angular coordinates

264 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

Q> X) v* Since R(r) is totally symmetric, the first of these integrals will
be non-zero if the product of the electronic wave functions transforms
under the operations of the group in the same manner as one of the
coordinates.

We consider first the transitions between two 2 states. For molecules
with unlike nuclei, belonging to the group C<x> v , the possible direct prod-
ucts of the irreducible representations are

2+2+ = 2+;

The only coordinate which transforms like any of these products is z,
which belongs to the representation 2+. The possible transitions are
therefore

2+ <-+ 2+ 2~ <<-> 2~

For these transitions the electric moment is along the z axis; the band
spectra corresponding to such transitions are known as parallel bands.
For molecules with like nuclei, belonging to the group D*^, the possible
direct products are

y+y-f _ y-t-

Zlg 2j0 = Zjg

2~2~ = 2+

2+2+ = 2+ 2+2+ = 2+ 2~ 2+ = 2~"
2u 2~ = 2+

Again the only coordinate which transforms like a 2 state is z, which
belongs to the representation 2+. The possible transitions are therefore

s+^s+ s,

The integrals over the angular functions associated with the integrals
over z are the same as those involved in determining the selection rules
for the hydrogen atom, where it was shown that the integrals vanish
unless

AJ - 1, AM - 0, 1

From Tables 14-1 and 14-2 it can readily be seen that the selection rule
AJ" == dbl, together with the selection rules + **-> , s <-> s, a <-> a,
give the same results for the allowed transitions between 2 states as the
above considerations bas^l on group theory.

When ono 01 more of the states has X 7* 0, then the integrals over the
angles give the following allowed transitions:

a:, y integrals ^0 AJ - 0, 1; AX - 1, AM = 0, 1

(perpendicular bands)

z integral 7* AJ = 0, dbl; AX 0, AM = 0, 1

(parallel baud

THE INFLUENCE OF NUCLEAR SPIN 265

These restrictions on J and X, plus the selection rules + <-> , s <-> s,
a <-> a, are sufficient to determine the selection rules for such transitions.
All allowed transitions for diatomic molecules 3 are listed in Table 14-3.

We note that, for homonuclear molecules, all allowed transitions
require a change in the electronic state; that is, pure rotation or rota-
tion-vibration spectra are forbidden.

TABLE 14-3
ALLOWED TRANSITIONS IN DIATOMIC MOLECULES

n ^S* u g +*'2+ n u <^s+

n <- S~ Hg <r+ 2~ n u ** s~

n <-> n u g <->> n u

A <~ n H0 <- A M n M <~> A0

A <-> A A fl <- A M

So far in this chapter we have not included the effect of electron spin.
As mentioned in Chapter XI, molecules may possess singlet, doublet,
triplet states, etc., according as the total spin angular momentum quan-
tum number AS is equal to 0, f , 1, . For molecules, just as for atoms,
the selection rule is AS = 0; this rule must be considered in applying
the results given in Table 14-3. Tables 14-1 and 14-2 are correct only
for singlet states, which are by far the most important. For a detailed
account of the way the states are modified when S ^ 0, the reader is
referred to a textbook on molecular spectroscopy. 4

14d. The Influence of Nuclear Spin. Since it is known that an
atomic nucleus has spin angular momentum as well as mass and charge,
we will have obtained an exact wave function for a molecule only when
the wave functions discussed above are multiplied by a nuclear spin
wave function <p. Designating the total wave function by \p t) we have
\l/t = \p<p, where ^ is the wave function previously discussed. We have
previously stated that a wave function must be antisymmetric in elec-
trons; the wave function must change sign when two electrons are inter-
changed. It has been found that the total wave function \l/ t must be
chosen so that it changes sign whenever any two like particles are inter-
changed; the behavior of \l/t for the interchange of two proton? or two
neutrons is the same as for the interchange of two electrons. If, in a
homonuclear molecule, the two nuclei are interchanged, the total wave
function will change sign if the number of particles (and hence the

4 G. Herzberg, Molecular Spectra and Molecular Structure, 1 , 231 , Pren tice- Hall, 1 939.

266 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

atomic weight) is odd, and will remain unchanged if the number of
particles is even. We thus have the following rules for homonuclear
molecules:

1. If the atomic weight is even, the combinations ^(s)<p(s) an d
$(a)(p(a) are possible wave functions.

2. If the atomic weight is odd, the combinations ^(s)^>(a) and
^(a)(p(s) are possible wave functions.

The symbols s and a denote functions which are symmetric and anti-
symmetric in the nuclei, respectively. Since it is known that protons
and neutrons, as well as electrons, have spin J, the resultant nuclear spin
is integral for even atomic weight and half-integral for odd atomic
weight. For atoms with nuclear spin equal to zero, we can construct
only one nuclear spin wave function ^o(l)^o(2), where ^o(l) means that
nucleus (1) has zero spin, etc. This wave function is symmetrical in
the nuclei. Since zero spin can occur only for even atomic weight, we
have the following corollary 'to the above rules:

3. For atoms with nuclear spin zero, only those states with wave
functions ^(s) are possible.

When we construct nuclear spin wave functions as indicated below,
we find that, if the nuclear spin is 7, then

Number of functions <p(s) _ I + 1 Number of " ortho " states
Number of functions <p(a) I Number of " para " states

For molecules in S states this will result in alternating statistical weights
for the alternating J values, giving alternating intensities of ratio

- to the lines in the rotational bands resulting from transitions

between these states.

We discuss first ortho- and parahydrogen. Since the proton has
spin J, the possible nuclear spin wave functions are <p + ^ and ^>_^, corre-
sponding to the spin wave functions a and /3 for an electron. We can
therefore construct three symmetrical wave functions

and one antisymmetric wave function

giving a ratio of symmetric to antisymmetric functions equal to

iii
3 = ^-~i . The ground state of hydrogen is l ^Sig. As hydrogen

THE INFLUENCE OF NUCLEAR SPIN

267

belongs to case 2, we see that the function ^ must be combined with
antisymmetric nuclear spin wave functions for even values of /, and
with symmetric nuclear spin wave functions for odd values of J. Since
the forces acting on nuclear spins are very small, there is little possibility
of a transition from a symmetrical nuclear state to an antisymmetrical
one, hence hydrogen exists in two quite distinct species : orthohydrogen,
with only odd J values allowed, and with nuclear spin statistical weight
3; and parahydrogen, with only even J values allowed, and with nuclear
spin statistical weight equal to 1. In hydrogen gas under ordinary con-
ditions the two species are thus present in the ratio 3:1. A possible
electronic transition for hydrogen is from the ground state *Z* to an
excited state 1 S^f". The symmetry properties of these states are given
in Table 14-4, with the possible transitions indicated by full and broken
lines. We see from this table that transitions involving odd values of J
in the ground state will be three times as intense as those involving even
values of J.

Deuterium has a nuclear spin of unity, so that the possible nuclear
spin wave functions are p +1 , <p Q , ^?_i. With these functions we can form
six symmetrical combinations

(2)

*-i (2)
and three antisymmetrical combinations

*_i (2) -

TABLE 14-4

in H 2

Excited State
J =

Nuclear Spin 1
Statistical Weight

268 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

Deuterium has the ground state 1 S^" and belongs to case 1, so that ortho-
deuterium has even J values, paradeuterium has odd J values; the
ortho-para ratio is 2 : 1. For a transition such as that illustrated for
hydrogen in Table 144, transitions involving even values of J in the
ground state are twice as intense as those involving odd J values.

The ground state of C>2 6 is 3 2^. O 16 has zero nuclear spin and there-
fore belongs to case 3. In the ground state, only even J values are
allowed.

14e. The Vibrational and Rotational Energy Levels of Diatomic
Molecules. The radial wave function jR(r) satisfies the equation

where E is the total energy, EQ(T) is the electronic energy, and E f (r),
the solution of 14-5, is, according to 14-12 and 14-14, given by the
expression

2

O7T [JiT

14-17

The electronic energy Eo(r), according to the results of Chapter XI, has
the following qualitative behavior. For r = 00, it has the value zero,
corresponding to dissociation of the molecule. For some value r r ey
the equilibrium distance, it has the minimum value Eo(r e ). For r 0,
EQ(T) -> oo. Regardless of the exact form of jEoO')* it can be expanded
in a Taylor's series about the position r = r e as

2!

Since r = r e is a position of equilibrium,

Equation 1446 may then be written as

x ' 8?r V
or

VIBRATIONAL AND ROTATIONAL ENERGY LEVELS 269

where

c = E - E<>(r e )
V(r) = #o(r) - E (r e )
A - J(J + 1) - X 2

The function F(r) acts in this equation as a potential energy; the energy
origin has been shifted so that F(r) =0 for r = r e . The energy c now

represents the energy of vibration and rotation of the molecule. If we
now let Jf2(r) = - <S(r), the function S(r) satisfies the equation

To a first approximation, we may write

that is, the potential energy curve, to this approximation, is that of a
harmonic oscillator. Using this value for the potential, the equation
for S becomes

14 - 21

h 2 A
Let us treat the term ; as a perturbation. Then, if we let

STT /x r

x = r r e , the equation for S is

~ S = 14-22

The boundary conditions for this problem require that R = for
r .= 0, r + oo , which is equivalent to the conditions that S = f or
r == 0, r == + oo ; or S = for # = r ej x = <*>. At a: = r e , the
potential energy is 7 (r) = ^fcrg. For those states for which c <C f fcrj,
the wave function will be quite small in the region x = r e from the.se
energy considerations alone. It will therefore be a valid approximation
for small values of c to replace the boundary condition S = for
x = r e by the condition S = for r = oo. Equation 14-22 thus
has the same form and the same boundary conditions as the equation for
the harmonic oscillator problem; the solutions are

_a

14-23

270 THE PRINCIPLES OP MOLECULAR SPECTROSCOPY
where

4V. 1 JT

- , " = ^V-

and the energy levels are e n (n + %)hv e .
The first-order correctioji to the energy will be

1 X

The term - - r^ may be expanded in terms of as
(x + r e ) 2 H r.

Only the first three terms in this expression being kept, the perturbation
energy is

h 2 A

/* f 1 T 2 1

- 2 \ S* n (x)\l - 2 ~ + 3 ~2 S n (oO <fc 14

r e ^-oo I r e rj

26

The first integral is equal to unity. Using the recursion formulas for the
harmonic oscillator (equation 4-99), we find that the second integral is

zero, the third is equal to 2 ( n + i)> so that the first-order perturbation

/ Oi e

energy is

aJ.-,.- 14.27

ar e hv e

Of equal importance to the term in a f e is the contribution to the second-

2x
order perturbation energy arising from the term -- B' e A. This gives

a contribution

Tf 12 ITT/

,
I

n 71 i CTJ

Using the results

(n|:r|tt + 1) =

VIBRATIONAL AND ROTATIONAL ENERGY LEVELS 271
of equations 848 and 849, this becomes

9Pt f2 A 2 4ft' 3

- -DM 2 - D r - =*- 14-29

~ ' ~

The coefficient of A in the energy expression is - ( -o ) For the

^v-

harmonic oscillator, ( -^ 1 is greater than -^ , so that a f e is negative. For
actual potential energy curves, the curve is flatter in the direction of

greater r than in the other direction; as a result, ( - ) is less than -

V / r 6

and e is positive for actual molecules. The part of the vibration-rota-
tion energy which is a function of A, or, as X is usually zero, of J( J + 1),
is usually written, in analogy with the above results, as

~ = B n J(J + 1) - D n [J(J + I)] 2 14-30

where

When the molecule is in the vibrationless state n = 0, its effective mo-
ment of inertia can be found from the equation

-Bo = Be

In general, a e will be positive; therefore BQ < B e and thus r$ > r e ]
that is, the average value of r, due to the zero-point vibrational energy,
is greater than the distance r e corresponding to the minimum in the
potential energy curve. For example, in HC1 the values of the constants
are B e = 10.5909, a e = 0.3019, giving r e = 1.2747 A, r = 1.2839 A. 5
The quantities a e , D e , & will usually be quite small in comparison to B e ]
their exact analytical form depends upon the exact form of the potential
energy curve.

The above results are based upon a harmonic oscillator potential
energy curve, which of course does not give a particularly good repre-
sentation of the potential energy curves for actual molecules. We
could include additional terms of 14-18 in the expression for the potential
energy; a more satisfactory procedure is to assume some appropriate

5 Reference 4, p. 488.

272 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

analytical expression for the potential energy curve. Morse 6 has written
the potential energy as

V(r) - D e {l - -Cf-*>}2 14-31

which has the correct qualitative form; the constant D e (not to be
confused with the constant used above) is equal to the depth of the
potential energy curve at r = r e . Using this value for F(r), the equation
for S(r) becomes

h 2 A
or, if r-g 2 k to k e treated as a perturbation

d 2 S 87T 2 M , ^ ( r ,i /,

-T5 + -TT ( - JM1 - <r o(r - r ' } } 2 )S = 14-33

If we now make the substitution x = e~" o(r ~ r ' ) , equation 14-33 reduces to
d^S 1 dS 8ir 2 /n /e - D e 2De ^ \ _

Making the further substitution

S(x) = e*~y*L(y) 14-35

where

- 47T^

ah

A ^

b = V2ju(D e e)
we obtain as the equation satisfied by L(t/) the result

^

d 2 L./b + l \dL , Vofc'^'- 2 ;

2 i" I ~~ I i

V2nD e -
+ ^2 1 LL = o 14-36

The associated Laguerre polynomials L^+i satisfy the equation
d a L# /2(Z + 1) WL^tf m-Z-1 21+1 _

-_ .. j_ i - . . _j L/m-l-l ~ vl l^t'Ol

1 *2 * \ ~ I 1 ' ~^Jrt"T"

dy * \ y / dy y

Equations 14-36 and 14-37 will be identical if we set b + 1 = 2(1 + 1)

{2 T _ fr i -n
^/ l 2fj,D e [ equal to some integer n. This
ah 2 J

6 P. Morse, Phys. Rev., 34, 57 (1929).

VIBRATIONAL SPECTRA OF POLYATOMIC MOLECULES 273
last step gives an equation which will determine the energy. We have

-n + ^ = ^ = -^-V2;uCD.-0 14-38

from which

where

According to the results of 14-39 and 14-30, we should expect the vibra-
tional-rotational energy levels to be expressible by a function of the form

+ B n J(J +1}- D n [J(J + I)] 2 1440

For example, the energy levels of HC1 (in the ground electronic state)
can be very accurately reproduced by the equation 7

f- = 2988.95 (n + J) - 51.65 (n + ) 2
he

+ [10.5909 - 0.3019(n + %)]J(J + 1) - 0.0004[J(J + I)] 2

The values of the constants D e and a in the Morse curve can be deter-
mined from the experimental values of co e and x e <u e ; however, the values
of the dissociation energy D = D e \hv e as determined in this manner
will usually be in disagreement with the values determined by more
direct methods.

14f . The Vibrational Spectra of Polyatomic Molecules. If we con-
sider a molecule as a system of point masses (the atomic nuclei), then,
according to section 2d, the kinetic and potential energies of the molecule
can be written in the form

, 1441

at

where N is the number of atoms in the molecule. Most of the simpler
molecules will have certain elements of symmetry; they will belong to
one of the symmetry groups of Appendix VII. Since subjecting a mole-
cule to a symipetry operation cannot change the potential or kinetic
energy of a molecule, the normal coordinates must transform in the

7 Reference 4, p. 121.

274 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

following manner. If Qk is non-degenerate, that is, if no other X is
equal to Xt, then the symmetry operation R acting on Q k must change it
either into itself or its negative, so that RQk = dblQ&. If Qk is degener-
ate, for example if Xj = X&, then the symmetry operation may change Qk
into a linear combination of Qk and Qi. - In general we would have

J-l

where the summation is over the / values of X for which \n = X^. If
S is another symmetry operation, then

fi fi fi

SRQik = E aikSQn = E E oifcMbfl,

Jl i-l m=l

and if T is the resultant of the successive application of the two opera-
tions R and S, T = SR, we hjave

fi

TQik = E CmkQim
tn-*l

The coefficients therefore obey the relation

fi

We thus see that the normal coordinates form bases for irreducible repre-
sentations of the symmetry group of the molecule in exactly the same
way as do the eigenf unctions of H. If we form a reducible representation
of the group based upon any arbitrary set of 3AT coordinates, and then
find the irreducible representations of which it is composed, there will be
as many distinct values of X (except for accidental degeneracy) as there
are irreducible representations in the reducible representation. The
degeneracy of a given X, that is, the number of normal coordinates which
have this value of X as their coefficient in the potential energy expression,
will be equal to the dimension of the corresponding irreducible represen-
tation. Further, these normal coordinates will transform in the manner
indicated by the matrices of the corresponding irreducible represen-
tations.

If we express the kinetic and potential energies in terms of an arbi-
trary coordinate system as

at at
then, according to 2-55, the values of X in equation 1441 are given by

VIBRATIONAL SPECTRA OF POLYATOMIC MOLECULES 275
the roots of the determinantal equation

If, however, we use a set of coordinates (which are not necessarily nor-
mal coordinates) that transform under the symmetry operations of the
group to which the molecule belongs in the manner indicated by the
matrices of the irreducible representations, then all cross products of
the type Q t -Q/, where Q$ and Q/ belong to different irreducible repre-
sentations, will vanish; this choice of coordinates will thus greatly
simplify the solution of equation 2-55. We shall return to this point
briefly a little later; first we wish to derive the selection rules for optical
transitions between the various possible vibrational states of polyatomic
molecules.

Using equations 1441 for the kinetic and potential energies of a mole-
cule, the vibrational wave function is found to be

3N 8 2 \l/ Sir 2 3N

E^ + ir^-**: ^+ = 1442

<=i dQi h* i~i

Making the substitutions

ZN ZN

*=nfc(<fc); E= ZEi

i=l 1

we obtain the set of 3N equations

+ (E t - faQfoto = 1443

each of which is the equation for a one-dimensional harmonic oscillator.
Some of the X/s may, of course, be equal; the vibrational wave function,
aside from a normalizing factor, may be written as

where H ni is the Hermitian polynomial of degree n in

According to the discussion in Chapter VIII, a transition from the
state #n lt n a . . . n 8 tf to the state $ n ' lt ^ . %y is possible only if

n' . . . n dr ^

276 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

(in this general discussion, x = x, y, or z). But, according to
section lOd, this integral will be zero unless the direct product
r(^ ni ...n 3 tf) r(^ ni '...n^) is identical with T(x). This is a general
statement of the selection rule for vibrational transitions in polyatomic
molecules; all the selection rules for the appearance of fundamentals,
overtones, or combinations can be derived from it. Here, however, we
shall be interested only in the selection rules for the fundamental fre-
quencies, that is, for transitions of the type

&>!, 2 , . 8 AT - ^O lf 2 , - l it - - - 3 *

The exponent in ^ ni . . . n ^ has the same form as the potential energy,
and hence is invariant with respect to every symmetry operation. The
function \l/ will thus belong to the same representation as the product of
the Hermite polynomials. We therefore have the results

I, o 2 , o 8 # ) = FI (the totally symmetrical representation)

so that the selection rule for the appearance of fundamentals in the infra-
red is: The frequency v* is infra-red active if F(Qt) = F(x), F(T/), or
F(z), where F(Q t -) is the irreducible representation to which the correspond-
ing normal coordinate Qi belongs.

According to section 8g, a frequency v a b is Raman active if one of the
matrix elements of the type (a|:n/|6) is different from zero. Proceeding
as above, we therefore have the selection rule for the appearance of fun-
damentals in the Raman effect: The frequency vi is Raman active if
F(Qi) = FCr 2 ), F(2/ 2 ),r(z 2 ), r(xy), T(xz), or T(yz\ where r(&) is the
irreducible representation to which the corresponding normal coordinate
Qi belongs.

FIG. 14-2. Symmetry properties of the H2O molecule.

We shall now proceed to the discussion of several examples.

EXAMPLE 1: H^O. HgO belongs to the symmetry group C^ the
symmetry elements are the identity J, C 2 (rotation about the z axis by
180, the xz plane is taken to be the H O H plane, the z axis goes
through the O atom), <r v (reflection in the xz plane), cr' v (reflection in the
yz plane) ; in Figure 14-2, the atom is above the plane of the paper,

VIBRATIONAL SPECTRA OF WATER

277

the hydrogen atoms are in the plane of the paper. The character table
for this group is reproduced in Table 14-5; the character of the represen-
tation generated by the possible motions of the molecule is included in
this table. The character of this representation is found as follows.

TABLE 14-5

E

tr v

*,y*,**

z

Ai

1

1

1

1

xy

R,

A 2

1

1

1

-1

xz

Ry

X

Bi

1

j

1

-1

yz

R x

y

B 2

1

j

-1

1

T m

9

-1

3

1

Each atom can move in any direction. We therefore imagine vectors
x, Ji, ii attached to each atom i, these vectors representing the dis-
placements of this atom from its equilibrium position, and see how these
vectors transform under the operations of the group. The operation E
leaves each vector unchanged; therefore \(E) = 9. The operation C 2
interchanges the hydrogen atoms, so that the contribution to x(C 2 )
from these atoms is zero. For the oxygen atom, z > z, y y,
x > x under this operation, so that x(C%) = 1. <*( interchanges the
hydrogen atoms, for the oxygen atom z z, y y, x x, so that
x(<r) = 1. For o-,, z >z, yy, x >x for each atom, so that
xfav) = 3. Breaking the reducible representation down into its irre-
ducible components, we have

+A 2

2B 2

Certain of the normal coordinates which belong to these irreducible
representations represent translational and rotational motion. Since
the vectors representing translational motion transform in the same way
as the coordinates x, y, and 2, and the vectors representing rotational
motion transform in the manner given in the table, we have for the
representation which has these motions as its basis

r, ff ^A l + A 2 + 2B 1 + 2B 2
Subtracting T tt r from F OT , we obtain

T vib = 2A l + B l

as the representation which has the vibrational motions as its basis.
There are therefore three distinct vibrational frequencies for H^O.
Two of the normal coordinates associated with these frequencies belong

278 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

to the totally symmetric* representation AI; one belongs to BI, and this
normal coordinate changes sign when the hydrogen atoms are inter-
changed. In order to find the actual normal coordinates we would have
to solve the secular equation with some assumed form of the potential
energy. We can, however, find by inspection certain coordinates which
have the necessary symmetry properties. If, for simplicity, we assume
that the mass of the oxygen atom is infinite, a possible set of such
coordinates would be

Qa = e'(n - r a )

where <p is the H H angle and TI and r 2 are the O H distances.
(Coordinates of this type must always be chosen in such a way that they
do not introduce any translational or rotational motion.) With these
coordinates, the secular determinant, which would in general be a three-
row determinant for H 2 0, is broken down into the product of a two-row
and a one-row determinant, since the products QiQz and Q^Qz may not
appear in either the kinetic or potential energies. Solution of the deter-
minant, with an assumed potential energy, would then lead to the
proper linear combinations of Q[ and Q' 2 which are the actual normal
coordinates. When this is carried out, it is found that one of the normal
coordinates belonging to AI represents (approximately) a stretching of
the H bond ; the other represents a bending of the O H bond. The
normal coordinates are thus approximately

Qi = top Q 2 = b(ri + r 2 ) Q 3 = c(r t - r 2 )

The modes of vibration associated with these normal coordinates are
illustrated in Figure 14-3. Actual calculations involving the solution

A of the secular determinant are

J^ Q^ a <f> rather difficult and are carried

out only if one wishes to find the
force constants in some assumed
1 2== * r ^ potential energy expression. By

inspection, diagrams of the type
B 3 Q 3 ^ c ( r i~ r 2) in Figure 14-3 can usually be

, , drawn. These will not in gen-

FIG. 14-3. Normal modes of vibration of H.2O. , i j-

eral represent normal coordi-
nates but merely coordinates with the proper symmetry.

From the selection rules for the infra-red transitions and Raman
scattering, and the transformation properties of the coordinates and

VIBRATIONAL SPECTRA OF ACETYLENE

27&

products of the coordinates, we see that in H 2 O all fundamentals are
allowed in both spectra.

In studies of the Raman effect, the degree of polarization of the Raman
lines is usually measured; this information often aids in assigning the
observed frequencies to particular modes of vibration of the molecule.
If the incident light is traveling in the y direction and the scat-
tered light is observed in the x direction, then, if the incident light
was polarized in the z direction, the degree of depolarization is defined

by the equation p = , where I(y) and I(z) are the observed inten-

l(z)

sities of light polarized in the y and z directions, respectively. The
theory of the polarization of Raman lines is given by Kohlrausch. 8
If the incident light is unpolarized, then the degree of depolarization p n

2p

of the scattered light is given by p n = . The degrees of depolari-

1 + p

zation are given by the following rules :

1. If the matrix elements of x 2 , y 2 , z 2 are all zero, then

P = f , Pn. = f

2, If not all the matrix elements of x 2 , y 2 , z 2 are zero, then

For H^O, we see that the frequencies belonging to A\ have p n < ^; the
frequency belonging to BI has p n = -f-.

EXAMPLE 2: Acetylene. In Table 14-6 we have reproduced the
character table for the group D Mhj to which acetylene belongs, and have

TABLE 14-6

E

C 2

x*+y\z*

A\

i

1

1 1

1

1

A\u

i

1

1 ~1

1

-1

AI O

l

1

-1 1

1

-1

Z

AZU

i

1

-1 -1

-1

1

(xz, yz)

( ,)

EI O

2

2cos^

2

2cos<?

(*,y)

EIU

2

2 CO8<f>

-2

2 cos <f>

(** - y\ xy)

E 2g

2

2 cos 2<?

2

2 cos 2<f>

Ezu

2

2 cos 2<p

-2

-2cos2^>

r m

12

4 + 8 cos <p

4

8 K Kohlrausch, Der, Smekal-Raman Effekt, p. 27, Springer, Berlin, 1938.

280 THE PRINCIPLES OF MOLECULAR SPECTROSCOPY

included in this table the character of the representation which has all
possible motions of the molecule as its basis. For the operation C^,
all z coordinates are unchanged, giving a contribution 4 to %(>) The
matrix of this transformation for the x and y coordinates has the charac-
ter 2 cos <p, so that these coordinates give a contribution 8 cos <p to
x(C^). The characters for the other operations are found in a straight-
forward manner. Breaking the reducible representation down into its
irreducible components, we have

T m = 2A lg + 2A 2u + 2E lg + 2E lu
Subtracting the representations for the external motions leaves

so that there are two non-degenerate modes of vibration belonging to the
totally symmetric representation AI Q , one non-degenerate mode which is
antisymmetric to inversion, and two doubly degenerate modes. In

AI# R.

Alg R t

AZU I.-R*

| S 5 ^ E* I.R.

FIG. 14-4. Normal modes of vibration of C2H2.

Figure 144 we present a schematic representation of possible coordi-
nates having the required symmetry, as well as the selection rules for the
corresponding frequencies.

EXAMPLE 3. Benzene. The results of a similar calculation for
benzene are given in Table 14-7. In determining the characters of the
reducible representation for the operations C 2 and C^ it is convenient to
let the axis of rotation be one of the coordinate axes x or y. Benzene is
seen to have twenty frequencies, of which ten are doubly degenerate.
Of these frequencies, those associated with the representations E\ u
and A 2u are infra-red active, so that the infra-red spectrum should con-
tain three degenerate frequencies and one non-degenerate frequency.
The frequencies associated with E\ g , E 2gi and A\ g are Raman active, so

VIBRATIONAL SPECTRA OF BENZENE

281

that the Raman spectrum should contain five degenerate and two non-
degenerate frequencies. The other frequencies do not appear (as funda-
mentals) in either case.

TABLE 14-7

Da (C 6 H 6 )
E C 2 2C 3 2C 6 3<?2 3Cg' iE iC

x* + y\ z*

A io

1 i

1

1

1

1 1

1

1

1

1 1

A\u

i 1

1

1

1

1 -1

-1

_1

-1

-1 -1

R,

A%

1 i

1

1

-1

-1 1

1

1

1

-1 -1

z

A'lu

i i

1

1

-1

-1 -1

-1

-1

-1

1 1

Big

l -l

1

_1

1

-1 1

-1

1

-1

1 -1

Biu

l -l

1

-1

1

-1 -1

1

-1

1

-1 1

&2g

i ~i

1

1

-1

1 1

-1

1

-1

-1 1

B 2u

l -l

1

-1

J

1 -1

1

__1

1

1 -1

(z 2 - t/ 2 , xy)

E%g

2 2

-1

-1

2

2

-1

_1

E%u

2 2

-1

-1

-2

-2

1

1

(xz, yz)

(R X , Ry)

Elg

2 -2

j

1

2

-2

-1

1

(*,y)

EIU

2 -2

-1

1

-2

2

1

j

r m

36

-4

12

4

-f

-f 2B 2u

-f

A 2o

-f

-f

-f

If the molecule has a center of symmetry, then x, y, and z belong to u
representations while their products belong to g representations. In
such molecules, therefore, a frequency cannot appear as a fundamental
in both the infra-red and the Raman spectrum.

CHAPTER XV
ELEMENTS OF QUANTUM STATISTICAL MECHANICS

Our study has thus far been limited to the discussion of the structure
and properties of individual atoms and molecules. The determination
of the properties of a system of atoms or molecules (" particles ") from
a knowledge of the properties of the individual particles requires the use
of the methods of statistical mechanics. We do not intend to treat this
subject in any great detail; but it is of interest at this point to see how
the connection between the properties of individual particles and the
properties of systems of these pa,rtieles is made.

Let us consider a system of n particles, all of the same kind. We
represent the coordinates of the n particles by the symbols qi, q^ #3 q n ,
where qi represents all the coordinates necessary to specify the state of
the ith particle. We assume that the Hamiltonian operator H of the
system can be expressed in the form

H = Hfa) + H( 92 ) + + H(g w ) 15-1

If we represent the wave function for the system by \l/ n , then the energy
levels E n of the system are given by the solutions of the equation

= Entn 15-2

Because of the assumed form of H, this equation may be separated into
the n equations

15-3

The function \f/ will be expressible in terms of the ^?'s; we must, however,
take a combination of the <p's which will give ^ the proper symmetry.
These symmetry requirements lead to three possible cases, which we now
discuss.

15a. The Maxwell-Boltzmann Statistics. We consider first the
classical case, where our system contains n distinguishable particles. In
this case there are no symmetry restrictions on ^; any combination of

282

THE MAXWELL-BOLTZMANN STATISTICS 283

the <p's such as

^n = <Pa(qi)<Pb(q2) ' *>m(n) 154

,will be an acceptable wave function for the system. Since the particles
are distinguishable, any interchange of particles among the occupied
states <p at <f>b ' * ' <f>m will lead to a new state for the system.

Let us now divide the <p's into the groups 1, 2 k so that the
eigenvalues for all the <p's in the fcth group lie between the limits &
and & + dejc, and let us suppose that there are g^ <p's in the fcth such
group. We now ask the question: How many ^'s correspond to a dis-
tribution of the n particles such that there are n\ in the energy region
corresponding to group 1, n 2 in the energy region corresponding to
group 2, etc. ? (If a particle is in the fcth energy region, it may be con-
sidered to have the energy e&.) In order to answer this question, let us
calculate the number of different ways in which we can distribute the n
particles among the various regions so that there are n\ particles in
region 1, etc. For simplicity, let us consider that we have only two such
regions. We readily obtain the following table:

n

HI

n z

N

I

1

1

1

I

2

2

I

1

1

2

2

1

3

3

1

1

2

3

2

1

3

3

1

4

4

1

1

3

4

2

2

6

3

1

4

4

1

(JV = number of ways of obtaining the given distribution.) In order
to see how this table was formed, consider the case where n = 4, n\ =2,
n 2 = 2. If the particles are numbered from 1 to 4, the six possible
distributions are:

REGION 1

REGION 2

1 2

34

1 3

2 4

1 4

2 3

2 3

1 4

24

1 3

34

1 2

284 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

Since the order in which we select the particles which we are going to put
into a given region is immaterial, the designations 1 2 and 2 1 are equiva-
lent and are counted only once. It is to be noted that, in each of the

n\
cases listed, N n , n lt n 2 = r~ ; (0! defined to be 1). If the above table

Wj i/2-2

Is extended to include more particles or more regions, we find that the
same expression holds. The number of ways of distributing the parti-
cles among the different regions will therefore be

ATn.n lf n....f ~ ~. - J - 1 - 15-5

12 ni!n 2 ! njfc!

We now wish to calculate the number of ^'s corresponding to the above
distribution. We denote this number by G Ut ni , n 2 . . . n k ____ Since
there are g k v's in the kth region, each of the n k particles can be in any
one of the g k different ^>'s, since there is, no restriction regarding the
number of particles that can occupy a given <p. The n k particles can
thus be put into the g k states in $* different ways, each corresponding
to a new ^. This gives us the relation

<?n,n lf n 2 ...n*... =0l'02 2 0J*"'tfn f n 1 . *,...!*... 15-6

or

r n , n lf n. n fc . . . = - ~ '

k n k \

This is the answer to our first question. We now ask the additional
question: If our system contains a fixed number of particles n, and a
fixed total energy E, what distribution is the most probable? In order
to answer this question, statistical mechanics makes the following
assumption: Each state of the system which is consistent with the
requirements that n = constant and E = constant has the same
a priori probability; that is, any two such states have equal chances of
occurring. It therefore follows that the probability of the occurrence of
a given distribution is directly proportional to the number of
states representing such a distribution; if we call this probability
P n , tit. n, ... it* ... then

* n, n lf fig tift == ^^n, n lt n a n$ lO f o

where C is a constant. If we consider P to be a function of the n k 'B,
then the most probable distribution will be that for which P is a maxi-
mum, or that for which dP = (where the variation is with respect to the
nfc's), subject, of course, to the restriction that tin = and 8E = 0.
Since log x is a maximum when x is a maximum, it will be equally valid

THE FERMI-DIRAC STATISTICS 285

and more convenient to determine the distribution for which d log P = 0.
According to 15-7 and 15-8,

log P n , ni . . . = log C + log n\ + En* log 9k - E log n*I 15-9

k k

For x sufficiently large, Stirling's formula states that log x\ =
x log x x. Using this relation, the condition d log P = gives

Uif

log dn k = 15-10

& 9k

Equation 1540 must be solved subject to the restrictions

8n = Z>* = 15-11

k

dE = L^nfc = 15-12

k

The desired solution can be obtained by the method of Lagrangian multi-
pliers; we multiply 15*11 by the parameter a, 15-12 by the parameter 0,
and add the three equations, obtaining

(log + a + 0e*) 5n k = 15-13

k \ 9k /

The variations drib may now be considered to be arbitrary; therefore
equation 15-13 can hold only if

nif

log + a + pe k = 15-14

9k

for all values of k. The most probable distribution will thus be that for
which each n& satisfies the equation

n k = g k e- a e-t" 15-15

Equation 1545 is correct only when each n k is large enough so that the
Stirling formula does not introduce any error. When this approxima-
tion cannot be used, the most probable distribution would have to be
computed directly from 15-9. The systems ordinarily considered in
chemistry contain a sufficiently large number of particles to make the
Stirling formula adequate.

15b. The Fermi-Dirac Statistics. If our system contains n indis-
tinguishable particles, and we require that the total wave function ^
be antisymmetric with regard to interchange of two particles, we are led
to the Fermi-Dii#c statistics. As we saw in Chapter IX, an antisym-
metric wave function can be represented by a determinant, or by the

286 ELEMENTS OF QUANTUM STATISTICAL MECHANICS
expression

} 15-16

This method of writing \f/ is equivalent to stating that no state <? may
contain more than one particle. Since the particles are indistinguish-
able, it does not matter which particle we choose to occupy a given state;
the number of ^'s corresponding to a given distribution will be equal to
the number of ways in which we can select occupied ^>'s for this distribu-
tion. As before, we divide the <p's into groups of essentially equal
energy. If there are gk ^>'s in the fcth group, then the number of ways
N 0kt nk in which we can select n k occupied v?'s is illustrated by the
following table:

gk n k N gkt nk

1

1

2

2

1

1

1

2

+ 1

= 3

2

1

+ 2

= 3

3

1

1

1

3

+ 1

S3 4

2

3

+ 3

= 6

3

1

+ 3

SB 4

4

1

1

1

4

+ 1

= 5

2

6

-1-4

- 10

3

4

+ 6

- 10

4

1

+ 4

= 5

5

1

Since a given <p can be occupied by not more than 1 particle, n k < g k .
The method of forming the above table can be illustrated by the case
g k = 5, n k = 2. If the first <p is empty we have 2 particles to be divided
among 4 states, which, according to an earlier section of the table, can
be done in 6 ways. If the first <p is occupied, we have 1 particle to be
distributed among 4 states, which can be done in 4 ways, so that, for
gk = 5, n k = 2; N gitt njb = 6 + 4 = 10. In a similar manner, each line
of the table can be obtained from some preceding lines, so that the entire
table can be built up from the trivial case g k = 2. It is to be noted that
in every case

AU, n k = - * r. 15-17

THE BOSE-EINSTEIN STATISTICS 287

a result which is perfectly general. The number of ways in which a given
distribution can occur will evidently be equal to the product of the num-
ber of different ways that we can select occupied states in the various
regions, that is,

G n n\, n n* = " ^fft. *k = " T7 - \~i 15-18

12 k k n k \(g k - 71*)!

The remainder of the analysis for the Fermi-Dirac statistics is identical
with that for the Maxwell-Boltzmann case. We find that the most
probable distribution is that for which

15 ' 19

15c. The Bose-Einstein Statistics. If our system contains n indis-
tinguishable particles, and we require that the total wave function \l/ be
symmetrical with respect to the interchange of two particles, we obtain
the Bose-Einstein statistics. A symmetric wave function may be repre-
sented by the linear combination

} 15-20

For this case of symmetric wave functions, there is no limitation on the
number of particles which we ca,n put in a given state <p. Aside from
this difference, the procedure is identical with that followed in the Fermi-
Dirac statistics. The analogous table is:

ffk
2

n*

"Ok, n

1

1

2

2

3

3

4

4

5

1

1

1+2 = 3

2

1+2 + 3

ft

3

1+2 + 3

+ 4 = 10

4

1+2 + 3

+ 4 + 5 - 15

1

1

1+3 = 4

2

1 +3 + 6

= 10

3

1+3+6

+ 10 =20

4

1+3+6

+ 10 + 15 = 36

288 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

For Qk 2, njc = 2, the first state can contain either 0, 1, or 2 particles,
the remainder being in the second state, giving N 2 , 2 = 3, etc. The
section of the table for gf& = 3 can be obtained from that for #& = 2 as
follows. For gjc = 3, n k = 4, the first state can have either 4, 3, 2, 1, or
particles in it. If the first state has 4, the remaining particles can be
distributed among the remaining 2 states in 1 way; if the first state has
3, the remaining particle can be distributed among the remaining 2
states in 2 ways ; if the first state has 2, the remaining 2 particles can be
distributed among the remaining 2 states in 3 ways; etc., so that

# 8 . 4 = 1 + 2 + 3 + 4 + 5 = 15 = L N*. i

i=0

In each case

so that

_(nfc + 0b- 1)1

The completion of the analysis shows that, for Bose-Einstein statistics
the most probable distribution is that for which

n k = e -^- 1 15-23

The expressions forn^ (15-15, 15-19, 15-23) contain two parameters,
a. and ft. The value of a may be determined by means of the require-
ment that

= n 15-24

The parameter ft may be evaluated by calculating some property of the
system, such as, for example, the pressure if our system is a perfect gas,
and comparing the calculated with the observed value. In this way it is

found that, for each of the three statistics, ft = , where k is the Boltz-

K JL

mann constant and T is the absolute temperature. For all actual
systems (except electrons in metals and gases at temperatures very close
to the absolute zero), e a eP* k ^> 1. The term 1 in the denominators of
15-19 and 15-23 may thus be neglected; the three statistics then give
the same result

g k e e

-
~ kT 15-25

STATISTICAL MECHANICS AND THERMODYNAMICS 289

_*.

Since ^n k = e~ a "g b e *r = n

_

we have = - 15-26

n

as the general expression defining the most probable distribution.
Equation 15-26 may with equal validity be regarded as the probability
that a given particle will be in the state with energy e&.

15d. The Relation of Statistical Mechanics to Thermodynamics.
The relation between the energy levels of a system and its thermo-
dynamic properties may be simply derived by the following argument.
Let us regard the " particle " in the above discussion not as a single atom
or molecule but as, let us say, a mole of any chemical substance; that is,
we now regard our " system " as a " particle. 7 ' An analysis identical
with that presented above then leads to the result that the probability

_L
e kT
that the system be in the state with energy Ei is equal to - ^7 . The

average energy of the system, the thermodynamic internal energy E, is
therefore given by the expression

~

_

E = -i - BJ- = UT log He w 15-27

I _*,v

{He w )
t '*

i

and the specific heat at constant volume will be

If we now calculate the entropy of this system, we have

By integrating by parts, we obtain

- | + k log {Ee~^} - k (log {l>~ **

* < \ i

15-30

T-O

290 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

The constant term may be identified with S ; we note that
/So = k log QQ, where <7o is the " statistical weight " of the ground state,
that is, the number of eigenfunctions corresponding to this state. This
is the statistical-mechanical formulation of the third law of thermo-
dynamics. If, as is usually true, the ground state of a system is non-

77' A
degenerate, then SQ = 0. Since S = - , we have the final result

A = -kTlogZ 15-31

where

_#
Z = e kT

n

and the summation is over all energy levels E n corresponding to the
allowed eigenfunctions \l/ n of the system. The effect of phase transi-
tions is the addition of the same term to TS and E\ equation 15-31 is
thus perfectly general. The quantity Z is known as the " sum-over-
states" or " partition function" for the complete system.

Where it is possible to express the energy E n as a sum of terms each of
which depends on one particle only, it is convenient to write Z as a func-
tion of the partition functions of the individual particles. If $ n is

15-32

then

E n = e (l) + 6 (2) + - + e m (n) 15-33

where a (l) signifies that particle (1) has the energy c . We now con-
sider a system where the available number of states <p is much greater
than the number of particles n a condition which is well satisfied for
most systems of chemical interest. It will then be extremely unlikely
that a given <p will be occupied by npiore than one particle, and the possi-
bility of the occurrence of such states can be neglected. From 15-33 we
note that a given value of E n can then be obtained in n\ different ways
which differ only in the numbering of the particles. For the Maxwell-
Boltzmann statistics, each of these n\ ways of obtaining E n corresponds
to a different function ^, since the particles are distinguishable. For
this case, therefore, we may write

_
e kT = D e kT 15-34

where the second summation is over all possible values of (!), ,(2),
etc. For the Fermi-Dirac or Bose-Einstein statistics, where the parti-

STATISTICAL MECHANICS AND THERMODYNAMICS 291

cles are indistinguishable, the n\ ways of obtaining E n according to 15-33
represent only one wave function \l/. If we were to form a sum of the
type 15-34 for these cases, we would have, corresponding to a given value
of E n , n\ terms in the summation on the right which differ only in the
numbering of the particles. Since our summation is to contain only
one term for each distinct ^, a summation of the type 15-34 is thus too
large by a factor nl. The correct expression for indistinguishable
particles is therefore

_En I (D+ (2) + +< (n)

6 kT = - f L e kT 15-33

n n U(l),t(2)---

If the particles are all alike, as we have assumed them to be in this
discussion, then

__.

e kT = (Ee kT ] =r 15-36

2),-" \ t /

where / is the partition function per particle. We have as our final
results:

For distinguishable particles Z = /* 15-37

For indistinguishable particles Z = f" 15-38

If our system is a perfect gas, the particles are obviously indistinguish-
able, and the correct partition function for the system is given by 15-38.
If our system is a perfect crystal, the particles are distinguishable
because of their fixed positions in space, and the correct partition func-
tion for the system is given by 15-37. Intermediate systems such as an
imperfect gas or, more important, a liquid, introduce considerable diffi-
culty. In the first place, it will not in general be very exact to write the
Hamiltonian in the form 15-1. If we make the approximation that the
interactions of the neighboring particles with a given particle can be
represented by some average potential field, this separation of the
Hamiltonian can be achieved; the question then arises whether we
should use 15-37 or 15-38. Since the particles are actually indistinguish-
able, and since they have a certain amount of mobility, it would appear

that the correct partition function would be Z = Tr/** where

1 < N < nl. It seems likely, however, that any exact treatment of
such systems must be based upon the use of a Hamiltonian which con-
tains terms involving the coordinates of more than one particle.

292 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

15e. Approximate Molecular Partition Functions. The partition
function for a single molecule, as defined above, is given by the relation

/=2>~^ 15-39

3

where the summation is over all the allowed energy levels of the mole-
cule. Equation 15 '39 may alternatively be written

/ = ff#~** 1540

where Qi is the degeneracy or statistical weight of the ith level. To a
good approximation, e may be expressed as the sum

i(n) 1541

where (), ^-(v), ^(r), ;(e), ti(n) are the energy levels associated with
translational, vibrational, rotational, electronic, and nuclear motions,
respectively. To this approximation, the partition function for the
molecule becomes

f^ftWJn 1542

where

ej(Q _i(t>)

O* * y , etc. 1543

We now consider these terms separately. As far as nuclear energy is
concerned, it may be taken equal to zero. The nuclear partition func-
tion then becomes f n = g n s, where g n8 is the nuclear spin statistical
weight. Since the allowed rotational levels are dependent upon the
nuclear spin wave functions, as we have seen in section 14c, it is conven-
ient to combine the nuclear spin statistical weight with the rotational
partition function, and write

fB = frfn = 0n.(0fc(r)~ 1544

i

The partition function for electronic energy cannot be further simpli-
fied; however, the first excited electronic state is usually so high above

_<i(e)

the ground electronic state that the term e M may be neglected in com-
parison with unity. Then the electronic partition function is go(e) 9
where 0o(e) is the degeneracy of the ground state.

According to section 14e, the vibrational energy of a polyatomic mole-
cule, to the harmonic oscillator approximation, is

1545

o, i, 2, 3

APPROXIMATE MOLECULAR PARTITION FUNCTIONS 293

where vi is the fundamental frequency of the ith vibrational degree of
freedom and there are k vibrational degrees of freedom. Now

nhv

1546

so that the vibrational partition function for a polyatomic molecule, to
the harmonic oscillator approximation, is

i-k

/.= nj

1 - e

1547

The translational energy levels of a molecule are given by the solution
of the wave equation for a particle in a box (section 5b). If the mass of
the molecule is m, and if it is constrained to move in a rectangular box of
edges a, 6, and c and volume V = abc, the translational energy levels are

1548
n x , n yy n z = 1, 2, 3, 4

The translational partition function is therefore

00 A 2 nj A 2 ^ A^nj

/* = 53 6~~ 8ma*kT 5Z ^ Sm^T* 53 ^ 8mc 2 A7' 1549

^ (, / V V X^rf " V ^_^ W

A 2

Now the quantity ^ 2 is very much less than unity for ordinary
~ /c jt

h 2 n 2

temperatures and reasonable values of a, so that - 91 changes only

Sma 2 kT

slightly as we vary n x . For this reason it is permissible to replace the
summations by integrations, and write

/ A 2 4 / Anj /* A^nj

e 8ma*kT dn x I e~*wrdn v I e*&*kTdn z 1550
J J

Q

according to Appendix VIII,

r^
e

^, 1KK ^

e sma^kT dn x = - - - a 15-51

294 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

so that the translational partition function becomes

, (2irmkT)^ (2irmkT)** __

f t -. _- a fo a- y 15-52

Classically, the energy of a system is given by the equation H(p, q) = E.
The classical analogue of equation 15-40 is therefore

1 / _gfag)

f ~ Tn I e kT ^l * * dpn ^5l ' ' ' ^2n 15-53

where the factor has been introduced so that / may be non-dimen-
sional. The introduction of this specific factor may be further justified
as follows. For a particle of mass m moving under the influence of no
forces, the Hamiltonian function is

JnL == ' \Px ~T~ IPy i" 'Pz) Io*v4

so that equation 15-53 gives for the translational partition function the
result

i r r r cp+py+p) r r r

ft = 73 I I I e 2mA;r dp x dp y dp 2 I I I dxdydz 15-55
If the particle is constrained to move in a volume 7, then

f f f dxdydz = V
Integrating over the momenta, we obtain

/, _ % r >" y !5.56

which is identical with 15-52.

We next consider the rotational partition functions for diatomic
molecules. According to section 14c, there is no restriction on
the allowed values of the rotational quantum number J if the nuclei are
different. If the two nuclei have spins s\ and s 2 the nuclear spin
statistical weight is (2si + 1)(22 + !) The rotational energy levels
are

where I is the moment of inertia, and each level is (2J + l)-fold degen-

APPROXIMATE MOLECULAR PARTITION FUNCTIONS 295
erate. The rotational partition function f R is thus

co A'JCJ+l)

1)6 ** T 15-58

/-o

For large values of / and T, we may replace the summation by an inte-
gration, and obtain

/oo AW4-D

fa = (2*1 + I)(2s 2 + 1) / e w (2J + 1) dJ

VQ

Q-.2TJL/T7

= (2 5l + I)(2s 2 + 1) ^ 15-59

When the nuclei are identical, there is a restriction on the allowed J
values. For hydrogen in the ground electronic state, even J values are
allowed for parahydrogen, with nuclear spin weight unity; odd J
values are allowed for orthohydrogen, with nuclear spin weight three.
If we consider hydrogen as being a single species, which is legitimate at
high temperatures, we have the rotational partition function

h*J(J+l) ft'W-fl)

/ = 3 (2J + l)e *** IkT +1 Z (2J + l)e ***& 15-60

./even

If we replace the summations by integrations, then, since the summations
contain only one-half the possible terms, we obtain, according to 15-59,
the result

fa = (3 + 1)

2A 2

In any homonuclear diatomic molecule, with nuclear spin s, the statisti-
cal weight of the ortho states is (s + l)(2s + 1); that of the para
states is s(2s + 1). The rotational partition function will thus in
general be

Rir^Tlf'T Rir^ fffffl

15-61

For all diatomic molecules, we may therefore write the rotational parti-
tion function as

f R = (2$i + I)(2s 2 + 1) rj 15-62

where cr, the " symmetry number," is 1 for heteronuclear and 2 for
homonuclear diatomic molecules; that is, <r is the number of indis-
tinguishable ways of orienting the molecule in space.

296 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

For polyatomic molecules, the analysis is similar to that given above.
For high temperature, the result is

f ^

fs = - - - ^ 3 n ( 2 ^ + 1) 15-63

where A, B, C are the principal moments of inertia and where the sym-
metry number a is again equal to the number of indistinguishable ways
of orienting the molecule in space; for example, a = 2 for H 2 0; a = 3
for NH 3 ; a = 12 for CH 4 and C 6 H 6 . The origin of the symmetry
number is the same as in the diatomic case; if there are n equivalent

orientations of the molecule in space, then only - of the possible energy

n

levels are allowed by the symmetry restrictions on the wave functions.

15f . An Alternative Formulation of the Distribution Law. 1 Consider
some definite atomic or molecular system A which is in thermal equilib-
rium with a system B composed of s harmonic oscillators. Suppose that
the combined system A + B possesses a total energy E. What will
then be the probability that the system A has an amount of energy e
distributed in some one exactly specified way among the various degrees
of freedom of A ?

In order to perform this calculation we make the usual assumption of
statistical mechanics : Any exactly specified way of distributing the energy E
in the system A + Bis as probable as any other exactly specified way of dis-
tributing the same total energy. The probability of a partially specified
distribution is consequently proportional to the number of exactly
specified distributions which are compatible with it. Thus the probabil-
ity of A having the energy in some exactly specified way is proportional
to the number of exactly specified ways of distributing the remaining
energy (E e) among the s oscillators of the system B. But this is

/ET _ \

just the number of ways of distributing n = - - quanta of energy

among s oscillators, where hv is the energy per quanta. Since the quanta
are indistinguishable, and since there is no restriction on the number of
quanta in a given oscillator, the problem is similar to that already -met
in the discussion of the Bose-Einstein statistics. The number of ways
N n ,* of distributing n quanta among the s oscillators is, analogous to
Equation 15-21,

N n .. - (n + '~ 1 , )1 15-64

n\(s 1)!

This result can be derived quite simply in the following way. We count
I 8ee E. U. Condon, Phys. Rev., 54, 937 (1938).

DISTRIBUTION LAW 297

the number of different ways in which n quanta and s oscillators can be
arranged in a line so that there is always at least one oscillator on the
extreme right; all quanta which are placed between two oscillators are
considered as belonging to the oscillator to the right. There are
(n + s 1)! such arrangements. Eliminating arrangements which
have been counted more than once then leads directly to 15-64.

Equation 15-64 is an exact answer to our problem but can be simplified
when n and s are very large numbers. Using Sterling's approximation

xl = (torxpafe-* 15-65

vvhich is valid for large values of #, equation 15-64 becomes:

If n s we have

, n(n- D(n-2) s - 1

+ 3i

, e -i

a result which becomes strictly correct as n oo . Also,

.-

i __ _ p

+ (s - 1)A*I

If the average energy per oscillator is 7, we can write
E + (s - l)hv = sy +g

where g will be much smaller than sy provided that n is large compared
with s and is much smaller than E. In this case

< + . _ ir ,

'E + (s - IJAA*" 1
hv )

_ . _

- 21

VJ 1

+ g/

298 ELEMENTS OF QUANTUM STATISTICAL MECHANICS

hv
Therefore

where C is a constant independent of e. But, according to our original
postulate, N n , 8 is just proportional to the probability P(e) that the
system A have the energy e in an exactly specified way. Therefore,
we have

' p( ) = Ce" 15-68

The constant 7 in 15-68 (which is the reciprocal of the ft in our earlier
discussion) can be evaluated by calculating some property of the system
A (such as the pressure if we let A be a perfect gas) and comparing with
the experimental results. As mentioned before, such an analysis gives
the value y = kT, where k is Boltzmann's constant and T is the absolute
temperature.

CHAPTER XVI

THE QUANTUM-MECHANICAL THEORY OF
REACTION RATES

16a. Formulation of the General Theory. In previous chapters we
were largely interested in the application of the principles of quantum
mechanics and statistical mechanics to the study of those properties of
chemical systems which are independent of time, that is, to the study of
structural chemistry. These same principles can be applied successfully
to the problem of the calculation of rates of chemical reactions. As the
first step toward the solution of this problem it will be profitable to con-
sider the method of representing reactions and changes in the state of a
system generally by means of a geometrical picture.

The state of any system is described in quantum mechanics by its
eigenf unction ^(g, )> where t is the time and q represents all the coordi-
nates which would be necessary, in classical mechanics, to specify the
positions of all the particles in the system completely. With some
reservations, these positional coordinates will be termed " degrees of
freedom/' although this notion is not so precise in quantum mechanics as
in classical mechanics. (Since the observable properties of a mechanical
system are completely determined by the energy of the system, it might
be said to have only one degree of freedom, even though it consisted of
many interacting particles. In the following discussion we shall use the
term only in the sense of classical degrees of freedom.) Given the eigen-
function ^, we can calculate all the physical properties of the system; if
R is the operator which corresponds to the property in question, then

/ ^*R^ dr
R = 16-1

I ty*\L elf

is the expectation value of the property, that is, the average value of a
large number of measurements of the property. If SF does not represent
a stationary state, that is, if ^*^ is a function of the time, these expecta-
tion values will be functions of the time.

The reaction between a hydrogen molecule and a hydrogen atom, such
as occurs in ortho-para hydrogen conversion, constitutes a transition in a

209

300 THEORY OF REACTION RATES

three-atom system. The properties of this system are completely deter-
mined by the Schrodinger equation governing it, that is, by its eigen-
functions. The system itself may be represented by a point in four-
dimensional space: three dimensions are required to specify the relative
positions of the nuclei, one additional dimension is required to specify the
energy. (We assume here that the motion of the electrons is so rapid
that the electrons form a static field for the slower nuclear motions.)
The three internuclear distances ri 2 , r 23 , 7*13 can be conveniently used to
specify the configuration of the system. At some time fo this system is
found in a configuration in which, say, r\% is much smaller than either
7*23 or 7*13; that is, atoms 1 and 2 form a molecule. The reaction in
question is represented by a transition from this configuration to one in
which, say, r 23 is much smaller than either ri 3 or ri 2 . The probability
of such a transition will be denoted by K.

As the internuclear distances change in such a transition, the total
binding energy of the system, that is, the electronic energy of the ground
state of the system, will also change. The problem of determining the
binding energy for any given values of the positional coordinates is
identical with the problem of the calculation of the binding energy of
stable molecules and is subject to the same limitations. The surface
which represents the binding energy of the system as a function of the
positional coordinates is known as the potential energy surface for the
system; its determination is usually the first step in the theoretical di&-
cussion of the rate of a chemical reaction. The general features of
potential energy surfaces may be illustrated by the surface for the reac-
tion H + H 2 . For simplicity we assunje the three hydrogen atoms to be
collinear; the system is then completely specified by the two coordinates
r J2 and r 23 . If the calculation of the binding energy is made by use of
the London formula (equation 1328), assuming the coulombic energy
to be 14 per cent we obtain the surface illustrated in Fig. 164, 1 where
the lines of constant energy have been plotted as contour lines. The
coordinate axes are inclined at an angle of 60 rather than 90, since for
this particular system this choice of axes diagonalizes the kinetic energy;
that is, the kinetic energy is expressible as 2

where m is the mass of a hydrogen atom. The equations of motion of
the system, from the classical viewpoint, therefore represent the friction,

1 H. Eyring, H. Gershinowitz, and C. E. Sun, J. Chem. Phys., 3, 786 (1935).

2 S. Gladstone, K. Laidler, and H. Eyring, The Theory of Rate Processes, p. 100,
McGraw-Hill Book Company, 1941.

FORMULATION OF THE GENERAL THEORY

301

less sliding of a mass point of mass n on the surface of potential energy.
(Strictly speaking, the analogy is not exact, since the motion of a mass
point on a gravitational potential surface takes place in three dimensions;
the discrepancy is not serious for our purposes.) We note that the
potential energy surface is made up of two long narrow valleys, represent-
ing the stable H 2 molecules, connected by a region of higher energy and
separated by regions of still higher energy. The potential energy sur-
faces for more complicated reactions are similar. The regions corre-

62 JZero point Clergy) ff^H / y

20

FIG. 16-1. Potential energy surface for three hydrogen atoms (after Eyring,
Gershinowitz, and Sun).

sponding to the configuration of the reactants (the initial state) will
usually be separated from the regions corresponding to the configuration
of the products (the final state) by a region of higher energy, in which
event the reaction has an activation energy. Classical mechanics pre-
dicts that, if the height of the barrier is FO> no systems with energy
E < VQ can react, and all systems with energy E > VQ which approach
the top of the barrier proceed to cross it and lead to reaction. If, as in
the H + H 2 example, there is a shallow basin with a second pass just
beyond the first barrier, a system may move into this basin and be
reflected into the valley from which it came. Figure 16-2, 3 illustrating
the classical motion of a system entering this basin, is interesting in that

8 J. Hirschfelder, H. Eyring, and B. Topley, J. Chem. Phys., 4, 170 (1936).

302

THEORY OF REACTION RATES

it shows that the motion of a particle in the basin tends to become com-
pletely random, and thus has an equal chance of leaving by either pass.
In this case only about one-half of the systems which enter the basin
will leave it through the pass into the valley corresponding to the.con-

.8 .9 1.0 1.1

* Distance between the atoms 6 and c in A

1.2

1.3

FIG. 16-2. Path of representative point on H H H potential energy surface
(after Hirschf elder, Eyring, and Topley).

figuration of the products. These predictions of classical mechanics are
modified when the problem is considered from the quantum-mechanical
viewpoint. In the first place, there is a finite probability that a system
in the initial state with energy E < VQ may nevertheless appear in the
final state at a later time. This phenomenon of " leakage " through the
potential energy barrier is important in the decay of radioactive nuclei

I

1

m

FIG. 16-3. One-dimensional potential energy barrier.

and in reactions involving only the transfer of electrons. In the secojid
place, even if E > y , there is a finite probability that the system will be
reflected in the region of the barrier and will hence not proceed to the
final state.

The method of calculating these probabilities may be illustrated for
motion in one dimension in the potential field of Fig. 10-3. The poten-

FORMULATION OF THE GENERAL THEORY 303

tial energy is

I

V(X) = - 00 < X < - -

2
F(x) = FO - l -<x<\ 16-3

2i

V(x)=0 \< x <+

The Schrodinger equation for the system is

n*)]* = 16-4

In region I, where V(x) = 0, ^ has the form of a plane wave (section 5-1),
the two independent solutions corresponding to motion toward the
barrier and away from the barrier. The general solution in this region
is then

fo = Ae<"* + Be"* 16-5

27T / -

where a = v2mE, and E is the total energy of the system. In
h

region II, where F(x) = F > the solution is of the same form if a is

2 V . -
replaced by ft = V2m(E - 7 ). When E < 7 , /J is imaginary,

fi-

and ^n corresponds to an exponentially decreasing probability of finding
the system in the region inside the barrier. In region III, ^m has the
same form as ^i, and so we have

+ De-* 9 16-6

Fe iax + Ge- 4 ** 167

The eigenfunction for the entire region must be continuous and have a
continuous first derivative; therefore the following conditions must be
fulfilled:

I

when x = - - 16-8

2

foi foil, -7 = when x = + x 16-9

ox ax 4

These four conditions enable us to determine the constants J5, C, D, and

304

THEORY OF REACTION RATES

F in terms of the constants A and G. The set of simultaneous equations
is readily found to be

*!

= Oi

whence

B =

08*5

.*

O i

ae

-fie-**

ae*

1640

16-11

with the analogous expressions for C, D, and F.

Let us suppose that there are no systems returning from the final
state; that is, there are no systems in region III with negative momen-
tum. In this case G = 0. Let the probability of finding a system in
region I with positive momentum be unity; that is, let A = 1. Then
\F 2 gives the chance that it be transmitted into region III and \B\ 2
gives the chance that it be reflected back into region I. For E > VQ,
it is seen that the transmission coefficient \F\ 2 is small but finite. More-
over, it is a function of the variable 6 = fil] that is, it varies with the
original momentum of the particle, the height of the barrier, and the
width of the barrier. When E > FQ, the transmission coefficient oscil-
lates with these variables. Thus, for a fixed barrier height and width,
particles of certain energies will be completely transmitted, while those

FORMULATION OF THE GENERAL THEORY 305

of slightly different energies will be less likely to cross the barrier, even
though they have enough energy.

The energy surface for an actual chemical reaction is always at least
two-dimensional and is usually many-dimensional, since it must include
a dimension for every internuclear distance for all the nuclei involved in
the reaction. In every case, however, there will be some initial configu-
ration in which the eigenfunction for the system can be well approxi-
mated in one dimension by a plane wave traveling toward the region of
configuration space which connects with the region of the products.
Also, there will be a region where the eigenfunction for the configuration
of the system representing products can be well approximated in one
dimension by a traveling plane wave. Thus it will always be possible to
expand the exact eigenfunction for the system so that it will represent,
asymptotically at least, a plane wave which we shall call the transmitted
wave, traveling away from the activated state down the valley repre-
senting products. Then this representation will, in general, reduce
asymptotically in the reactants valley to the superposition of plane waves
traveling to and from the activated state, that is, an incident and a
reflected wave. The ratio of the amplitude of the transmitted wave to
that of the incident wave defines the transmission coefficient. Similarly,
the ratio of the amplitude of the reflected wave to that of the incident
wave defines the reflection coefficient.

It often happens that motion in degrees of freedom orthogonal to
that in which the reaction takes place is quantized, and the system
exists in discrete energy levels corresponding to vibrations of the react-
ants or products. These vibrational quantum numbers may or may not
change during the course of a reaction. A change in the energy of some
quantized degree of freedom requires a corresponding change in the
energy of some other degree of freedom. Thus, reflection or transmis-
sion at a barrier in the energy surface may induce a transfer of energy
between the various degrees of freedom of the system. Let \// n represent
the eigenfunction of the incident wave, where n denotes the set of quan-
tum numbers specifying its vibrational state. Let p n be the trans-
lational momentum of a wave in the vibrational state n. The reflected
wave may have any vibrational state consistent with the available total
energy. Let the eigenfunction of that fraction of incident systems
initially in the vibrational state n which are reflected in state m be
Rnmtm' Their momentum will be p m . Similarly, let T n m$m be the
eigenfunction of the wave transmitted with change in vibrational state

from n to m. Then \\f/ n \ 2 is the density of incident systems and -~
their velocity. (M is their mass.) Since all systems are either reflected

306 THEORY OF REACTION RATES

or transmitted, and since there is no piling up in the region of the barrier,
we have the relation

/ \2\ i \2 P& 1/310

- 16 ' 12

where the sums are over all values of m and k which are consistent with
the total energy of the system. When the ^'s are normalized to unity,

the ratio |JS!nm| 2 ~ = Pnm is defined as the reflection coefficient, and the

Pn

ratio I Tnk\ 2 = K n k is defined as the transmission coefficient for the tran-

Pn

sition from level n to level fc.

Let us now formulate the rate of a chemical reaction in terms of these
coefficients. We consider a region in the reactant valley which is far
removed from the potential energy barrier. Let C n (p) be the number
of systems in length of path dx in this region which have momentum
between p and p + dp along the reaction coordinate and energy E n
in the other degrees of freedom. The number of such systems which
pass a given point in the reaction path per second moving toward the

(^ ^T)^ T) T)

barrier will be -~ -- , since is the velocity of these systems, where
dx m m

p is considered to have only positive values. The fraction of these
systems which react will be

n ^ P 1A1Q

*T m 16 ' 13

where K n fc(p) is the transmission coefficient as defined above. Assuming
an equilibrium distribution in this region, the concentration C n (p) will
be

C n (p) p p, "* kT e ** T ?-^ 16-14

where (Ai), (A 2), are the concentrations of the reactants, FI, F^,
are the partition functions for the reactants, and w n is the statistical
weight of the level with energy E n . The net rate is obtained by summing
1643 over n and integrating over the momentum p. The general expres-
sion for the rate of a chemical reaction is therefore

~ 16-15

(A 2 ) - - - r

This expression cannot be further reduced unless certain simplifying

FORMULATION OF THE GENERAL THEORY 307

assumptions are made. We obtain an approximation to the classical

P 2

case if we assume /c n &(p) = for - 1 - < VQ, /c n &(p) equal to some average

2m

P 2

value K for > VQ. Equation 1645 then becomes
2m

_tfn

The quantity Z)co n e * r has the general form of a partition function and

n

may be denoted by F*. We therefore have for the specific reaction rate,
to this approximation, the result

,_.- , 6 .17

FiF 2 h

This is the familiar equation originally derived by Eyring, 4 using a
slightly different method of approach, and is a generalization of the
results obtained by Pelzer and Wigner 5 in their study of the H + H 2
reaction. In the derivation by Eyring's method, it was shown that F*
is the partition function for the activated complex (not including the
degree of freedom along the reaction path). The investigation of the
specifically quantum-mechanical effects in chemical reactions, as dis-
cussed later in this chapter, have confirmed the validity of equation 1647
for most cases of interest. Most of the applications of the theory of
absolute reaction rates have been based directly on equation 1647;
the results are discussed in detail in the recent textbook by Glasstone,
Laidler, and Eyring. 2

It will be noted that in equation 1645 all explicit reference to the
nature of the potential energy surface in the region of the barrier has
disappeared; the effect of the barrier is contained implicitly in the trans-
mission coefficients K nk (p) . In this formulation of the theory there is no
assumption of equilibrium between the initial and activated states. It
is necessary to assume equilibrium between the various vibrational
states of the initial configuration, which appears always justifiable, since
the initial configuration can be chosen in such a way that the interactions
between the reactant molecules are as small as desired. When reaction

4 H. Eyring, /. Chem. Phys., 3, 107 (1935).

6 H. Pelzer and E. Wigner, Z. physik. Chem., B15, 445 (1932).

308 THEORY OF REACTION RATES

induces transfer of energy from translational to vibrational degrees of
freedom in the system, there will be, as we have seen, a different K n k(p)
and hence a different rate for each vibrational level in the final state. In
general, the distribution of systems over the vibrational levels in the final
state will not be the equilibrium distribution, but is given automatically
by adding the rates of reaction for each level. Thus there will be a rate
k' m for reaction in which the products are in the vibrational level m
given by

1 /*

-Fy J

F i? 2 ^0

16-18

Integrating this expression with respect to time will give the concen-
tration of products in the rath vibrational state (disregarding changes
after reaction) . This will be the equilibrium value only for special forms
of K.

The above formulation is completely general and applies immedi-
ately to bimolecular and higher-order reactions. Unimolecular reac-
tions require further consideration. They are usually considered to
involve two steps, an activation by collision followed by a decomposition
of the activated molecule. The first step, activation by collision, can be
treated in principle by the method given above. Here we are interested
in the calculation of the probability that systems approaching the
barrier will be reflected with a change in vibrational energy. Analo-
gously to equation 16-18, we have for the rate of formation of molecules
in vibrational state I the result

I / En P 2 ft Jft

ki = - / Lpn^pW^e a-w52 I .i9

F it* 2 ' t/o n Win,

where p n i(p) is the reflection coefficient. The rate of deactivation by
collision is given by an analogous equation. Activated molecules can
disappear either by decomposition into the products or by deactivation
by collision. The usual theory of unimolecular reactions, as developed
by Lindemann, Hinshelwood, Kassel, Rice, Ramsperger, and others, 6
assumes that the concentration of activated molecules is constant.
This steady-state assumption leads to the equation

where (AT) is the concentration of normal reacting molecules, (N*) is
the concentration of activated molecules, (A) is the total concentration

6 L. S. Kassel, Kinematics of Homogeneous Gas Reaction, p. 93, The Chemical
Catalog Co., New York, 1932. C. N. Hinshelwood, The Kinetics of Chemical Change,
Oxford Clarendon Press, 1940.

FORMULATION OF THE GENERAL THEORY 309

of molecules which can cause activation or deactivation by collisions.
ki and fc 2 are the specific rates for activation and deactivation by colli-
sion, respectively, and fc 3 is the specific rate for reaction of the activated
molecule. The actual rate of production of the products will be

or, in the high-pressure limit

Kf - *

A?2

The rate constants fci and k 2 are calculable by the equation given above;
however, in the high-pressure region, where only a negligible fraction of
activated molecules disappear by reaction, we may write

F N

where FN* is the partition function for activated molecules and FN is
the partition function for normal molecules. According to equa-
tion 16-17, fca may be written as

kT

fc 3 = * -f
ti

where FN* is the partition function for the activated complex of the
reaction N* > products. (In FN* and FN* the zero of energy is taken
to be the ground state of the normal N molecules.) The high-pressure
rate is therefore

,/

k =

that is, all explicit reference to the activated molecules N* has disap-
peared, and we are back to our familiar rate equation.

There are two alternative ways of formulating the reaction rate prob-
lem which are more useful for some purposes. In practice it is not
possible to obtain the exact eigenfunctions for an n-body system when
n is greater than 2 because of the fact that the equations of motion are
not separable. They can be computed approximately, however, by first
calculating the eigenfunctions for an approximate Schrodinger equation
which is separable and then treating the terms which were neglected in
order to make the separation as a perturbation on these approximate
solutions. It is then found that the density of systems in any particular
unperturbed level is a periodic function of the time. The perturbation

310 THEORY OF REACTION RATES

induces transitions from one approximate level to another. For uni-
molecular decompositions the approximate levels are those of the mole-
cule which is decomposing, and the lifetime of an energy-rich molecule
is determined by the probability with which transitions to continuous
levels (corresponding to dissociation) are induced. Rosen 7 has used
this method to calculate the mean life of a linear triatomic molecule.
For more complicated molecules, even the separable problem is difficult
to solve. Kimball 8 has treated the question classically by calculating
the time elapsed before the amplitude of harmonic oscillations exceeds
a critical value such that the energy of the oscillation is equal to the
dissociation energy of the molecule.

A third method which is often useful in treating problems where the
rate is determined primarily by leakage through an energy barrier may
be called the method of beats. If two single minimum potentials are
connected by an energy barrier, it is found that the energy levels which
lie below the top of the barrier split into two levels of slightly different
energy as the width of the barrier is decreased. The amount of split-
ting is small compared to the energy difference between pairs of levels.
The corresponding eigenfunctions have different symmetry with respect
to inversion in the origin. The one corresponding to the lowest level
of any pair is antisymmetric, that is, it changes sign when the coordinate
changes sign; the other eigenf unction is symmetric with regard uo
inversion. We then see that the function |^ a + \l/ 8 \ 2 is located almost
entirely on the left side of the barrier; the function \\f/ a ^] 2 is located
almost entirely on the right side of the barrier. The subscripts s and a
refer to the symmetric and antisymmetric eigenfunctions, respectively.
We therefore represent a system moving from one side of the barrier to
the other by the eigenfunction

Et
h * 16-20

The density is given by

When t = 0,

*** = |*. + *.| 2

and the system is on the left side of the barrier: when t = - >

2(E t - EJ

7 N. Rosen, J. Chem. Phys., 1, 319 (1933).
8 G. E. Kimball, /. Chem. Phys., 6, 310 (1937).

BEHAVIOR OF TRANSMISSION COEFFICIENT 311

and the system is on the right side of the barrier. The frequency

1 2(E, - E.)
__.. _

is therefore the rate of penetration of the barrier. As the barrier width
increases, (E 8 E a ) decreases and v decreases. This method is appli-
cable only for small splitting of the energy levels and thus cannot
be used to determine the rate of passing over the barrier. Of the three
methods presented, the method of transmission coefficients, leading to.
equation 16-15 for the rate, is the most satisfactory from the theoretical
viewpoint. The applications of the approximate equation 16-17 have
been adequately discussed elsewhere; we wish at this point to investigate
the behavior of the transmission coefficients K n jc(p) more closely.

16b. General Behavior of the Transmission Coefficient. The trans-
mission coefficients for a number of special types of energy barriers can
be calculated exactly; this calculation has already been carried through
for a sharp rectangular barrier. A less drastically idealized case has been
considered by Eckart. 9 The Eckart potential is

v &> = ~ i4fj - (i ? e) 2 = ~ e ~^ 16 ' 21

and the corresponding Schrodinger equation has exact solutions in terms
of hypergeometric series. This potential is a step function of the coordi-
nate x\ its exact shape is fixed by specifying the constants A, B, and I.
We have

y(_ oo ) = 0, F(+oo) = A 16-22

Foi; < \B\ < \A\, the potential has a maximum whose height is

(A + B) 2

45

16-23

The rise in potential is accomplished practically in the distance 21.
The Schrodinger equation with this potential is

Changing variable from x to gives

'3+?+Wh +<*+}-

9 C. Eckart, Phys. Rev., 35, 1303 (1930).

312 THEORY OF REACTION RATES

This equation is of the hypergeometric type and has solutions in the form

Since for large positive or large negative values of x, the potential 16-23
reduces to a constant, the solutions 16-26 should be asymptotic to plane
waves with wavelength

and

X = f or x >

X' =5 -====== for x > + oo

V2m(E - A)

At x = +00 there will be a single transmitted wave:

e~ = (-&* 16-27

At x = oo there will be an incident and a reflected wave:

The requirement that the exact solution reduce asymptotically to the
forms 16-27 and 16-28 determines it uniquely and also determines the

I 12

coefficients a\ and a 2 . The reflection coefficient is then p =

M
A solution of 16-25 which converges for large values of (x > 1) and

reduces asymptotically to ( ) x/ as | | becomes very large is given by

16-29
where

-i + f( - /S - 8), 1-2^,

That this reduces to ( f)*^ for large is evident since F(a, 6, c, 0) = 1.
Its value for small cannot be determined immediately, however, since
F(a, 6, c, 1) does not converge. But it is possible to express 16-29 as the

BEHAVIOR OF TRANSMISSION COEFFICIENT

313

sum of two series which do converge for small values of . We thus find
the analytic extension of ^ in the region where its representation by
16-29 does not converge.
Using the known formula, we get

Ol

, -~

(- - 18

16-31

where

6) }

For small values of , equation 16-31 reduces to equation 16-28 with the
values 16-32 for ai and a%. The reflection coefficient is then given by

When the constant C is small, 5 is real and applies when the wavelength
of the incident particle is short compared to the region where potential is
changing. If the wavelength is long compared to the length Z, C will be
large and 5 will be imaginary. The expression for p is expressed differ-
ently in the two cases. For d real, we have

For 5 imaginary,

cosh 27r(a ft) + cosh 2w8
cosh 27r(o: + j8) + cosh 2ird

cosh 27r(a ff) + cos 27r|g
cosh 2ir(a + /3) + cos 2ir|$

1634

16-35

The transmission coefficient is given by K = 1 p.

This value of p or K may be used for any potential barrier which is
sufficiently well represented by equation 16-23. The general method, of
course, is applicable whenever the exact solutions of the Schrodinger
equation corresponding to the given potential are known. However,

314 THEORY OF REACTION RATES

many barriers are flatter than that represented by equation 16*23. The
question then arises: Under what conditions is it permissible to treat the
rising side and the falling side of the potential as independent changes,
each having a reflection coefficient given by 16-34 or 16-35? This
question has been investigated by Hirschf elder and Wigner. 10 Any
irregularity in the potential surface, besides inducing transitions in the
reflected and transmitted waves, also sets up non-propagating " diffrac-
tion patterns " which may be considered to be exponentially damped
standing waves in the vicinity of the irregularity of the potential. If
the top of the barrier is wide enough so that these diffraction patterns
do not overlap, the transmission coefficients at the two edges may be
considered independent. The progressive wave representing the react-
ing system on the top of the barrier then loses all traces of the effects of
one edge before it reaches the other. It then exists in a fairly well-
defined quasi-stationary state while it is on the top of the barrier.
Under these conditions, it will be permissible to treat the activated
complex as a definite molecule with properties which can be calculated
from the shape of the potential at the top of the barrier. The potential
surface must be flat enough so that the density of reaction complexes, as
given by the Boltzmann factor, is sensibly uniform over it, and the flat
portion must be wide enough so that the uncertainty in the velocity due
to the relationship Ap Ax ~ h is negligibly small. If the potential along
the reaction path through the activated state is approximated by the
parabola

V = V Q - ax 2

these conditions may be expressed in the inequality:

T 16-36

M

where M is the mass of the complex. It appears very probable that
most reactions satisfy this criterion, and it is consequently justifiable to
treat the transmission coefficients at the two ends of the activated state
as independent.

We may .suppose that there is a probability p that a system entering
the activated state will be reflected and a second probability p/ that a
system leaving the activated state will be reflected. The transmission
coefficient K is the fraction of those systems in the activated state moving
from the initial to* the final state at thermal equilibrium which originally
came from the initial state and which will proceed directly to the final
state without returning to the activated state. At thermal equilibrium,

10 J. Hirschfelder and E. Wigner, /. Chem. Phys., 7, 616 (1939).

BEHAVIOR OF TRANSMISSION COEFFICIENT

316

let there be AT systems in the activated state, A of which arrive in unit
time from the initial state, N A = B of which arrive from the final
state. Of the A systems, Ap/ will be reflected at the boundary of the
activated state and A (1 p/) will be transmitted. Of those reflected,
Apip/ will be reflected at the other edge of the activated state and will

Initial
state

Transition

state

A

Final
state
! A(l-Pf)

_ A Pf (1-

\

B

&

FIG. 164. Calculation of transmission coefficients (after Hirschfelder

and Wigner).

recross it toward the final state. Figure 164 illustrates the flux of
systems of various types. Adding up the number of systems crossing
from left to right, we find the total number to be

16-37

" PiP/} 16 -38

At equilibrium, equal numbers of systems will be moving in each direc-
tion, that is, Ni-+ r = N r -*i, so that

16-39

= A(l + piPf + pip} + )+ B Pi (l + Pi pf
Similarly, the number crossing from right to left is

(1 - Pi)

316 THEORY OF REACTION RATES

Substituting this expression into 16-37 gives

Ni^r = ~~~ 1640

(1 ~ Pi)

But the number of systems originating in the initial state which proceed
to the final state is, by addition,

1641

^

(1 - Pip

and the transmission coefficient is the ratio

- PiP/

16 . 42

This expression for K is the average for a great many systems in thermal
equilibrium, whose energies consequently differ slightly. The trans-
mission of a single quantum state will depend critically upon the energy
of the system, as we shall see, but averaging over a range of energies
gives an average transmission coefficient which agrees with equa-
tion 1642.

We can complete the calculation of K for the Eckart potential. Taking
B = reduces the barrier to a simple step of energy A with no maximum
in V. For this special case the reflection coefficient is

P = -aw e h 1643

where p = V2mE and q = \/2m(E A) are the momenta in the initial
(or final) and activated states, respectively. For an abrupt energy
change, I = 0, and the transmission coefficient becomes

16.44

. ^ ,

(r + PIP/) (pi + Pf)

where p-, q, and p/ are the momenta of the system in the initial, acti-
vated, and final states, respectively. This value for K may be compared
with the exact result for a rectangular barrier, as determined earlier in
this chapter. If we take the expression analogous to 16-1 1 and reduce the
determinants, we find for the transmission coefficient the result

_ * __
K = -5- - ; - TO - o - ; . o ', - To T~5 J 9 = T~ ** 1 0-4:0

2 (P< + P/r cos 2 ? + ( a + Pip/) 2 sin 2 <f> h

where d is the barrier width. If this equation is averaged over <p, that is,
over a range of momenta in the activated state, the result is identical

BEHAVIOR OF TRANSMISSION COEFFICIENT 317

with equation 1644 for Jc. /c covers a cycle in its oscillation as <p goes
from to 2ir or as q goes from to - . Thus, for wide barriers, the oscil-

lation in K as the momentum in the activated state is increased becomes
more rapid, and averaging over a small range of momenta smooths out
the oscillations more completely. Thus the treatment of pi and p/
as independent is justified in a statistical sense as giving the average
transmission coefficient over a band of energies when the barrier is
sufficiently wide.

In an n-atom system a decrease in momentum as the system passes
from the initial to the activated state means that some of the initial
translational energy has been converted to vibrational energy. To
consider such a possibility, we must treat the problem in more dimen-
sions. An abrupt increase in potential in a straight channel of para-
bolic cross section will illustrate the essential features of the two-dimen-
sional case. Let us designate by X the coordinate along the channel,
and that perpendicular to the bottom of the channel by x. We consider
an abrupt change in the potential at X = 0, i.e., for X < 0, let F =
A + a& 2 ; for X > 0, let V/ = a/x 2 . In the left-hand region, the
Schrodinger equation

with potential V,-, must be satisfied. On the right side of the potential
drop, the equation to be satisfied is

with potential V/. The p's may be expanded in terms of the ^'s as

1646

x

An incident wave from the left has the form $i(x)e h l and gives rise
to a reflected beam in which the systems have all possible quantum
numbers m for their motion in the x direction. Thus the wave function
for X < is

2r y oo 27rt _

tie^* 1 + Z Ri>nt, n e~^ qm 1647

m=0

and the total energy of the system is given by

*-jt + I, .648

318 THEORY OF REACTION RATES

For X > 0, there is only an outgoing wave

E Tumi*" = E T lk u km t n e-* Pk 16-49

fc=0 k,m

and the energy is

For high quantum numbers, the momenta pk and qk are imaginary, and
the corresponding wave function represents a local non-propagating
disturbance in the neighborhood of the discontinuity. These imaginary
momenta, when divided by i, must be positive to represent exponentially
decreasing disturbances.

Both the wave function and its derivative must be continuous at
X = 0. Equating 1647 and 1649 and their derivatives with respect
to X at -X" = 0, and comparing coefficients of \[/ m , gives

8lm + Rim =

* 16-51

k

These equations may be solved for Ri m and Tik by the methods of matrix
algebra. The results are

R= ft -!)(* + I)- 1 1AW

T - '

where is the infinite square matrix == quT l p~~ l u. From the wave
function 1647 we see that the number of systems with quantum num-

ber I which are incident in unit time is , that is, density times velocity.

M

Similarly the number reflected with quantum number ra is |ftzm| 2 77 , so

M

that the probability of reflection with quantum number m is |/2j m j 2

<li
In the same way we see that the probability of transmission with quan-

tum number k is I T tk \ 2 ~

qi

Equations 16-52 are only a formal solution, since we have not evalu-
ated the matrix elements of . This can be done for the case where the
total energy E is much greater than the energy of the highest vibra-
tional state F& to which there is likely to be a transition. This means
that the translational energy in the final state will be large compared

BEHAVIOR OP TRANSMISSION COEFFICIENT

319

with the vibrational energy, though this need not have been true in the
initial state. The results of this calculation show that the reflection
coefficients for no change in vibrational quantum number are slightly
smaller than those calculated for one dimension, owing to the possibility

1.0 r

1.0 2.0 3.0 4.0 5.0

Translattional Energy in Initial State -

kcal.

6.0

7.0

2.0 3.0 4,0 5.0 6.0 7.0

-Translational Energy in Initial State kcal.

FIG. 16-5. Probabilities of reflection and transmission at abrupt drop of 9 kcal. in

potential energy when vibrational frequency changes simultaneously from hi>i *= 4 kcal,

to hv f as 2 kcal. (after Hirschf elder and Wigner).

for reflection with simultaneous change in vibrational quantum number.
To the approximations made, such changes can be either or db2 for
the reflected wave; the transmitted wave may have any quantum num-
ber differing by an even integer from that of the initial state.
In Figure 16-5 the calculated transmission and reflection probabilities

320

THEORY OF REACTION RATES

are plotted. It is evident that the probability of a vibrational transi-
tion is not large and that it decreases with increasing initial translational
energy. It thus appears that the interchange of translational and
vibrational energy has little effect on the transmission coefficient, at
least as long as the translational energy is large in the final state.

The effect of curvature of the reaction path on the transmission
probability may be investigated in the same manner. A convenient
energy surface to consider is that formed by the channel between two

FIG. 16-6. Hyperbolic reaction path.

confocal hyperbolas, Figure 16-6. The potential energy inside the
channel is taken to be V = 0; outside the channel, V = . The
Schrodinger equation is separable in the elliptic coordinates /* and <p
given by

x = R cosh ^ sin v? y == R sinh /* cos y? 16*53

into two equations : the equation in /t describes the motion parallel to
the reaction path, and the equation in <p describes motion perpendicular
to the reaction path. These equations are

d?U

h 2
8w 2 MR 2 (E

-cosh:

16-54

16-55

BEHAVIOR OF TRANSMISSION COEFFICIENT 321

where R is the focal length of the hyperbolas, E is the total energy of the
system, and X is the separation parameter to be determined by the
boundary conditions. The lines <p = <PQ are the hyperbolas

x y _ #2 1ft K A

*\i lO'OO

. 2 2

Sin <PQ COS (f>Q

whose asymptotes have the slope = cot <p Q , so that <p is the angle

ax

between the asymptote and the y axis. If the angle between the
asymptotes of the two bounding hyperbolas is denoted by A, the bound-
ary condition is that $ = on the two hyperbolas whose asymptotes

A A

are given by<p = <PQ + - and <p = <p Q -

Equation 16-55 is now solved for <3>, and the allowed values of X are
determined. When these values are substituted into equation 16-54
it becomes a one-dimensional equation for the translational motion of
the system through the channel. For low values of E and for A < 30,
the characteristic values for X may be developed in a power series in
A. 11 Up to terms in A 2 , the allowed values of X are

cog2 <Po - cos 2<p Q 16-57

where W is the width of the channel at its narrowest point. With this
value of X, equation 16-54 is a one-dimensional equation for motion
parallel to the reaction path <p Q . It is not in the usual form, since the
energy E is multiplied by a function of the coordinates. Making the
transformation

X = U V sinh 2 ju + cos 2 <p

/.,. _ 16-58

s = R I Vsinh 2 /* + cos 2

^o

equation 16-54 becomes

where

fe 2 "| | cosh 2 2jit + 2 cosh 2/i cos

|"
\_

(cosh 2/i + cos
11 R. Langer, Trans. Am. Math. Soc., 36, 637 (1934).

322 THEORY OF REACTION RATES

h 2 n 2 cos 2 <f>n

A 8MW 2 | (cosh 2 M + cos

The coordinate s is measured along the reaction path with s = at the
narrowest point of the channel.

Classically, every system which began the journey through the hyper-
bolic channel would get through. However, equation 16-59 shows that
the quantum-mechanical wave packet encounters an effective barrier.
This barrier may be ascribed to two causes. The largest part of it is 7\,
which arises from the increasing zero-point energy of vibration as the
channel narrows. In order to get through the neck of the channel, the
system must have sufficient translational energy initially to overcome
the increased zero-point energy of vibration in the narrow neck. The
second part of the effective barrier is much smaller, of the order of
0.3 kcal. per mole at most. At large distances, the potential V ^ is
attractive, but it goes through a maximum at the activated state, adding
to the effect of V\. It arises from the transformation of coordinates
made in equation 16-54 and may be considered analogous to the centrifu-
gal potential which appears in discussions of the motion of a particle
which is constrained to move in a circular orbit. From the wave view-
point, the attractive minimum corresponds to a focusing effect similar to
that observed with light reflected from concave mirrors. In regions
where V^ is low there is a concentration of systems, due to the Boltz-
mann distribution, similar to the concentration of light intensity at some
points in front of a curved mirror.

We may conclude from this example that the effect of curvature of the
reaction path is to introduce a virtual potential barrier at which there
will be a reflection probability given by equation 16-44 or 16-45. How-
ever, it is important to note that for a hyperbolic channel there can be no
interaction of vibration and translation, since the Schrodinger equation
is exactly separable in the coordinates corresponding to these motions.
This, as we have seen, is never completely realized in actual reactions,
although for many reactions it may be a good approximation.

The transmission coefficient for a surface where vibrational transitions
do take place may be calculated for a channel having vertical parallel
walls which makes a 90 turn. Here, as in dealing with the straight
parabolic chaijnel, it is necessary to join solutions of the Schrodinger
equation for each of the three regions shown in Figure 16-7. If the
width of the channel is I and the origin of coordinates is at the outside
corner, then, for x > Z, < y < Z, there is an incident wave moving to
the left on which is superposed a number of reflected waves moving to

BEHAVIOR OF TRANSMISSION COEFFICIENT 323

the right, so that the wave function is

oo 2rt

' sin Y y 16-60

i

For 0<#<Z, Q<j/<Z, the wave function must satisfy the boundary
conditions

Such a function is

*? = E U w sin ^ 2 sin ? y + B w sin ^ x sin ^ yl 16
j L * * ft J

61

(0,0

in

ii

(0,0) (1,0)

FIG. 16-7. Reaction path with 90 turn.

In region III, where y > 1, < x < l y there are only the outgoing waves

~PI/ TTJ
*, h P ' y sin T x 16-62

At x I we must have

Similarly, at y = Z we must have

16-63

16-64

If we expand sin -7-^ x in terms of sin -r x, we can write down the four
h I

ir
equations indicated above and compare coefficients of sin x. This

324 THEORY OF REACTION RATES

gives

Sir*

= A,, p , cos z + (- 1)- B kn c ni

h h n I

T kj = B ki sin fe- Pj ' 16-65

- B W p, cos

where

2 C l . 27Tp,' . TTJ

sin x sin - 2 da:
ft /

The solution of these equations gives for the matrices of the coefficients
Akj and B k j the results

A = -2i(l - G 2 )- 1

- -2i(l - G 2 )- 1 ^ 16-66

where G is a matrix with elements given by

2-7T J***

sin p kl e h '

16 ' 67

For a given total energy p k can be calculated for different values of k
and the matrix elements evaluated. Equations 1665 then give the
amplitudes of the reflected and transmitted waves.

P'l

Inspection of equation 16-67 shows that, if is equal to some

h

integer n, there is no transmission. Such energies correspond to reso-
nant frequencies for region II. The transmission coefficient is conse-
quently an oscillating function of the energy of the system, as it is for a
ono-dimensional barrier. Figure 16-8 shows the density of transmitted
systems in the first and second vibrational states if the incident wave is
in the lowest vibrational state. In contrast to the results for a straight
channel, it is seen that the probability of a vibrational transition upon

BEHAVIOR OF TRANSMISSION COEFFICIENT

325

turning the corner is quite high. That this is not a peculiarity of the
sharpness of the corner is evident from the classical consideration of two
straight channels joined by a circular section, as in Figure 16-9. Let

1. 3. 3. 4. _ 6. 6. 7.

FIG. 16-8. Density of transmitted systems for 90 turn.

the last collision with the walls of the straight section be just at the
point where the circular section begins. Inspection of the geometry
of the figure shows that, unless the circular channel has a certain very
definite length, the component of velocity perpendic-
ular to the reaction path will be altered in passing
around the corner. This is equivalent to a change
in the vibrational quantum state of the system.

From these results, we are led to the conclusion
that, though a barrier does not very often cause a vi-
brational transition in the reacting system, a bend in
the reaction path nearly always induces transitions
to the highest vibrational state consistent with the
total energy of the system. A system moving slowly
in the initial state has less chance of surmounting
the barrier but more of turning a corner than one FI 169
with a high initial velocity. Actually, it is seldom
that the sharpest bend in the reaction path occurs just at the activated
state; usually it occurs just after the system has passed through the

326 THEORY OF REACTION RATES

activated state and consequently is moving slowly. It thus seems
unlikely that large reflections due to a bend occur in practice.

It will* be noted that the energy unit in Figure 16-8 is inversely pro-
portional to the effective mass of the reacting system. Consequently it
will be different for different isotopic compounds. Thus if 6 2 = 3 for
the complex H + HC1, it will be 6 for D + HC1. The transmission
coefficients are markedly different. The extent to which this difference
is realized in practice depends upon the degree to which the maxima in
the transmission coefficient average out as the result of the thermal
energy distribution. For reactions involving light atoms, the zero-point
energy is so high that averaging over the thermal spread may not affect
the peaks to any great extent. However, the potential used in calculat-
ing Figure 16-8 is highly idealized. If the incoming channel is narrower
than the outgoing the width of the peaks is much diminished. Further-
more, any smoothing of the sharp corner ought to increase the rapidity
of oscillation of the transmission coefficient, so that it appears that only
exceptionally should there be any noticeable experimental result from
these peaks. In reactions involving isotope separations, the slight differ-
ences in transmission coefficient of the isotopes may well be important.
It should also be observed that the classical transmission coefficient
rapidly becomes equal to the average of the quantum-mechanical trans-
mission coefficient. For all ordinary cases it thus appears justifiable to
calculate the transmission coefficients on a classical basis.

16c. Transition Probability in Non-Adiabatic Reactions. 12 * 13 We
have considered so far only those reactions which proceed on one energy
surface. Frequently, however, two such surfaces come very close
together or, in fact, appear to cross each other. Actually, at the point
of the apparent crossing the system is degenerate, since the two different
electronic configurations have the same energy. This introduces a
resonance energy which separates the surfaces slightly so that they never
actually intersect but only approach each other closely, as has been
explained in section lid. This state of affairs is typified by a diatomic
molecule which has both homopolar and ionic states. Figure 1640 is a
plot of the energy as a function of the interatomic distance for such a
molecule. The electronic eigenf unctions are ^i and ^ 2 . For r^ro,
\l/i is ionic in nature and ^ 2 is homopolar; for r-Cr , the roles are
reversed. If the molecule is initially in state ^ 2 (homopolar) and the
internuclear distance r increases infinitely slowly, the molecule will
remain in state ^ 2 for r ^> TQ. But if r changes with a finite velocity,
there will be a finite probability that the molecule will change from

12 C. Zeneiy Proc. Roy. Soc., London, A137, 696; A140, 660 (1933).
18 L. Landau, Physik. Z. Sowjetunion, 2, 46 (1932).

TRANSITION PROBABILITY

327

fa to fa as it passes r = r so that its final electronic state will be repre-
sented by a linear combination

* = Ai(r)fa + A 2 (r)fa 16-68

For convenience in calculating AI and A 2 , however, we shall express fa
and fa in terms of two other wave functions <?i and <?2> defined so that

E

FIG. 16-10. " Crossing " of energy surfaces (schematic).

<pi is equal to ^i for r ^> 7*0, that is, ionic in this example, but <pi remains
ionic in character for all r. Similarly, <p% is equal to fa for large r (homo-
polar) and remains homopolar in character for all r. The energies ei

Fia. 16-11. Energy relations at point of " crossing."

and 2 corresponding to <pi and <>% intersect when plotted as functions of
the internuclear distance r (Figure 16-11). Consequently <f>i and <&
are not exact eigenfunctions of the complete Hamiltonian for the

328 THEORY OF REACTION RATES

system, and their eigenvalues ei and c 2 are only approximate; actually
they are eigenvalues for the Hamiltonian which does not include the
interaction energy ci 2 between the two states <p\ and ^> 2 . At the point
of crossing we have EI = i e i2 and E 2 = e 2 + ^12, where ci 2 is the
difference between the exact eigenvalues EI and E% and the approximate
eigenvalues ci and 2 . If we choose <pi, <? 2 , \l/\, fa to be normalized and
(orthogonal we must have

to = Tr fo + **) J to = -7= G*i ~ *>2) 16-69

V2 V2

Then, since

16-70

we readily see that

H<pl = iy?i 12 ^ 2

16-71

We now consider the state of the system in terms of <pi and <p 2 as it
passes through r = r$. Suppose that initially it is in the state fa. The
final state will be given by equation 16-68. But this can also be ex-
pressed in terms of <pi and <p 2 , if we consider the state of the system to vary
with the time rather than with the internuclear distance. <p\ and <p 2
are not functions of t; they do not, however, represent steady states,
since systems are jumping from <p\ to <p 2 at the crossing. We must
therefore write for the final state of the system the expression

c 2 (tf)0 h <p2(r) 16-72

At any time t$ y there are |ci( )l 2 systems in the state <p\ and |c 2 (o)| 2 in
the state <? 2 . It is thus necessary to consider the time rate of change of
Ci and c 2 at any convenient fixed position r\. Since we know the rela-
tion of <pi to fa and #? 2 to fa for r ^ 7*0, it is convenient to take TI at < .
We now study the time variation of ^ given by 16-72, using the time-
dependent Schrodinger equation:

(T n\ f 2*-t . 2ir* . )

^. H O \ I / \ "T"*i* /\ /\ "T"'^

jj 1 1^1 vO^ ^i v ) "f* c 2 (i)6 ^jO/ 1 == 16'73
2ri a^/ 1 J

Substituting equation 16*71, satisfied by <p\ and ^ 2 , we have

j- (!-,) ^

,. ^ 16-74

<l * 'ci

TRANSITION PROBABILITY 329

These equations must be solved simultaneously subject to the boundary
conditions

Cl (-oo)=0 |c 2 (-o)|=l 16-75

which correspond to our knowledge that initially, t = > , for r ^> TO,
the system is in the state <p 2 , equivalent to ^ 2 - The probability of a non-
adiabatic transition is |Ai(rro)| 2 in equation 16-68. Denoting this
by P, we have

P = |A 1 (rr )| 2 = jc 2 (>)i 2 = 1 - | Cl (>)l 2 16-76

Equations 16-74 need be solved only for their asymptotic values of c\
and c 2 . Eliminating c 2 from 16-74 gives

d 2 d J2irf 1 d 12 l &i , /27T 12 \ 2

^ + | T ( l - 2 )-- J-^ + ^jc^O 16-77

We now simplify this equation by means of the following assumptions.

(a) i 2 (ro) <C the relative kinetic energy of the two systems. When
this is true, the motion of the centers of gravity of the two atoms is so
fast that this motion will have a negligible effect on 12 during a crossing.

(6) The transition region is so small that we may regard ei 2
as a linear function of the time, and e 12 , <pi, and <p 2 as independent of the
time. This is true if i 2 is sufficiently small.

We therefore write

27r / ^ *

~r (*i ~ C 2J = <*t

\ , * 16-78

3*12 _ d<Pi _ d<?2 _ ^

dt "" a^ "" a

This means that <=i (r) and 2 (r) are the asymptotes to the curves EI (r)
and E 2 (r) in the region of the crossing. The smallest separation of EI
and E 2 is #i(r ) - JS 2 fro) = 2 12 (r ).
The assumptions 16-78, together with the substitutions

16-79

fl

reduce equation 16-77 to the form

-o

The solution of this differential equation, subject to the boundary con-
ditions 16-75, gives for the asymptotic value of |ci ( oo ) | 2 the result

| Cl (oo)| 2 = l - e ~ 2 ' r 16-81

THEORY OF REACTION RATES

12

330

where

7= l

Therefore

The denominator in 16-82 can be expressed as
d ,

at

dr

16-82

16-83

16-84

dr .

where V = is the velocity with which a system crosses r = TO and
at

\s\ 52 | is the difference of the slopes of the two crossing potential
surfaces at r = r . We have-

i 16-85

for the probability of a transition, that is, of " non-adiabatic " behavior.
The probability that a system stay on the initial energy surface is then

P' _ 1 y> AF|1 2 f 1fi.fi

JL 1 ~* / ' L *' 1O OO

16d. Thermodynamics of Reaction Rates and the Effect of Applied
External Forc'es. Returning to equation 16-17, we may formally
write it

JLA77 "fp+ '0 IpHP

Iff .. ... />~kf ._Trt 1A.87

J,/?^ 6 ~" K J,^ i0 '

/I r ir 2 ' ft

where K* is the constant for the equilibrium between the activated
complex and the reactant molecules. Using the thermodynamic relation
- AF = RT log K, we may write 16-87 as

h

16-88

where AF* is the difference in free energy between the activated and
normal states and is called the " free energy of activation." Since
AF = Aff - TAS, we also have

.

k' = K^-

16-89

THERMODYNAMICS OP REACTION RATES 331

Equations 16-88 and 16-89 have proved to be very useful in the interpre-
tation and correlation of experimental data. 2 ' 14 ' 15

Equation 16-88 may readily be extended to include the effect of an
applied external force. Let AF* be the free energy of activation for a
given process in the absence of the external force. Then if an applied

force has a cojnponent / along the reaction path, and if this force acts
y

through a distance - from the normal to the activated state, the free

2

fx
energy of activation will be decreased by an amount - . The specific

forward rate in the presence of the external force will thus be

t Nfx\

~*~>
RT = 2kT 16-90

where fco is the rate in the absence of the external force and N is
Avogadro's number.

In the same way there is a backward rate against the field of

fx

k b = k e 2kT 16-91

This gives a net rate of

/ A. L\ fx

ft' = k/ - k b = fc \e 2kT - e 2kT ) - 2fc sinh + 16-92

ntJL

and hence a net forward velocity of

Xfc' = 2\fc sinh - 16.93

Here X is the average distance traveled per jump, which may, or may not,
equal x. lQ

Elaborations of 16-92 and 16-93 have been successfully applied to the
study of a wide variety of physical problems, including viscosity, diffu-
sion, plastic deformation, and electrochemical phenomena. 2

14 M. G. Evans and M. Polanyi, Trans. Faraday Soc., 31, 876 (1936).
16 W. F. K. Wynne-Jones and H. Eyring, J. Chem. Phys., 3, 493 (1935)i
16 H. Eyring, J. Chem. Phys., 4, 283 (1936).

CHAPTER XVII
ELECTRIC AND MAGNETIC PHENOMENA*

17a. Moments Induced by an Electromagnetic Field. In order to
discuss the phenomenon of optical rotatory power, as well as certain
other phenomena, we need expressions for the electric and magnetic
moments induced in an atomic or molecular system by an electromag-
netic field. The procedure to be followed in calculating these moments
will be quite similar to that in Chapter VIII in the discussion of radiation
theory.

The wave functions for the unperturbed molecule will be of the form

Let the normal state of the molecule be ^ Then, in the presence of an
electromagnetic field the wave function for the molecule may be written
as

where

^ = - -
dt n

and

In equation !?! the assumption has been made that the coefficients of
all wave functions except ^2 are very small.

In developing the principles of radiation theory we made the approxi-
mation that A could be regarded as constant over the molecule; this
assumption gave the result that the probability of a transition between
two states was proportional to the matrix element for the electric dipole
moment between the two states. For the discussion of optical rotatory
power this approximation is insufficient. The value of A at any point
in the molecule may be expressed by a Taylor's series expansion in terms

* See General References.

MOMENTS INDUCED BY ELECTROMAGNETIC FIELD

333

of the value of A and its derivatives at the origin of a coordinate system
fixed in the molecule. For the x component of A at the position occupied
by thejth particle in the molecule we will have, neglecting higher terms:

^ +y i (^) +z i (^} 17-2

To this approximation the perturbation H' may be written as

Terms 1

17-3

moment

The first and second terms in H 7 include the operators for the electric
and magnetic dipole moments, respectively. The electric quadripole
moment will be of no importance for our purposes ; hence this term in H ;
will be disregarded. The matrix element (^*|H'|^2) is therefore

:^P-

(*?*

*2j (V*A)o[ 174

or, since

d

1 dt

= - *- (6|R|o) - A + (6|M|a)

17-5

where

E ba = E b -E a
(6|R|a) =

We now introduce explicitly the time dependence of A as

A

17-6

334 ELECTRIC AND MAGNETIC PHENOMENA

where e = hv and v is the frequency of the electromagnetic field which is

(L(*i*

perturbing the molecule. The coefficients are thus

at

( ffba-H, ..ffba-e,]

\e % ~*~ + e ~*~ / 17-7
Integrating with respect to time, we have

T ( 6 1*1) A8 + (6|M|o) (VXA) j x

h ^ - 1 + Constant 17-8

The constant in this expression for c& is not a function of the time-
dependent perturbing field and may for convenience be put equal to
zero in this problem.

According to section 8g, the electric dipole moment of the system
represented by the wave function ^ is

R = Re (**|R|*)
Neglecting terms of the order of c 2 , this is equal to

{-it}
(alR|a)+2Lc6(a|Rlb)e * 17-9

b J

The first term in 17-9 represents the permanent dipole moment /x of the
unperturbed molecule; the second term represents the dipole moment
RO which has been induced by the perturbing field. Substituting 17-8
into 17-9, the expression for the induced dipole moment becomes

(a\R\b) (b|M|a) (VxAg) r + T 17 ' 10

or, once

,

e g e ^bae^+e ft e ft e

+ ^6. - t ~ El, - e 2

MOMENTS INDUCED BY ELECTROMAGNETIC FIELD 335

Z(a|R|6)(6|M|a)

From the relations E = --- -A, H = V*A between the fields E and H

c at

and the vector potential A, we have

*--* (.*-.-*)

17 . 12

dH . .

~

Using these relations, equation 1741 reduces to

- (a|Rl&)(6|R|a)

lR|b) (MM|a) . 17-13

We will be interested in the components of R in the direction of the
fields. To obtain these components, we average the expressions of the
type RR E over all orientations of the molecule with respect to the field
directions, assuming all orientations to be equally probable, and obtain,

336 ELECTRIC AND MAGNETIC PHENOMENA

as in Chapter VIII, the result R RE. Equation 1743 thus becomes

( lRlb) ' (6lRla)E

17 * 14

The induced magnetic moment M is obtained from 17-14 by replacing

(a
(a

R|6)-(6|R|o) by (o|MJ6) (bJRJa) and (a|R|b) (6JM|a) by
M|b) (6JM|a), as is apparent from the derivation. In taking the

real part of 17-14, we see that, since

(a|R|6)-(6|R|a)^|(a|R|6)| 2

dE

is real, the coefficient of r- is purely imaginary, so that its real part is

at

JTjO

zero. Similarly, for M, the coefficient of is zero.

at

(a|R|6) (6|M|a) can be complex, so that

Re{i(a\R\b) (6JM|o)} = ~7m{(a|R|b) - (6|M|a)|

where Im means that we are to take the imaginary part of the term in

dE

brackets. In calculating MO, the coefficient of can be further

at

reduced, since

|M|b) (6|R|a)

El

.> ^ba

in

- - 2 E(a|M|6) (6|R|o) + Z (v ^ (a|M|b) - (6|R|a)

e 6

= - 2 (a|M - R|a) + E lv _ 2 , (a|M|b) (6|R|a)

6

The first term is purely imaginary, so that its real part is zero. We
note further that, owing to the Hermitian character of R and M,

[(a|M|6) - (6|R|a)] [(a|R|&) (&|M|a)]*

DIPOLE MOMENTS AND DIELECTRIC CONSTANT 337

so that

Jm[(a]M|b) (&|R|a)] = -/7n[(a|R|&) - (&|MJa)J

Making use of the various relations given above, we have the final
expressions for the induced moments:

U' a = *JB? + 7 oE' + p a 37 1745

c at

The superscripts on the fields have been dropped, but the fields have
been written as E', etc., to denote the fact that they are the total effec-
tive field at the molecule, which may not be the same as the external
applied field. The quantities a' aj K a , y a , f$ a are, as may be seen from 17*14

"i, a v

17-16

, a .

wnere v& a = -7- ana v = 7
/& A

17b. Dipole Moments and Dielectric Constant. As shown in
Chapter VIII, the matrix element (a|M|&) for the magnetic moment will
in general be very much smaller than the corresponding matrix element
(o|R|6) for the electric moment. Of the quantities a 7 , K, /3, 7 listed
above, K, p, y will thus be much smaller than a , so that, to a first approxi-
mation, and one which is sufficient for many purposes, we may state
that the effect of an electromagnetic field on a molecule which is in the
state a is merely the induction of an electric dipole moment

R = a' a E' 17-17

where a' a is the polarizability of the molecule in the state a. If the matrix
element (a|R|a) is different from zero, that is, if the molecule possesses a
permanent dipole moment ju a , there will be an additional contribution to
the polarizability arising from a partial orientation of the molecules with

338 ELECTRIC AND MAGNETIC PHENOMENA

their dipole moments pointing along the field direction. We may calcu-
late this contribution to the polarizability in the following manner.

Let us consider that the molecule is subjected, not to the influence of
an electromagnetic field similar to that associated with light, but to a
constant electric field in the z direction, of strength E z . For this case,
the vector potential A may be taken equal to zero; the scalar potential p
is zE z . The classical Hamiltonian function for a system of charged
particles will then be

H = H - 1MB* 17-18

where HQ is the Hamiltonian function for the system in the absence
of an external field. For this system, the dipole moment is

dH

Quantum mechanically, the relation analogous to 17-19 will be

17-19

where c a is the energy of the molecule in the state a, this energy being
expressed as a function of the perturbing field,

The quantum-mechanical method of calculating the contribution to
the polarizability arising from the orientation of the permanent dipole
moment of a molecule may be illustrated by the simple example of a dia-
tomic molecule with a permanent moment ju along the internuclear
axis ; this calculation, in fact, leads to results of quite general validity.

We treat the diatomic molecule as a rigid rotator in space; the wave
functions, in the absence of a perturbing field, are

*j, M (0, <f>) = P^ (cos Q}e iM<p 17-21

and the associated energy levels are

J(J + !)/*

60 ~

The perturbation is fj^E z cos 0, so that the first-order perturbation
energy is

' = (**, jf | -H&* cos e\*j. M) = 17-23

as may readily be seen from the known properties of the wave functions
\l/j t M. In order to evaluate the second-order perturbation energy we
need the values of the integrals

H' Jt M, j>, M ' = (**. jr| - MO#. cos 0|*/. f M 17-24

DIPOLE MOMENTS AND DIELECTRIC CONSTANT 339

These integrals are similar to those involved in the determination of the
selection rules for electronic transitions in the hydrogen atom, and are
zero unless M = M', J = J f 1. From the recursion formulas 4*85,
we find for the squares of the pertinent integrals:

1*7' |a 2p , 2 -

\H J , M .. J+l , M \ -MOB. - - 17-25

The second-order perturbation energy will be

ITT/ 12 I Tjt 12

// \Hj t M; /-.I. Af . /!/, M ; /+!, M

j M - .) --

J J_! tj

For J = 0, M = 0, we have

600 - ~~ W "

1727
~~ "" 17 ' 27

For J 7^ 0, we obtain

" _ 8y 2 J/ipgf f 3M 2 /(/ + !) ]
CJ.M = 2 | J(J + 1) (2J_ l)(2J+3)j

According to 17-20, the z component of the dipole moment when the

d tf
molecule is in the state \f/j, M is M/, M , z = ^ - Since the energy

of a state, to" a first approximation, depends only on /, it will be con-
venient to calculate

MJ.*= Z M/,Jf f . 17-29

E M 2 = 2(1 2 + 2 2 + 3 2 + + J 2 ) =

Af-_J d

so that
[(3M 2 - J(J + 1)] = J(J + 1)(2J + 1)

MJ

- (2J + 1)(J)(J + 1) = 17-30
We thus see that MJ, = for J ^ 0. For J = 0, we have

The average z component of the dipole moment is thus just MO. z multi-

340 ELECTRIC AND MAGNETIC PHENOMENA

plied by the probability that the molecule will be in the state with
J = 0. According to 15-26, this probability is equal to

No _ 1
N ^ --

where c; is the energy of the ith state measured from the state with
/ = (). If we neglect the second-order perturbation energy, we obtain,
according to 15-59, the result

ft

~

so that the contribution to the polarizability arising from this effect is

Mo
- ; . This result is identical with that obtained from the well-known

ofCJ.

2

classical theory. It is to be noted that the final result is valid only

3/cl.

for low field strengths (neglect of 6 t " in c^), and high temperatures
(replacement of summation by integration), but these conditions are
adequately fulfilled in the usual measurements. Although this result
has been here derived only for a diatomic molecule with a permanent
dipole moment, a similar treatment shows that the result obtained is of
general validity.

The above contribution to the polarizability arises from the partial
orientation of the molecules with their permanent dipoles along the field
direction. In deriving the expression for a' a , we assumed all orientations
to be equally probable. The effect on a a of this partial orientation is of a
higher order than the effects considered above and will be neglected.
Our above derivation assumed the field to be stationary. If this is not
true, the results must be modified. The nature of the necessary modifi-
cation may be seen as follows. If the frequency of the field is much less
than the frequencies associated with molecular rotation the molecule
will be oriented in the manner discussed above; if the frequency of the
field is much greater than the rotational frequencies the molecule will
not be oriented with respect to the field, and there will be no contribution
to the polarizability arising from this effect. Since molecular rotational
frequencies correspond to wavelengths in the far infra-red, there will be
no molecular orientation when the perturbing field has the frequency of
visible light. We thus have the final formula for the dipole moment
induced by an electric field

Ro

DIPOLE MOMENTS AND DIELECTRIC CONSTANT 341

with

2

ct a = 57 + a (stationary fields)

oK J.

<*a = o (visible light)
where

I, |p| M. / 2 , bQ |(a|R|6)| 2
Ma = (oRa); a = ^ri; 2

o/l 6 Vba ~~ V

To complete the discussion of the relation between dipole moment and
dielectric constant, we consider a material medium containing N\
molecules per cubic centimeter, all of the same kind and in the same state
a (the extension to mixtures is obvious; if more than one state need be
considered, it must be weighted with the appropriate Boltzmann factor).
The polarization of the medium will be P = N\R. For an iso tropic
medium, we may use for the effective electric field the Lorentz field

E' = E + yP
If E' is expressed by this relation, the polarization of the medium will be

1 -

3

From electromagnetic theory, we have the relations

D = cE = E + 47rP

where c is the dielectric constant of the medium and D is the electric
displacement vector. In terms of the polarizability, the dielectric
constant will be

c= 1+-

i

3
or

e- 1 4irJVitt
+ 2 ~ 3

For stationary or almost stationary fields, we have

( - 1
+ 2 =

17-31

17-32

342 ELECTRIC AND MAGNETIC PHENOMENA

which is the relation used when dipole moments are determined from
measurements of dielectric constants. The usefulness of a knowledge of
the dipole moment in investigating the structure of a given molecule is so
well known as to require no further comment at this point.

17c. The Theory of Optical Rotatory Power. In order to under-
stand the origin of the phenomenon of optical rotatory power, we must
include the higher-order terms when calculating the moments induced
by an electromagnetic field. We therefore write, for the fields associ-
ated with visible light,

17.33
17 ' 34

where a = ]pa<*a> & = Zpaa, * = Z)pa*a, and pa is the probability that

a a a

the molecule will be in the state a. For the effective electric field we have

47T

again used the Lorentz field E' = E + P; for the effective magnetic

o

field we have set H' = H, since the magnetization is in general very
small. The terms in y a have been neglected, since it may be shown that
their inclusion would have only a second-order effect on the optical
rotatory power. 1 The electric induction D and the magnetic induc-
tion B may, with the aid of 17-33 and 17-34, be calculated to be

R,

-K 17-35

at

B - H +

(1 + 4ir#iK)H + r - r-r: -- E (to the first order in 0)
3

= KH + g - E 17-36

dt
1 E. U. Condon, Rev. Modem Phys. 9, 444 1937.

THEORY OF OPTICAL ROTATORY POWER 343

where

"i

6- 1

e + 2 ~ 3
K = l +

17 . 37

The nature of the electromagnetic field in a region free from real charges
or real currents is specified by Maxwell's equations

V-D = V-B =

i a i a 17 * 38

V xE = --~B VH =--D
C dt c dt

For any particular medium, these equations must be solved subject to
relations of the type 17-35 and 17-36. Let us suppose, for the moment,
that the parameter g is zero, so that D = cE, B = KH. From the equa-
tion for V X E, we obtain, by taking the curl of both sides,

V*V*E = - V 2 E + VV.- E = - - - (V*B) = - ^ -^E

C ut C ut

or, since V E = 0,

V'E.f^E ,7.39

In the same manner, we find

These equations represent waves propagated with a velocity

c c
v =

where n is the index of refraction. In non-magnetic media K ^ 1, so
that n 2 = e, which is the familiar relation connecting index of refraction
with dielectric constant. We will now investigate the solutions of
Maxwell's equations for the case where g is not zero.

For our present purposes, we will investigate the nature of the solu-
tions representing plane waves propagated along the z axis. The equa-
tion V D = may be written as

'-S. + 'Dl + '&.O 174 l

dx dy dz

344 ELECTRIC AND MAGNETIC PHENOMENA

dD x dDy

For our plane wave, = = 0, and so we conclude that D z = 0.

ox dy

In the same way, the equation V -B = requires that B z = 0; since
D z = B z = we must have E z = H z = 0. We now specialize our
desired solution even more; we require that the solution represent right
circularly polarized light. For this type of wave we have

E = E(i cos ^ j sin ^) 1742

where E is the amplitude of E and ^ = 2wp It 1 is the phase of the

wave of frequency v propagated with a velocity v = - . At t = 0,

Tl

z = 0, E is directed along the x axis. As t increases, E rotates toward the
y axis, or, to an observer stationed so that the light enters his eyes,
E rotates clockwise. From the curl equations, we have

_ . dE y . dE x

VE == -1^ + j
dz dz

_ , . . . , 1 #
E{ i cos ^ + j sin \f/\ = ~-B

C dt

or

B = nE{i sin ^ + j cos ^} 17-43

In 17-36 we make the approximation that K == 1. Then

or

H = ( n ~> *| B = ( n + 2intf)J(i sin ^ + j cos t) 1744
\ n /

From the curl equations, we have

V*H = - (n + 2irvg)E(-\ sin ^ - j cos ^) = - -
C c dt

or

D = n(n + 2irvg)E( i cos ^ j sin ^) = n(n + 2irvg)E 1745

But, according to 17-35 and 1744

at n

or

- - E 1746

n + 2irvg

THEORY OF OPTICAL ROTATORY POWER 345

In order that 17-45 and 1746 be consistent, we must have

ne

n + 2irvg

n(n

n R = - 2irvg 1747

where n R is the index of refraction for right circularly polarized light.
For left circularly polarized light, we may write E as

E = E(i cos ^ + j sin ^) 1748

Proceeding in exactly the same manner as above, we find that in this
case the index of refraction is

n L = c w + 2-Kvg 1749

We therefore write

ife = ft> + 5, fa = ^ - 5 17-50

( nz\ .

(-7)-

where ft) = 2*-i' ( t -- J is the phase of a wave propagated with the

mean index of refraction n = ** and 6 = 4w 2 v 2 g - . The superposition

c

of right and left circularly polarized light waves of equal amplitude
gives a plane polarized wave. Adding 1742 and 1748, we have

E - #{i[cos (* + 5) + cos (ft, - 5)]

+ j [- sin (ft, + 6) + sin (ft, - 6)]}
= 2E cos ^ {i cos 5 j sin 6} 17-51

For S = 0, E is along the x axis; for 5 > 0, E has been rotated through an
angle S in the clockwise direction as viewed by an observer looking along
the z axis. A medium for which 6 is positive is therefore dextro-
rotatory. The rotation in radians per centimeter is thus

6

z c
or, according to 17-37 and 17-16,

, 4*-V tvNi f + 2

17>52

where fl*. = 7m{(a|R|6) (6JM|a)} is the "rotatory strength" of the

346 ELECTRIC AND MAGNETIC PHENOMENA

transition a > b. From the theoretical viewpoint, the calculation of the
optical rotatory power of a molecule is thus reduced to a calculation of
the matrix elements (a|RJ6) and (6|M|a). Since the eigenfunctions of
complex molecules are not known to any high degree of accuracy, it has
not as yet proved possible to determine the absolute configuration of any
molecules by actual calculation. For a discussion of the attempts
which have been made in this direction, as well as of related topics from
this viewpoint, the reader should turn to one of the review articles in
this field. It is possible, however, by a study of the symmetry proper-
ties of the rotatory strengths, to state the conditions necessary for optical
activity.

The operators R and M in rectangular coordinates are

M. h

x y ]\ 17-53

k dy dx/j

If the molecule has a center of symmetry we may classify the states of
the molecule as odd or even according as the wave function for a given
state changes sign or retains the same sign when subjected to an inver-
sion at the center of symmetry, that is, when each coordinate is replaced
by its negative. Since the operator R changes sign upon inversion, we
have a non-vanishing value of (a|R|&) only between odd and even states.
The operator M does not change sign upon inversion; hence we have a
non-vanishing value of (a|MJ6) only between two odd or two even states.
The scalar product Xa|R|b)-(bJM|a) will therefore be identically zero for
all states a and b, and the optical rotatory power will be zero. If the
molecule has a plane of symmetry, we may again classify the wave func-
tions as odd or even with respect to reflection in this plane. Let the
plane of symmetry be the yz plane. Then, with respect to reflection in
this plane, the x component of R is odd and the x component of M is
even. There will be no pair of states a and 6 for which the x components
of (a|RJ6) and (b|M|a) are both different from zero. The same is true
of the y <and z components. A fundamental requirement for optical
activity is thus that the molecule possess neither a plane nor a center of
symmetry.

We now consider the values of (a|R|b) (b|M|a) for a given molecule
and its mirror image. A molecule may be transformed into its mirror
image by reflection of its coordinates in any plane; this reflection is
equivalent to changing from a right-handed to a left-handed coor-

DIAMAGNETISM AND PARAMAGNETISM 347

dinate system. If we reflect a molecule in the xy plane, the new
value of (aJRJb) (6JM|a) for the molecule is obtained by replacing
z by z in the wave functions a and b in the corresponding expression
for the original molecule. If we change z to z in both eigenf unctions
and operators, the product of the matrix components will remain un-
changed, since the values of the integrals involved, being pure numbers,
are independent of the particular coordinate system in which they are
evaluated. From the form of the operators, we see that R M changes
sign when z is replaced by z; therefore (a|R|b) (i>|M|a) must change
sign if we replace z by z in the eigenfunctions only. An unsymmetrical
molecule and its mirror image must therefore have equal optical rota-
tions, but with opposite signs. If a molecule is identical with its mirror
image, its optical rotation must vanish. These conditions for the exist-
ence of optical activity are, of course, identical with those which have
long been known.

17d. Diamagnetism and Paramagnetism. According to equa-
tion 845 the Hamiltonian operator for a system of charged particles in
an electromagnetic field is

H = Ho + H'

where H is the Hamiltonian operator for the system in the absence of
the field and

H' = Z [^ ( I V< A, + 2in* A; V; + ~ | A,| 2 ) + e^] 17-54

Let us consider a uniform magnetic field of magnitude H z along the
z axis. The potentials are

Axi = ^Hzyi] A yi = \R z Xi\ A z = 0; <p =
From these values of the potentials we readily find the relations

17-55

where M 2 is the operator for the z component of the angular momentum
of the system. The perturbation H' can thus be written

+ *

If we include the interaction of the magnetic field with the electron spin,

348 ELECTRIC AND MAGNETIC PHENOMENA

the perturbation H' becomes, according to 941,

H' - - ~ H Z (M, + 2S Z ) +iE (*? + y?) 17-57

2mc *

The term in //f will in general be much smaller than that in H z and will
be of importance only where the eigenvalues of M 2 and S z are zero. We
will discuss the term in H%, which is responsible for diamagnetism, later,
and will for the present concentrate our attention on the term in H z ,
which is responsible for paramagnetism.

For atoms or ions, a portion of our discussion will parallel the discus-
sion of the Zeeman effect in Chapter IX. The paramagnetic term, for
an arbitrary magnetic field H, is

H' = - ~ {H M + 2H -S) 17-58

2mc

For an atom in the state characterized by quantum numbers L and S,
the first-order perturbation energy will be

p ij
'(L, S) - 7^- {H -L* + 2H S*} 17-59

where L* is a vector of magnitude v L(L + 1) and S* is a vector of
magnitude VS(S + 1) The magnetic moment for this state will be
given by the relation analogous to 17-20, that is,

= (L* + 2S*)Mo 17-60

eh
where MO "; - is the " Bohr magneton." The component of |x(L, S)

along the vector J* representing the total angular momentum will be
|ty(L, S) = MofL* cos (L*, J*) + 2S* cos (S*, J*)|

= ffJVo 17-61

where

, J(J + 1) + S(S + 1) - L(L + 1)

(7

2,7 (.7 + 1)

In the presence of a magnetic field, the atom is so oriented that the pro-
jection of J* along the axis of the field has the quantized values M = J,
j _ i . . . _ j f The average value of the component of the magnetic
moment of a given atom along the direction of the field, provided that

DIAMAGNETISM AND PARAMAGNETISM 349

only the ground electronic state has an appreciable chance of being
occupied, is

17-62

M

For small values of H, we may expand the exponential and retain only
the first two terms. Then

/* = ~ 2: fL_ 17.53

El JL ~T~ I
\ I*'/"* /

Since

Jlf M Af

equation 17-63 reduces to

The contribution to the magnetic susceptibility arising from the pres-
ence of permanent magnetic dipoles in an atom or ion is thus

- - - , an expression which is very similar to the contribution
oA/7

to the electric polarizability arising from the presence of the permanent
electric dipole. This paramagnetic term vanishes for atoms with
J = 0; for atoms in S states we have J = S, and the paramagnetism
arises entirely from the unpaired spins. Most molecules have zero
orbital angular momentum and no unpaired spins in their ground states,
and thus show no paramagnetic effects. A notable exception is 62,
which has the ground state 3 2^~, and thus exhibits a paramagnetic
effect arising from the unpaired spins. The exact value of the para-
magnetic susceptibility for molecules depends upon the nature of the
coupling between the spin, orbital, and rotational angular momenta.
All atoms or molecules, regardless of whether or not they are para-
magnetic, show diamagnetic effects. The perturbation energy arising
from the second term in 17-56 is

a 2 J(*? + 2/?) 17-65

8mc .

350 ELECTRIC AND MAGNETIC PHENOMENA

where (x 2 + y 2 ) is the average value of (x 2 + y\). Since, to a first
approximation, all directions in space will be equivalent, x 2 = y 2 = z 2 =
^rf, where r? is the distance of the electron from the center of gravity
of the system, so that

e 2

The associated magnetic moment will be -- oHY,r?, and the corre-

orac i

e 2

spending susceptibility is - X)rf, which we note to be negative.

orac

The complete magnetic susceptibility for an atom or ion will thus be

6mc 2

CHAPTER XVIII
SPECIAL TOPICS

18a. Van der Waals' Forces. The weak attractive forces between
atoms which are not connected by ordinary valence bonds have long
been known by the name of Van der Waals' forces. If the particles
possess permanent dipole moments, certain forces will arise from this
cause;- these forces can be calculated from classical theory. Van der
Waals' forces exist even if the particles are symmetrical; the origin of
the forces may be most simply seen by consideration of the interaction
of two hydrogen atoms when the interatomic distance is large.

For the hydrogen molecule, the wave function, in the Heitler-London
approximation, may be written as

18-1

If the internuclear distance is sufficiently large, we may assume the
electrons to be definitely located on one or the other hydrogen atom, and
we may write the wave function for this system as

/, = &k(l)\fo>(2) 18-2

The first-order perturbation energy will then be

E f = I ^o(l)^&(2)HVa(l)'/'6(2) &TI drg 183

where

\ r ab 7*12 7*a2 r bl /

the symbols , etc., having their usual significance.

FIG. 18.1.

If, in the coordinate system as illustrated in Figure 18-1, we let
(a?i, 2/1, Zi) be the coordinates of electron (1) relative to nucleus a, and
fej 3/2, 22) the coordinates of electron (2) 'relative to nucleus 6, the

351

352 SPECIAL TOPICS

analytical expressions for the distances are

(72

- * 2 ~ R) 2

Since the atoms are assumed to be far apart, R x\, R x 2 , etc.
Since

we obtain, upon expanding the expressions in 184 and retaining only
the first two powers of the coordinates of the electrons, the results

= 1

r ab ~ R

7*12

1

1

02
1

61

J.

_ n o - O ^.-r-v~n

^ 1 fli x^ 1 O /?'9 1

R
1

_ x 2/i; 2 J

R

L 2fl 2 J

18

18-

Combining these terms gives, for the perturbation H 7 , to this approxi-

mation, the result

e 2

H/ * ^ XlX * + yiy * "" 2ziZ ^ 18 ' 6

which is the " dipole-dipole " interaction.

For ^o(l) and ^&(2) we take the Is eigenfunctions of hydrogen.
Using the perturbation above, we immediately see that the first-order
perturbation energy is zero, since H' is an odd function of the coordinates
while the fs are even functions of the coordinates. According to
equation 7-27, the second-order perturbation energy is

lo7

where the summation is over all the states k of the hydrogen atom
except the ground state. The denominator of this summation is equal to

VAN DER WAALS' FORCES 353

e 2 / j\

-- f 1 -- J ( s i nc e there are two hydrogen atoms), where n is the
a \ n /

principal quantum number of the state k. The denominator of the

3 e 2 e 2

summation thus varies from -7 to - ; in order to evaluate the

4a

e 2

sum, we set the denominator equal to -- for all terms. This gives an

a

approximate value for the second-order perturbation energy equal to

'<***'

18-8

since H ' w = and (#^#0) = (ff' 2 )oo. In evaluating (H /2 ) 00 , the

k

cross terms are zero for the same reason that H' OO vanished ; we therefore
have the result

18-9

Since the Is state is spherically symmetrical,

xf = 2/? = F = H etc. 1840

SQ that

18-11

According to equation 6*33, r 2 = 3a. The second-order perturbation
energy is thus

By an actual summation of the series 18-7, Eisenschitz and London 1
obtained the more accurate value E" = 6.47^ ; an exact pertur-

1 R. Eisenschitz and F. London, Z. Physik, 60, 491 (1930).

354 SPECIAL TOPICS

bation calculation 2 gives 6.499 as the value of the numerical coefficient.
If the expansion of H' is extended to include higher powers of the

coordinates, the van der Waals energy contains not only the term in ^

but also terms in g > -^ , etc. By a procedure similar to that carried

out above, Margenau 3 obtained for the van der Waals energy the
expression

6e 2 al 135e 2 ag 1416e 2 ag

~ ~~ 18 ' 13

The van der Waals energy may be expressed in terms of the polari-
zability by an argument due to London. 4 Instead of writing

E" = - -y- (#' 2 )oo, we write E" = - (# /2 ) o, where 7 may be
c /ct>o &1

taken to be the first ionization potential with the same degree of accu-
racy as we formerly took it equal to the ionization potential. According
to 176 the polarizability of an atom in a stationary field (v = 0) is

= ^ (a|r|6).(6|r|a) 18-14

Oil b V a b

An approximate value for the polarizability is therefore

where r a & is an average value of *> a &, and hv a ^ has been replaced by I.
(H /2 )oQ is thus e
Waals energy is

3 I 2 a 2

/2 )oQ is thus equal to -- <p ; to this approximation, and the van der
2 R

E"- 3 18 16

E - - - R s 18-16

London has shown that, if the atoms are not equivalent, the analogous
expression is

3 I A Is

-

17

' 17

8 L. Pauling and J; Beach, Phys. Reo., 47, 686 (1935).
8 H. Margpnau, Phys. Rev., 38, 747 (1931).
4 F. London, Z. Physik, 63, 245 (1930).

QUANTUM-MECHANICAL VIRIAL THEOREM 355

where I A and <XA are the ionization constant and polarizability of atom
(or molecule) A and IB and a.% are similar quantities for atom B.

For many-electron atoms, the perturbation energy is a sum of terms
of the type 18-6, with one term for each pair of electrons. Further
details on the van der Waals forces in complex atoms may be found in
the review article by Margenau. 6

18b. The Quantum-Mechanical Virial Theorem. 6 In classical
mechanics, the virial theorem states that

(j x \ 2 _
*) = -E*iF*i 18-18

at/ i

where the bars denote time averages and F x . is the x component of the
force on the ith particle; the left side of this equation is 2T, where T is
the kinetic energy of the system. This theorem also holds in quantum
mechanics, as may be seen from the following argument. Schrodinger's
equation for a system of particles is

+ (V ~ * )* = 18-19

Operating on equation 18-19 by x$* , we obtain

OXj

while from 18-19 we have

flifr h 2 dt d 2 ^*

**<? -*>-***,'-&

Substituting 18-21 in 18-20, and summing over,;, gives

It is possible to integrate the first term in equation 18-22. We have

dXjdx

,8.23

6 H. Margenau, Rev. Modern Phys., 11, 1 (1939).
6 J. Slater, J. Chem. Phys., 1, 687 (1933).

356
Therefore

SPECIAL TOPICS

-r^-rVl 18-24

or

/,*2.

18-25

When 18-25 is integrated with respect to x,-, the last term vanishes
because the term ^* 2 is zero at . Integrating 18-22, we thus obtain

2-

i

dV
Since F x . = - , this may also be written as

OXj

18>26

or

18-27

18-28

where the bars now represent quantum-mechanical averages.
If the potential energy arises entirely from the interactions of charged

particles, the term

takes a particularly simple form. The

potential F for such a system may be written as V = V mn with

^mn = ~^ 9 where c m and e n are the charges on particles m and n a dis-

Tmn

tance r mn apart. We will now calculate the quantity # -- ~^

OXj

for this pair of particles. Since

(y m - y n )

we have

etc.

QUANTUM-MECHANICAL VIRIAL THEOREM 357

We thus obtain the result, since - = - -- ^ * etc-

dr mn dx

mn m

= _ Sf2 {( Xm -

= -F wn 18-29

7*mtt

or, summing over all pairs

. - = 7 18-30

y toy

For this particular case, equation 18-28 becomes

2T = -? 18-31

for a system of charged particles.

For a diatomic molecule, the virial theorem can be put into the
following useful form. If the potential energy of the nuclei (the elec-
tronic energy) is E(r) 9 the external force required to hold the nuclei

n Tf> / \

fixed at a distance r is . If we apply this force to the nuclei, the
virial theorem 2T = ^XjF Xf becomes

2T 7 = -F 7 - r 18-32

dr

where T f is the kinetic energy of the electrons only and V f is the poten-
tial of the electrostatic forces. Since the nuclei have no kinetic energy
in this case, the energy E is^equal to T' + V. Using this relation, we
may solve 18*32 for T r and V f separately, obtaining

AW 18-33

V' = 2E + r
dr

The potential energy curve for a diatomic molecule can be repre-
sented with reasonable accuracy by the Morse curve

B(r)

or

#(r) - D{ -2e- a(r - f ' ) + e***-** } 18-34

358

SPECIAL TOPICS

where the energy zero has been chosen so that E(r) = when r = >.
For purposes of illustration, let us take D = 100 kcal., a = r, = 1.
The resulting_curves^ f or E, T f , V' are plotted in Figure 18-2. At
r = oo, E = T f = V f = 0. As the interauclear distance is decreased,
the kinetic energy first decreases, then increases rapidly as the nuclei are
brought closer together. This increase in kinetic energy is more than

ou

Af\

\

40

nn

\

y'

6V

\

^

*^^*"

n/\

\

/

/

E-

^

~

-

-zo

A(\

V

f

"Xfc^^

^ -

^

,.

*****

^ *

^-^

^- <

-- ***

4U

Cl\

\

/

/

^

>*

~

-

r'

-bO

E Qf\

\

^

^

oO

i f\t\

\

s "~1*_

i^

^

100

19fi

/

i^U

1 Af\

/

JL40

1 Rf\

/

IbO
Iflfl

/

m\t\

/

ZOU

99n

/

240

uw

\

^s

/

)

1,

2

'

r

3

.0

4.

FIG. 18-2. Energy relations for a diatomic molecule.

compensated for by the decrease in potential energy, leading to the for-
mation of a stable system. It is of particular interest that at the equilib-
rium position 2T 7 ' = F'; that is, the kinetic energy of the electrons is
J the electrostatic^ potential energy, so that the binding energy E is

equal to \V r = T'.

18c. The Restricted Rotator. Of considerable importance for the
study of the internal motions of complex molecules as well as for the
study of the motion of molecules in crystals is the problem of the energy
levels of a rotator moving under the influence of a potential field. As
the simplest example, we consider a rigid rotator whose moment of
inertia is /, moving in a sinusoidal potential field of the form

18-35

THE RESTRICTED ROTATOR 359

where v is the angle of rotation. The wave equation for this system is

Making the substitutions,

rup
-

9 = f^f 18-37

"Po>

, a r ad = 18-39

dor

this equation reduces to Mathieu's differential equation

d 2 4(x)

~~- + (a r + 26 cos 2x)^(x) = 18-38

the solutions or which are far from simple. We may, however, consider
the limiting cases, where the motion resembles that of a free rotator or
that of a harmonic oscillator. For a r ^> 26 the equation becomes

which has the solutions

where r is a positive integer, including zero. From the symmetry of the

potential field, we must have ^(<p) =^(<pH -- 1 . This requires that

\ n/

r be restricted to even integers. Since a r = r 2 the energy levels are

- = 0,246... 1841

For the various values of n, we have

-0,1, 2,3,4

n = 2 E m - = m2 ' " -0,2, 4,6-..

360

SPECIAL TOPICS

For n > 1, only - th of the energy levels for the free rotator are allowed.

n

The connection between this fact and the origin of the symmetry number
(section 15e) in rotational partition functions is obvious. These energy
levels are doubly degenerate.

For the opposite case, a r <C 20, the wave function will have appreciable
values only for re ^ 0, so that we may write

20(1 -

1842

which is the wave equation for the harmonic oscillator. The energy
levels are given by the relation a r = 2(2r + 1)V0 20, so that

E r = (r + %)hv , r = 0, 1, 2, 3 - 1843

where VQ = */ For the intermediate case a r ~ the energy

levels will not be well approximated by either of the above equations.
However, the Mathieu equation has been solved exactly for certain
values of 0; the eigenvalues a r are tabulated by Wilson 7 for values of
from to 40. We reproduce in Table 184 the exact eigenvalues a r for
the first seven levels. for = 4, 9, 16, and 36, together with the values
calculated by the above expressions for the limiting cases (in parenthe-
ses). Values above the line were calculated by the harmonic oscillator
approximation; those below the line were calculated by the free rotator
approximation.

TABLE 18-1

EIGENVALUES a r OF THE MATHIEU FUNCTION

9

-4.2805
2.7469
6.8291
16.452
16.150
36.229
36.230

4
(-4.000)

e -

-12.262
- 1.3588
7.9828
17.303
20.161
37.157
37.21

9

(-12.000)
( 0.000)
( 4.000)

e =

-24.259
- 9.3341
4.3712
16.819
26.009
39.315
40.22

16

(-24.000)
(- 8.000)
( 4.000)

( 4.000)
( 4.000)
( 16.000)
( 16.000)
( 36.000)
( 36.000)

( 16.000)
( 16.000)
( 36.000)
( 36.000)

( 16.000)
( 16.000)
( 36.000)
( 36.000)

6 =

-60.256
-37.303
-15.467
5.1467
24.379
42.118

36

(-60.000)
(-36.000)
(-12.000)
( 12.000)

( 16.000)
( 36.000)

f E. B. Wilaon, Chem. Reva., 27, 31 (1940).

APPENDIX
L PHYSICAL CONSTANTS

These values of the important physical constants are taken from the
tabulation of R. Birge, Rev. Modern Phys., 13, 233 (1941). Many of the
numerical values used in the text are based on slightly different values
for certain of the physical constants; these differences are not significant.

Velocity of light
Charge on electron

Ratio, charge to mass of electron

Planck's constant

Ratio

Mass of electron

Mass of proton

Mass of hydrogen atom

Ratio

Boltzmann constant
Gas constant

Avogadro's number
Rydberg constant for H
Rydberg constant for oo mass

Bohr radius

Bohr magneton

/ eh

eh \

c = 2.99776 X 10 10 cm. seer 1
e = 4.8025 X 1(T 10 abs. e.s.u.

= 1.7592 X 10 7 abs. e.m.u. gm." 1
m

h - 6.6242 X 10~ 27 erg - sec.

- -= 1.3793 X 10~ 17 erg sec. e.s.u." 1
c

m - 9.107 X HT 28 gm.
M p = 1.6725 X l(T 24 gm.
M H - 1.6734 X 10~ 24 gm.

^ = 1836.5.
m

k = 1.3805 X 10~ 16 erg degree^ 1 .

R = 8.3144 X 10 7 erg deg." 1 mol." 1
= 1.9865 cal. deg." 1 mol." 1

N - 6.0228 X 10 23 mol.~ l
R H = 109,677.581 cm.- 1
#00 = 109,737.303 cm."" 1

o = 0.5292 X 10"" 8 cm.

= 0.9273 X 10" 20 erg gauss" 1

I electron volt = 1.6020 X 10~ 12 erg molecule"" 1 = 23,052 cal. mol."" 1
= 12395 X 10~ 8 cm. = 8067.5 cm."" 1

L = 2/goo = 27.205 e.v.

a

II. VECTOR NOTATION

The concept of a vector as a quantity having both magnitude and
direction is quite familiar, as is the parallelogram rule for vector addition.
If A is any vector, and AI, A 2 , and A 3 are the projections of this vector

361

362 APPENDIX

along the x, y, and z axes, respectively, A may be written as the vector
sum

II-l

It is convenient to define a set of unit vectors i, j, k, which are vectors of
unit length directed along the x t y, and z axes, respectively, of a rectangu-
lar coordinate system. If we now let A x be the magnitude of AI, we
have AI = iA x , etc., so that, in terms of the unit vectors and the com-
ponents A X) A y , A z we have

A = iA x + jA v + bA g II-2

There are two types of products of vectors, the scalar product and the
Vector product. If \A\ and \B\ are the absolute magnitudes of the two
vectors A and B, and 6 is the angle between them, then the scalar prod-
uct A B (" A dot B ") is defined as

AB = AB cos0 11-3

In terms of the components of A and B, the scalar product is

A B = A X B X + A v By + A Z B Z 114

As an example of the use of the scalar product, consider the motion of a
particle through a small distance ds, subjected to a force F which makes
an angle with the direction of motion. The work done by the force is
the product of the component of the force along the direction of motion
times the distance through which the particle moves, or, in vector nota-
tion, dW = F ds.

The vector product A*B (" A cross B ") is a vector C perpendicular
to the plane of A and B. If A is rotated into B through an angle less
than 180, the direction of C is that in which a right-handed screw would
move if given a similar rotation. The magnitude of C is defined to be
|(7J = \A\ \B\ sin 9, where 6 is the angle between A and B. In terms of
the components of A and B the vector product is

A*B - i(A y B z - A z By) + J(A Z B X - A X B Z ) + k(A x B y - A V B X )

i J k
A x A y A z

B X By Bg

From the definition of the direction of C = A*B, it is obvious that
(A*B) = (B x A). Angular momenta can be quite simply expressed in
terms of the vector product. If r is the vector distance from a fixed
point to a particle of mass m moving with the velocity v, it can readily be
seen from the definition of the vector product that the angular momen-
tum of the particle relative to the fixed point is r*wv = w(r*v),

THE OPERATOR V 2

363

Many physical laws assume a particularly simple form when written
in terms of the vector differential operators. We define a vector opera-
tor V (" del "), which, in rectangular coordinates, is

V = i~+j~+k4 II-6

Operating on a scalar function tp, this operator generates a vector called
the " gradient " of (p:

dx

dy

dz

II-7

The scalar product of V and a vector A gives a scalar function known as
the " divergence " of A:

dx

dy

dz

II-8

The vector product of V and a vector A gives a new vector called the
" curl" of A:

k

dx

A*

j

dy dz

-"L A-Z

II-9

Of particular interest in quantum mechanics is the operator formed by
taking the scalar product of V with itself:

VVsV 2 ==-^2 + ^0 + ^-2 11-10

dx 2 dy 2 dz 2

The operator V 2 (" del squared ") is usually called the Laplacian
operator.

IU. THE OPERATOR V 2 IN GENERALIZED COORDINATES

In rectangular coordinates, the operator V 2 is given by the relation

d 2 d 2 d 2

V = H 5 H o Because of the frequent occurrence of this

dx 2 By 2 dz 2

operator in quantum mechanics, we will show by means of a physical
interpretation how this operator can be transformed to coordinate
systems which are not rectangular. The only coordinate systems that
we shall consider are those in which the three coordinate surfaces cut
each other at right angles. Such coordinates are known as orthogonal
coordinates. Let these othogonal coordinates be <ji, #3, #3. The dis-

364 APPENDIX

tance ds\ perpendicular to the surface q\ = constant, between the two
points (gi, g 2 , g 3 ) and (gi + dqi, q 2 , ffs), will not in general be equal to
dqi but may be written as ds 1 = Mgi, where hi may be a function of
the coordinates. The distance between the points (q\, q 2 , #3) and
(qi + dq\, #2 + dq 2 , #3 + dgs) will therefore, since the coordinate sur-
faces are perpendicular, be

(ds) 2 = (<fei) 2 + (<fe 2 ) 2 + W 2

III-l

Comparing this with the corresponding expression in rectangular
coordinates

(da) 2 - (dx) 2 + (dyY + (dzf

fdx , , dx dx \ 2

= I dqi + dq 2 + dq z 1
\dqi dq2 d<}3 /

in-2

V^ffi ~ ^2 ^gs /

we find

dx \ . i vy 1 t | v ~ 1 _

the other coefficients being necessarily equal to zero.

Now suppose that the coordinate space is completely filled with a fluid
whose density at the point (#1, q 2 , #3) is p(q\, q 2 , tfs)- Suppose further
that the motion of the fluid at any point is determined by a velocity
potential V (qi, q 2 , g 3 ) such that the velocity in any direction is given by

dV

the value , where ds is a displacement in the given direction. Let

ds

us calculate the rate of accumulation of fluid in a small element of volume
bounded by the surf aces q\ = g?, gi == g? + dq\] q 2 = g,g2 =* <& + dq 2 ]
?3 - Qz, 2s ^ ffa + d 3 (Figure III-l). Since the sides of the volume
element are infinitesimal, we may assume that V is constant over any
one surface. Now consider the surface qi = g?. The rate of flow

THE OPERATOR V 2 365

perpendicular to this surface is

_g = _I^ m . 4

and the area of the surface is h 2 h 3 dq 2 dq$. Hence the amount of fluid
flowing through this side in unit time is

h 2 h 3 dVj J

/ dF

I "" T"

\ asi

The rate at which fluid is flowing in through the other side is
V , 7 , d \ h 2 h 3 dV} J J J

p

m-a

IH-6

FIG. III-l.
so that the rate of accumulation of fluid due to this pair of surfaces is

toi dq 2 da* III-7

i
The other two pairs of surfaces contribute

dq 2
and

hih 2 dV

\p | dqi dq 2 dq 3 III-8

dqi dq 2 dq$ III-9

so that the total rate of accumulation is

HMO

366 APPENDIX

The rate of accumulation of fluid per unit volume at any point is just the
rate of ch'ange of density at that point. Hence, since the element of
volume is

dr = dsi ds 2 ds^ = hih 2 h% dq\ dq 2 dq% III* 11

the rate of change of density is

dp__l _ r~_d_/ hJ^dV\ d_i hih 3 dV\
dt "" hih 2 h 3 Ldqi \ P hi dqj dq 2 \ h 2 dqj

m, 2

If we carry out the same analysis in rectangular coordinates, we find

= ( p ) H ( p 1 H ( p I

dt dx\ dx/ dy V dy/ dz \ dz /

= V (pVF) IIL13

Since the value of is independent of the particular coordinate system
dt

we use, we conclude that, in general,

* f w j r d ( h ^ dv \ , s ( h ^ dv \

V (pVK) = 7-7-7" I P "7 " I + T~"l P ~; )

hih 2 ri3 [_dqi \ hi dqi/ dq 2 \ h 2 dq 2 /

h ^~\] m-14

Taking now the special case p = constant, we find the expression for the
operator V V s V 2 to be

V 2 = 1 T fl fhfa d \ + d /hih s d \
hih 2 h 3 [_dqi \ hi dqj dq 2 \ h 2 dqj

+ (ia JLYI m . 16

dqs \ h z dqJl
In particular, if we use rectangular coordinates, equation III45 becomes

IIL16

In rectangular coordinates, the operator for the kinetic energy of
particle is

THE OPERATOR V 2

367

We thus conclude that, in generalized coordinates, the operator for the
kinetic energy of a particle is

T - - r4- V 2 111-18

where V 2 is given by equation III45. Equation' 111*18 may also be
verified by a direct transformation of III- 17.

We give below a few of the most important coordinate systems,
including the equations for the transformation from rectangular coordi-
nates, the explicit expression for V 2 , and the value of the volume element

1. Polar Coordinates.

x = r sin 6 cos <p y = r sin 8 sin <p z = r cos

1 d 2

dr = r 2 sin 6 dr dd d<p
2. Cylindrical Coordinates.

x = r cos y = r sin
1 d / d'

r 2 sin 2

z =i

(0,0, -a)

(0,0,a)

Fia. III-2.

3. Elliptical Coordinates (Figure III-2). (<p measured from xz
plane.)

x =

C "

- v) cos

- ,, 2 ) sin

r a r b

R
(R - 2a)

368

APPENDIX

a 2

R 3

dr - G* 2

o

d/i dv d(p

IV. DETERMINANTS AND THE SOLUTION OF
SIMULTANEOUS LINEAR EQUATIONS

A determinant, it will be recalled, is defined by the equation

011

012

013

' 01n

021

022

023

* * * 0*271

031

032

033

* * * Qln

'

0n2

0n3

' * ' 0nn

=- L (-

0nn) IV-1

where P,, is the operator which permutes the second subscripts and
v is the number of interchanges of pairs of subscripts involved in P v .
From the definition, it is readily seen that the determinant vanishes if
two rows or two columns are identical, since interchange of two identical
rows or columns changes the sign of the determinant but leaves its value
unchanged. If we denote the above determinant by A, we can define a
^et of numbers A t - m by the relation

n
A = 2J a im^im IV-2

n

From this definition, it is obvious that ]T <*ikAim = for k 7* m, since,
if a;* = a im for all i, the determinant is identically zero, and, by IV-2,

n

]C Oifc^im is the determinant which has a^ = di m for all i.
If we have a set of simultaneous equations

"T~ 012*^2 i"~ 013*^3 "i "-p- CL^fiXfi ** Cj

+ 022^2 + ftza^a + + Oan^n = Cj

IV-3

a nn a: n c n

THE EXPANSION OP l/r< ; 369

we can obtain a solution in the following manner. We multiply the first
equation by AH, the second by ^21, etc., and add, obtaining

' ' ' + E CLinAnXn = * CjAji IV4
-l t-1 <-l *

n n

But 53 a n AH = A and EamA t i = Oform 5* 1, so that

or #1 = - - IV'5

A

The quantity Ec*Aii is just the determinant in which a^ has been

i

replaced by c. The values of the other x's are found in exactly the same
manner. The general result is

* = 1^- IV.6

A solution exists only if A ^ 0, or, if A = 0, a solution exists only if all
the c's are zero. Conversely, we may say that, if all the c's are zero,
then a solution (aside from the trivial solution x\ = x% = = x n == 0)
exists only if A =0. This last statement is the one of particular impor-
tance in quantum mechanics.

V. THE EXPANSION OF

*V

In terms of the distances r and r ; - and the angle 7 between the vectors
from the origin to the two particles, the distance r t -j between the two
particles is

nj = Vr? + rf 2r t r ,- cos 7 V-l

If we let r> be the greater of n and r,-, and r< be the lesser, then

r7 r>Vl + a; 2 - 2x cos 7 V-2

T 1

where a? = . Let us now look for an expansion of of the type

(cos 7) V-3

x
TV/ r> Vl + a; 2 2x cos 7 **> n

If we square both sides of this expansion, multiply by sin 7 dy, and inte-

370 APPENDIX

grate over 7 from 7 = to 7 = TT, we obtain, by 4-53, the result

But, by an expansion, we have

ifc.O+'J.z; *

x (1 x) n 2n + 1

nclude that
P n (cos 7) is

so that we conclude that a n = x n , and the expansion of in terms of

- = - x p n ( COS T ) V6

Tij T > n

We must now express P n (cos 7) as a function of the 0's and <p's of the two
particles. Considering this as a function of 0; and <pt, we may expand it
in terms of the orthogonal functions P\ m \ (cos t -) e imvi . P n (cos 7) is a
solution of the equation

sin 0;

since this equation remains unchanged under any rotation. The general
solution of this equation is a linear combination of the functions
Pj^' (cos 0i) e imvi , so that we may express P n (cos 7) as

where, by 4-53,

2nH

V-8

(cos 7) Pr 1 (cos 0,) e-*< dr V-9

4?r (n + m

\ * I i ,

We also can expand Pjj"' (cos 0) e~ imvi in t^rms of the functions
(cos 7) e ik * as

&+n

PJT 1 (cos 0) e"" l ' mv> = L BnfcPlf 1 (cos 7) e** V-10

where

(cose,-) **"* (COST) e- <ft "dr V-ll

Equation V-10 must hold for 7 = 0, that is, for 0< = 0,-, < = *>j- Then
PJT 1 (cos ,-) e- im ^ = ESniPjf'a) e'' fc " = B n0 P2(l) V-12

ORTHOGONALITY RELATIONS 371

since F*(l) = for fc ^ 0. P(l) = 1; from V-12 and V-ll we have

o/i i i /*

/ PlT 1 (cos 0,-) e-*"*P n (cos 7) dr = P!?' (cos 0,-) a-*"** V-13

TiTT t/

so that

^-^THy^^-""' v ' 14

Combining V-6, V-8, and V-14, we have the final result

1 5 m =+n ( n ___ | m |\ f n

-=E L . ; -4i H" 1 ' (COSffi) PJT 1 (COS gy) e*"<*^> V-15

r iV n=0m=-n ( n ~T \>\) ^>

Equation V-15 may alternatively be expressed as

i o tw = +n O y n

-=L L r rT-uzieL" l| ( < )e!, m| (^) <m( ' rnf ) V-ie

T\j n=0m=-n^ n "T 1 ^>
i oo tn=-fn j_ r n

Tii . 0w =l2n + 1 r^ 1 ^

VI. PROOF OF THE ORTHOGONALITY RELATIONS

The most important single theorem in group theory is that giving the
orthogonality relation between the irreducible matrix representations
of any group. As stated in Chapter 10, this theorem is

^(awJ-M-^ VI-l

where Z and lj are the dimensions of the representations and h is the
order of the group. We wish to give a simple proof of this theorem,
following essentially the method of Speiser. 1 Before we can begin the
actual proof of the orthogonality relations, we need several preliminary
theorems.

THEOREM 1. If we have two sets of variables x( x' n and y{ y f m ,
then every bilinear form

n m
f ^ YV--r'?/ VT-2

J '--' / -* ^IJ^i yj v JL a

of these variables can be reduced to the normal form

/ - E ** VI-3

1 A. Speiser, Theorie der Guppen von endlicher Ordnung, Springer, Berlin, 1927.

372 APPENDIX

where r < n, r < m, by a suitable linear transformation of the variables
x't and t/y .

Proof. The product

in m

"" '^ "^ '^ VI-4

contain all the terms in / which involve either x( or y[. If we make the
substitutions

1 n ./

VI-5

we may write / as

n m

/ I Y" 1 y /7..f'?/' VTfi

<-2S2

We can, without loss of generality, assume that n < m. After (n 1)
substitutions of the type VI-5, we will have obtained the result

f I i ^ I I <<r- /r'lf' VT7

We now make the final substitutions

m
= f __ y / 77. = 77' ^7 *> W^

Equation VI-7 then reduces to

f _ y r^/- VT-R

y / , *ui,y\ A o

which is the desired result. As the determinants of the transformations
on the o/'s and y''s are different from zero, the transformations have
reciprocals. In certain special cases the normal form VI-8 may co'ntain
less than n terms.

Let us now take a set of variables x[ x f n which form a basis for an
irreducible representation IV of a group. We also take a set of vari-
ables y( y' m which form a basis for an irreducible representation T y ,
of the same group. We have then

THEOREM 2. If IV and T yf are two irreducible representations of a
group there is no bilinear form of the variables x( and 2/y which is always
invariant when both the x\ and the y'j are subjected to some operation R
of the group unless IV is identical with iy .

ORTHOGONALITY RELATIONS 373

Proof. We shall prove this theorem only for the type of groups in
which we have been interested, namely, those representing transforma-
tions of coordinates. The corresponding matrix representations involve
only unitary matrices; for simplicity, we will assume that the matrices
are real. According to Theorem 1, any bilinear form

/- ZE^X.*/; VI-Q

-iy-i
can be reduced to the form

VMO

k-1

by a suitable transformation. We consider that the matrices of the
representations IV and Y y > have been subjected to the same transfor-
mation, so that we have obtained the corresponding new representations
T x and T y which have the x's and y'& as their bases. We now require
that / be invariant when both the x's and the y's have been operated on
by some operation R of the group.

If we operate on the x's only in equation VI- 10, we have

f VI-12

3=1 =1 -l

Arranging this according to the x's we have

/ = XiJ:T x (R) lk y k + + x n J:T x (R) nk y k VMS

A; = l *-l

When we operate on the y's, equation VMS must reduce to equa-
tion VI- 10. This requires that

yi t =
which is equivalent to the requirement that

or

B~ l Vi - g r*OR)ay* < - 1 r VM4

374 APPENDIX

For the real unitary matrix representations we have been considering,
the matrix of the inverse transformation is obtained from the original
matrix by simply interchanging rows and columns. By definition,
therefore

BT l y< = L r y (/J)*ifr i = I m VI-15

*-i

Comparing VI-14 and VI-15, we see that VI40 will be invariant only if
m = r. By interchanging the order of operations, we could prove in
the same way that VI- 10 will be invariant only if n = r. Again com-
paring VI45 and VI14, we see that VI- 10 is invariant only if
Tx(R)ik T v (R)ik for i, k = 1 r. Equation VL10 is, therefore,
invariant only if T x and T y are identical.

We now have the necessary background for the proof of the orthogo-
nality relations. Taking the sets of functions Xi x n , yi y ny which
are the bases for the irreducible representations T x and F y , we have

THEOREMS. If T x is not identical with T v , then

=

for all values of i, j, fc, I.

Proof. Referring to the definition of the direct product given in
Chapter 10, we see that the mn functions x 8 y t form a basis for a repre-
sentation F^r^ of the group of dimension mn. If we denote these mn
functions by 2i z r (r mn) and the corresponding representation
by T z = T X T V , then the r 2 matrix elements of T 9 (R) are T x (R)ijT v (R) k i
(i, j = 1 ft; fc, I = 1 m). If we now operate on z s by one of the
operations R of the group, we have

VI-16
Summing over all the operations R of the group, we have

VM7

R R tl

Let A be any operation of the group. Then

Af = EARz, = ERz 8 /

R R

since AR is always an operation of the group and the operation by A
merely changes the order of the summation. Now / is a linear form of
the z's, tod hence a bilinear form of the z's and y's. But we have just
seen that there is no such form / which is invariant under an operation
of the group. Since Af = /, we must, therefore, conclude that / is

ORTHOGONALITY RELATIONS 375

identically zero. This can be true only if r,(12)< is identically zero

R

for all values of t and s. Referring to "the correlation between the ele-
ments of r 2 CR) and those of T X (R) and T V (R) 9 we thus see that the
theorem is true.

Now taking the sets of functions x\ x r , yi y r , which are bases
for the representation r, and noting that according to Theorem 2 the
function

i
f =

is invariant under operations of the group, we can prove
THEOREM 4.

w = o (tj) * (mn)

where h is the order of the group and I is the dimension of the

representation.

i
Proof. The function / = ]C#A#fc is invariant. We operate on / with

some operation JK, sum over all operations of the group, obtaining

ft/ VMS

R R A=1U-1

The coefficient of x 8 y t must vanish if s ^ t, so we immediately have the
relation

Er(/2). jk r(JB) fc = s ^ VM9

Now H-fiT" 1 / = H^f, since each operation is contained once and only
R R

once in both summations. Recalling that the inverse matrices are
obtained by interchanging rows and columns in the original matrix,
we have

Z/?/ = EJT 1 / = E E L r(R)k*r(R)ktXsyt

R R R fc=l#l <=1

which gives us immediately the relation

= o a ?* t vi-20

R

Equation VI- 18 is thus reduced to

- E Z Zr(B)/r() / n!^ - V vi-21

376 APPENDIX

This requires that

Z Er(fl)#r(fl)y h (j = 1 Z) Vl-22

# *-l

Considering the inverse transformation, we also obtain

r(/2)*/r(fi)*/ = A (j - i Z) Vi-23

5 Jkl

From VI-22 and VI-23, we see that Y,T(R) jk T(R) jk is independent of

R

both k and j. Therefore

Combining the results of Theorems 3 and 4, we have the desired relation

S r<(fl)m i r y (fi)*/n' jf = dij8 mn >d nn , VI-24

/l

VII. THE SYMMETRY GROUPS AND THEIR
CHARACTER TABLES

Most simple molecules will have a certain degree of symmetry; that
is, there will be certain transformations of coordinates which leave the
atoms of the molecule in a configuration in space which is indistinguish-
able from the former configuration. The possible transformations of
this type will be either rotation about an axis of symmetry, reflection
in a plane of symmetry, inversion in a center of symmetry, or various
combinations of these transformations. If two such transformations
are carried out successively, the configuration will be that which could
be obtained from some other transformation. The set of transforma-
tions which do not alter the configuration of the atoms in a molecule
thus form a group, the group of the symmetry operations for the mole-
cule. We include in this appendix the character tables for most of the
symmetry groups which are likely to occur in problems of molecular
structure. 1 * 2 ' 3

The notation used for the operators which transform a symmetrical
molecule into itself is the following:

E = the identity operation, which leaves each particle in its original
position.

1 J. Rosenthal and G. Murphy, Rev. Modern Phys., 8, 317 (1936).
8 H. Sponer and E. Teller, Rev. Modern Phya., 13, 75 (1941).
8 E. Mulliken, Phys. Rev. t 43, 279 (1933).

THE SYMMETRY GROUPS 377

2ir
C n = rotation about an axis of symmetry by an angle In protn

Tl

lems of molecular structure, the values of n which will be of particular
interest are n = 1 (no symmetry axis), 2, 3, 4, 5, 6, and oo . (n = 7, 8,
etc., are possible values but probably occur only occasionally; these
groups are not included in the tables.)

<r = reflection in a plane of symmetry. The symmetry planes are
further classified as follows. If the plane is perpendicular to the princi-
pal axis of symmetry (the axis with the largest value of n), reflection in
this plane is denoted by a h . If the plane contains the principal axis,
reflection in this plane is denoted by <r v . If there are axes with n = 2
perpendicular to the principal axis, and if the plane contains the princi-
pal axis and bisects the angle between two of these 2-fold axes, reflection
in this plane is designated by o- d .

2ir

S n = rotation about an axis by followed by a reflection in a plane

?i

perpendicular to the axis of rotation.

i = inversion in a center of symmetry.

The symmetry groups are designated by the following notation. We
may divide them into three main types:

I. The Rotation Groups. These are symmetry groups which have
one symmetry axis which is of a higher degree than any other symmetry
axis. This axis is considered to be the z coordinate axis. The following
groups belong to this general class :

A. The molecule possesses an axis of symmetry only. The groups
of interest are Ci, C 2 , C 3 , C 4 , C 5 , C 6 . The possible operations are found
as follows. For the group C 6 , with a 6-fold axis, all powers of Ce are
likewise symmetry operations. We thus have the operations

C/nr2 /nr /nr3 /nr /nr4 />2 /nf5
6, O 6 ~ O 3 , G 6 O 2 , O 6 G 3 , O 6

B. The molecule has the symmetry operations C n and <r v . The
manner in which the operations are found will be discussed later. The
possible groups are C 2r , C 3v , C 4t) , C 5v , C 6v .

C. The molecule has the symmetry operations C n and o^. The
possible groups are Cu(C t ) , C 2 *, C 3 *, C 4A , C &h , C Qh .

D. The molecule has the symmetry operation S n . The possible
groups are 82 (CO, S 4 , S 6 (C 3t -).

E. The molecule has 2n 2-fold axes perpendicular to the principal
n-fold axis. These axes are denoted by C", C^, etc. The notation for
this type of group is D n . The possible groups are D 2 (V) , D 3 , D 4 , D 5 , D 6 .

F. The molecule has the symmetry operations D n and <r<j. The
possible groups of interest are D 2< *(V d ), D 3d .

378 APPENDIX

G. The molecule has the symmetry operations D n , v d , and cr h .
The possible groups are D 2 *(VjO, D 3A , D 4A , DSA, I>6A-

A number of the above groups can be expressed quite simply as the
combination of some other group of symmetry operations plus the
inversion i. These are

S 6 =

II. Groups of Higher Symmetry. These are groups which have no
unique axis of high symmetry but which have more than one n-fold
axis where n > 2. The groups of interest are

T = the group of operations which sends a regular tetrahedron into
itself. T h - T*f.

O = the set of operations which sends a cube or a regular octahedron
into itself. O^ = O*i.

Id = the symmetry group of CH.

III. Groups with the Symmetry Operation Coo.

Co,,, = the group of the symmetry operations of a heteronuclear
diatomic molecule.

DOOA = the group of the symmetry operations of a homonuclear
diatomic molecule.

The symmetry operations for the rotation groups can most easily be
found by means of the stereographic projection diagrams. For the
group C 4v , this takes the form of Figure VII- 1. The square in the center
represents the 4-fold axis. Starting with point 1, where the + indicates
that it is above the plane of the paper, we can ob-
tain the points 3, 5, and 7 by applying the opera-
tions C 4 , C 4 , and C\ to this point. Applying the
Cit, operation <r v to point 1 we obtain point 2; from
this point we obtain the points 4, 6, and 8 by the
rotations. These eight points are all that can
be obtained from a single point by any combina-
Fio. VIM. Stereographic tion of C 4 and cr v . From the diagram, we see
projection diagram f qr the that thig get of points algo possesses the sym-
group 4i>. metry elements <r<j, where ad represents a reflec-

tion in a plane which bisects the angle between the original <r v planes
of symmetry^

Diagrams of this type are given in Figure VII-2 for all the rotation
groups. The principal n-fold axis is represented by the n-sided shaded

THE SYMMETRY GROUPS

379

i. c,

4. C 4

C 6

11.

12. .

1/1

13.

14.

'}

18.

19.

20.

'^rV

^-^^-^

*3

21.

22.

23.

24.

25.

'th

29.

'9k

'M

32.

FIG. VII 2. Stereographic projection diagrams for the simple point groups.

380 APPENDIX

figure in the center of the diagrams. S n axes are represented by open
n-sided figures. If the symmetry operation a^ is present, the large circle
is full; otherwise the large circle is dotted. The planes cr v and ov* are
denoted by full lines. Two-fold axes perpendicular to the principal
axis are denoted by the symbol for a 2-fold axis placed at the ends of the
line through the center of the circle. The symbol + represents a point
above the plane of the paper; the symbol O represents a point below
the plane of the paper. All these points can be obtained from any one
of them by application of the various symmetry operations. As the
exact significance of the various symmetry groups can be better seen by
means of examples, in Table VII4 we show molecules belonging to the
various symmetry groups.

In the character tables, the operations of the various groups are
collected into classes. For example, the group C 3t , has three classes:
the class of E, the class of the two S-foM axes, and the class of the three
planes of symmetry. This may readily be verified from the definition
of a class and the stereographic projection diagrams.

The characters of the irreducible representations of the groups of
most frequent occurrence are integers. For certain of the groups of low
symmetry, particularly the C n groups, complex characters occur.
When they do, the irreducible representations can be taken in pairs, the
characters of one member being the complex conjugates of the corre-
sponding characters of the other member of the pair. Such a pair of
representations is essentially equivalent to a single doubly degenerate
representation. 3 When applying the orthogonality rules to such
representations, the rule

must be replaced by

R

M

etc. Also, it is to be recalled that, if co = e 6 , for example, then
1 + co + co 2 + w 3 + w 4 + co 5 =

Non-degenerate representations are designated by the letters A and B.
The A' & are symmetrical and the B'a are antisymmetrical with respect to
rotation about the principal axis of symmetry, that is, about the z axis.
Doubly degenerate representations are denoted by E and triply degener-
ate representations by T. The characters of the groups of the type
D0A = De^' are not given explicitly. They may readily be found in the
manner illustrated by Table 14-7 for the group D 6 &. These representa-

THE SYMMETRY GROUPS 381

tions are denoted by the symbols A\ and AI U , etc., the g representations
being symmetrical and the u representations being antisymmetrical with
respect to inversion. The transformation properties of the coordinates
x, y, z, the products of coordinates x 2 , 2/ 2 , z 2 , xy, xz, yz, and the rotations
R Xy R y , R Z) about the x, y, z axes, respectively, are also given. If the
operation i is present, the coordinates belong to the u representations
while the rotations and products of coordinates belong to the g represen-
tations, as may be seen from Table 14*7.

TABLE VIM
EXAMPLES OP MOLECULES BELONGING TO VARIOUS SYMMETRY GROUPS*

DOO* H 2 , 2 , C 2 H 2 , CO 2

Coo,, CO, HC1

C 2v H 2 O, SO 2 , H 2 S,

C 3w NH 3 , PC1 3> CH 3 Ci

D 3d " Staggered " C 2 H 6

D 3 A " Eclipsed " C 2 H 6

DeA C 6 H 6

T d CH 4

A SF fl

*See also E. B. Wilson, /. Chem. Phys. t 8, 432 (1934).

The characters themselves are easily found for any group by applica-
tion of the four rules given in Chapter X. It may perhaps be of value
to determine the transformation properties of the coordinates and rota-
tions in several instances. For C 2v , for example, the various operations
on the vector r, with components x, y, z, give us the matrix equations

1 1 o o\/zN

= -1 1 y
1/W

'-1 0'
1
.0 01

<r v = reflection in xz plane.
<T f 9 = reflection in yz plane.

Each coordinate therefore belongs to an irreducible representation,
x to J5i, y to JS 2 , and ztoAi. The products of the coordinates belong to
the representation of the direct product of the irreducible representa-
tions involved; for example, xy belongs to BiB 2 - A 2 . The various

382

APPENDIX

rotations may be indicated by curved arrows as indicated in Fig-
ure VII-3. Operating on these arrows with the symmetry operations,
we have

/R x \ I 1 0\/#A IR\ /-I 0\/R X \

#(#] = ( 1 OllRy] <r v [Ry ) = [ 1 }[Ry)

\R Z / \ 1/W W \ -1/W

/R x \ /-I 0\/R X \ /R x \ / 1 OWflA

Cal/Zjsl -1 ][Ry] *', [RV} = \ ~1 )[R y ]

\Rj \ 1/W W \ -1/W

so that R x belongs to B^ R y to B\, and R z to A%.

FIG. VII-3*

When we have degenerate representations the procedure is sometimes
less simple. For example, for D 3 &, we have the matrices

E

/I 0\
(0 1 0]

\0 I/

^1

{o

1

\
-J

1

1

2

/

x/i
2

\

Vi

2

~2

I/

/

1

V3

\

o

2

2

v

1

__

o

2

~2

\

-I/

fl

\

-1

o

(o

-I/

<r,

/I 0\
[0 -1 0)
\0 I/

THE SYMMETRY GROUPS

383

We thus see that z belongs to the non-degenerate representation A",
and that the pair of coordinates (x, y) belongs to the degenerate repre-
sentation E f . The product z 2 belongs to A* A* = A(, the pair (xz, yz)
belongs to A'z'E' = E". The products x 2 , y 2 , and xy must belong to
E'E f = E' + A( + A' 2 . If we consider any vector of unit length, then
x 2 + y 2 + z 2 = 1. Since z 2 belongs to A[, the combination x 2 + y 2
must belong to A[. This leaves us xy and the combination x 2 y 2
to be assigned to E f + A' 2 \ we conclude that the pair (x 2 y 2 , xy)
belongs to E'. These conclusions may be verified by working out the
transformation matrices for the vector with components x 2 + y 2 ,
x 2 y 2 y xy. In a similar manner, we find that R z belongs to A& the
pair of rotations (R x , R y ) belongs to E".

2.

Ci E

A 1

C 2

E C 2

x 2 , y 2 , z 2 , xy

R> z

A

1 1

xz,yz

\

Rxt $y J

B

1 -1

c,

E C 8 C|

x* + y\ z*

ft,*

A

111

(xz, yz) \
(jcP ^2^ xy)\

fo y) \

'(

1 w w 2 (w = e^)

1 ft) 2 W

C 4

E C 2 C 4 Cl

x*+y\z*

Rz

A

1111

y? - j/ 2 , y

B

1 1-1-1

(xz, yz)

(*, y) \
(R*,R V ) ]

'{

1 -1 i -i
1 -1 -t t

5.

C 8

JS? C 5 C| Cf Cj

* + y*, z 3

&, )

^'(

11111

1 ft) ft) 2 ft) 8 ft) 4 2_rl
1 ft) 4 ft) 8 ft) 2 ft) (ft) 5 )

1 ft) 2 ft) 4 ft) ft) 8
1 ft) 8 ft) ft) 4 ft) 2

384

APPENDIX

C 6

JJT /"Y /If ft f*W> /""*

El Ug 03 v/2 ^o ^6

* 2 + y\ z*

R*,z

A

111 11 1

B

1-11 -11 -1

(xz, yz)

(x, y) 1

(R X ,R V ) I

E' |

1 CO CO 2 CO 8 CO 4 CO 6 (CO = "*")
1 CO CO CO CO CO

(x 2 - y\ xy)

T?" }

E (

1 co 2 co 4 1 co 2 co 4

1 CO 4 CO 2 1 CO 4 CO 2

C 2v

E Cz <TV 0*

999

* 2 , 2/ 2 , 2 2

2

A!

1111

xy

R,

A t

1 1-1-1

xz

R v ,x

Bi

1-1 1-1

yz

R x ,y

B 2

1-1-1 1

7.

C 3v

E 2C 3 3ff v

x 2 + y\ z*

z

AI

1 1 1

R*

A 2

1 1-1

(x*-y\xy}\
(xz, yz) }

(*, v) 1

(RX.RV) /

E

2-10

c*

E Ct 2C 4 2<r v 2od

x 2 + y\ z*

z

Al

11111

R.

A t

11 1-1-1

z 2 -^

B!

11-1 1-1

xy

B 2

11-1-1 1

(xz, yz)

(*, v) \

(R*,Ry) /

E

2-2

9.

c*

ET O/^ O/^2 e?_
^v ZL/5 ^SO5 Odt,

:r;^,

(x, y) \

"2

Jb 1

11 11

11 1-1 ,-2=

5
2 2 cos z 2 cos 2z
2 2 cos 2rr 2 cos 4z

10.

THE SYMMETRY GROUPS

385

11.

12.

13.

14.

Co,

E C 2 2C 8 2C 6 3<r d 3<r,

x* + V \ z*

z

Ai

111111

R.

At

1 1 1 1-1-1

Bi

1-1 1-1-1 1

B*

1-1 1-1 1-1

(xz, yz)

(x, V) 1
(R*, ,) }

Ei

2-2-1 1

(*' - y\ xy)

E%

2 2-1-1

Cu

E cr h

s 2 , y* } z z , xy
xz t yz

Rz t x, y

R x , RV, Z

A'
A"

1 1
1 -1

c*

E (/2 &h 1

x 2 , y 2 , z 2 , xy
xz t yz

z

RX, Ry

A\t

B

1111
1 1-1-1
1-1-1 1
1-1 1-1

CSA = C3 x <Tfc

TJI /-y x-y2 or / /nr2\
& 03 C/3 ffh 03 wA^3/

x 2 + y\ z 2

R*

A'

111 111

z

A"

11 1 -1-1-1

(x 2 - y 2 , xy)

(3, y)

E'l

12 -g 2 2irt
W 0) 1 W CO ~

co Ct) L (i) 0)

(xz, yz)

fT% "D \
\K X , Ky)

B"l

1 O) O) 1 O) CO

\

1 to 2 co 1 -co 2 co

16.
16.
17.

18.

S-t

E i

XX t/2

#a -^yi /2f

A Q

1 1

M/C/i yl

X, /, 3

A

1 -1

386

APPENDIX

84

E C 2 S 4 Si

* 2 +j/ 2 ,z 2

R,
z

A
B

1111
1 1-1-1

(as, yz) 1
(x* - y\ xy)l

(x, V) \
(, #y) j

'{

1-1 i -i
1 -1 -t i

19.

20.

S 6

21.

D 2

fit rii fiU fiX

xz Ry, y #2

1111
1 1-1-1
1-1 1-1
1-1-1 1

D 3

E 2(70 3C/2

Rg, z A*

(xz, yz) } (x, y) \ E

1 1 1
1 1-1

2-10

D 4

E Ci 2(?4 2(7' SC'o'

(xz, yz)

(x, y) }

Ax

E

11111
1 1 1-1-1
1 1-1 1-1
1 1-1-1 1

2-2000

23.

24.

D S

* 2C 6 2CJ 5Ci

5^

(, y) \

^A 1

A 2

11 11
11 1-1

/ 2A

2 2cos* 2cos2 ( x == )

2 2cos2 2cos4x

THE SYMMETRY GROUPS

387

D

E C 2 2C 8 2C 6 3C 2 3C 2 '

z 2 + V 2 , 2

R., t

A!

A t

BI

R

111111
1 1 1 1-1-1
1-1 1-1 1-1

(xt,yz)
(** - V*, *)

(x, y) \

(RX.RV) /

-02
#1
J0 2

2-2-1 1
2 2-1-1

25.

D M

E C 2 25 4 2Ci 2v d

x s + y\ z*

AI

11111

R,

A*

1 1 1-1-1

x*-v*

Bi

1 1-1 1-1

xy

z

B*

1 1-1-1 1

(xg, yz)

(*, y) \
(R*,R V ) )

E

2-2000

26.

27.

28.

D 8 *i

D s = DjXffjk

E a h 2C 8 25 3 3C 2 3o- 9

x 2 + y 2 , z 2

^J

111111

<

fi.

^i

1 1 1 1-1-1

AC

1-1 1-1 1-1

2

A "
^1 2

1-1 1-1-1 1

(x 2 - y 2 , xy)

(*,y)

7'

2 2-1-1

(zz, j/z)

(x, )

E"

2-2-1 1

29.

30. D4& = D 4 x t

31. D = D s v

32. DM = De*i

T

E 3C, 4C, 4Ci

Active

A

1 111

33.

Active

E |

1 1 f (--")

Active

( ' { f

T

3-100

34.

388

APPENDIX

o

E 8C 8 3C 2 6(7 2 6C 4

Active
Inactive
Active

A \
A 2

E

11111
1 1 1-1-1
2-1200

Active
Active

(x,y, z) }

ll

3 0-1-1 +1
3 0-1+1 -1

35.

36.

37.

= O x i

T d

E 8C 3 3C 2 6<r d 6S 4

Active

A l

11111

Inactive

A 2

1 1 1-1-1

Active

E

2-1200

Active

(R x , RV, Rz)

Ti

3 0-1 1-1

Active

fe V, *)

T 2

3 0-1-1 1

38.

CQQV

Tjl O/^" ._
Et &\j <p <f\f

x 2 + y\ z*
(xz, yz)

z

(x, V) 1

Al

1 1 1
1 1 -1

2 2 cos <p

(a; 2 - y\ xy)

E t

2 2 cos 2<?

D k

E 2C V !> i 2iC* iC' 2

x* + V\ * 2
(xz, yz)

Z

/ n n \
\Kx, Ky)

(x,y)

I

11111 1
111-1-1 -1
11-11 1 -1
11-1-1-1 1
2 2 cos v 2 2 cos v?
2 2 cos <p 02 2cos^
2 2 cos 2<p 2 2 cos 2<f>
2 2cos2v> -2 -2cos2v>

39.

/"

Jo

VIII. SOME SPECIAL INTEGRALS

sV"

-s/V-

aj

B da? = --r7, n>~l,a>0
a*"*" 1

GENERAL REFERENCES 389

/*oo 1 /,.

3.

o

4.

.
o 4\o

,./

Jo

/oo

' J. *

/oo

B. / r

Jo

-.

2a

i

2
2o 2

5 da; =
a

10. f xe- ax dx=-j
a

-r

f+l 1

13. I 6-* dx = - (e a - <T)
J 1

a

r+i

14

*+l

/+l i

xe-*dx = - {e a - e~ - a(e a -f e" )}
i a

16. /
J-i

(~a) - A n (a)

IX. GENERAL REFERENCES

A. Classical Mechanics

W. BYERLY, Generalized Codrdinates, Ginn, 1916.
G. Joos, Theoretical Physics, Blackie, 1934.
H. B. PHILLIPS, Vector Analysis, Wiley.

J. SLATER and N. FRANK, Introduction to Theoretical Physics, McGraw-Hill,
1933.

B. Quantum Mechanics

P. DIRAC, Quantum Mechanics, Oxford, 1935.

S. DUSHMAN, Elements of Quantum Mechanics, Wiley, 1938.

H. HELLMANN, Einfuhrung in die Quantenchemie, Deuticke, Leipzig, 1937;

390 APPENDIX

E. KEMBLE, Fundamental Principles of Quantum Mechanics, McGraw-Hill,
1937.

L. PAULINO and E. B. WILSON, Introduction to Quantum Mechanics, McGraw-
Hill, 1935.

V. ROJANSKY, Introductory Quantum Mechanics, Prentice-Hall, 1942.

A. SOMMBRFELD, Atombau und Spektrattinien, Vieweg, Braunschweig, 1939.

C. Atomic Structure and Spectroscopy

E. U. CONDON and G. SHORTLEY, Theory of Atomic Spectra, Cambridge, 1935.
S. DUSHMAN, Chapter II in TAYLOR'S Treatise on Physical Chemistry, Van

Nostrand, 1942.

G. HERZBERG, Atomic Spectra and Atomic Structure, Prentice-Hall, 1937;
L. PAULINO and S. GOUDSMIT, The Structure of Line Spectra, McGraw-Hill,

1930.
H. WHITE, Introduction to Atomic Spectra, McGraw-Hill, 1934.

D. Group Theory

J. E. ROSENTHAL and G. M. MURPHY, " Group Theory and Molecular Vibra-
tions," Rev. Modern Phys., 8, 317 (1936).

A. SPEISER, Theorie der Gruppen von endlicher Ordnung, Springer, Berlin, 1927.
E. WIQNER, Gruppentheorie, Yieweg, Braunschweig, 1931.
E* Molecular Structure and Molecular Spectroscopy

D. DENNISON, " Infra-Red Spectra," Rev. Modem Phys., 12, 175 (1940).
G. GLOCKLER, " The Raman Effect," Rev. Modem Phys., 15, 112 (1943).

G. HERZBERQ, Molecular Spectra and Molecular Structure, Prentice-Hall, 1939.

L. PAULING, Nature of the Chemical Bond, Cornell, 1940.

O. K. RICE, Electronic Structure and Chemical Binding, McGraw-Hill, 1940.

H. SPONER, Molekulspektren, Springer, Berlin, 1936.

H. SPONER and E. TELLER, " Electronic Spectra," Rev. Modern Phys., 13, 75

(1941).
J. H. VAN VLECK and A. SHERMAN, "Quantum Theory of Valence," Rev.

Modern Phys., 7, 174 (1935).

F. Statistical Mechanics and Reaction Rates

R. H. FOWLER, Statistical Mechanics, Cambridge, 1936.

R. H. FOWLER and E. A. GUGGENHEIM, Statistical Thermodynamics, Cambridge,
1939.

S. GLASS-TONE, K. J. LAIDLER, and H. EYRINO, Theory of Rate Processes, McGraw-
Hill, 1941.

R. B. LINDSAY, Physical Statistics, Wiley, 1941.

J. MAYER and M. MAYER, Statistical Mechanics, Wiley, 1940;

R. C. TOLMAN, Principles of Statistical Mechanics, Oxford, 1938.

G. Electric and Magnetic Phenomena, Optical Activity

E. U. CONDON, " Optical Rotatory Power," Rev. Modern Phys., 9, 432 (1937).
R. H. FOWLER and E. A. GUGGENHEIM, Statistical Thermodynamics, Cambridge,

1939, Chapter XIV.
W. KAUZMANN, J. WALTER, and H. EYRING, * Optical Activity," Chem. Revs.,

26, 339 (1940).
J. G. KIRKWOOD, " Polarizability Theory of Optical Activity," J. Chem. Phys.,

5, 479 (1937).
C. P. SMYTH, Dielectric Constant and Molecular Structure, Chemical Catalog

Co., 1931.

E. STONER, Magnetism and Atomic Structure, Dutton, 1926.
J. H. VAN VLECK, Electric and Magnetic Susceptibilities, Oxford, 1932.

INDEX

Activated state, 305
Activation energy, 301
Angular momenta, 39

commutation rules, 41
Angular momentum operators, 42
Angular momentum, total, 134
Antisymmetric eigenf unctions, 130
Associated Legendre equation, 55
Associated Legendre polynomials, 59
Atomic orbitals, 129

order of, 131

radial part, 162
Atomic units, 101

Benzene, bond eigenfunction treatment,
249

molecular orbitals treatment, 254

vibrational spectra of, 280
Black-body radiation, 1
Bohr radius, 83
Bohr's theory of hydrogen, 3
Bond eigenfunctions, 234

for four electrons, 238

for six electrons, 239

matrix elements between, 241

Rumer's theorem, 240
Bose-Einstein statistics, 287

Character of a representation, 182

orthogonality relations, 183
Character tables, 383
Class, definition, 177

number of classes, 182
Commutation rules for angular mo-
menta, 41
Complex atoms, angular momenta of, 133

closed shells, 131

electronic states, 128

energy levels, 132

energy matrix, 143

fine structure, 151

selection rules, 159

"terms "of, 141

vector model, 155

Compton effect, 21
Conjugate variables, 15
Coordinate systems, 367
Coulombic energy, 214

for complex molecules, 247
Coulombic integral, 149
Co valence, normal, 170
Covalent bond, 218
Covalent double bond, 223
Crossing of potential surfaces, 327

Degeneracy, accidental, 186
Degenerate eigenvalues, 96
Degrees of freedom, definition, 9
Depolarization of Raman lines, 279
Determinantal eigenfunctions, 130
Diamagnetism, 349
Diatomic molecules, correlation dia-
grams, 207, 210

electron configurations, 207

electronic states, 190

heteronuclear, 209

homonuclear, 203

nuclear spin, 265

selection rules, 262

symmetry of rotational states, 263

vibrational and rotational energy, 268
Dielectric constants, 337
Dipole moments, 337

matrix element, 112
Direct product, 187
Directed valence, 220, 227

stable bond arrangements, 231

Eigenfunctions, antisymmetric, 130

determinantal, 130

definition, 26

linearly independent, 96

of angular momenta operators, 42

of commuting operators, 34
Eigenvalues, definition, 26

of angular momenta operators, 42

of Hermitian operators, 27
Einstein photoelectric hypothesis, 3

392

INDEX

Einstein transition probabilities, 114
Electromagnetic field, 108
Electron, charge and mass of, 1

dual nature of, 7

magnetic moment of, 125
Electron configurations, and directed
valence, 231

of complex atoms, 128

of diatomic molecules, 207

of the elements, 168

Electron density distribution, by the
Hartree method, 166

in hydrogen, 87
Electron-pair bond, 218
Electron spin, 124

angular momentum, 125

quantum numbers, 125
Electronic states, of complex atoms, 128

of diatomic molecules, 190, 203

of H 2 + ion, 201

of heteronuclear diatomic molecules,
209

of homonuclear diatomic molecules,

205

Electronic structure of complex mole-
cules, 232

coulombic and exchange energies, 247

four-electron problem, 245

London formula, 246

" semi-empirical " method. 246
Eulerian angles, 259
Exchange energy, 214

in complex molecules, 247
Exchange integral, 149
Exclusion principle, 129

Fermi-Dirac statistics, 285
Fine structure, 151
Free particle, 68
Functions, class of, 26

Generalized coordinates, 8, 365
Generalized momenta, 14
Group, Abelian, 177

definition, 176

element of, 177

order of, 177

representation of, 178
Group theory, 172

and quantum mechanics, 184

Group theory, and vibrational spectra

274'

proof of orthogonality relations, 371
Gyromagnetic ratio, 127

Hamiltonian, classical, 14
Hamiltonian operator, 29

for charged particle in electromagnetic
field, 108

in generalized coordinates, 39
Hamilton's equations, 14
Harmonic oscillator, 75
* selection rules, 117, 123
Hartree method, 163
Heitler-London theory, 213
Helium atom, perturbation method, 101

variation method, 104
Hermite equation, 61
Hermite polynomials, 61

normalization of, 63

recursion formula for, 62
Hund's rule, 151
Hybridization, 220
Hydrogen atom, 80

continuous spectrum of, 90

energy levels of, 82, 90

selection rules for, 116
Hydrogen molecule, Heitler-London
treatment, 213

singlet and triplet states, 216

variation treatment, 216
Hydrogen molecule ion, 194

electron density distribution, 198

electronic states, 201

energy levels, 204
Hydrogen quantum numbers, 85
Hydrogen wave functions, 85
Hydrogenlike atoms, 84
Hydrogenlike wave functions, 89

Indicia! equation, 51
Induced dipole moment, 118, 333
Integrals, table of, 388
lonizationpotentials, 168
Irreducible representations, 180

Lagrange's equation, 9
Lagrangian function, 11
Laguerre equation, 66
Laguerre polynomials, 63

INDEX

393

Lande interval rule, 168
Laplacian operator, 363
Legendre equation, 52
Legendre polynomials, 52

recursion formula for, 59
Light, dual nature of, 6
Linear differential equations, 48
Linear equations, simultaneous, 368
Linearly independent eigenfunctions, 96,
Lorentz field, 341

Mathieu's equation, 359

eigenvalues of, 360
Matrices, 172

unitary, 174

Maxwell-Boltzmann statistics, 283
Maxwell's equations, 343
Molecular orbitals," 192

quantum numbers, 203

separated atom notation, 207

symmetry properties, 205

united atom notation, 203
Molecular spectroscopy, 258
Moments, induced, 333
Morse curve, 272
Multiplet structure, 135

Neutron, 1

Non-adiabatic reactions, 326

Normal coordinates, 16

of acetylene, 280

of water, 278
Normalized functions, 31
Nuclear spin and diatomic molecules, 265

Operators, angular momenta, 42

commuting, 25

electron spin, 126

Hamiltonian, 29

Hevmitian, 27

linear, 27

magnetic moment, 127
Optical rotatory power, 342, 346
Orbitals, atomic, 129

bonding, 208

molecular, 192

Orthc^ and paradeuterium, 267
Crtho- and parabydrogen, 266
Orthogonal functions, 31
Overlap integral, 197

Paramagnetism, 348

and electron spin, 349
Particle in a box, 70
Partition functions, 290, 292
Pauli exclusion principle, 129
Penetration of potential barriers, 302
Periodic table, 168
Perturbation theory, 93

for degenerate systems, 96
Perturbations, time dependent, 107
Photoelectric effect, 2
Photon, 3

Physical constants, 361
IT bonds, 224
Planck's constant, 2
Polarized light, 110
Polarizability, 118, 121
Potential energy surfaces, 301
Promotion, 221
Proton, charge and mass, 1

Quantum mechanics, postulates, 29
Quantum theory, old, 5

Radiation, induced, 110

spontaneous, 114
Radiation density, 113
Raman effect, 121, 123
Rayleigh scattering, 121
Reaction rate theory, approximate equar
tion, 307

external forces, 331

general equation, 306

thermodynamics, 330

unimolecular reactions, 308
Reduced mass, 81
Reflection coefficient, 306
Representation of a group, basis of, 185

dimension of, 181

irreducible, 180
Resonance, 248
Resonance energy of benzene, 249, 251,

254

Restricted rotator, 358
Rigid rotate-, 72, 75
Rotational energy, diatomic molecules,

268

Rotatory strength, optical, 345
Russell-Saunders coupling, 155
Rydberg constant, 4

394

INDEX

Scalar potential, 108

Schrddingdr's equations, 24, 25, 29

Screening constants, 163

Secular equation, 98

Selection rules, for complex atoms, 159

for diatomic molecules, 262

for harmonic oscillator, 117

for hydrogen, 116

for vibrational spectra, 276
Self-consistent field, 165
Separation of wave equation, 190, 258
<T bonds, 224

Similarity transformation, 179
Singular points, 50
Slater orbitals, 163
Spherical harmonics, 47, 58
Spin functions of hydrogen, 215
Spin theory, complex molecules, 232
Spinning electron, 124, 125
Spin-orbit interaction, 151
State function, 28
Statistical distribution laws, 283, 285,

287
Statistical mechanics, 282

and thermodynamics, 289
Statistical weight, 290
Stereographic projection diagrams, 379
Sum-over-states, 290
Symmetrical top, 260
Symmetry groups, 376
Symmetry number, 295
Symmetry operations, 176

Tetrahedral orbitals, 223
Time-dependent perturbations, 107

Transformations of coordinates, 174

Transition probabilities, 114

Transmission coefficient, 304, 306
for Eckart barrier, 311
for special cases, 317, 320, 323

Trial eigenf unctions, 100

Uncertainty principle, 21
Unpaired spins, 170

Valence orbitals, 218

of N, O, C, 220, 221
Valence structure, of acetylene, 224

of benzene, 249

of water, 225

Van der Waals' forces, 351
Variation method, 99
Vector model of atoms, 165
Vector operators, 363
Vector potential, 108
Vectors, 175, 361
Vibration theory, 16
Vibrational spectra, and group theory,
274

of acetylene, 279

of benzene, 280

of water, 276

selection rules, 276
Virial theorem, 355

Wave number, 4

Zeeman effect, 162
Zero-point energy, 77
Zeroth-order approximation, 97