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A §TUDY OF ‘t'HE
BORN=OPPENHE£MER APFRQXEMAUGN

T519515 f0? the chmo of M. S.
MICHEGAN S?ATE UNIVERSI‘E‘Y
Alvin R. Hagler
1965

A STUDY OF
THE BORN-OPPENHEIMER APPROXIMATION

Alvin R. Hagler

A Thesis

Submitted to the College of Natural Science
of Michigan State University in Partial
Fulfillment of the Requirements for
the Degree of

Master of Science

Department of Physics

1965

Abstract

A STUDY OF
THE BORN-OPPENHEIMER‘APPROXIMATION

Alvin R. Hagler

The Born-Oppenheimer approximation for molecules is
presented in detail. Up-to-date terminology and notation
are used. Also presented is a survey of work that has

been done in molecular physics in which the Born-Oppenheimer
approximation is assumed or known to be insufficient.

ii

ACKNOWLEDCMfiNT

The author wishes to thank Dr. P. M. Parker for suggesting

this work and for his assistance in carrying out the work.

iii

Chapter

TABLE OF CONTENTS

The Born-Oppenheimer Approximation ........
(1.) Notation and Definitions ..........
(2.) Electronic Mbtion with Fixed Nuclei .
(3.) The Perturbation Equations. . .......
(4.) Solution of the Perturbation Equations
of Zero and First Order; Nuclear
BqUilibrimn O O O O O O O O 0 O 000000
(5.) Solution of the Perturbation Equations
of Second and Third Order; Nuclear
Equilibrium ................
(6.) Solution of the Perturbation Equations

of the Fourth and Higher Orders; Rotation
and Coupling Effects. . . . . . . . . . . .

(7.) Special Case of Diatomic Molecules .....

(8.) Independent Treatment of the Diatomic
Molecule .......... . .......

Errata ......................

'Magnetic Interactions Between Molecular
Rotation and Electronic Metion ..........

The Jahn-Teller Effect ..... . ........
The Renner Effect ................

The Interaction Between Electronic Metion
and Nuclear Vibrations ..............

Critiques and Refinements of the Approximation. .

Bibliography ...................

iv

Page

13
15
17
23

27

31
34

35
44

SO
52
S8
63

LIST OF FIGURES

Figure Page
1 A DUplicate of the First Page of the
Original Paper of Born and Oppenheimer ...... 1
2 HUnd's Coupling Case (a). ............ 36
3 Hund's Coupling Case (b) ....... . . . . . . 37
4 Hund's Coupling Case (c) ........ . . . . . 38
S Hund's Coupling Case (d) ....... . ..... 39

1927 M 20

ANNALEN DER PIIYSIK

VIEBTB FUNK. BAN]! 81

I. Zur Onanmmheorie dcr Molekeln:
uon M. Burn and R. Oppen h dimer

Es wird gczeigt. 4.8 due bekanmen Ameulg der Tfl‘me vimr Hold-ml
, die def Energie der Eldklmnanbcwq‘up‘. dcr Kernschwin‘un‘ev‘ and
dc!- Rom'mncn enu'orechen. spasm-Huh als div Guilder emcr Patch:-
ontvhklung Bach der vierlen Wurzel des Verimltnissvs Elektrnmnmuu
“(wattle-er) Kemrnmso gewonnen werden kb‘nnen Dat Vérfahnn
50hr? u. a. pine Chic-hung Fur die Routiomen. dic- cine Vemllgemqin.
fun, dc: Ahntzu Von Kramer! and Pauli Kniscl mi? ringebautcm
Bchwungnd danlcllt. Former ergibt sick cine Recl‘tfcrtigung der wn
Franck and Condor: angwtellten Betrachtun‘en fiber die Intenmlt
van Bondcnliaicn. Die VerhSltniuc warden an 80'5ka der twai-
Remix-"Holden trunk-rt.

s Einleitung

Die Terme' der MolekelSpektren setzen .s'ich bekarmtlich
IUS Anteilen vet-achiedener Gr'dbenordnung 7USamMeh; der
038m Beitrag r'uhrt van der Eleklronenb‘ewcgung um die
Kern: her. dann folgt em Beitrag der Kemschwingungcn'.
endlic‘h dre ~won den Kernroutionen erzeugten Anuile. Der
Grand fir die Mfiglichkeit emer sou-hen Ordnung liegt offen-
tiCHIick in der Grade der Masse der Kerne. vet-flicker: mit
dcrder Elektronen. Vom Standpunkte def alteren Quanten-
theorkfim die sutionlren Zustz’mde mit Hilfe dcr klassisclnen
Mechan‘uk bererhnet, is! dies" chanke wm Born und
H¢i5¢"bel‘g‘) durchgcluhn worden; es wurde gczugt. daB
die aufigeultcn Energieahlelle a]: die Clieder Wachsgndtr

Ordnung hi'ntichtlich des Verhfilthiut‘s I/ Hi erscheinen, “'9 W

dig thct'm’e‘m‘usfiv M eine mittiere Kernmasse 3st. Dabci
'(zweiten)Kernschwingungeu und Rotatiunen ‘m der gle'xchm

mm"? 3‘14. Was dcm empi'isdaen Behind (bei'kleirwsc
‘ Rotationulmcnzahlen widerspricht.

' ”n.39rn “I W. Heistn‘firfi, Ann. i P3178. 74. 5.1- 1924.
Anna!“ it? PM!“ W.Fc~lgc. 94. 30

Fig. 1

INTRODUCTION

The Born-Oppenheimer approximation was formulated in a famous
paper published by Bonn and Oppenheimer in 1927 Ann. Physik, Vol. 84,
page 457 . It is the most fundamental and.basic work in molecular
physics. The first page of that paper, which is the subject of this
study, is reproduced as Figure l of this thesis.

The prOblem.that Born and Oppenheimer investigated may be
quite simply stated. A molecule consists of a semi-rigid nuclear
framework which is free to rotate in inertial space. The nuclei are
in vibratory motion about their equilibrium positions. About the
nuclei, electrons move in considerably more rapid motion. To compute
the allowed energies fer the electronic motion along with those of
nuclear vibration and rotation from first principles would provide
accurate prediction of all molecular parameters as well as the complete
emission and absorption spectrum of the molecule. The difficulty, is
however, that the Schroedinger wave equation is impossible to solve
exactly for all but the most simple molecules.

Born and Oppenheimer, in attempting to find an approximate
method of solution of the molecular wave equation, assumed that the
electronic part of the wave equation could be solved fer various
nuclear configurations with the nuclei held fixed. The wave equation
for the nuclei could then be solved by successive approximations with

the electronic energy making up part of the potential field in which

the nuclei move. This procedure is now known as the Born-Oppenheimer
approximation.

Theoretical work done in molecular physics can be generally
classified as falling into one of a few categories. The main divisions
can be taken as electronic structure calculations, vibration-rotation
calculations, and vibronic calculations. Among these divisions the
number of papers involving electronic structure calculatims, (with
the nuclear framework fixed in its equilibrium configuration), and
those involving vibration-rotation calculations (assuming no direct
electronic interaction with the vibration-rotation motion) is vast.
These papers are inportant to the understanding of molecular spectro-
sc0py and structure. This study, however, is limited to papers in
which the Born-Oppenheimer approximation is not considered sufficient,
i.e. , cases that deal with interactions between nuclear and electronic
motion which is beyond the scope of the Born-Oppenheimer approximation.

The original paper by Born and Oppenheimer is not readily
available in English. It has been translated and a microfilm copy
was obtained from the Oak Ridge National Laboratory. The quantum
mechanical language and notation used is characteristic of early
quantum mechanics and does differ somewhat from modern works in that
respect. This difference makes the original paper more difficult to
read. In this study modern terminology and notation has been used
in presenting the original theory of Born and Oppenheimer. Some
typographical errors were located in the original paper and these will
be pointed out at the end of this chapter.

As was previously stated, the vast body of work which assumes

complete validity of the Born-Oppenheimer approximation will not
be considered here. .A survey of work which does not assume the
Born-Oppenheimer approximation sufficient was made. It is
possible to classify this work in several more or less distinct
categories. These were taken to be: rotational Spin uncoupling,
the Jahn»Te11er effect, the Renner effect, general vibration-
electronic interactions, and miscellaneous improved.methods of
calculation. Each of these is discussed in this work in turn.

The method used was a library search of physics abstracts
from 1926 to 1964. The abstracts were copied on cards and grouped
according to category. There is, as might be expected, some over-
lapping of categories particularly among the Jahn-Teller and Renner
effects, and vibration-electronic interactions since it could be
argued that the first two are simply special cases of the third.

Definitive or basic papers were identified, separated, and
given Special emphasis. The most basic papers were then looked up
and read, whereas only the abstract was read for the others except
when the abstract was not clear or sufficiently explicit for our
purpose.

It is hoped that this study will make a contribution toward
identifying and clarifying those problems which are connected with

the Born-Oppenheimer approximation.

CHAPTER 1
THE THEORY OF BORN AND OPPENHEIMER -

The theory appearing in the famous paper by Bom and Oppenheimer
will be presented in this Chapter.

One seeks to solve the Schroedinger equation for the motion of
the electrons of a molecule located in ferce free space. Each electron
moves in the force field provided by the nuclei and by all the other
electrons. The nuclear framework is semi-rigid and thus admits
vibratory motion. This in turn provides a non-constant force field for
the electronic motion. Also, rotation of the molecule about its center
of mass is possible. Occurence of coupling among the various types of
motion needs, of course, to be considered as well. The translational
motion, as in classical mechanics, separates completely from the other
types of motion if one takes the coordinate origin at the instantaneous
center of mass. This will be done, and the translational motion will
not be considered further.

Since exact solution of the SChroedinger equation is out of
the question, an expansion method of approximation is used. In this
expansion, the zeroth order of approximation relates to the electronic
energy. Terms of the second order correspond to the harmonic portion
of the vibrational energy (normal modes portion), and the fourth
order of approximation is associated principally with the rotational

motion. As will be seen later, energy contributions from orders one

and three vanish. The vanishing of terms of first order is due to
the existence of an equilibrium position fer the nuclei. For this
situation the electronic energy of the molecule is a minimum. The
vanishing of terms of order three is more difficult to interpret.

It is, however, indirectly due to the same fact which is responsible
for the vanishing of the first order tenms. The theory also shows
that in order to determine the complete eigenfunctions in zeroth
order (and with them the transition probabilities in zeroth order)
one must calculate energies through tenms of fourth order. Continu-
ation of the calculation beyond the feurth order has not been carried
out. It is neither simple, nor does it yield results of fundamental
significance. The diatomic molecule is treated as an example of

the general theory.

1. Notation and Definitions.

 

The mass and rectangular coordinates of the elections will be

denoted by

m,Xk,ij,zk, k=l,2,3,°°-,YI
and that of the nuclei by

ML, XL, YL,ZL, 1.= 1,1,3,"‘,N

Let P4 be the average of the nuclear masses P45”. Let

E

(l) K =
then
-l’l = m
(2) ML_NL Ell/71'

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

The “1. are dimensionless and on the order of unity.

The potential energy of the system is

U(x"|j‘Iz19 79,31,112”. XHYHZHXUYIUZPJH): UOQXL
Here X denotes the totality of the election coordinates and X
that of the nuclei. The potential energy function depends only
on the relative positions of the particles, but here no use
is made of the particular fbrm (Coulomb's law).

The kinetic energy of the electrons is represented
by the operator
”V a: it. at.
E -. fm %1( BXR‘F 33:..- 51R)!
and the kinetic energy Operator of the nuclei is
9. ‘i 31 5“ + 1
R‘_% 1,:1N1‘(-a—Xi+a;1‘l 657:).
The Hamiltonian is given by

H = H0+K4 H‘,
where

W's-Lu = H.(7<, $7“)

‘7; = K} *h ( Rik)-

It will be convenient to use the13hr-6 independent relative

nuclear coordinates

1% = €300 , i= 1,1,3,'“(3N‘6).

which specify the instantaneous distances between the nuclei,

plus an additional six coordinates:

9'1 3 9‘; (X) y '1' 1,1,324y‘5267

(11)

(12)

which are the three Euler angles (9 , CD, 11/) and the three coor-
dinates of the center of mass (xu Yo, z.) , these six coordi-
nates serve to locate the position and orientation of the
molecule in inertial space. The new coordinates g and 9 are
obtained from the nuclear coordinates X by a linear transform-
ation, a fact that has been recognized in writing (9) and (10) .
This transformation does not separate the Hamiltonian

into translational, rotational, and relative motion. However,
H1 - representing the kinetic energy of vibration and rotation -

may be written in three parts as follows:

H1: H“ + ng-i- H69:
(1)

and developments of general vibration-rotation Hamiltonians
show that H§§ is linear and homogeneous in 51/6 €16 g} ; Hg 9
involves the as: i ; and H 99 is independent of Q . Some
general conclusions may be drawn about these operators. If the
Operator H, is applied to any function of the relative coordinates
Hg) , the resulting quantity must be independent of the position
in space and therefore independent of the 9i . In particular,

in Hgg the coefficients of the 3,7553%. cannot depend on 61 .

On the other hand g, , 91, and 95—91 must appear in ng in
addition to %—§i , H99 will contain 23%, , aghand e, in addition
to 6% e. a 9;, -

The Schroedinger equation of the molecule is
(H.+K‘H, —w>w = o,

where the eigenfunctions are denoted by \V and the allowed

energies by W .

(13)

(14)

(15)

Electron Motion with Fixed Nuclei.

 

If in (12) one sets K: o , then a differential equation in the

1., alone is obtained and the X1. appear only as parameters:
[H.cx, 95.7.. ,x>-w1\v=o

This equation represents the motion of electrons about fixed
nuclei. The relative nuclear distances and Eulerian angles
appear as parameters. We assume that this equation is solved.
If we let the origin of coordinates move with the center of
mass of the molecule, translational motion will separate off
completely from all other types of motion, the center-of-mass
coordinates X. , Y., Z. will be superfluous if translational
motion is not of immediate interest, and 91 will now repre-
sent only the three Eulerian anglesCG, 4), W) . Translation
will be disregarded for our purposes and will no longer be
considered explicitly.

Let the nth eigenvalue and its corresponding eigen-

function be denoted by

w=v.(§) , V = ‘Pnhc, gone-1).
Then (13) becomes
[H.(x,%7(,§,e)— v,(§)l\0,.(x,§,o)=o

All V35 are here assumed to be non-degenerate eigenvalues.
This is not usually true for all electronic states of a given
molecule, and if it is not so true, then special considerations
apply. These degeneracies were not, however, considered by

Born and Oppenheimer and will not be considered in this chapter.

10

It will be shown below that the function V..(£_,) plays
the part of the potential energy for the vibration of the nuclei.

Instead of taking derivatives with respect to 8,; directly,
let us replace 5-, by g, + K g, and expand with respect to K .
The coefficients of a given power of K will then be a homogeneous
polynomial in C1 whose coefficients are the derivatives with
respect to Ci . Thus the expansion starts with a given nuclear
configuration specified by g and expands about this configura-

tion, the deviations being given by c . We have then,
(M) w<€+wn=\C+KMY+KM?+W,
where
(a) v: = V..( §) ,
(17) (b) VI." = Z Q %%‘,
(c) V?) 3’22. Cigj—a—‘L— a; 5,5 ,
V

. . 5V _ 3V 5 +’ =
ThlS 15 obtained by taking 572 - m) J—g—KEQ C b §+K§)

and evaluating at K = O , etc. Similarly,

m) th=FC+KHT+KWfiH-

(18) o I) 9.)
w)‘fi=%+KR+fiR+

Now we expand the quantities “Pi” and flu in terms of the

eigenfunction ‘9: (x, §,e) :
(a) ‘22). 'Z kit)“, Up: ,
19
()(b) To” Zu‘flfi‘?

where Limo is a homogeneous polynomial of the Y‘tb degree in Ci .

(20)

(21)

(22)

(Z3)

(Z4)

(25)

11

For exainple, multiply (18b) by { 9,: Y and integrate over all

electron configuration space:
HWY ‘PL"dx= 11‘." HRS") LPf. c1122 11‘" A . u,

since the \fi’s are orthogonal functions. Now

Op
S— =; Sza—é18=|_—-§"”|<iDSEI-m-IKSZL)35:. a §i+K 1.§; which if

evaluated at K: 0 becomes ”P5." = ; 4‘55 FE: - Making use
i

of this equation (20) becomes:
= ms) 2 c g3 d. =3: entirgfidx.
By a similar treatment:

me-vezqqnwfi—éng

Thus as asserted, Uni/V is a homogeneous polynomial of degree Y‘
in C1 . The integrals given here, in which Jar represents
the volune element in configuration space of the electrons, are
independent of the orientation of the nuclei in space and there-
fore independent of 91 . They may be evaluated in the princi-

pal axes system.

If F is any Operator Operating on X1 then we will define
0 Y‘
We? F v.“ Ax a a.

as the matrix element of the Pit order of F’ . For 1‘ = 0 this

becomes the ordinary matrix element
x
“e" = th'_ ‘ I< Yrs) F ‘9: &7(
In general, according to equation (19) ( win: 2’ Um" (a: )
:

FW‘) = (Y‘
““1 Z UnJfiF Fn" n! O

12

Now it follows from (15) for K=O , and for F=(H2-V.°) that
(H: -v )3: 2-: Hemp-v.3) n".

j( KP,‘.’)“(HZ-\/.°) g. of}. “P“: out

-2; ”3;. [ H «mm: as: - k now: 19.314.

.1; 11.3..(vxu d...» —V° den»)= utE’lO/J'VK).

Thus
0 o ) O o
(26) (H° - v“ :3! = USN ( W! "' V» ).

Now by substituting (16) and (18) into (15) and equating coef-

ficients of similar powers of K=owe obtain
(a) (Ht-W) *PI
(27) (b) (H;- v:) ‘95." +(H3’ -vfl’) ‘9: = o
(c) (H: -v.:)‘R‘."’ + ( 142’ - vf’Wt.” +(H?’—v.‘.“) LP.“ =
Due to the orthogonality of the eigenfunctions, multiplication

by UPS)" and integration over It will, with the use of (25)

provide:
(a) “at“ C V; " Vn)+( Hm)nn’ " V3) Amn' = O,

(28) I
(b) UMJ Vn' "‘ V: )+ ( H? VBLM’ +( H?) )’nn “Va’énflw

The expansion (18a) for the Hamiltonian may be treated similarly

(7,) . .
in terms of (Hg nn' ,(H, )M. , .. . which are the matrlx elements

. H '
( B H 1),,“ 7 (Walt,

These will be used later.

13

3. The Perturbation Equations.
An arbitrary configuration of electrons and nuclei cannot be
dealt with by means of a general perturbation treatment. Only
those conditions that conform to a stable molecule can be
considered. For the unperturbed system the electron motion
will be considered for an arbitrary, but fixed nuclear con-

figuration E1 . Next, all quantities will be developed in
terms of small variations of g, which will be designated

KC, . We assume that for a stable molecule the amplitudes

of vibration. go to zero as K goes to zero, an assumption

that is justified by the results.

As in paragraph 2, equation (18) , we have the expansion
(29) HA1. $7, §+Kc,e)=H:+KHL’+K*H‘:’+~-,
where

H§=H.(7<, 5-.— ’5‘),

o_ _§_fl.°_
H.)- Vzlzj Cigjagiagé ,

and by (11), since '3'; {1(§+K§) =fi §¥(€+K§),
where {CE +K§) is some function of g + KC

(30) K‘H.(X.%7)

‘
K“ R‘ H96 +1K’Hce + Hoe)

K“ ch+ K’( H§e+ H31,» K‘( H;.+ Hg’,+ ""

)+...

)

(31)

(32)

(33)

(34)

(35)

14

where

(a) HZC Hg? ( g 35615 5C3 )7

(b) pizza: (:1 o__HZ-< ’

(a) H;9=H;e(§a 9 21—52%):
(b) H1119 -é:: €15 LH:9,

(a) H... =H§e(§ 9:33-25?)
(b) H“) =2 qi 314:9 .

The argument £1 is from here on to be considered as a constant.

The Hamiltonian becomes
H = H.+ K4 H, = H2 4- KH‘f-l- KN Ht.“+ HZ. ) +K3(H‘§’+H;9+

after substitution and collection Of the coefficients of the
various powers of K .
Now we expand the eigenfunctions and energies in the

following manner:

- o (I) 3. (2) O
(a) \V-\V +K\V +I< W + H ,
(b) w= W°+K W“’+ K‘w‘”+m .

Then, by equating the coefficients of the different powers
of K, in the resulting wave equation, to zero, the following

set of perturbation equations result:

(36)

(37)

(38)

15

(a) (HZ-W°)W°=o,
(b) <H2— W°)\I’“’=(\/~I"’ - H?) W,
(c) (H: - W°)\V“’=(w""- H‘f’ - HZ?) \V° + (w“’- H?) w‘”,
(d) (H? - W°W“;(w“’- H‘Z”- 39' Hg; ) w

4" Wu" H?" Hic) W + (w — Ht? ) W‘“,
(e) (H: - WW": ( w“’-— H2”— H3, - Hg; .. H2; ) Ly"

HW— H‘f’—H;, - Hg.) W

+< ww— Hirh H; > W.» (ww- Hi" ) W.

Solution of the Perturbation Equitions of Zero and First Order;

Nuclear Equilibrium.

 

The zeroth order perturbation equation (36a) was presented in
paragraph 2 in the discussion of electron motion for fixed nuclei.
From the normalized eigenfunctions \Pn°(’x, E , 6) introduced in
paragraph 2, and the associated eigenvalues V: = Vn(§), one

obtains the general solution of (36a) in the form
‘1? = X3(c,e> \Efm §,e>,

where X: is an (as yet) arbitrary function of C5. ,9;
These functions are needed to obtain the solutions of the higher

perturbation equations. The next perturbation equation (36b) is
(HE-WINK") = (wt? —H:;’W.°.

It may be solvediif \K‘" is imagined expanded as a series in
\K" . Then it may be seen that the left side of the equation

is orthogonal to W: . Therefore, for non-trivial solutions

(39)

(40)

(41)

16

to exist the right-hand side must be orthogonal to W; with

respect to the x1 , i.e.,

JW.°>*< wt? - H?) w: dx = o,

J<X§)*( LPf)‘ \A/‘J.’ XI.” “Pf.” - ((XZVOFS)’ Hfl’Xi “Pictwo,
or

wfi‘IXZI‘JWS)” kP.‘ dx - IXZI"( Hil’)... = O.
Thus,

[(HS’)... - wit’] IXSI‘ -- 0,

where (Ht?)M is the diagonal element of the operator HEX Xi)
and according to (39) it must be a constant, since for equation (39)
to be satisfied, we must have that (Hfhn ‘-' W3: ; the function
IX: ‘1 must be non-zero, otherwise ‘44" would be identically

equal to zero. It thus follows that
w‘" = <H‘”) =0

Now, from (17) and (28a), and taking n’: n , we have that

 

(H? mm: V;u = Z: (:11 ‘35—}? = 0 ;hence
1
3g :0 ,forall i.
1

Equation (41) shows that the relative coordinates gi cannot
be arbitrarily chosen but must correspond to an extreme value
of the electron energy V“(§) . It will later be shown that
this extreme value must be a minimum value. The function

X:(§, G) is as yet unspecified. If in (38) one sets

17

W: =V.(§) = V: ,W‘:=O , and W:=X: KP: , then

the defining equation for W“: is obtained:
0
(42) (Hi-VDW’ =44? LPVI’X...
A particular solution according to (27b) is of the form

(43) Wt.” = Xn ‘95.”,

where ‘93) is the function (19a) defined by (18). The
general solution of (42) is obtained by the addition of a
general solution UPS) of the homogeneous equation with

>
another as yet undefined factor Xi“ g ,9) ; thus:
() - ° (” <0
(44) W3. - X. “P. + X, W.

5. Solution of the Perturbation Equations to Second and Third

Order; Nuclear Vibrations.

 

The second order perturbation equation (36c) reads:
'(H: — wxw‘“ = ( w‘” — H‘Z" - H2; W°+<wm- H?) \V‘”.

After introduction of the first order solutions (44) this

equation becomes
(45) ( H2 - V3) ‘14.“ = (W3) - HED- H;;)X: ‘9: - Hg“ X3 waxy?”

which has solutions only if the right-hand side is orthogonal
to \P: . Multiplying by ( ‘13:)” and integrating over cl x

results in
o I)
(46) u H?’+ch)m. +< ((2):... — w?1x:=o.
Now it follows from (28b) for vat-'0 that

(47) (HEW... + (H3533. = V5.” ,

(48)

(4 9)

(50)

18

and since Hg; is independent of X- , one has that, (Ha, )Ml - 44.9..

Now it follows that
1) O
IH§§+ Vi. - w‘hx, :0.

Since Hg. is the vibrational Hamiltonian, and since V?
depends on the displacements (:1 of the nuclei from their equi-
librium positions g1 , (48) must represent the equation for
nuclear vibration. More explicitly we have for (48) ,

1H:¢(§, 3%)+1/2i23 €14: ab}; —_:y§'_”§3 1X3): Wm DC.
It is seen that the second order portion of V“(?,) plays the
part of the harmonic potential energy of vibration of the
nuclei. This provides a further condition for the existence
of a stable molecule, viz. , that the extreme value of V“( 5:)
defined by (41) must be a minimum; for the quadratic form
VS“ E) must be positive definite if all normal modes are to
be stable and vibrations about the equilibrium point are
possible. Equation (49) would be separable if a linear trans—
formation of the C1 to normal coordinates were to be performed:
If G'n°5(€') is an eigenfunction of (49) with eigenvalue Wt: ,
then the general solution to (49) is

(a) Wm‘ Wns , X:- “X‘s , where

O

(b) X... = (0:.cem-fscq').

The index 5 specifies the set of 3N-6 vibrational quantum
numbers. The function ’0“; (9) is as yet undetermined and
must be analyzed when continuing to higher orders of approxi-

mation. As is well known, the functions O':S( C) are

19

linear conbimations of products of Hermite orthogonal functions
in the normal coordinates. The property of the Hermite ortho-
gonal functions, that they are either odd or even will be used
later.

If Q is any Operator dependent on 4-, , then the matrix
(51) ”Egg: = Rainer)" <1 0‘35 dd; ,

may be formed, where d C is the volume element in C1 space.
In order to solve equation (45), (48) is substituted into

(45) in the form
(W53; -H;,)X.°, = V? X35,
and then (45) becomes
(52) (H:- v:) w: = M? — mm LP: - (ms, 9:")- Xii: W),

O
K ’

which has as the solution to the reduced equation X33)
. . o w )
and the part1cular solution X“ \P“ + X35 LP: . The general

solution is therefore;
u) 0 a.) a) u) (z) o
(53) ‘1’. = X... LP. + X“, ‘P. + X.5 LP. ,
2.)
where Xés is another, as yet unspecified, function of 4'1
and e; , since (53) is the solution to the equation of vibration.
Next, the perturbation equations of third order (36d)

may be studied. Introducing functions that have already been

determined , (36d) becomes

(54) (H:-— W) W = (W’- H‘f’- ng— H2’,)X.°, kP:
+(wtf; - H‘3- H;,)(x.°. wen: kRf)

- (42% XS. kP‘.Z"-+— X it; KP1? + X13; ‘83).

(55)

(56)

(57)

(53)

(59)

20

If the right-hand side is considered to be expanded as a series

in ‘9: , (54) may be written
(Ho- v: )WO): WC” XY‘OS 1P): - 2"; Rafi) LP;

where

8

Fa) =F-(Jm X024. +F43,2) xii; +F-(35 3) X0
m

Equating the right-hand side of (54) to the right-hand side of

n5-

(55) after substituting (56) provides
Z}( 53ow ‘83 + F3813; ‘8); + .31) ”XI... 8° )
=< Hz? 4- Hg, 4- Hg" )st LP° +(- wit; + H234 H;,)( X5, ‘8'
+X‘”, ‘8 °)+ H‘;’(x:, we. ‘95,"4- +_x::; w; ).

From (57) it is seen that

(a) 2; E?" 532’ ‘8; = H‘BXL? Lat,

(b) 2;; F‘fi?’ X“; ‘83 [<-w:.'-; + H9.“ + H2.) ‘P:+ H‘L’ ‘8."JXSL,

(c) 4:)?" Faj’st LPnn = [(H‘§’+ ng +H "’ “)‘P" +(- M.” + H§”+ H243 “P:
+ H? LP? 1X5; ,

Multiplying (583, b, c) from the left by (W:..)" and integrating

over Jar gives

(a) F3? = (H‘l’)..v,
(b) Etta)... (H; + Hm__ W3: )vm’ + (H(;))(l)

)mn' )
= (I) a)
(C) “R,” ( H?) + ng+ H"); “H, _ (W(:)- Hw‘ H; +(H¢ )VWI’
where "313) is a homogeneous Operator of odd degree in 4':

and 274-, (see equations 31 and 32). Multiplication of (SS)

(60)

(61)

(62)

(63)

21

from the left by (\P: Y and integration over 47! yields
Wm - R? = 0.

Substituting (59a) into (56) and using (60), we find
F""Xf,'; = C W- WW“

and substituting (47) into (59b) , we have
Rf?“ = H5, + VJ,” - :33.

Equation (61) is the inhomogeneous equation related to the
vibration equation (48) through the functions X :5 . Since
(48) has solutions X5, = ,0“: (6)031“) by (50b) with
eigenvalues W535). (61) has solutions only when the right-
hand side is orthogonal to 01:, with respect to <7 Space.

Because of (50b) this provides a differential equation for
0
do (9) ,
(3)3) _ u) 0 .-
.. (3,3) . .
However, as has been established, Em 15 odd 1n 4‘;
and g2. . If #3:) is transformed to normal coordinates
1
Y1. , 3—. , then it will also be odd in the normal coordinates,
1 v1,
because the transformation from Ci to Y“ is a linear trans-
formation. Furthermore, the O'“°5( §) , when transformed to

0’“: (Y1) will be products of harmonic oscillator wave func-

tions, so that the integrands of
( ) _ .0 ° 0
Fug?! " loco-1:0“) Etc:3>(rls%—fi) dhs(yl) le

will be odd in Vii for all i = 1,2,3,”‘, 3N-6. Hence,

“‘3: = O for 3 z 5’ . Then for (63) to have non-zero

SS

(64)

(65)

(66)

22

solutions (0:: (9) (which we must have in order for the total

wave functions \V to be non-vanishing) , we must have that
w(3) = o .
The functions a: remain undefined. Now (61) becomes

1) (I) __ 33)
Fm an - PC»! Xnosr

mm

and its solution has the form

 

m _ w
5 P: ,
where 50) is the following operator (with respect to e ):
“S #33) 1
’ 0,6
Sm- FER: Gins'u.) .
as :2 w:1;_“s,

Equation (65) can be obtained by the following steps:
'13.,»ng : affix“: I'd?) [3:5 (Twas .9
(H; + vr—w win.) X‘” = -Fj:’”p;s on?“
[wi‘é- -< H£<+v3’>1><‘” Fff’fl‘; cris
we now let X2): = (a: 2:; ans" 0:50» and have
my; x25; g, awaits» -—< ng+vf)a°,z ansuoam
=[ :3: f3; E5; ans" 01‘2" ,_ Ion: 25;: ans” Wm s” n;”]= R3190; us a.”

We multiply from the left by (0120* and integrate over 4 Q" to

 

obtain
(2.) (2.) (3,3)
Wns Owns' " any Wns’ = Fun ,°
55'
thus (3,3)
F“
a a: 83' O _
us’ " ( (“’Wyf? 3 (JIM-o

Since we had X3: : 5:: a: ;- fl‘: gamuqzusubstitution of
s

(67)

(68)

(69)

(70)

(71)

23

(66) into this establishes (65). Now (55) becomes
(H: -v:>\l":’ =- g at"? 493..
If W?) is expanded as a series in ‘P: , then (67) may be
multiplied from the left by ( ‘P:.)* with nh’t' and integrated
over do: to give:
, ‘ .5." “PS.

“’33": V:-v:

no )

which can be written as
X( ) (3 m
V“ 2; ( ‘3?" " ML+G ’I’MX .‘3+ 6&3 X302).

This expression may be treated in a manner rsgflar to (S6) and
one finds that G13“? is a number, d:”= (Va—V ’ ,), and is a
differential operator with respect to 4‘1 , whereas Gm: is an
operator with respect to {1,6, Now using (28b) ,

.> (P. ’(H‘flnnofi'o w _ ’
2G 3' (”=25- Vn°"'Vm an ‘éu Um nI‘Pn'X‘nz;= “Ran33

Gnn'
thus finally we can write
w:”= up? x53; +Z< ii’fi’fx: a: +— of? x:, LP: >.
Solution of the Perturbation Equations of the Fourth and Higher
Orders: Rotation and Coupling Effects.

Equation (36c) , after substitution of the quantities that have

already been found, becomes
(H: warm? = <w“’- H:“’— H:;. - gg—ngxs. LP:
~( Hf,”+ H°e + H") XX":- LP° +Xn°s (”‘9‘”)
+< wS; - H‘?- chxxtfi W°+ LP" LPUG; W”)
HS’WS’ .+Z’<cr";?’x:.':~9:. +a‘33’x:.a0:).]

(72) ‘

(73)

(74)

(75)

(76)

(77)

24

The right-hand side can be expanded in terms of ‘9: to obtain
0 a (4) O
(H. —v,) V,” = W’Xfis LP: -% Ev LP“,

where

(4) _ (4.2) m (4,3) m (4,4) 0
En’ " my ms + va’ ms + va’ n5 ,

and where

(4, > u)
Em' 2 :( He}; + H?) — wY‘la; )vm’ +(H(<:))nn’ 0

This is found by multiplying (71) from the left by ( ‘49:)" and

o o ‘ o o o )
1ntegrat1ng over 0‘1 . "(,4)” 1s 1dent1cal to as)”, (59b), “:7,
is of odd order in (’1 and 332:1 , and E21015 of even order in d,“
a
and — .
ac,

Equation (72) has non-trivial solutions only if the right-
hand side is orthogonal to “P: . Multiplying (72) from the left

by (“PS )7 and integrating over six provides
0 _ (4) _
Wm an Em " 0'
Substituting this into (73) gives

(4 (1) (4) ( 4) ° - (4 3) (I)
51;”an =(W -R1:>an‘ t ’ an.

nu

The left side, because of (74), is identical to the vibration
equation (48) , and therefore the right-hand side must be orthogonal
m

to 02°, . If the values of X; and “3 from (50b) and (64) are

introduced along with the definition

’(§ FHA?)
(6):; = Mania“? 53:: a: =§ ;; _” :3:

then it follows that

(4‘14) 8,3) (I) (4) o _
[F' +(F'M )5, -W ijs .0.

VM
SS

(78)

(79)

25

These equations at last define A: (e) , and therefore the
motion of the principal axes of inertia in inertial space.

The most interesting term of the operator in (77) is the one
that contains the second derivative with respect to 6 1 ; it

is formed from H39 X; KP: as may be seen in (71) and therefore

corresponds to the term in F5" ,
(113;). = loan" H; ( tan->47:

where in the position of the dots the function on which the
operator acts in introduced. Instead of the simple Operator
H39 , the more complicated (EL, occurs. Physically, this
corresponds to the inclusion of a coupling of the rotational
motion of the nuclei with the electron motion. This is the
effect which Kramers and Pauli(2) attempted to describe in
diatomic molecules by mounting a flywheel on the rotating
nuclear framework. Then there is in (77) the term which is
derived from the Operator H“ . This corresponds to a coupl-
ing of the rotation with the angular momentum which results
from the nuclear vibrations. Finally, there is a term that
does not affect 61 . It gives an additional contribution on
the order of K4 to the vibrational energy and represents
anharmonic correct ions .

We now consider the rotation. If r represents a
suitable set of rotational quantum numbers, then one has for

the solution of (77)

W“’= “‘2’. ;fi‘; =flé’M9).

(80)

(81)

26

Now (75) and finally (72) can be solved. It is not instructive,
fer our general considerations however, to write out the solu-
tions explicitly.

The treatment could be carried on, but nothing fundament-
ally new would result. The higher perturbations describe the
coupling between rotation, vibration, and e1ectron.motion. No
new quantum numbers that have not already been introduced would
result.

we now summarize our principal conclusions. It has
been demonstrated that to completely define the eigenfunctions,
even to zeroth order, the solution of the perturbation equations

is required to fourth order, since
W... (at, 4‘, e) = \P:('x, gem-,5, (¢>/€.§r(e)+---,

where ‘9: is the eigenfunction of the electronic motion for
fixed nuclei, 0'}: is the eigenfunction for harmonic nuclear
vibration, and F“; is the eigenfunction for rotation of the
nuclear framework. The vibration coordinates 4'; are measured
from the equilibriun positions g, , which are defined as the
positions for which the energy of the electronic motion V“( 3;)

is a minimun. The energy to the fourth order is

- — 2 L2.) (4)
WWSV-v;+K whs+K4 Wn5r+°",

o
where V“ is the minimum value of the electronic energy
which is characteristic of a situation in which the nuclear
. . . . . . (2.) .
framework 15 rigid and at rest in inertial space, WM, 15 the

. . . (4) .
harmonic nuclear Vibrational energy, WM? gives the energy

(82)

(83)

(84)

(85)

27

of rotation plus anharmonic corrections to the vibrational energy.
Up to and including the fourth order of approximation, the
vibration-rotation motion is independent of the motion of the
electrons.

Special Case of Diatomic Molecules.

 

As an example of the method, a short treatment of diatomic
'molecules will now be given. Since degeneracies will not be
considered, this will be for the special case when the axial
component of the angular momentum of the molecule is zero.

In the case of two nuclei there is only one § coordinate
for the nuclear separation, and five 6 coordinates , which are
the coordinates of the center of mass X, , Y, )z, and the polar
coordinates of the orientation of the nuclear axis, 9 and w .

The kinetic energy of the nuclei is given by
- .. 4 h). I14: 9.... 2 §___ -‘ ,g
1;, - K i?” [Ag-F g: 3§(§ a§)+ gAe]

where

vim—fin, , ,uJ/HM)2

WM;

and
.b‘ ‘ b‘
(a) A. = 57: +373+gzg ,

1 " ____1__ .
sun‘s 655" + 51116 3‘6 (51"‘9 3‘9):
1

1

.5?

(b)

(a) Hg; :"2.m N 5%" i
{‘1
(C) H99 :- 7%Wt(A°+ M Ag).

(86)

(87)

(88)

(89)

(90)

(91)

(92)

(93)

28
Replacing g by § +K§ and expanding in terms of K , we find

(a) ”Sshgainf" 52-1.
(b) H' = o, P=1,2

(a) Hcl =-‘t‘r 2-5 L

(b)

If:
gig:
WE

n
$1110!

:9 =

2.
(a) H39= - 7’57” (130+ ’5 A9),
.(I) _ ’- Q,
(b) Hee- 7%“ {i CAe.
The equilibrium nuclear separation is determined by

\/
,=——"=O.
VQ (J 5

The equations of nuclear vibration (48) are

big, 3‘2"” ‘a 4“ V.”<§> - wif’lx: =o.
Let

a=%,WS’ ., hfiaw, n a :45,-

then (90) becomes

[fizi-(VQ‘E'WZNX: =o,
with eigenvalues

V213 = 25H , s = 0,1,2,---
and eigenfunctions

on: = 8'“ HM)

where H$(Y\) is the 5th hermite polynomial.

(94)

(95) '

(96)

(97)

(98)

i (99)

(100)

29

The vibrational energy is then;

2. W53 :03“: 57:42" :(QS +1) b fly/$5 =(S+VQ)$TTW
or

K‘ wi‘; = cs+a> m;
where

#JE4/qé1v: 3‘ HT «M14731. ” =Vo

The rotational equation (77) exclusive of the anharmonic

 

 

correction to the vibrational energy is given in (97) . Since
”:9 does not contain derivatives with respect to 63 , according
to (85), all terms in (77) except (H 39)“ will be represented

by a constant, Cns ,
_‘ O
“ng». 4' C705 _ Wow-1&5 =0

The translational part of H99 may be omitted. According to

(78) and (88), for an arbitrary function {(9):
(17°) He) ="1mgzi(‘Pf)A(\P° max,
and by (84)

A9( ‘93) = “P: A94: + (A810: *2 (si1nze 36:23 53+§§°§él

 

Therefore
(T41), Hen-fin [A £+H(~P,:)*A 8:4:
4' 517; e 57’ SOP: ig—
Now since + 7' Ba"; 50%)" ¥“o;:].

A9:'§§ +Cot9—B+-—-1-—teaw2

30

it is convenient to let:

a. J<¢:)*%‘§dx, 21.. Ref-a—fgtdx,

(101)

GB: =](\P,,°)* g—gJOK; 31‘“

I083)" 33-5,": .14.

The quantities (101) are the diagonal elements of the matrices

of the angular momenta L9 , L...) as well as E: , E: (except for

a factor of n2 or 4152) , where e and w are Euler angles, and
-2.

1.9, LN» are squares of the mean values. Using this notation

in (97) provides
(102) ((39.» 29.%g3+@:)+cote(5 +éw)

+ $>i1)'u"65)(8123?.)1 +231)». boo 4‘ .n. )+ 15(WW- Cnsflfoo n: 0.

According to (102), the magnitude of 3%?35 ’ is equal to
Vil‘

a numerical function of the rotational quantum numbers, 3“;(r) ;

for the rotational energy it then follows that

K4 2.
(4)
(103) K Wu“. =§£§E 345(7‘) = 1'3. jmhr)

where

(104) = fi‘ g2 =M1+Mzz g2,
and is the moment of inertia of the nuclei in the equilibrium
configuration.

Higher approximations will not be considered in this
section. It will be shown next that the diatomic molecule may
also be dealt with by a different perturbation treatment where
the unperturbed motion will consist of the electronic motion

plus unifonm nuclear rotation, rather than nuclei at rest.

(105)

(106)

(107)

(108)

(109)

(110)

(111)

31

Independent Treatment of the Diatomic MOlecule.

 

Substituting equation (11) into (12) we obtain
[H.+ “(Hgg +H§e + H99) -w]\)f= o.

In the diatomic molecule ng is independent of G , therefore
it is again possible to separate translation and rotation.
Making use of (8S) and omitting the terms corresponding to

translational motion we have
[Hr 4?” <%-g+ gig-g? J§-».A.,)-\/J1HJ=O.
Now let
V= Y,(G,co)\I/r('x,§),

where ‘Yr is a spherical harmonic of"rt5 order, i.e. a fUnction

which satisfies
A6 Y, +r(r+1)Yr‘-‘
Then it follows that
(tn-$.35. (s'g+%5Ԥ seen-mine
Now we replace § by '5' + K ’3’ to provide an equation for the

vibrations with uniform rotation. The energy of these states

is given by

2 z
12‘ +3 K NEH) ,__ 2.1 run-1).
Now we set
vv = E3+-12

and equation (108) becomes

(Ho4-KH(”+K2H(2)+"° ’E)\K.:O,

(112)

(113)

(114)

(115)

(116)

32

where
(a) H°= 3,
|)_ n o
(b) H‘ - H2+<R,
2 a
(C) Hm= H?“ 1/z qzkl‘jffi‘ $232- 2

II 2.
(d) H“’=H‘f’+%438"% ‘3’ 3‘5; 7

o ) . . . .
and 14.,143,"“ are the operators defined earlier in this paper.

Following the procedure that has already been established, the

perturbation equations are
(a) (H°'E°)‘Yr° =0:
(b) (H'- E°)‘I’.‘"= (5“- H‘”) ‘14",
(C) (Ho- E~)\y§>.(1=.‘”- H‘“) \1/;+(£"’- WWI/g”,
(113a) has the solution

5°: v,,< 1:) , 91:: \an = 011nm) flu, g),

where Vn( E), LP,,°(7(,§) have been defined previously and O}: is
arbitrary. The condition of integrability (making use of

orthogonality relations) for (113b) is
(E"’ — H22“) 0”,: c c) = 0,
Now according to (28a), section 2,
Hi1; =(H2’),m + 512': v,:”+<78’= (ff;- (v.+12);
therefore it follows, as in section 4, that

7

E(l):o Ell—g(vn+R):O.

This means that for the undisturbed rotation, the centrifugal

33

force must be equal in magnitude but opposite in direction to
the restoring force which is due to electronic motion. The

centrifugal force is given by

__ _ _L_.-_ J. .1. 2 < )
(117) “(M+M,,)§ ' (MI+«M1)JB YE? 2

 

where L,=V¥‘(r+1)¥\ is the angular Immentun of the system.
From (119) the equilibrium displacement 5' r may be computed
in terms of the rotation quantum number Y‘ . For small values

of the rotational energy, g. may be expanded in powers of {3 ,

where
(118) fl = K4 4Y4“ 5‘2 r(r+1)= flair/14))? Mir-+1)
to provide
1 m
(119) §.= ‘9' + 37.13“ fed” 5/2 {7 )(8’+---.

Since (5 is on the order of K4 , only as many terms of this
series may be used as will correspond to the degree of the
approximation in the perturbation treatment.

Proceeding by the method used to solve (113b) , we have
(120) ‘i’fiut' =01?“ “Pm-*- 6"” ‘R 9
and the condition of integrability of (112c) is
(z) (I) (I) _ <2) 0 _
(121) [HM +(H >M [-3M 10‘” - O.
Now the vibration equation becomes
121 1 1 1. u , (2.) o _
(122) [*gfi %?a+/i C (WNW-En Jove-O.
Using the method of section 7, we obtain

(123) KZ’E‘QS = (5+ ‘6.) hv. 7

(124)

(125)

34

where the frequency 1),. is given by

= 417T ((81,381 W + R")

and is still dependent on the rotation quantum number 1" that

 

appears in R . Further as in section 7 ,
a 8-1/1 '1
G-rns = H 5W)
where

«if

is the 5‘5 Hermite polynomial.

bl“; (Vn"+R”) .9 and! H5(Yl)

If this treatment is continued in the manner previously
established it will be found that E‘3’=o,anol Emis of the
character predicted by the general treatment.
£11223:-

We conclude this chapter with a list of errata that we have
found in the Born-Oppenheimer paper. Corrections have been
made in this thesis.

In (54) [(51) of the original paper] a superscript in
the first part of the last term was corrected.

Equation (59b) [54b of the original paper] hfif’ should
appear with a positive sign.

On page 16, after equation (59) , it was stated in the
original paper that F3“. is of t___hird degree in 3:1, ,5-21 .

A correct statement is that F'ci’”

is of odd degree in
€1,343 In (71) [(63) of the original paper] several errors in
both superscripts and signs were corrected.

In (90) [(86) of the original paper] a negative sign was

omitted in front of Wm.

CHAPTER 2
MAGNETIC INTERACTIONS BETWEEN MOLECULAR ROTATION
AND ELECTRONIC MOTION

As we have discussed, the Born-Oppenheimer approximation assumes
no direct interaction between electronic and nuclear motion. However,
when the electronic state is not one of zero angular momentum, then —-
aside from generally small electrbstatic and gyroscopic interactions
of higher than fourth order of approximation -- it is necessary to
consider magnetic interactions between the rotating nuclei, orbiting
electrons, and electron spins. This will result in vector coupling
of various types depending on the relative strength of the predomina-
ting interaction. The effect is most pronounced in diatomic molecules.
For these, F. Hund in a series of papers from 1926 to 1928 distinguished
four ideal cases of coupling. It has been found that molecules usually
exhibit predominantly one of these cases. Generally, no actual case
corresponds completely to one of the ideal coupling schemes. It is,
however, found that most actual cases approximate one of the ideal
cases fairly closely. In some molecules one form of coupling may
go over into another if the vibration or rotation energy of the
molecule is changed.

we describe now the four coupling cases as given by Hund.

35

36

Case [a]: Here the spin-orbit angular momentum with quantized pro-
jectionn‘h along the molecular axis couples with the rotational

angular momentum on of the nuclei to form the total angular momentum

J‘h as shown in Fig. 2.

 

Fig. 2

We have that S*fi is the resultant spin angular momentum of all
electrons with quantized projection Sh along the molecular axis,
L*h is the resultant orbital angular momentum of all electrms with

quantized projection All along the molecular axis, and

A+Z

L(L+1)

5*: Js<s +15
J"; JJU +15

r;
ll

37

Actually, on is not simply the rotational angular momentum
of the nuclei, but it is a sum of this and rapidly fluctuating
components of the spin and orbital angular momenta perpendicular
to the molecular axis due to their precession about this line. If
A i: O . the electronic states exhibit a two-fold degeneracy which
is removed by the rotational-electronic interactions giving rise to a
splitting of levels that is called A -doub1ing. (Before 1930, this
was called a'-doub1ing). The splitting is due to a small difference
in the energy depending on the sense of the over-all rotation relative

to the electrmic motion.

Case : If Ato , or if L* is small, then 8* cannot be considered

as tightly coupled to the molecular axis; then A and 0 form a result-
ant N* to which 8* couples, forming the total angular momentum

vector J*. In this scheme the spin is thus uncoupled from the molecular
axis, as shown in Fig. 3. When L* is small but A #0 , an electronic
degeneracy is again re:noved by the rotational-electrcnic interactions

giving rise to A-doubling similar to case (a).

 

 

38

Case [c]: If the axial field is not strong enough to break down
the spin-orbit coupling of the individual atoms that form the
molecule, then the total angular momenta J i*h of the individual

atoms will precess about the molecular axis with quantized projec-
tions along the molecular axis that add to form :11: = IZ J1“ (axial
component)‘h which then is added to the rotational angular momentum 0h
forming the resultant total angular momentum J*h of the molecule.

Case (c) is illustrated in Fig. 4.

 

 

Fig. 4

Case [d]: If one electron in a molecule mows in an orbit that is
large canpared with separation of the nuclei then this electron,

which carries most of the orbital angular momentum, will be influenced
by axial field polarization, so L* will not have a quantized projection
along the molecular axis but will add to the angular momentum of

rotation (with quantized magnitude R*) to form N* which then adds to

39

5* to form the total angular momentum J*h of the molecule, as in
Fig. 5.

Rotational-electronic interactions cause each rotational
level to be split into (2L + 1) components. As in case (b) the

Spin is uncoupled from the molecular axis.

 

Fig. 5

The development of these ideas is given in a series of papers
published by Hund3:‘*-505-7 in 1926, 1927, and 1928. Experimental
work since that time has demonstrated that cases (a) and (b) are the
mes which principally occur along with a coupling scheme inter-
mediate between (a) and (b).

Further work was done by Hill and Van Vleck8 (1928) who
developed a theory of A-type doubling. They considered the Spin-
rotation interaction and showed that case (b) can be converted to
case (a) with an adiabatic increase in coupling energy. The coupling
energy was found to be prOportional to the cosine of the angle

between S and the molecular axis. The treatment forms the basis of

40

all further work on spin-uncoupling phenomena in coupling inter-
mediate between Hund's cases (a) and (b).

An experiment by Weizel9 (1928) in whiCh he discovered nine
bands in the He,L spectrum consistent with the angular momentum
becoming uncoupled from the rotation helped verify Hund's ideas.

The theory of the distortion of spin multiplets produced
when the molecule rotates along with A-type doubling was developed
by Van Vleck10 (1929)o He predicted that singlet 4T states would
exhibit A-type doubling proportional to j(j + 1) where j is the
rotational quantum number, ,A doubling, he f6und, is modified by the
spin in ‘11 states, while ,0 doubling in is states is due to rota-
tional coupling with the fluctuating components of spin and orbital
angular momentum perpendicular to the molecular axis.

Rotational uncoupling of the orbital angular momentum from
the molecular axis was discussed by Watson11 (1929). He concluded
that the rotational energy is limited by rotational instability due
to the uncoupling.

Kronig and Fujioka12 (1930) outline a method by which
constants that measure the decoupling may be determined from.the
spectrum,and the manner in which they affect intensitieso Using this
method Fujiokal3 developed several intensity expressions.

Mulliken and Christyl“ (1931) refbrmulated Van Vleck's equation
for A -type and spin doubling in 1W ,‘Z , and ”'11 states; the results
were applied to several molecules (including intermediate coupling
cases)o They found that agreement was particularly good for CaH.

They revised some doubtful J values and identified a new branch in

41

the spectrum.

Rotational uncoupling was discussed by Davidson15 (1932)
who fOund band values and intensities that agree fairly well with
theory for bands in the hydrogen molecular spectrum.

From the work of Van Vleck and that of Kronig the conditions
for the occurrence of perturbations were investigated in detail by
Dieke16 (1935). He found that when the electronic motion can be
described.approximate1y by the precession of a constant angular
momentum about the molecular axis, the elements of the perturbae
tion matrix can be calculated completely.

Van Vleck17 (1936) demonstrated that the coefficient of
J(J + 1) and (v + 1/2) in the energy terms of hydrogen or deuterium
must contain small corrections due to L uncoupling. The magnitudes
were estimated.

The explicit form.of the interaction between rotation and
spin was given by Kovacs18 (1961) for cases intermediate between
(a) and (b) in the “11 , 1A ,311, 8A , and 4'n' states. Previously
the explicit form of the interaction between rotation and spin was
known only for Hund's case (b)°

MOst of the major theoretical work on Hund's cases was
completed in 1936. An outline, again in historical order, of other
work done in this area will now be given.

Mulliken19 (1927) derived equations for the intensity in '
case (b) doublet states in diatomic molecules.

Hund20 (1928) showed that rotational uncoupling has only

limited application when the molecule is in strong electric or

42

magnetic fields.

Mulliken21 (1929) published a table smarizing the experi-
mental work that had been done on A -type doubling up to that time.
Diekezz (1929) discussed the influence of a progression

between cases (a) and (b) on the band spectrum of helium.

Mulliken23 (1930) gave rules for the determination of .n.
values and symmetry properties for case (c) coupling.

Mullikenz“ (1930) published a table of spectroscopic notation
proposing some revisions. One change was the decision to give the
name x-type - doubling to the effect that had previously been called
0' ~type doubling.

Van Vleck25 (1951) reviewed the theory of rotational uncoupl-
ing in modern formulation relying heavily on angular momentum
conmutation relatims. He then applied this to:

(a) Polyatomic molecules with no internal angular momentum.

(b) Electron spin coupling in diatomic molecules not in

2 states.

(c) A -type and p-type doubling.

(d) Coupling of electron spins in polyatomic molecules.

(e) Coupling of nuclear spins in molecules with S - 0.

(f) Intensities and behavior in external fields.

Mann and Hausez‘S (1960) studied rotational uncoupling by .
measuring the magnitude of the infrared Faraday rotation (rotation
of the plane of polarization of incident plane polarized light) in

the N0 molecule in "Try and "11' states.
2 39.

43

Hougenz" (1963) obtained expressions for the rotational
energy levels in vibronic states using spin uncoupling concepts.

Chiu28 (1964) studied predissociation of diatomic molecules
in case (b) coupling.

Flygare29 (1964) derived equations relating spin-rotation
constants and magnetic shielding.

Raynes3° (1964) presented matrix elements for the Hamiltonian
of a nonlinear polyatomic molecule in a multiplet electronic state
which include magnetic interactions between unpaired electrons

when the multiplicity is non-zero.

CHAPTER 3
THE JAHN-TELLER EFFECT

Degenerate electronic wave functions are possible only if
all the atoms of a molecule lie on a straight line, according to
a theorem proved by H. A. Jahn and E. Teller31 in 1937. In all
other cases involving degeneracy of the electrmic energy, a more
stable configuration will be one in which the nuclear framework
is distorted by the electron cloud. This has the effect of removing
the orbital electronic degeneracy by the perturbing influence of
the distortion. The degenerate states will be split in energy by
an amount approximately determined by first order perturbation

theory as:
EC. -E.. = I‘V'H'Volr

where H is the perturbation potential, and E; "En is the amount
by which the electronic energy is shifted if it is in state ‘0’,

in the absence of the perturbation. The perturbation takes the
form of a distortion vibration which is usually called a Jahn-Teller
vibration. Since degeneracy in the electronic wave function was
ignored in the original work by Born and Oppenheimer, the Jahn-
Teller effect lies thus beyond the scape of the Born-Oppenheimen

approximation.

44

45

A distinction.must be made between the so called ”static Jahn-
Teller effect," and the "dynamical Jahn-Teller effect." The former
refers to distortion of the nuclear framework by a given electronic
configuration, i.e. the equilibrium configuration of the nuclei
is permanently changed by the electronic distortion. The latter,
which is the situation that obtains when the electron-nuclear
coupling is weak deals with the distortion of the nuclear motion
by the electronic motion without the nuclei taking up a new equi-
librium position.

The Jahn-Teller theorem holds rigorously only if the wave-
fUnctions considered are for all electrons involved in molecular
binding. It can further be shown that spin degeneracy is also
necessarily removed by the distortion of the nuclear framework if
a stable molecule is to result.

The Jahn-Teller theorem was introduced in 1937. However,
further contributions to the basic theory included in the general
discussion above were made by H. A. Jahn32 (1938) and by Sponer and
Tellersl+ (1941). Further contributions in the area follow below.

V'an'Vleck33 (1939) in developing the theory for the structure
of a molecular cluster of the fOrm X’6H20 (X 8 Ti, V, Cr) fbund
that stability is achieved only if the H20 groups are distorted
from a cubical arrangement in the manner required by the Jahn-
Teller theorem.

The ideas of Jahn and Teller were enlarged upon by Opik and
Pryce3“ (1957) in a survey article in which they discuss the linear
molecule as well as octahedral complexes.

MOffitt and Liehr3S (1957) discussed the implications

46

of the Jahn-Teller theorem for the case that the distortion forces
tending to lower the electronic symmetry are of the same order of
magnitude as the vibrational restoring forces. They show that this
leads to direct coupling between electronic motion and nuclear

modes of vibration in the dynamical Jahn-Teller effect. Extensive
Calculations were perfbrmed. These are of importance fOr the general
understanding of vibrational effects in the ultraviolet spectra

of molecules .

It was fOund by Thorson"5 (1958) that some forbidden transi-
tions become allowed when the electronic degeneracy is removed by
the Jahn-Teller effect.

Longuet-Higgins, Opik, Pryce, and Sack37 (1958) provided a
survey of the dynamical Jahn-Teller effect and also found that
several "ferbidden" electronic transitions become allowed transitions
when the nuclear framework is distorted to remove electronic de-
generacy.

Clinton and Rice36 (1958) reformulated the Jahn-Teller theorem
with the aid of the Hellmann-Feynman theorem (which states that
many properties of molecules may be explained by considering the
electron cloud as a classical charge distribution rather than
relating it to the customary quantum mechanical interpretation).

In this way they were able to formulate the problem in terms of
forces rather than in terms of the usual energy approach.

Child“9 (1960) feund by group theoretical arguments that
under certain conditions there is no Jahn-Teller coupling between

vibrations of the same symmetry type.

47

Clinton38 (1960) outlined a dynamical treatment of the Jahn-
Teller effect in which effects in the moving nuclear framework
are treated as perturbations on the orientation of the charge
density in inertial space. The Hellmann-Feynmann theorem was used.

It was shown by Hobey and M'cLachlanso (1960)\that the Born-
Oppenheimer calculation can be adapted directly to a degenerate
electronic state. They used a dynamical Jahn-Teller treatment to
set up the equations of motion and discussed symmetry-forbidden
transitions that become allowed through Jahn-Teller distortion.

Clinton and Hamilton51 (1960) used the results of Clinton
and Rice36 to calculate force curves for 0: and NO.

A dynamical Jahn-Teller treatment was used by Liehr52 (1960)
to calculate vibronic intensities in electronically forbidden
bands.

Zalewski39 (1961) presented the results of a study of the
static Jahn-Teller effect in the CGHG+ ion. Bond lengths were
computed.

Child and Longuet-Higgins“° (1961) deve10ped the theory
needed to interpret infrared, Raman, and microwave spectra of
‘molecules in electronic states with orbital degeneracy. Conclu-
sions were drawn about a molecular dipole moment that would be
symmetry-forbidden in a non-degenerate electronic state, and
Jahn-Teller active vibrations that give rise to overtones.

Child“1 (1962) investigated the general case of a four-fold
degenerate octahedral molecule for strong vibronic coupling with
a Jahn-Teller vibration and made predictions about the spectra

that should result.

48

The Jahn—Teller effect in the particular case of aromatic
ions was studied by Coulson and Golebiewski“2 (1962). They feund
a potential function more general than that of Moffitt and Liehr35.
It is in good agreement with theoretical electronic structure
calculations for benzene and triphenylene.

Coulson and Strauss"3 (1962) computed potential energy

4.

curves for an arbitrary displacement of the atoms in CHu+, CF“ ,

and the excited states of NH3+ and NH3 by using the Hellmann-
Feynman theorem. They were able to make predictions of the
magnitude of the displacement of a nucleus in the static Jahn-
Teller effect.

Child““ (1963) deve10ped formulas for vibronic energy levels
of electronically degenerate molecules that exhibit a weak Jahn-
Teller effect.

The static Jahn-Teller effect in octahedral and tetrahedral
molecules was examined by Birman53 (1963). ,Several modes of
instability were found, and suggestions were made for further work.

Forgman and Orgel“6 (1959) measured the infrared spectra
of the tris-acetylacetonates of chromium, manganese, iron and alumi-
num to 400 cm-) The nature of the bands which they attributed to
the vibrational motion of the oxygen atoms relative to the metal
was, they concluded, consistent with the Operation of the Jahn-
Teller mechanism in the manganic compound.

Claasen and Weinstock“7 (1960) failed to detect a Jahn-Teller
effect in IrF6 similar to that found in OsFG.

Snyder"8 (1960) attributed poor resolution in the E.S.R.

49

spectra of aromatic negative ions to the Jahn-Teller effect.
Experimental work done on the Jahn-Teller effect that comes
under the general heading of solid state physics is not covered
here although.much has been done in that area.
Further references to the Jahn-Teller effect will be found

in another section dealing with general vibronic interactions.

CHAPTER 4
THE RENNER EFFECT

The structure of the n-term in linear, triatomic molecules
was investigated in 1934 by Renner55. The two-fold degeneracy
in the n-term could be removed, he found, by a perturbation
produced when the molecule was "bent." This bending is produced
by two-fo1d degenerate vibrations, the "bending modes."

A two-feld degenerate electronic wave function of the
form “qr: hi.“ en” describes the 1: state in a linear triatomic
molecule when all of the nuclei lie on a straight line. (Where ¢>
is the azimuthal coordinate) and .K is the component of the total
electronic angular momentum along the molecular axis. (The angle
4) is mearured around the figure axis.) Under the perturbing
influence of a distortion vibration, the degeneracy is split such
that even (gerade) and odd (ungerade) non-degenerate wave fUnctions
result. These may be written:

(6” + e Wb)

.3 - .L.
2M7!-
w= 51,—,(62 m— 6"”)

Renner was able to compute the difference in energy in the'Hfi
and. 46 vibronic states.
There was no further work on this effect until Dressler and

Ramsay 55 (1957) suggested that the transition between two states

50

51

with large vibronic splitting in the electronic absorption spectrum
of NH2 should be written 2A1 ”u "231 according to Mulliken's defini-
tion of B1. They expected spectra of the same type fer HCO, CH2+,
BHZ, and H20+.

The ideas of Dressler and Ramsay lead to predicted frequencies
that are in agreement with experimental observation, according to
P0ple and Longuet-Higgins57 (1958). Pople58 (1960) extended the
theory of vibronic interaction in linear triatomic molecules in n
electronic states to account for coupling between an odd electron
spin and orbital angular momentum. He obtained expressions for
splittings and shifts in energy.

Hougen59 (1962) considered the effect of Fermi resonance on
the vibronic energy levels of linear triatomic molecules in n
electronic states when the Renner, and spin-orbit interactions are
small compared to the distortion vibration frequency. The results
agree with experimental data on the Azn vibronic states of 802.

Hougen and Jesson5° (1963) give expressions for anharmonic
corrections to the energy of vibration of linear triatomic molecules
in 1 electronic states with very small Renner effect and spin-orbit

interaction.

CHAPTER 5
THE GENERAL INTERACTION BETWEEN
ELECTRONIC MOTION AND NUCLEAR VIBRATIONS

In the Born-Oppenheimer approximation, interactions between
nuclear and electronic motion are completely neglected. The wave
function e is assumed separable: w = we on, where w is the com-
plete wave function, we is the purely electronic wave function,
and on is the wave function corresponding to nuclear'motion.
Actually the wave function u contains an interaction term
neglected by the Born-Oppenheimer approximation, i.e., w -
(we ¢n+ wen). This function is termed the "vibronic wave function"
if on s uv (vibrational wave function) and wen s wev (an inter-
action term that cannot be factored). The function u is termed

" . . . n . = =
the rOV1bron1c wave functlon 1f on - wv or and wen - w where

evr’
or is the rotational wave function and wevr is an interaction term
involving vibration, rotation, and electronic terms that cannot

be factored.

The Jahn-Teller and Renner effects are seen to be special
cases of this more general treatment of interaction terms ignored
in the Born-Oppenheimer approximation. At times it is difficult
to distinguish cases which should be classified Jahn-Teller effect
or Renner effect from the more general vibration-electronic inter-

actions.

The first significant paper published in this area - which

52

53

remains a very active one to this day - was by Condon61 (1927)

in which he pr0posed that energy eigenvalues should be the sum of

a function depending on electronic quantum numbers and a function
depending on vibrational quantum numbers, i.e., w = wne’ where wne
is a function of electronic quantum numbers, electronic coordinates,
and nuclear coordinates. Little appears in the literature fellow-
ing Condon's contribution until 1956. (WOrk on the Jahn-Teller
effect and Renner effect is not considered here.)

Wu and Bhatia62 (1956) found it necessary to include the
coupling between electronic and nuclear motion when considering
Vander waal's interactions, since these are of the same order of
magnitude.

A survey of non-empirical and semiempirical calculations of
vibronic interaction was provided by Liehr63 (1957).

Liehrs“ (1957) showed that the complete molecular wave
equation must be modified, if approximate electronic wave functions
are used, in order for the Born-Oppenheimer approximation to be
applied prOperly. He provided a reformulation of the Born-Oppenheimer
calculation that incorporates the needed modifications.

The interaction between nuclear and electronic motion in
degenerate electronic states of octahedral molecules was investigated
by MOffitt and Thorson65 (1957).

Liehr55i57i58 (1958) performed calculations for several
rovibronic intensities. He found, however, poor agreement with
experiment at 50,000 cm'l. and 39,500 cm'l. This paper was followed
by another in which he evaluates a number of integrals that appeared

in the first. A third paper was published to provide some numerical

54

corrections to the intensities of vibronic transitions previously
calculated.

Dressler and Ramsay69 (1959) measured the electronic absorp-
tion spectra of NH2 and ND2 and were able to verify vibronic
structure in an excited vibronic state.

Liehr7° (1960) discussed the variation of electronic energy
as the nuclei of the C6H6+ molecule are displaced. Vibronic
constants for the determination of energies were computed, and also
vibronically allowed intensities were calculated. Only fair
agreement with experiment was obtained.

Large vibronic interactions in complex molecules should not
be expected, but radiationless transitions are highly probable
according to El'Yoshevich71 (1960).

Fulton and Gouterman72 (1961) presented a general mathematical
treatment of the vibronic coupling of two electronic states. They
show that spectral distribution in bands differ considerably from
the expected if vibronic coupling is present.

Liehr73 (1961) derived formulae for non-degenerate electron
distributions by using the Born-Oppenheimer approximation and first
order perturbation theory. He was able to compute vibronic ab-
sorption intensities of benzene, the cyclopentadienide ion, and
the tropylium ion. Configurational instability is found for these
in agreement with the Jahn-Teller theorem. A.mathematical and-
pictorial description of the nuclear dynamics of molecules exhibit-
ing the Jahn-Teller effect is given. Suggestions for future work

were made.

55

An essay concerned with the classification of vibronic
interactions in molecular systems was written by Liehr7“ (1962).

He used the pictorial model provided by Jahn and Teller31 and by
Sponer and Tellers“.

Second order perturbation theory was used by Bader7s (1962)
to determine the change in electronic charge density due to nuclear
vibration. He found a particular type of electronic distortion
is energetically favored over other possible distortions.

Merrifield76 (1963) showed that the vibronic Schroedinger
equation may be solved numerically if it is assumed that electronic
excitation influences the nuclei only in changing their equilibrium
distance but not their frequency of vibration.

Kolos and WOlniewicz77 were able to compute energies and
expectation values for the vibronic ground states and the first

vibrational excited states of H2, D and T2 using the complete

2.
non-relativistic Hamiltonian and l47-term variational wave functions.
Hougen78 (1964) found that in vibronic interactions in
molecules with a fOurfold symmetry axis the Jahn-Teller active
vibrations are non-degenerate, whereas degenerate vibrations are not
Jahn-Teller active. The Renner effect does affect the position of
the energy levels, however. In limiting cases of the Jahn-Teller
effect and the Renner effect it is possible to write vibronic
wave functions as a product of vibrational and electronic wave-
functions even though there are degenerate electronic states and

vibrations capable of removing the degeneracy.

General vibronic equations for the COUpling of two electronic

56

states in terms of previously given adiabatic potentials72, in
which the coupling terms depend only on the coordinates, were
deve10ped by Gouterman79 (1965). This is the last major theoretical
contribution as of this writing. Other work pertinent to the
theory of vibronic coupling will now be presented.

Herzberg and Teller80 (1933) discussed the selection rules
fOr the vibration quantum numbers during an electronic transition.

Using the fact that with group theoretical calculations it
can be shown that all degenerate electronic states in non-linear
polyatomic molecules are unstab1e31, Narumi and Takano81 (1950)
computed the vibronic interaction energy, with the assumption that
it is on the order of the potential energy of the vibration.

McDowell82 (1954) discussed the fbrmation of different
vibronic states in methane under impact of electrons with known
energy.

Sidman and McClure83 (1956) studied the absorption and
emission spectra of'azulene in which, they decided, one of the
transitions is perturbed by a vibrational-electronic interaction.

A theoretical discussion of the weak bands of formaldehyde
was presented by Pople and Sidmane“ (1957). They showed that the
intensity of the perpendicular bands can be accounted for by
vibrationally-induced mixing of excited electronic states.

Witowski and Moffitt85 (1960) derived the Hamiltonian that
represents the vibronic states of a dimer formed by two identical
molecules.

Albrecht86 (1960) suggested that the major part of the

57

electric-dipole allowedness in the phosphoresence of benzene is
due to vibronic mixing of 3Blu and 3Blu states.

A formula was derived for line shape contours of a band due
to an electronic-vibrational transition by Rebane87 (1960).

DeVOe88 (1962) used first order perturbation theory to explain
the electronic absorption spectrum of chromophores by considering
the interaction between vibronic lines of the same electronic
band or others.

In a conference on luminescence the Fourier representation of
vibronic bands was discussed by Stepanov89 (1962).

Albrecht90 (1963) used second order perturbation theory to
bring dipole allowed character into a spin-forbidden transition.

He considered spin-vibronic coupling in this treatment.

Hougen91 (1963) discussed nearly degenerate vibronic states.
He points out a loose analogy to the problem of spin uncoupling in
the 2n state of a diatomic molecule in his treatment.

Read92 (1964) discussed the effect of vibronic interactions
on the differential cross sections for excitation of a molecular

state by electron collision.

CHAPTER 6
CRITIQUES AND REFINEMENTS OF THE APPROXIMATION

Presented in this final chapter is significant work of
fairly recent origin pertaining to (a) critical evaluation of the
Born-Oppenheimer approximation, (b) consideration of correction
terms neglected in the Born—Oppenheimer approximation, and (c)
methods of solution that attempt to not introduce the Born-
Oppenheimer approximation at all.

In 1951 Born93 devised a method which permits the direct
inclusion of terms representing interactions between nuclear and
electronic motion. It was assumed that w total (x,X) = 2% wn(X)-
¢n(x,X). Here w total is a function of both nuclear and electronic
coordinates, X and x, ¢n(X) depends on nuclear coordinates (X), and
¢n(x,X) depends on electronic coordinates (x) with the nuclear
coordinates (X) entering as parameters. This treatment, however,
still makes use of the adiabatic approximation in that the nuclei
are considered clamped for the computation of ¢n(x,X), the electronic
wave function.

Aroeste9“ (1953) found that expanding the molecular wave
function by the Born-Oppenheimer method in terms of the parameter
r = (m/M)V“ provides perturbation matrix elements of the first
order in K. Some of these contain non-adiabatic terms which may,

in principle, be calculated.

58

59

The coupling between electronic and nuclear motions of two
helium atoms has been considered by wu95 (1956). He concluded that
this interaction is not negligible when compared to the vander Waal's
interaction.

Dalgarno anndCarroll96 investigated the coupling between
electronic and nuclear motion in diatomic molecules, particularly
for cases of large nuclear separation. They found that the
nuclear-electronic coupling energy is on the order of % times the
electronic kinetic energy if the molecule were imagined separated
adiabatically to two atoms in S states. Hence,'for this situation
the nuclear-electronic coupling may be ignored. If one or both of
the separated atoms are in non-zero orbital angular momentum states,
then the coupling cannot be ignored since it will not be negligible
compared to the vander Waal's interaction. Quantitative results are
given for the 150:; and 1P0; configurations of H: and the 12;
state of'Hz. An estimate of the error involved in the use of the
Born-Oppenheimer approximation is also given. This work was con-
tinued and enlarged upon in a later paper97.

A variational method was formulated by Kolos and Roothaan98
(1960) fOr calculating the exact electronic wave function by explicitly
accounting for the interelectronic distance in the hydrogen molecule.
Excellent agreement with experiment was attained for energy calcula-
tion. A comparison with various approximate methods of solution‘
was made in order to obtain a better evaluation of those methods that
must be used fer more than two-electron systems.

Jepson and Hirschfelder99 (1960) present a review of the

60

work that has been done on calculation of the coupling terms
neglected in the Born-Oppenheimer approximation. They suggest
that more accurate wave functions are attainable if center-of‘mass
coordinates are used rather than the usual coordinate system

fixed in space.

The portion of the binding energy of the hydrogen molecule
that may be attributed to interaction between nuclear and electronic
motion was calculated by Kolos and Wolniewiczloo (1961). The
correction to the computed binding energy which ignores this inter-
action cannOt be checked directly, however, as it is smaller than
the experimental error of the best available value.

Froman101 (1962) devised a method for the calculation of
"reduced electronic energy" which is found by assuming finite
nuclear mass for molecules similar to the manner in which it is
done for atoms. He uses the Born-Oppenheimer approximation in
this calculation and claims to have attained results in slightly
better agreement with experiment than heretofore available for
H2, HD, and D2. The effect, as expected, is largest for H2.

Kolos and Wolniewicz102 (1963) discuss the shortcomings
and limitations of the Born-Oppenheimer approximation. They
assert that greatly improved results will not follow from attempts
to improve the approximation. Complete accuracy will come only
from exact solution of the complete wave equation for all the
particles involved. A variational procedure is formulated which
does not at any point introduce separation of nuclear and electronic

motion. The wave equation used is the Schroedinger equation.

61

This equation is, however, not completely satisfactory to the
accuracy desired, because such matters as spin effects and rela-
tivistic corrections are not included. An extended discussion of
these effects and their expected contributions to the energy is
given. Application of the procedure to the hydrogen.molecule is
outlined and gives satisfactory results.

The computational work on the hydrogen molecule is extended
and further refined by Kolos and Bolnevich103 (1963).

The Franck-Condon principle was re-formulated by Tavgerlo“
(1963) so that it would be more consistent with the quantum
mechanics of electronic transitions in polyatomic molecules.

Further calculations on the hydrogen molecule were performed
by Kolos and Wolniewicz105 (1964) by using the methods given in
a previous workloz.

The interaction between electronic and nuclear'motion was
analyzed by'Micha106 (1964) with the aid of time-dependent quantum
mechanics in a study of molecular systems. A comparison was made
with the results of the Born-Oppenheimer approximation.

Fisk and Kirtman1°7 (1964) investigated sources of error
involved in the Born-Oppenheimer treatment. An effective potential
function for nuclear motion was found in terms of vibrational
momentum and the internuclear distance. A method was presented for
computing the energy shifts due to nuclear-electronic COUpling‘
which was then applied to H2 by obtaining a numerical solution of
the Schroedinger wave equation for the effective potential. An

energy shift of about 0.2 cm“1 was computed due to the interaction.

62

It is now possible to solve the Schroedinger wave equation
to experimental accuracy for the hydrogen molecule, but for more
complex molecules for which exact solution is virtually impossible
at the present time(even with numerical techniques), the Born-

Oppenheimer approximation continues to be of value.

10
11
12
13
14
15
16
17
18
19
20
21

I.

B
H
F
F
F
F
F
E
W.
J
W
R
Y
R
P
G
J

.H.

.S.
.M.
.H.
.H.

.T.

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. Hund, Zeits f. Physik, 4_0, 742, (1927).

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