^ecknicaL ^2©te

PB 161568

67

ON THE NATURE OF THE

CRYSTAL FIELD APPROXIMATION

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^f\ote

OCTOBER 1960

ON THE NATURE OF THE CRYSTAL FIELD APPROXIMATION

Henry Goldberg and Charles Herzfeld

This research was supported by the Free Radicals Research Program at the National Bureau of Standards,

under contract with the Department of the Army.

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TABLE OF CONTENTS

. CHAPTERS

I. INTRODUCTION 1

II. THE INTERACTION HAMILTONIAN 6

III. THE MATRIX ELEMENTS 21

IV. THE SPLITTING OF LEVELS 55

V. DISCUSSION 67

VI. ACKNOWLEDGEMENTS 89

APPENDICES

A. TAYLOR SERIES EXPANSION OF (1 + x + y ) 2 70

B. THE EXPANSION OF Eim (1,2,3) FOR r u

+ r2B

< rAB

71

r

1A + r3C < rAC

C. THE EXPANSION B + B' OF SECTION II.

81

D. N. NORMALIZATION CONSTANTS 83

l

E. CRTHONORMALIZATION OF THE WAVE FUNCTIONS 85

F. PROOF OF THE FORM OF THE MATRIX ELEMENTS 86
FIGURES

1 . THE SYSTEM H - H

2

(2p UBC ) 2

2. PLOT OF

1 - (2pJ02 x1

BC ;

LIST OF TABLES

II. 1. EXPANSION OF H

INT (1,2,3)

111.1. CLASSIFICATION OF H

INT

TERMS

111. 2. T MODEL, H' , DEPENDENCE ON 2- AND 3-CENTER INTEGRALS

111. 3. H' , DEPENDENCE ON 2- AND 3-GENTER INTEGRALS (T MODEL)

III. A. H

INT

COEFFICIENTS OF H£

2

(T MODEL)

111. 5. H»„ DEPENDENCE ON 2- AND 3-CENTER INTEGRAL (T MODEL)

111. 6. LINEAR MODEL H£

± MATRIX ELEMENTS

B.1. LINEAR TERMS IN ATOMIC COORDINATES (SECOND ORDER TERMS)

B.2. TERMS LINEAR IN ATOMIC COORDINATES (THIRD ORDER TERMS)

B.3. TERMS QUADRATIC "IN ATOMIC COORDINATES (THIRD ORDER TERMS)

B.A. TERMS LINEAR IN ATOMIC COORDINATES (FOURTH ORDER TERMS)

B.5. QUADRATIC TERMS IN ATOMIC. COORDINATES (FOURTH ORDER TERMS)

B.6. CUBIC TERMS IN ATOMIC COORDINATES (FOURTH ORDER TERMS)

B.7. COMPLETE EXPANSION OF Eim FOR T AND LINEAR MODEL
ON THE NATURE OF THE CRYSTAL FIELD APPROXIMATION*

Henry Goldbergs and C. M. Herzfeld

National Bureau of Standards

ABSTRACT

A new method is developed for the treatment of molecular interactions, and is apDlied

to a system consisting of a hydrogen atom in a 2p state and a hydrogen molecule in the ground

state. The interaction of these two species is

calculated using ordinary crystal field theory

and also the new method. A comparison of the

results shows some of the shortcomings of the

conventional crystal field theory, and provides

corrections to it. The new method consists of

1

)

expanding all electron terms of the total

Hamiltonian for the system which involve interactions between the atom and the molecule, thus

transforming the interaction Hamiltonian into

sums of products of one-electron operators, and

2) of using properly ant i symmetrized wave functions made up of products of atom and molecule

eigenfunctions. The calculations show the

effect of the neglect of overlap and exchange

in ordinary crystal field theory.

All calculations and results are presented

in full detail. Transformations cf three-centei

to two-center integrals are given explicitly.

*Based in part on a thesis submitted by Henry Goldberg to

the Faculty of the Graduate School of the University of

Maryland in partial fulfillment of the requirements for

the degree of Doctor of Philosophy in Physics.

+Present address: University cf Buffalo, Buffalo, New York.
ON THE NATURE OF THE CRYSTAL FIELD APPROXIMATION

Henry Goldberg and CM. Herzfeld

I. INTRODUCTION

1 Apparently Freed and Spedcling were the first to point out that

the electrostatic fields in ionic crystals would lift the degeneracy

of states of ions in the crystal, and thus affect their magnetic and

2 optical properties. Shortly thereafter Becquerel , and Brunetti and

3 L

Ollano studied the effects of such crystal fields, and Bethe gave

an extended group theoretical discussion of the effects. A little

' , 5 i

' later Van Vleck and his students laid the foundations for the

practical use of crystal field theory. The development of paramagnetic

resonance and the revival of optical spectroscopy of solids after 194-5

gave strong impetus to further developments of crystal field theory.

6-9

These developments are summarized in several recent reviews .

Crystal field theory has been extraordinarily useful in explaining and cserelating a wide variety of magnetic, optical and chemical

properties. Relatively little attention has been paid, however, to

the basic validity or meaning of the method. In this paper we

examine some of the fundamental properties of crystal field theory.

In particular we examine some of the approximations implied by the

theory.

Usually, crystal field theory proceeds as follows. The Hamiltonian

10 of the species of interest in the solid includes the terms appropriate

for the free reference ion, together with some terms which express its

interactions with the solid. These interactions are assumed to be
largely electrostatic , and are assumed to arise chiefly from the

neighbors of the reference ion. The application of the theory has

usually two parts:

1) A Hamiltonian for the perturbed reference ion is set up,

using- such information of the crystal structure as is available.

The arrangement of neighboring ions or molecules determines the

symmetry of the elctrostatic field in which the reference ion finds

itself. In the absence of definite structural information plausible

assumptions about the geometry must be made. The matrix elements

of this Hamiltonian contain parameters which give the ( initially

unknown) magnitudes of the interactions. The eigenvalues and

eigenfunctions are calculated as a function of these parameters.

The values of the parameters are adjusted to give the best agreement

between predicted and observed properties. In this way the magnetic

susceptibility, paramagnetic resonance absorption, optical spectrum,

thermodynamic, and magneto-optical properties of the material can be

accounted for. If adequate agreement can be obtained using fewer

parameters than independent sets of data, the general assumptions

about the nature of the system and the symmetry of the structure are

probably correct.

2) One attempts to calculate the values of the parameters numerically using various detailed models, to see if a model can be found

with numerical parameters equal to those obtained from the fit with
experiment. This problem is more difficult, and the results much less

certain than those of part one. The reasons for attempting even crude

numerical calculations of the parameters are these: Such calculations

serve to check the reasonableness of the general model, and they may

shed some light on the details of the structure of the center.

Attempts to calculate the parameters in crystal field theory

from electrostatic models have had mixed success. A recent careful

11 calculation by Kleiner of the crystal field splitting constant in

3+

chrome alum, Cr(H 0)/ , gave the wrong magnitude and sign. Tanabe

12 and Sugano have carried out calculations of the crystal field

parameter on the basis of a molecular orbital treatment of the complex

made up of the reference ion and its nearest neighbors. They find a

parameter of correct sign and magnitude. Their chief improvement

over Kleiner's calculation is the careful treatment of exchange

between the electrons on the reference ion and those on the nearest

neighbors

.

It is clear that it must be possible to give a theory of the

perturbations of the reference ion in the crystal which is basically

like crystal field theory. The Wigner-Eckart theorem guarantees

that such a theory will have a finite number of scalar parameters

if a finite number of electron configurations of the reference ion

9- are sufficient to describe its properties . It would, however, be

interesting to see the exact nature of the crystal field approximation

by comparing it with a. theory which is both relatively clear and successful, such as molecular orbital theory. In this paper we carry

out such a comparison between crystal field theory and MO theory,
and are able to make explicit the nature of the crystal field

approximation. The basic approach is, in some ways, similar to

13

Moffitt's method of atoms in molecules . To make the problem as

simple as possible, we study the case of an H atom perturbed by an

H_ molecule. Systems of this type have been studied successfully

with crystal field methods: N atoms perturbed by N« molecules have

been treated by Herzfeld , and atoms perturbed by N. molecules

' 15 by Goldberg and Herzfeld . The hydrogen system exhibits all the

features needed for an application of crystal field theory: The

H atom is taken in a 2p state, which has orbital degeneracy, and

the fU molecule has an electric quadrupole moment whose field

lifts the orbital degeneracy of the atom. Furthermore, this

system gives rise to a three-electron problem, which can be treated

with molecular orbital techniques without exorbitant complications.

The following procedure is used: The total Hamiltonian for

the three electrons and three nuclei is written out. The total

Hamiltonian is then expanded in such a way that it appears in a

form that is directly comparable with the usual form of the crystal

field Hamiltonian. Antisymmetrized wave functions for the 3 electrons

of the system, which are products of atom and molecule functions, are

used to calculate the matrix elements of the expanded Hamiltonian.

The removal of degeneracy of the 2p state, is compared to that

obtained from crystal field theory, and in this way the relationships
are brought out between these matrix elements and the ones of ordinarycrystal field theory. The expansion of the interaction Hamiltonian is

crucial to our approach. It results in producing the two-particle terms

of the interaction Hamiltonian as products of ana-particle Hamiltonians

.

Hence the matrix elements are functions of one-electron integrals only.

The effect of the Pauli exclusion principle appears, therefore, in

relatively simple form. The particular method of expansion of the

interaction Hamiltonian used also eliminates the necessity of

evaluating 3-center integrals. This feature may make it feasible to

obtain numerical results relatively simply. A descriptive designation

of our method could be EH-SA-MO (Expanded Hamiltonian Seperated Atom

Molecular Orbital) Theory.

Expressions showing the explicit dependence of the energy levels

of the system on overlap are given and compared with conventional

expressions obtained by crystal field theory methods. In our treatment a new complication appears: The matrix obtained from the expanded

16 Hamiltonian and the product wave functions is not always hermitian .

1

3

This type of difficulty has been observed by Moffitt in a problem

related to ours. The hermiticity can be restored by modifying the

matrix elements in a standard way.
CHAPTER II

THE INTERACTION HAMILTCNIAN

1 . The Physical Approximations of the Present Theory . The

physical system under investigation consists of a hydrogen atom

(with nucleus A) in a 2p electronic state, and a hydrogen molecule

(with nuclei B and C) in the ground state. The interactions of

these two species will be studied. We assume all nuclear positions

to be fixed, thus ignoring the important effects of vibrations and

rotations on certain aspects of the problem, such as transition

probabilities.

A special problem is presented by the overlap of the electron

clouds. In problems where it is appropriate to treat the interactions

of two species as the interactions of two non-overlapping charge distributions, a straight forward and simple procedure exists. If the

charge distributions do overlap, then the calculations are usually

separated into several parts which correspond to different regions

of convergence. This approach is discussed exhaustively by

17 Hirschfelder, Curtiss and Byrd . In the present treatment we wish

to make a compromise, for the sake of simplicity. Cn the one hand

we use quantum mechanical wave functions for the atom and the

molecule, and these wave functions certainly overlap. In fact

some of the basic results obtained depend on this overlap. On the

other hand, we wish to expand the Hamiltonian expressing the interaction between the atom and molecule in a way which exhibits its

similarity with the crystal field Hamiltonian. We, therefore, limit
ourselves in the expansion to a region which is equivalent to the

zero-overlap region, as discussed by Hirschfelder, et. al. In this

way we obtain a manageable result, at the cost of some consistency.

It should be pointed out that this same inconsistency pervades the

whole of conventional crystal field theory.

When we carry out the expansion of the Hamiltonian in Section II. 4.,

we make the following assumptions: (See Figure 1 for definition of the

variables.)

• For all expansions we assume that r . •_ and r . „ are each larger

than r~~, i.e. that the atom is well separated from the molecule.

In the expansions of the electron coordinates we assume:

Forr

1A

! r

1A <

rAB

0r rAC

Forr

12

: ru

+ v^ < rAB

II.

1

. ^A + r3C < rAC

•

The assumptions for r

12 are the usual ones made in considering Van der

17 Waals interactions of non-overlapping charge distributions . The

assumptions for r^ . are somewhat weaker than the former. The consequences of these assumptions are worked out in Section 4, and

analyzed briefly in Section 5.

2. The Hamiltonian for the System H - H

2

. We write the

Hamiltonian in the following convenient way:

EL. = The Hamiltonian of the free atom

EL = The Hamiltonian of the free molecule

H-J-, = The interaction Hamiltonian between the atom and

the molecule
II.

2

Hjj - --nV2m % - e7ru

B^ = -TiZ/2m V

2

- h

2/2m V^ - e2/r

2B

- e2/r2C

" e2/r3B " e2/r3C + e2/rBC

+ e2/r

23

H

INT = "e2/r2A " e2/r

3A " e2/r

1B " e2/r

1C

+ ?*'*& + ^13 + e2/rAB

+ e2/rAC

An interchange of any 2 electrons does not leave HTM, invariant. But in

the calculation of matrix elements ve only use the total Hamiltonian

and this is properly invariant under interchange of any 2 electrons.

Let us write the Hamiltonian in a slightly different way where the

dependence of the parts on the electron label has been emphasized:

"TOTAL V1) + V2 ' 3) + H

IOT<

1 i 2 >3> • JI - 3

We expand the interaction Hamiltonian in a certain way which is

described in Section II. 4-, making use of the inequalities of

equation (1.1). Then we group together the expansions of different

order in the coordinates of the electron of the H atom. The matrix

obtained from this expansion shows directly the relationship between

crystal field theory and molecular orbital theory.

The solution to Schroedinger ' s equation without the interaction

term is, by definition, denoted as the zeroth order eigenfunction.

If we eliminate H

INT (1;2,3) from H-,--.. then the solution to this
zeroth order Schroedinger equation would be / „(1) r M

(2,3)> where

y R

(1) and ^(2,3) are eigenfunctions of E

R^) , and 1^(2,3)

respectively. ty-a can take on the specific forms 2p , 2p and 2p ,

i.e n = 2, I = 1, with three components.) This is not an acceptable

solution, since it is not antisymmetric on interchange of electrons

1 and 2, for instance. We can, of course, construct a wave function

antisymmetric in the three electrons:

^x = \3^ [2px

(l)<TM(2,3) + 2px(2)^M(3,D + ^(3)^0,2)] 114

where r M is the molecular zero-order eigenfunction, which is assumed

n

antisymmetric in its two electrons and N is a normalization factor.

Two other functions, Z and Jr , formed with 2p and 2p„ respectively, also

y z y z occur. We note that jf cannot strictly be a zeroth order eigenfunction

since i „ and ri. are not orthogonal. We are not really applying

perturbation theory; instead we ask: Given i„ and T as correct

eigenfunctions of the Hamiltonian of the atom and the molecule when

they are far apart , how are the states changed when one brings the

particles together? How do the energies of atomic states change as

a function of distance and orientation? We do not investigate in

this work how the molecular states are affected, but reserve this

18 for later work. Margenau has investigated the H

?

- H problem,

using a zeroth order eigenfunction completely different from ours,

namely functions of spin bond type, constructed of 1s orbitals.

In this treatment the H atom is in the ground state, and no comparison with crystal field theory is possible.
10

3. Notation. In this section, we introduce special notations

to facilitate the discussion, x.^ is the x coordinate of the ith

electron with respect to nucleus M. Similarly the coordinate x-.

represents the x coordinate of nucleus M with respect to nucleus N.

X

BC^= rBC^ XAC' yAC' yAB* r

AB* rAC

are constants » Tne siSn conventions

for the vectors r

1A , r^g, r,c

, etc. are indicated in Figure 1.

- *

• The three atoms are assumed to lie in the xy plane. The following

relations are typical:

rAB

= {

(xA

- ^

(yA." yB

} ' (zA

" *B> }

X

1B = X

AB

+ X

1A

X

1C = X

AC +

*1A

y1B = yAB

+ 71A y1C = yAC + y1A

Z

1B

=Z

1A

Z 1C=Z

1A

X

2A ~

"X

AC * X2C y2A = " yAB

+ y2B

II.

5

X

2A "

"X

AB

+ X

2B y3A "

"yAC + y3C

Z2A = Z2B . - Z3A = Z3C

x

12 = X1A

+ X

AB " X

2B x

13 = X

1A + XAC ~ X3C

IX "*

y12 = y1A

+ yAB " y2B y13 = yU

+ yAC " y3C

Z

12 = Z1A" Z2B z 13 = Z1A" Z

3C
11

2 2 2 ""^ '

rr = r + r + 2r • r

1B r

1A rAB ^r

1A rAB

P

1C = 4 + TtC + 2r

1A ' rAC

II.

6

r

12 ' (X1A " X

2B + XAB

)2 +

<7lA " y2B

+

^AB^-

+ (Z

1A " Z2B

)2

r2A = r2B + rAB " 2r

2B " rAB

r3A ~ r3C + rAC " 2r3C * rAC

We 6.1so denote by special symbols important functions of the coordinates:

a = x

AB

rAB

e^AB'AB

_. -2

Y ~ X

AC

rAC

6 = 7AC

rAC

U = X

1Ar

1A

II.

8

V = y1A

rU

V

1A = ^U +

^1A

V

2B =

<«2B

+ &2B
12

U

3C

s ^30 + *3C

II .9

U

1A * ^1A + *U

W

AB = XABU +

^AC*

W

AC

X

ACU + W *

11.10

W

2B

S

*2BU + 72B

V

J

W

3C

S X

3CU + y3Cv

A. The Expansion. We give here/expansion of Hj._ for the conditions of equation II. 1. Ve recall from sections II. 2, and II.

3

the notation and relationships given there, and the expression for

HTMi and proceed to expand H-.— in the following way. We first

rewrite some of the expressions of section II. 3 (cf. Eq. II. 6):

*

'To

B I'U + r

AC

+ *1» ' ^C^ " rM t

1 + 4rIc * *tt ' trf&* "-1

'» " 14 + 4 " *2B ' ^" *a [1 > 4rf " *2B * Vi$4
13

T

ll = [r3C +

*AC

" ^Tc * ^1"* " rlJ [1 + r3CrIc " *^C • Wlc^

r

^2 = rAB t

1 + < rU " ^B> • <^ " ^B> Tll

+ 2(rU

"^)

* ^BrAB^ n«"

r

13 = rAO [1 + <C * ^o' • (rU -^c' ?AC

+ 2( *U - ^ • WAC^ •

To make clearer the nature of the expansion for the first U expressions,

-1 we write out r

1B in full detail. Thus,

r

1B = rll [1 + (x1A

+ A + At) Vll

+ 2(x

1A

X

AB

+

yiAyAB} rAB

]4 '

2 —2

We now assume r... < r.g (cf. Eq. II.1); hence, r

1ArAB is a second

— 1 —1 o —t^- order quantity in r

1A A\B* Let us write r~ ••= r~ [1 + y + x] ,

where y

2 = r^r"2 and x = (x^x^ + y^^) rj2

. We expand

2 —— 2

[1 + y + x] 2 in a power series of (y + x), and we group the terms

2 according to their order of magnitude. Although y and x represent

2 quantities of different orders of magnitude, y + x < 1 assures the

2-4-

convergence of the Taylor Series expansion of (1 + y + x)~ in

2

powers of y + x. The general result is given in Appendix A. In

the expression for r

12 , we regard (ru

- r2B)

• (r^ - r2B) r~B

as

a second order quantity in |r1A - r__j r~R

, and

(x1A " X

2B) xAB

rAB

+ rAB (y1A " y2B) 7AB

as a first order <J«M»tity

in |i» - r2B | r~ (cf. Eq. II.1). The result of the expansion is

given below, where we have grouped together terms of the same order

of magnitude.
H

To simplify the discussion we adopt the following convention: We

group terms in the expansion o*\H

INT by their order in the expansion,

and then collect the terms of a given degree in the electron coordinates

measured from the atom. We shall use order and degree only in the

sense just given. The zeroth order term:

*2 Kb - *ac - rIB - *ll

+ 2r

IB + *$ = ° •

The first order term: t

. &Z

[r1A * rAB

rAB

+ r

1A

- rAC

rAC ~ r

2B * WaB " r

3C * rAC

rAC

"

(rU " Kb\ ' WAB " ( C " ^C} ' Wlc3 =

°

•

We have shown explicitly the grouping of terms for the zeroth and 1st

order of the expansion. The physical significance of their vanishing

means that up to, but not including, second order terms, the hydrogen

atoms appear to the hydrogen molecule and vice-versa as uncharged

objects without dipole moment. We must go to second order terms to

find any interaction at all. The various orders of the expansions

are presented in vector notation in Table II. 1. The detailed cartesian

form of these expansions which show the resemblance of our treatment

to the expansions of crystal field theory is shown in Appendix B.

It is striking that the 2nd order terms of the expansion contain only

linear terms in x^.. In conventional crystal field theory they would

contribute only through configuration interaction. To find terms
15

TABLE II. 1

Expansion of H

INT (1,2,3)

Second Order Terms:

(?1A ' *5b)pAB

+ (rU ' ^C)rAC " 3V

1A

V

2BrlB " 3U1A

U

3CrIc

Third Order Tarns:

- 2

i rAB ^V1A

r

2B " T2Br

1A

' 2* V

1A " V2B^

r

1A ' r

2B^

L+ r

Io t D

iAr3c- Via- 2(<tia " V^a '^c1

+15/2 [r^Ta, (Vu

- Va) + r-JnlA

l^ (Hu

- l^)] .

Fourth Order Terms:

.-3/4

r

AB ^U - r

2B^ r

1A

+ r

2B " r

1A ' r2B^ " r1A

r

2B^

r

AC ^1A " r3C^r

1A

+ r

3C " r

1A " r3<P " r1A

r

3C^

rIi [ 2r

1A • r2B<

VU " V

2B>

+ 4V

2B<

2V

1A " V

2B>

+ r

2BV

1A

(2V2B " V

'In C2rt. • rtnCB,, - 0,„)

2

+ r?,U,-(2n1 , - U-„) + r

2 V,AZO,. - 0,.)],

rABV

1A

V

2B<

2V

?A " 3V1A

V

2B

+ 2V

2B>

^^uSo^^A-^IA^O* 2^)
16

quadratic in x.. one must go to third order terms. Then we find that

in the latter only linear functions of the molecular electrons coordinates Xtv«> and Xq. arise. Hence, averages of these over unperturbed

molecular functions will vanish. Thus, we must go to the fourth

order expansion to find large effects on the atomic levels.

5. Critique of the Expansion of Section II. 4. » In expanding

o — 1 —1

HTW-(1;2,3) one must sum the following terms: e [2 - r7« .+ 2 r7<]*

^is

INT i,M

m

i>j iJ

clear that the first 'sum has different converging properties than the

' -1

second. For instance, the expansion for r_ given in the last section

converges for < r... < r.g, while that for r~2 converges in the region

< r- . < r.

fi -pr^B' -^us, ^e ©^cpansion of HIN_,(1;2,3) given in

Section II.A is strictly correct only in the region of < r

1

. < rAB - r«B.

We summarize now the converging expansions of attraction and

repulsion terms appearing in H

IN_(1;2,3). We consider the typical sum

-" rl. - rig + ri2 ( r

2B a&ain is understood always to be less than r.-)

and define the regions of converging expansions:

a) r^ and r^

Expansion A converges provided r

1A < r. R

Expansion B converges provided r1A > r. D 1A Ad

b) r-J

Expansion A' converges provided < r... < rAB- r-B

Expansion B

1 converges provided < r. n - r_- < r„ . < r iri Ao <jo \k ka
Expansion

A + A

1

AA+ B'

B + C

B + D»

17

Expansion C converges provided rAB < r^ < r^ + r^

Expansion D

! converges provided rAB + r2B < r1A < »

We now describe the 4- regions and indicate how the different sums

should be added.

Region Condition

Region I < r^ < r^ - r2B

Region II rAB- r

2B

<r

1A <

. rAB

Region III rAB

< r^ < rAB

+ r2B

Region IV rAB

+

^ < r^ < -

In the detailed calculations of the interactions of the H atom

and H

? molecule, we confine ourselves to the "easy" region I of the

expansion. The results for the region I can be related in detail

to ordinary crystal field theory. For the sake of illustrating the

complexities which arise- in the expansions for the other regions,

we present in the next few pages details for the expansions B and

B'. The attraction and repulsion terms are rewritten so that the

expansion is readily made in the region of interest here. The factor

e has been omitted for simplicity. Then, H

INT (1;2,3) contains the

terms:

" rw = -^ + feu + *u • feu^

- t

?c - -^ + fet + *u • v;?4

.

- r'A

-Tll^ + fell - ^2B • fe&*
18

i

_ r

- 1 =-r

" 1 n + r

2 r~

2 - 2?" • r** r~

2

l~2 r

3A AG L ' r3CrAC ^r3C

rAC

rAC J

+ r i2 = r

ll [1 + 2(?Ib " *W * *Ur

1A + ( Cb " ^ ' ( ^B " ^B)rU]

+ r

13 = r1A

[1 + 2(rTc " ^C> ' ^ArU

+

<*a"c ~ ^C> * (?AC 7

?3C)rU] "T

+ rIB + rIc

'

The last 2 terms represent the nuclear repulsion between the atom and

the molecule and are constants in the problem. The zeroth order terms

are:

-1 -1 -1 -1 -1 -1 -1 -1 _ n

" r

1A " r1A " rAB " rAC

+ r

1A + r1A + rAB

+ rAC " ° "

The first order terms:

r

1A^1A ' [?2B +

*3C ] " rAB^B * ^B " rll^C ' ^AC * °

*

Second order terms: In order to perform the expansion more easily,

we use the notation of section II. 3. The expression for H

TNT thus

becomes:

- r"

1 ='-r~1 h + r

2

r"

2 + 2W 1"' r

1B

r

1A U rAB

r

1A ^

W

AB J 2

" rVc =

"rU t

1 + rfcrU * *uF~2

" r

2A = ~'u t

1 + 4rAB - 2V2B^
19

+ *« = r

il [1 + <C " 4> • ( ^ " 3ffi> *U +2(WAB >

W2B»

+ *« = ru [1 + (rTc

"^ • (rtc " *& ru

+ *ac " V1

"4 '

Using the Taylor series, Appendix A, we find the 2nd order expansion

to be of the form -£ y

9

+ 3/8 x

9-' . We collect the various terms, and

simplify to obtain:

- i 7

2 = i r'l [viz + r

2

,, - 2^B

• 7*, - 2^ • ^ * $ r^ + £ f'^

3/8 x

2 = 3/2 r;| [(W2B) 2

+ (V30) 2 - W^ - SW^J - 3/2 r"^ - 3/2 r^ .

The third and fourth order expansions will be found in Appendix C. Some

conclusions can now be reached. We see here that the first order terms

no longer vanish. If we look at the members we find that r

1A appears

in the denominator. Thus, as we integrate from (r. B

± r-B

) to « the

integral is damped quite considerably beyond r. R (average of the 2

lower limits of integration). The fact that rr. occurs in the denominator,

makes the overlap integrals between the 2p. and 1s_ function (out of which

\. is constructed) behave beyond the region r... = r.g as if the 2p electron

is in a 1s state. In such a case, the overlap integral from rAt> to °° Ad
20

in the r-integration does not contribute very much. This argument

applies even more strongly to the higher order terms in the expansion.

We therefore conclude that reasonable accuracy could be obtained if

we cut off the r-integration at a distance where the overlap is not

appreciable between 1s electrons belonging to the different systems.

This is discussed in Reference 16.

As was pointed out above, in our detailed considerations

throughout the paper we ignore the difficulties just discussed, by

limiting ourselves to the "easy" region of the expansion. It is useful to point out that similar problems of convergence are ignored in

ordinary crystal field theory. There the considerations are also

limited to the "easy" region. Such a procedure lacks, of course,

any real mathematical rigor. The aim of this paper is to shed light

on the nature of the crystal field approximation. We feel, therefore,

justified not to reformulate the expansions used in the theory with

greater rigor, but to carry out our discussion on the level of rigor

customary in the field.
21

i

CHAPTER III

THE MATRIX ELEMENTS

1. The General Result . We wish to know how the H„ molecule

removes the orbital degeneracy of the 2p state of the hydrogen atom.

We have to investigate a 3 x 3 matrix, if we ignore the spins of the

atom for the moment. Before we obtain the matrix elements for this

problem, we define a set of 3 orthonormal functions for the system:

2p , 2p , and 2p are normalized hydrogen atomic orbitals. They are

centered about nucleus A.

a(i) - spin eigenfunction for ith electron with m = '+£- s

(3(i) = spin eigenfunction for ith electron with m = -•£

1sB

(i) "= ground state hydrogen atom function for ith electron about

nucleus B

1s-,(i) = ground state hydrogen atom function for ith electron about

nucleus C.

The choice of molecular functions poses a certain problem. It is clear

that one could take here the point of view that any molecular function

can be used which gives good numerical results, for example an elaborate

self-consistent field function. We wish, however, to get a simple

result whose significance can be understood readily, and we are not

primarily interested in numerical accuracy, at least at present.

Therefore, we have limited ourselves to a molecular function which is

built up from a product of one-electron functions. This would be

achieved either by a valence-bond or a molecular orbital function.
22

The valence-bond function is simpler, and would give better results

at long separations of the nuclei of the molecule. But the molecular

orbital function gives slightly better results for the actual molecule.

An additional reason is the following: The MO type function may give

rise to terms in the final result which are of interest (i.e. those

arising from the ionic structures for the molecule), even though

these terms are not given the proper weight in the result. A valence

bond function would omit such terms altogether. In the derivations

given below the molecular wave function is kept general as long as

possible. We use

^M (2 ' 3) = NM

[1SB

(2) + 1s

C

( 2

^

1s

B

(3) + 1s

C (3)] 2

^ [ a < 2 W3 > " <*(3)0(2)]

J^d',-2,3) =3^N

x

[2px

(1)a(l)^M

(2,3)+2p^a(2^(3,l) + 2px(3)a(3)£ (1,2)]

K,f N , etc., are normalizing constants. See Appendix D.

/ and tys are obtained by replacing x by y, and z, respectively.

V^. stands for V , jp , and ^ , as i = 1,2, and 3 respectively. i * x y z

<E>j, $-r-r> and ^-r-r-r are an orthonormalized set of functions constructed

out of the 3 independent functions j , IP , and If/ . Details are

found in Appendix E. To avoid confusion: ( V^ • | ^K .) refers to

integration over one electron orbital only, tys . refers to

A, 1

the ith atomic orbital about nucleus A. ^ v> c is defined as the

function (1sr + Is^) \~ , which has the normalizing factor NTj .

III.
23

III. 2

The following is an important product of overlap integrals:

a

ij =

^ 2pA il^BG^^Bc' 2pA ^' As the indeX i runS fr0m °ne t0

three, 2p. .refers to the 2p , 2p , and 2p atomic orbitals

a j 1 x y z respectively.

Throughout the present calculation ve shall take the spin of the

H atom to be +J-. The doublet spin degeneracy is, of course, not

lifted in this problem. The set of orthonormalized functions is

chosen as follows:

i *K

•n - < ai2 *i

+ KH' - ^

*iii=JV

The general result of the calculation of the matrix elements of

the total Hamiltonian, using the expansion of Section II. 4- (Region I),

and the functions of equation III.1 is this:

~

(EH

+ y<6 ij- a

ij>

N.N.

i J
2A

Here: HV is the i-th term of a homogeneous polynomial of k-th degree

in coordinates centered on A;

hS is the i-th term of a homogeneous polynomial of m-th degree

a in coordinates centered on B, and multiplies the -t-th term HV ;

H^ is the i-th term of a homogeneous polynomial of the m-th

degree in the coordinates centered on C, multiplying the

'L-th term Hv . A

Each of the polynomials contains constant coefficients involving

powers of internuclear distance in such a way that the total product

has the correct dimension of energy. The details of this are developed

below. , The proof for this result up to and including fourth order terms

of HTM will be found below, and in Appendix F. The simple result of

equation III. 3 depends strongly on the following argument. Examination of the result of the expansion of H.^-, (Section II. 4 and Appendix B)

shows that the result of the expansion may be summarized as follows:

H

INT = £ Kf < + #> • **•*

k,l,m

The nature of the terms H? , etc., is explained above. The details

of their structure can be seen in Appendix B. No 'electron labels appear

in this expression for the interaction Hamiltonian, because all interaction terms are expanded into one-electron Hamiltonians . Furthermore,

these one-electron terms are transformed in such a way (by means of

expressions of type II. 5 and II. 6) that the coordinates of electron 1

always refer to the origin at A, those of electron 2 to the origin at B,
25

and those of electron 3 to the origin at C. The wave functions are

products of one-electron wave functions. Therefore all integrals can

be transformed into one-electron integrals, and these are, of course,

independent of electron labels. It should be noted that this great

simplification arises from the method of expansion used here. The

Pauli principle is still taken into account, however, since the total

Hamiltonian is a symmetric function of the electron labels, and the

wave functions are antisymmetric functions of the electron labels.

The general significance of this very interesting simplification is

not fully worked out at present.

We can give the matrix elements ($ . | EL } <3>

. ) in terms of the

(\|> . j EL | v • ) de sc ribed above

.

(OjlHplGj) = (^IH^)

(*iI HtI*II ) = (1 - a ?2

)4 [a12 C+il^l+i)"' + (^|H

T IV ] (^H^) = (^|H

T N3

)

(©nlHjI^jj) = (1 - a*

2

)"* [a

12 (tjlBljl^) - (t|>2 l%H>3

)3 .

The influence of overlap in the expressions for the matrix elements

should be noted. The overlap between the atomic and molecular functions

causes the mixing of the atomic orbital s regardless of the strength or

nature of the interaction Hamiltonian.
26

There are 2 special orientations of the H atom with respect to

the molecular axis of symmetry of the H« in. which (C>.|HT J0.) reduces

to (\J>. ! H_j^. ). This affords a great simplification in cur analysis

and therefore they will be discussed next. In one orientation the H

atom is situated along the molecular axis of symmetry (linear model).

Here, only a

1 ^ ^ because (2p i^BC ) = and (2p

z HBC ) = 0, from

symmetry considerations. We call this model the linear model.

In the other case the H atom is situated on the perpendicular

bisectrix of the internuclear axis of the molecule (T model). Here

11' has the same functional dependence on the coordinates centered on B

as Hp has on those centered on C. Again, only a~p \ because

(2d hLp) = (2p is^n) = from symmetry considerations. In these 2

cases the matrix elements of IL, can be written as follows:

Wi|hR,) -^ -

hl ) h . * H . K . 2k>m>l ( 2 pA>i |nf |2pA(3 )(*bc !< Hf UBC )

Let us further write the matrix H. . as the sum of 3 other matrices to

facilitate the discussion.

H. . - H°. + N.N.H?. - i N.N.H.' . Ill

o where K. . = (Ep + K,) b. . III
27

r kli - w . i„ml , „ml;

Hi, »S , (?pA)i iHf ^!-W(vX''! ?

PA,.i>

III>9

'^ K,m,t

< ?PA ,ii V«bc< +

^*I*bc><*bcI h"1 ?Pa,j>

o

F. . reoresents the contribution to the Hamiltcnian if the atom and molecule

ij are st an infinite distance from each other.

KT . represents terms which we call crystal field type terms: They will

be discussed later on in Sections 3a, 3b, and in Chapter IV.

II .' . re ore sent s the effect of the Pauli principle on the 3 interacting

electrons, involving products of 2- and 3-center integrals.

The matrix elements, as they appear in Equation III. 3 are not always

hermitean . This is at first surprising since we are evaluating

matrix elements cf the hermitean operator H_. The difficulty arises

because we are satisfying both the Pauli exclusion principle for the

electrons and the fact that we assume the chosen molecular wave function

to be an eigenfunction of II, (cf. II. 2). Mcffitt encountered the

same difficulty in treating a similar problem. But in the special

1«ses we consider, up to and including <Cth order terms of the

interaction Kamiltonian, HI, = for i f j, so that the problem of

hermiticity does not arise.
28

2. Kore Notation. Certain expressions can be simplified very

considerably by adopting additional special notations.

We define, for use with the T model:

p£ = k

B

±kc

. If k=x, P^=x

B

4-x

c

. III. 10

Qkl = Vb ± kcV If k = y

and > = z

> then %z = ¥B + yG

Z C

* m' 11

^Im = Wb ± kcVc m' 12

The symbols k,-i and m stand for variables , and can range independently'

over x,y, and z. If the superscript signs are left off, the + sign

will be understood to be present. aT, a, . , a , a. , a,., and cu

are constant parameters which appear in the interaction Hamiltonian

for the T model. These parameters multiply the following terns

respectively: P. r . , Q1p r., R r, s. , T, , r. , &,r.s.« and P.r.s.t,. ^ J k A' *kl A k A A' klm A* kI A A k A A A

The superscripts for the parameters a are related to the variables in

12 the factors which the parameters multiply. Thus a, multiplies

3 r _±

Pk

X

AyA'

and a

122 mul'tiPlies T

xyv

ZA*

The exPression a FT is an

abbreviation for a, P. + a, P. , etc.
29

The interaction Hamiltonian for the T model can then be written

in the following special form:

Him - £ ( \

pk V + £j€i <C V

k,r,s

I rst ±

, t K *k r

A

SA V • m- 13

k,r,s,t

+ I '-rs ** (ak* Set

rA

S

A

5 k,l,r,s

K,i,m,r

2

A common factor of e has been omitted in every term.

It can be seen from equation 13 that the interaction Hamiltonian

is a sum of terms, in ascending orders of the expansion, and in various

degrees in the electron coordinates. In the above expression all

electron labels are again superfluous. For the reason see the discussion after equation III. A..

For use with the linear model we introduce the following notation:

The constants Opk

, Gpkt , Opk

, Opk4m , otpkl > Opk multiply certain functions

of electron coordinates. The subscript P ranges over the two nuclei B
30

and C of the molecule. Thus cc^ multiplies k

firA

, and acklm multiplies

kp-tja-r.. In these expressions k,£,m,rand s again range over x,y, and

z as above

.

It is then possible to write the interaction Hamiltonian for the

linear model in the form:

- '?. f 4k k

P rA

) + IW „f. f'&t^W

I (aPk "P

rA3

A> P,k,r,s

I (aPkto

,k

P *P "p rA

) III.U

P,k,l,m,r

+

P t\ ,

{ 4ll k

?

1

? rA

S

A

)

"+ L (cff kp rA

sA

t

A

) + ... .

2

A common factor of e has been omitted in every term.

3. Analysis of the Matrix Elements . The form of the matrix element

(Eq. III. 3) together with the expressions for the Hamiltonian (Eq. III. 13

and III. 14-) enable us to make a direct comparison with the matrix elements

obtained by crystal field theory methods. As was already explained

above (Eq. III. 6, 7, 8, 9)> we are able to show that the matrix elements

consist of 3 distinctive parts. The first part represents the zeroth
31

order energy when the 2 interacting systems are at infinite distance

from each other. The second part contains terms which are analogous

to crystal field theory matrix elements, which can be shown to

include a dependence on the overlap between the atomic and the

molecular charge clouds (see Chapter IV). The third part depends on

2- and 3-center integrals. Because of the form of the Hamiltonian,

the 3-center integrals can be reduced to 2-center integrals by means

of simple linear transformations like x_ = x

fi + R $

.

. . -

(For the sake of brevity we write R instead of rBC .) As pointed out

above, we are going to limit ourselves to the 2 special orientations

(the T and linear models), thereby effecting great simplification in

the analysis of these integrals. These simplifications have already

been pointed out in Section III.1, namely, all a. . = 0, i ^ j, and ! H.^O.i/j.

The present analysis is divided as follows: subsections 3a and

3b will deal with the H. . part of the matrix elements for the T and

linear models respectively, while subsections 4a and 4-b will be

concerned with the H! . part of the matrix elements of the T and

J linear models. The HI. matrix elements are presented in the form of

tables. As can be seen from table III. 2 through III. 6, these matrix

elements are dependent on a large number of overlap and two-center

integrals, but because of the nature of the expansion used, no

three-center or exchange integrals occur. Most of these integrals

have not been evaluated before, because they arise only in this type

of analysis. The evaluation of these integrals would be a major

computational task, and is not carried out in this paper.
32

3a. T Model, hJ° . We classify the various order terms of HTM up to and

including 4-th order terms in Table III.1 to give perspective to the

(c) discussion for both sections 3 and 4-. Thus, we can write H:,' in

the following manner.

Here IL. has the same functional dependence on Xq as H-, has on

the x„ except for a possible change of sign. This can be best

understood by noting that x.. x^p + x.. x. c = 0.

This expression occurs in r.. . • r. B + r.. ,. r. c

. The Hamiltonian

E\el 5 m lift Hv (EL + EL ) has only relatively few non-zero matrix elements

k,l,m A B. C

•of the crystal field type H.. . The non-vanishing terms must involve

only even polynomials in H. , because they occur only in matrix elements

of the form (2p..|H7 | 2p. . ) . The selection rules for the matrix

elements (\|>bJhb + HjiJ)__) are more complicated. The molecular wave

functions t|>Rp are even functions under all reflections in planes

perpendicular to coordinate axes which are centered in the molecule.

To derive the necessary selection rules, the terms H_. and H^, which

a Li are functions of (x„, yB

, zJ), and (x

c

, y., z_), respectively, must

be transformed to the "center of molecule" system. When this is

done, and the inversion symmetry of the matrix elements examined, it

is found that the only non-vanishing elements are the following:
33

TABLE III.1

Classification of Terms in H-^TM, in Expansion in Appendix B.

H

INT = I hJ* (iff1 + iff1

) k,l,m A V "B

Pkl

h

A (4

m

+ up

2nd Order Expansion:

linear in x

A

linear in x„, x, B C

3rd Order Expansion:

linear in x.

quadratic in x,

quadratic in x_,, xn linear in x-, xn

£5 L»

4th Order Exoansion:

linear in x.

ouadratic in x,

cubic in xA

cubic in xD , xn

quadratic in xD , x~

linear in x^, x„
34

V^BC* HO-

(c)

The non-vanishing matrix elements of H

N .'. are, therefore, the

following: (See Table B.7 for the terms in the expansion, and use

the selection rules of equation 111.16.)

H

ic

o>=e2|k

^i^v K

r «Boi p;>Bc>

+ <& tv>i«^iw} \i • m- 17

No terms mixed in r . s occur because they do not occur in the expansion

of IL, . ., in combination with the required terms in H^ and H„. See

Table B.7. Here the (iJj-Jq,. |^t}„) are proportional to definite comi5L> iCK Jjw

binations of the components of the quadrupole moment of the molecule

(cf. Ref. 17 pp 839, 840). This relation is striking and will be

further commented upon in the next chapter.

(c) 3b. Linear Model, H. . . The analysis is here somewhat more

involved than above, because the functional dependence of H« and

H^J on their respective coordinates is not the same. This imolies

that in general 0|>bc|h£1 + H^I^J V? even if H^ and H^* have the

same odd power dependence on their respective coordinates.
35

When the selection rules for the matrix elements (^BC I H

B + Hc^BG^

are determined, it is found that the only non-zero elements are the

following (P stands for either B or C)

:

^BC^V = 2 XBC*°

^BC^V = -^ x

BC V ° ^BC^V * ° ^BC^V k °

<^BclVpl V * °

III. 18

Because only even powers of r can produce non-zero matrix elements

of the form (2p.. |H.| 2p. .), which are the multipliers of the matrix

elements of equation III. 18, only the matrix elements of the form

H>BC

|x

p

|^BC ) and U>BC |r^BC ) will occur in H<°\ thecrystal field

part of H

INT

.

(c) The matrix elements of H. . for the linear model are, therefore:

"S"'

2

p.L.. ^cl^plV^A.iK -aI*m)

+ { %cHL Xp Xpl V(2pA,i |rA "l}

2

^

Since there are no terms in HTM of the form (cf . App. B, all such terms

are multiplied by y _ which = in this case):
36

°Pk *? rA

S

A

(r ^ s) ' and °fi±

^ ^

rA

S

A

(r ^s) then (cf * Section

III. 3a) we obtain: ,

^r°\l ^B0l«S *FHBC>

+ «BCI°PL "p MSc" < 2PA,i |rAl 2pA^ »„

r,JC,r

III. 19

These quantities represent averages over the molecular coordinates

which are not, as was true in the previous case, simple linear combinations of components of the quadrupole moment of the molecule.
37

4a. T Model, HI. Matrix Element. To facilitate the discussion,

•^ -^——^

we write (cf. Eq. III. 10, 11, 12, 13):

H

INT = . .^ «k

^

rA

+ °^

S

*k

rAS

A

+ "W ^l r

A

m',t' r ' s ' rs r

m.»

+

°k *£ rA

S

A *A + akt \l rA

8

A

+

"klm \l* rA

'

2

A common factor e has been omitted. The following selection rules

of 2- and 3-center integrals are used in deriving the expressions for

HJ, in the T model case.

(2PXWBC ) = (2PZI*BC ) = 111.21

(2p

y

UBG ) / 111.22

Equation .111.23 is to be interpreted as meaning: (2p. .|P. |^Dn ) - 0,

unless i - k. Similarly, the equation below:

< 2

*A, iMW = [ ^ 6i2 + 6i1

6

k1

6

12

+ 6

i3

6

k3

^ ^i'Sa'IV '

III. 24

This last term vanishes for each i = 1, 2, or 3 unless the conditions

on k and £ given above are satisfied.

(2- A,il TkJV = 6 ik

b m ^a.AJV ' ln- 25
38

^PA,il rAi^Bc)- & ir(^A,il rAlV m' 26

^A^Va'V = £ 6 i2

6 rs + 6 i1 Vl *s2 + 6

i3

6

r3

5 s2^ ^.llVi'V

III. 27

The same remark that was made after equation III. 21, applies here too.

^A.iK S

A *a!V = 6

ir "st ^A.iK h \\%J ' 111M

The results in equations III. 21 - 28 are easily derived by considering

the symmetry properties of the matrix elements, particularly their

behavior under reflections in planes perpendicular to the coordinate

axes, with origin at the center of the molecule. The results for the

elements HI. are presented in Tables III. 2 - 5. The elements HI. (i V j)

vanish identically, as can be shown by use of the selection rules and the

explicit expressions for H-.-,. We only present the HI. matrix elements,

since it can be shown that HJ . =0, i\ j in the T case (Appendix I, Reference 16

These complicated results still contain 3-center integrals.

They are presented in such a form that they do not depend on the

specific form of the one-electron molecular orbital. As such, they

are of general interest. We can, by means of linear transformations

like x„ = Xp + R, transform these 3-center integrals into linear

combinations of 2-center integrals if we substitute

the specific function which stands for i{)_.~ (cf. Equation III.1)

19 and transform the integrals to standard form
39

TABLE II I.

2

T Model. Matrix Element H^

H

ii = ^{to,

(2P,

(2P.

(2P,

(2P,

(2p.

21

a

^V^Va^x'

11

Txxx

+ a

122

T

xyy

+ am T

xJV'.M ^x*

28

pxI*bc><*bcIVaK>

^WVaK'

°k <*J Vl+BCM*Bcl«ll^ + a

12 ^V1 V^W2^}

1 a

1 \

r"3 - 3r-^c2 rAB J' rABx

AB

2i

a

i "

-5 -7 2 "3r

AByAB

+ 1 5r

AB

X

AByAB

1 -(3/2)r-5 + l 5r-^

B -(35/2)r^B

a

122 = ^ " C105/2)p^ ^b

^

a

133 =

"

(3/2)rlB

+ ( 15/2

^" 7x

2

AB-AB

111

a,"'. -(3/2)^^5,-^.(35/2)^^

Q

1 22 = . W*Kl - (35/2)(3x^

A

2

B

) r"?

a^= 133 _ -(3/2)r2 + (15/2)r--7 2

ABX

AB

a

12 = - 15rlBWAB + 3rAB^AB

S* = '^ll + '°5 «3 4&AE
40

TABLE III.

3

T Model. Matrix Element Hi^.

H

22

" < 2P

y

l V«BC

I«?1 Sac + 4 «yy +

°33 ".JW ^BC^'V

11 22 + < 2P

y

l«2 ^V^W+ (2p

y

l02 W^A

bo^+bcISi %x * 4z V

+ a

33 'UV^bc'**

+BO )( *Boh

2

1 Sac +

°22 V +

°33 Qzzl+BC )( *Bol yA

*BO>(*Bol^1 Sex + 41 «yy +

°33 «JV*Bcl*A

+ 1:2Pyl

+ 1:2Pyl

+ 1:2Pyl

+ : 2Pyi

+ [2p

y

l

+ (2p 1

+ [2p 1

+ [2p 1

+ :2p

y'

+ (:2Pyi

+ ( 2pF I y'

+ ( 2p 1

2Py)

2p

y

) V

2Py)

2P7

)

°112 T

xxy + a222

T

yyy

+ a332 WV^BC^a' V

11 11 11 a

11 Sac + G

22 Sy

+ a

33 Q

zz' V^xaI 2V

22 22 22 a

11 Sex + a

22 V

+ G

33 Qz Z

l V^W

33 33 33 a

11 Qxx + a

22 V +a33" Qz Z

l V^Bc' Z

A> 2py>

a

2

12

P

7l'*BC^+Bclvfl 2V

+ (2p

y

|a

2

22

Pyl*BC^+Bcl^l V.

^^y^BC^^BclVAlV

a

"l Sex + a

22 Sy

+

°33 Qzz^BC )(ry^ 2V

222 _-

1 x

112 _ V^i^ p;i*bc»*boI»aIV

v<+bcI°i p;i+Bc)(vivA2

i 2-°y )}-
v\

TABLE III. 3 (cont.)

+ (*yi

+ (2py

l

+ (2Py|

+ (2p

y

|

+ (2p

y

|

2 _- *BC)( ^B0l a

1 ^bc)<*bcI»aI 2V

11 _- v<+bcK ^v^w

22 „-, Be)( +Bcl°r p!ilV ( *Bcl»AlV

.33 „- V<*Bcl af ^V^BcKK'

.233 -- bc'<+bcI°i ^*bc>(*bcIVaI 2V
42

TABLE III. U

H

INT

COEFFICIENTS OF H^. T MODEL.

2a

2

2

a

11

a

22

11 a

2

22 a

2

rAB Wb " 2yAB^

<3/2) r]^ - (15/2) r]^^

< 9/2 > rIi>AB " < 15/2

> rI^B

a

33 = (3/2) rI^AB

( 3/2 > rIi>AB

+ < 15/2

> ^aVaB

af = -(3/2) r^

(9/4) r-5 - U5/2) r"^

B + (105A) r^

^+ (105A) r^xf^

(3A)r-|-(l5A)x2

B

r^

-3^ + d05A)r^fB

x2

B

(9A)r^.- U5/2 ) r-^

B+ ( 10 5A)r^B

(3A) r^ - (15A) r-^

(3A)r^-(l5A)r^

11 a

11

11 a

22

11 a

33

22 a

11

22

a22

22

a

33

„a

33 -

11 "
43

TABLE III. 4 (cent.)

a

33 _ to U\ „-5

2a

112

2a222

a

22 = (3A) rAB - (15A) r«4

(9/4) r^

6r

AB " < 105/2 > rAB

xAB4

(3/2)^+15^.(35/2),-^

(3/2) rll + (15/2) r^

B

6 r^ - (105/2) rftfafa

(3/2)r-B+ 15r^

B -(35/2)r-^B

(3/2) r;B+ (15/2) r-kykB

15/2) 'IbWaB - < 35/2

> rU?U?L

(15/2) r-kl

3xAByAB

- (35/2) 3r^rAB

, -5 " J AB

xAByAB

a

1

1 =

-< 3/2 ' rAB

3x

AB

+ (15/2) 'I&U

f

= (15/2) r'

kl y

2

B

x

AB

I 3 = -<3/2) rABxAB

af3 = (15/2) rAB

xAByAB

2a

233

112 _

a

2 - 6

222

a

2

233 _ a

2 "

222

a

1 =

112

a

1

2a

1

a

1

a

1
U.

TABLE III.

5

»33 DEPENDENCE ON 2- AND 3-OHTTEH INTB3RALS. T MODEL.

H33 = e2

{(2p, ^> p, I VbcX V> *A\ «P.)

+ C2Pzl««', *x« I^BCJC^ I *Al 2P«)

+ ftP.I <&> Tml tB0K^c\«Al 2P»)

+ &P.UJ,, *i»| "KBcJOlie |u|2p.)

+ C2P,

J ^» FU.V*

^Wl «*>

+ (2p,i<

y P,lt»KV» «ky*-\2p,)

OWST'P.-'l tBC)C VI «J I 2pt

)
45

TABLE III. $ (cont.)

HINT COEFFICIENTS OF H33. T MODEL.

*1- P

AB

«? = -3«S&TiB

^-O/arjg+dS/^pjJyj*

°23 = 3r

AB7AB

°23 - 3r

AB " 15WaB
46

Thus the only integrals that must be evaluated are relatively easily

evaluated double integrals. Then it is just a matter of using linear

substitutions to obtain the final numerical values of HI..

4-b. Linear Model, HI, Matrix Elements. -The analysis proceeds

'

in similar fashion to the one given in the previous section. To

simplify matters, the interaction Hamiltonian is repeated in the short

notation (cf. Equation III. 14-).

Xr..p" ^ ^ ?A + ^ ^

""A '* J

+

P k I . r , t\^Mm "P **"? r

A + °FM "p lP rA

SA

r,K,t,m,r,s,t|

2

+C^k

tk

P

r

A

S

A M'

A common factor e has been omitted. y

Before we present the H! . matrix elements for the linear case, we list

J » some of the symmetry properties .of the many-center integrals which

occur. These properties are readily established in the same way as

equations III. 21 - 2S.
47

f2o IO t

III. 29

x' YBC

(2p

y

UB0 ) <= (2P2 I^ BC ) = 111.30

^A.A W =

, m- 32

< 6

i1

6W

+ 6

i?

6

k2

6 n

+ 6

i3 *k3 KJ&k.fo -WThis last expression vanishes fori = 1, 2, or 3 unless the conditions

on k and I are satisfied.

(SPi.iUptp-pl+aj) =

>ik »lm (2PA ;il*p l

? ^\% \ . m'»

< 2?A,il r

A aI*B6') " IIX - 35

[5 H

6

fs + 6

i2

l

r2

l

s1

+ 4

i3

l

r3

&

S1 ](2pA,il rA

sa' V •

The same remark applies here as the one made after Equation III. 32.

<?P

A( il rA

S

A **IV = 6

ir

&

St <*A,±I*A SA h^' HI - 36

Similar selection rules can be derived for integrals over 4) R

_ only.
AS

With these properties at our disposal, it is easy to derive the matrix

elements H).. The Hj., just as in the T model, are identically zero.

The non-vanishing matrix elements are presented in Table III. 6. The

coefficients dependent on the geometry of the problem are somewhat

more complicated than those of the T model, due to the fact that in

general al" t a'" . D • • • U • • •
49

TABLE III.

6

LINEAR MODEL MATRIX ELEMENTS.

H

11 = e t

(2p

(2p,

<2p.

" (2p,

(2p,

(2p.x

(2p,

(2p,

G

P1 x

p

IW( *bcI*aI 2^

BC^BC^pU Vp + V2 Vp + Q

P33 Vp'W^^a' 2^

a

p11 Vp + G

P22 Vp + a

p33 VpI *K)WBc'

xaI 2px

) V »l>'*B0 >( *BC l ^|2px

)

.

G

pr

xplV^Bcl z

fl 2Px).

*BC^*B0l°i

1

1 Xpl^Bcl$ 2Px>
50

TABLE III. 6 (cont.)

+ C2 Px ltBc)C^cM,P

,

l| Vp**m Vp+*p«VpI V( ^Bfcl x#*\ 2Px)

+ (2Px| ^B0)^ec| <, XpXp I

^

ypyp

V

^

sy^ ^X^c t Vk7k \ *Px>

+

(2Px | 4Bc>< ^d **» XpXp t 4L ypyp

f «4\a Vp 1 ^bc>< V\va \

2p*>

* (2Px(^pin Wp * ^Pua. WP * ^Plfc VpM +B0^

^ I *A | 2px

)

+ (2 P*Kn VP * °C VP * ***»* Vpl ^BC>< ^fc I XA*A | 2Px>

> (2PxK>„ Vp + *pLw+*p» Vp1^bc)(^cIVaI 2Px) I

:"+ (2p*Ku Vp * *& Vp + *S> VpI ^boX^ W«a| 2p*>

' I

* C2Pxl(* P.) *p |

l kBc)C V I Wa| 2Px ) I

* C2Px|d'£ *p |tB0)< **- I WA I 2 Px>

*

*C2PxW? *pI+bc><VIWa| 2Px>

+ (2px UBC )(^BC l411l x| + ct1 pi22 Xpy

2

+a1pi ^ ^^|^BG )UBC

|xA

|2px) I

+ (2Pxl^BC ) ^Bcl a

Pl

1xP^BC ) ^Bcl x

ll 2Px

) + (2^l V^4f xp' V^Bc' Va' 2-Dx> + ^^^BC^^BgI^-pI^C^^BcIVaI^x)
51

i TABEE III.6,Ccont.)

COEFFICIENTS IN H{r

<•! «-*£*£ « ZxAB

^Bu = "3*AB

d^*6fflkfl

*1**M*£b

BU - AB

°W J3CAB

of" = -3 x"5

*Btt

J X

AB

*B«I **' X

AB

•C«»M*m

.*

B "» 4 X

AB

'din dL---*\l

,i *«* = +* x"|

-i*;£

AB

*/' - +3x w

*««- ^AC

<*- -(3/2) xT*

AC

4= -(3/2) *$

: <***&' .

**» J AC

°C

U

=.-3 x"5

til- J X

AC

<--<*/»*,£

J»» - , -5
1 a

C111

s "ixIo

1 a

C122

s +6xIo

1 a

C133 = **o

111 a

C1 = "4xIo

122 a

C1

' S 6

*Ic

52

TABLE III. 6 (cont.)

133 _ A

-5 a

C1 - 6 XAC

11 o -U

"Bl = 3XAB

«B1

=-<3/2)xlB

11 - --4 a

C1 " "3 X

AC

°ci = + < 3/2 > xl£
53

TABLE III.6 (cont.)

tt, _ LINEAR MODEL Hvu n

22

e

LINEAR MODEL Hv

% K^4A y

p \^BC )(tuc|yA l 2p

y

)

(2Py V 4U y

p

xp| ^bc )(%l l yA | 2^)

+ C2Py \ J& y

p

| tBc><^c |xAyA | 2Py )

+ C2py I 4iu Vp^ MPa:u ypVp * "*» ^"p^ \ B0) < ^ | 7A \

2V

* C2%l°C Vp* +bc )( ^c UAyA \ 2Py )

t (2Py |< y

p

| +«,)<+*. |yAVA | 2Py)

* C2*y\< y

P I %>K%\ Wa\ V

t

C2pyl < i yp I *w)C ^ I wa\ 2Py )]

*tt - X

AB °<C4 « XAC

* <= 3xAB ^=-3 x

Ic

•4ns

" &/2) x

^

°£*r

" 0/2) xj^

«4m - - 6/2) XAf <ujr - (3/2) xAg

*li, -

-12xa1 <i = -12*^

B* - 6x

AB *« =6xAC

*£ s -G/fe)xi! <^---(3/2)x^

«*2? "^-AB ^"^if
5A

E>

TABLE III .6 (cont.)

3? —

+ <

2P.K> SI^BC^BcIWaI^

*»p,U£ y»V( ^1 Wa| *.> ;

+ <2P,\< ^plV^f^lVA'Al^)]

J| = X

AB

To obtainfl*8*... replace

^1* -r 3x^g subscript B by subscript

<**} --3xAB C - (e *«* *<* xAc)

^BU3 = ^XAB except that

«m =

- O/2) "if ?

_' -4

^=-°/2)xAf

^=-0/2)^
55

CHAPTER IV

COMPARISON OF CRYSTAL FIELD SPLITTING WITH THIS THEORY

In this chapter we compare critically the application of both

crystal field theory and expanded Hamiltonian molecular orbital (EHSAMO)

theory to the interaction between the H« molecule and the H atom. The

comparison will consist of comparing the splittings obtained by means

of both methods in the two special cases, the linear and T models

Sections l and 2 treat the linear case by means of crystal field

theory and EHSAMO theory respectively, while sections 3 and 4 treat

the T model.

1. Linear Model, Crystal Field Calculation. We apply crystal

field theory in the usual way. We replace the H

2 .molecule by a

linear point charge quadrupole and compute the. splitting of the

2p state of a hydrogen atom due to this quadrupole field. We expand

the quadrupole potential about the nucleus of the atom, to obtain a

series expansion of the form

The x. are electron coordinates with respect to the nucleus of the

H atom, i.e. center A. The meaning of the parameters is :
56

O

. - E r—

J

- -2 + 3 1 iv.2

o ' o

22 33 (x + a) 3 < (xn - a)

J

o o

A

12 " A

13 " A

23 ~ °*

Here the coordinates of the H atom in the center of molecule system

are x » y 1 z , with y = z =0. The coordinates of the nuclei of

o 000

the molecule, in the same system, are x = -a for nucleus B and

'x = a for nucleus C. The point charge distribution is the following:

a charge of e on each nucleus, and one of -2Z at the origin, e is

adjusted so as to make the quadrupole moment of the charge distribution

equal to' the observed value of the quadrupole moment fcr the molecule.

We calculate the eigenvalues of the Hamiltonian

H = eV . IV.

3

We take as our basis the three atomic functions: ?p , 2d and ?p . ^x ^y * z H is diagonal in this basis. The matrix H has the following form:
57

2p*

2P7

2pz

2

?x : HH

2p

y

H

22

2PZ

H

33

Obviously, EL^ = H,,, and the three eigenvalues are

E

11» H

22' and ^tv

As is well known, we obtain 2 levels, one doubly degenerate and one

non-degenerate level. We exhibit them here explicitly so as to make

the comparisons with the results of Chapter III, in terms of the

parameters G, a-, A.., and A^.
58

The matrix elements are:

Hn •- 9An J 4 < 2

*x> z * + eA22 O

2

. <*«£? *.- K <*,>

a *3

H22-= eA

!1 J

X

A <*?/ * + ^22 [K < 2P

y

> 2 * * J *A < V* *J "'*

H

33 " *11 J

X

A

(2p/ * + ^22 [M '«*•>' * + j «A < 2p/ *»] '

r 2 2

Let the integral jx. (2p ) dx be denoted by I , etc. Symmetry provides

the following identities for the integrals:

xx ~ yy

r zz

and an elementary calculation shows that I = 31 . Using these

results we obtain:

Hn *ii V

H

22 " H

33 "

~A 11 V

IV. 6

The splitting is then:

H

11 " H

22 = ^xy <A

11 " A22> e = 3A

11 V • IT - 7
59

The crystal field splitting has the usual two factors: A factor

which depends on the geometry of the problem and on the details of

the charge distributions of the neighbors of the reference species,

in this case A^, and another factor, which is essentially an average

of a polynomial of electron coordinates averaged over the wave functions of the reference species, in this case I .

2. Linear Model, Expanded Hamiltonian Molecular Orbital Treatment ,

Since H. . •= 0, i £ j, the 3 hew levels arising from the original

degenerate 2p state under the presence of the electrostatic H,.-,

are given by IL.., Hgp, and H-^ respectively. More explicitly, they

are (Equation III. 6 and III. 19 and Table III.1):

H

11

=EH

+ EM+

2 2

+ ire*

X

P,k,r

I, [(^BCI^I *pl V +

<+Bcl fl &k $V>PX

i rA^x> ' *

N

x

H

11

H

22 = EH

+ \

+

IV.S

H

33 = E

H

+ IW+

+ ^% I

[^BOI^T *pl+BC> * WboI'Sc 4%C^y^^ - * «£ H33-

r ,iC,r
60

If we write out these matrix elements in a more explicit way the

relationships with crystal field theory matrix elements can be

brought out more clearly. Thus,

%

+ ve2

J, [<*dcI<# "pi+Bc* "

+ (%A #*bc>] ^j4^ 4

P,k

N

2

X

i N

2

H' . ^ x 11

Let us for convenience denote e L [(*BC |a]£ XpUBC ) + (*Bc'

cSlc ^"Wj

bv A

rr

« From the. form of the electrostatic -coefficients (cf. section IV. 1)

it is clear that A^

2 = A' . Thus, we obtain:

H

11 = hi

+ V^li**^1 *^ + A22(2Pxlyf|2Px) + A|

3 (2PX

I «f|2px )] - i A

H

22 = ^

+ V^lV^tfV + A

22

(2Pyl^l 2V

+ HS^^V]-*^*

H

33 - % + \

+ *9!1il(*«l^l*1

) + A

22

( 2PZ I^I 2P2

) + A^CZP.I'IlZP,)]-?

^3

i
61

Using the properties of the integrals (2p \ r

.

| 2p ) = I of equations

o a S L o

IV. 5, we can now simplify the expression for the energy levels of the

atom in the field of the molecule.

H

11 = EH

+ V^ll V

+ A

22

J xy

+ A

33 V Nx

" K H

11

H_ = EL. + E+e(A' I + 3A' I + A' I ) N* - ^N2HJ 22 H m 11 xy 22 xy 33 xy' y 7 * IV.

9

= E„ +' E, +e(A' I + A' I + 3A' I ) N

2

- |-N

2

H' . H m

v 11 xy 22 xy 33 xy

7 z ^ z 33

N , we recall, is equal to 1 - (2p |*I> BC ) I > while N = N - 1

in this case (App. D) . It is convenient to write

(2pJV) 2 a?

j2 _ „ . 1 ; E :''.._. ., x

N = 1 + r = 1 + ———- . There are here 2 degenerate

1 - (2PX UBC) 2

1 - •*,

levels just as in the previous section, and the splitting is given

as follows:

2

a<

- H ' = 2^^;.. A',) * (-U-g) Ixy

e( 3A'

1 + 2AA

2

) - ^ (h^ Hfe). IV.10

n

11

Three important terms are present in the splitting as it is obtained by

the EH-SA-MO technique. The first term is analogous to the expression (IV. 7)

obtained in the previous section. The contribution from the first term to the

splitting is proportional to kl* - Ai

2

, which is analogous to A., - A

22 « While

the A., were proportional to the quadrupole moment of the molecule, the
62

A'., are proportional to quantities which involve constants and the

quadrupole moment of the molecule. The second term, although it

contains the same parameters as the first term, has a different

dependence on them. Also it contains a factor which is stronglydependent on the overlap between the charge distributions of the

2

molecule and the atom. This factor 11 is plotted in figure 2

1 2

" a

11 as a function of (2p |^ nr.). The approximate .meaning of this is x dl> -

. _ ______

as follows: When r B = 2.5 A°, (2p Ms-J = 0.4-. Using

(2p lis,..) as an estimate for (2p j^-) , for purposes of illustration, we obtain (2p__HBC) 2

. [1 - ( 2 Px kBQ ) 2]~ 1 " °- 2 >

which is quite appreciable. For the purposes of the analysis given

here this rough estimate is indicative of the relative importance of

these terms. The third term contains a variety of many-center integrals

which have no analog in the results obtained by means of ordinary

crystal field theory. No analytical simplification results when the

term H' - Rl

? is analyzed as one, since most of the integrals ocurring

in H' are different from the ones appearing in H' .

3. The T Model, Crystal Field Calculation. As in section IV. 1,

we present the results of crystal field theory methods to determine

the removal of degeneracy of the levels, so as to provide a basis for

comparison in the next section. The interaction Hamiltonian is again

written as follows:
63

Y

v

V = C + L. a k....._, + L. A r. s A

. IV. 11

k A rs A A

k

The k. are electronic coordinates with respect to center A. The meanA

15

ing of the parameters is given below:

C

A

- Ui/*± a1= a

2 = 2y

o

£

[

(&

2 +

1 y2) 3/2 " ^3"] .

11 "'{(a2 ^)5'2 "

t

(a

2 + y

2 )^ " ~^j

A22' e |

3 yo[

(a

2. +

1 y

2

)5/2 " ^"] *

t (a

2 +

1 y

2

)3/2 " ^]j

,17.12

A

33 = ^

(a

2 +

1 ^)

3/2 " ^

A

12 ~ A

23 = A13 = °*

We write down the matrix for H = eV, remembering H. . = for i ^ j,

f w because A. , = for i ^ j.
64

The Hamiltoniaii H = eV has the following non-zero matrix elements

(using the notation of the previous section):

H

11 = eIxy

(3A

11 * A22 + A

33 5 = 2eI

xy

AH

H

22 = eIxy < A

11 + 3A22 + V = 2eIxy

A

22

H

33 = eI

xy

(A

11 + A

22 ' 3A33» =

"2eV (A

T1 * A

22» :

These are also the eigenvalues. The splittings are

IV. 13

H

11 " H

22 " 2eI

xy

^ A

11 " A 22^

H

22 " H

33 = 2eI

xy

< A

22 " k

33 ] *

2eI

xy

(A

11

V ^22> '

As expected, in this model all the orbital degeneracy is lifted.

4-. T Model, Expanded Hamiltonian Molecular Orbital Approximation.

This case is of more interest than the previous one, because the

spatial degeneracy of the 2p state is completely removed and one

has 2 parameters which can be compared in the two theories. Here

again H. . = 0; i ^ j . Before we present the eigenvalues of this

problem, the following notation is adopted:

IV. 14.
65

A

il

=e ? <*BcKk Sdcl+BO 5 + ^' (*Bcl Pxl+BC J 11 „ I , ^ . 11

A

22

=e £ WbcI'G QJ*BC> + «f<*d p;l*BC> IV - 15

k

A

33 =4 ( *BC |cS «JW + 'iXc^'V '

(Note that the integral (t}) RJ P~|r|> RC ) = constant. See Section III. 3a.)

The three eigenvalues are as follows (cf. sect. III. 3a, 4a, and IV.1):

H

11 = %

+

"M + Kx VNl + A

22 + A

33]

" *^ H

11

H

22 = E

H

+ ^

+ HyVLA

11 + 3A22 + A33]-*N

y

H

22 w- 16

H

33 = %

+ ^ .+ N

z Vk'l + A

22

+ 3A

33]

- &z H

33 •

.

2

Hx

= H

z = 1

' while N

y "[

1 " ( 2Pyl+B0 >2]" 1 = 1 + ~22_T

(of - App - D) • 1 " a

22

Nx = 1, because (2p UfiC

) = in the T model ease. The same argument

2 applies to N . The level splittings are:

2

H

11 - H

22 = V(A

11 " A

22 } " (fH-) V[^i + ^

2 - Ai3]-*(H

l'l " H

1 " a

22 $ 22

2 IV - 1 ?

H

22 " H

33 " 2VU22 " A

33> +

< ^

2 > V^'l + 3A22 +

A

3;d

"*(N

y

%-H^).

i ~ a^
66

Here as in the case of the linear model, there are 3 important terms.

The first one is clearly analogous to the crystal field theory result

2el (A.. - A..). The analogy is even closer here than in the

linear case. For instance, the A!, (cf. Equation IV. 15) are directly

related to the quadrupole moment of the molecule Except for a constant).

The second term appearing in these level splittings, although dependent

(a ) 2

on the parameters A!,, has a factor v 22' which is strongly

1-(a2/

dependent on the overlap between the charge distributions of the

molecule and the atom. The magnitude of this contribution is quite

appreciable for rAO •= 2.5 A° (cf. Section IV. 2). This term has, of

Ac

course, no analogue in the crystal field approximation. The third

term (cf . Section III. 4-. a) depends in a very complicated manner on

the multicenter integrals, and does not contain any parameters

similar to the ones encountered in the crystal field approximation.
67

CHAFER V. DISCUSSION.

The nature of the crystal field approximation has been

investigated by studying a complex consisting of a hydrogen

atom in the 2p state, and an K„ molecule in its ground state. This

system exhibits the chief phenomenon discussed by crystal field theory,

namely the removal of orbital degeneracy. In the investigation, this

removal of orbital, degeneracy was calculated with ordinary crystal

field theory, and with a molecular orbital theory having special

features, and the two results compared. These special features are

the following: 1.) All terms in the interaction part of the Hamiltonian

are expanded in a certain way, which is appropriate for non-overlapping

charge distributions. The expansion has the important consequence that

the two-particle terms in the Hamiltonian are transformed to sums of

products of one-particle Hamiltonians. 2.) The wave functions used

to build up the MO's for the whole system are products of eigenfunctions

of the separated atom and molecule. Therefore the parts of the

Hamiltonian referring to the free atom and molecule are treated easily.

3.) The wave functions for the whole complex are properly antisymmetric

under the exchange of any two electrons.

The theory using the special features enumerated above allows

one to display the interaction as a sum of two types of terms.

(Equation III. 3 and III. 9). The first type of term is essentially a

•

classical type of term, involving only the charge distribution of the
68

molecule and the multipole moments of the atom. This type of term is

exactly analogous to the usual crystal field approximation. The

second type of term involves the overlap of the atom and molecule

wave functions and the three center integrals explicitly. In a

certain sense this is a correction term to the crystal field

approximation. It arises from overlap and the fact that the total

wave function of the system is properly antisymmetrized. It is,

therefore, clear that the chief formal shortcoming of the crystal

field approximation is this neglect 'of the Pauli exclusion principle.

12

The same conclusion has been arrived at by Tanabe and Sugano " and

90

by Phillips .

It is interesting to point out that. a method of approximating

\ the interactions in the many-electron problem analogous to ours has

21 22 been used successfully by Pluvinage and by Walsh and Borowitz

With the general result in hand, the comparison between the

crystal field approximation and the EK-SA-MC approximation was

worked out in full detail for twc special geometries. These are

the linear model, with the atomic nucleus on the molecular axis, and

the T model, with the atom nucleus on the bisectrix of the mclecular

axis. In both cases the splittings involved in the lifting of the

orbital degeneracy, as obtained in the EK-SA-MO approximation, have

three types of terms (Equation IV. 10 and IV. 17). The first term is

again the classical term, and is analogous to the crystal field

splitting, though somewhat different in detail. The second term
69

involves the same parameters as the first term (i.e. those analogous

to the crystal field parameters), but also involves the overlap

between the atom and molecule functions. The last term involves the

three-center integrals introduced by the Paul! exclusion principle,

as well as the overlap.

In this analysis it is, therefore, possibTe to see in some detail

the "corrections" to crystal field theory which are needed to take

into account overlap and exchange. Unfortunately the present

formulation of these corrections is tied very much to the particular

system studied, and has therefore the status of a special case, rather

than a general formulation. It is to be hoped, however, that this

special case is sufficiently clear to indicate the general situation.

The theory presented here seems applicable to one other type of

situation than the one contemplated here. This is the so-called

strong field case in crystal field theory, where the crystal field is

assumed to be so strong as to uncouple the electrostatic repulsions

cf the electrons in the reference ion. Each electron in the reference

ion is then assumed to move independently in the crystel field, and

its orbital degeneracy' is lifted. The mechanism of this lifting of

degeneracy is entirely analogous to the case treated here.

Several tasks remain unfinished. The results cf the method

should be generalized, the method should be applied to other systems

of interest, and the method should be tested by numerical computation.
70

APPENDIX A

Expansion of

(1+ x + y*-)** =

f

(1 * x + y*) * 4 Cl * a)

Collecting terms of the s ame order together, we obtain?

Zeroth order term 1

1st order term —| x

2nd order term '-§ y* 4 3/8 x

x

3rd order term 3/U xy1, - j£ *}

Uth order term 3/8 y

f -j£ x

x y

l

4- J£f x + -
71

APPENDIX B

THE EXPANSION OF Hjj^-^ 3) FOR r1A + r2e < r^, rM 4» f}c <rAC' REGION I

TABLE B.l

LINEAR TERMS IN THE SECOND ORDER PART OF THE EXPANSION

| [*2Br

AB* * X3CrAC " 3* V2BrAB " * U

3C" rAC ]

X

1A

!+ Cy2BrA^ * y3CrAC - 3 PV

2BrAB " 3*> U3C 'ao] ^1A

+ [

z 2Br

AB « z 3Cr

AC*] Z1A
72

TABLE B.2

LINEAR TERMS IN x 1A OF THE THIRD ORDER EXPANSION

xu[3/2

I

**§(*!& * 2V2B*2b) * r£(Y rfG

* 2U3c -x3c ) J

- l*/2 Cr^V^Vr^r U^fj

yu[3/2 I

rAB<P r

2*B * 2V2By2B) *

r

AC< 6 r

3C * ^C^c) } * ^

<*&> V

2VrAC

6 U

3C

>J

z ut3CV2B^ rAB

J + U3C^3CrA^J
73

tf *a

N

*N

<~-*—

)

t CO

o

o

n*A-i

CM.

PQCM

"UN

.a. *x'

£

o

CM

03

CM 79

o

cm

pq

•o

CVJ

o

(VlO

>?

00.

CM

PQ

+

o

PQ

+

y-o

CO.

\AH

o

.T* X

•o . •*O

iT^ »

,>

7*

r-A«-» r-^n

'.b o <«-N 4- 8

*~» >- N

pq •0 ff «\0 *w< CO. tf

•+ 4-

4-

PQ <%' /"*

i? PQCM

•* N N

V ca.

*•-»* <w 19 ^

^H ^-r-' V^V-'

1 T ?

4-
n «&o

Pn

_rf

W • rn

CO

h

w

M « o § e

«»«* ««j «# 4- >-+ O o -T^

.JT* |3

!=> o o J^ t*\ >» X

<-x <w o o _f>

>_r^ !=> t= wsQ «q I «JJ ».< *H

CQ

<M Xm

CM >TMH o

ta

PQ

CVJ NnH oX

1*

uoN

PQCM 1

r»CQ

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CM Nu+
•o

75

o

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fcv

*»c^

**CM H

+

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L

¥

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H

** CM >

n

Is

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r—i

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o

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+ —

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eni

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* CO.

CQ PQ .TM CM >~ >~

t9 7£§

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W-»-J ^rJ

VN \r\ H H

1 1 c*>~ •-*-» o o r> r"> H N O o o ..<*>

h k

PQCM N

PQCM & •£ -S -Su

-
76

m4 C>S P3

o ' O '

J- •o r-TS ; r-~7 »r * *^*

.+• 4- ' O

P3 cq ^ CM CM > t>

•o % cfL J-

-sa -9 O

fs fc M

vor- CO~»

(M CM \ "^

lA 1A .o m CO 5-CM

O

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4-o

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p/

UN

Jb

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ecu •

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C30-

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8

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<3

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ca

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CQ

<*2-

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3-

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O

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»

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CQCM

ta

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9

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V-.PQ
77

i

a

V

S

>t

T9 $

r—I r^>

•£ »Q

h 1^

fc « ' <+• V

m CO. £ .8

**9

^

vv*

V? £^

•P* ^* <; ***. \A UN

o

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•O

m

M'

0ftO

* V

H

Mj

I
78 4e-i i 8B -0 ^

CM

WS*

cl:

-9 I

CM *9 I

.x

Pi -*-x

.9 M

cux

CMJ

X

C\J + •

*9XH

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CM

ft

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Pi

"LA HX

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79 <X >7 1n§ |sB

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NN

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PQ CV «aj X ~ «£m

cv <s X+. «

CV«*

<t

4-, HI NN

Or

J-. --

ft J< i 1 59

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4-J4

cv

ts) N

or

PQ

o*

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PQ %%

vg

NOHI a i

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4-

cv

N

r—tm

cv <; 1 'a § •-fe

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or<

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afr

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cv^+

,?

««

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r- cy
80 « g

PQS 3

'J*9HI \9

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C*N I

U\i9 IK9v9

UN

T *-
81

APPENDIX C

THE EXPANSION B + B'

We present the third and fourth order terms in the expansion B + B

1 .

The first and second order terms were given in Section II. 5.

1. THIRD ORDER TERMS (B'+ B')

3/2 ru[r

2BW

AB

+ r3CW

AC " *2B"*ABW

AB " *3C# ?AC

W

AC " <*AB

" *W ' (?AB

' ?

2B)W2B

" ^ rAC " r3C^rAC " r

3G^

W

3cJ

+ (3/2 ^Br2B

V

2B + 3^2 rIc

r3CU

3C

+ 5/2 r;

A[3(W

AB ) 2 W

ffl OW^+ W^ + 3W

2

c„3c

. 3WAoM

2

c + w

3 o]

- 5/2 rA>2B " 5/2 rliU

3C
82

APPENDIX C (continued)

2. FOURTH ORDER TERMS (B + B

1 )

3/8 ru[4 + ^AB' ?

2B

)2 " ^AB^B^B + 4) + 2l ABr2B

•3C

+ ^

(rAC^3G

)2 " ^AC^3G)(rAC

+ P

3C ) + 2rAC

r3c] + r.

- < 3/8 ' rll4 - 3/8 rAC

r3C

- 15^ ru[( 'AB " *fflM?

AB " ^B^-^AB^B + 4'

+ ( *AC " '30 ),( 'AG * ^o'^VSc +

"3c'

( ^AB-'2B " T

2B>

WL " <*AC'*3C " r3C> "fc ]

* (15A) r^4V

2B * (15A) r^D*.

35/8 r;][-4^

B

W

2B .+ 6 i^ - AW^ + ^

-*W

lcW

3C

+ <0W

3C " iWAC

W

3C * ^C

<^/8) pIXb - ^/8 > rli4c
83

APPENDIX D

NORMALIZATION CONSTANT Nj

The function to be normalized is

fc s L

2fA,i Cl)o^^*M^3) * 2^i (2)o<(2) ^M (3,D + 2^*0) ^rfl,2)]V^

-*i<1 > 2'3>" A i In' (D * First we show that ( YC (1,2,3) 1 2f^

f± (lW(D TM (2,3);

z ( <fc (1,2,3) | 2flk, 1 (2)«(2J+ ll (3,l)) (0-2)

r (

^ (1,2,3)| 2^^(3)0^(3) f M

(l>2) ( D.3)

The proof is as follows. We interchange the labeling of electrons 1 and

2 in Eq. D.2. This leaves the integral invariant.

;'

(<fc (1,2,3) \2PA>i (l)oCCl)^M (2,3)) H^i (213)| 2%t

(2) *(2) fM (l,3))

(D.4)

* - (ti(123)l 2PA)i (2)ot(2)tM

(l,3)) » (-l)(-l)(^i (123)|2PA)i

(2)o< (2)fM

(3,l))

(D.5)

The proof of eq. 0.3 proceeds in the same way as the one for eq f Q.2. A

consequence of eq. 0.2 and. 0*3 is that

(fc \ 4> ±

) - 3^(^(123)1 2PM(lW(l)fM (2,3j) «>-6)

But (^(123)) 2PA>i (l)o((l)tM

(2,3)) *

(2pA,i (l)<* (1 ^M(2 ' 3) I 2PA,i (l) * ^M^2 ' 3 ))

y%

\ ¥ (

2 *A,i (2)€( ( 2 )

V

(/

M

(3 ' 1)

I 2PA,i (l)* (1)^(2,3))

+

(

2 PA,i< 3

^

C3) V1' 2 * I ^i^1^^^

2 ' 3

))

* 3* [i - (xn,i\^ BC

)*J

<»•?>
84

APPENDIX D (cont.)

This last result was obtained by recalling the- definition of Ty(k,l)

(Eq. III.l)i Using Eq. 0.7, ( $l \§±

) s [l - (2fA>i |f ^fj

Thus % - [l - (2pAji | ^bc)

1] "V2, and ^-* N. ±

, where foji^) = 1
85

APPENDIX E

ORTHONORMAIIZATTON OF THE f^' 5

-

From Appendix t>. we know that ( v^ \ v|/g) , ( ^

(V

y

) = -(2px \ ty bC)(Mh \ 2py) - ~*

v< Also from symmetry consideration (2p I t jjq) « Oj thus, «jj r 823 s 0.

We use the well known Gram-Schmidt orthonormalization process*

Let fx

«

'4>M

then In^Njj fiifc.4 fy

)

We determine at

(fcl Jin) r - a 1 ( +k |f

y

). Thus a * a^

" (feil.fei) *i--Nn d-*J2 >

Let ^jjj * (bj)j 4 cf^j f ^ 2 )Nm • We determine b and c.

Then ($j\ Jjjj) r (b - a^M^ t. 0, and since a^ = 0, b * 0.

Now ($nl 5m) r c + (^ l V B

) =. 0. Since 823-0, c s 0;

thus, ^jjj a^8

v

'
86

APPENDIX F

PROOF OF THE FCRM OF THE MATRIX ELEMENT EQ. Ill .3- .

We show explicitly how we arrive at the result III .3* for the diagonal

elements H^, Then we show how to generalize the result when i £ J. We

• Uat

make use of eq. B.o, and the fact A H_ is invariant nnderrelabelingY the

electrons to obtaint

( VfrWI tj) r 3^Nj(^\ht | 2pA^(l)«Cl)+M(2,3)) C*0

r 3^N

j

(vVv' I HH

(1) H„(2,3) * %htC1j2,3)| W^jCD* M*ifiM):

< F *0

Making use of the fact that the wave functions 2jDa,i and 4^ are eigenstates

of H^ and H^ respectively, we obtain the following t

?*^[%(^l*%i<l>*<l)f*&.3>) «fc0*4 12PA>1 (1)« (1)^(2,3))

, ^(^|HIHTCl,2,3)l 2 %1(l)«((l)fH

(2.3))] CF-Jj

Using the results of eq. F«l , and f.* ,-j. Pi becomes (for i = j)

\^\^ ^\ (>v I Hn|TCli2»3>| 2pM

(l)* (1)^(2,3)) (**)

= Efl * \ f 3YtNi(^ I £ Hf (D(h^(2) 4- £(3>) | ^(1)*(1)^(2, 3))

F-S

Ccf. Table III.l). F.£ is correct up to and including lith order terms in the

expansion of Hjnt. Recalling the expression tor 4j (Eq. II 1.1), Eq.F.S is equal to
87

APPENDIX F (cont.)

Since N^ x £l - (2PA i\ ^BC-M"' » we fin<1 the re8Ult (HI..3)., The

factor -2 arises because of the spin dependence of the wave function , and

the resulting integration over the spin coordinates • thus, when evaluating

(2PM.(2)0f (2)tM

C3,l) | H^(1)(H^(2) + H^C3)) | 2PM(l)o< (l)f M(2,3))

we must recall that tM

(2,3)-- ^BCC2)f BC C3) [<* C3)p\CD - <* (l) $ (3)Ja^

) Therefore, we must perform the following integration over the spin coordinates :

f c( ft) [ *(3)Pfa)-«ft)Pf3)

j

|a(M|r^fg^B(^-c(fnBf« .

' j\ 4>

since the only nonvahishing integral is

-J (* (2) o( CD p C3) | o( CD ot (2) p (3))

The proof for H^ ., for i f j is identical except that one replaces in Eq. F- 6

"i »y N

is3.

' % < * *>r Ob J«)[yM - (2PA,i I + bc)OPa>j | *

BC)] »i»j»

and the second index i by J*
88

i REFERENCES AND FOOTNOTES

1. S. Freed and F. H. Spedding, Nature, 122, 525 (1929).

2. J. Becquerel, Z. fur Physik, £8, 205 (1929).

3. R. Brunetti and Z. Ollano, Nuovo Cimento, 6, 34-5 (1929).

' 4.. H. A. Bethe, Aun. der Physik, 2, 133 (1929).

5. J. H. Van Vleck, "Electric and Magnetic Susceptibilities", Oxford

(1932); W. G. Penney and R. Schlapp, Phys. Rev., £1, 194- (1932);

R. Schlapp and W. G. Penney, Phys. Rev., £2, 666 (1932).

6. B. Bleaney and K. W. H. Stevens, Rep. Prog. Physics, 107 (1953).

7. W. Moffitt and C. J. Ballhausen, Ann. Rev. Phys. Ghem., 7, 107 (1956)

8. W. Low, "Paramagnetic Resonance in Solids", Academic Press, New

York (1960).

9. CM. Herzfeld and P.H.E. Meijer, "Group Theory and Crystal Field

Theory", Solid State Physics, Ed. F. Seitz and D. Turnbull, Vol.

XII. In press.

10. Here after called "reference ion".

11. W. H. Kleiner, J. Chem. Phys. 20, 1784. (1952).

12. Y. Tanabe and S. Sugano, Jour. Phys. Soc . Japan, 11,, 864. (1956).

13. W. Moffitt, Proc. Roy. Soc. London, AJ210, 245 (1951).

14. C. M. Herzfeld, Phys. Rev., 107, 1239 (1957).

15. H. Goldberg and C. M. Herzfeld, Bull. Amer. Phys. Soc. II, 3,, 180

(1958). Publication in preparation.

16. H. Goldberg, Ph. D. Thesis, University of Maryland, 1959.
89

17. J. 0. Hirschfelder, G. F. Curtis, and R. B. Bird, "Molecular

Theory of Gases and Liquids", Wiley, New York (1954.) , Chapter 12,

18. H. Margenau, Phys. Rev., 66, 303 (1944).

19. C. A. Coulson, Proc. Camb. Phil. Soc, £8, 210 (194-2).

20. J. C. Phillips, Jour. Phys. Chem. Solids, 11, 226 (1959).

21-. P. Pluvinage, Ann. de Phys., 12, 145(1950).

22. P. Walsh and S. Borowitz, Phys. Rev., 11j§, 1206 (1959);

AFCRC-TN-59-657, Research Report CX-45, Institute of

Mathematical Sciences, New York University.

VI . A CKNOWLEDOEMENTS

We thank our colleagues for many helpful discussions, and in

particular we are grateful to Dr. Rosemary M. fiercer for checking

these results in detail, and to Miss Sara Baialardo for preparing

this manuscript for publication.
90

Figure 1. The system H - Hp. The nucleus A belongs to the H atom,

the nuclei B. and C to the H_ molecule. The coordinate

systems used in the calculation are indicated. Thus x„

,

" • 1A

is the x-coordinate of electron 1 measured from nucleus A,
X

ll

CVJ X

>- M

1 X

E

CO>s

CO

-C

\ CD \ CM \ t

6

^"^

CO

CM

" M
92

Figure 2. An analysis of the effect of overlap. The scaling factor,

(2px

|t|>BC ) 2

[1 - (2pxUBC) 2

r

1 is plotted against the

overlap integral ( 2PX

I^BC)»
c\ o

m3-a

cv

cv o

.05 .1 .15 .2 .25 .30 .35 .40 ,45 ,50 .55 .60 .65 .70 .75 .80 .85 .9

Fig. 2 An analysis of the effect of overlap

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Office of Weights and Measures. BOULDER, COLO.

CRYOGENIC ENGINEERING. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Gas

Liquefaction.

IONOSPHERE RESEARCH AND PROPAGATION. Low Frequency and Very Low Frequency Research. Ion- osphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. RADIO PROPAGATION ENGINEERING. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements. Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

RADIO STANDARDS. High frequency Electrical Standards. Radio Broadcast Service. Radio and Microwave

Materials. Atomic Frequency and Time Standards. Electronic Calibration Center. Millimeter-Wave Research.

Microwave Circuit Standards.

RADIO SYSTEMS. High Frequency and Very High Frequency Research. Modulation Research. Antenna Research. Navigation Systems. Space Telecommunications.

UPPER ATMOSPHERE AND SPACE PHYSICS. Upper Atmosphere and Plasma Physics. Ionosphere and

Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.
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