Title Page
 Table of Contents
 List of Tables
 Hartree-fock and beyond
 Reduced density matrices
 Atomic properties
 The projected general spin orbital...
 Appendix 1: Values of some physical...
 Appendix 2: The computer progr...
 Appendix 3: Conventions for spherical...
 Appendix 4: On the cusp condit...
 Appendix 5: Solution of the matrix...
 Appendix 6: List of abbreviati...
 Appendix 7: Some theorems on direct...
 Biographical sketch

Title: General spin orbitals for three-electron system
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097596/00001
 Material Information
Title: General spin orbitals for three-electron system
Physical Description: viii, 238 leaves. : illus. ; 28 cm.
Language: English
Creator: Beebe, Nelson Howard Frederick, 1948-
Publication Date: 1972
Copyright Date: 1972
Subject: Atomic orbitals   ( lcsh )
Lithium   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 194-237.
Additional Physical Form: Also available on World Wide Web
Statement of Responsibility: Nelson H. F. Beebe.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097596
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577230
oclc - 13943039
notis - ADA4925


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Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
        Page viii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
    Hartree-fock and beyond
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
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        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
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        Page 30
        Page 31
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        Page 36
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        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
    Reduced density matrices
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
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        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
    Atomic properties
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
    The projected general spin orbital calculations
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
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        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
    Appendix 1: Values of some physical constants
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
    Appendix 2: The computer programs
        Page 135
        Page 136
        Page 137
        Page 138
    Appendix 3: Conventions for spherical harmonics and spherical tensors
        Page 139
        Page 140
        Page 141
    Appendix 4: On the cusp conditions
        Page 142
        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
    Appendix 5: Solution of the matrix Schrudinger equation
        Page 148
        Page 149
        Page 150
        Page 151
    Appendix 6: List of abbreviations
        Page 152
        Page 153
        Page 154
    Appendix 7: Some theorems on direct product matrices
        Page 155
        Page 156
        Page 157
        Page 158
        Page 159
        Page 160
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        Page 231
        Page 232
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        Page 236
        Page 237
    Biographical sketch
        Page 238
        Page 239
        Page 240
Full Text

General Sp;n Orbitals for Three-Electron Systems


Nelson H.F. Beehe





I would like to thank all the members of the

Quantum Theory Project for helping to provide such a

stimulating environment for research, and especially

Per-Olov L8wdin, for having the wonderful idea of an

international group working in quantum science.

I am very grateful to Sten Lunell, who first

suggested the problem examined here and with whom I

have closely worked, he in Uppsala, and I in


Yngve Ohrn and Charles Reid have often helped

when problems arose that I could not stumble through


Support of the Computing Center and the

Chemistry Department of the University of Florida, and

of the Air Force Office of Scientific Research and the

National Science Foundation through grants AFOSR-

71.1714B and NSF-GP-16666 is gratefully acknowledged.

Finally, I wish to dedicate this thesis to my

wife, Thesa, for her sacrifice over the last four years

has greatly exceeded mine.



LIST OF TABLES . . . . . . ..

ABSTRACT . . . . . . . . .

INTRODUCTION . . . . . . . .


1.1 The Hartree-Fock Method . . .

1.2 Extensions to Hartree-Fock Theory

Relaxation of Restrictions ..
The Symmetry Dilemma .. . . .
On Spanning the Angular Momentum
Spaces . . . . . . .

1.3 Other Methods . . . . . .

The Configuration Interaction
Method . . . . . . .
Interelectronic Coordinate Methods
Bethe-Goldstone Perturbation
Theory . . . . . . .


2.1 Introduction to the Density Matrix
Literature . . . . . . .

2.2 Construction of the Reduced Density
Matrix . . . . . . . .

Definition of the Reduced Density
Matrix . . . . . . .
The Reduced Density Matrix for a C
Wavefunction . . .. ..
The Reduced Density Matrix for a
Non-CI Wavefunction . . .

2.3 Properties of Density Matrices . .

CI Expansion Convergence . .
Bounds on Occupation Numbers . .


* ii

* vi

. vii

S 1

. 10

. 10

. 23

. 23
. 25

. 30

. 39

. 39
. 42

. 44

. 46

. 46

. 47

The Carlson-Keller-Schmidt Theorem . 71
Symmetry Properties . . . .. 73
Density Hatrices of Some Special
Functions . . . . . ... 78

2.4 The N-Representability Problem ..... 85


3.1 Introduction . . . . . ... 88

3.2 Energies . . . . . . ... .89

3.3 Specific Mass Effect . . . . .. 90

3.4 Relativistic Mass Increase . . .. 92

3.5 Transition Probabilities and Oscillator
Strengths . . . . . . . 94

3.6 Fine and Hyperfine Structure . . .. 95

CALCULATIONS . . . . . . 104

4.1 Introduction . . . . . ... 104

4.2 Matrix Fornulation of the PGSO Method 105

4.3 Choice of Bases and Initial Orbitals . 108

4.4 Evaluation of the 1-Matrix . . . 112

4.5 Evaluation of the 2-latrix . . . 118

4.6 The Hyperfine Analysis . . . ... 119

4.7 Comparison with Other Methods . .. 126

4.8 Hyperfine Structure Results by Other
Methods . .. . . . . . . 128






EQUATION, HC = SCE . . .. . 148


MATRICES . . . . . ... 155

BIBLIOGRAPHY . . . . . . . ... 194

BIOGRAPHICAL SKETCH . . . . . . .. 238


Table Page

1. Energy conversion factors . . . ... 159

2. Comparison of the convergence of some
properties with convergence of the
energy for various basis sets . . .. 161

3. Sample bases, properties, and natural

analyses for Li 2 2P and 2 2S ....... 171

4. Contributions to one-electron properties
by NSO . . . . . . . . . 177

5. Comparison of energies and Fermi contact
terms from different methods . . . . 180

6. Angular parameters for the evaluation of
hfs matrix elements . . . . . . 184

7. Experimental hyperfine structure
parameters . . . . . . . . 185

8. A and B hfs parameters for several states
of atomic lithium . . . . . . 186

9. Summary of hfs parameters for Li
calculated by different methods . . .. 188

Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy



Nelson H.F. Beebe

June, 1972

Chairman: P.O. Lw.d din
Cochairman: NI.Y. thrn
Major Department: Chemistry

The use of general spin-orbitals in a spin-

projected Slater determinant for a number of states of

several three-electron atomic systems is studied. The

spin-optimized self-consistent-field method, which is

an alternative way of treating the spin degeneracy

problem, is shown to be a special case of the projected

general spin-orbital approach, and comparisons are made

between the two methods.

Computation of the hyperfine structure

parameters has been carried out for all the systems

studied, and the results should be a useful guide to

future experimental work.


The method of construction of a general p-th

order reduced density matrix from a configuration

interaction wavefunction built from non-orthogonal

orbitals is derived as part of the work.

The general conclusion of this work is that

while the projected general spin-orbital method is a

conceptually satisfying approach, particularly for

treating the spin degeneracy problem, it is not

competitive with configuration interaction calculations

of comparable accuracy.



The time-independent Schridinger equation,

H/ = E (1)

in principle describes most of the physics and

chemistry of matter. Unfortunately, neither the

Hamiltonian -operator, H, nor the wavefunction, Z are

ever known exactly, and consequently, neither is the

energy E. For the non-relativistic case of a single

particle, such as an electron, moving in a central

field of force, such as that provided by a nucleus, H

is known fairly accurately, and (1) can be solved

exactly. The series of hydrogen-like atoms, H, He+,

Li++, etc. fall in this category. The relativistic

counterpart of (1), the Dirac equation, cannot be

solved exactly, even in this simple case, unless the

nuclear mass is assumed infinite; a covariant many-

particle Dirac Hamiltonian is not even known.

As soon as another particle is introduced,

the equation, like the classical Kepler three-body

problem, is insoluble, and approximation methods must

- 1

- 2 -

be resorted to.

The non-relativistic Hamiltonian is known

sufficiently accurately to describe chemical

properties, so that determination of the wavefunction

is the principal problem. An approximate relativistic

Hamiltonian may be written for a many-particle system;

in the limit of low electron velocity, it can be made

to reduce to the usual non-relativistic Hamiltonian.

The physicist may be interested in studying the

construction of the proper Hamiltonian, while the

chemist is more concerned with the results, that is,

the wavefunction and the properties that can be deduced

from it.

Although the calculations'described in this

thesis use a non-relativistic Hamiltonian, many ideas

and corrections are borrowed from the relativistic

case. These will be introduced as we go along.

The non-relativistic atomic Hamiltonian that

we shall use may be written

7 e2 + e + (2)
H iri

where the summations go over electrons, and

- 3 -

-e = electronic charge,

h = Planck's constant,

J> = h/2n,

ri = distance of the i-th electron from the

center of mass,

r.. = distance between the i-th and j-th

/ O= (1/m + 1/M)

= reduced mass,

m = electronic rest mass,

M = nuclear rest mass,

and Z = atomic number.

Since it is inconvenient to introduce

experimental values of h, e, and p, we shall convert (2)

to atomic units. To distinguish between ordinary units

and atomic units, we will temporarily put primes on

quantities expressed in ordinary units. We begin by
introducing some atomic unit of length, a. Then, since V

is homogeneous of degree -2, we replace it by -2 72 and

r' by ar. Multiplying by a2/A 2, we obtain

1-1'- ji ,2

-2: (ez #z



and E = E'( A2/ 2). (4)

We can remove the experimental factors now by requiring

which gives

which gives

a = )2/(, 2),



E = 2/ (/2) = e2/a = ,e4//2, (6)

we see that the atomic units of energy and length, E

and a, both depend upon u, and hence upon the nuclear mass

M. We shall call the atomic unit of energy, E, the

Hartree (H.) and the unit of length, ., the Bohr (B.).

If we let M become infinite, the reduced mass becomes

equivalent to the electronic rest mass, and we find

E 2/me4, (7)

or E = (1 + m/M)E0, (8)

and a = /me2 (9)

or a = (1 + m/M)aO. (10)

It is also convenient to introduce the fine structure


= e2/Ac = 1/137.0888


We then find that the Hartree is also given by

- 5 -

e4/2 = dc2 (12)

That is, the Hartree corresponds to the rest energy of

the reduced mass times the square of the fine structure

constant. We observe that the Hartree is therefore on

this scale a rather small unit, although Appendix I,

which tabulates the most recent values of the

fundamental constants and conversion factors, shows

that for chemical energies, it is a large unit.

According to (6), the energy levels of two

isotopes of a given atom will differ by a slight amount

given by

_____ %I (-I

(M'- N) +
= --+ m ( - / *

The spectral lines of the heavier isotope will

therefore appear at higher energies (higher

frequencies, shorter wavelengths). This effect is most

pronounced for the three isotopes of hydrogen and is

called the isotope shift, or normal mass effect. Even

then, it is small, since /XD//ZH = 1.000817. Its existence

- 6 -

led to the discovery of deuterium in spectra of

residues of liquid hydrogen in 1931. Its importance in

spectroscopy lies in its very simplicity; since the

shift can be exactly predicted, it can be used as a

tool for identifying spectral lines. In the case of

the isotopes of hydrogen, for example, it may be used

to separate the specific mass effect, or mass

polarization, which will be discussed later.

Our assumption of a point nucleus, which was

implicit in writing (2), is not entirely correct.

Nuclei of different mass will have different finite

volumes and shapes, and this will influence the energy

level separations. This effect is small since nuclear

dimensions are of the order of 0.00001a0, and will be

discussed in'more detail later.

In (2), we used the reduced mass rather than

the electronic mass. This is an approximation that

comes from the transformation to the center of mass

(CM) coordinate system. It is usually ignored in

textbooks, although a few, such as Shore and Ilenzel /1/

and Bethe and Salpeter /2/ do treat it. The

implications for molecular calculations have recently

been discussed by Fr8man /3/. le will sketch the

results for an atomic system here.


If P is the momentum of the nucleus and -i is

the momentum of

the i-th electron,

then, by

conservation of momentum, in the CM system, we have

P t



=e k 2 pe T, is-

The kinetic energy, T, is


Making the usual

transformations from classical to

quantum mechanics,

p -i1f

we obtain



- i ) .-z

* .
+ Z, l.



S, i '-~ D 2

- 8 -

T -= .2 (15)

The second tern is generally omitted since it is of

order p/i/, which is small, and also, it is a two-electron

operator, which is more difficult to calculate than the

first term. The correction due to the second term is

usually called the specific mass effect, or sometimes,

the mass polarization effect, in contrast to (11),

which is usually called the isotope shift. In the

relativistic case, it is not possible to perform a

rigorous separation of internal and translational

motion by going to a CM system. This sort of

difficulty is quite general, and consequently, one has

to be extremely careful about attaching physical

significance to correction terms derived from non-

relativistic theory. This difficulty is compounded by

the fact that experimental data is frequently

interpreted on the basis of these approximate

corrections. A simple example of this is the spin-

orbit effect, which is really a two-electron effect,

and yet experimental data is usually reported in terms

of a separation parameter based upon the one-electron

effect found in the one-particle Pauli extension to the

Schrodinger equation. In view of our non-relativistic

approach in this work, it may be better to view

- 9 -

calculated small terms as empirical parameters; they

contribute to level shifts and splitting, but they in

general represent only part of the physics.

Nevertheless, such quantities are often useful in

intercomparison of wavefunctions and can help to point

out deficiencies in the description of various parts of

a given wavefunction.

The reader who is interested -in the

relativistic CM problems is referred to Bethe and

Salpeter (Ref. 2, section 42) and elsewhere /4,5/.

Finally, we remark that for one-electron systems,

Garcia and Mack /6/ have given theoretically calculated

tables of energy levels containing all known

corrections to the relativistic Dirac Hamiltonian.



1.1 The Hartree-Fock Method

Let us assume that the total wavefunction for

an N particle system is built up from a set of N

orthonormal one-particle spin orbitals, q5 (x), where x

= (, O ) is a combined space and spin coordinate. The

Pauli exclusion principle requires that the

wavefunction be antisymmetric, so we write it as an

antisymnetrized product of spin orbitals, i.e. a Slater


_L L L, ) X... ?j (1)

where a (- P (2)

is the antisymmetrizer, P is a permutation of

electrons, and p is the parity of the permutation.

Application of the variation principle to the

Schridinger equation then leads to the well-known

- 10 -

- 11 -

Hartree-Fock equations (see for example, Ref. 7,

Chapter 16) which may be written

S,. - ,.) (3)

where it has been assumed that

-0. 64 ^[(4)

The -*.are Lagrange multipliers, and by a unitary
transformation on (1), which leaves the wavefunction
invariant, one can diagonalize d and obtain

A. (5)

where i= -
and the #i in (5) are now linear combinations of those
in (3). The one-particle effective Hamiltonian, h, is
given by

; --Z -/ 2 Z
r> (6)

)j Sd 5^( (2) f-,)D V^U)^

- 12 -

where P12 permutes the coordinates of electrons 1 and

2, and g12 = /r12. The Hartree-Fock (HF) equations have

received extensive study in the literature, and we will

refer the reader to the extensive bibliographies in

Slater's books /7,8,9/.

To simplify writing, we will let

f-- (7)


(.1, -,) (8)

so that



-i < ilRili> (10)


The total energy is given by

- 13 -





?Note that = 0, so that in (11),


Substituting (9) into (11), we


j Z

/1212 IJ>




- 2, < /If.i>
> J

so that the total

energy is not given by the sum of the

orbital energies.
Let us see what happens if we remove an
electron from orbital m without changing any of the
remaining orbitals. We find



6 <2mI 12/2II M>





~ = c 4' iHl 4'> ~ 4J 1 41~

-! 57
- 2

- 14 -

- Z / (Z> / /<
i-j u Z Cm


(A) < -l- ,z, 2 <)inm ,Ivm >

- Z < YYj/ /-/,mj/>
(X4 j

= 61


Equation (14), known as Koopnans' theorem /10/, says

that the ionization energy is approximated by the
negative of the one-electron energy of the removed

electron. If we allowed the orbitals to relax, then

this would no longer be true. Equation (13), which is

not generally remarked upon, implies that the orbital

energies are decreased by the term

< ,'J2,2 / I'M >

which is always a positive quantity.
follows simply from the the fact that

(i(Pj)1 .- ( & / 1)


= i )

- Z -

Z' _L f-P,.

- 15 -

so (1 P12)/2 is a projection operator, and is therefore

positive, and since g12 is also positive, .,is positive,


< -0Z im -- ILM ?

> 0

If we assume that the orbital energies are ordered

6, 62 63 -- !S-N

and then we successively ionize electrons N, N-1, N-2,

...,1, without allowing orbital relaxation, we obtain

S= E. -1 6

IA/- E N-Z -

-< +A < 1 2 A-1I >

- EZ-2





- 16 -




* < N-2, A/ I2,, /A/-2, n/>

A= /


() -

'Z= < J 1A N ?,0

and finally,

'N- k

-E_ -E

- 17 -

- 7 6 (A )


This is of



course expected since the sum

energies must be the same

of all the

as the total

Similarly, we can study the effect of adding

an electron to the system in spin orbital k = N + 1.

Assuming again that the N occupied orbitals do not

change in the process, except that an orbital N + 1 is



-' < 1 i,/z-> i




Z Ik



- 18 -


r c^


- ,.,<.> 2 y'
y J J1^



-z: <)tv 'j//AiLf


However, in order to determine this last quantity,
which is the electron affinity, we need to solve the

additional equation


/4/ 41t/

where h is determined by the original l orbitals only.

The third quantity of interest in addition to

the ionization potential and the electron affinity, is

the excitation energy of an electron in orbital rn going

to an unoccupied orbital, say U + 1. A similar

" A.


- 19 -

analysis as before gives

e I/ p > i l

er' "- ,,


J- t < 1vp/ 1 2 z V/ A l

and again, we must solve (19).

We have emphasized these three quantities --

the ionization potential, the electron affinity, and

the excitation energy, because they are often cited as

the major use of HF theory. Ideally, these should all

be computed by making accurate total energy

calculations on the two states involved and then taking

the difference of the two total energies. The

assumption of no orbital relaxation in the final state

enables one to approximate these quantities by a single

calculation (for the ionization energy), or a single

total energy calculation and one single-particle

calculation, (19), which involves less computational


The orbitals, however, do relax

processes, and the effect may be significant.

the difference of total energies approach has

practical drawback of accuracy loss due

subtraction of large numbers to obtain

in these


the very

to the

a small


- 20 -

difference, schemes have been introduced to enable the

energy difference to be expressed in terms of an

orbital energy and a correction term (see for example,

Ref. 13 and references cited therein). Mention should

also be made here of Slater's statistical exchange

approximation /11/ to the HF equations because simple

equations can be derived for excitation energies,

ionization potentials, and electron affinities, which

do not require total energy differences. Instead, one

makes a single calculation in the so-called transition

state /12/ which is a state halfway between the initial

and final states. The method also has the advantage

that it is computationally much easier than HF, and

work is in progress in extending the calculations to

molecular systems.

The HF approximation and related empirical

and semi-empirical one-electron theories have carried

chemistry and solid-state physics a long way. In

particular, the mechanistic approach which has been so

useful in synthetic organic chemistry is based

essentially upon the one-electron picture, and

certainly chemical concepts of molecular geometry and

bonding are modelled upon the localization of electrons

into regions of space called orbitals. For atoms and

molecules, we can go beyond these ideas, but for the

- 21 -

solid state, they are still very much a computational


However, from a purely theoretical

standpoint, the HF approximation is unsatisfactory in

many ways. Consequently, there have been suggested

many different schemes for improving it. Before we go

into any of these and show how this leads into the

subject of this thesis, we will give the four major

restrictions that are made upon the orbitals in what is

(naturally) called the restricted Hartree-Fock (RHF)


a) The space and spin dependence is


b) The radial and angular dependence is


(see Appendix III for definitions of

the spherical harmonics, Ylm).

c) R(r) is taken to be independent of ml;

this is not a restriction for states

with L = 0, if L-S coupling holds.

- 22 -

d) R(r) is taken to be independent of ms;

this is not a restriction for states

with S = 0, if L-S coupling holds.

All of these are made for computational and

conceptual simplicity. The first and fourth permit the

determinant, for certain special cases, to be an

eigenfunction of S2, and the second and third, of L2.

The assumption in going from (3) to (5) was that a

unitary transformation could be applied to the

determinant (1). Because of this, a) introduces a

further restriction -- double occupancy of the

orbitals; that is, each spatial function occurs twice

in the determinant, once with c( spin, and once with /spin.

This restricts the method to closed-shell systems.

Since the latter are of less interest chemically than

open-shell systems, various methods have been worked

out to deal with these. The most important are

discussed in the next section.

- 23 -

1.2 Extensions to Hartree-Fock Theory

Relaxation of Restrictions

In the general open-shell case, with the

restrictions noted in the preceding section, the off-

diagonal Lagrange multipliers cannot be eliminated, and

the total energy expression becomes considerably more

complicated. We will refer the interested reader to a
review of open-shell methods by Berthier /14/.

If we reexamine (9) and carry out the spin

integration, we find that the effective Hamiltonian,

h, is given by

Oz (- R2 Z

.z fJ = (^.( + 27 C )f(1ij*(2)
d J

~ O(D ( 1)


Because of the delta-function in the second term, which

is the exchange tern, only those orbitals having the

same spin as orbital i will cone into the operator. If

there are differing numbers of d and 3 spins, i.e. S

f 0, then there will be different effective operators

for electrons in the same (n,l)-shell if they have

different spins. lWe have already noted the double-

- 24 -

occupancy restriction, and it was just this observation

that led to the introduction of what is called spin-

polarized HF (SPHF), or different orbitals for

different spins (DODS), in which orbitals from the same

(n,l)-shell are allowed to have different radial parts

if they have different spins. This is a relaxation of

restriction d) in the last section. Similarly, for

states with L 1 0, one finds different effective

Hamiltonians for orbitals of different m1 values. Relaxing

restriction c) then gives orbital-polarized HF (OPHF).

In principle, there is no reason to require

that orbitals have a fixed value of 1. A partial

relaxation of b) and complete removal of c) and d)

leads to the unrestricted HF (UHF) method. A UHF

orbital will have the form

U 4 Y i 4, (23)

The term unrestricted is not particularly good, because

the orbitals are still restricted to a single ms and a

single ml value, so that they are cigenfunctions of sz

and 1,, and to definite parity (which is why only odd

or only even values of 1 occur in the expansion (23)).

The SPHF/DODS and OPHF methods are relatively easy to

implement starting from an existing RHF program, but

- 25 -

the UHF method is much more difficult. As is usually

the case, the more we generalize, the harder the work


The Symmetry Dilemma

We are still far from removing the HF

inadequacies. One of the major difficulties is the

famous "symmetry dilemma" /15/. This has to do with

the fact that the eigenfunctions of the Hamiltonian

operator should also be eigenfunctions of the normal

constants of motion, that is, 'the normal operators

which commute with the Hamiltonian. This is required

because the wavefunctions and energy levels of a system

are classified by their symmetry properties spin and

angular momentum values, parity, rotational or point-

group symmetry, translational symmetry, time-reversal

invariance, and so on. The four restrictions noted

previously are all chosen to make determination of

these symmetry properties simple. For example,

restrictions c) and d) guarantee that the wavefunction

is an eigenfunction of Sz and Lz. In fact, since most

N-particle symmetry operators can be written as a

simple sum or product of single-particle symmetry

- 26 -

operators, the temptation has always been strong to

restrict the one-particle functions to be

eigenfunctions of the single-particle symmetry

operators. This has had unfortunate consequences,

particularly in chemistry, where one often hears

discussions of the importance of d or f orbitals in

molecular bonding or of the importance of hybridization

in the determination of molecular geometry /16/.

It was thought for a long time that if a

variation were carried out on a wavefunction that the

end result would be a function corresponding to the

lowest energy, and therefore would automatically have

the correct symmetry properties. This is not

necessarily so, and it has been .shown by L8wdin /17/

that if a wavefunction has mixed symmetry, then it has

at least one symmetry component which will have an

energy at least as low as the mixed symmetry function.

By the introduction of symmetry production operators,

L8wdin /18/ showed that one could select a specific

symmetry, component of the wavefunction. The

wavefunction for a particular symmetry k is then

written in the form

1' =(9k D (24)

- 27 -

where D is the Slater determinant and the projection
operators Ok fulfill the usual relations:



0 Ok
0-^ C9 /-

That is, they are idempotent, Hermitian, bounded by the
zero and unit operators, and form a resolution of the
identity. The energy then takes the form

E /- < /IH/> //O/Z>

S< O)lHl/O >/< < /OIOD>



This may be regarded either as a modified expectation
value of the operator OH = OHO = HO, or as the
expectation value of a wavefunction that is a sum of

- 28 -

single determinants, since this is the form that OkD has

in (24). L8wdin /17/ then carried out a variation of

the individual orbitals in the determinant and showed

that one could still obtain HF-like equations, which he

called extended Hartree-Fock (EHF); these differed from

the HF equations only in having a more complicated

effective one-electron operator. These equations are

very much more difficult to solve, and it is only

recently /21,22/ that attempts have been made to put

them into a computationally feasible form. Instead, a

number of simpler approaches have been tried.

The first was simply to take a RHF or UHF

determinant, project it, and obtain the total energy.

This is not difficult to carry out, and the energy

improvements found are quite small for the state of

lowest energy. This is to be expected since the

unprojected variation does approach a minimum or a

saddlepoint, although it need not be the absolute

minimum. The name applied to this method starting from

the UHF determinant is projected unrestricted HF

(PUHF). This approach may be criticized because the

variation is carried out before the projection. It was

emphasized /23/ that the variation should be done after

projection, and this has recently led to the spin

projection of a Slater determinant followed by orbital

variation, the spin-extended HF (SEHF) method. Kaldor

- 29 -

/24-27/ and Sando and Harriman /28/ perform direct

variations of the total energy, abandoning the use of

the one-electron effective Hamiltonian for

determination of the orbitals. Goddard and Ladner /29-

33/ solve the one-electron equations and determine the

SEHF orbitals. The two approaches are equivalent for

the total energy calculation since the final orbitals

in each case are related by a linear transformation,

which leaves the determinant invariant (except possibly

for a constant factor which vanishes in the

normalization). Both are difficult because a large

number of non-linear parameters must be determined.

Solution of the one-electron equations is particularly

difficult because of the very complicated effective

operator, so their neglect in favor of a total energy

minimization is understandable. If we abandon the one-

electron equations in the computations, we lose the

simple pictures of ionization and excitation energies

and electron affinities, concepts which can be

generalized from the simple HF results given earlier to

the case of these more complicated one-electron

equations. In one sense, this is regrettable, but in

another, the fundamental limitations of the independent

particle model (IP I) -- that N orbitals describe the

motion of N electrons -- presently do not justify the

- 30 -

cost and effort of obtaining the results. There are

better methods available.

On Spanning the Angular Momentum Snaces

Aside from the inadequacies of the IPM, one

of the major problems remaining in the projected

Hartree-Fock methods is that the projected determinant,

even though it is a pure symmetry component, generally

does not span the complete spin or angular momentum

space. This has recently been emphasized in a note by

the author /34/, but since it i.s directly concerned

with the subject of this dissertation, we will repeat

some of it here.

For- N electrons and total spin S, the number

of linearly independent spin functions is given by


f = (2S + 1) N! < 2N (27)
(N/2 + S + 1)! (N/2 S)!

As N increases, f becomes very large. For example,

with N = 10 and S = 1, f = 90. The previously

discussed projected Hartree-Fock methods obtain only

one of these. The choice of which spin function to use

- 31 -

is not immaterial, even if the Hamiltonian is spin-

free. For example, for lithium 2 2S, E(RHF) = -7.432725

H, while E(SEHF) = -7.432813 H, an improvement of only

0.000087 H. There are two independent spin functions

for this system, one of which SEHF uses. If the other

one is used, one obtains E = -7.447560, an improvement

of 0.014835 H, nearly one hundred seventy times the

SEHF improvement over RHF! The reason for this

difference is clear. The SEHF (Goddard's GF)

wavefunction for doublet lithium is

A ( )s)l s (c a^ -p (28)

and consequently, the first spin factor forces the

inner shell to have unpaired spins which is

energetically unfavorable. Instead, it would be

preferable to have paired spins in the closed shells.

This scheme is known as maximally-paired HF (MPHF,

Goddard's Gl). For doublet lithium, the MPHF

wavefunction is

4 (s) (S/))(2 )s (~gca ,'J)/vi (29)

Although the energy of this function is better, it does

not correctly describe the hyperfine structure. The

MPHF function gives zero contribution to the Fermi

- 32 -

contact term for closed shells, thus predicting a zero

contact term for closed shell systems such as He, Be,

and Ne, in contradiction to experiment.

A way of taking into account all the spin

functions, but still remaining within the IPM, was

first suggested by Kotani /37,38/ in 1951 and has

recently been applied by Lunell /72/, Ladner and

Goddard /33/ and Kaldor and Harris /39/. This is to

use as a wavefunction a product of N spatial orbitals

and a linear combination of all the f independent spin

functions 0k for the value of S desired:

4S 0 (1) 0 (N) (30)

Ladner and Goddard call this method spin-optimized GI

(SOGI); Kaldor and Harris name it spin-optimized self-

consistent field (SOSCF). The results that have been

obtained with this method will be discussed later.

There is an alternative way to span the spin

space, however. We recall restriction a), that the

spin orbital be represented by a product of a spatial

orbital and a spin function d or /3. This obviously can

generate only one spin product in the determinant for

a projection operator to act upon. However, we have a

choice of one of two spin functions for each of the N

spin orbitals, so we could generate 2 different spin

- 33 -

products, which completely span the N-electron spin

space, since there are no other possible products. If

we carried out a spin projection on this set, we would

obtain all f linearly independent functions. We then

recall that the solution of the relativistic Dirac

equation for a single particle is a single-particle

wavefunction, or orbital, with four components. Two of

these correspond to electrons, and two to positrons.

Each pair has one component with a-spin, and one with

/-spin. Since we are dealing with non-relativistic theory,

we can ignore the positron components, and write a

general spin orbital (GSO) for an electron as

40 (") = # (): (c F;)/ (31)

where 0 and are independent spatial functions. This

possibility seems to have been first noticed by L8wdin


A projected determinant of these orbitals is

a more general function than the SOSCF one (30),

primarily because there is more spatial flexibility.

Since there has been some disagreement about this in

the literature /34,11/, we shall sketch the proof.

If we restrict (31) to the case

- 34 -

OK k X, 5 (32)

where Ak is a constant, the projected GSO (PGSO) function

takes the form of (30). Since there are N A's to be

determined, the SOSCF function is a special case of the

PGSO function, provided f does not exceed N. The first

case where this happens is for N = 6, S = 1, when f =

9. For f > N, the condition (32) is too restrictive.

The PGSO function consists of a sum of orbital

component products multiplied by spin function

products, which may be written in the form

Y ^ /(1u ... (33)

where ;K will in general be a sum of orbital component

products, and @k is the spin function used in (30). At

least g different orbital components must appear in the

PGSO function, where N g 2N. Now, suppose that for

computational purposes, we expand the orbitals in (30)

and (32) in the same M-function basis. The SOSCF

orbitals will then be functions of M variables each,

and the GSO, functions of 2M variables. Since we can

perform a linear transformation on the orbitals in a

single determinant without changing its value, except

for a constant multiplicative factor which vanishes in

- 35 -

the normalization, we can without loss of generality

orthogonalize the GSO orbitals, but not the SOSCF

orbitals. Note however, that this does not imply that

the sets

are individually orthonornal. The number

independent variables for the SOSCF function is

= NM + f 1

and for the PGSO function,

nPGSO = gM N

since the 'orthonormalization has removed one

independent variable from each orbital in the PGSO

function. We can always satisfy npGSO nSOSCF provided

we choose H sufficiently large that


= (g N)H N f + 1 0

M (f + I 1)/(g N)




- 36 -

The case with g = N corresponds to that state of

highest multiplicity, for which f = 1, independent of

N; in this case, the PGSO and SOSCF functions are

equivalent. In this case also, since f = 1, the SOSCF,

EHF, and PGSO methods are equivalent. Except for one-

and two-electron systems, these states of highest

multiplicity lie high in the continuum and only

recently have become of interest. For most chemical

applications, they are of no concern.

The PGSO and SOSCF methods are substantially

more difficult to carry out than the other methods that

we have discussed. PGSO is essentially an EHF method

with general spin orbitals rather than pure spin

orbitals. In both cases, one has to deal with non-

orthogonal orbitals, and this makes the integral

computation considerably more time-consuming.

Computations using the PGSO method have been carried

out on helium and its isoelectronic sequence by

Lefebvre and Smeyers /42/ and Lunell /43/. Lefebvre

and Smeyers however, after expanding and projecting the

GSO determinant, orthonormalize the components, which

does not leave the original determinant invariant.

Consequently, their wavefunction is not a true PGSO

wavefunction. Lunell is therefore the first to

actually carry out the solution of the EHF one-electron

equations for GSO. For all but the smallest bases, he

- 37 -

encountered convergence difficulties in their solution

and consequently abandoned them in favor of a total

energy variation for his larger basis sets. He also

made preliminary calculations on lithium using the

restriction (32) which makes the PGSO function

equivalent to the SOSCF function for this system. At

his suggestion, the author began work on the general

case for three-electron systems work that forms the

main results reported in this dissertation. These will

be discussed in Chapters 3 and 4.

Thus far, we have emphasized the spin

degeneracy problem in the projected Hartree-Fock

methods. There is also, however, an orbital angular

momentum degeneracy problem which has not yet been

satisfactorily treated in the PHF approaches. This is

much more difficult, because unlike the spin problem,

where we had a finite space of f 2 2N spin functions,

we have an infinite space to deal with. In the spin

case, the result of applying a spin projection operator

to an arbitrary product of spin functions is well-known

/36,91/. The result is a sum of spin products with

fixed coefficients, called the Sanibel coefficients,

which can be obtained from closed expressions without

having to carry out the rather tedious operation of

applying the spin projector directly. For the case of

a product of an arbitrary number of spherical

- 38 -

harmonics, closed forms for the coefficients are not

known /45/. Of course, one can still work out the

result using the projection operator. One approach

that has been used to avoid this is to set up a matrix
2 2
eigenvalue problem for the operators L L, S and S

and solve for the eigenvectors and eigenvalues /46/.

!e have ignored this problem in this work for

two reasons. One, it introduces many additional

complications in a method that already is perhaps too

complicated for practical everyday use with our current

computer capabilities. Two, the problem can be handled

more readily in the configuration interaction method

discussed in the next section. The angular problem is

important, however. For the ground state of lithium,

the RHF energy is -7.432 H, the best PGSO energy, using

orbitals restricted to 1 = 0, is -7.448 H, and the

exact non-relativistic energy is -7.1478 H. The correct

treatment of the spin space has yielded only about 350

of the correlation energy, the difference between the

RHF and exact non-relativistic energies.

- 39 -

1.3 Other Methods

We shall ignore all the empirical and

semiempirical theories. These have of course a great

deal of use in systematization and preservation of

simple concepts, but their reliability for any

particular system is generally questionable. Rather,

we are more interested in the ab initio methods -

methods which could be trustworthy, if we could carry

them out to a sufficient degree of accuracy.

The Configuration Interaction Method

The first, and most important of these, is

the configuration interaction (CI) method. It was

shown by LB.wdin /18/ that any antisymmetric

w.avefunction may be written in the form of a linear

combination of determinants formed from a complete set

of one-electron functions; the CI idea itself goes back

to Hylleraas /17/ in 1928. The fact that all the

complete sets of one-particle functions of interest in

quantum chemistry are infinite means that the CI

wavefunction is an infinite expansion, since a basis of

MI functions for an (l-electron system can generate (1)

- 40 -

determinants. For a fixed basis, variation of the

total energy with respect to the configuration (i.e.

determinant or projected determinant) coefficients

leads to a secular problem of the same dimension as the

length of the CI expansion. The variation principle

and the separation theorem guarantee that the

eigenvalues of the secular equation will be in order

upper bounds to the exact eigenvalues of the

Hamiltonian used. This fact is extremely important,

for it means that we can treat excited states,

something that cannot be done easily with the HF

methods (unless the state is the lowest one of its

particular symmetry). In addition, the CI function can

be optimized for each state, while -the RHF function can

be optimized only for the ground state. The virtual

(i.e. unoccupied) orbitals that come out of the HF

problem can be substituted into the HF determinant to

obtain an approximation to an excited state; the

resulting determinant by Brillouin's theorem /48/ will

be orthogonal and non-interacting with respect to the

ground state determinant, and therefore its energy will

be an upper bound to some excited state energy, but not

necessarily the one desired.

With a sufficiently large basis for the CI,

one can span a much larger section of the spin and

angular momentum spaces, and the determinants can all

- 41 -

be projected. Consequently, these difficulties of the

PHF methods can be minimized. There are two general

problems in CI calculations however. The first is that

the number of configurations for even a moderate basis

is usually much more than can be handled, so that one

has to make a selection of configurations. This aspect

of the method is currently more of an art than a

science. The second problem is the selection of the

basis. If the functions are fixed, and if one can

include all possible configurations, then any linear

transformation on the basis only alters the

configuration coefficients; solution of the secular

problem yields the same energies. For a truncated set

of configurations, there exists an optimal linear

transformation of the basis which will give the lowest

energy for a particular state, although finding it is

non-trivial. This is discussed further in Chapter 2.

An alternate approach would be to use a sufficiently

small basis so that all configurations could be

handled, and then to vary the orbital basis. Lowdin's

EHF equations /18/ can then be used to obtain the

optimum orbitals. This has been implemented in a

somewhat restricted fashion, and is known as the

multiconfigurational self-consistent field method.

- 42 -

Interelectronic Coordinate Methods

A difficulty with the CI method is the slow

convergence of the expansion. This is partly due to

the difficulty in satisfying the cusp condition on r 12

that is, the proper behaviour of the wavefunction as r
--> 0. This is discussed further in Appendix IV. This

difficulty can be avoided if interelectronic

coordinates are introduced into the wavefunction

because the cusp conditions can be satisfied exactly.

The use of interelectronic coordinates was first

introduced and used by Hylleraas /49/ in 1929 for the

helium atom; wavefunctions containing interelectronic

coordinates are now generally called Hylleraas

functions. James and Coolidge /50-52/ in 1933 used a

thirteen-term Hylleraas function on the hydrogen

molecule. The helium calculations were carried further

by Kinoshita /53/ and then in a monumental piece of

work, the wavefunction was extended to up to 2300 terms

for the singlet and triplet S and P states of the

helium isoelectronic sequence by Pekeris and coworkers

/54-56,103,104/, with an accuracy of about 0.001 K,
about one part in 10 The method has been extended

to lithium by Burke /57/, Ohrn and rordling /106/,

Larsson /58/, and Perkins /59/, and to beryllium by

Szasz and Byrne /60/ and Gentner and Burke /61/. The

- 43 -

Ohrn and Nordling, and of Perkins, is notable in that

excited states have also been treated. These

calculations are currently the most accurate available

on two and three electron systems. The method has not

been carried out for atomic systems with more than four

electrons, or for molecules other than hydrogen,

primarily because of the difficulties in integral

evaluation. (See however, ref. 107 and references


Recently, Sims and Hagstrom /62/ have shown

that a combined Hylleraas/CI wavefunction of the form

2J= Z cV k


.. G 0 g ,; ... ,l

is feasible for atomic calculations with any number of

electrons and have reported results for the beryllium


For atoms at least, the combined Hylleraas/CI

method seems to be the most promising, if it can

compete with ordinary CI in terms of computer time.

For small diatomic molecules, several Cl programs are

currently in use, written by Kouba and thrn, by Harris,

- 44 -

Michels and Schaefer, by Bender and Davidson, and by

Hagstrom and coworkers. The IBM research group in San

Jose, California is running a linear triatomic CI

program called ALCHEMY, and work is currently in

progress there on the production of a general

polyatomic CI program using the Gaussian orbital

integral package from their polyatomic RHF program,


Calculations with such programs are currently

rather expensive; assuming that computational costs

continue to fall and computer facilities continue to

expand, these programs may be in general use in a

decade or two. In the meantime, we shall undoubtedly

continue to work on simpler methods to try to increase

our understanding of atoms, molecules, and solids.

Bethe-Goldstone Perturbation Theory

Finally, we should mention a method brought

over into quantum chemistry from nuclear theory,

principally by Brueckner, Nesbet, and Kelly /97/, which

they call Bethe-Goldstone (BC) perturbation theory. A

great deal of work has come out lately on atomic

calculations of excitation energies and hyperfine

- 45 -

structure. The method relies heavily on the pseudo-

physical (i.e. intuitive) pictures that one has in the

Feynman diagrams which represent the terms in a

perturbation series, based usually upon a Hartree-Fock

starting point. The method, while giving good results

by careful workers, does not seem to provide the simple

conceptual pictures of electron densities which are

desirable in chemistry. Further, the wavefunction, if

indeed one exists, is not exhibited, making comparison

with other methods difficult. In the author's opinion,

one of the main functions of quantum chemistry is to

provide chemists with a language that can be used to

systematize and if possible, quantify, chemical ideas.

We feel that the BG methods, however close their

results agree with experiment, do not succeed in this


With these closing remarks, we shall now pass

over to a discussion of density matrix theory.



2.1 Introduction to the Density Matrix Literature

The concept of a density matrix goes back to

von Neumann and Dirac in the late 1920's, but the more

useful concept of the reduced density matrix was

introduced by Husimi /63/ in 1940 for use in the

Hartree-Fock problem. Husimi's work was significantly

extended to arbitrary wavefunctions by LBwdin

/18,19,20/ in 1955, who should be credited with

demonstrating the quantum mechanical utility and

significance of reduced density matrices. For reviews

of the subject, see IlMcWeeny /64/, Coleman /65/, Ando

/66/, Bingel and Kutzelnigg /141/, and the proceedings

of two recent conferences on density matrix theory

/67,68/. In her Ph.D. dissertation, Ruskai /69/ has

given an excellent survey of all the known theorems in

reduced density matrix theory, with a bibliography of

the principal theoretical papers.

- 46 -

- 47 -

2.2 Construction of the Reduced Density Matrix

Definition of the Reduced Density Matrix

We define the p-th order reduced transition

density matrix as

J' ^ 1. -..,A) w I :'.'. p' ,,; (34)

The term density function would be preferable,

particularly since the term density matrix will also be

applied to a quantity with discrete indices introduced

later. However, the dual use of the word matrix is

well-established, so we will continue to use it. The

primed variables always come from the complex conjugate

of the wavefunction; also, when K = L, we shall drop

the subscripts K and L and the word "transition". It

is often convenient to economize the notation by

letting x = (1,2, ... ,p) and y = (p+l,p+2, ... ,N) so

that (34) becomes

[I' IT 0-' rJ -

The factor ( ) is chosen so that the trace operation,

r --7p)

gives the number of groups

I and J are orthogonal,
transition p-matrix is zero.

(34) is 18wdin's normalizat

use are Coleman's,

of p particles. If states

then the trace of the
The normalization used in

:ion. Two others in common

D p) (A) /-7(P)

and McWeeny's,

For our purposes, the L8wdin normalization is most

convenient. The wavefunction in (34) may be symmetric,
for bosons, or antisymmetric, for fermions. Many of

the theorems can be developed for both cases. However,
for most quantum chemical applications, we are

interested only in fermion systems, so in the

following, antisymmetric wavefunctions are to be

- 48 -

Z frI^wa/


- 49 -



We shall also use the usual notation

We obviously have

" .- p-)l .
i- (2P' 112'"~1

f7 -p -
Pi I

and the p-th order reduced density matrix, or simply,

p-natrix, is antisymmetric with respect to interchange

of any two indices on the same side of the vertical



where P is any two-particle permutation operator such

that P 0= -z. We also have


r I ') -

When K = L, the p-matrix is therefore Hermitian, and

in this case, it is also positive, since for arbitrary

functions f(x),


rOp) (xl P') =- F ( Ix)


- 50 -

f&(P) -/'~(x/x) j ) -

J rJ*' c/i-


The diagonal elements have the interpretation that

fl(x 1)cA',.. C/Y

is the probability of finding any p particles in the

volume element dx dx2...dx regardless of the positions
of the remaining particles.

Suppose that we have an arbitrary operator,
Q, which is symmetric in the particle indices, so that
we may write

ZQL 3 Z'Q3 /
'fXJ2,t J


o, I~e :L

The primes in the first set of summations indicate that
we omit terms with two or more equal indices. Then

I < > I/CI> <4 >_
< a >,j

QZ' :


- 51 -

ffz -A 4ch J Y.2


- .

where we follow the convention that the operators act
upon the unprimed variables only, and after this the
primes are removed and the integration carried out.

The Reduced Density Matrix for a Cl ,lavefunction

Following LBwdin's original work /18,19,20/,
we introduce a CI expansion for the wavefunction of
state I:

, i


The index K represents an ordered configuration defined
by the one-particle function indices (kl, k2, ...,kN )
where k1 < k2 < ... < k ; that is,

SQ, r, (121r~~ dr, dr,

- fQ,'(llw, ,

613Z- !QJ)^ ( v 4 42

-- Z, c,<
v S1

- 52 -

-A^#b-- (A"

'1? a'e(2)

The one-particle

orthonormal. We i


basis is not assumed to be


I4 T -





For notational convenience, we introduce "fat" symbols,
g= (k1, k2, .. ,k ) and = (k < k2 < ... < k ). We
- 2 p 1 2 p
also need

)( V, r; 1, ,; /p IL,; .. ,...P)- T;CKI; L/,,PJ; ) (45)

the cofactor of the determinant (43), which is formed
by deleting rows containing functions kl, k2, ...,k and
columns containing functions 11 ...,1 and evaluating
1 2 p


d- f S IT, KLI

c brlp~~~

- 53 -

the resulting determinant and applying an appropriate

parity factor. The parity factor is -1 if the sum of
the cardinal positions of the function indices in the
original determinants is odd, and +1 if the sum is
even. With these definitions, we obtain

JW (i\.I p ) f- 21 1 2 (LI CL ** LI))


The notation 4 above a summation means that all
configurations containing the orbitals in set A are to
be included. We can simplify (46) by introducing a
discrete p-matrix,

E < (< ms i l l1 U-

[^ LI C^( M LI) ^ LJ,/J


The presence of a tilde under a matrix signifies that
it is indexed by the non-ordered sets k, f. We drop
the tilde wthen we mean the part indexed by ordered sets
of indices, b and We then have

- 54 -

(?) 7 X (.. (') ('T

,K ) ~(48)

The peculiar reversed form of the indices I and J Is
simply to make the matrix form of this equation
simpler. The discrete transition p-matrix satisfies

r ---- r (49)

Pr k(j/i ) (50

where P permutes any two indices in a set. That is, it
is antisymmetric with respect to interchange of any two
indices on the same side of the bar, and is Hermitian
if I = J. The summations in (48) are over all values
of the indices. With a little study, one can use (50)
to reduce (48) to

_L 3 (i... 51)
ft p) (51)

where the summations are now over ordered sets of
indices. This is extremely important for computational

- 55 -

use, since we will only need a small part of the full

p-matrix in (48). In fact, for M orbitals, the full p-

matrix in (47) has dimension M ; the part with ordered

indices has dimension ( ) which for large M goes as
rP /p!.

Thus far, we have essentially followed L&wdin

/18/, except that we have used the transition matrix

throughout and introduced Slater determinants in (51).

At this point, LBwdin specialized to orthonormal

orbitals (Ref. 18, eq. 59); we shall avoid this


Now we would like to study the properties of

the discrete p-matrix. Using (44) and (48), for I = J,


For orthonormal orbitals, S.. = .., and (52) reduces

^ r(P- i(V Ps Fej(P)ij (53)
'S < '

- 56 -

.z EF )(k 1)


In the general non-orthogonal case,
re r i tten

(52) can

& rF"


f S = s ...

is the direct product of p overlap matrices. The
product matrix in (54) is not Hermitian, which is
rather unpleasant to deal with computationally. We
note that since the trace is invariant under
permutations of the product matrices that

Lt = C(fPIRYM ']- tb(li


so we define


PiI tr ~B~

- 57 -


Y C^'ff"


fR2 'TY2

and then in the general case, we will have

ir 61(/

-/ N

From now on, we shall refer to the primed Hermitian
matrix in (56) as the p-matrix, because it has the nice
properties that hold for the case of an orthonormal
To proceed, we need some theorems on direct
product matrices, which we state here, and prove in
Appendix VII.




(A x B)(C x D) = (AC x BD)

If AB = CD and EF = GH,
then (A x E)(P x F) = (C x G)(D x H)



If r is a rational number, then
(AI x A x x A )A r (Ar x Ar .. x ANr
1 2 ,iI 1 X .. X )





- 58 -

provided that all the powers on the right exist.

t t t t
Theorem: (A1 x A2 x ... x A) = (A1 x A2 x ... x AN) (61)

From these easily follows the results




_ [ S S ... J (63)

z. ( xLi.... )r )XX )

Since S
for the
so we can

> 0, all rational powers r exist, and we can
compute (56) from the eigenproblem solution
basis overlap matrix. Now (56) is Hermitian,
diagonalize it:


(c17u) A lfYJJ)

... < J) = ( x ...x -U)(AlXA -- xA) (62)


= },uI

r"' /m-~2-Y2 6'"~`"

- 59

For I = J, (48) can be written

-S X .

- 0 / x j ... x '-

S# r.... x

,a-^Os-Yj['yf--'-16 ]





r 1


There is-an infinite number of ways of orthonormalizing
a basis. It is interesting that the particular one
(66) comes into this quite naturally. It is the well-
known symnetric orthonormal ization /70,71/ which has
the particular property that if the basis set is
symmetry adapted, the orthonormalized set is symmetry
adapted also to the same symmetry operations.

Fr7 (x/x) =

[ (f)X (U .. (,] [.(ti Y (2)... 'p

0-= :y'

t(r/'c = S ^^/^ S1">

- 60 -

Substituting (63) into (65), we can bring the expansion
of the p-matrix (48) to diagonal form:



~i (69a)

f I (69b)

The general form (69b) is again more convenient for
computation because we need deal only with ordered sets
of indices.
Practically speaking, the operators that one
is usually interested in contain at most two-particle
interactions, so according to (41), we need at most the
2-matrix. Let us therefore specialize (68) to the
cases p = 1 and p = 2. We obtain for p = 1,

) oL J~)il

--- \ ) < 4(n'\ (70)

?I [( 6',%- c a ] (5-6-...r)dI]

^= ^ L

#Vz LA

and i is obtained from

and for D = 2,

r1l) 21'2')

C de (6k, 6, (1L (11

*^^~ z IO

.,n (J tz(,) A* (1


whe re

(,,(1,2) =


The orbitais defined by (71) are cal led the natural
spin orbitals (NSO); the functions defined by (73) are
called the natural spin geminals (NSG). The p-matrix

- 61 -


I -3
7- 2j Jez^) m.

- 62 -

eigenvalues are called occupation numbers, and the

eigenvectors, natural p-states.

Obviously, had we used the orthonormal basis

(66) to begin with, the primed p-matrix in (56) would

have been obtained directly from (47) for I = J. For

this case, for the diagonal elements we find

so that

O -a _r i o f ( ) 1 (74 )

The diagonal expansion of the p-matrix in

terms of the occupation numbers and natural p-states is

particularly convenient for the computation of

expectation values. We obtain from (70) and (72)

Z < ,I >, X > (75)


- 63 -

ce/>- L Zr

; 2 A,-1o <;/&^ ^,,Z> (76)

The NSO and NSG have the physical significance that

they are the set of functions for which expectation

values are strictly additive. The occupation number

factor in (75) and (76) is computationally important

because it means that the sum may usually be truncated

after the larger occupation numbers.
The discussion of the eigenproblem has thus

far been centered on the non-trans'ition p-matrix case.

To the author's knowledge, no work has been published

on the transition matrix eigenproblem. In fact, Bingel

and Kutzelnigg (See ref. 67) seem to be the only

authors who have carried out derivations in terms of

the more general transition p-matrix.
As we noted earlier, according to (49), the

transition matrix is not Hermitian. It is well-known

/73/ that an arbitrary matrix may be brought to triangular
form T by a similarity transformation with a non-singular

matrix :

W TW= T (77)

The eigenvalues of Fare the diagonal elements of A.
The eigenvectors are found as follows.

J r WT Vh'

t 1 (78)

This is a triangular set of equations for the vector
JV from which V may be determined according to

VV') (79)

Since we do not have a diagonal form of the matrix ,
the simple results (75) and (76) do not hold. There is
consequently no value in determining eigenvalues and
elgenvectors of the transition p-matrix because doing
so does not simplify the situation. Instead, one

- 64 -

- 65 -

simply computes transition values directly from (41)

and (51):

<^Ef ^-
^>^' Qo frI4>( rl}~4

< Z

lur 1i

J, ii


Thp RPnduired Density Matrix for a lon-CI


If the wavefunction does not have the Cl form
(42), one can still obtain a reduced density matrix
front (34). Obtaining a representation of it in a
discrete basis is not difficult. The p-matrix may be
considered the kernel of an operator such that

f F (x/x, ,)/'xdx' /


for arbitrary functions
eigenfunction of then


If f(x) is an

1 _X ~__~_Y___ ______ _I _~_ _~_ _

(Jz, P 0

<,.O IQ, O 0 k,( f ^

F' f() =

- 66 -

r"F ? ANW


Larsson and Smith /74/ have recently used this relation
to derive NlSO's of the 1-matrix of Larsson's Hylleraas-
type wavefunction for lithium. They introduce an M-
function basis and expand the NSO's in this basis:


Using this in (82) gives

(8 i)

which leads to the secular problem

<^/ >C Cr< ^ ~




? = 4 C


IC sck

- 67 -

- f e)f (y ) k 4& (87)

J4. ,
-<, jI >-

f07 (x) C&


The secular problem (86) may be solved by the methods
discussed in Appendix V.
The occupation numbers calculated this way
will be lower bounds to the exact occupation numbers.
The proof is not difficult. Let the set of exact
normalized ISO be with eigenval.ues A :

9',% K- A


Now let us order the exact and approximate solutions
according to

14z 7

>- 0


and then construct the operators

- 68 -

9x, 21 (91)

P E 2 IZ<>1 / (92)

which are projection operators satisfying the usual

relations. From the theory of outer projections /75/,
we know that for an arbitrary projection operator 0,

and any operator I bounded from below, the eigenvalues
of O0O are upper bounds in order to those of /.

Now oSO= 6 (93)

and 01 O i (Z' /)

so () n^ -z v : (94)
so that 9j has eigenfunctions X and eigenvalues and

we have immediately

,i /i (95)

The approximate occupation numbers are therefore lower

bounds as stated. The sum of the approximate
occupation numbers approaches ( ) from below and provides
a convenient measure of the adequacy of the chosen

- 69 -

2.3 Properties of Density latrices

Cl Expansion Convergence

In Chapter I, we mentioned the convergence

problem in the CI method. In his original paper on

density matrices, L8wdin /18/ showed that the natural

spin orbitals are actually the orbitals which give the

most raid convergence of the Cl expansion, the HSO of

highest occupation number being the most important. Of

course, one needs to know the wavefunction to begin

with in order to obtain the p-matrices and the natural

p-states. However, if a truncation is made of the CI,

one can obtain NSO for this truncated function, put

these back into a new CI, perform a new truncation

based on the size of the NSO occupation numbers or

other criteria, obtain new NSO, and so on. This

natural spin orbital iteration technique has recently

become quite a popular tool in Cl calculations, but the

convergence of the scheme does depend on the quality of

the initial truncation.

- 70 -

Bounds on Occupation Mumbers

Since the p-matrix is positive and of finite

trace, its eigenvalues obviously satisfy

o ~ ()


Coleman (See Ando, ref. 66) showed that


p = 1 and p = 2 takes the form



L Ni ( )

1 CMn> !)

A/L ( >3)

it can be shown that the upper bounds are never

attained except for p = 1 and p = N 1. Sasaki /76/

obtained better bounds than these, the first few of

which are

which for



- 71 -


21. '2)



_ 1+ 3.[i V-3)]

where [x] is the integral part of x.
He also proved that the bound for p = 2 is the best

The Carlson-Keller-Schmidt Theorem

Carlson and Keller /77/ showed that the non-
zero eigenvalues of the p-matrix are identical to those
of the (N-p)-matrix, and if the number of non-zero
eigenvalues is finite, then these two matrices are
unitarily equivalent. In addition, if


p (Ac~l~f(jt 21 c (i ~ 1G~ry) i



. 1 <'/P > --

,/<.?, > (102)

- 72 -


",^-A (ovfSY )^ (104)


7 y) (A/ (105)

If and were derived from an antisymmetric wavefunction,

then the resolution (103) of the wavefunction is

automatically antisymmetric already. The

eigenfunctions of the p-matrix are called natural p-

states, and those of the (N-p)-natrix, co-natural p-

states. Coleman /65/ later pointed out that this

theorem had already been discovered more than fifty

years earlier by Schmidt /7'/. Schmidt's results, in

the terminology of density matrices, show that the

expansion (63) gives optimal convergence in the least

square sense to the wavefunction; this, coupled with

the fact that the natural n-states can always be

expanded in terms of the ISO /65/, leads to the CI

convergence theorem independently obtained by L8wdin

which we referred to earlier.

- 73

The Carlson-Keller-Schmidt theorem is of

particular significance for N = 3, since the 1- and 2-

matrices then have identical non-zero eigenvalues, and

the NSO and NSG can be obtained from each other by

virtue of (104) and (105).

Symmetry Pronerties

We mentioned earlier that the wavefunctions

should be required to be eigenfunctions of the group of

the Hamiltonian, and the question of how the symmetry

properties of the wavefunction carry over to the p-

matrices and the natural p-states has been extensively

studied. We shall merely list some of these results

here which have significance for our own work.

Theorem 1: If 1 is an N-electron Hermitian operator

of the form

4"- z (106)

or a unitary operator of the form

.n j=4 (107)

or an antiunitary operator of the form

- 74 -

(Al) IT fi(108)

where, in (107) and (108), R ;is unitary

and K denotes complex conjugation, and

if P is an eigenfunction of then

the natural p-states can be chosen as

eigenfunctions of .

Theorem 2: If 4 and J transform as the irreducible

representations 9 and d respectively of

some group, then /7 transforms as the

direct product representation ex X

The particular significance of these results

is best illustrated by a few examples. If the

wavefunction is an eigenfunction of L S or rarity,
z z
theorem 1 applies, and the p-matrix blocks by ML, M'S

or parity value, and the natural n-states are

eigenfunctions of L S or parity. If the wavefunction
9 2
is an eigenfunction of S2 or L2, the natural p-states
2 2
can generally not be chosen eigenfunctions of S or L

except when S = MS = 0 or L = ML = 0. Of course, for

special choices of approximate wavefunctions,

additional symmetries may be introduced. Garrod has

shown for example that if the wavefunction is taken as

- 75 -

an average of M components with identical space and spin

parts, then the NSG's can also be made eigenfunctions

of L2. Theorem 2 is probably more useful for molecules

and solids; for atoms it essentially duplicates theorem


In order to better see the structure of the

1- and 2-matrices, it is sometimes useful to expand

them in terms of separated space and spin parts. If

the wavefunction is an eigenfunction of S it may be

shown that

l^')c o 9 i

,-~ ,-~


F 61 W'~~i:7bd

Scd / ./rr, cc

, dc/^' rj-dr,

dc I /r, dc



a-c d

C r -I{CLA^^CL)


dJ L (c-e
_V-2 Ig -sc

The presence of the cross terms cd* and dc* in (110)

shows that the 2-matrix is generally not an

F Cx, 'iY/2

- 76 -

eigenfunction of S2. Also, one sometimes introduces the

charge-density 1-matrix,


the spin-density 1-matrix,

?d (riy)


zfS'z6- I/'C')

and the chare-density 2-matrix

and the charge-density 2-matrix,


The eigenfunctions of the charge-density matrices are
called charge-density natural orbitals (CDNO) and
charge-density natural geminals (CDM'G), or simply,
natural orbitals (NO) and natural geminals (NG). The
eigenfunctions of the spin-density 1-matrix are called
spin-density natural orbitals (SDIO). To find bounds

on the eigenvalues of (112) and (113), we can use the
matrix representations

S(^ ')- = f I )d6;
a ^ a ^

fo; r6i C ) cla- r'

I~~vEi ~rL ;j

- 77 -

dr7 c

t P/I.



We then use the result that if matrices /4, and C have
eigenvalues a,, /, and respectively, arranged in non-
increasing order, and if

--= /f *(117)

then /73/

nnu- (L //4/ c// A,/A


From this result, we obtain the following bounds.

0 x40- x M7.2 "i>)

l\m~ ^:;jc. I )~~ ~ S d

-fc, /)Z 1


The interest in the CF!O and CDING is two-fold. First,
if the wavefunction is an eigenfunction of S2 and S
and if MS = 0, then nd are identical, and the NSO's
are NO's with l or spini; F and /'vanish, and the I!SG's
are NG's with one of the four spin functions (111).

,,,_ ~

75' --

- 78 -

Second, for an t-function basis, there are 2M NSO, only

ft of which can be spatially linearly independent.

Consequently, the Il linearly independent NO's have

sometimes been suggested for the CI iteration scheme

discussed earlier. In general, both the NSO's and N!O's

will have mixtures of either odd or even values of

angular momentum; that is, s orbitals will have s, d,

g, i, ... admixture, and p orbitals, f, h, j, ...

admixture, and so on. This mixing poses a

computational difficulty in that most programs are set

up to deal with orhitals of a single (l,ml) value rather

than of a single mi value; angular-momentum projections

become considerably more involved if 1-mixtures are


Ihile we have not made explicit use of them,

we have generated the CDHO's and SDNO's for all the

systems studied in this work.

Density Matrices of Some Special Functions

For a single Slater determinant of N

orbitals, the 1-matrix has N occupation numbers equal

to 1, and the remainder equal to 0. If the orbitals

are orthonormal, the 1-matrix is diagonal directly from

- 79 -

(46), and the orbitals are the NSO. This is a
particularly important case and has been discussed
extensively by L8wdin /18,19,20/. In this connection,
it is worthwhile to introduce the extended Hartree-Fock
(EHF) equations which LBwdin derived for an orthonormal
basis set. We mentioned these briefly in the last
chapter, but deferred a derivation because the density
matrices provide a particularly convenient tool for
this. We begin with the expression (41) for K = L,
where Q is now the Hamiltonian operator.

= Ho

* J.h-/, ( )1 ,

9. rwd, ..


Varying the expression (51), we obtain

S z

yi'^ ..- ) s iPrWk) 7 / <-( j

, ,. (ii.%.. p &

. c ... c i 0

+ Complex coyj'j._ e. (122)

- 80 -

Using this result, we find

So t d et )

where we have introduced a Hermitian matrix of Lagrange
multipliers to maintain orbital normalization. By the

usual argument, the expression in brackets must vanish;

we then multiply by 8() and sum over Y obtaining the
EHtF equations:
EHF equations:

- 81 -

P, 1iir) < .j^ r [7 d'), > ...

+ f P ,,....p ^ ('< ...rl ,,z,... ^') d,.. dx

w= .(( (124)


(l/'J = /l ) < } (125)

Note that nowhere have we assumed an orthonormal basis

or a particular form of the orbitals; unless we start
with an orthonormal basis, there is no need even to
introduce the Lagrange multipliers, and the right-hand

side of (124) then vanishes. For a non-orthonormal
basis then, there is no need to determine Lagrange
multipliers, but we have a more difficult p-matrix to
compute. In general, it is not possible to

simultaneously diagonalize the 1-matrix and the
Lagrange multiple ier matrix, so we essentially lose the
concept of orbital energies.
It is often useful to introduce a quantity,
called the "fundamental invariant", defined by

- 82 -

p 0

which satisfies


where M is the number of orbitals in the basis.
invariance follows from the fact that a nonsi

linear transformation on the basis leaves

IrA> T V' < >Tk TI

-- /


For the case M = N, L8.wdin /19/ showed that o determines
all the p-matrices, and these are given explicitly by



ngu ar

< ^ '<^

! > '+14>' 4

- 83 -

S= J (/...(p,)) (130)

The fundamental invariant therefore contains all the

information contained in a single-determinant

wavefunction, regardless of the form of the basis

orbitals. This noint has lead to some confusion in the

literature. In an often-quoted paper, Bunge /44/

arrived at the result that for a PGSO wavefunction, the

EHF equations do not yield unique orbitals; i.e. the

fundamental invariant is not invariant. This result is

incorrect; the error in the paper is the omission of

the factor <>> in ,p ; this simplified form holds for an

orthonormal basis. Bunge then proceeded to vary the

orbitals, destroying the orthonormality. The EHF

equations are perfectly well-defined, even for GSO.

That p determines all the p-matrices for a single

determinant is true, even for a Drojected determinant,

OD. The occupation numbers are 1 and 0 if OHO is

considered the modified Hamiltonian and D the

wavefunction; however, if OD is considered the

wavefunction, the occupation numbers are in general not

0 and 1 because the projection introduces new orbitals.

In this case, the fundamental invariant must be

constructed from the complete set of orbitals,

including all the ones introduced by the projection.

- 84 -

The p-matrices must still be determined by the

fundamental invariant, but the form of the natural p-

states and occupation numbers is not obvious. For the

case of a spin-projected determinant of pure spin

orbitals, Harrinan /79/ has derived explicit formulae

for the 1-matrix, NSO, and occupation numbers,

Hardisson and Harriman /80/ derived a formula for the

2-matrix, and this has recently been extended to point-

group and axial-rotation symmetry projection by Simons

and Harriman /81/ to obtain formulae for the 1- and 2-

matrices. The first two /79,80/ are derived for a

projected DODS determinant; in the last /81/, the

orbitals are only assumed to be orthonormal. The form

of the occupation numbers and the natural p-states for

a PGSO wavefunction is not known in analytic form,

although we have calculated the 1- and 2-matrices

directly from the projected determinant treated as a CI

expansion over non-orthonormal orbitals. The formulae

for the DODS case are already very complicated; in view

of the great increase in complexity in going to GSO, we

feel that an attempt at obtaining an analytic formula

for the p-matrices of a PGSO wavefunction would not be


- 85 -

2.4 The N-Representability Problem

The Schr8dinger equation, (1), has never been

solved exactly for a system with more than one

electron. As the number of electrons increases, the

approximate wavefunctions become increasingly complex.

The Hylleraas coordinate functions discussed in the

last chanter have not been extended beyond four-

electron systems; the CI programs mentioned are limited

to less than forty electrons. Systems of chemical

interest frequently have hundreds or thousands of

electrons which we have so far been unable to treat

accurately. One can imagine Colenan's excitement in

1951 when he first observed the significance of the

equations (41) and (121); since the usual Hamiltonian

employed contains at most two-particle terms, the

energy, and all one- and two-electron properties defend

at most upon the 2-natrix, from which the 1-matrix can

be derived according to (36). The 2-matrix is a

function of only four particles. Thus, by varying a

certain four-particle function, one should be able to

obtain practically every result of chemical interest

for any system, no matter how large. Rather than

launch a calculation on DNA, Coleman contented himself

at that time with a calculation on lithium, a three-

electron system. The calculation gave an energy 30%

- 86 -

below the experimental value, in seeming violation of

the Rayleigh-Ritz variational principle. The

difficulty was that the four-particle function had been

varied over too wide a class of functions. This

problem has since become known as the "N-

representability" problem -- the problem of finding the

conditions under which a 2p-particle function, such as

a p-matrix, can be shown to be derivable from an N-

particle antisymmetric (or symmetric) wavefunction

without actually exhibiting that N-particle function.

This problem has received a great deal of study in the

last two decades. The indications so far

pessimistically are that either the general solution

does not exist, and therefore cannot be found, or that

if the solution exists, and is found, implementing it

will be at least as difficult as carrying out a

calculation with the N-particle wavefunction. This

thought is rather depressing, considering that a

feasible solution has the strong possibility of

revolutionizing a good part of chemistry, physics, and

biology. More optimistically, one might hooe for an

approximate solution so that variation of a reduced

density matrix could be implemented in such a way as to

provide a useful alternative to ab initio, semi-

empirical, or even empirical theories. Some progress

87 -

has been made along these lines by a number of authors

/82 90/.

In the meantime, reduced density matrices

provide a convenient tool for interpretation of

wavefunctions and nronerties.



3.1 Introduction

Reading the current quantum chemical

literature gives one the feeling that a total energy is

the only property atomic and molecular systems possess.

Since the total energy, like the thermodynamic enthalny

and free energy, is meaningless except when compared

with another total energy, one miFght even begin to

question the motivation of the calculations. In fact,

of course, there are a good many oronerties of interest

which we can in orinciole compute. A recent hook by

Mfalli and Fraga /92/, although somewhat concise and

uncritical, does at least give an idea of some of the

orooerties of interest. A review article by Doyle

/105/ discusses relativistic and non-relativistic

corrections to atomic energy levels and a number of

numerical tables with these corrections is given. We

will content ourselves in this chapter only with givinr

a short indication o4 some of these properties with

- 88 -

references to work where greater detail may be found.

3.2 Enprries

The calculation of the energy determines the

wavefunction. Fxcept for one-electron systems, which

can be solves exactly, and Pekeris' work cited earlier

on two-electron systems, calculations of energy levels

cannot corrnte with exoerimnnt in accuracy.

Consenuently, xr-ent for determination of the

wavefunction and com)nrison with other theoretical

results, for atoms, calculation -of energies is of

little interest because the exnerimental data is so

much better. Cor nolocules, even small di-tomics, this

is not the case, and one can often cet better

characterization of nntential curves by theoretical

comnutat ions than current exner mental n thds can

five. It is perhaps one of the sad facts of quantum

mechanics that determination of the energy is the only

route to the wavefunct ion, ann that even if an

aonroximate wavefunction gives a good enerTy, other

properties calculated from it may be rather Door.

In this work, in addition to the energy

determination, we have also evaluated the scale factor

- 89 -

- 90 -

and scaled energy given for atoms by /98/


E V< > __ (132)
2 4 =c-'/

An atomic wavofunction may always be scaled to satisfy

the virial theorem; if the unsealed wavefunction

satisfies it already, then the scale factor is

necessarily unity. We have found this useful in that

a scale factor differing from unity by more than about

0.001 indicates that the basis is poorly chosen.

3,3 Snecific Mass Fffect

In thp introduction, we derived the snpcific

mass effect, or mass nolarization, correction to the

kinetic energy, en. (15). FrmHan /99/ has estimated
the effect from Pxnprimental snectra to be about 10

!1 (0.2 K) for Li 2 S and 10-7 H for Li 4 2S. He also

states that tho effect should be approximately

independent of Z, so that the same estimates apply to

the rest of the isoelectronic sequence. However,

- 91 -

Prasad and Stewart /100/ have recently evaluated the

effect from Weiss' 45-term CI wavefunctions for the 2

S and 2 2P states of the sequence from Z = 3 to 8; for

the 2 2S state, their data gives the shift proportional

to 1.29; or the 2 2P states, the shifts decrease with

increasing Z, becoming negative for Z > 4. The shift
9 7 6
for the 2 'S state of Li is 2.587 K, and for Li6, 3.017

K, a difference of 0.430 K. By contrast, the normal

isotope shift, (13), causes the same level of the two

isotopes to differ by 21.353 K. The specific mass

effect is therefore small for light atoms, but

important for accurate determination of energy level

separations. For heavier elements, the specific mass

effect can be many times larger than the normal isotope

shift. Dalgarno and Parkinson /102/ have estimated the

specific mass effect in lithium by perturbation theory

applied to the results of Pekeris and coworkers /54-

56,103-104/ on two-electron systems, and obtain results

in agreement with Prasad and Stewart.

According to Kuhn /101/, theoretical

determination of the specific mass effect would be a

valuable contribution, and we therefore intend to

compute the effect with our wavefunctions at a later


92 -

3.4 Relativistic !ass Increase


the electron

mass varies

with velocity according to



and m is
energy is
energy is

the electron rest mass.

The relativistic kinetic

-T= -t' M 0o C

3 i

- . .

and the relativistic four-momentum is

p= (-YM, imc)

but its magnitud- is constant:

pp D- = -III C

We therefore take t-e non-relativistic momentum,





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