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General spin orbitals for three-electron system

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Title:
General spin orbitals for three-electron system
Creator:
Beebe, Nelson Howard Frederick, 1948-
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Copyright Date:
1972
Language:
English
Physical Description:
viii, 238 leaves. : illus. ; 28 cm.

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Subjects / Keywords:
Atoms ( jstor )
Eigenfunctions ( jstor )
Eigenvalues ( jstor )
Electrons ( jstor )
Energy ( jstor )
Hyperfine structure ( jstor )
Lithium ( jstor )
Matrices ( jstor )
Orbitals ( jstor )
Symmetry ( jstor )
Atomic orbitals ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Lithium ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 194-237.
Additional Physical Form:
Also available on World Wide Web
General Note:
Manuscript copy.
General Note:
Vita.
Statement of Responsibility:
Nelson H. F. Beebe.

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University of Florida
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General Sp;n Orbitals for Three-Electron Systems


by

Nelson H.F. Beehe












A DiSSERTATiO;J' P2ESENTEP TO THE GRADUATE COUiNCIL OF
THE' UNIVERSITY OF FLORIDA I'i PARTIAL FULFiLL!' ENT OF
THE REO'UIJ EMU TS FOR THF DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1972












ACKNOWLEDGEMENTS


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

Gainesville.

Yngve Ohrn and Charles Reid have often helped

when problems arose that I could not stumble through

myself.

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.








TABLE OF CONTENTS


ACKNOWLEDGEMENTS . . . . . .

LIST OF TABLES . . . . . . ..

ABSTRACT . . . . . . . . .

INTRODUCTION . . . . . . . .

CHAPTER 1 HARTREE-FOCK AND BEYOND ..

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 . . . . . . .

CHAPTER 2 REDUCED DENSITY MATRICES . . .

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 . .


Page

* 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

CHAPTER 3 ATOMIC PROPERTIES . . . . . 88

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

CHAPTER 4 THE PROJECTED GENERAL SPIN ORBITAL
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

APPENDIX 1 VALUES OF SOME PHYSICAL CONSTANTS . 130

APPENDIX 2 THE COMPUTER PROGRAMS . . . .. 135

APPENDIX 3 CONVENTIONS FOR SPHERICAL HARMONICS
AND SPHERICAL TENSORS . . ... .139

APPEND IX 4 0N' THE CUSP CONDITIONS . . . . 142

APPENDIX 5 SOLUTION OF THE MATRIX SCHRUDIDINGER








EQUATION, HC = SCE . . .. . 148

APPENDIX 6 LIST OF ABBREVIATIONS . . . .. 152

APPENDIX 7 SOME THEOREMS ON DIRECT PRODUCT
MATRICES . . . . . ... 155

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

BIOGRAPHICAL SKETCH . . . . . . .. 238










LIST OF TABLES


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






GENERAL SPIN ORBITALS IN THREE-ELECTRON SYSTEMS

By

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.


vii






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.


viii












INTRODUCTION


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
IJ
electrons

/ 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
'2
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
227


-2: (ez #z
217


ly


(3)










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

We can remove the experimental factors now by requiring




which gives

which gives


a = )2/(, 2),


(5)


Setting

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

constant,


= e2/Ac = 1/137.0888


(11)


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


(13)
(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.




-7-


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



pt


/


=e k 2 pe T, is-



The kinetic energy, T, is


~2pi2


Making the usual


transformations from classical to


quantum mechanics,



p -i1f


we obtain


T7


PE


- i ) .-z
AiD


* .
+ Z, l.


(14)


a~vr1_


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.
















CHAPTER 1


HARTREE-FOCK AND BEYOND



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

determinant:


_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)
Jy


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)



and

(.1, -,) (8)


so that



S(9)


and


-i < ilRili> (10)



j=/


The total energy is given by




- 13 -


(11)


21

(#


i.C212Ij>


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


w


Substituting (9) into (11), we


find


-x..
j Z

/1212 IJ>


+2


<7


11212/j>


- 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


(Al-I)


(13)


6 <2mI 12/2II M>


I.


2:t


-z


(12)


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


-! 57
- 2
iy




- 14 -


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


-Z


(A) < -l- ,z, 2 <)inm ,Ivm >
tf770m


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


= 61


(14)


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)


Positiveness


= i )
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,

and


< -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


(v-a)


c/,-/


hr-j


11-.2




- 16 -


N-2



A/-2


S






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


(A/-k)
A= /


(15)


() -
A/-kC



=k-
'Z= < J 1A N ?,0


and finally,


'N- k


-E_ -E




- 17 -


- 7 6 (A )
(k)


~2
2>*
J





This is of

ionization

energy.


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

added,


(A//)
a'


-' < 1 i,/z-> i


(17)


(4)


+


kt
Z Ik
kZi


(16)


AI'/
AN J




- 18 -


2--




1
r c^


61
I,/


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


A/tI

1/


-z: <)tv 'j//AiLf


(18)


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

additional equation


(19)


/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.


<0/




- 19 -


analysis as before gives



e I/ p > i l


er' "- ,,


(21)


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

effort.


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

Because

the very

to the

a small


(20)




- 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

necessity.

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)

method.

a) The space and spin dependence is

separated.







b) The radial and angular dependence is

separated.







(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)

(22)

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

becomes.







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:




(25)

(25)


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>


/


(26)


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

/35,36/



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:

3-
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

/40/.

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


nPGSO nSOSCF


= (g N)H N f + 1 0


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


ILI


nSOSCF


or,




- 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)
F




- 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
12
--> 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,
10
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

therein.)

Recently, Sims and Hagstrom /62/ have shown

that a combined Hylleraas/CI wavefunction of the form


2J= Z cV k
K


where


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



is feasible for atomic calculations with any number of

electrons and have reported results for the beryllium

atom.

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,

IBMOL.

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

respect.

With these closing remarks, we shall now pass

over to a discussion of density matrix theory.
















CHAPTER 2


REDUCED DENSITY MATRICES



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,
p









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/


(35)




- 49 -


understood.

conventions


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

bar:


(37)


where P is any two-particle permutation operator such

that P 0= -z. We also have


(38)


r I ') -
XT J


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

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

functions f(x),


(36)


rOp) (xl P') =- F ( Ix)
FI IIT XI


(P)




- 50 -


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


J rJ*' c/i-


(39)


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


(40)


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


(41)


- .


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


(42)


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


D])(CVIL;T) J


basis is not assumed to be


introduce

I4 T -
V ST L


(43)


where


(011)VL)
Y


(44)


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


r


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))

K L


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


(47)



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
p
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

restriction.

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
Ij IJ
to




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




- 56 -


.z EF )(k 1)

-LK


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


(52) can


& rF"


where


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


(55)


so we define


(54)


PiI tr ~B~




- 57 -


f7T')


Y C^'ff"


P")


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
basis.
To proceed, we need some theorems on direct
product matrices, which we state here, and prove in
Appendix VII.


Theorem:


Theorem:




Theorem:


(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)


(58)


(59)


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 )


(60)


(56)


//rA'


(57)




- 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


fl(LUX LX


SLJL U


and


_ [ S S ... J (63)

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


Since S
therefore
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:


(64)


(c17u) A lfYJJ)


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


T'P''


= },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 ]


(65)


where


(66)


and


r 1


(67)


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:


r~iui=


(68)


~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]









where
^= ^ L

#Vz LA


and i is obtained from






and for D = 2,


r1l) 21'2')


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


*^^~ z IO
rz-Pr


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


(72)


whe re


(,,(1,2) =


(73)


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 -


(71)


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




- 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)
K


and




- 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

(80)


Thp RPnduired Density Matrix for a lon-CI


Wavefunction


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' /


(81)


for arbitrary functions
eigenfunction of then


f(x).


If f(x) is an


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


(Jz, P 0


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


F' f() =




- 66 -


r"F ? ANW


(82)


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:


(83)


Using this in (82) gives


(8 i)


which leads to the secular problem


<^/ >C Cr< ^ ~


(86)


where


(85)


? = 4 C


a~-~~


IC sck




- 67 -


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


J4. ,
-<, jI >-


f07 (x) C&


(88)


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


(89)


Now let us order the exact and approximate solutions
according to


14z 7


>- 0


(90)


and then construct the operators




- 68 -


9x, 21 (91)




P E 2 IZ<>1 / (92)
k.-M+/


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
N
occupation numbers approaches ( ) from below and provides
a convenient measure of the adequacy of the chosen
basis.




- 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 ~ ()


(96)


Coleman (See Ando, ref. 66) showed that


(97)


p = 1 and p = 2 takes the form


OLX






(2)
O"i1L


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


(98)


(99)




- 71 -


;11


21. '2)

;1
'3


(100)


_ 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
possible.


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


and


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


(101)


1,11~4


. 1 <'/P > --


,/<.?, > (102)




- 72 -


then




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

and


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)
-I


or an antiunitary operator of the form




- 74 -


i=/
(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

1.

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


,-~ ,-~


(109)



F 61 W'~~i:7bd


Scd / ./rr, cc


, dc/^' rj-dr,



dc I /r, dc


(110)


where


a-c d

C r -I{CLA^^CL)


(111)


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 -


2
eigenfunction of S2. Also, one sometimes introduces the

charge-density 1-matrix,


(112)


the spin-density 1-matrix,


?d (riy)


(113)


zfS'z6- I/'C')


and the chare-density 2-matrix

and the charge-density 2-matrix,


(114)


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.


(115)


(116)


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


(118)


From this result, we obtain the following bounds.


0 x40- x M7.2 "i>)


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


-fc, /)Z 1


(120)


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

allowed.

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, ..


(121)


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)





where


(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
unchanged:


IrA> T V' < >Tk TI


-- /


(129)


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


(126)


(127)
(128)


The
ngu ar
(126)


< ^ '<^


! > '+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

worthwhile.




- 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.
















CHAPTFR 3


ATOMIC PROPERTIES



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/



(131)
.2



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
-6
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

date.




92 -





3.4 Relativistic !ass Increase


Relativistically,


the electron


mass varies


with velocity according to


eWo




where
/V_
C


and m is
0
energy is
energy is


the electron rest mass.


The relativistic kinetic


-T= -t' M 0o C


3 i
C2


- . .


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,


(133)


(134)


(135)


(136)




Full Text

PAGE 1

General Spin Orbitals for ThreeElectron Systems by Nelson H.F. Beebe A DiSSERTATlOM. PRESErJTEP TO THE GRADUATE COUfiCIL OF THE Uf.'l VFRSiTY' OF FLORIDA i'l PARTIAL FULF i LLf lEfiT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHI LO SOPHY UN ! VERS i TV OF FLORIDA 1972

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ACKNOWLEDGEMENTS 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 LHwdin, 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 v;hom I have closely worked, he in Uppsala, and I in Ga inesvi lie. Yngve Ohrn and Charles Reid have often helped when problems arose that I could not stumble through myself. 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 AFOSR71.171'iB and NSF-GP-16665 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. I I

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TABLE OF COriTEMTS Pa^e ACKNOWLEDGEMENTS H LIST OF TABLES vi ABSTRACT vii INTRODUCTION 1 CHAPTER 1 HARTREE-FOCK AND BEYOND ........ 10 1.1 The Hartree-Fock flethod 10 1.2 Extensions to Hartree-Fock Theory ... 23 Relaxation of Restrictions 23 The Symmetry Dilemma 25 On Spanning the Angular Mom.entum Spaces 30 1.3 Other Methods 39 The Configuration Interaction flethod 39 I nterel ectron ic Coordinate Methods . . '2 Bethe-Gol dstone Perturbation Theory Ulf CHAPTER 2 REDUCED DENSITY MATRICES h6 2.1 Introduction to the Density Matrix Literature 'B 2.2 Construction of the Reduced Density Ma t r i X f» 7 Definition of the Reduced Density Matrix k7 The Reduced Density f'atrix for a C! VJavef unct i on 51 The Reduced Density Matrix for a Non-CI V/avef unct ion 65 2.3 Properties of Density Matrices 69 CI Expansion Convercence 69 Bounds on Occupation Numbers 70 I I I

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The Carl son-Kel ler-Schni dt Theorem . . 71 Symmetry Properties 73 Density Matrices of Sone Special Functions 78 2.ti The n-RepresentabI 1 1 ty Problem 85 CHAPTER 3 ATOMIC P^.OPERTIES 88 3.1 Introduction 88 3.2 Energies 89 3.3 Specific Mass Effect 90 3.k Relativistic flass Increase 92 3.5 Transition Probabilities and Oscillator Strengths 94 3.5 Fine and Hyperfine Structure 95 CHAPTER 1+ THE PROJECTED GENERAL SPIN ORBITAL CALCULATIOf.'S lOt* J+.l Introduction 101+ l|.2 flatrix Fornulation of the PGSO Method . 105 k,3 Choice of Bases and Initial Orbitals . . 103 ti.U Evaluation of the 1-M,atrix 112 h.5 Evaluation of the 2-Matrix ....... 113 I+.6 The Hyperfine Analysis 119 '+.7 Comparison v;ith Other Methods 126 4.3 Hyperfine Structure Results by Other Methods 123 APPENDIX i VALUES OF SOME PHYSICAL COflSTANTS . . 130 APPENDIX 2 THE COflPUTER PROGRAMS 135 APPENDIX 3 CONVENTIONS FOR SPHERICAL HARMONICS AND SPHERICAL TENSORS 139 APPEr,'DIX h ON THE CUSP CONDITIONS 142 APPENDIX 5 SOLUTION' OF THE MATRIX SCHRODIN'GER

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EQUATION, HC = SCE 1U8 APPENDIX 6 LIST OF ABBREVIATIONS 152 APPENDIX 7 SOME THEOREMS ON DIRECT PRODUCT MATRICES 155 BIBLIOGRAPHY 194 BIOGRAPHICAL SKETCH 238

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LIST OF TABLES Table Page 1. Energy conversion factors 159 2. Comparison of the convergence of some properties with convergence of the energy for various basis sets 151 3. Sample bases, properties, and natural 2 2 analyses for Li 2 P and 2 S 171 h. Contributions to one-electron properties by MSO 177 5. Comparison of energies and Fermi contact terns 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 VI

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Part i al Ful f i 1 Inent of the Requirenents for tne Decree of Doctor of Philosophy GENERAL SPIN ORBITALS IN THREEELECTRON SYSTEMS By Nelson H.F. Beebe June, 1972 Chairnan: P.O. LBwd i n Cochairnan: N.Y. t5hrn Major Department: Chenistry The use of general spi n-orb i tal s in a spinprojected Slater deterninant for a nunber of states of several three-electron atomic systems is studied. The spin-optimized se 1 f-cons i s tent-f i e 1 d 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. V 1 1

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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 v/ork. The general conclusion of this v/ork 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 vnth configuration interaction calculations of comparable accuracy. VIII

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INTRODUCTIOri The timeIndependent Schr8dinger equation, H^ -E• (1) in principle describes most of the physics and chemistry of matter. Unfortunately, neither the Haniltonian operator, H, nor the wavefunct Ion, ^, are ever knov/n exactly, and consequently, neither is the energy E. For the non-relat ivi st Ic 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 knov/n fairly accurately, and (1) can be solved exactly. The series of hydrogen-like atoms, H, He+, Li++, etc. fall in this category. The relativlstic counterpart of (1), the Dirac equation, cannot be solved exactly, even in this simple case, unless the nuclear mass is assumed infinite; a covarlant manyparticle Dirac Hamiltonlan is not even knov;n. As soon as another particle Is introduced, the equation, like the classical Kepler three-body problem, is Insoluble, and approximation methods must 1

PAGE 10

2 be resorted to. The non-relativl Stic Hamiltonlan is known sufficiently accurately to describe chemical properties, so that determination of the wavefunction is the principal problem. An approximate relativistic Hamlltonian 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-relat i v i s t i c Hamiltonian. The physicist may be interested in studying the construction of the proper Hamiltonian, while the chemist Is more concerned v;ith the results, that is, the wavefunction and the properties that can be deduced from i t. Although the ca 1 culat ions ' descri bed in this thesis use a non-re1a t iv i st i c Hamlltonian, many ideas and corrections are borrowed from the relativistic case. These will be introduced as we go along. The non-relat iv ist ic atomic Hamiltonian that v/e shall use may be v/ritten (2) J w here the summations go over electrons, and

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3 -€ = electronic charge^ h = Planck's constant, Y\ = h/27r, r, = distance of the i-th electron from the center of mass, r.. = distance between the i-th and j-th U electrons /U= (1/m + l/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 /^, 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 r7'2 introducing some atomic unit of length, a. Then, since v is homogeneous of degree -2, we replace i t by a. V and r' by ar. Multiplying by ^IfP-, we obtain (3)

PAGE 12

« and E = E'( a^/J^^). ik) We can remove the experimental factors now by requiring whi ch gi ves a = K/(^e^), Setting 2-2. (5) 1 = /i2/(/ia2) = e^/a = /ue^/J^^^ (6) we see that the atomic units of energy and lengthy E and ^/ both depend upon ju, and hence upon the nuclear mass M. We shall call the atomic unit of energy, E, the Hartree (H.) and the unit of length, a., the Bohr (B.). If we let M become Infinite, the reduced mass becomes equivalent to the electronic rest mass, and we find Eg = l^2/ne^ (7) or 1 = (1 + m/M)EQ, (8) and Bq = Y^'^/me'^, (9) or a = (1 + m/M)aQ. (10) It Is also convenient to Introduce the fine structure constant, OC = e^/y^c = 1/137.0888 (ID V/e then find that the Hartree is also given by

PAGE 13

5 /ue**/}^^ = a^/Mz^ (12) That Is, the Hartree corresponds to the rest energy of the reduced mass times the square of the fine structure constant. Vie observe that the Hartree is therefore on this scale a rather snail unit, although Appendix 1, 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 vn 1 1 differ by a slight amount given by i^ ^ f + m d' ^)(t' ^) M'M MM' (13) The spectral lines of the heavier isotope v/i 1 1 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 /^q//^h "" 1.000817. Its existence

PAGE 14

6 led to the discovery of deuterium in spectra of residues of liquid hydrof^en 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, v/hich will be discussed later. Our assumption of a point nucleus, v/hich 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 O.OOOOla., and will be discussed in'more detail later. In (2), vie 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 fev;, such as Shore and flenzel /I/ and Bftthe and Salpeter /2/ do treat it. The implications for molecular calculations have recently been discussed by Fr8man /3/. We will sketch the results for an atonic system here.

PAGE 15

7 If "p Is the momentum of the nucleus and Pj Is the momentum of the 1-th electron, then, by conservation of momentum, in the CM system, we have P . Zp. ^ The kinetic enert^y, T, is ilk) Makin;^ the usual transformations from classical to quantum mechanics, ~p -^ i'K'^ we obtain

PAGE 16

8 T--4^zr l|^-g (15) The second tern is generally omitted since it is of order /^/n, which Is small, and also, it is a two-electron operator, which is more difficult to calculate than the first tern. 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 relativlstic case, it is not possible to perform a rigorous separation of internal and trans la t iona 1 motion by goin» 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 nonrelatlvistic 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 spinorbit 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 Paul i extension to the SchrBdinger equation. in view of our non-rela t I vi s t i c approach in this work, it may be better to view

PAGE 17

9 calculated snail terms as empirical parameters; they contribute to level shifts and splittings, but they in general represent only part of the physics. Nevertheless, such quantities are often useful in interconpar 1 son of v/avefunct ions and can help to point out deficiencies in the description of various parts of a given v/avefunct ion . The reader who is interested -in the relativistic CM problems is referred to Bethe and Salpeter (Ref. 2, section k2) and elsev/here /k,S/, Finally, we remark that for one-electron systems, Garcia and Mack /6/ have given theoretically calculated tables of energy levels containing all knov;n corrections to the relativistic Dirac Hamiltonian.

PAGE 18

CHAPTER 1 HARTREE-FOCK AND BEYOND 1.1 The Mar tree-Fock Method Let us assume that the total wavefunction for an N particle system ?s built up from a set of N orthonormal one-particle spin orbitals,
PAGE 19

11 Hartree-Fock equations (see for example, Ref. 1, Chapter 15) which may be written ^3^^i J j/ " "^^3 (3) where it has been assumed that <^M = '^.(r)^ ; ^[^ (It) The A.are Lagrange multipliers, and by a unitary transformation on (1), which leaves the v/avefunctlon invariant, one can diagonalize ^ and obtain Ai4>^ = ^,^ (5) where 6•" -^n and the 4>i in (5) are now linear combinations of those in (3). The one-particle effective Hamiltonlan, h, is given by A -JrV. •1 (6) , i: J^U^)j'(^)^.(^-^JA('^iu)'^x, ^^(0 i CO

PAGE 20

12 -*!• where P^ „ permutes the coordinates of electrons 1 and 1, and Zi2 ~ ^/'lo* ^^^ Hartree-Fock (HF) equations have received extensive study in the literature, and we will refer the reader to the extensive bibliographies in Slater's bool*a)4'UV (9) and ^. = (10) A/ The total energy is given by

PAGE 21

13 E <^lHl^>/<^\^> (11) Note that 2h'i> (12) i ^ 1>j J J so that the total energy is not siven by the sun of the orbi ta 1 energi cs . Let us see what hapoens if we remove an electron fron orf)Ital m v/ i t h o u t changing any of the remaining orbitals. V/e find .'"-" r i"" <2yr>in.J:y7,> (13)

PAGE 22

Ik ^v-^., ^^r' l<^ji^Ju> -z e. (tj-/) 7 ^ " ^*"' 7>/ ^ l/ Equation (lU), knov/n as Koopnans' theorem /lO/, says that the ionization energy is approximated by the negative of the one-electron energy of the removed electron. If v/e allov/ed the orbitals to relax, then this v;ould no longer be true. Equation (13), which is not generally renarked upon, implies that the orbital energies are decreased by the term < ivn ifliz I I'y^ > which Is always a positive quantity. Positiveness follows simply from the the fact that ± ~ 2.

PAGE 23

15 so (1 Pio^/^ is a projection operator, and Is therefore positive, and since g^^ is also positive, A^is positive, and o v*. 'II if we assume that the orbital energies are ordered e, ^ ^^ ^ ^, ^ ^ 6. n and then v;e successively ionize electrons TJ, N-1, N-2, ...,1, without allowing orbital relaxation, we obtain J-fj ' '^N-l ^V " ^ ^fj,l ' ^N-X ^-/ e ^-0 N'l = -C^ ^ JlfUl^'1. /J> ^N-l ^N-h ~ ^fJ'. 6

PAGE 24

16 (tJ-') '' ^7-2 ^ ^^'^' ^^^ ^'^ I ^-^' '^'' > " '^tL ' ^^'2.^'iinj^/'2,AJ-n -^< N-2.N Ifla I ^-2,A/> I fj-k. E t --e (M'k) (15) ^ A/-/C fe-' I--0 and f i na 1 1 y.

PAGE 25

17 Zl k--l Zi (k) ^ £ V (16) This is of course expected since the sum of all the ionization energies must be the same as the total energy. Similarly, v;e 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 added. (17) (^j) £/' +

PAGE 26

18 2,/ J ^ ^ *•' OJ) V ^ Z. A/i^/ 2-/ ^
r £ (18) However, in order to determine this last quantity, wiiich is the electron affinity, v/e need to solve the additional equation -^v./' t., 6 4> (19) wliere h Is determined by the original M orbitals only. The third quantity of interest in addition to the ionization potential and the electron affinity, is the excitation enerj^y of an electron in orbital m ^^oing to an unoccupied orbital, say N + 1. A similar

PAGE 27

19 analysis as before gives it* ^ Cf ' ^ (20) E^''£^'= ^Z, £5 ^
PAGE 28

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 /ll/ to the HF equations because simple equations can be derived for excitation energies, ionization potentials, and electron affinities, v/hich do not require total energy differences. Instead, one makes a single calculation in the so-called transition state /12/ which is a state halfway betv^/een 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 scm i -empi r i ca 1 one-electron theories have carried chemistry and solid-state physics a long way. in particular, the mechanistic approach v/hich 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 orbltals. For atoms and molecules, we can go beyond these ideas, but for the

PAGE 29

21 solid state, they are still very much a computational necessi ty . 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 hovv 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) method. a) The space and spin dependence is separated. b) The radial and angular dependence is separated. 4>ir) ^ ^(r) X, (^, ^) (see Appendix ill for definitions of the spherical harmonics, Y, ). im c) R(r) is taken to be independent of m, this is not a restriction for states with l. = 0, if L-S coupling holds.

PAGE 30

22 d) R(r) is taken to be Independent of m^; this is not a restriction for states v/i th 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 el genfunct ion of S^, and the second and third, of L . The assumption in goinj; from (3) to (5) v;as 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 v/ith c
PAGE 31

23 1.2 Extensions to Hartree-Fock Theory Relaxation of Restrictions In the general open-shell case, v/i th the restrictions noted in the preceding section, the offdiagonal Lagrange multipliers cannot be eliminated, and the total energy expression becomes considerably more complicated. V/e will refer the interested reader to a reviev/ of open-shell methods by Berthier /lU/. If we reexamine (9) and carry out the spin integrations, we find that the effective Hamiltonian, h, is given by A, n. ' (22) Because of the del tafunct ion in the second tern, which is the exchange tern, only those orbitals having the same spin as orbital i will cone into the operator. If there ?^rQ differing numbers of d. and /3spins, 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. I7e have already noted the double-

PAGE 32

2U occupancy restriction, and it v/as just this observation that led to the introduction of v/hat is called spinpolarized HF (SPHF), or different orbitals for different spins (DODS), in v/hich orbitals from the same (n,l)-5hell are allo'./ed to have different radial parts if they have different spins. This is a relaxation of restriction d) in the last section. Similarly, for states v;ith L ?^ 0, one finds different effective Hamiltonians for orbitals of different m, values. Relaxing restriction c) then gives orb I tal -pol ar I zed HF (OPflF). 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 4>. OHF ^^ ^». ^ X_ "" -^'2 />/;>.« '^ 'VV.^ ^Ii-4,yr>'^ (23) The term unrestricted is not particularly good, because the orbitals are still restricted to a single m^ and a single m, value, so that they are e i genf unct i ons of s^ and 1^, and to definite parity (v/hich is v.-hy only odd or only even values of 1 occur in the expansion (2?)). The SPHF/DODS and OPHF methods are relatively easy to implement starting from an existing RHF program, but

PAGE 33

25 the UHF method is much more difficult. As is usually the case, the more we generalize, the harder the work becor.es. The Svr!n
PAGE 34

26 operators, the temptation has always been strong to restrict the one-particle functions to be eigenf unct ions 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 bondin;^ 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 lov/est energy, and therefore would automatically have the correct symmetry properties. This is not necessarily so, and it has been shown by LBv/din 111 I that if a wavefunction has mixed syninetry, then it has at least one symmetry component which will have an energy at least as lov/ as the mixed symmetry function. By the introduction of symmetry production operators, L8wdln /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 ^^e,T> '^'"

PAGE 35

27 where D is the Slater determinant and the projection operators 0, fulfill the usual relations: k ^k 0, 'Ok S,i <9, . d)/ (25) Z 0, J That is, they are idempotent, Hermitian, bounded by the zero and unit opprators, and form a resolution of the identity. The energy then takes the form f .< iMhi•> /<^i^> < (9Dlf^/6D>/ /<'OiOID> C26) This may bo rcf^arded either as a modified expectation value of the operator GM = OHf) = HO, or as the expectation value of a wavefunct i on that is a sum of

PAGE 36

28 single determinants, since this is the form that O.D has in (.2k). 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 Hartroe-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 aporoach 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 apnroach may be criticized because the variation is carried out before the projection. It v/as 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 (SF!^F) method. Kaldor

PAGE 37

29 llh-11 I and Sando and Harriman /28/ perform direct variations of the total enersV/ abandoning the use of the one-electron effective Hamiltonian for deterni na t ion of the orbitals. Goddard and Ladner /2933/ solve the one-electron equations and determine the SEHF orbitals. The two approaches are equivalent for the total ener.TV calculation since the final orbitals in each case ^r& related by a linear transformation, which leaves the determinant invariant (except possibly for a constant factor v;hich 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 oneelectron equations in the computations, v/e lose the simple pictures of ionization and excitation energies and electron affinities, concepts which can be generalized from the simple FIF 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 (IPM) -that M orbitals describe the motion of W electrons -presently do not justify the

PAGE 38

30 cost anrl effort of obtaining the results. There are better methods available. On Spanning the Angular Momentun Soaces 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 fbhf, but since it is directly concerned with the subject of this dissertation, v/e will repeat some of it here . ForN electrons and total spin S, the number of linearly independent spin functions is given by /35,36/ f = (2S + 1) NJ ^ 2''' (27) (rJ/2 + S + 1)! (M/2 S)! As N increases, f becomes very large. For example, v;ith 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

PAGE 39

31 is not immaterial even if the Hamiltonian is spinfree. For example, for lithium 2 ^S, 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.U47560, 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 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 X, ( (s) ( kO Us J U/3a /9acc)//2 (29) Although the energy of this function is bettor, it does not correctly describe the hyperfine structure. The MPHF function gives zero contribution to the Fermi

PAGE 40

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 v/ay 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 1111, Ladner and Goddard /33/ and Kaldor and Harris /39/. This is to use as a wavefunctlon a product of N spatial orbitals and a linear combination of all the f independent spin functions @. for the value of S desired: (30) Ladner and Goddard call this method spin-optimized G! (SOGI); Kaldor and Harris name it spin-optimized selfconsistent 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. IVe recall restriction a), that the spin orbital be represented by a nroduct of a spatial orbital and a spin function oi or /^. This obviously can generate only one spin product In the determinant for a projection operator to act upon. However, v/e have a choice of one of two spin functions for each of the N spin orbitals, so v/o could generate 2 different spin

PAGE 41

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 wavef unct ion, or orbital, with four components. Tv/o of these correspond to electrons, and two to positrons. Each pair has one component with ^-spin, and one with p-spin. Since we are dealing with nonrela t i v i s t i c theory, we can ignore the positron components, and write a general spin orbital (GSO) for an electron as 4>U)'4>'(r)oc ^
PAGE 42

3I* 4>: /]^ ^K (32) where A, is a constant, the projected GSO (PGSO) function takes the forn 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 riiSO L )e.I (33) where ^k vvi 1 1 in general be a sum of orbital component products, and 8. is the spin function used In (30). At least g different orbital components must appear in the PGSO function, where U ^ g ^ lU, flow, 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

PAGE 43

35 the normalization, we can without loss of generality orthogonal ize the GSO orbitals, but not the SOSCF orbitals. Note however, that this does not imply that the sets f^;/, /i7 are individually orthonornal. The number of independent variables for the SOSCF function is n_-^_ = MM + f 1 SOSCF and for the PGSO function. "PGSO = ^^^ ^^ since the or thonornnl i za t ion has removed one independent variable from each orbital in the PGSO function. We can alv/ays satisfy npQ5Q ^ f^soSCF Pfovi'^ed we choose M sufficiently lar^e that "PGSO "soSCF ^ ^^ '^^" f^ f -^ 1 ^ or, M ^ (f + M l)/(g N)

PAGE 44

36 The case with g = N corresponds to that state of highest nul t ipl ici ty, 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 oneand 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 orbltals. In both cases, one has to deal with nonorthogonal orbitals, and this makes the integral computation considerably more time-consuming. Computations using the PGSO method have been carried out on helium and its i soel ect ron i c sequence by Lefebvre and Smeyers /k2/ and Lunell /k5/. Lefebvre and Smeyers however, after expanding and projecting the GSO determinant, or thonorma 1 i ze the components, which does not leave the original determinant invariant. Consequently, their wavefunction is not a true PGSO wavefunct ion. 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

PAGE 45

37 encountered convergence difficulties in their solution and consequently abandoned them in favor of a total enerjiy 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 v/ork on the general case for three-electron systems v;ork that forms the main results reported in this dissertation. These will be discussed In Chapters 3 and k. 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 v/hich 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 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 sun 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

PAGE 46

38 harmonics, closed forms for the coefficients are not known /U5/. Of course, one can still v/ork 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 , I , S , and S and solve for the eigenvectors and eigenvalues /k6/. Me have ignored this problem in this v/ork for two reasons. One, it introduces many additional complications in a method that already is perhaps too cornplicated 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 RflF energy is -7.U32 H, the best PG50 energy, using orbitals restricted to 1 = 0, is -7.U!;8 H, and the exact non-rel at i vi st i c energy is -7.1(78 H. The correct treatment of the spin space has yielded only about 35!"^ of the correlation energy, the difference between the RUF and exact non-re 1 at i vi s ti c energies.

PAGE 47

39 1.3 Other flethods ',;e shall ignore all the empirical and semi enpi ri cal theories. These have of course a groat deal of use In systenat I za t Ion 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 trus tv;or thy, i_f v;e could carry them out to a sufficient degree of accuracy. The Conf I .":urat Ion Interaction fiethod The first, and most important of these, is the conf I <::urat Ion Interaction (CI) method. It was shown by LBwdIn 718/ that any antisymmetric wavefunctlon may be v.'ritten 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 I hll in 1D28. The fact that all the complete sets of oneparticle functions of Interest in quantum chemistry are infinite means that the C! wavefunctlon is an Infinite expansion, since a basis of M 1 functions for an N-electron system can generate (.,)

PAGE 48

kO determinants. For a fixed basis, variation of the total enerf^y vvi th respect to the configuration (i.e. determinant or projected determinant) coefficients leads to a secular problem of the sane 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. unoccuoied) orbitals that come out of the l-IF problem can be substituted Into the HF determinant to obtain an approx Inat Ion to an excited state; the resulting determinant by Brillouln's theorem /U8/ v/I 1 1 be orthogonal and nonin teract I n;^ v;ith 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 largo basis for the CI, one can span a much larger section of the spin and angular momentum spaces, and the determinants can all

PAGE 49

Ul be projected. Consequently, these difficulties of the PHF methods can be minimized. There are two general problems in C! calculations hov;ever. 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 oroblen 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. L8wdin's EHF equations /IS/ can then be used to obtain the optimum orbitals. This has been implemented In a some'..'hat restricted fashion, and is known as the mu 1 1 1 conf i gura t iona 1 self-consistent field method.

PAGE 50

U2 I nterel ectron i c Coordinate Methods A difficulty with the CI method Is the slov; convergence of the expansion. This is partly due to the difficulty in satisfying; the cusp condition on r , that is, the proner behaviour of the v/avef unct i on as r 12 --> 0. This is discussed further in Appendix IV. This difficulty can be avoided if i ntere 1 ectron i c coordinates are introduced into the wavefunction because the cusp conditions can be satisfied exactly. The use of i nterel ectron i c coordinates v/as first introduced and used by Hylleraas /U9/ in 1929 for the helium atom; wavef unct 1 ons containing i nterel ectroni c coordinates are now generally called Hylleraas functions. Janes and Coolidgo /50-52/ in 1933 used a thi rteen-tern Hylleraas function on the hydrogen molecule. The helium calculations were carried further by Kinoshita /53/ and then in a monumental piece of wor!',, the v/avef unct ion was extended to up to 2 300 terms for the sinrjet and triplet S and P states of the helium i soel ectron i c sequence by Pekerls and coworkers /5tt-55, 103,104/, with an accuracy of about 0.001 K, about one oart in 10 . The method has been extended to lithium by Burke /57/, Ohrn and flordling /106/, Larsson /58/, and Perkins /5G/, and to beryllium by Szasz and Byrne /60/ and Gentner and Burke /51/. The

PAGE 51

' k5 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 therein. ) Recently, Sims and Hagstrom /62/ have shov/n that a combined Hylleraas/CI wavefunction of the form K where M. -A <9s. 0,^ 0,. 0,^ ^y''A.<'>\<=' fij^> is feasible for atomic calculations with any number of electrons and have reported results for the beryllium atom. 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 diatonic molecules, several CI programs are currently in use, v/ritten by Kouba and flhrn, by Harris,

PAGE 52

kk ' 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 ALCHE'lY, and work is currently in progress there on the production of a general polyatonic CI program using the Gaussian orbital integral oackage from their polyatomic RHF program, IBMOL. 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. BetHo-no 1 ds tone Perturbation Theory Finally, we should mention a method brought over into quantum chemistry from nuclear theory, principally by Brueckner, flesbet, and Kelly /97/, which they call Bethe-Goldstone (BG) perturbation theory. A great deal of work has come out lately on atomic calculations of excitation energies and hyperfine

PAGE 53

U5 structure. The method relies heavily on the pseudophysical (i.e. intuitive) pictures that one has in the Feynrnan diagrams which represent the terms in a perturbation series, based usually upon a Hartree-Fock starting ooint. The method, v/hlle giving good results by careful v.'orkers, does not seem to provide the simple conceptual oictures of electron densities which are desirable in chemistry. Further, the wavefunct ion, 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 v/Ith a language that can be used to systematize and if possible, quantify, chemical ideas. V/e feel that the BG methods, however close their results agree v/ith experiment, do not succeed in this respect . With these closing remarks, v/e shall now pass over to a discussion of density matrix theory.

PAGE 54

CHAPTER 2 REOUCEn DENSITY MATRICES 2.1 Introduction to thp 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 v;as introduced by Husimi /63/ in 19U0 for use in the Hartree-Fock problem. Husimi 's work vias significantly extended to arbitrary v/avef unct ions 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 McV/eeny /Gk/, Coleman /65/, Ando /56/, Bingel and Kutzelninn /I'd/, and the proceedings of tv;o recent conferences on density matrix theory /G7,63/. In her Ph.D. dissertation, Ruskai /69/ has given an excellent survey of all the knov/n theorems in reduced density matrix theory, with a bibliography of the principal theoretical papers. i»6 -

PAGE 55

U7 2.2 Construction of thf? Reduced Density Matrix Definition of the Reduced Density Matrix We define the p-th order reduced transition dens i ty matr i x as 2J [ M H^r S^ 4V.7 "'^ The term density function would be preferable, particularly since the term density matrix v/i 1 1 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 wavefunctinn; 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, ... ,M) so that (3'}) becomes LS^'•sc/t s^^^;jr] V N The factor ( ) is chosen so that the trace operation.

PAGE 56

k8 i. r'^' = S r'uu)j, ^it) (35) gives the nunber of j^roups of o particles. If states I and J are orthosonal, then the trace of the transition p-natrix is zero. The noma 1 i zat ion used in {l)h) Is LBwdin's norna 1 i za t ion . Tv/o others in common use are Coleman's, d'^'= Ciy'r'^' and McV-Zeeny ' s. ^c,. =^^.^;./r'^' For our purposes, the L8wd i n normalization is most convenient. The v/avefunctlon in (34) may bo 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 fermlon systems, so in the following, antisymmetric v/avef unct ions are to be

PAGE 57

' k9 understood. V;e shall also use the usual notation conventions We obviously have 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 bar: where P is any t\/o-par t i cl o permutation operator such that P ^= -•. We also have rj'(^u')' r^l' U'U}* (38) When K = L, the p-natrix is therefore Hermitlan, and in this case, it is also positive, since for arbitrary functions f(x).

PAGE 58

50 The diagonal elements have the interpretation that is the probability of finding any p particles in the volume element dx.dx-...dx , regardless of the positions 1 / p of the remaining particles. Suppose that we have an arbitrary operator, Q, v/hich is symmetric in the particle indices, so that we may wr i te ^Q^^ZQ.* 1^^' f,S^'The primes in the first set of summations indicate that we omit terns v/i th two or more equal indices. Then

PAGE 59

51 Q. ya ^ ^aJ^JihUx, ' jQ,^r^,cnir2U.,j,, ' 1^173 Q,'"(i23irr3')J>iJfJ,j + . . (Ul) where v/e follov/ the convention that the operators act upon the unprined variables only, and after this the primes dive. removed and the integrations carried out. The Reduced Density flatrix for a CI V/avef unct 1 on Following LBwdin's original work /18,19,20/, v/e introduce a CI expansion for the wavefunctlon of state I : KI (1*2) The index K represents an ordered configuration defined by the one-particle function indices (k,, k . ... k ), 1 2 N where k.^ < k2 < . . . < k^^; that is.

PAGE 60

52 The one-particle basis is not assumed to be orthonormal . VVe introduce -j)LUIjLj) _J^^ 5, ^r ^ Jet [5 / (U3) v/here (iiU) For notational convenience, we introduce "fat" symbols, ^= (k^, k^, ... ,k ) and .^ = (k^ < k^ < . . . < k ). W also need the cofactor of the determinant (U3), which is formed by deleting rows containing functions k, , k„, ...,k and 12 p columns containin;^ functions 1,, 1 , ...,1 / and ovaluatin.'x 1 2 p

PAGE 61

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. I7ith these definitions, we obtain ^ i The notation j^ above a summation means that all configurations containing the orbitals in set ^are to be included. We can simplify (1*5) by introducing a d i s r r e t F! p-matrix, (U7) The nrosence o'' a tilde under a matrix signifies that it is indexed by the non-ordered sets ^, £ , We drop the tilde when wo mean the part indexed by ordered sets of indices, ^ and ^ . We then have

PAGE 62

51* (fK 4>tii')... ^ICp') («t8) 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 where P permutes any two indices in a set. That is, it is antisymmetric with respect to interchange of any tv/o Indices on the sane side of the bar, and Is Hermitian if I = J. The summations in (48) are over all values of the indices. V/l th a little study, one can use (50) to reduce (ftS) to f,_miiij...^;(f>) (51) where the summations ^re now over ordered sets of Indices. This is extremely important for computational

PAGE 63

55 use, since we will only need a snail part of the full p-matrix in (U8). In fact, for M orbitals, the full pmatrlx in (47) has dimension M ; the part v/i th ordered M indices has dimension ( ) v/hich for larse M goes as P m" /p!. Thus far, v;e have essentially followed LHv^din /18/, except that we have used the transition matrix throughout and introduced Slater determinants in (51). At this point, L8wdin specialized to orthonormal orbitals (Ref. 18, eq. 59); we shall avoid this res tr i ct ion. Mow we would like to study the properties of the discrete p-natrix. Using (kh) and (48), for I = J, For orthonornal orbitals, S.. = o . . , and (52) reduces iJ ij to

PAGE 64

56 if? hr r if) In the general non-orthoconal case, (52) can be rewr i tten I, r^^ . ± i, (Qr^'j pi (5tt) where /I = S '^S < . . . ^5 Is the direct product of p overlap matrices. The product matrix in (54) Is not Hernitian, which is rather unpleasant to deal v;ith computationally. V/e note that since the trace is invariant under permutations of the product matrices that so v/e define

PAGE 65

57 (56) or y^ r(^' ny^ r ' ff-rn and then in the general case, we will have i.r'',.'(P ; '^ ^"'=(% (57) From nov/ on, we shall refer to the primed Hermitian matrix in (55) as the p-matrix, because it has the nice properties that hold for the case of an orthonormal bas i s . To proceed, we need some theorems on direct product matrices, which we state here, and prove in Appendi x VII. Theorem: (A x B)(C x D) = (AC x BD) (58) Theorem: I f AR = CD and EF = GH, then (A x E)(B x F) = (C x G)(D x U) (59) Theorem: If r is a rational number, then (A x A X ... X A,)'' = (A,'' X A^*" x ... x A,,'') 12 \\ 1 I N (60)

PAGE 66

58 provided that all the powers on the right exist. Theorem: (A, x A^ x . . . x A ) 1 2 N (A^ X A2 X ... X Aj^) (61) Fron these easily follov/s the results fliU^U^ ... < d) (Li'^iix...^ii)U^A^---^X) (62) if and (63) Since S > 0, all rational powers r exist, and we can therefore compute (56) from the eicenproblen solution for the basis overlap matrix. Mow (55) is Hermitian, so we can dia,
PAGE 67

59 For I = J, (US) can be v/ritten r''"UU') = [4'(')'^4'U)k.. 'i -yz„tfi'^-Yi .^ f = [(DA^K.-.x^jr [l^^xdF^^ .. . x^ '^J (55) whore and r -V2, = ^^"^ (66) <&-i = ^~ ^<4'l 4> $~ ^ r i (67) There Is an infinite number of ways of or thonorma 1 i z i ng a basis. It is intcrGStin<^ that the particular one (66) cones into this quite naturallly. It is the wellknown symnetric or thonorma 1 i za t ion /70,71/ which has the particular property that if the basis set is symmetry adapted, the or thonorma 1 i zed set is symmetry adapted also to the same symmetry operations.

PAGE 68

60 Substituting (63) into (65), we can bring the expansion of the p-matrix CtS) to diagonal form: or J^ i ^' (69b) The general form (69b) is again more convenient for computation because v/e need deal only v/ith ordered sets of indices. Practically speaking, the operators that one is usually interested in contain at most two-particle interactions, so according to ('*!), we need at most the 2-matrix. Let us therefore specialize (68) to the cases p = 1 and p = 2. V/e obtain for p = 1, /(fll'j^i\][iyi[l\\r}

PAGE 69

61 where (71) and U Is obtained from fd^ iin and for d = 2, r(i2i rrj = Z Z 4 i'i K «:.J -^'^^'-^^ 4 ^'i ^i€^ k £ ^^y^ fe,^g,) v^ = Z g..(t^^ ^vnni ^!n<^^'/2''^ (72) v/herc ^.Jt2; 4 Z JcHc^^ii) (r,^C2)j LJ^^, (73) The orbitals defined by (71) are called the natural spin orbitals (MSO); the functions defined by (73) are called the natural spin geminals (MSG). The p-natrix

PAGE 70

62 eigenvalues are called occupation numbers, and the eigenvectors, natural p-states. Obviously, had \ic used the orthonormal basis (55) to be,'^in with, the primed p-matrix in (55) would have been obtained directly from (U7) for I = J. For this case, for the diagonal elements vie find so that Lj>)/ 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) K and

PAGE 71

63 <^.>'if Q,,r (76) The MSO 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 eigennroblem has thus far been centered on the nontransi t ion p-matrix case. To the author's ';nov/l edrte^ no v/ork has been published on the transition matrix e i genprob 1 em. In fact, Bingol and Kutzelnigg (See ref. 67) seem to be the only authors v;ho have carried out derivations in terms of the more general transition p-matrix. • As we noted earlier, according to ih9), the transition matrix is not Hermitian. It is well-known /73/ that an arbitrary matrix i may be brought to triangular form 4 by a similarity transformation with a non-singular matrix Vl :

PAGE 72

6U U/"TW= T (77) The eigenvalues of i are the diagonal elements of " . The eigenvectors are found as follows. rviTvi -I r y = ^^ (78) This is a triangular set of equations for the vector \J^~ \j , from which \i may be determined according to VJiv^"^) (79) Since we do not have a diagonal form of the matrix i , the simple results (75) and (76) do not hold. There is consequently no value in determining Gip;envaluGS and eigenvectors of the transition p-matrix because doing SO does not simplify the situation. Instead, one

PAGE 73

65 sinply conputes transition values directly from (fil) and (51): (80) The Roduced Density flatrix for a Mon-CI Viavef unct i on If tho v/avefunct ion does not have the CI form {hi), one can still obtain a reduced density matrix fron (34). Obtaininj^ a representation of it In a discrete basis is not difficult. The p-matrix nay be considered the kernel of an operator ' such that r^'7^;J r(Kl^'):t(x')d)C' (81) for arbitrary functions f(x). If f(x) is an rtlf) e i genf unct ion of / , then

PAGE 74

65 r'7w A/w (82) Larsson and Smith llhl have recently used this relation to derive fJSO's of the 1-matrix of Larsson's Hylleraastype vvavefunct Ion for lithium. They introduce an Mfunction basis ^ , and expand the NSO's in this basis: ^ = ^t (83) Using this in (82) gives 1% /. i (8t») which leads to the secular problem <^\l!\h C <^li> Cji (85) or U^ s^j^ (85) where

PAGE 75

67 S^.<4;l^j> ffMi^MdK (88) The secular problem (86) may be solved by the methods discussed in Appendix V. The occup^ition numbers calculated this way will be lower bounds to the exact occupation numbers. The proof is not difficult. Let the set of exact normalized NSO be A , with e i f^enva lues ^: K^ -^id ^ki (89) Now let us order the exact and approximate solutions according to ^ ^ (90) and then construct the operator:

PAGE 76

68 M 6Z IX.xXJ (91) 00 (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 & bounded from below, the eisenvalucs of ^vu arc upper bounds in order to those of 2f . Now (9^0= ^0' ^ ^S (93) and 01^ 4 (1^^ ^) so that ^'^ has e igenfunct ions A. and eigenvalues A , and we have immediately X, ^ /. (95) The approximate occupation numbers arc therefore lower bounds as stated. The sum of the approximate M occupation numbers approaches ( ) from below and provides P a convenient measure of the adequacy of the chosen bas i s .

PAGE 77

69 2.3 Properties of Density Matrices CI Expansion Convergence In Chapter I, we mentioned the conver.'^ence problem in the CI method. In his original paper on density matrices, LBwdin /18/ showed that the natural spin orbitals are actually the orbitals v^hich give the most ranid conver rcence of the CI expansion, the MSO of highest occupation number being the most important. Of course, one needs to knovi the wavef unction to begin v/ith in order to obtain the p-matrices and the natural p-states. However, if a truncation is made of the CI, one can obtain MSO for this truncated function, put these back into a new CI, perform a new truncation based on the size of the MSO occupation numbers or other criteria, obtain new NSO, and so on. This natural soin orbital Iteration technique has recently become quite a popular tool in CI calculations, but the convergence of the scheme docs depend on the quality of the Initial truncation.

PAGE 78

70 Bounds on Occupation Numbers Since the p-matrix is positive and of finite trace, its eigenvalues obviously satisfy ^ A/^^ ^ (^; (96) Coleman (See Ando, ref. 66) shov/ed that which for p = 1 and p = 2 takes the form Q . X'"' ^ ^^^m.. a.Tl^J (99) It can be shown that ttio 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 ?\rQ

PAGE 79

71 ^r / (100) w here (xj is tho intcj^ral part of x. Ho also proved that the bound for p = 2 is the best po s s I b 1 e . The Car Ison-KnlTor-Schni dt Theoron Carlson and Keller /77/ showed that the nonzero 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 unitarlly eoulvalent. In addition, if r^7x//j ^ ^l^p(x)^"M ^ -& (101) 7^ ^^ and f'-^'ljlf] Z X, Gi, hj] <5/y^ , ^ ^^ <»^'

PAGE 80

72 then (103) and XZ T -V^ ^,^y ^ ^/; '^:v^.^'<^^^^y^^ . ^105^ If ' and ' \io.rG derived from an an 1 1 symrnetr I c wavef unct ion, then the resolution (103) of the wavefunction is automatically an t i symnet r J c already. The eigenf unct ions of the p-natrix are called natural pstates, and those of the ( N-p)-na tr i x, co-natural pstates. Coleman /65/ later pointed out that this theorem had already been discovered more than fifty years earlier by Schmidt /7C/. Schmidt's results, in the terminology of density matrices, show that the expansion (G3) f^ives optimal convergence in the least square sense to the wavef unct ion; this, rounled with the fact that the natural n-states can always be expanded in terms of the NSO /65/, leads to the CI convergence theorem independently obtained by LHwdin which we referred to earlier.

PAGE 81

73 The Carlson-Keller-Schmidt theorem is of particular significance for N = 3, since the 1and 2matrices then have identical non-zero eigenvalues, and the NSO and MSG can be obtained from each other by virtue of (104) and (105) . Synn^trv Properties We mentioned earlier that the wavef unct ions should be required to be e i j^enfunct ions of the group of the Hamiltonian, and the Question of how the symmetry properties of the wavefunction carry over to the pmatricos and the natural p-states has been extensively studied. We shall merely list some of these results here which have significance for our ov/n vjork. Theorem 1: I f Ai. is an N-electron Hernitian operator of the form or a unitary operator of the form /]'"= HA (107) or an antlunltary operator of the form

PAGE 82

7k n""' K Jin. (108) 1=/ where, in (107) and (108)/iiils unitary and K'^ denotes complex conjugation, and If ^ is an e i genf unct ion of il , then the natural p-states can be chosen as ei genfunct ions of J ^ . Theoren 2: if ^ and ^j transform as the irreducible representations ^ and Qj. respectively of some group, then /7r transforms as the direct product representation '^ X J The particular significance of these results is best illustrated by a few examples. If the wavefunction is an e igonf unct i on of L , S , or narlty, z z theoren 1 apolies, and the p-matrix bloclcs by M|^, M-, or parity value, and the natural n-states are ei genfunct ions of L , S , or parity. If the wavefunction ? 2 Is an eigenf unct ion of S or L , the natural p-statos 2 2 can generally not bo chosen e i genfunct ions of S or L , except when S = M^ = or L = M|_ = 0. Of course, for special choices of approximate v/avef unct Ions , additional symmetries may be Introduced. Garrod has shown for example that If the wavefunction is taken as

PAGE 83

75 an average of M components with identical space and spin parts, then the MSG's can also be made e i genf unct ions 2 of L . Theoren 2 is probably more useful for molecules and solids; for atoms it essentially duplicates theorem 1. In order to better see the structure of the 1and 2-matrices, it is sometimes useful to expand them in terns of separated space and soin parts. If the vvavef unct ion is an eigenf unct ion of S , it may be shown that /(x,)y;jr^c/rT'j UoC f^i^ir/)/^^ (109) rU.hUu-) -r'^(f:rjr:rj)a.a** f'VifPl/nVrjfci^ JJ /-^ * r^cn^jfjvrjcc * r"(r,7jrX}dd where /r^ (r,ri lr'rj)cd^ -f f '(r,^Jr'rj)dc' (110) (111) The presence of the cross terns cd* and dc* in (110) shows that the 2-matrix is generally not an

PAGE 84

76 ei genfunction of S . Also, one sometimes introduces the charge-density 1-natrix, (112) the spin-density l-matrix. (113) and the charge-density 2-natrix, Q(Hl^'ri')'!r(U^6-Jr:
PAGE 85

77 I r ' /'^^ I : rr VJe then use the result that if matrices n, B, and <^ have eigenvalues c(^, ^f^,, and ^respectively, arranpied in nonincreasing order, and if €~~ 4^5 (117) then /73/ '^^(^.^Ay^^^AJ ^ ^^ ^^^i'A^^x'A) (118) From this result, v/e obtain the following bounds. f ^ -« (Ar/, jf/; . x: -' M„ (ir, ;if ; -/ < The interest in the COfin and CDriG is tv/o-fold. First, if the wavofunction is an e igenf unct ion of S^ and S and if Mg = 0, then and av q identical, and the NSO ' s are flO ' s with a or /spin; T and /^vanish, and the MSG': ^x-^ MG's with one of the four spin functions (111).

PAGE 86

78 Second, for an fl-function basis, there are 2M NSO, only M of v;hich can be spatially linearly independent. Consequently, the f1 linearly independent NO ' s have sometimes been suggested for the CI iteration scheme discussed earlier. In ireneral, both the f-.'SO's and N'O's will have mixtures of either odd or even values of angular momentum; that is, s orbitals v/i 1 1 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 vyith orbitals of a single (l,m.) value rather than of a single m, value; angular-momentum projections become considerably more involved if l-mixtures are al lowed. 'i/hile we have not made explicit use of them, we have generated the CDIlO's and SDriO's for all the systems studied in this v;ork. Density .Matrices of Some Special Functions For a single Slater determinant of U orbitals, the 1-matrix has N occupation numbers equal to 1, and the remainder equal to 0, If the orbitals are or tiionormal , the 1-matrix Is diagonal directly from

PAGE 87

79 (U6), and the orbitals are the NSO. This is a particularly important case and has been discussed extensively by LHwdin /18,19,20/. In this connection, it is worthwhile to introduce the extended Hartree-Fock (EHF) equations which LBwd i n 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. I7e begin with the expression (.kl) for K = L, v/here Q is now the Hanlltonian operator. (121) Varying the expression (51), we obtain <5Z ^.J:i[^,,...h,)t'mk}-L,,!i;^... < fe,^ '' '''' + Complex CO'^joocoii. (122)

PAGE 88

80 Using this result, we find •tCoy>nplc)( Con jOaecte, M 9 '^ ^ where we have introrluced a Hernitian natrix of Lacranj^e multipliers X to maintain orbital normalization. By the usual argument/ the expression in brackets must vanish; 4'' we then multiply by "J^^vand sun over ^ , obtaining the EHF equations:

PAGE 89

81 ^Z ' =: XiW^O <12U) where XL\\\') -li'(-^)> <^^^^J'^i (125) Note that nov/here have v;e assunod an orthonornal basis or a particular form of the orbitals; unless v/e start with an orthonornal basis, there is no need even to introduce the Lagrange multipliers, and the right-hand side of (12U) then vanishes. For a non-or thonorna 1 basis then, there is no need to deternine Lagrange multipliers, but we have a more difficult p-natrix to compute. In general, it is not possible to simultaneously diagonalize the 1-matrix and the Lagrange multiplier matrix, so we essentially lose the concept of orbital energies. It is often useful to introduce a quantity, called the "fundamental invariant", defined by

PAGE 90

82 p. |^>^^/4>'^^l (126) which satisfies (127) (128) where M is the number of orbitals in the basis. The Invariance follov/s fron the fact that a nons i n,'^u lar linear transforna t ion on the basis leaves (126) unchan/^ed: t-'nrt JT' <^i^>^^ ^^^^ f'><^l'^> < f (129) For the case M = W, LHv/di n /19/ showed that /? determines all the p-natrices, and these are ,^Iven explicitly by

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83 ^'^^^. Jd p I (f(lll')^C2U'J....^(plp')) (130) The fundamental invariant therefore contains all the infornation contained in a single-determinant wavefunction, re.n;ardless of the form of the basis orbitals. This noint has lead to some confusion in the literature. In an often-quoted paper, Bunge /hk/ arrived at the result that for a PGSO wavefunct ion, the EMF 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^ ; this simplified form holds for an orthonormal basis. Bunge then proceeded to vary the orbitals, destroying the or thonorma 1 i ty . The EHF equations are perfectly well-defined, even for GSO. That ^ determines all the p-matrices for a singldeterminant is true, oven for a Projected determinant, on. The occupation numbers B.r^. 1 and if OHO is considered the modified Hamiltonian and D the v/avef unction; however, if 0D is considered the wavofunction, the occupation numbers are in general not and 1 because the projection introduces new orbitals. In this case, the fundamental invariant must be constructed from the comolete set of orbitals, including all the ones introduced by the projection.

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&k The p-matrices must still be determined by the fundamental invariant, but the form of the natural pstates 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 nointgroup and ax i al -rotat ion symmetry projection by Simons and Harrinan /SI/ to obtain formulae for the 1and 2matrices. 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 v/avef unct ion is not known in analytic form, although vie have calculated the 1and 2-matrices directly from the projected determinant treated as a CI expansion over non-or thonorma 1 orbitals. The form.ulae for the DODS case are already very comnlicated; in viev/ of the ,?;reat increase in complexity in ^oing to GSO, vie feel that an attempt at obtaining an analytic formula for the p-matrices of a PGSO wavc^unct i on would not be wor thwhi lo .

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85 2.tt The M-RenrR5ent;=^bn i tv Problp n The SchrHdin;^er equation, (1), has never been solved exactly for a system v/lth more than one electron. As the number of electrons Increases, the approximate wavef unct Ions become i ncreas I n?;l y complex. The Hylleraas coordinate functions discussed In the last chanter have not been extended beyond fourelectron 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 ina^^lne Coleman's excitement in 1951 when he first observed the significance of the equations (41) and (121); since the usual Hamlltonian employed contains at most two-particle terms, the enemy, and a1 1 oneand two-electron properties deoend at most upon the 2-matrlx, from which the 1-matrix can be derived accordinj^ to (55). The 2-matrlx Is a function of only four particles. Thus, by varylnri 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 D\'A, Coleman contented himself at that time with a calculation on lithium, a threeelectron system. The calculation gave an energy 50%

PAGE 94

86 VJ below the experimental value, in seening violation of tlie Raylei ,p;h-R i tz 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 "Nrepresentabi 1 i ty" problem -the problem of findinf^ the conditions under v/hich a 2p-particle function, such as a p-matrix, can be shown to be derivable from an Nparticle antisymmetric (or symmetric) wavefunction ithout actually exhibiting that N-particlo function. This problem has received a great deal of study in the last tv/o 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 v/avef unct Inn. 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 hone 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, semlpirical, or even empirical theories. Some progress em

PAGE 95

87 has been made alon;; these lines by a number of authors /82 90/. In the mcantine, reduced density matrices provide a convenient tool for interpretation of wavefunct ions and nrooerties.

PAGE 96

CHAPTFR 3 ATOMI C PROPFRTI FS 3.1 I nt roquet ion Readino; the current quantum chenical literature ^ives one the feellnr that a total ener^ry is the only property atonic and molecular systems possess. Since the total ener.^^y, like the thermodynamic enthalny and free energy, is meaningless except when compared with another total energy/ onr> mlpht even Serin to question the motivation o^ the calculations. In fact, of course, there are a <^ood many oronerties of interest which we can in nrinci^le compute. A recent hook by Malli and Fraga /92/, althouf.h somev/hat concise and uncritical, does at least Kive an idpa o''^ some of the properties of interest. A review article by Hoyle /105/ discusses relativistic and nnnre 1 a t i v i s t i c corrections to atomic enerry levels and a number of numerical tables ",7ith these corrections is j^iven. Vie will cont«^nt onr«;elvr.r, in this chapter only vnth ftivinn: a st^ort indication o''^ some of these properties with 88 -

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;o reff^rences to wor'c wHere p:reater Hptail nay be found, 3.2 Fnor^ies The calcul^ition of the onerry detemlnos the wavef unct ion . Fxcent for one-electron systons, which can be solved px?ctly, anH Pekeris' vtork cited earlier on tv;o-el ectron syster^s, calculations of ener^^y levels cannot cor'-.'^te with exnerinont in accuracy. Consenuently, "xrent for detern i na t ion of the v/avefunction and ronnarison with other thporetical results, for atons, calculation of pppf-rip^s is of little interest hocauso the exnerinental data is so much better. ^^or nolorules, even snail diptonics, this is not the cas^^, and one can often ";et better characterization of notential curves by theoretical comnutations than current exnerinental n^thnds can '^'V'^. It is nerhaps one of the sad facts of nuantun mechanics t^^at de tern i nat ion of the enorry is the only route to the wavefunct ion, an-^ that even if an aonroxinate wavefunction rives a r'ood ener^^y, other pronertif^s calculate'^ fron it nay be rather ooor. in t^'is work, in addition to the enerp;y deterninat ion, v/e have also evaluateH thp scale factor

PAGE 98

90 and scaled energy givpn for atons by /98/ 7 £, 4 (131) (132) An atonic v/av^f unct i on nay aU-zays ho scalod to satisfy tho virial t^oorf^n; if the unscalod v/a vof unct i on satisfios it already, thon the scale factor is necessarily unity. V/e have found this useful in that a scale -factor di-P-Ferin?^ frnn unity hy more than about 0.001 indicates t'^at the basis is poorly chosen. 3.3 Specific Mass Fffect in tho introduction, v;o derived the snpcific mass effect, or nass polarization, correction to the kinetic ener,f?;y, en. (15). FrHnan /OO/ has estinated the efff^ct fron exonr inenta 1 snectra to be about 10 !J (0.2 K) for Li 2 ^S and 10~^ M for Li if S. He also states t'^at t^^«^ effect st-iould be apnrox in?i tel y independent o^ Z, so that the same estinntes annly to the rest of f^e i soe lect ron i c seouence. However,

PAGE 99

91 Prasad and Stewart /lOO/ have recently evaluated the effect from Weiss' i+5-term CI wavef unct ions for the 2 ^S and 2 ^P states of the sequence fron Z = 3 to 8 ; for the 2 ^S state, their data ^ives the shift proportional to z"*"'^^ ; for the 2 P states, the shifts decrease with increasinfi 2, beconins negative for Z > U. The shift for the 2 "^S state of Li^ is 2.5S7 K, and for Li , 3.017 K, a difference of 0.430 K. By contrast, the normal isotope shift, (13), causes the sane 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 he 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 Pekorls and coworkers /5k55,103-lCU/ on two-electron systems, and obtain results in agreement with Prasad and Stev/art. According to Kuhn /lOl/, theoretical determination of the specific mass effect would be a valuable contribution, and we therefore intend to compute the effect with our wavef unct i ons at a later date.

PAGE 100

92 3.h Relativistir '"ass Increase Relat ivi St Iral ly, the electron mass varies v/I th velocity accorHinp; to ?»7/, m ' ff^ (133) where /= c^ (13t() and m is the electron rest mass. The relativistic !',inetic o enerf',y is T=^ ync^yn^a^ (135) i-m.M'* i ^i^ ^ J. '"t> and the relativistic foiir-nonentiin is p= i-YA^^imcL) (13G) but its na-rnitudo is constant P^p^ ---^.o We therefore takp t*^e non-rel a t i v i s t i c norrntun.

PAGE 101

93 P = -yr)\j (137) expand (I3t;) in powers of n, havln;^ ellninateH^ by usin(153) and (137), anH ohtain T = P' _ C ^yvtC i^3 + cp' ILrr,fc ^/i5 (138) Putting p --> -i)^V ^ ^nd transforming to atonir units, we find 7-f7^^>^_ 4V (139) This result is obtained by Shore and Menzel /I/, but thoir derivation is in prror, so wp have sketcHf^d it ^ore. Fron Anpendix I, v/o find ^t'8 = 6.6513 X lo"^ so v;e exnect the corrPction to be snail for li^ht atoms. For Li \, v/e have found the net e^^ect to be about 0.005 H, but it Increases with Z to about 0.6 H for Me VIII. The mass increase is the major relativistic effect in li^ht atons. The derivation above is only oeda.-TOKical; the correct way, procppdin;, ^ron the Dirac equation, gives the sane result to order u"^.

PAGE 102

9i» 3.5 Transition ProHahilitiPS nnH Oscillatnr Str^nrrths Shore anH flenzel /I/ give a good discussion of this subject, so we will omit specific formulae. Atonic transition nrohabi 1 i t ies hpve recently become of strong interest due to extensive theoretical work at the U.S. national Rureau of Standards by Weiss and coworkers /lOS/, coupled with the experimental advance of beam foil spectroscopy /109/, and the experimental measurement o^ absolute, rather than relative, transition probabilities /115/. The reason for this interest is that knowledge of atomic transition probabilities is necessary for accurate determination of stellar and Interstellar elemental compositions, which In turn affects astrophysics, as trochemi s t ry, and theories o^ the origin of the universe. In addition, the oscillator strength is related to bulk properties such as polari^ibilities /117/, electrir and magnetic susceptibilities, dielectric constants, absomtion coefficients of el ect ronagnet I c radiation, pho to i on I za 1 1 on cross-sections, an'^ so on. For tho lithium sequence In v;hlch we are interested, wor!; by V/elss /llO/, All and Schaad /111/, and Flannery and Stewart /112/ should be cited. Important contributions have been made by l/einhold

PAGE 103

95 /113/ v/ho has ohtalneH methods for bounds on oscillator streni^ths, and by Staracp /lli+/ who has denonstrated that the len;rth fomula, rather than the velocity or acceleration formulae, should be used for the conoutation of transition probabilities from aonroxinate v/a ve^'unct ions . Theoretical evaluations of the oscillator strengths are dif-^icult because the wavef unct ions of a larf^e number of excitoH states are reouired. In fact, Fano and Coooer /116/ noint out that after hydrot^n, the continuum contributes the bulk of the oscillator strength. For t^-is reason, oscillator strengths have be'-n little treated, and tf-en usually only v;ith i^artree-Fock or nerturbed ^artree-Fock wavef unct ions . 3.6 Fine anH "vner^ine Structur*^ There are two important areas of atomic spectroscony wh^rtheoretical calculations can assist experiment. One of these is the calculation of transition nrobab i 1 i t i es, discussed In t^e last section. The otbpr is the c-lculation of hynerfine structure (hfs) seoarations.

PAGE 104

96 Fine structure is due to spin and orbital angular nonontun interactions, '.vhile hfs is due to interactions of the electrons with a non-spherical nucleus. ."'easurenont of aton.ic hfs gives data which depends upon the electron distribution and the nucleon distribution. Usually insufficient independent data is available from experinent to separate the electronic and nuclear effects, so theoretical evaluation of the electronic contributions coupled v/ith the experimental data enables one to obtain the nuclear contributions in the form of the nuclear spin and g factor, magnetic dipole nonent, electric quadrupole moment, magnetic octupole nomont, and so forth. The knov/n nuclear quadrupole and octupole moments have nearly all been determined by study of the hfs in optical spectra. Knowledge of these moments gives information about the shape of nuclei 'which is important because despite the huge effort In high energy and nuclear physics of the last three decades, the forces which hold nuclcons together (and therefore deterr-.ine nuclear shapes) are still not understood. While hfs has received a great deal of study in the last decade, fine structure has probably been one of the most neglected fields. This is probably due to the fact that although treatment of the angular symmetry in fine structure has long been understood

PAGE 105

97 /118/, the radial interactions for many-electron systems have not been successfully treated until the recent v/ork of Blume and Watson /119,120/. These authors showed that the spin-orbit interaction is given by a rather complicated two-electron operator, and this greater complexity has effectively discouraged theoretical calculations. Hfs is comparatively v;el 1 -understood, thanks to the work of Trees /121/ and Schwartz /122,123/, and has been extensively discussed in some recent books and review articles /12U-12S/. One of these, the book by Armstrong, is particularly notable In that the entire discussion derives from an approximate relatlvistic Hamiltonlan rather than from a patched-up nonrelatlvistlc treatment. Schaefer, Klemm, and Harris /129/ have recently give a convenient tabulation of the principal hfs formulae v/hich vie will summarize here. Including the hfs splitting, the energy of a given J-state nay be v/ritten \'l = W + AK/2 + BK(K+1) r J ilkO) where K = F(F+1) 1(1+1) J(J-l) (141)

PAGE 106

98 and F, I, and J are the total, nuclear, and electronic angular momenta resn^^ct i vel y. The constants A and B are rJven by ^ 2U2i-\) "^l ^ and n = nuclear electric nuadrupole moment, g = elp>ctron t factor, 2.002290716 /I/, /l^= Pohr mar;neton, JUx = nuclear nap:netic dinole moment in nuclear ma«^netons, /A(/= nuclear marrnetnn The narameters in ^ and p. are .f^Iven hy J^xAi ' <»^) JA,«. <^^1rlZ '^^s.,\7^r> <»=> Tic^ U5)

PAGE 107

99 J = L + S and Mj = J are understood In these four matrix elements. The ct's are called the orbital, dipolar, contact, and quadrupole parameters; the A's are angular factors which 3re tabulated in Table 5 for some states of interest. The anf!:ular parts of the inte^'^rals (144) (1U7) are most conveniently worked out with spherical tensor algebra /I/. Three of these integrals have a factor r , and this has led many people to attempt to use a single value in all of these, so that the hfs parameters are then related by known angular factors. This is an approximation, as may be seen easily by considering the evaluation of these quantities from the NSO expansion of the 1-matrix, (75). The riSO will make different contributions to the integrals depending upon their spin and angular momentum values. As an example, Lyons and [iesbet /130/ have recently pointed out the necessity of treating the three as independent parameters for a proper description of the hfs of Li 2 in this area too, the orbital picture and the I PM have often led to an incorrect statement about A and B, namely, tfiat for S states, B is zero and A is determined only by the contact interaction. If the 'ISO 2 ivere eigenfu notions of L'', then the orbital, dipolar, and quadrupole parameters would be zero for S (L = 0)

PAGE 108

100 states, since only s-type NSO would appear. Then the contact term v;ould be the only one non-zero. This is a fairly r;ood approximation, but the presence of different values of 1 in the MSO expansion shov/s that in general all but the orbital term, cLj^ , will contribute to A and B for S states. The orbital term does not contribute because for a pure L state the NSO are eigenfunctions of 1^, which in this case corresponds to an eigenvalue m, = 0, which gives a zero factor in the i ntegral . The Integrals in (141;) to (lU?) may be considerably simplified if the f.'SO are eigenfunctions of L (as they are with our wavef unct ions ) . They nay be wr i tten CI n^ >^^, ^A -= z {2Ji^i){2l-l) '« ' ri S = < Z i^!!!il^> ^.'

PAGE 109

101 a. = < ^HScrj s^y ^^i^liz. 47tr > = j2 '^s, ^^ I2'j6)))' il Z^j:^^^/Z-^J2:,^^i/' ^.OC ^^ The requirenent .J = L + S leads in our case to n, = 1, so that the factor in the second and fourth redures to ^ _Finally, v/e note that the Fermi contact contrihution (1U5) to the hfs solittinjr is usually

PAGE 110

102 reported in terns of the oarameter /= fTT <^s(v:)s^.> ^ /^(O) ^ ^ 7^^(o] lJJo)l' Zn^ \Xjo)\" (1U8) which riops not involve any exnerinpntal quantitips. Since the contact trrn is a no i n t orooerty, wo should anticinate difficulty in ?^ettinp: consistent results for it fron dif^'erent calculations. This difficulty has heen lanented in a recent survey article /131/. Tl-ie ef-=ect of the finite nuclear size on the electronic energy levels ray he shown /2,125/ to he l^iven to a «^ood aoprox imat ion by ^B = '^z^'(^(o)| in Hartroes, an'^ is the mean snuare nuclear radius Since nuclear radii are of the order of 10 Rohr, the -10 enerjry shift should be of the order of 10 Z Hartree, or a few MHz. For li?:ht a tons, this effect is too small to bo observed in ordinary optical s^ect roscony . Mowever, it can be measured v.'itH nBssbauer snectroscony

PAGE 111

103 for atons in a solid. If the electronic matrix elenent can be calculated theoretically, ti^en inforr.ation can he obtained about nuclear sizes fron measured shifts. Hov/ever, the ^act that the techninue can only be anolied to solids means that our results for t^is quantity, which is essentially the electron density at the nucleus, !^re not directly apnlicable to available exoerinental data. The hfs and the nuclear volume effect t^us r;ive us information about the total electronic charre density in the nucleus and about the total electronic spin density at thp nucleus, the dlffrrence between char,^e densities of each snin. We remar!'s in closing t^pt the contact interaction in t^e solir^ is known as the Knight shift, and also that the spin density is often sloonily referred to in t»^e literature as \^Ho)]^^ w^^Ic^ may be confused v/it^ the total charpie density at the nucleus.

PAGE 112

CHAPTER It THE PROJECTED GENERAL SPIM ORBITAL CALCUIATIOVS U . 1 I nt ro^urt i on As we discussed in Chapter 1, the objective of this v/or'c is to carry out a study of the projected flieneral spin orbital method in order to determine what imorovement over other methods may be expected, and whether the method mi^bt prove useful for ]arrer systems. The ef'Pect of completely snanninp; the snin space is of particular interest for spi n-denendent properties, particularly the Fermi contact term. Since Lunell /hli/ has already carried out PGSn calculations on some f'/o-e 1 ect ron systems, he and the author bepian ivritin?^ a nro^ram for the doublet states of threeelectron systems, naively thinkinn; the worf' could be comnleted in just a few months. The computations proved to be less than simnle, and we shall discuss some of the nrohlems encountered as we go alonr;. lOU -

PAGE 113

105 li.2 f'atrix Formulation of the PH Sn Method \7e st^rt with a PGSO deterninant for a threeelectron system: ^ ' Os'O.X^.'ii'^s (11+9) •.. = Vv . (150) The GSn product is exnanded into 2=8 products of pure snin orhital cormonents, an^ two of these are elininated by the S^ projection. In order to have sone way of conveniently varyinrr the GSD, wo introdiire an Mfunction baS'is^ and expand the HSn in terns of this basis ii^ ,^c*^oC ^ i^ G50: = <^//C> -s. (152) lOL

PAGE 114

106 this can be carried out by the v;el1-knov/n Schnidt process. Since we work v/ith the GSO conponents, which are non-orthoo:ona 1 , no advanta«;e is rained by or thonormal i zat ion of the GSn in the cornnuta t i ons ; this is done only for nublication and for plottin.pourooses. We then assump tb=it t'-/o of the HSn are knov/n, and Insert t^e expansion (151) for the third into (lUO), ohta i n i n.^ (153) This has t^e form of a CI expansion for the wavef unct ion, which leads to the usual secular orohien He = S C E (i5f,) 6^ -[/ (157) €'i f ^ I (158) c Jlaiy i E^J (159)

PAGE 115

107 which may be solved hy methods discussed in Appen'^ix V. The matrix elements (155) and (156) are s tra i :^h tforv/a rd hut tedious to derive. Since there is nothinp; worth comment inp; about then, we s^^all not tabulate the lengthy exoressio'^s obtained for them. in practice, v/e have found that t'^e overlap matrix '-ir>finr.r! by (IS^i) is nearly alv/ays singular, havine one or tv/o very small eigenvalues. The reason for this seems to be that one of the components of t^e valence orbital is generally quite small. This is straightforwardly dealt with by canonical orthonormal i za t ion, discussed in Apnendix V. The net effect is tbat t'//o secular nro^^lems must be solved, one for H, and one for S, making the nrocess more time consumi ng. Having o^^tained C anH E, v/e have unner bounds to as many states as we have 'eigenvalues, althouf^h in practice, we only compute the few lowest ones. I7e have thus obtaine'^ tho ti^ird HSO, so wo nut it back into (153), and expan-^ the second orbital. Constructing and solving a new spcular nroblem gives a new second orbital. Proce^j^in^ this way, v/e imnrove each o-^ the orbitals every third iteration. Although it is not obvious that this process should converge, we have aU.'ays found it to dc so v/hen tho starting estimates of the first two orbitals w^-rp reasonable.

PAGE 116

108 An alternative v;ay to proceed, v/hJcb would avoid the iterative process, mi^ht be to expand all the orbitals at once, which would p;ive a single secular problem of dir^ension (2M) for M basis functions; the 3 eigenvectors would each define (2M) equations to determine the 3 X 2.M = 6M orbital coefficients. Aside from these equations beinj^ p;reatly overdeterni ne^, this soon becorres prohibitive. For M = 15, for examnle, which is the larr:est basis we have used, we would have a secular problen of dimension 27,000. This would require the computation of 729,027,000 matrix elements over nonorthoj^onal orbitals (although v/e could use an orthonornal basis to simolify construction of these), and on our computer, solution of the secular problem alone v/ould take annrox ima te 1 y twelve years! For M = 10, we could solve the secular problen in about fortytwo hours, still •'^a r too much. h.3 Choice of Has'^s and Initial Orbitals Our method has been plas^ucd with slow conversience; examples are riven in Apoendix II. 17e have tried Aitken's h oroccss and an extrapolation procedure based upon the pattern search method /136/; both

PAGE 117

109 diverged. The prospects for extrndln? the calculations to larr.er systems thus annear rather disnal unless a way of .trroatly snop'^inr f^n converrence can be found. Because of th^se difficulties, it has been necessary alv/ays to have rather ?;noH startin;^ orbital estinates, of MPMF, SOSrrr, or S^PHF /[,!/ quality; an RH^ starting point, which v.-ould bp useful because of the extensive tabulations available, v/as -Pound to be inadequate. K'e have used Slater-type orbitals (STO) in our bases; they are defined by ^ = ^r"-' e' ^'" X T;>f (160) where ^/ Vz (161) The exponents may be chosen in a number of ways. Most of our results t^ave used bases renorted by others; this facilitates co-narison an-" enables one to use the renorted orbitals as startinooints. The cuso condition nay be used to set one orbital exnonent for each i-sheii; tMs is discussed in Appendix IV. One nay use t'^'^ •f'art th^t

PAGE 118

110 = 2.C IS tbf^ avera?:p vr1u9 of r for a sinfle ST'^. Thp exponents nay t'^'^n be chosen according to the anticipated values. !7e found that t^^e exponents reported by Kaldor and Harris /39/ for tbpir SOSri^ calculations on Li \, Re II, and B 111 could be plotted linearly ar^alnst Z, and exponents for C IV through Ne VI 11 were obtained in this way. Another way, which is peculiar to ato'-is v-ritb a sin-^le valence electron, conos fron an observation of Moseley /137,138/ in 1913, now known as flosele^'s lav/. This states that the X-ray spectral ener'^ies Br(i^ related to the atonic char.o:e, Z, by I^ ' ^^^-s) (162) where R is the Rydber.r; constant (0.5 Hartreo), n the principal quantum nunber, and s a screenin;^ constant, about 2 ^or K lines, and 10 to 20 for L lines. The justification -^or this formula was found in both the Rohr tbeory and in Gch r^5d i n Ter ' s solution of the wave equation for onr>-e 1 ect ron systems, namely, that the enerp.y is pronortional to (Z/n) . A valence electron in lithium sees a screened nucleus, V'/i th effective

PAGE 119

Ill charge closer to unity the higher Its energy. From the experimental snectra, one can easily ohtain the effective char^re, Z s, from (162), and then simnly use the fact that for the one-electron SchrHdinrPr equation, the solutions have orhital exponents given by c ^i4 n This enabled us to obtain quite satisfactory bases for the D and F states of lithium, once we had the core bases well -Hetermi neri from calculations on lower states. Another v/ay, nerhaps the most desirable in principle, v/ould bP to ootimlze the exponents for each state by direct minimization; however, the slow convergence precluded this. Since we bv^ not employing tHe EHF enuations for determination of the orbitals, numerical orbitals would be out of the question because there would be no way to determine thon; we do have to introduce a basis. Most of the basos we have used could have been improved considerably by exponent ont Im i za t i on . To see the effect of this, we have augmented some of the bases to fourteen or fifteen functions, which was the larrest set we could use in core on our comouter; larger bases would have required considerable restructuring to do the i nte'^ra 1 handl i ng pa r 1 1 y i n

PAGE 120

112 core and partly on external storage. In addition, the large basis set calculations become very expensive to carry out. The orbitals, natural analyses, basis sets, and one-electron properties, because of their volume, will be made available separately as a Quantum Theory Project Technical Report. We give a sample tabulation of these in Table 3. tt.U Evaluation of the 1-Matrix As v/e discussed in Chapter 1, we need to determine eigenvalues and eigenvectors of where / is obtained from (ti7). The expansion of the PGSO function nay be written, for S = S = +1/2, z

PAGE 121

113 ^ J? ( r^ ^:/ 4J/^ ) ir^^.ytc) V/e renarkpd earlier t^^^t even if t'^e wavefunction Is an 2 2 ei^^onfunct inn of S , L , S , and L , that the natural p-states are e i Pien^'^unct i ons only of S and L , unless M = or *^, =0 (v;hich is not the case ho re). fov/, for the P/ D, and F states of three-electron systems, v/e can p-tiarantee that the wavefunction hp an e i r:enf unct i on 2 of I_ if v/e choose t'^e tv/o core orhitals to he s-tyne and the valence orhltal to he nure p, d, or f type, respectively. We see then that for this special choice, if v;e inte,'t;rate over tv/o coordinates in (16^0 to ohtain the 1-natrix, v/e v/ i 1 1 have only pure s or Dure p (or ^, or f) orhitals in the exnansion, so that v/ith this restriction, the "SO are also e i "-enf unct i ons ? of L'. Accor^in": to (70) an'^ (71), vi<^ have

PAGE 122

111+ fU^ iin : U'UUW1 lUhj '\^^)>Um > A~''f/i"'^<4l = /^>A"'^/Jl n UlW'^< t> n<-t\ t '^A^'U (165) (166) Note that so that tho "SO am orthonornal. We have pncoun t" reH practical conpu ta t i ona 1 nrnblens duo to the near singularity of A, v.-Hirh in our case frenerally has one or two nearly zero e i .r^nva 1 ues . in this case, on'^ has to be careful in the cc^niita t ion of <^ and ^ w^irh is fornally done from

PAGE 123

115 (167) -Vi -V. , / 1 A"^^ \/ r 1/ As an r^xapnl^?, snnnoso on<^ olrrenvalue Is of th^ ordpr of 10 / n > 0. Then its sauarp root is o-p t^p ori^er of 10 ' , so t^'at i nacriirar ! es In t^p sra 1 1 el<*envalues propat^atP to the \p:^\., rialcinp; the He ter^ii nat i on of the hal f -powers o^^ the overlao matrix unstable. In practice, v.'^>a t rve have done is to renlare the halfpowers r>^ eiTenvaliies helo\-f 10 ~ by zero before con^^utine the ha1f-nowers o-*^ the overman r^a t r i x . This affocts the orthonornal i ty of the f>'SO, (16G). Since ortbonor'na 1 i ty nust be naintainc'-' "Por the conoutation of exp<^ctation values fron t^^e natural expansion, we must irnrove this. One cannot sinoly SchniHt ortho.rtona 1 i 20 t'"'e MSO because the occuoation numbers are not defrenerate. Instead, v/e let

PAGE 124

IIG 6= 'f'A-'^U -Xe (168) Then S-< 01 ^ ^Util>e = eU (169) where wr? use U ^or the aprjroxinate NSO and <^ for the "exact" ''SO, If v/" ^ssuno that the error natrix 6 is Herni t i an, thpn 5^ 6^ (170) (171) We can conoute ^ , an'^ it v-/ i 1 1 differ fron the unit natrix / by a snail nuantitv rp : vSi /7^ (172) Usin,"; the wel1-kno'-/n nnv/er-ser ! es oxnansion, we find

PAGE 125

117 -V2 , , , -y. (173) so t'-'at After an;»lyz'nt sonp o"" the error matrices if fron our results and v/oririnpout the relative e-^fort nee'^ed to apniy (17!+) if^ratively or directly, denppHipo; unon the numher o^ terras o'^e^e'^, v/e inn! en^on f^d a sinelo-steo second-order corrrct'on: / § Lii^ r jy^^J (175) This has v/or'e cor-pijte^ 1—atrix fmn ; ts trace. This should h^ t^^ree in our case, and the connutations have alv/ays f^ivon a result di^^f Print -^ron this onlv in

PAGE 126

118 thp sixtpentH '^i'^It. Finally, fron thp pxnansion of the wavef unct i on (1(1(4), one can sbovv that if the tv/o cor^ orhitals have 1 = n. =0, an'^ t'^p valence orbital has 1 '=' m. = 1^, thon there are ^our s oL NSO, four s^ NSD, two oC f!Sn wit*^ 1 = m. = 1,, and tv/o A MSn with 1 = m^ = '3k.5 FivaTi la 1 1 on n^ the ?-"atrix Were vie nepH eirrenvalues and eigenvectors of r' (a''''AV f (d'^'A^^) \1e have actual Iv cor^nuted these for sone o-*^ our v/avef unct t ons, but the results are rather unwieldy. The non-zero e 1 rrenva 1 ues ae;rep vnth those of the 1matrix to fifteen dijrits, as they should by the narlson-''el ler-Schnidt theorem. The exoansion thporer^ (105) allows us to write dov/n the ^orr-' o-*^ the ^'!^^ once we have the ^'SO. The resulting pxoression does not seon to he particularly instructive, so we on i t it here.

PAGE 127

iiq If thp two core orbitrils have 1 = m = 0, and the valence orhital has ^ "" "^i ~ N ' then one can shovi that there are four MSH with f' = 0, ^^ = 1,, four v/ith r< = 1, M, = 1,, tvro with M = M, = 0, anH tv/o v/Ith M^ = 1, r< = 0. Tf^e rpnaininf: fiSn, n^ v/hich there are (2) 12 = ( ) 12 = 5U, hav« zero occun?>t!On nunher and 2 are conputat ional 1 y useless. h.B The ^'yner-^ine Analysis There has been a ,r;ood deal of interest in the quality of t^e hyner'f^inf^ naraneters conputed from approx ina te '.-.•avf^unct i ons . These so far hpve always heen evaluated hy direct calculation of the exoectation values o^ t^p '">fs on^r^^tors over v/avef unct i ons w'^'ich are not e i r:*^nf unct ions of the '-'ami 1 ton i an if the hfs operators are inclu'-'ei^. Since the enerr^y differences between dl^forent h-^s levels in the sane (L,S,J) state are so snail, t'lis first-order perturbation theory approach is certainly nuitr> ^Horuiate for the accur^icy desired. In Table 2, we ,iive thpso as a function of the PnSO ener
PAGE 128

120 part Icul^ir 1y constant *or rii^^pront hasns. Fxanination of Table 2 shov/s thnt we can exnect i t to be consistently i^eterf-i i neH to about t'^ree fi<^ures for S states, "^ut for P, n, an'^ F states, there is* cons i r^ornb 1 e variition. For exannle, t^e best three bases -for Li 2 P ":ivp ener.tries rli-Pferinp; by 0.000002 '*, while thp contact terns rJi-^-Per by about 0.03. The orbital, Hinole, anr' onadriinole nara^^oters, which are not ooint oronprt'f^s, are considerably better deterni neH. For t^ree-el ect r-^n systens, bfs has been 2 exoer inental 1 y resolve'^ only for t^>e Li 2 S /1I|2,143/, 2 ^P /lH!t,l't?/, 3 ^^ /lUf)/ and k -^P /1!;6/ states. Exnerinenta problens so far have nrevonted measur'^f^f^n ts on any of the ions in thp Li seonence, even though t'^n bfs snlittinfTS are considerably ]^rrf^r in the heavier ions. 2 Table 2 shows' that the snlittinjj; in ^'e Vill 2 S s^^ould 2 be two orders o-*^ na^-nitude ?:reatf^r than in Li 12 S. Mowevf^r, so far it has not been nosslhle to produce sufficient concent r--" t: i OPS o-'^ ions to measure a snectrum In a reasonahlo lenf^th o-^ tine. Fxcited state measurements su-f^^^r •'^rom t^^e samp di -^^ i '~u 1 1 i es . The measurements on t^*^ excited states mentioned above were all done by ootical numoin;r the introduction of a rad i o-f rpnunor*/ s'Tnal or a laser beam having the frequency npof^ed to cause excitation anH significant

PAGE 129

121 denoniilat ion of the lov;er states. For the heavier Ions, the lowrr energy level separations becone lar,o:er, an'l currently, no tnnahle lasers arf^ available in the required frequpncy ran.Pie. because of the dinole radiation selnction rules, even-even transitions are ^orbl^den, so t*^at nonulation of excited S or H states requires an S-p excitation, followed by a P-O excitation. PouHlo nunninr has so far allov/ed an accurate determination o* the fine structure of Li I 3 2 n state /162/, but tbe hfs could not be resolved. In addition, all t*^e excited state moasurerents have been with lev<^l -crossinn; snect roscopy . Since this is perfor-^'=>d in a non-zero na'^netic field, t^p zero-field enerry is determlneH by ^ittint the o'^serve'^ levelcrossinr f i o 1 H to t*^e oxpress'on ^or the energy in a na,5^netic f i e 1 '^ doter^^ined fron a treatment of the Zeenan effect. Tonseouent 1 y, the snlittin^ is not directly deternin^d. The snlittinn: determines the constant A in (lit2). For S^^^ states, the cuadrunole ef'ect is zero, and only the contact tern is non-zero, so tbat t^^e contact tern alone is dpternine-^. If ^ach of the hfs 1 in'^s couV bo accurately resolved, then the individual Daranet^rs a^, a^, a^, and a^ could be deternine^, This ^^as not yet been achieve^' ^or any th rop-e 1 er t ron system. However, t^e constants A and 3 in (lUO) denend upon J, \!^i]e tbe 's srp in^enen-'ent of J. Conspouen 1 1 y.

PAGE 130

122 measurcnents on statos v/ith the same L aor^. S values/ but Hlffrront
PAGE 131

123 a A '/£ ^H yz (180) (^d, yz. -/Oa ^.3/z (181) «c, V2 a c.i/z (182) so /I '/Z Iff^ {2^,,^ -I0a,,y^ -a,^J < Similarly, we ^i n-^ 185) ^ i^ljj , SJhl^ rf «.,.^ I*,./, -«,.J n Sii) g(%J= |S^%J (185) /7P/^/J = i^£-Af ^1^^^^^ _ 'f^d.V,^ ~ ^c,yj (185) 3^V./J/ hL'F. % (187) Unless one of t^e a's is nuch smaller th^n tHe r'^st, we cannot det'^rr-i no any of t^^en fron only tv/o nea surenents. f^ron Table 2, w^ ^ipd a c,3/2 •0.0227, 0.0036 for the 3 •-^. State. '.-/e f^erefore conrlu^^o, as HiH Lyons and ^'es^et

PAGE 132

l-^lt l\l)^l , t^at three in^p^ipn^ent n^iranoters ^rt^. necpssary to '^etPT'^ine A ani^ A, , . To Hat^. t^^ero is i nsii'^f I r ? on t 3/2 1/2 exper i'^pp ta 1 ^^ata to dpternino the three. The ahove discuss'on has ne^-'Iecte'^ the OLiaHrimole o'^^^'^pot '^n tfie '^^s f^nlittinT. The quadrunnlp tern, ", In (1U3) is zpro for statps v/i th J = 1/2 or for niirlpi '-/ith I < 1. It rlonpnr's unon the elpctron r! i str i ^iJ t I ^p t^rou"-h a apH itoon f^e purlpar nuaHrtinolp n nomppt n. The naramp>tpr a is oft<^n rr-'rorrf"^ to as t^^e elpctr'c '^IpI'^ f^rar'ipnt hf^rausp thp opprator ip (147) is 5C0S-P-/ 3^^-r^ >r(188) In a one-p 1 Pct ri^p nirture, V = 1/r, rrivipr rise to ap electric ^ i p 1 -i "? = -V V, V-F = -7*7, apH (i:^'>) is Just ope conoopept o'^ t'-'is. Thpro Is po v/ay to rfptPrnipp this fron experinppt, so nuclear cua^rupole mor^ppts cap oply he HetprninP'' ^ron R if a is Hptprri i ppc' t*^f^orp t i ca 1 1 y . n Ip t^e repent tahulation o^ pu clear snips apd mo'^epts hy Pullpr ;^n^ Pohpn I W\ I , \7hirh collpcts data reported prior to vIupp, IHG'', sevep atomic apri nolecular b^a"-" exni-r inpp ts ap'-' opp 1 eve 1 -cross i pfr experinppt ip v/'^ic^ Q values havp bppp dptern i ppi-' ^r^. listp-^. Of thos"^, c^ly on'^' v/as pi^r"'^orned op atoni-^ Li, hy "^ror:, V.'iodpr, ap^ '^ck /I'tF/. These authors renort

PAGE 133

125 a Li^ 2 ^P separation equivalent to B = -0.0075(50) MHz, and from an approximate value obtained from their A value, find Q = -0.03(2) barn. Using our result from Table 5, B = 0.92838Q, we obtain Q = -0.032(21) barn, in accord vnth their result. Isler, .6 Marcus, and Novick have recently measured A for LI and 7 2 7 Li in the 3 P state and also obtained B(Li ) = 3/2 ^ -0.00079(91) MHz, f ron which they found 0(Li ) = -0.0117(123) barn from a value of determined from the A value. Usinf; this with our result in Table 5, v;e 7 • find Q(Li ) = 0.0113(131) barn, a^aln in accord with thei r resul t . These may bo compared with the measurement by V/harton, Gold, and Klemperer /163/ on LiH obtained from a molecular beam electric resonance experiment, which gives the quadrupole splitting directly, coupled with a CI calculation by Kahalas and Mesbet / Ikl / of the electronic contribution. Their reported result Is Q(Li ) = -0.0!}5(5) barn. This result Is probably the most reliablemeasurement available. Evaluation of the quadrupole moment of light nuclei is particularly valuable for checks on the predictions of nuclear model ca 1 cu lat I ons . The hfs parameters A and B/Q computed from our best wavef unct i ons are summarized in Table 5, and the available experimental data is given in Table 7.

PAGE 134

126 For non-S statps, t^o exnrr ?"ir?n ta 1 contact interaction Is obtaineH from a two-narapptor fit of A, v/hic'^ v/e have pointer' out is not rip:orous1y correct. We also have enohasiz'^d t^^t t^e comute'^ contact terns are not exnecte'^ to ^e np rt i cui ar 1 >/ stahin; in soite o^ these objections, t^e atre^^^ent hetv/een our results in Tables 1 and 5 and t^e exner 'n^^n ta 1 results in Tahlo 7 is rena r'ca*^ 1 e . in a^Hitiop, t*^e exner i^^enta 1 results ^or the excito'' stat'^s, pxce-it for thp 2o • 1/ state, vere Heternino^ f^ron lev^ 1 -c ross i n"snprt roscony, v/hicH Is not exnecto'^ to yiel'^ nar t i cii 1 n r 1 y Pirnirv^tP h-Ps pa ranetor s. It . 7 <^onnpir ' son V/'t*^ Othor Met*''ods One o-^ t'^e notives ^or '^oin?^ ti^e PGSO calculations v/as to investirate thr. pf-^ort of t'^e nrooer treatnent n^ t'^e snin snace, hot^ on the enerry ani^ on ato^^ir ^rn'^orties. The e'ffect on thp contact tor'-" is particularly well illustrateri by an SDHl 2 C^O^rr) calculation on Li 2 S rennrto^ by Ladner and ^odda rd /3?/ in w'mc^ thpy varied f^e cop-^^^ i c i on t s t-, and t„ in (3^) suoH t^at thp i /avo^'^nnc t i on '-/ent fron the rP^-"^ case to f-e SE'^r case. They found that the

PAGE 135

127 contact tprn vario^ fron -1.00 to +9.k2; the f.P^^ and SFMF va1u(^s hr,->c''.Pte^ tH^ oxnerinnntal value of 2.00^ w^ile the SOni value was 2.8U6r;. KalHor nn'^ i^arr's /3n/ ohtained 2."'f63 in their Snsrt: calculation, w^ile v/e obtained 2.802 v/ith the sano hasis in our pnso calculation. Because o^ the hasis sensitivity o-F f^e contact tern, it is probably not nean i nr;''^ti 1 to na''e clos<^ conoarison of values obtained fron dif-*^pront bases. In Table 5, we cor?pare cner^^ies and contart terrs •'^or various svstens connuted by various netbo'^s v/it^ the sane bases. ^or the Li 2 '-S state, at lr';5st, it apnpars that tho SOSTP nethod ,'^ives a better contact tern. For other states, the experinental H,Tta is not ^ooi^ enou?:h to 'determine v/l^ich is better (i.e. closer to exoerinent). however, in every case, t'^e ^GSO contact terns are snaller than the SOSCF ones in t'^e sano basis. The PGSO pnersries are better than tho SO?^'^^ ener.f^ios \:\t^ d i -f^^^r rences usually in the fourth to sixt^^ (^eciral places. Ther" is one cc^nnnt that we ourHt to make about t'^'e exnorinnntal contact t^rns for Li 2 ^S p;Iven in Table '7. Tb.e contact terns ^or Li^ and Li differ by about ^.01";. The difference Is nuite real, anH Is called t^n. hfs anonaly. It is dlsciissc^ In some detail by ficColm /lU?^/ aoH is -^ue essentially to four factors. Pirst, the wavefunction at the nucleus is modified by

PAGE 136

128 the differing: distribution o^ nurlnar cbarre in t^n two isotones; this is kno'-zn as the Rre i t-f^osentha 1 nf^ect. Second, t^n nuclpar na'Tnotir>n is contained in a finite volune; t^is 'S calle'^ t'^'^ "^ohr-V/e i sskop-P effect. Third, nuclear notion nolarizes the electron distribution near t^r> nud'^tis; tbis is tt^e "obr n^*^<^ct. f^ourtb, the soecifir rass e'F^ect leads to slitbtly different '-"an i 1 ton i ans '^or the tv.'o isotones, v/bicb j?:ives sli,thtly dif-^erent v/avef unct ions . t* . 8 Mypor-Pine Structur" Resi'lts '•^v Otbor M^t^nris There are a number of surveys of contact term 2 evaluations by di '^'"e ren t net^oris for !_ i 2 S in the literaturr /7 2, 1 5 2, 1 53, 15 U/ ; rat^^er than renoat t'^esp, and since we have already o-iwon some connarisons in Table 5, v/e concentrate mainly on the excite^ state b-^s results. These am sunmarizc^ in Table 9. There n.re sone v^ry accurate calculations on Li in t^^n literatur*^. l/eiss /no, 153/ t-as rpnorto'-" CI calculations on t^^e 2 ""S anH 2 2 P states, and tbere are ^-^ylleraas calculations on the 2 ^S state by Larsson /S"/ F\n^ on t^^e 3 an-^ h ^S states by Perkins /S^f . ''atural an?>lyses of t^^esn results have also boon recently nublisbe<^ for t^^e 2 "S state /lh,l^.^/

PAGE 137

129 nnH thp2 ^f st^tr> /15n,l'^^/. Unfortunately, no h^s analysis was rarrif^H cit on t'^e P states. A contact tern, f 2.901, fro'^ tHe ^irst three s-tyne MSO, and f = 2.915 fron t^e ^11 n set o^s + n + H+ f MSO v/as o'-'tal ppH fron Larsson's 2 S calculation, renarl'ahly close to the exner Irnental value 2.91P p;iven in Tahle 7. Larsson's calculation is notahln In that both snin functions v/ere inrluHprt, so thp results s*-iou1H he ouite reliable. Pert'Jns' excite^ state calculations are not sninprojecte'^, hut until b^s calculations are Hone for these, we cannot na''.e as assessment o^ the imoortance of this. This r-.'oulH he interestingto Ho, because sone of tbe Hisr.ussion in t'^i^ literature has centered around the nuestion o-p \/hether it is tho nron^r treatment of the soin-snaop or t^-^e i norove^on t of t^^'= '^^ner^y that is more innortant for the b-^s. Our feel inr^ is that the PnSO and SOoi^t^ ra 1 cul ^-t ' ons have shov.'n tb^t the nroner trf~-a tnent o-^ t'^e snin so^^c" is the n-iore innortant. Little 'for'' has been done beyond the f^irst tv/o S and p statnr, o^ Li. i^oddard /ISf/ has briefly reported Rwr^, LIMf^, r^ni^t^, and S'^i-'^^ calculations of the 3 "^ state, and Poo'-, 7^n^ Ali /l^l/ have rr^^ortP'^ mdut o ener.":ies -^or th" 3, k, an-^ 5 "0 states of the lithiun sequence. Our wor!-. on the F states is the first to our knov/1 ed,q:e .

PAGE 138

AppFMnix 1 VALM^s OP SO'^'" PWYSICAL C^^'STAMTS The follo\7inr riata is ta'''^n fron tbo most recent conpM^tion of Cohen an^ riuMond /132/. The rJi^its in narnnthosis reorpsent t^n stapH?5rH Hoviation error in t'^e '"inal ^irrits. h = 6.62559(16) x lo"^^ er? spc -27 ^ = 1 .05t+'inU(25) X 10 er^ sec e = 4.80298(7) x lo"'^'^ esu n = 9.10008(13) x 10 j^n = 5.l|85 97(3) x 10 anu 2.907025(1) x lo""-"-^ cn/spc C< = 7.29720(3) X 10 -3 13 7.03^8(6) 130 -

PAGE 139

131 H = 2in(j7i+.62(2) K 5.2nir,7(2) X lO"'^ c p rt = 2.002 200 716 (electron p;--Partor) Table 1 reives ennr.fry convprslon factorr, h^spd on this H,Tta. It is also convonient to have atomic masses for conn.jta t ions . The ^ollo'-'indata Is ta'
PAGE 140

132 LI^ 7.ni6 00t» 8 Re^ 9.012 182 8(5) Rp'^^ 10.012 93^ 5(k) .11 11.009 3f^5 33(30) 12 12. (exact) M Ik 1U.003 07'j !jn(i3) 16 15.09U 915 02 .IH 18.000 937 0(10) 19 18.998 4 Oil 6(7) 20 Me 19.992 li'jO SCO Fullor an^ Cohen /13';/ rrnort thn follov/in?^ nuclear naf^nptic '^ioo1r« r.ononts, /^ / ( '^ ^ nnclpar rar'notons ) / anH nuclear electric nuadrun^ile nononts, '^ , (In hrsr^s).

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133 I sot'-^oe Snip, I Li +0.82202 -0.0008 Li 3/2 +3.256ff -0.0/j 3/2 -1.1776 +0.05 .10 +1.8O07 +0.08 11 3/2 +2.6885 +O.Olt 13 1/2 +0.7024 ,,lk + 0.1(036 +0.01 ,15 1/2 -0.2S31 17 5/2 -1.3037 -0.026 .10 1/2 +2.6 288 Ne 21 3/2 -0.6618 +0.09

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13^ 12 IG 20 22 The Isotones C , , Me , an<^ Mp h^vo zrro snip, Hlnole/ and ou^dnioole monents. Fuller and Cohen caution that t^e oua'^r'jnole monents nay he unrcliahle Hue to "adj US tnen ts" ^or nol ar i 7.a t Ion effects, Sternhoinpr s'^ieldTp"', etc.

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jMF rnMoiiTpn pRonf'A'^9 The r;' 1 cu U-1 1 'on<; vjrrp carripri out nr> an IBM 360/65 ronniitpr, in Hou^1r> nrociston (53 s i <^n i -^ i c-nn t hit rantissa, n^^out 16 Hpcinal firuros). Thp nrnrrrans v;ore v/rittpn in r'L/l, t^D^yAM, anr' ^ssenbler 1anp-ua^^'^ .lAC'^f'l, an Assrnhior 1ann dote rn i nan ts, reniiircd ^or the construrt'on o^ t^o 1^nr\ 2-'^atrires, v/ere evaluated in an A5sem^->1er lan^-iinro rotitino because o^ the f^reat increase in e^^iriency o'^tained. The atonic intp"^ra1s 135 -

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136 were TenernteH hy a j^eneral atonic inteirral nackaj^e which v/p intend to suhn't tn nrPF. A few small routines for tinln'^/ siippressTon of underflows, and interfacing i^nRT"/^/' with PL/1 were written in Assenhler 1anruao;e. The rerain'-'er o^ t^e pro.piram v/as written in PL/1 ^nr a nu'-'^er o^ reasons. The dynanic storaj^e allocation -"^ac i 1 i t i f^s nade nossihle ohtaininp; Pirray stora'^e and thpp '^reeincr it when it v/as no lon^^er needed. This '/^s i^nort-int hecanse of the lar,p-e arnoimt of intetral stora^^e r^ntiire'-' we needed ^our di •'"f^err^nt two-electron natric'-'S in core at the sane tine for the construction o-^ n-'tr'x elr.nents. It also allowed us to use only the amount o-^^ core required for the particular hasis set, result'n'^ in a reHnction in cost and turnaround tine. PL/l's inout/outnut facilities are nuch nore -^lexihln t^^an FORT'^AM's, and allov/ed oa.^e headinp:s, free-'^orn innut, and editing of numerical output -^or readability. T^^e tinp o^r iteration for 'I hasis functions is ropresente'^ renark-ible accurately hy the forntjla tC^') = 1.0!j1 p^-236M spf,^nH5; For exanple, t(6) = h .3 seconds, t(lO) = 11. n seconds, and td'j) = 2^.5 secoP'^s.

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137 Starting: fron thr> Snsrp results, vie could usually obtain convergence In the enereiy to six fipiures in about twenty interations, altbou,":b sone cases 2 require^ nor^. Por tbp seven-function ^'e VIM S basis obtained by ext ranolat ion of tbp orbital exnonents for the Li i, ^e II, an^ ^ III bases, the enerp-y v/as still chanorinf? in t^e s^^vent^ *^i"-ijre after ninety iterations. 2 The seven-function SOS^n basis for Li I 2 S tool-, nore than two hundred iterations starting fron t^^e SOS'^F orbitals to conver'^p t*^ t«*n fi"-ures in the energy, and even t*^en, t*^e orbitals had converrxed to no more than four •Pi^'ur'^s. ^Jow^ver, onc^ \-'e bad a rood dpscrintion of the coro orbitals, t^e excited state calculations usually conver^e^ in about ton iterations. TKp l-f-i'-? t r i X, char '^e-dpns i ty 1-natrix, and soin-density l-r'atrix could h" constructed, thpir e I frennrob 1 e"^s solvo'^, and the one-e 1 p-ct ron pronprties --^eternine^ in about 5.9 seconc's. Tbp 2-ratrix, with 8712 elenents, could be constructed in 3.0 seconds, but the trans'^orr^ t i on an^ e i p-^nsol ut i on '^or fourteen roots reouireH 'i secoo'^s. V/hen synnotry blorlriptr v/as int^oduce'^, t^^ calculation too'r hh sf^conds, desoite t^^e Sf->allor blor'-s; t'^is is due to t^e rather connlicate'^ inr'oxin'^ nor"^r>H to extract th'^ blocks fron thp '^ul 1 r>r>t r * X .

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138 The connlptp prop:ram is about 3500 cards lonfT, o^ v.'^^ich ahrmt a t^Ird Is comnpnt rards. For connarlson, t*^*^ connlpt" Koiiha-Hhrn diatonic C! prorrran written in POR"!"f?AM |s about UOOO text cards ionft. Hc^dard's rrn<^r^} S0<^| nrorran (arbitrary nunber of elprtrops an^ cr'Pters) Is r^^nort'^d /13n/ to bp about 55,000 PO^^Tr^.A'i cards lon^!

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ApDF^!n|v 3 rOf'VPMTIOMS COD c^ourn^r^i uadmomipj; ;^^,n spf'F'^irAL TENSORS The so^nriral harnnnJcs tb?it wo uso are those r^e^inp'' hy Shoro an^ '^enzel (Ref. 1, p. 151, en. 9.27; note that a cor^olex copju^af^ s ' rn is nlssin^r^ in t^-e i r enuat ion ) . y,. = ('lyf) (A3.1) yyi < (A3. 2) These ^onov/ tHe stan-i^nr-' Pondon anH Shortley /118/ nhase ropvpntion. They satisfy the rolations X. (f t. (A3. 3) 139 -

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Kin <%j%...> ijy^r. y,„s.n9c/ej^ ^W Om-yr<' (A3.I;) 7 f y ^-^ 'im 'Xvn YV1 (A3. 5) .Many aut'^ors on? t t^p hn factors in (A3.1) and (A3. 2), so that tHp intp^ral (A3.'j) is 4% 6f g< dyy,rr,' il£' ^y^ This factor annoars an-^ H i sa'->"par5, anH sinc*^ authors seWon ^<^-*^Ine t^oir nor'^a 1 i za t ion convr-nt Ions, one mist he 'Xtr'^p'-'l" c^r"ful in talcinr: results -^ron t' 1 i terattire . Per pvaltiatiop of Intor;rals, it is copvoniont to v.'ork with thr> "aca^ snhprical tensors /I/ HofipoH hy i ZZ^l X lyYi (.Al.6) v/b i ch sa t i sf y

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lul (A3. 7) 12.)"^ r(!>^ Z C C yn *yi ^^ y*i (A3. 8) !n terns of these, the Gaunt coefficients, or Condon and Shortley coefficients, (see, e.g. Slater, ref. 7, p. 232; Slater's spherical harmonics are actually the sane as ours, although his definitions look different) are (k) c'Um-i'>.')
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APPEMDIX h ON THE CUSP cor.'DiTions The exact solution to the Schr9dinger equation has (Ati.l) constant everywhere. However, the atonic Haniltonian, //-^zrz z t>j J (A4.2) has s i nsul ar i t i es as r. --> and as r.. --> 0. The wavefunct ' 'J must therefore have certain properties at these singular points. These have been derived by Kato /9h/ in a rather complicated mathematical paper. However, v/e will give a simplified derivation here of the singularity at r. --> 0, and then give Kato's general f ormu 1 ae. li>2

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1U3 We inarine elpctrnn 1 boinp: closp to the nucleus, anH t^^e ot'ier N-1 electrons far repoveH. The major contriHution to t "^e enerp-y then cones fron the anoroxinate '^'ani 1 ton ian ^, ^r," ' cY, (Alt. 3) 7 sinco r is nearly zero. The ennr<:!;y is constant, so The exact e i "-en-^unct ion o* the "aniltonian (AI4.3) may he v/r i t tpn where ^^ is a nroHuct o^ a nolyno'-'ial and pn exnonential. '.'.'e finH H,ip I U.'Jr, j^ ^ '' 4 ^K dji J (AU.5) and from <.-/^ich r%asi1y -^ollov/s

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Ikk M (ill) a' _ ^u^^ji^7 /// II ^ ' If ^^i ^ ^ ^/ rr r/ (A!+.8) ^*u 1 1 i nl y i n": ^>y r , ta!:!ne: the 1 inT t as r --> 0, and using the ^act that r^f^'' anH r*^f^''' then vanish, v/e find .?ri ^ ^ r-^iS? (AU.9) or Ju/n -J X l^ 1 (Aij.lO) This is t'le cusn co'^'^ition v/^lrh is stated, hut nnt nrove'^, hy Root^^ian an'^ Kellv /HS/. The case o^ nost interest is ^'or t'^e exoansion <^^ ^n orhltal in terns of Slater-type or'^itals; t^en f. nay he v/ritten iZA -''-'" z f e ^i'^ r. (rtr-^'O (A(+.ll) frnn '/hich c.asily follows

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litS (1) (At+.i2: Z c-'A If one cHnosos t^e> h^s'\z STO's such that thppe is only one v/It^ n = 1 + 1, nnnp with n = 1 + 2, and any nunhpr wif^ n > 1 + 2, t'-nn t*^is ro^ucns to the sirnlo rosult ^/ . or < C^/V ^*/ /«; cry^c = -C (AU.13) which ^Ixos thr^x. orhit^l oxoonont. This choicf^ has been na^fSy nanv aut^^ors, but ohvioijsly, it is not the only v-ay to satisfy V-^pr-isn ron^ition (A(|.10). Ope ar,-unent ^or sti^K ^ r^o'ro is t^^^^t it Pnahles natchinp; at Ipast one nro-^ertv of thp annroxirate wave^iinr t i on wit*^ t*^e px^rt '/avp^unct ion . Another is that satisfaction o^ tho c-isn condition at t'^e nuoipus should be o^ 'noort^nco ^or tHo F^rni contact torn, which Honen^s uoon ti^e hoh^viour o^ t^o v/ave^unct ion at the nucleus. He d'srute both those ar-u-^ents for a nunSor of roa<;ons

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1U6 Pirst, t^p r.hn'\r.p loadinoto (Aft. 13) Ip^ds onr> to r^nrpsrrit a Is orHit?>1 by n sinrlp Is ST'^, no 2s, and any 3s or hi,rr:y, one nust descri'^p' then ODtirally ^or f^oo^ tot^i enerr^y results, and t'^is requires nore t^^n a s'n"-le Is ST^ in tl^e basis. jKir'^, f-o Marniitontan (^h.2) is ohysically not exact, na r t i cii la r 1 y at t^e rejrions of these cusos, since the nucl^^us an'^ elr^ctrons are really finite nartlclos. In -^act, v/'t'^ a uni'^or'-'ly c'-'^rred ball as the pod"l a nuclei's, it is a classical electrostatics nroNlen to s'^o\.' tb^t t^*^ notential inside t^^o ball is pronnrtion^l to r" rat'^^er tb^n to 1/r, wbicf^ v/ould lea'^ to "" i ' / -^ ^ = in (^h.^). Pinall'', our results inHi^-nte t^^at the corr'^ct tr^atn^nt o-^ snin via SO^'^^ or PC?'^ is "-luch morp Innortant t'^an t^p cusn con^'itlon In o^^taininp; rpasopabln values o'' t'^o '^erni cont-ict ter^. Kato's "-eneral cusp conriitions pay bp written

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1U7 = Zh^(r=o) (Alf.l^) (AU.15) where the horizontal bar indicates that the spherical averaf^e of the v/avef unct ion with fixed r. or fixed r.. is taken before the differentiation. The correct behaviour as electrons approach each other, (Afi.15), is actually more difficult to ensure, and this difficulty is partly responsible for the slov/ convenience of CI expans ions .

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ApDFf'^IX 5 SOLUTIO" nn TWF '"MRIa SC'J^'^ni Mnrn E'^nATinn^ up = SCF If S > 0, v/e can use t^rt Tholpsky Hpconnos i t ion /73/ of S into a nrodiict of a lower trianfTU Tar r=itrix and its aHloint: S i L (A5.1) Tho rfpro^^oos i t ion is sta'^ip, an^ so is t^o inversion o^ L. If S is not oositivp de'^initp, t^is is easily HetecteH Hurip"f^e cor^utat'on as an attennt to ta'':e the squar'^ root o-^ a nerative niinKpr. If t^p deronnos i t ion Is surcess^'^ul , the inverse matrix is calculate'-^ anH one oStains (L-'HL-"jfrt) = itOE (A5.2) v/hic'i has t^e stan'^n^^' -f^orn IkS -

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lun /lU ^ UA (A5.3) v;hic^ one c^n snlv^ w't^ existln-^ proo-r^ns which inolpnpnt t^p JacnM, Hivr-ns, or '-"oiisphnl Hn r rrthods /73/. If S IS not nos'tn"^ Hn-PInite, the rholf'sky dpconoos i t i on '^ii'^t ^p ahan'-'onn^, anH v;g t^en use iHwdin's ranoniral or t^-^nrirmp 1 i za t i on /06/ v/hich nrocpo'^s as •^ollo-fs. I'o first solve 50 U^ (A5.U) aP'^ t'^en '^israr'^ all t'^-^ n i "-onva 1 ti'^s v.'h i rh aro pf^^'ative or KpIo'-.' sono tolorancf^ and ohtai'^ rp'^ijc^d r'atrims, denoterl by a horizontal bar. S = A'^^U SUA" V2 (A5.5) f'ov.'. /^ = <^\Hl4^> J (A5.6) so if v;e let

PAGE 158

isn / = 4>ur'\ (A5.7) then ^-Vi H - -X'lAlrlWl 1-Vi (A5.8) '7f> tHen solvr> t^p rp'^'.'rp'^ nrn^lpn. W C -C E v/hich f^ives a s«^t O''^ solutions ^ -li -•k\^^ C for t*^p ori'^inpl nroSlp"^. E I ^pnv^ 1 'IPS c^n hp r'ptprninoH only to soj^o aI-)so1iJte arriir^rv. If wo h^vp for oxar^plo, an elfronvalup o''^ 1^ , in Hom^Ip nrorision on our n.^rhine/ it ^vT s no norp t^'^n t'.-'o correct '^i";urps; on ta'cin"tHo square root, v.'p intm^Licp unco r ta i p t ' p? into thn ei.Thth olaro. It is t^nrpforp i^^nortant to discard not only thn pprativp o "pnva 1 ups, but also s-^^a 1 1 positive e i crenva 1 UPS . In nractice, wp discar'^ all eirenvaluos Tess than lO" . This affects our rpsults at worst in the nlpvpnt'^ ^*<^urp, an-^ t'^p" arc usually sc^ov/hat

PAGE 159

151 bettor than this. l/r> c'^ock our results by coroaring the sun of t^<^. pi»envaln*^s v/it^ the trace of the oriTinal natr'x; t'^'^se sel^on d]^^r>r by nore th?»n one unit in t^*^ ^ i '^t^'^nt "^ nlac. ''/e also occasionally conoute HC SC^: a^'^ter t^e canonical orthonornal i ^a t i on has been uso'^ to solvo t^^e oririnal nrohlem; the norn /7?/ of t^^e resultant n^ t r i x is tynically of t^^e order -12 ID or less, so v/e •^eel cop-^id'^nt t^^at our results are nunerically cnrrr^rx. to at least tv/elve firures.

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APpFMHix 5 LI <^T oc ^nnDt:\J\/\J]n^\f; R rnMr; CI ni nsn M MF Mz 1 P'1 K Bohr; atonic unit of lon^-t^ "nth'^-^o 1 (^s tonp norttirb^ t ion thnory cha r TP-r'pns i ty natural T^ninal; an p i mnfunct i op of t^r^ sn'nlnss 2-r'atrix charr^p-'-'ens i ty natural orbital; an e i epnf unc t i on o^ con-^^i r'Jra t ion intf^rartion pxtpn'-'Tl u^jrtreo-Foc^' variant o' '^nHrl?i rH ' s HI; pouivalont tn SF"C GoHdnrd's not^^ori^ essentially t^o usp o^ the \-t^ snin '^iinrtion •'^ron tHe synnetrir nrouo variant o-^ HoHri,-. rH ' s f^ I ; eniiival«^nt to Mpnr ren^ral snin or*^ital ^-'artr^'^; atonic unit n^ energy ^artre"-Fock Hert/'; cv'~l o/s^con^ irr^^'^'^n^^nt particle noi^el Kayser; en 152

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153 MPWF MG MO f'SG MSO PGsn nuc SC^ SHMO Scur max Iral 1 y-m i rpH ^^ar t rpf^-Fock ; the use o'F a snip ^unrt'^n havinp; na I re^^ s'^ins in natural '^eninal; an e i nen^unct I on of the sninless 2-^-iat'-ix; a C^^'f^ nati!r?i1 or'-'ltal; an e i t^en-^unct inn of the sninle<;s 2-'^">trix; nom nrecise1\', a C^^'O natural <;n*n o-p'^'.n^]^ an c t in t'""^ radial functions projeote^ general snin orhital orojortpd 'Jn r t ree-Forh ms t r ' otori '^n r t renFork snif-ronc;'';tont Fiel^ sni n-'-'ons ' tv natural orhital; an e i Tenf unr t i on of snTn-ext^n-^'^^' '^artrop-f^oo'-. ; t^o ijso o''^ snin-oroiort'on ho'^or'^ orhital variation in t'^n '^a r t roe-'^or': schene sni n-o^t 'r-ii 70^ r: I ; S'^'^'^f" Snin-ontir^i^or' sol'^-COPSiStent field; t^r> use o^ an an t i s v'~"~'o t r ' zod nroduct of M or'^it^ls an'^ a linear rorhinntion o-*^ nil

PAGE 162

15Ji spun S PMF 7. STO IJMF nossi'^l'^snin^unrt'rins sn I n-oo'' ^ r i 7*"^ ^'a r t rr«^-Por'', ; t^o usp of c!l p^^^r-^>Pt orSitals for di*'-^oront snins; POns S proirrtion of '"^ f^SO dpto rn i nan t , v/i t'^ no z 2 S nro ' '^r t ' on S la t'^r-*:vnp. orbital unr^s t r i rtr>H "ar t rpo-Foo'^ ; tHp usp of Hlf-Perent 1 valu'^s in thn exnansion o-P t*no or^itals, and at f^e sann tine allov/ipir dlf-f^ront orhitals for di-p^crpnt snins

PAGE 163

APPENDIX 7 SOME THEOREMS ON OIRFCT PRODUCT MATRICES The direct product of tv/o matrices is defined by (A X R)., . , = A..B, , (A7.1) Note the particular order of the subscripts. The direct product r.atrix has rov/s indexed by the pairs (i,k) and colunns indexed by the pairs (j,1). In Chapter 2^ we referred to some theorems which v/e here prove. Theorem 1 is piiven by V/i^ner /93/, but all have been worked out independently. They are. undoubtedly known in mathematics, although v;e have not been able to find any discussion beyond VH, oner's. Theorem 1: (A x R)(C x D) (AC X BD) (A7.2) Proof : (A X B) ij ;ki A B Ik j 1 C,._D (C X D), . v., ,., Kl ;mn km 1 n ((A X B)(C X n)).. =2 (Ax R).. ,,(C X D) ij;mn iji ij;kl kl;mn = y A.,B.,C._D. ^^ I K J I Km I n 155 -

PAGE 164

15G ( 2 A C )( 2 B n ) K i k kn i j 1 1 n (AC). (RH). im J n (AC X RP) I J ;mn Theorem 2: If AR = CO and EF = GH, then (A X F)(R X F) = (C X n)(^ X H) Proof: From Theoron 1, (A X F)(R X F) = (AR x EF) = (Cn X HH) = (C x G)(n X H) Theorem 3: If r is a rational nunber, then r (A7.3) (A, X AX ... x A.,) v., 12 M 1 r r r (A X A ^ X ... A ) ( A 7 . 4 ) orovidod that all the pov/ers on thp rirht ex i St . ' Proof: If p is any positive intej^er^ then from Theorem 1, (A" X n^) (AA''"^ X RR''"^) (A X R)(A^"X B^~^) (A X R ) ( A X B ) ... (A X P ) = (A X R) \7e then ohsorve that (A"^ X R"^)(A X B) = (A7.5) (a"-^A X r'-'-B)

PAGE 165

157 = I X I = I (A7.6) where I is a unit natrix of appropriate dinension, so that , -1 -1 -1 (A ^ X B ^) = (A X B) V/e then replace A by A~ and B by B~ , and find that (A7.5) holds for negative inte.fter p. Sinilarly, if q is a positive inte;^er, then A X B = (A A ... A ) X (B^ ' 1/c ...B ) (A 1/q 1/n X B )((A 1/n 1/n . . . A ' ) X (B^/^ ... B^/^)) or (A X B)^/^ = A^/^ X B^^' (A7.7) As before, the result is easily obtained for q negative. Combining (A7.5) and (A 7. 7), we find that (A X B)^/^ =' (A^/" X b'^''") (A7.8) and finally, by successive substitutions for B, v/e obtain (A7.rO, for r = p/q, a rational numbe r . Theorem k: (A x R)' = (A^ x B^ ) (A7.9)

PAGE 166

158 Proof: {(A X B)' ) ij;ki (A X R) kl;ij k I 1 J (aM Ik (r'") (a'*' X r'') J1 fj;k1

PAGE 167

159 4-1 (U tc

PAGE 168

160 i-H C I o E (J CM 1-^

PAGE 169

161 -

PAGE 170

162 err art o o C C o u CI o< in Jt-l 00 tr cc o ct^ C3 i-H cKl

PAGE 171

153 c I oo

PAGE 172

164 X c C C s < I-

PAGE 173

165 c c c o V <

PAGE 174

166 c 3 C c c u CM < «M O rc c~ en cr. co rH t-H CnI r-rH C C C~ C C c c(r cC-. en C-1 ccccC r-1 I C I

PAGE 175

167 c o C C c u ccccU3 rH

PAGE 176

168 •c o a c c o u CM < l-l

PAGE 177

-169 •c o CI co o cCI c er: c~ o en co

PAGE 178

170 -

PAGE 179

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172 •-40000000 lAOOOOOOO tOCOOOOOCO • l-l I t III ocot-iCT'C: coinoooooo O r-*> oo r* cr O CO LA lh r*. ' O o r* c* in I cr C3 o o o o o L.-1 O O O O C3 o c: o c c o o o or^.jr**u^oooooooooo 0r^t-«03^l^-ti I I I I C I lOtrtCMoorjevjutooo ooo LOfAcMf-tr^r^^esrOOO ooo mcoa^K\csjooK%ooo ooo OOCSU^CNiOCOr^OOCD cdo^ r*Cn00C0U-\OOC3OOOOC3 CMi^P^OiOl^rHOOOOOO O I i CM -( I I O I CMJOC^OCOO KNCDOC^OOOO CSI00C30000 t-HlAOOOOOC r^f-toocooo eoiooc'oooo Or>tHOOOOOO III III en 5 trtr*(X>»^.:?CN*-*000000 ^^r^Ic^b•^^'^^~lOc:>oocc^o inminiDunocoooooo^ ^s,T-.^^lf-^C•CO."^000000 or^^cr^oooc'sCJOoooo oc^r^r^c:r-*roooocoo OOLnincoomoooooo O i-H I K\ CM r-l I I O I I J-Ors-.-IJ-OOOOOOOC? K\LAi-H.-iC^inoooo ooo mj-rofoj-ooi-tc^ooooo r*.oor^or^crio ooo ooo oc^Jcsr>..r-^fO cooo ooo o-3•^a-:rK^o^*ooo ooiO Of-400C)mOOOOOOC3 O I 1-H I I I o C C (J LlJ -J < iAr».oooooo c^-s-oc^oooc; CNI^OC)0000 occoc^oooo lOLTvcococrc: r^occDoooo ooeooooooo I »-H ooOf~*r*.«cci-H ocjincT'jTC^crtr^ oorjomcocnto oocci^cccor^cc c^ o o ij:> L-v ^^ cr 1— I C5 o en Cs! en r J LTi o C30000000 I I I I I < r) <_> c c. < to -J < c c: K^oooo oocMLp. or^tnto r^COOCCJCCSCMCTCILACrv CTCOCOOCcocnr^tooco .:r o irio csjor*cnooc:0)00 cocNcnr^mcr OOOCDOOOCOC^r^COOOO jroocoooLAOuncM^coocooooc^o.:rc^f-^c:: COOOCJC^OfjO^-r-Ot^C: OOOOOOOi— (Oi-HOOO O II I I I I o .d-otcmc^K\f-iooo ooc; o-r-rAu-icr. r^o coc: ooo i-imt-tuai-aou^ooc ooo cmor^csii-tcM ooo ooo 0(.c^in»-4r*-.r-ocoooo Of-(or^i-r>L-ir^ooo ooo oco.i'or^trcooococo ocsjt-ccc-a-ii-Hi I I I • o I >o o rj cTk o c e^.r-COcccvicnr-LAo oooocrif^ooco OOLAiOLncM-r-CTj OOOO JC"^-^0 (X;00)CCf^t^O»-( OOOOOO O I I C-J r-l I I oocococc o^oocooc li-»f^COOOtOO OC00C)OC:OO r-< Cn: co o c o c r-b"\ooooc5C 0^•^L:;oc::.ooo^Ooc>oo r:coc^^^ocM^/^c; ooooo tn00.=rrHCNJC0OOOOOOO r^rHf— lOK-evicrc-c oooo in oOi-KOf^ooooc oo i-HOOOCOOOCOCOO ^OOOOOC^f-Hf-HCr^O COOOOC>OOC^— U~. inrHh^ f*^coooc o^^^^f'^u^^*•M^ cnoooooo^r^.rl.'^-^ln•-l 100000:00 r^OL"CC.-r-'%ooooo>o r^ o oj fH c en I o oooo oooo 00)000000000 oo O I I I 1 I I I mor-^ sc oo*— tfNj •-(Cs;fA-:riAOr^ccoOf-trj t-ic-jrA.:ftnir;r>.ccoo»-^cN

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173 ooooooc ooo I I + + + + 1 I LUUJUJLJUjLlJuJUJ— uj 3 C C o u I+ + + + + +I •Z2 UJ ._!.. ...... o • r) »• < • >^ ter a » lij « * • . . ^ y .^ ••••UJU^ fc C tJ O < to lO I^ HU. L^ LI. (/) {_ /N J—I I— /N /\ ^N f-t * — t• CT — >rccr UJ :i c:r a UJ UJ c^ + + c^ o c to c /^ r _ I 1— i;-. rC — I < o < •• UJ
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17ti CMOi-HOOOOr^OOOO ooooo^oocaooo I I I lACOCNJf-Hf-tH^C3C~:N"\COr-400 CMOCJOO.— IC>CMC:.COO oooooooooooo III II c c c c u 02 < Uf O C CM o — CCM CO < cocMCMO»MCsjr-*c~rsicr i-hjt f-(OOCOOOC>JC1.000 ooooooooc^ooo I I I I I I I K\rOj00t-ICMKtC3fACCgCT' coococMmcviorAr-o udcm cMooc:c:;dCrsioooo OOOOOCJOOOOOO I I III! lArHr^CCCOJ-rAiJDCDCJr* r-. CM ^'^ rsi u-\ C" t-t ^"^ i-H CN I— t c>j lA o r>; c o i-H c m cr o cs; cr OOOCDOOOOOOCCJ oooooocooooo I i 1 I I t I I I I mrHOOC. tn-s-r-^LDccrACNi r-^c^J^-^^^^LJ"^^— tccoocc c-r-cn .rr r-< ^L'^ c: CNj r-"^ rj c: cc _r e:; COOCf-iCOOCOr'-. OK% ooccocooco oc I 1 I I I I I t I t I CO Ll02 CO (/)0•-1K\J«WLr\c^lC:)f-^K^rA^*•cs^ oo^oooooooo»-LA.3-COK>J-0 toror^j-Of-HCNiL-^t-Hp^cOf-HLa f-IOOOOOOOOOCDOO O I 1 • I I I I o .Ln. »o CM O f*~l JCO CM CO O O f^ C3 r^ O W\ t<\ iS> t^ O i-l CM K\ O III tr: o K> CO C3 ID D. o CO f-« o r*H-

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175 .:r»-HciooOoo.:yK>ooci^HtDOO 03 CnO^U^CTJ'lDN^Csir«*.K^K\03 K\LAeOOOOC3f-4 I I Or-HLnoo«-tp^coomi-DLr»j' U^CNOOOOCK^fHOCNJCO K\00»-H000000r^0r-« O I It II o K% fH I K> Csi r^ • I 00 ,r•u^^-^ or*»r*incviOj*p^OLnpor*-o oc^OLr\C3cvjLrvCMK-. ooooro otr. f^f-iocNioocoootn.*mcQ O III -a C c^lAtnL'^K^cocl-l in LTi r^ CO c~i '^ ^^ en CM CM O ^'^ CO .^ C~i CO cr*oc~»^ioincncNi »H
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176 -a c c o o OiHO.:rcsjf-«OCNJO^H oooooocooo ••+ + + + t lit II • til III 111 til ] I I M n 1 til CslfHt— («f-«C0^r'4OO OC^-^OOfOm-T'tXtOtA OJ-OOCOOi— too.-H •••• + + VI • r) ut • _J. ••••••• o •3 Z •< •>S »er• "— UJ • • • • * ^ ^ • 2 .....UJUJ U.' t1-1cr o o c <. (r> ui H r (^ Llu< CC • • • • ^ U_ CO O /^ J*J I— /N ^\ /N i-H * c -:* u". r-*! K^ i-"\ O O mu^ooc^^"\.r-:rou'^oo t^r^Of-tt-icJ-a-iAr^oo ,^ ^ c; ^ L-t c^ c^j CJ -5c o .:J^Lr\^^cJ.;^co.r•oo I I+ + + + + + + + + •UJ 3 _J U r3 t; <: t~ < >•IV — H • CO — V E CT o UJ :r: C C UJ UJ c 2 =: UJ — :: CO ^ >e !.'> c /N /^ . csi 1 C UJ i— u-i rA I < CJ « « • « U-' < hUJ * * ,-\ * 2 <_) li' _i cc rrc: UJ to CO Ul V V V -^ : c c

PAGE 185

177 w (/)

PAGE 186

178 o o o o o c o o o o o c c o o o c c c c o c c c c c o u < o o o o c o c; o o o o c o o o o o o o o o c c c c I I in (/) CNI

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179 o o o c o c c o c c o o o o o c c o o o c o c o o o o o c C3 o o o o c o c o o o o o c o o o o o o o o o o o o o c c o c; c c o u JUi ec < »lO

PAGE 188

180 D C •—

PAGE 189

181 CO (S o in U3 C3 to o I a: C3 c 3 C C3 I c 8 cs < CI C9 CM CC O o CM c en ID I o CO O to to ec cn: to |0. c o I c I c I I to CM »o

PAGE 190

182 en o lA O <71 3 C < C-. o C

PAGE 191

183 C C o u in liJ < »CM CO -303 C-4 i-< o r> u. o 2 c 3 C c 3 C c 3 o C4 en r^ in C u c v E c •c a: o u c i. m 3 in

PAGE 192

ISfi 0) c a. > o c o to > 0} x:

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185 1. o ec < E I. 0} 4J V «c 4-1 c c o E V. 0) c N X V. o »-' c E ro irc C. c 1. 3 .;-) O r? L. 4-1 Vl Q) C U (U c re o c L. or c

PAGE 194

185 TABLE 8 A and B hfs pararp^ters for several states of atomic lithlun. The values are calculated from the physical constants ?;ivon in Appendix 1 and the results of the best (i.e. lowest ener.^y) bases in Table 2. The electron ^ factor is taken to be exactly 2, and the mass-dependence of the Bohr unit is ir,nored; this should not make any difference since the values were computed in. double precision (16 digits) and rounded to four fi<^ures for the table. State Li ,6 A(M'^z) LI .7 B/0 (riMz/harn) Li Li 2

PAGE 195

187 TABLE 8 (continued) 5/2 7/2 0.059 51 0.157 2 0.002 021 0.000 673 6 0.O07 635 0.O20 16 0.001 123 0.000 374 2

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188 TABLE 9 Sunmary of hfs parameters for Li calculated from v/avef unct ions obtained by various nethods. The physical constants in Appendix 1 are used to derive the conversion factor 2 ArA^/',/! = 207.128 MHz. The electron g, factor is taken to be exactly 2 in our PGSn results, but a different value may have been used in the others; this v,/oul'^ Introduce a difference of only 0.01% in the quoted values. The mass dependence of the Rohr unit has been i,<^nored. The PGSO values in the table v/ere obtained in this v/ork. State Parameter Value (f'lHz) Method Reference 2 2 S 1/2 289.3 2 S '} . n 578.02 379.02 390.00 323.7 RHF 33 RMF lfi9 RHF + perturbation l'i9 theory RHF + per turbc-'t ion 149 theory + relativistic effects UHF 150 PUHF 155

PAGE 197

189 TABLE 9 (continued)

PAGE 198

190 TABLE 9 (continued)

PAGE 199

191 3/2 c TABLE 9 (continued) + 6.5

PAGE 200

192 TABLE 9 (continued) 3.16 SOSCF 33 3.1335 PGSO 33 35 35 33 35 ^1

PAGE 201

193 TARLF 5 (continued) 5/2

PAGE 202

BIP.LIOGDApMY 1. B.V/. Shore an-^ P.M. flenzel, PriPC'n1»^s pf Atorr>;c 9,oe:ct ra , (.John V/Mey and Sons, Inc., New York, 1958). This is an excellent book covering; the mathematical hackr^roiind o'f^ quantun mechanics with emphasis on the Racah tensor alj^ebra, an^i-ilar momentum, nerturbatinn theory, relativistic and nuclear effects, and electromagnetic fields. Unfortunately, the typesett i n.p; v/as poorly proofread, and there are extremely many errors in the equat i ons . 2, M.A, Bethe and E.^.. Sal peter, O uantnn Mec ha nics of Oneand T'"n-F 1 ect r-^n AtO"''S , (Academic Press, Inc., flew York, 1957 ) . This is a very v/el 1 -'.-/r i t ten text coverinn: a r,reat deal o^ quantum mechanics, both relativistic and non-rel at i v i s t ic theory. The material treated is most certainly not limited' to only oneand two-electron atoms. 19lt -

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195 3. A. FrBman, "Isotope Effects and Electronic Ener;;y in Molecules", J. Chem. Phys. li, 1^*90 (1962). Transformation of the SchrBdlnj^er equation to the center of nass systen in nolecules is rigorously treated, and an extensive analysis is "^iven of its significance for the nolecules M„^ n and HP. k. E. Bre i tenber^er, "Magnetic Interaction betv/een Charged Particles", Am. J. Phys. 16, 505 (1968). Magnetism in classical relativistic mechanics, the transition to quantum mechanics, and the center of mass transformation are discussed. 5. F.F. Close and ». Osborn, "Relativistic Center-of-Mass M.otion and the Electromagnetic interaction of Systems of Charged Particles", Phys. Rev. PI, 2127 (1070). Relativistic effects on CM motion in the presence of electromagnetic fields is tr'^ated for tv/o-part Icle systems. This paner is a logical, and more detailed, extension of Ref. k. 6. J.D. Garcia and J.E. Mack, "Energy Level and Line Tables for One-Electron Atomic Spectra",

PAGE 204

196 J. Opt. Soc. Am. 11, eSk (1965). Theoretically-computed energy levels and spectral line tables are given for H I through Ca XX. The energies consist of the Dirac energy plus corrections for a) the non-separability of the Dirac equation in terms of reduced masses, b) finite size of the nuclear charge distribution and finite nuclear mass, and c) quantumelectrodynamical corrections (Lamb shift). The energies are believed accurate to 10 Z . Typical values of these corrections for H i are: E(a + b + c) = +0.27 K (Is) = +0.0013 K (6s) E(c)= +0.0 35 K (2s) 7. J.C. Slater, Quantum Theory of Matter , (McGraw-Hill, Mew York, 1968), Second Ed. 8. J.C. Slater, Quantum Theory of Ator'ic Structure , Vol. I and il, (McGraw-Hill, New York, 1950). 9. J.C. Slater, Quantum Theory of Molecules and Solids , (McGraw-fUn, Flew York, Vol. I, 1963, Vol. !!, 1965, Vol. Ill, 1967, Vol. IV, 1972 (to be published)). 10. T.C. Koopmans, Physica i, lOi^ (1933).

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197 11. J.C. Slater and K.H. Johnson, "Self-Consistent Field Xa: Cluster f'ethoH for Polyatonic Molecules and Solids", Phvs. Rev. ^, ^kk (1972). The Xo( nethoci as ainlled to rolecules and the concept of thp transition state are discussed in detail. Extensive references are (^iven to earl ier 1 i tera ture . 12. f.'.H.F. Reebe, "On the Transition State in the Xoi flethod", to ho puhlished. An analytical expression for ener":y differences in the XcC method v.'hich involves only one-electron energies and a fev/ simple intej^rals over products of densities is described. 13. L, Hedin and A. Johansson, "Po 1 a r i 7.a t ion Correctinps to Core Levels", J. Phys. (London) ill, 1336 (1969). Innrovenent of ionization ener?:ies in the "artreo-Fock anprox ina t ion is surrested by the introduction of a polarization potential correction to tf-e orbital ener.f^y. The techninue is in princinle exact v/ithin th^HF annrox ima t ion, and avoids the canc<^ 1 1 a t i on from subtraction of total ener<^ies.

PAGE 206

198 Ik. G. Rerthier, "Sel ^'-Cons i stent Field Methods for Open-Shell Molecules" in f 'ol eru1 a r Orh i t?> 1 s in Chenistr", Physics and Bioln'^v A Tribute to R.S. f^ul 1 i!-en , Ed. P.-O. LHv/din and t^. Pullnnn, (Academic Press, flev/ Yorl;, IDHIO/ P. 57. Bcrthier reviews various approaches to open-shell SCF, with particular erphasis on the methods of Roothaan and of McV/eeny. 15. P.O. LBwdin, "Piscussinn on the Martree-Fock Approxinat ion". Rev. Mod. Phys. 51, ^^^^^ (1963). The syr^mptry d i l er^na in the Ha rt recFoci', schene is discussed. 16. M. Ratner and J.R. Sabin, "Synrnetry Considerations Concerning d-Orbital Participation in Chemical Bonding of P.econd-Row Elements", J. Am. Chen. Soc . £3, 55'j2 (1971). Emphasis is made of the fact that it is the molecular symmetry rather than the isolated atomic orbital symmetry v/hich is of imnortance in understanding?; chemical t^ondin.f^.

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199 17. P.O. LtiwHIn, "The Normal Constants of fiction in Quantun Mechanics Treated by Projection Technintje", Rev. flod. Phys. 31, 520 (19^2). IS. P.O. LBv/din, "Quantum Theory of Many-Particle Systems. I. Physical i nternrota t ions by fleans of density Matrices, f!atural Spi n-Orb I ta 1 s, and Conver.'^ence Problems in the Method of Conf i rura t iona 1 Interaction", Phys. Rev. 22^ l't7'i (19 55). This is tbe ori,
PAGE 208

200 all the h i
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201 The EHF equations v/ith the diagonal Lagrange rultlpllors replace'^ by an average one are derived. 22. I. flayer, "Herlvation of the Extended Hartree-Fock Equations", Chen. Phys. Letters H, 3^7 (1971). The f^HF equations are derive<^ with all diagonal La.f^ran^e nultlnliers included. The offdiaj^onal elenents may be elininater' by a unitary transfor^^at ion . 23. P.O. Lyv/Hin, "Hand Theory, Valence Pond, and TI .Tht-B i nd i nr Calculations", J. App. Phys. 11, 251 (1962). In addition to the subjects of the title, the treatrent o''^ trans 1 at lona 1 symmetry by projection ooerators, the extended HF schene, the non-or tho-^ona 1 I ty problem, an'-' the pa r t i t i on i n,?^ techninuo arf discusser'. 2I(. U. Kaldor, "Calculation o"^ Extended Hartree-Fock V/avefunct ions", J. Chern. Phys. M/ '5 3 5 (19 6^2). EHF calculations are carried out on the lithiur.-^tom, obtainin,F^^^j. = -7.^32^132 vs. Fp,,p = -7.'t32722U for the sane basis. The orbitals are not detemlned fron the EilP equations, but fron a

PAGE 210

202 CI involving: all projected si ngl y-exci teH detorninants . At each Iteration, nev/ orbitals are deterrined by collapsing the part of the CI involvin?^ the particular orbital, and the process continiiod until the unsubs t i t uted projected detorninant is suf -^ i c i ent 1 y dominant. In the final wavef unct ion, all the CI coefficients but the first v/ere less than l.E-6. 25. U. Kaldor, "Spin-Extended Har tree-Foe': Functions For Atomic Roron", J. Chen. Phys. M/ ^ (1968). The r^ethod of rcf. 2k is anplied to the deterr I nat ion of the SEM^ enertry and snin density of B (2 P). The orbital exponents v;ere conpletely or>tIni?.cd by a steepest descent method. The SEHF results v.-ere E = -24 . 5 296n'i, f = 0.611; the RMF results v;ith the same basis vie re E = -2 U. 52006, f = 0. The exnerimental contact term is f = 0.0029. 26. U. Kaldor, "Spin-Extended '-'avof unct ions for First-Row Atoms", Phys. Rev. 17^, 10 (1068). SEHF calculations of the eneri^ies and Fermi contact terms o^ C, M, 0, and F are reported and compared v/i th Rwr an^ UMF results. The method used is the same as note^ in ro-*^. 2'v above.

PAGE 211

203 27. U. Kaldor, "Spin-Extended Wavefunction for Atomic Sodium", Phys. Rev. 176, 71 (1963). SEHF calculations using the method noted under ref. 2k are ,iiven for Na and compared with RHF and U!^F resul ts. 28. K.M. Sando and J.E. Harriman, "Spin-Projected and Extended SCF Calculations", J. Chen. Phys. kZ, 180 (1967). Spin projection before and after variation is carried out on pi-electron radicals v/ithin the Pariser-Parr approximation. The EHF orbitals are not determined. Instead, the projected determinant is expanded in terns of the basis set, and the resulting energy expression minimized with respect to the orbital coefficients by a steepest descent method. 29. '.;.A. Goddard Ml, "Improved Quantum Theory of Many-Electron Systems. I. Construction of 2 Einenf unci ions of S l/hich Satisfy Pauli's Principle", Phys. Rev. 15J_, 73 (1907). A projection operator, G., based upon the i symmetric group is introduced which, when applied to any space-spin function, produces an ant i symmetri zed function which is also an

PAGE 212

20i| 2 ei genf unct I on of S . The superscript u. corresponds to the /U-th Irreducible representation of the symmetric group of order N (i.e. to a particular value of S), and the subscript i can take on any value from It. u. 1 to f , where f is the ^&^^rQ.& of the /M-th irreducible representation (i.e. the number of linearly independent spin functions). if a particular i is chosen, the method is called Gl, e.g. I = 1, Gl, \ = 2, G2, I = f, OF. For a particular Gj, variation of the total energy Is shown to lead to an effective Hamiltonlan (In general, different for each orbi tal ) . 30. IJ.A. Goddard III, "Improved Quantum Theory of f iany-El ectron Systems. II. The Basic f'ethod", Phys. Rev. 151, 81 (1967). The Gl method is applied to H2, using Gl, and LIH, using GF, and It is shown that the method does give correct molecular dissociation, v;hlch HF in general does not. It is shown that the GF method corresponds to spin-extended Hartree-Fock (SEflF), and the Gl method to maximally paired Hartroe-Fock (MPHF). 31. W.A. Goddard III, "improved Quantum Theory of Many-Electron Systems, ill. The GF Method", J. Chom,

PAGE 213

205 Phys. M/ 1+50 (1968). The GF method is discussed and the firstand second-order reduced spatial density matrices are derived for arbitrary spin and nur;ber of el ectrons . 32. U.A. Goddard III, "Inproved Quantun Theory of Many-Electron Systems. IV. Properties of GF V/avefunctlons", J. Chem. Phys. hS, 5337 (1953). GF calculations are reported for Li„, CH . 2 k and CH,. Their contention that one cannot introduce orbital an.-^ular momentum projections without leaving? the I PM is incorrect; see ref. IS, where the method is derived for arbitrary projection operators . 33. R.C. Ladner and V.'.A. Goddard 111, "Improved Quantum Theory of Many-Electron Systems. V. The Spin-Coupling Optinizod Gl llethod", J. Chem. Phys. H, 1073 (1969). By taking a linear combination of all possible G., a more general operator G = il, c . G . I i ' ' is obtained. One-electron equations can still be derived; the resultant method is called SOGI (spin-optimized Gl). Applications are presented for LI, Be"^, B"^"^, LIH, H,, and H, .

PAGE 214

205 5k. N.H.F. Beebe, "A Note of Space Spanning", Int. J. Quant. Chem. M, 0000 (1972). The problem of spanning the spin and angular momentum spaces is discussed and the spinoptinized SCF (SOSCF) or spin-optimized GI (SOGI) methods are shov/n to be special cases of a projected r^eneral spin orbital v/avef unct ion . See al so ref . kl . 35. F. Bloch, Z. Physik, 57., 5fi5 (1929). 36. R. Pauncz, Alternant Molecular Orbital r^,etbod , (17.8. Saunders, Philadelphia, 19G7), p. 16. In addition to discussions of the AMO method, this excellent little book has discussions of the construction of spin e i genf unct ions, projection operators, and density matrices. 37. t1. Kotani, Proceedings of the Conference held at Shelter Island, New York, 1951. 38. I'i. Kotani, A. Ameniya, E. Ishiguro, and T. Kimura, Tables of flolecular Interrrals , (M.aruzen, Tokyo, 1955), p. 13. 39. U. Kaldor and F. Harris, "Spin-Optimized Self-

PAGE 215

207 Consistent Field Wavef unct ions", Phys. Rev. 183 , 1 (1969). SOSCF, MPHF, and SEHF calculations (see no te under ref. 2k) are reported for Li 2 S, E(SOSCF) = -7.!4U7 565, 2 ^P, E(SOSCF) = -7.380 082, Be"" 2 ^S, E(SOSCF) = -Hi. 291 520, and B"""^ 2 ^S, E(SOSCF) = -23.389 919. UO. P.O. L8wdin, "Correlation Problem in Many-Electron Quantum Mechanics. I. Reviev/ of Different Approaches and Discussion of Some Current Ideas", Advances in Chemical Physics , (Ed. I. Prigogine, Interscience, New York, 1959), Vol. il, p. 207. (See Eq. I I .50). kl. S. Lunell, "On the Use of General Spin Orbitals in the Projected Hartree-Fock Method", Chem. Phys. Letters 13, 93 (1972). The relation of the PGSO functions to SOSCF functions is discussed and results of a sinple S -projection of a GSO function are presented for helium and lithium. See also ref. 34. k2, R. Lefebvre and Y.G. Smeyers, "Extended HartreeFock Calculation for the Helium Ground State", Int. J. Quantum Chem. 1, ti03 (1967).

PAGE 216

208 The EHF method is applied to He S. The one-electron equations are not solved; instead, a GSO determinant is set up, expanded, and projected to obtain a CI v;avef unct ion, and then all the orbitals are orthogonal i zed and varied independently. This last step means that the function can no longer be considered a projected single determinant of GSO, although the EHF method still applies also to the CI v/avef unct i on . Several interesting points are noted about the stability of the solutions around local minima obtained by restrictions to pure spin orbitals, or fixed 1 values, or fixed m, values, or doubly-occupied orbi tal s . k5. S. Lunell, "Spin-Projected Hartree-Fock Calculations on the He-Like Ions Using General Spin Orbitals", Phys. Rev. M/ 360 (1970). The one-electron EHF equations for both DODS and PGSO wavef unct i ons are solved for H-, He, Li+, and ne++. Some of the discussion centers on a result by Bunge (See ref. hk) implying the nonuniqueness of the EHF one-electron operator for the GSO case, which v/e believe to be incorrect. Iterative difficulties in the EHF one-electron equation solutions are noted for moderate (more

PAGE 217

209 than four functions) basis sets. kk. C.F. Bunge, "Unrestricted Projected Hartree-Fock Solutions of Two-Electron Systems", Phys. Rev. 151, 70 (1967). The EHF method for a PGSO wavefunction for the He series is discussed. Hov/ever, we believe that the results of Sections IM.(c) and (d), and Section V.(c) are Incorrect. See text, p. 83 for discussion. 'U5. F. Harris, private communication. U6. H.F. Schaefer and F.E. Harris, "Construction and Use of Atomic L-S E i genf unct i ons", J. Comp. Phys. 1, 0000 (1063). The program is available from the Quantum Chemistry Program Exchange as program QCPE 129. h7 . E.A. Hylleraas, "Uber den Grundzustand des Hel iunatones", Z. PhysikM/ ^69 (1928). I}8. L. Rrillouin, "Les Champs ' Se 1 f -Cons i s tents ' de Hartree et de Fock", Actual ites Sci. et Ind, f.!o. 159.

PAGE 218

210 U9. E.A. Hylleraas, "Neue Berechnung der Enerp;ie des Heliums in Grundzustando, sov/ie des tiefsten Terns von Or tho-Hel i un", Z. Physik BJL, 5h7 (1929) 50. H.fl. Janes and A.S. Coolidge, "Ground State of the Hydrogen Molecule", J. Chen. Phys. i, 825 (1933). 51. H.M. James and A.S. Coolidge, "Improved Calculation of the Ground State of H2"/ Phys. Rev. M/ 588 (1933). 52. H.f'i. James and A.S. Coolidge, "Correction and Addition to the Discussion of the Ground State of H2", J. Chem. Phys. 1, 129 (1935). 53. T. Kinoshita, "Ground State of the Helium Atom", Phys. Rev. 1115, 11(90 (1957). 5'j. C.L, Pekoris, "Ground State of Tv/o-Electron Atoms", Phys. Rev. ill, 1GI(9 (1953). The Hylleraas method is applied to the helium sequence, obtaining an accuracy of 0.01 K. A number of oneand tv/o-el ect ron properties are tabulated.

PAGE 219

211 55. C.L, Pekeris, "1 S and 2 S States of Helium", Phys. Rev. Hi, 1215 (1959). The Hylleraas method Is pushed to 0.001 K and a number of properties are given. 56. Y. Accad, C.L. Pekerls, and B. Schiff, "S and P States of the Helium I soel ect ron i c Sequence up to Z = 10", Phys. Rev. M/ 516 (1971). This is an extension of the work in ref. 5'} and 55, with extensive tabulations of energies and properties. 57. E.A. Burke, "Variational Calculation of the Ground State of the Lithium Atom", Phys. Rev. HO., 1871, (1963). A Hylleraas function calculation on Li gave E = -7. (+7795 H; cf. Weiss' t»5-term CI, E = -1 jnii. 58. S. Larsson, "Calculations on the S Ground State of the Lithium Atom Using Wavef unct ions of the Hylleraas Type", Phys. Rev. 169, 49 (1958). A 100-tcrm Hylleraas function v.-ith a linear combination of the two spin functions gave E = -7.1+78069 H and f = 2.90G.

PAGE 220

212 59. J.F. Perkins, "Calculations of Some Excited S States of Li in Hylleraas Coordinates", Phys. Rev. A5 , 512 (1972). Unprojected Hylleraas function calculations give E(3 ^S) = -7.3535 H, and Eik ^S) = -7.3175 H. 60. L. Szasz and J. Byrne, "Atomic Many-Body Problem. Ill, The Calculation of Hyl 1 eraas-Type Correlated Wavef unct ions for the Beryllium Atom", Phys. Rev. 158 . 3 4 (19 67). A 26~parameter Hylleraas function gave E =--14.6565 H. 61. R.F. Centner and E.A. Burke, "Calculation of the S State of the Beryl liun Atom in Hylleraas Coordinates", Phys. Rev. 1,76, 63 (196S). A 25-parameter Hylleraas function gave E =. -14.6579 H. 62. J.S. Sims and S. Hat^stron, "Combined ConfigurationI nteract ion-Hyl leraas-Type V/avef unct ion Study of the Ground State of the Beryllium Atom", Phys. Rev. Ak, 908 (1971). A CI with each orbital product multiplied by a pov/er of rrbefore an t i symmetr i za t ion

PAGE 221

213 is used to obtain E = -lii.eseBU H for a 107-term CI (cf. 180-term CI, E = -1 fi. 561+19 ) . The exact nonrelat i vi sti c energy is -14.6657 H. The method Is said to be easily general i zab le to an arbitrary number of electrons. 63. K. Husini, "Some Formal Properties of the Density Matrix", J. Phys. Math. Soc. Japan 21, 26k (1940). 6k. R. McV/eeny, "Some Recent Advances in Density Matrix Theory", Rev. Mod. Phys. H, 335 (1960). 65. A.J. Coleman, "Structure of Fermlon Density Matrices", Rev. Mod. Phys. 15., 558 (1963). 56. T. Ando, "Properties of Fermlon Density Matrices", Rev. Mod. Phys. 15., 590 (1953). 67. A.J. Coleman and R.M. Erdahl, Ed. "Reduced Density flatrlces with Applications to Pliysical and Cheinical Systems", Queen's Papers in Pure and Applied Mathematics (Queen's University Press, Kingston, Ontario, 1968). 63. A.J. Coleman and R.M. Erdahl, Ed. "Report of the Density Matrix Seminar", Queen's University,

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21I» Kingston, Ontario, 1969, unpublished. 69. M.B. Ruskai, "The N-Representabi 1 i ty Problem", Ph.D. Thesis, VHsconsin, 1959. Available as V/1 S-TC! -3it5 from the Theoretical Chemistry Institute, University of V/isconsin. 70. R. Landshoff, Z. PhysiklOl/ 201 (1935). 71. P.O. L8v/din, "A Quantum Mechanical Calculation of the Cohesive Energy, the Interionic Distance, and the Elastic Constants of Some ionic Crystals", Arkiv. Mat. Astron. FysiklSA, No. 9, 1 (19ii7.). (Thesis, Almqvist and Wiksolls, Uppsala, 191(8). 72. S. Lunell, "flagnetic f-Iyperfine Structure of Lithium Usin,'^ an Approximate Projected HartreeFock Method", Phys. Rev. 173, 85 (19G8). A PGSO calculation, restricted to the SOSCF case, is carried out on Li, obtaining E = -7.UU753G06 and f = 2.0I46. A natural analysis is carried out on the v/avef unct ion . 73. J.H. V/il!;inson, The Algebraic Eigenvalue Problem , (Clarendon Press, Oxford, 19GS).

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215 This is a marvellously well-written text, replete with examples, with rigorous theoretical treatment, detailed error analysis, and designed for practical computational appl i cat ions . 7h. 5. Larsson and V.H. Smith, Jr., "Analysis of 2 the S Ground State of Lithium in Terns of Natural and Best Overlap (Brueckner) Spin Orbitals with Implications for the Fermi Contact Term", Phys. Rev. 173, 137 (1969). A natural analysis is carried out on Larsson's Hylleraas function for Li (ref. 53). 75. P.O. L8wdin, "Studies in Perturbation Theory. X. Lov/er Bounds to Energy Eigenvalues in Perturbation Theory Ground State", Phys. Rev. 119, A357 (1965). 76. F. Sasaki, "Eigenvalues of Fernion Density flatrices", Phys. Rev. US, B1338 (1955). Upper bounds for the eigenvalues of an arbitrary fermlon p-matrix are given, and it is shown that for p = 2, the bound is the least upper bound. 77. B.C. Carlson and J.M. Keller, "Eigenvalues of

PAGE 224

216 Densi ty Matrices", Phys. Rev. ill, 659 (1951). The non-zero elj^envalues of the p-matrix are shown to be Identical to those of the (M-p)matrlx, and the two are unitarily equivalent if the number of non-zero eif^envalues is finite. The expansion of the wavefunction in terms of the pand (M-p)-states is denons tra tod. 78. E. Schmidt, Math. Ann. 5J./ ^^^ (1907). 79. J.E. Harrinan, "Natural Expansion of the First-Order Density flatrix for a Spin-Projected Single Determinant", J. Chem. Phys. M/ 2827 (1954). The 1-matrlx, NSO's, and occupation numbers of a general spin-projected DODS determinant are derived. 80. A. Mardlsson and J.E. Harrlman, "Second-Order Density Matrix for a Spl nProj ected Single Deterni nan t", J. Chem. Phys. M/ 3539 (1967). The 2-matrlx of a spin-projected DODS determinant is derived. 81. J. Simons and J.E. Harrlmon, "Firstand Second-Order Density flatrices of SymmetryProj ected SingleDeterminant V/avef unct Ions", J. Chem. Phys. SX,

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217 296 (1971). Expressions for the 1and 2-matrices of point-group and ax ia 1 -rota t i on group symmetry projected single Slater determinants are derived. 82. T.B. Grimley and F.D. Peat, "Bopp's Approximation to the Energy of an M-Electron System", Proc. Phys. Soc. (London) 81, 2U9 (1955). 33. F.D. Peat and R.J.C. Brown, "The Ant i symmetr i za t ion of Geminal V/avef unct ions", int. J. Quant. Chem. IS, fi55 (1967). A variational problem for the 2-matrix vvith approximate antisymmetry conditions is proposed. Sh. F.D, Peat, "Ant i symmetr i zat ion of Geminal K'avef unct ions Ml. Calculations of the Density Matrix and Energy of a Three-Electron Atom", Phys. Rev. 113, 69 (1968). Approximate fi-representab i 1 i ty conditions on the 2-m3trix have been applied on a calculation of the energy of Be+. The results arc E(2,3,ti ^S) = -l't.302ii, -1!;.016, and -13.t;81 vs. the exact results -lit. 327, -13.927, and -13.801. 85. L.J. Kijewski and J.K. Percus, "Paul i -Pr i rici pi e

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218 Restriction on the Two-Matrix of Bopp for Atomic Ground States", Phys. Rev. 179, (+5 (1969). 86. L.J. Kijev/ski, J.K. Percus, I.H. Pratt, and J. P. Tranchina, "Some Restrictions on the Use of Reduced Density Matrices In Atomic Calculations", Phys. Rev. A2/ 310 (1970) . 87. L.J. Kljewski and J.K. Percus, "Lower-Bound Method for Atonic Calculations", Phys. Rev. A2/.1959 (1970). 88. J. Simons and J.E. Harriman, "Construction of Approximately M-Representabl e Density Matrices", Phys. Rev. AI/ 103ti (1970). A computational scheme for variation of a 2-matrix is laid down by imposing the constraint that <0 > --> 1; this can be computed without knowledge a s of the wavef unct ion . Mo numerical applications are given. 89. L.J. Kljewski and J.K. Percus, "Numerical Calculations for C++ Using Reduced Density flatrlcos". Int. J. Quant. Chem. 5^, 67 (1971). 90. L.J. Kljewski, "Test of the G-Matrix Condition on the Two-i-latrix Satisfying the Paul I Restriction",

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219 Int. J. Quant. Chem. 6^, 0000 (1972) (to be publ i shed) . 91. V.H. Smith, Jr. and F.E. Harris, "Projection of Exact Spin E i genf unct ions", J. Math. Phys. 10 , 771 (1969). The various formulae derived by various people for the Sanibel spin-projection coefficients are shown to be equivalent, and the corr.putat iona 1 suitability of the fornulae is discussed. The construction of matrix elements over spin-projected wavef unct ions is also di scussed. 92. G. f'ialli and S. Fraga, Many-Electron Systems: .Pro perties and Interactions , (Saunders, Phi ladelphia, 1968). Pvather concise and uncritical and slanted toward the authors' own research, but nevertheless to a certain extent fills the void of texts about properties of atoms and molecules. 93. E.P, Kigner, Groun Theory . (Academic Press, New York, 1959). 9't. T. Kato, "On the E i genf unct ions of Many-Particle

PAGE 228

220 Systems in Quantum Mechanics", Comm. Pure App. Math. iO, 151 (1957). This complicated and lengthy paper contains a rigorous mathematical treatment of the behaviour of the ei genfunct ions at the singularities of the Hamiltonian. 95 . C.C.J. Roothaan and P.S. Kelly, "Accurate Analytical Sel f -Cons i stent Field Functions for Atoms. III. The Is 2s 2p States of Nitrogen and Oxygen and Their Ions", Rhys. Rev. HI, 1177 (1963). 95. P.O. L8wdin, "Quantum Theory of Cohesive Properties of Solids", Adv. Physics 5., 1 (195G), pp. 49-56. 97. H.P. Kelly, "Correlation Structure in Atoms", Adv. Theor. Physics 1, 75 (1968). 98. P.O. LBv/din, "Scaling Problem, Virial Theorem, and Connected Relations in Quantum Mechanics", J. Molec. Spectr. 1, i<5 (1959). 99. A. FrBman, "Comments on the Relativistic Corrections and the Electronic Correlation in the Li-Like Ions", Technical Note 63, Uppsala University Quantum Chemistry Group, 1951 (unpublished).

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221 100. S.S. Prasad and A.L. Stewart, "Isotope Shift In Li and B++", Proc. Phys. Soc. (London) 82, 159 (1966). Isotope shifts are computed from the Weiss i*5"tern CI wavef unct ions (ref. 110). The results in Hartrees for < 2^i* 12'> aro Li ! Be I I Bill C IV ri V VI 2 ^S 0.1507 0.2267 0.3035 0.3800 . i> 5 8 5 0.5355 V ' ^ 2 h 0.1180 0.06185 -0.08i;25 -0.3123 -0.61325 -l.OOli The effect on spectral lines is u,/l\ times this. 101. f!.G. Kuhn, Atonic Spectra , (Academic Press, Nev/ York, 1962). 102. A. Oalgarno and E.M. Parkinson, "Properties of the Lithium Sequence", Phys. Rev. 115, 73 (1968). Perturbation theory estimates, using the results of Pekeris' v/ork (ref. 5U, 55, 56, 103, lOU ) on tv;o-el ect ron systems, are made for relativistic corrections. Isotope shift, specific mass effect, and transition probabilities of three-electron systems.

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222 103. C.L. Pekerls, "1 "^S, 2 ^S, and 2 ^S States of Li+", Phys. Rev. 115, li+3 (1962). A t;4ft-tern Hylleraas function is used to obtain ionization energies, relativistic corrections, and the specific mass effect for Li+, and tlie assignment of one of the lines of the Li + spectrum is shown to bo incorrect, lOtt. C.L, Pekeris, "1 S, 2 S, and 2 ^S States of Hand of He", Phys. Rev. ilG, 1U70 (1962). A ittjtt-tern Hylleraas function is used to obtain ionization energies, relativistic corrections, and the specific mass effect for Hand He, and a search for a bound excited state of Hgives negative results, 105. H.T. Doyle, "Relativistic Z-Dependent Corrections to Atomic Energy Levels", Adv. Atomic Molec. Physics .5, 337 (1969). 105. Y. Uhrn and J. Nordling, "Correlated V/avcf unct i ons for Three-Electron Ions", Arkiv Fyslk H, U71 (1956) Hylleraas calculations are reported for 2 the 2, 3, and k S states of LI I, Be 11, B III, C i V, U V, and VI .

PAGE 231

223 107. Y. Ohrn and J. Mordlin?^, "On the Calculation of Sone Atonic Integrals Containing Functions of r^2. r^j/ and r^^", J. Chen. Phys. 19, 1864 (1963). 108. W.L. \7iese, M.l/. Smith, and B.M. Glennon, At nnir Transition Probabilities. Vol . I. Ilvdro^ren throur^h Neo n, (U.S. Nat. Bur. Stand. MSRDS-MBS-U, 1956). 109. Proceedings of the Second International Conference on Beam-Foil Spectroscopy, Nucl . instr. Methods 91/ (1970). 110. A. 17. V/eiss, "Wavef unct ions and Oscillator Stren.o:ths for the Lithiun I soel ect ron i c Sequence", Astrophys. J. UA, 1262 (1963). Oscillator strengths and total energies have been conputed fron Mar tree-Fock wavef unct i ons for the 2 ^S, 3 ^S, 2 ^P, 3 ^P, and 3 ^D states of the Z = 3 to 10 three-electron ions. A ii5-term CI is carried out for the 2 P state of the Z = 3 to 8 three-electron ions. 111. M.A. Ali and L.J. Schaad, "'Har tree-Fock flultiplet Strengths of Is^tJl ^L --> Is^Ul' "L' Transitions of the Lithium 1 soel ect ron i c Sequence", J. Phys.

PAGE 232

221* (London) B2/ 1301* (1969). Hartree-Fock values of <'tl|rU1'> are used 2 2 to compute oscillator strengths for k S — > k P, 2 2 2 2 h P --> (4 D, and k ^D — > k ^F transitions for Z = 3 to 9 in the i soo 1 ect ron i c sequence. 112. M.R. Flannery and A.L. Stewart, "Oscillator Strengths for Lithium and Other Three-Electron Systems", Monthly Notices Roy. Astron. Soc. 121, 387 (1963). The MPHF method of Ritter, Pauncz, and Appel (ref. 135) Is used to obtain enerp;ies and 2 oscillator strengths of the 2, 3, and h P states of the three-electron sequence from Z = 3 to 9 . 113. F. V/einhold, "Variational Upper and Lower Bounds to Dipole Transition Moments", Phys. Rev. Letters 25, 907 (1970). Upper and lower bounds to the length formulation of the dipole moment which require only approximate wavef unct ions are given. llii. A.F. Staraco, "Length and Velocity Formulas in Approximate Oscillator-Strength Calculations", Phys. Rev. M, 12i»2 (1971). Starace demonstrates that of the length, velocity, and acceleration formulae for the

PAGE 233

225 electric dipole moment, the length formula is the correct one to use for HF, CI, and BG wavef unct ions. 115. M.H. Miller, R.A. Roig, and R.D. Bengtson, "Absolute Transition Probabilities of Phosphorus", Phys. Rev. M/ 1709 (1971). 116. U. Fano and J.V;. Cooper, "Spectral Distribution of Atomic Oscillator Strengths", Rev. Mod. Phys. M. ^^l (1908), kl, 721; (1969). This is an extensive review article of both the theoretical and experimental state of oscillator strength determination. Current theoretical work is limited entirely to HartreeFock and perturbed MF since the virtual orbitals are used as approximations to excited states. Another difficulty Is that after hydro-en, the continuum contributes the bulk of the oscillator strength. 117. A. Dalgarno, "Atonic Po la r i zab i 1 i t i es and Shielding Factors", Adv, Physics U, 2S1 (1952). 118. F.U. Condon and G.H. Shortley, The Theory of Afnmir lUfXtra, (Cambridge University Press, 1935).

PAGE 234

226 119. M. Blune and R.E. VJatson, "Theory of Spin-Orbit Coupling In Atons. I. Derivation of the Spin-Orbit Coupling Constant", Proc. Roy. Soc. (London) A270 , 127 (1962). The spin-orbit effect is properly treated for a many-electron systen for the first time. 120. M. Blume and R.E. V/atson, "Theory of Spin-Orbit Coupling In Atons. 11. Comparison of Theory with Experiment", Proc. Roy. Soc. (London) A271 , 565 (1965). 121. R.E. Trees, "Hyperflne Structure Formulas for LS Coupling", Phys. Rev. M, 308 (1953). 122. C. Schwartz, "Theory of Hyperflne Structure", Phys. Rev. 92, 380 (1955). 123. C. Schwartz, "Theory of Hyperflne Structure", Phys. Rev. lOi, 173 (1956) . 121*. A.J. Freeman and R.E. V/atson, "Hyperflne Interactions in Magnetic Materials", flagnot i spi . Vol. I! A, Ed. G.T. Rado and H. Suhl, (Academic Press, New York,

PAGE 235

227 1965), p. 158. 125. A.J. Freeman and R.B. Frankel, Hyperfine I nteract i on.*; . (Academic Press, New York, 19G7). 126. B.R. Judd, Operator Techniques in Atonic Spect rosconv . (McGraw-Hill, New York, 1963). 127. L. Armstrong, Jr. T heory of the Hyperfine Structure of Free Aton^;. ( W i 1 eyI nter sc i ence. New York, 1971). 128. S.fl. Blinder, "Theory of Atomic Hyperfine Structure", Adv. Quant. Chen. 2, '+7 (1965). 129. H.F. Schaefer ill, R.A. Klenm, and F.E. Harris, "Atonic Hyperfine Structure. I. Polarization V/avefunct ions for the Ground States of B, C, N, 0, and F", Phys. Rev. 125, 49 (1958). A convenient tabulation of hfs formulae is given. 130. J.D. Lyons and R.K. Nesbet, "Hyperfine Structure 2 in the 2 P State of Atomic Lithium", Phys. Rev. Letters, 2A, (|33 (1970).

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228 The necessity of describing the contact, spin-dipolar, and orbital hfs interactions by independent parameters is demonstrated. 131. P.S. Bagus, B. Liu, and H.F. Schaefer III, "Study of the Contact-Term Contribution to the Hyperfine Structure Obtained from Spin-Unrestricted Hartree-Fock Wavef unct i ons", Phys. Rev. kl, 555 (1970) 132. E.R. Cohen and J.V/.fl. DuMonri, "Our Knov/ledge of the Fundamental Constants in 1965", Rev. Mod. Phys. 37, 537 (1965). 133. A.M. Wapstra and N.B. Gove, "The 1971 Atomic Mass Evaluation", nuclear Data Tables, A9/ 265 (1971). 134. G.H. Fuller and V.17. Cohen, "fJuclear Spins and Moments", nuclear Data Tables, A5, 433 (1959). 135. Z.17„ Ritter, R. Pauncz, and K. Appel, "Approximate 2 2 Analytical V/avef unct ions for the Is ns ^^i/o States of Li and Li-Like Ions", J. Chen. Phys. 15, 571 (1961). MPHF calculations using one Is STO for Is, a different Is STO for Is', and a linear combination of STO's for 2s were carried out. The

PAGE 237

229 exponents and orbital s were optimized for each state separately. The SEHF case was Investigated, and also the SOSCF case, but the latter only lowered the MPHF energy by 0.005%, so it was not pursued. The negative energies in Hartrees are: 2 2 2 2 2 S 3 S US 5 S Li I 7.i|ti50 7.322U 7.2869 7.268tj Be I! 14.2889 13.8893 13.7557 13.6973 B II! 23.3870 22.5592 22.3079 22.1607 C IV 34.7358 33.3609 32.9130 32.6584 N V 48.3373 45.2541 45.5809 45.1904 VI 64.1884 51.2787 60.3113 59.7555 F VII 82.2895 78.4044 77.1043 76.3557 135. O.J. Wilde, Optintim Seeking Methods , (Prentice-Hall, Englewood Cliffs, M.J., 1954), pp. 145 ff. 137. H.G.J. Moseley, Phil. Mag. 15, 1024 (1913). 138. H.G.J. Moseley, Phil. Mag. 22, 703 (1914). 139. T. Surratt, private conmun i cat ion, Sanibel, 1970. 140. N. Sabelli and J. Hinze, "Atonic Mul t i conf i gura t iona 1 Self-Consistent Field V.'avef unct i ons", J. Chem. Phys. 5^, 584 (1959).

PAGE 238

230 lUl. VI. h. Blngel and 1/. Kutzelnigs, "Symmetry Properties of Reduced Df^nsity flatrices and Matural p-States", Adv. Quant. Chem. S, 201 (1970). Iti2. P. Kusch and H. Taub, "On the f. Values of the Alkali Atoms. The Hyperfine Structure of the Alkali Atoms.", Phys. Rev. TS, Ihll (1949). 2 For the Li 2 S state, the hyperfine separation betv/een states with F = I + 1/2 and F = I 1/2, J = 1/2, is W(F + 1) W(F) = A (F + 1). J The separc-^tion for Li was found to be 228.208(5) MHz, with g,/g, = -ii'472(3); for Li^, It was 803.512(15) MHz., with j^ /g = -1693.7(8). The results are J I from an atomic bnam exper iirent'. The data gives A^/2^Li^) = 1+01.756(3), and A^/2^'-' ^ " 152.139(3) MHz. 1U3. R.G. Schlecht and D.V;. McColm, "Hyperfine Structure of the Stable Lithium isotopes. I.", Phys. Rev. Ikl, 11 (19G5). Atomic beam experiments as In ref. I't2 give hfs separations of 223.20528(8) MHz and 6 .7 . , . 803.50UO!|('i8) MHz for Li and Li respectively in n the 2 S state, giving ^2/2 values of 152.13585(6) 7 6 and 401.75202(24) respectively. Using g|(Li )/g (Li ) = 2.64090538(20), they evaluate the hfs anomaly.

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231 given by ^C7 = ^ ^ ^ HjlL! to be 1.06 't X 10 . lUtt. G.J. Ritter, "Hyperfine Structure of the Level 2 ^P of Lithium-6 and Lithiurn-7", Can. J. Phys. ill/ 770 (1965). Optlcal-radio-frequency double resonance spectroscopy gave A;^/2^Li^) = 45.17(35) MHz, and A (Ll^) = 17.48(15) MHz for the 2 P state of 1/2 -i-/ ^ Li. The hfs energy separation is given by A. K.C. Brog, H. Wieder, and T. Eck, "Fine and Hyperfine Structure of the 2 ^P Tern of Li and Li ", Phys. Rev. 153, 91 (1967). 2 Level-crossing exporirents on the 2 P states of the tv;o isotopes of Li, combined with Ritter's neasurements (ref. 144), gave, for Li , J = 3/2, a = -10.53(23) !U!z c a^ = +7.13(3) MHz d A = -3.40(25) MHz B = -0.0 07 5(50) MHz Q = -0.03(2) barn

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232 where the hfs separation is given by (1/2)AK + BK(K+1). 146. R.C. Isler, S. Marcus, and R. Novick, "Hyperfine 2 2 Structure of the 3 P and h P States of Lithium and Lifetime of the 3 ^P State", Phys. Rev. 187 , 76 (1959). Level -cross i ng experiments give (in MHz) 7 2 Li 3 P: a^ = 2.11(3), a = -3.08(3), A = -0.97(6) d c 6 2 Li 3 P: a , = 0.80(2), a = -1.20(2), A = -0.40(U) d ' c fortheJ=3/2state,and Li'^ li ^P: a^i ^^^ = 0.89(2), a^ = -1.30(2), A^ = 2.19(U) for the J = 1/2 state. A,, = a^ + a , A = a, " 3/2 1/2 a^. They also find B = 0.0079(91) MHz for the Li 2 3 P,,„ state. Note that this a, value contains 3/2 a both our a , and a , . ci 1 1U7. S.L. Kahalas and R.K. Nesbet, "Electronic 7 Structure of LiH and Ouadrupole Moment of Li ", J. Chem. Phys. 19, 529 (1963). A CI calculation on LiH is used to obtain the electric field gradient at the Li nucleus. See ref. 165. ItiS. D. McColm, "Hyperfine Structure of the Stable

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233 Lithium Isotopes. 11." Phys. Rev. IkZ, l^ (1966). The orif^in of the different Fermi contact terms in Li and Li (ref. 1U3) is discussed. 11*9. T. Tteriikkis, S.D. Hahanti, and T.P. Das, "Theoretical Analysis of the Hyperfine Structure 'of Alkali Atoms", Phys. Rev. JJl, 10 (1968). Hartree-Fock, perturbation theory, and relativistic corrections are applied to the 2 7 determination of A(2 S) for Li . The result is: A = 2St*.00 (HF) + 94.02 (core polarization) + 1.00 (relativistic effect) = 379.02 f'iHz. 150. D.A. Goodings, "Exchange Polarization Effects in Hyperfine Structure", Phys. Rev. HI, 1706 (1961). Numerical RHF and UMF calculations are reported for Li, Be, B, M, F, Na, CI, and K, and hfs parameters are given. The analsis is in terms of the old two-parameter fit of the hfs constant A. 151. J.D. Lyons, R.T. Pu, and T.P. Das, "Many-Body 2 Approach to Hyperfine Structure in the P State of Lithium", Phys. Rev. 178., 103 (1969)

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231+ A Bethe-Goldstone calculation on Li 2 P gives the following results a = -9.5G83 c a^ = +8.6725 a^ = -1.89t+3 A^ •= -2.7906 ^1/2 " +'^5.8563 B = -0.0197 (All val uos In MHz . ) 152. J.B. Martin and A.W. Weiss, "Calculation of the flyperfine Splitting for the Lithium Atom", J. Chem. Phys. 19, 1618 (1963). A summary of contact terms by various 2 methods for Li 2 Sis given. 153. D.L. Hardcastle, J.L. Gammel, and R. Keown, "Hyperflne Splitting of the Ground State of Lithium", J. Chem. Phys. 19/ 1358 (1968). An extensive survey, more recent than that of ref. 152, Is given of the contact term 2 results for Li 2 S, 15U. D.L. Hardcastle and R. Keov/n, "Extended HartreeFock Calculations for Li and Li-Like Ions", J. Chem. Phys. 51, 598 (1969).

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235 Numerical MPHF-EHF calculations of 2 energies and contact terms of the 2 S states of the Li sequence^ Z = 3 to 9, are reported. 155. S. Larsson, "Har tree-Fock V/avef unct ions V/Ithout Symnetry and Equivalence Restrictions and the Calculation of Hyperfine Structure Expectation 2 Values for the Lov;est P States of Boron and Lithium", Rhys. Rev. A2/ 1248 (1970). Larne basis set (12 s, 12 p, and 7 d) DODS (SPHF) and UHF calculations are reported. The hfs A and B parameters, and the orbital, contact, dipolar, and quadrupolar contributions 2 2 are reported for the 2 Pi/o ^Vid 2 Pt/o states of Li^ and B^^. 156. 17. A, Goddard, ill, "Ma?^netlc Hyperfine Structure and Core Polarization in the Excited States of Lithium", Phys. Rev. 176., 106 (195S). 2 2 The hfs A parameters for the 3 3, 2 P, 2 2 3 P, and 3 D states of Li are reported, calculated from SEHF wavef unct ; ons . 157. fc'.S. Chang, R.T. Pu, and T.P. Das, "Many-Body Approach to the Atonic Hyperfine Problem. 1. Lithium Atom Ground State", Phys. Rev. ITji,

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236 1 (1958). A Bethe-Gol ds tone calculation for Li 2 2 S is reported, v/ith E = -7.tt78 and f = 2.887. 158. A.W. Weiss, "Conf i p;ura t ion Interaction in Simple Atomic Systems"^ Phys. Rev. 112, 1826 (1961). CI ground state calculations are reported for the He, Li, and Be sequences, Z = 2 to 8. 159. R.E. Brown and V.H. Smith, Jr., "OneE lect ron 2 Reduced Density Matrices for Li ( P) and B (^P)", Phys. Rev. A3, 1858 (1971). Natural analyses are carried out for the Weiss CI for Li 2 ^P (ref. 110) and the Schaeffer, 2 Klemm, and Harris C! for B 2 P. See also rof. 150 150. K,E. Banyard, M. Dixon, and A.D. Tait, "Electron 2 2 Correlation of the S and P States of Li-Like Ions", Phys. Rev. M/ 2199 (1971). NatLiral analyses arc carried out for the V;eiss CI of the 2 ^? (ref. 110) and 2 ^S (ref. 158) states of the litliiuni sequence for Z = 5 to 7 . See also ref. 159. 161. D.B. Cook and M.A. All, "Analytical Wavef unct ions

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237 of D States of Lithium I see 1 ectron i c Sequence"/ vJ. Chem. Phys. kh, kill (1966). 2 MPHF calculations of the 3, h, and 5 D states of the Li sequence, Z = 3 to 8, are reported. 162. R.L. Smith and T.G. Eck, " Level Cross i ng Slj^nals in Stepwise Fluorescence", Phys. Rev. A2, 2179 (1970). The fine structure splitting of Li i 2 3 n is found to be 1081.69(33) MHz; the hfs could not be resolved. 163. L. V/harton, L.P. Gold, and 17. Klemperer, "Quadrupole nonent of Li^", Phys. Rev. ill, B270 (196t+). A molecular bean electric resonance spectrum of Li F, coupled with results for the quadrupol coupling constant of LiM, and a theoretical calculation by Kahalas and Nesbct (ref. I'i7) for the electric field gradient at the Li nucleus, gave Q(Li^)/0(Li'') = +0.0176, Q(Li^) = -0.00080(8) barn, and Q(Li^) = -0.0;»5(5) barn.

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BIOGRAPHICAL SKETCH Nelson H.F. Beebe was born July 2U^ 19U8 in Barrhead^ Alberta, Canada. After attending schools In various parts of Alberta, he graduated from high school in June, 1965. He attended McGill University in Montreal, Quebec from September, 1955 to May, 1968, graduating with first class honors In Honors Chemistry. From June, 1968 to the present, he has been a graduate assistant in the Quantum Theory Project and Department of Chemistry at the University of Florida. ? 238

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, scope and quality, as a Doctor of Philosophy. n dissertation for the degree of Per-01o\>! LBwdin, Chairman Graduatfi Research Professor of Chemistry and Physics I certify that i have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. N. Yr^lgv^ tJhrn, Cochoirman Professor of Chemistry and Phys 1 cs I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the def,r(ie of Doctor of Philosophy. Charles E. Reid Associate Professor of Chemi stry I certify that I have read this study and that In my opinion it conforpis to accaptable standards of scholarly presentation and i r, fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ja^nes K. BrooPfs /ssociate Professor o1 la theriat i cs

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I certify that I have read this study and that in ny opinion it conforms to acceptable standards of scholarly presentation and is fully adequate. In scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associate Phys i cs Professor of This dissertation was submitted to the Department of Chemistry in the Collerre of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Juno, 1972 Dean, Graduate School