The reduced partitioning procedure in configuration interaction studies

MISSING IMAGE

Material Information

Title:
The reduced partitioning procedure in configuration interaction studies
Physical Description:
ix, 110 leaves. : ill. ; 28 cm.
Language:
English
Creator:
Bartlett, Rodney Joseph, 1944-
Publication Date:

Subjects

Subjects / Keywords:
Perturbation (Quantum dynamics)   ( lcsh )
Approximation theory   ( lcsh )
Configurations   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida, 1971.
Bibliography:
Bibliography: leaves 106-109.
Statement of Responsibility:
By Rodney Joseph Bartlett.
General Note:
Manuscript copy.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000953381
notis - AER5837
oclc - 16919167
System ID:
AA00003960:00001

Full Text













The Reduced Partitioning Procedure in
Configuration Interaction Studies














By

RODNEY JOSEPH BARTLETT


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE tIr'I RSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE PCQ.'IRL.LMN:TS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA
1971




























DEDICATED

to

MY PARENTS












ACKNOWLEDGE E; I- NTS


It is my pleasure to gratefully acknowledge my chairman, Prof. Yngve

Ohrn, for the unfaltering assistance, encouragement, and supervision he

has given me during the period I have spent in the Quantum Theory Project.

Without his cooperation in all manner of things, my stay here would have

been much less rewarding. Also, I wish to sincerely express my apprecia-

tion to Prof. Per-Olov Lwdin, my co-chairman, for his intellectual stim-

ulation and for several conversations in which he made numerous fruitful

comments which improved the quality of this dissertation. To Dr. Erkki

Br2ndas, I owe a special word of thanks, since this research was undertak-

en with his assistance and would have been impossible without his guidance.

To say that the exchange of ideas between us was of great benefit to me

is an understatement. I also wish to express my appreciation to the other

members of my supervisory committee for their continued cooperation.

All of the members of the Quantum Theory Project have been helpful to

me, but particularly I would like to thank Dr. C. E. Reid and Dr. Joe

Kouba for assistance in the use of some of their computer programs, as well

as Nelson Beebe and John Bellum for additional computational aid. I am

also indebted to Dr. Micheal Hayns for his careful drawing of the figures.

Lastly, I am especially grateful to Beverly Bartlett for her tireless

efforts in the typing of this dissertation as well as her constant encour-

agement during its preparation.












TABLE OF CONTENTS


AC -,O'.I-LFDGEMENTS . . .

LIST OF TABLES . . .

LIST OF FIGURES . ... . .

ABSTRACT . . ..

Chapter

I. INTI ODiJCTION . . .

II. THE PARTITIT.O;IiG TECHNIQUE AND PERTURBATION THEORY

II.1 The Partitioning Technique . .

11.2 The Partitioning Technique in Operator Form

11.3 Brillouin-Wigner Perturbation Theory .

11.4 Rayleigh-Schr'"dinger Perturbation Theory .

11.5 Inner and Outer Projections .

III. PADE A :.'. LA TS . .

11.1 Derivation and Definitions .

111.2 Connection with Moment Expansions and
Inner Projections . .

IV. THE REDUCED PARTITIONING PROCEDURE .

IV.1 Non-Linear Summation of Perturbation
Expansions . .

IV.2 A Model Hamiltonian for a Finite Basis .

IV.3 The Fundamental Equations for the Reduction
Process . . .

IV.4 'i. Method of Steepest Descent and the
Coometric Sumrule . .


Page

. iii

. Vi

* vii

. .. .viii


I I I I






V. EXCITTED STATES . . 50

V.1 The Super-Matrix Formulation . 50

V.2 The Selective Reduction Procedure . 56

V.3 The Geometric Formula for a Multi-Dimensional
Reference Space . . 59

VI. NUMERICAL APPLICATIONS OF THE REDUCED
PARTITIONING PROCEDURE . . 62

VI.1 Summary of the Computational Procedure 63

VI.2 Results and Discussion: The Single
Partitioning Case . 69

VI.3 Results and Discussion: The Selective
Reduction Procedure for Multiple States 87

BIBLIOGRAPHY . . . ... 106

BIOGRAPHICAL SKETCH . . . 110












LIST OF TABLES


Table Page

1. Reduction Energies for H2 with the Coulson
Reference Function . . 71

2. Reduction Energies for H2 with a Hartree-Fock
Reference Function . .. 73

3. Reduction Energies for HeH with a Hartree-Fock
Reference Function . . 75

4. Reduction Energies for the X E State of HeH as a
Function of Internuclear Separation . 77

5. Spectroscopic Parameters for Different Approximations
to the X17- State of HeH . 85

6. Illustration of Selective Reduction for Multiple
States of Hel . . 88

7. Reduction Energies for the A Z+ State of HeH as a
Function of Internuclear Separation . 91

8. Comparison of Rigorous and Approximate Multiple State
Reduction Energies for HevH as a Function of 97
Internuclear Separation . .

9. Reduction Energies for the 1 Excited States of HeHl
as a Function of Internuclear Separation . 101














LIST OF FIGURES


Figure


1. A Comparison of the HF and Full CI Potential Curves
for the Ground (X1L+) State of Hell .

2. Low Order Reduction Energies for the X E
State of He' . . ...

3. Full CI Results for the Ground and First Excited
States of HeH . .

4. Low Order Reduction Energies for the A E
State of HeH . . .

5. Improved Excited E1 States for HeH .

6. Reduction Energies for the BI E State of HeH .


95

100

104


* O


Page


* .


. .













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

THE REDUCED PARTITIONING PROCEDURE IN
CONFIGURATION INTERACTION STUDIES

By

Rodney Joseph Bartlett

June, 1971

Chairman: Prof. Yngve Ohrn
Co-Chairman: Prof. Per-Olov LVwdin
Major Department: Chemistry

Within the configuration interaction (CI) framework, a higher order

perturbation approach has been developed which allows a drastic reduction

in the number of n-particle functions required to give the energy of a

CI wavefunction to a desired numerical accuracy. By means of this method,

significant improvements over the standard Hartree-Fock (HF) and truncated

CI calculations are obtainable. Since the procedure does not require

solutions of large secular equations, the technique also has implications

for full CI calculations. As this procedure complements the natural orbi-

tal methods, the two techniques can be used in conjunction for additional

flexibility.

The theory derived to effect the reduction process, i.e., the reduced

partitioning procedure, employs the techniques of inner and outer projec-

tions and Pade approximants to yield a tractable computational scheme with

known convergence properties. The theory is formulated both for a single

unperturbed reference function and for a multi-dimensional reference space

to use in applications to excited states. In the latter case, a method

viii






which uniformly treats the excited states as well as one which permits a

selective reduction for one specific state is obtained. These reduction

equations are also shown to be related to the method of steepest descent.

The first order solution is of particular significance. In the

examples studied, this result, which corresponds to the application of a

geometric sumrule to the perturbation expansion, is found to give sixty

to seventy per cent of the possible energy improvement obtainable from

the specified basis. This solution also has some computationally simpli-

fying features when the reference function is of the HF or CI variety.

Preliminary applications to the H2 molecule and the HeH molecular

ion tend to justify the reduction procedure for more general cases. In

these examples, no more than a twelfth order solution is required to give

essential agreement with the full CI result in each case, even though as

many as eighty-two configurations constitute the full n-particle basis.

In addition, it is found that only a third order solution is required to

account for more than ninety per cent of the possible correlation improve-

ment over a HF calculation. These calculations presented here also in-

clude potential curves for the HF, the full CI, and the perturbation

corrections for selected states of HeH .












CHAPTER I. INTRODUCTION


The majority of theoretical studies of atoms, molecules, and solids

has its origin in the time dependent, non-relativistic Schr6'dinger equa-

tion [I],


HI = ifi (1.1)
6t

More commonly, though, one focuses on its progeny, the time-independent

form


HY = E Y (1.2)
n n n

which defines the "stationary state" eigenfunctions, ([Y The complete

Hamiltonian operator for a given problem is an enormously complex entity,

whose eigenfunctions would contain all electronic, vibrational, and rota-

tional motions that are possible for the system as well as a variety of

hyper-fine interactions. This enigma is never considered in its entirety,

but, subject to the problem of interest, one makes certain assumptions as

to the form of H. Even then the equations obtained are probably still

partially coupled and require another approximation to effect a separation.

It is this process and the Born-Oppenheimer, or "clamped nuclei ," approxi-

mation [2] that lead us to the usual Hamiltonian for the electronic motion,


H = -E2 + 1/r.. > Z /r + E Z Z /R (1.3)
1. i

The quantities in (1.3) are the kinetic uaL.';/ operator and terms arising

from the couloibic potentials for electron-electron, electron-nuclear, and

1






nuclear-nuclear interactions. Within the clamped nucleii approximation

the last term is merely a constant. When additional interactions are to

be considered, they are customarily treated as a perturbation of (1.3).

Even though (1.3) is a distinct simplification, it is still impossible

to solve (1.2) for a system with more than one electron, necessitating

approximate solutions as the only recourse. In this respect, powerful

methods have been developed to approximately solve (1.2). These fall pri-

marily into two categories, variational methods, based on the Ritz varia-

tional principle [3], and perturbation theory. The former has tradition-

ally been the more important of the two, with the latter essentially

limited to the effects of external fields. However, a renewed interest in

perturbation methods is very much in evidence currently.

Since (1.3) is bounded from below, the variational theorem allows one

to approximate Y by some Y which contains several parameters to be opti-

mized, form the functional


= <(_lHji)/<(i> ,) (1.4)


and by minimizing E with respect to the parameters embedded in % determine

an approximate wavefunction and an E E. With a sufficiently flexible

expression for Y, one can thusly obtain as good a solution as desired.

In this manner the energies of a few simple two-and three-electron systems

have been calculated to experimental or better accuracy.

For less simple systems, two relatively general variational methods

have replaced the "special case" solutions of the previous period. These

are the Hartree-Fock method [4], which for molecules is invariably imple-

mented as formulated by Roothaan [5], and the method of superposition of

configurations or configuration interaction. The ordinary Hartree-Fock

(HF) wavefunction for an atom or molecule with n electrons is assumed to






be a single Slater determinant,


HF =((1)(2(2) ... *n(n)) (1.5)


composed of n one-particle functions (spin-orbitals) where each spin-

orbital is a spatial orbital i. associated with either a! or P spin. In

the Roothaan procedure, the functions ({.3 are assumed to be approximated

by expanding in a given set of functions [X },


Z = E X c (1.6)


with arbitrary expansion coefficients c The coefficients can be

determined by the variational principle (1.4) to give the lowest possible

energy for the single determinant form. Besides being conceptually satis-

fying, a wavefunction of this form possesses a degree of justification

for the prediction of ionization potentials and one-electron properties

due to the Koopmans [7] and Moeller-Plesset [8] theorems. On the other

hand, all single determinant wavefunctions are unable to properly treat

the instantaneous electron interactions, i.e., "correlation," which arises

from the second term of (1.3). This has several consequences, one of which

is that a standard Hartree-Fock wavefunction for a molecule cannot general-

ly be used to obtain reasonable potential energy curves since it will not

dissociate properly except in a few very special cases. Furthermore, one

encounters difficulties with symmetry in proceeding to open shell studies

resulting in the near impossibility of treating excited states.

The configuration interaction (CI) procedure effectively eliminates

the complications inherent in the Hartree-Fock scheme, but not without


*A common basis set is the Slater type orbitals [6], X(l,m,n) =
n-1 -a'r
r e Y(l,m), where l,m, and n are the usual quantum numbers., r the mag-
nitude of the radial coordinate relative to an arbitrary origin, Y(l,m) the
spherical harmonics, and a the orbital exponent which is to be selected in
some manner.






causing some new problems of its own. If one considers the expression

(1.6) and if there are m expansion functions [X 3, then with spin, it

follows that 2m(i.) are possible. Consequently, if 2m > n, many more

than one determinant may be generated from these ({.* by forming the
r3-
( ) combinations. In the CI method one can consider these determinants,
n
[Dk}, to be the basis for a trial wavefunction,


E D kCk (1.7)
k

with the (Ck) to be determined from (1.4). Other than the multi-determinantal

nature of the CI approach, which permits electron correlation, the primary

distinction between it and Hartree-Fock theory is that the expansion coeffi-

cients in the ([.j are not usually determined via an energy criterion but

rather fixed -- such as by orthogonalization of the Nx ) -- thereby placing

the flexibility of the wavefunction exclusively within the [Ck}. (Exclud-

ing the multi-configurational Hartree-Fock techniques, the one exception to

this statement involves natural orbital iterations as discussed below.)

Immediately, a problem with the expression (1.7) becomes apparent,
2m
since the number of determinants ( ) rapidly becomes astronomical even
n
for a relatively modest basis set, (x }. If one is interested in a state

or states of a certain symmetry, many of these determinants can be excluded

by group theory since they will contain no component of that state. Even

better, the individual determinants which do contribute to the state of

interest can be grouped into linear combinations, (BL such that each BL

is a pure symmetry function. By so doing, the expansion (1.7) can be

substantially shortened, but,even so, the number of configurations is

still likely to be very large.

If one forms the approximate wavefunction

S= BLCL (1.8)
L






by including every configuration that can be constructed from the given

basis (X 1, then the result obtained from the variational principle is

termed the "full" CI solution for the problem. This solution has some

convenient properties. First, it is invariant to any transformation

among the basis functions {X } and thus is independent of the explicit

form for the [(i.; or, equivalently, the final result is solely determined

by the initial basis (X 3. The second advantage is that by using all pos-

sible (BL}, no dubious selection of configurations needs to be made. Un-

fortunately, though, the number of configurations needed for the full CI

is still usually much too large to enable one to use all of them, a fact

which leads to severe diagonalization problems among others. In order to

avoid most of these problems, it is necessary to make a somewhat arbitrary

selection of the configurations thought to be important and then solve

this "truncated" CI problem. If so, the full CI features are no longer

generally true. In particular, the explicit combinations [.} are impor-

tant to the results obtained.

A second problem encountered, the choice of basis (X }, is a universal

one for all methods with the exception of purely numerical solutions. In

some ways, however, the selection is more crucial for CI studies than in

others, such as the HF. The reason for this is two-fold. First, contrary

to a HF calculation, if the CI procedure is employed, one is generally

interested in excited states as well as the ground state. This usually

requires a basis that can do both adequately, a task which is e:.:cec'dingly

difficult to manage. Also, there is little or no "chemical intuition"

which can u;u:' ;t orbital exponents that will allow a satisfactory de-

scription of excited states. Of course, if the basis set were large enough,

then the choice of the (X } would not be as crucial, and this is a fairly






reasonable alternative in HF theory, but in CI one is again led to the

problem with the number of possible configurations.

One has essentially attempted to resolve this dilemma with two meth-

ods, both used separately and in conjunction. These are orbital exponent

optimization, which is most effective when undertaken for each state in-

dependently, and natural orbital iterations. The first of these is self-

explanatory, although still largely prohibitive due to the extreme cost.

The second is a potent tool for improving less than full CI calculations

by exploiting the flexibility in the selections of the linear combinations

(i}. This technique makes use of the concept of "natural spin orbitals,"

introduced by Lowdin [9], which are defined as the eigenvectors of the

first order reduced density matrix,


y(xIx') = ? (x x2...xn) Y(x'x2 ..x )dx2...dx (1.9)


in the sense of


jY(xJx)cpk(xp)dx = nkk(xl) (1.10)


where nk is the occupation number associated with the natural spin orbital

ckp. From a theorem due to Schmidt and later Coleman [10], it is found that
this set of orbitals has a certain optimum property. That is, for a full

CI wavefunction, the selection of the natural orbitals with largest occu-

pation numbers as the spin orbitals from which the configurations are

constructed results in a much shorter approximate expansion which has the

property of being in maximum possible coincidence with the full CI [11].

Natural orbital iterations further utilize this feature, except now per-

taining to a truncated CI. Since such an approximation is not invariant

to the choice of the ({i), by obtaining a set of approximate natural






orbitals from the density matrix associated with this CI wavefunction, one

can select a set via an occupation number criterion from which another

trial CI wavefunction is generated. This set is expected to be an improve-

ment over the previous ({.}. Consequently, by continuing the process and

assuming convergence, the mixing of the basis functions (X 3 is thusly

"optimized" [12,13]. This technique is very useful, but it also has de-

ficiencies, one of which is the dependence on the initial selection of

configurations.

Having outlined the essential components of the two most important

quantum chemical methods, given a satisfactory basis set, the predominant

problem that emerges in progressing beyond the Hartree-Fock approximation,

as well as utilizing configuration interaction to maximum efficiency, lies

in the exorbitant numbers of configurations that need to be considered.

The treatment of this standard problem in various contexts is the objective

of this investigation. By appealing to the partitioning technique [14,15]

and perturbation theory, we shall develop an approach to this problem that

differs from the natural orbital methods, although it is entirely com-

plementary. This process, the "reduced partitioning procedure," shows

promise of being useful in a variety of applications within the configu-

ration interaction domain.

The basic idea of the procedure is to reduce the dimension of a CI

problem by grouping N configurations into M functions, such that the energy

can be obtained to a desired numerical accuracy even though M << N. The

main distinction between this "reduction" procedure and the natural orbi-

tal methods is that the M functions obtained are linear combinations of

the previous configurations, whereas a natural orbital "reduction" replaces

one set of confi.,~I' ILitio; with a second. Thus, since each to.chiiiique em-

phasizes different aspects of the problem, they can be used in conjunction

for additional flexibility.




8

The method employed to achieve this result is essentially perturbation

theory, but in a more general framework than is customary. Traditionally,

the feasibility of applying perturbation theory to problems has depended

on three elements: the existence of an exact solution to an unperturbed

Hamiltonian; the convergence of a linear perturbation expansion; and,

due to inherent computational problems, the suitability of first and sec-

ond order solutions. These are impractical restrictions for arbitrary,

many-electron systems, however, and preferably variational principles

should be incorporated with perturbation techniques to give more powerful

methods for the calculation of molecular properties and energies. The

perturbation approach we have formulated includes the possibility of using

any approximate wavefunction as the unperturbed solution, guarantees con-

vergence via non-linear summations, and allows higher order solutions to

be easily obtained.

The reduced partitioning procedure (RPP) will have several advan-

tages over the more standard methods. For instance, a full CI calcula-

tion is seldom soluble since the diagonalization of large Hamiltonian

matrices, extremely time consuming at best, is usually impossible due to

near linear dependencies. Although the reduction process would generally

require that all the matrix elements are constructed, it would only require

a diagonalization of an M x M Hamiltonian matrix for M << N, thereby pro-

viding a feasible alternative to the full CI solution. Even though a full

CI may not be the objective, one may be interested in certain intermediate

solutions which would give a substantial improvement over a Hartree-Fock

or truncated CI wavefunction. In this respect, also, the RPP suggests

some possibilities. One of these, the first order solution, yields a

dramatic energy decrease while possessing important computational simpli-

fications. This first order solution can also be connected with the






"geometric" approximation and the method of "steepest descent." For

excited states, the RPP will provide two alternative approaches which

have different advantages. One of these allows a selective treatment of

the particular state of interest. The question of properties other than

the energy is also a significant one, since an ordinary perturbation

treatment of second order properties from a CI perspective would require

a set of N-1 excited states, or, equivalently, the inversion of an N-1 x

N-1 matrix. Consequently, here also, a systematic reduction of the

space is warranted, as this would permit a reasonable approach within the

CI framework.

After presenting background material on the partitioning technique,

perturbation theory, and Pade approximants in the next two chapters, the

general formulation of perturbation theory and the reduced partitioning

procedure will be developed in Chapter IV, with the discussion of excited

states following in Chapter V. In the final chapter, numerical applica-

tions to the 112 molecule and the HeH molecular ion will be described.













CHAPTER II. THE PARTITIONING lECHNiQUE AND
PERTUR'BATION TH-EORY


The partitioning technique, as developed by LUwdin [14,15], has been

shown to be of primary importance in elucidating the inter-relationship

between perturbation theory and the eigenvalue equation as well as leading

to a theory of upper and lower bounds to energy eigenvalues. In the fol-

lowing, the partitioning technique will be briefly presented from which

the resolvent formulation of perturbation theory will arise as a logical

consequence. In addition, the important idea of inner and outer projec-

tion will be described in the last section.


II.1. The Partitioning Technique

The paramount objective of molecular quantum mechanics is the solution

of the stationary state Schr'dinger equation,


HY = EY (2.1)


where H is the Hamiltonian for the system and Y an eigenfunction. If we

assume an expansion for Y in terms of an orthonormal n-particle basis

fE) = Iflf...f 2 n..), then instead of (2.1) we obtain the matrix form of

the eigenvalue problem


1BC = G E (2.2)


where the expansion coefficients for each state are grouped into I =

( n ...). If i is complete, then (2.1) and (2.2) are equivalent.

For a finite basis, assumed to be of the order N+-I1, the eigenvalues

10






(EE ...E ) are respectively upper bounds to the exact solutions. By

defining M = H 8*1, (2.2) can be written


M = 0


(2.3)


for M = <(f MIff).

If we now consider IfE)to be partitioned into two subsets ja )and[lb)

of order na and nb, respectively, then it follows that
a b


Ml = MI
aa
ba


ab
Mbb


(2.4a)


From (2.3) and


a0 al" an
bO T bl bn

(2.4), we have the system of equations


NI C + C ,
aa ai ab bi


ba ai bb bi


(2.5a)


(2.5b)


for each i. By solving (2.5b) for bi one obtains


Cbi = -1 b ai
bi bb ba ai


(2.6)


-1
as long as Mb exists. The substitution of (2.6) into (2.5a) gives the

expression


(M M M I ) M = C ,D
aa ab bb ba ai a

or, equivalently, if written in terms of the Hamiltonian matrix,


[(11 ) + (& 1bb (11 ) )bb) ] ba ]C a = (D
aa aa ab bb bb ba ai


(2.7)


. (2.8)
a


(2.4b)






Consequently, by defining the modified Hamiltonian matrix

)-1
aa aa ab bb bb ba (2.9)

we have the altered eigenvalue problem


S= S ., (2.10)
aa ai ai 10

which yields n roots for F = P. = E..
a 1 1
The choice of partitioning is completely arbitrary, and, in the course

of this work, various choices will be made. For the present, however, let

us restrict ourselves to the case when n = 1, and call this single func-

tion f = c with C 0 0. From (2.10), we have


= H = H + H -(1 H ) (2.11a)
o oo oo ob bb bb) bo' (2.11a)

or, with E = H =. .(polcPo )l


8 = Eo + (oHj Ib) (1 bb ( bj H Ib)) -1(b H ) (2.11b)


This implicit function of F, (2.11), is the "bracketing function" which is

crucial to the theory of upper and lower bounds [15]. The terminology

comes from the fact that for any e > E then 8 < E which allows one to
So o

"bracket" the eigenvalue E .
o
The connection between (2.11) and Brillouin-Wigner [16] perturbation

theory could be obtained directly by separating the inverse matrix into

its diagonal and non-diagonal parts and then expanding with respect to the

latter by means of the relation

(A- A)-1= A-1 -1 -1 -1 1 -1+ ....(2.12)
(A (B)" = A" -A ^ A + A IBA-IBA ....(2.12)

The relationship with perturbation theory, however, is more conve-

niently presented in terms of the operator formulation of the partitioning

technique.






11.2. The Partitioning Technique in Operator Form

In order to discuss the operator formulation it is necessary to in-

troduce the projector


S= oo) (' (2.13)


which projects the reference function po from the space of interest. oD:

idempotent and self-adjoint, i.e.,


&2 = f, = e0, (2.14)


where W' indicates the Hermitian adjoint. In addition, the projector for

the orthogonall complement" to cp the space previously spanned by | b), is


P = 1 0", (2.15)


with P2 = P and P = P as before. Also, since


aP = PO'= 0, (2.16)


the projectors U and P are said to be "mutually exclusive."

In order to treat (2.1) we shall also need the definition


H = PHP (2.17)


which is an "outer projection" of H with respect to the subspace of P

(see 11.5). The outer projected Hamiltonian (2.17) satisfies the relation


Hik= EkPk (2.18)

where Yk and E are its eigenfunctions and ci', v.'.ilues, respectively.

The reference function cp is also a trivial ei: r..'t Lion of H with

e iLi.'lu e zero. All the other eigenfunctions [(k] are restricted to be

in the subspace of P. From a simple theorem about outer projections, it







is easy to see that E-k Ek in order, where [Ek) are the eigenvalues of

H for the eigenfunctions Yk [15(PTX)].

Let us now introduce the complex variable, and subtract g from

both sides of (2.18) to give


( H)k = (e Ek (2.19)


Using the fact that PY = Yk and defining the "reduced resolvent,"


T = ( )-1P (2.20)


it follows that (2.19) becomes


T k = (- Ek)'lk (2.21)


demonstrating that T and H have the same eigenfunctions. Therefore, the

eigenvalue problem (2.21) is completely equivalent to (2.18), but T also

contains the complex variable 6 which adds to the flexibility of the

problem. It can be easily shown that


T = TP = PT = PTP (2.22)


and from (2.20), one obtains the useful relationship


P(& H)T = P (2.23)


which constitutes a connecting link with the eigenvalue problem (2.1).

Let us now define a "trial" wavefunction TY which is dependent on

the variable & by


Y = cp + T He (2.24)

Since T H8cp is in the orthogonal complement to cp Ye is composed of the

reference function and a term from the complementary space. With the






definition (2.24) we also have the "intermediate" normalization condition

satisfied, that is


(po IY =

The intermediate normalization is a convenient choice to make since it is

applicable to both the discrete and continuous parts of the spectrum. As

long as the normalization integral (T Yl ) exists, Y is said to belong

to the discrete part. Using (2.23) it is seen that the trial wavefunction

satisfies the algebraic identity


P(H &)Ng = 0 (2.26)


for all the values of 8. Also, by using the fact that &+ P = 1, we have


(H 8)Yg = (p 0 ) (2.27)


where 8 is defined as
o

o = = (

Expression (2.28) is the operator form of the "bracketing function" that

is equivalent to (2.11). Hence TY satisfies an inhomogeneous equation

which becomes identical to the eigenfunction in (2.1) when & = P = E .
o o
The trial function (2.24) and the expression (2.28) are essential to the

following treatment of Brillouin-Wigner and Rayleigh-SclirUdin;'i- r pertur-

bation theory.


11.3. Brillouin-Wigner Perturbation Theory [16]

Assume that the Hamiltonian for the system can be separated such that

H = H + V, (2.29)

where V is a perturbation which may or may not be small. hli,,n if the




16

previously defined reference function cp is assumed to be the unperturbed

solution with T the eigenfunction to the entire Hamiltonian, one may

arbitrarily define


Y = W(p (2.30)


where W, often termed the "wave operator," takes op into Y. Again, as

throughout this paper, we shall require the intermediate normalization

(poI>) = 1. (2.31)


By multiplying the eigenvalue equation (2.1) by cp (the complex conjugate

of cp ) and integrating we have


E = (oI H 7) = (cpjo Ho+ ) = E + (cpjvo j), (2.32)


giving the en,.rgy difference between the unperturbed result and the exact

solution as an expectation value over cp of the quantity VW. As yet we

still have no realization for W; but, by appealing to the previous defini-

tions (2.24) and (2.28) and using the relationship H Co = E 'p we obtain


Y = (1 + T V)%o (2.33a)


and


S= ( V + VT V cp). (2.33b)


For the case of 8 = o = E we have complete agreement with (2.32) for

W = 1 + TE V, which defines the primary formulas of perturbation theory.

In all perturbation treatments, we want to have expressions dependent

on H and cp o. Thus it is convenient to define the reduced resolvent T
o O
associated with H in an analogous manner to (2.20), i.e.,

T = (& PH P)1 P. (2.34)
O o




17

T can be related to T by means of the closed form of the identity (2.12),


(A B)-1 = A-1 + A-B(A B)-1, (2.35)


now pertaining to operators rather than matrices. With the separation

(2.29) and using (2.35), we find that T and T are related by


T = T + T VT, (2.36)
o o

allowing (2.32) to be written


8= E + ( IV + VT V + VT VTIp >. (2.37)
0 0 0 0 0 0

By repeated use of (2.36), an infinite geometric expansion for Tg in the

quantity T V may be obtained,
0

T = E (T V)kT, (2.38)
0 0
k=0

leading to


S= To + E (TV) o '(2.39a
k=l

and


& E + (CPoV[ E (T V)k ]cp). (2.39b)
0 0 k=O 0

To further illustrate the connection with the more standard inhomogencous

equation approach to Brillouin-cT.1n:r (BW) perturbation theory, we can

define


k = (V)kcp (2.40a)


and


S o I V I Pk) ;


(2.40b)







then once & = P = E (2.39) becomes
o


YE =o 0 + E Pk (2.41)
k=l

From the recursion formula,



k1 = (ToV)k (2.42)


and (2.23), we have the set of inhomogeneous equations


(E Ho)Pkl = VPk k+1o. (2.43)


Consequently, as long as it is possible to solve the equations (2.43), one

can construct all the ({ck) from the unperturbed solution cp and the succes-

sive evaluation of the kl i's.


11.4. Rayleigh-Schrodinger Perturbation Theory [17]

In most applications of perturbation theory, the Rayleigh-SchrUdinger

(RS) type has been used since it generally has better convergence proper-

ties than the BWU variety, as well as not yielding equations that are im-

plicit functions of the energy, 6. This latter simplification is a conse-

quence of the unperturbed :nc1rgy appearing in the RS reduced resolvent

rather than the 8 as occurs in T Instead of expanding the fundamental

resolvent T in terms of V as before, we can instead consider


V' = V (& E) (2.44)
o

as the quantity to be used in the expansion. If so, from (2.20) and

(2.29) we find that


T = R + R V'T (2.45)
o o


with the RS reduced resolvent defined as







R = (E PH P)1 P. (2.46)


In analogy with the BW case, the resolvent and wave operator become


T = E (R V')kR (2.47a)
k=0

and


W = 1 + (R V')kR V. (2.47b)
k=0

The connection with the conventional inhomogeneous equations of RS

theory is somewhat more complicated than in the BW case due to the remain-

der terms which arise in each order from the & = E in (2.44). However,

from the definitions (2.30) and (2.32) with (2.47), it follows that


T = 0 + E (R V')k (2.48a)
k=l

and



k=O

If the expressions (2.48) are arranged after powers of V in the normal

perturbation series expansions,


E = $ + E V (2.49a)
E o k=l

and


E E0 + E 6 (2.49b)
0 k=l k

then the lower order terms may be identified as


S= R V$ (2.50)
R (V o -
32 = R (V 6 )RoV = R (V I )$
2 o l o o o 1 1






SR (V E)R (V E6)R V E R 2V
3 o 0 1 0 0 2 0 o
: = R (V I)2 2 R o
o 1 2 2 o 1

and


Cl = < o|v| o>
E1 o V o
E2= (2.51)

E3 0 2 ( 0 0 0(-1)R 0v0)

E4 = (1V3> < VRo(V-E6)Ro(V-E6)R0oVI

: 2<( VR2V l o>



Eventually the pattern emerges that the RS perturbation wavefunctions $k

can be defined recursively as

k
k+ = R (V 61 )k E R 6o E (2.52)
+1 o- 1 ~k o k-A A

with the RS energies simply


E1 =
These are the usual working definitions of the RS quantities. Using the

intermediate normalization condition, (2.52) is easily framed into the

inhomogeneous equation form by means of the simple relation


P(Eo- H )R = R (Eo H )P = P (2.54)
0 0 0 0 0 0

that is obtained analogously to (2.23).


11.5. Inner and Outer Projections

Another important technique which is often used in connection with

perturbation theory and variational studies is the method of "inner" and

"outer" projections. Since we shall have several occasions for their use,

it will be worthwhile to briefly mention some of their properties.






An outer projection of an operator A is defined as


A = 8 A'f (2.55)


for 6f a projector characteristic of the space spanned by the basis f E).

For a basis of either finite or infinite order, af may be written explic-

itly


f = )ff Iff)-l(^, (2.56)


which is easily seen to satisfy the relations 0' = 0' and 0'f = 0 In

the event that the infinite basis is complete, then O' is simply the

identity operator. For a moment let us define A as the Hamiltonian oper-

ator for the system and assume that


\k = I f[)k (2.57)


is an eigenfunction to H. Then from the eigenvalue relation


HYk= Ek (2.58)


we have


IE) ((flfif)-1 H fk Ek) = 0. (2.59)


Since it is assumed that the basis if) is linearly independent,


HfI k f T k (2.60)


where the metric matrix /f= (fE ) was introduced. Therefore, the outer

projected Hamiltonian H has an exactly soluble eigenvalue equation, which

is identical to (2.1) if If) is complete and corresponds to the varia-

tional solution if the basis is finite. In the latter case, the set of

;-.v'.'lu..- ([k} are guaranteed to be upper bounds to the exact results




22

in order, since diagonalization of Ilf insures that each Yk is orthogonal

and noninteracting with respect to those of lower energy [18,19]. A

Hamiltonian of the form H is convenient since it allows one to formulate

quantum chemical problems in a consistent fashion even though an approxi-

mate solution may be involved.

If F and G are two Hermitian operators which fulfill the condition

that


(XIFIX) > (XIGIX) (2.61)


for all possible X in the common domain of F and G, then it can be said

that

F > G (2.62)

Consequently, if F > 0, F is positive definite.

Now let us require that the operator A be positive definite. If so,

we can define the inner projection of A as


A' = A f A. (2.63)


Since 0 Or f 1, A' satisfies the operator inequality


0 A' A. (2.64)


By making the transformation


i > = A Jf) > (2.65a)


and


go>= A affI) (2.65b)


we obtain two alternative forms for A', namely the Bazley projection [20],




23

A' = h) (1hI A-1 h)1 I

and the Aronszajn form [21],


A' = Agg) (. AJ-g)-1(glA. (2.66b)


The inequality (2.64) can only be concluded for A > 0, but expressions

(2.66) will still converge to A in the limit of a complete basis, consti-

tuting a reasonable approximation even though the operator of interest

has no unique sign.

One important example of the use of an inner projection is found in

the even order terms of a perturbation series expansion. For instance,

in the RS case for R < 0, from (2.46) and (2.51),
o

E2 = (olVRoVIo. (2.67)


With Pl )= Iln), A = (-R ), and (2.64), we have R' ; R giving


E6 = (JoVR'V[% 2', (2.68)


which gives an alternate derivation of the Hylleraas variational princi-

ple [22,23]. It can be similarly shown that as long as %,1l ,..., n-

are exactly known [24],


S (2n (2.69)
2n 2n

for any n, a result obtained by Scherr and Knight in a different way [25].

(See also [26].)












CHAPTER III. PADE APPROXIMANTS


In all applications of perturbation theory one is concerned with

various series expansions for functions or energies which quite often

converge slowly or not at all. In particular, in modern perturbation

theory the restriction that the perturbation be small has been essentially

eliminated, often leading to very serious convergence difficulties that

require more powerful summation techniques in order to extract useful

information. One possible approach to this type of problem that has been

shown to be fruitful is the Pade approximant summation procedure [27].

Although all the convergence properties of Pade approximants are not

yet known [28,29], in the cases where mathematical justification is avail-

able, the range of convergence for a Pade' approximant has been found to be

vastly superior to that of an ordinary power series. In numerous other

practical applications where a rigorous mathematical proof is lacking, the

results seem to indicate that the range of convergence is actually greatly

in excess of that for the situations in which it has been proved. In the

following, the basic concept and definitions will be briefly presented

including a convenient relationship with the inner projection technique

which will subsequently be of use.


III.1. Derivation and Definitions

Consider a function f(z) which has a formal series expansion


f(z) = E a z. (3.1)
1=0

The [N,M] Pade approximant to this series is defined as the ratio of two

24






polynomials P(z) and Q(z)


[N,M] = P(z)/Q(z),


(3.2)


which are respectively of degree M and N and coincide with the series

expansion (3.1) through the (M + N)th power. By requiring the latter prop-

erty, the coefficients of P(z) and Q(z) are determined by the condition

that


f(z)Q(z) P(z) = (zm + n+ 1)


with O(m+ n

To prove that

another ratio


+ 1) indicating zero up to terms of order greater than M + N.

the ratio (3.2) gives a unique [N,M] approximant assume that

(call it P'/Q') is possible, then


f(z)Q(z) P(z) = O(zm + n+ 1),

f(z)Q'(z) P'(z) = O(zn + m+ 1).


(3.4a)

(3.4b)


Left multiplying (3.4a) by Q' and (3.4b) by Q, the first terms may be

eliminated to give


P(z)Q'(z) Q (z)P'(z) = O(zm + n + 1).


(3.5)


Since PQ' and QP' cannot be of higher order than M + N, to that order we

have


P/Q = P'/Q'


(3.6)


which establishes the uniqueness of the Pade"approximant [30].

If we write


k
P(z) = bk z
k=O


(3.7a)


(3.3)









N k
Q(z) = E c k (3.7b)
k=0


then be performing the multiplication in (3.3) and equating like powers of

z, one gets the set of equations


ac = b


a c + ac = b1
oo o 1 1


ac + a c + ... + a c = b
no n-1 1 o n n


(3.8)


a c + a c + ... +
mo m-l 1

a ,c + ac + ... +
mi-b1 o m 1


a c = b
m-n n m

a M c = 0
m-nH1 n


a c+ + a c + ... + a c = 0.
m-n o m-n-1 1 min


Since the last N homogeneous equations involve the N + 1 unknowns, {c.},

there is an infinity of solutions possible; but since the Pade approximant

is only determined to within a constant factor, by arbitrarily choosing

c = 1, the remaining (c.} can be obtained. With these c.'s the [b.) are

determined from the first N + 1 equations of (3.5) and thus Q(z) and P(z)

are specified. The explicit Pade solution to (3.3) is








a
am-nr* 1

m-al2



a
m



m a. z3
=n-n
3=n .


m-n+32

a
m-in-3






n-1
z


nil-i

a i


a
m-rn3






m j

j=n-1 -n+1 Z


* am.I1
mi-1






ri-n

1


subject to the convention that a. = 0 when j < 0, and any sums which have

the initial element of higher index than the final element are excluded.

Another form of (3.6) that is particularly convenient for computation

is the inner projection form [15(PTX ),31,32]. (See also III.2.) If we

define


a = (a.a ..a -)
i i i+N-1


A. [a i ...a -]
1- i i-l i+N-1


(3.10)


(3.11)


then for M = N + j, (3.8) can be written [33]


[N,Ni-j] = Za + z l A j+l- zAj+2 -la j-1 (3.12



where the poles are given by the solutions of A\ j+ zA\ .j 0.

The expression (3.12) could be shown to be equivalent to (3.9) by


[N,M] =


(3.9)


m
E a.zJ
j=0 I


am-n+l1

a
m- n+2



a
m

n
z






standard determinant manipulations, but the equality can be established

more efficiently by expanding the inverse in (3.12) by (2.12); then

Ij 2N-1 1
[N,NFj] = a z + zj+1 a aiA --(zA2 A .
=0 "higher order terms" (3.13)

+ "higher order terms" (3.13)


If we write


-1
Aj+l


bj+1

b j+2


bj+N


where each b. is a row matrix of N

of the inverse


(3.14)








elements, then from the properties


bk a = 6k


(3.15)


With the notation Ik meaning a column matrix with the kth element unity

and the others zero, we can establish the relations


-1
A- a = 1 1 k N
ji1 j+k k


(3.16a)


A j+2k j+k+1


1

from which


(A -2 A ) Ia-1 =
j+2 j+1 j+l j+.,


-I -I1(
Aj01 j+ 2 j+ )+=1 1


and


(3.16b)


(3.17)


(3.18)






From (3.18), (3.13) finally becomes

2N-j
[N,N-j] = E a z + "higher order terms" (3.19)
-=0

By the uniqueness theorem, it necessarily follows that (3.12) is equiva-

lent to (3.9) [33].


111.2. Connection with Moment Expansions
and Inner Projections [33]

Let us consider two operators R and A related by

-1
R(z) = (1 zA(z)) (3.20)


and the expectation value of R with respect to a normalized function *,


f(z) = (*IR(z) I). (3.21)


If R(z) is expanded by (2.35), it follows that


f(z) = E z (flA%). (3.22)
2=0

Defining a = (fIJAA I), a, is the Ith moment of A and (3.22) is the moment

expansion of f(z). For any basis |h ), an inner projected approximation

to f(z) can be given in the Bazley [20] form by


f(z) (f h) ( <| 1 zA h)>1 (allJr> (3.23)


Since the basis Ih) is arbitrary, we are at liberty to choose the functions

| ) = AA24...) (3.24)


from which it is found that


f(z) a [A zA Il 0 (3.25)

The result (3.25) can be identified from (3.12) as the [N,N-1] Pade'







approximant. Furthermore, since the expansion of (3.20) can also be

written as

-1
R(z) = 1 + zA(1 zA)-1, (3.26)


it is possible to give another approximation


f(z) ao + zat[ zA12 ]1a1 (3.27)


which is seen to be the [N,N] approximant. Similarly, the higher Pade'

approximants,


f(z) [N,N+j]'= E az + z j+(A+ zA ) j+ (3.28)
l a +_1 j+02 j+

are derived. In the event that R is positive definite, from the inner

projection property (2.64) it can be further concluded that


[N,N] 2 f(z) [N,N-1]. [33] (3.29)


In this case the moment expansion is a series of Stieltjes and many addi-

tional conclusions may be reached. For other operators that are not of

a definite sign, one can usually only treat (3.28) as an approximation

to f(z). However, since in the subsequent discussion, the [N,N-1] Pade'

approximant will be identified as a partitioned eigenvalue problem, in

this instance, it necessarily gives an upper bound for any finite basis.













CHAPTER IV. THE REDUCED PARTITIONING PROCEDURE


By a consideration of perturbation theory within the configuration

interaction (CI) domain, we shall develop a procedure that allows an

adequate representation for a CI wavefunction with only a small number of

perturbation corrections to an unperturbed reference function. This pro-

cess, termed the reduced partitioning procedure, leads to several advan-

tages over the more standard approaches, one of which is found in the

treatment of the full CI problem. The theory as formulated incorporates

variational principles with perturbation methods to yield a scheme that

has known convergence properties and one which is also very convenient

from the computational viewpoint. In connection with the non-linear

summation techniques invoked, a modification of ordinary perturbation

theory is obtained from which an alternate set of perturbation corrections

to the energy and the wavefunction are defined.


IV.1. Non-Linear Summation of Perturbation Expansions

In order to approximately solve the Schrb'dinger equation,


HY = EY, (4.1)


via perturbation theory, one usually assumes a separation of the Hamilton-

ian operator


H = Ho + XV, (4.2)


with H chosen to have an eigenfunction 0 and its associated eigenvalue
o O
31






So, i.e.,

H $ = & 4 (4.3)
0 0 0 0

Within the Rayleigh-Schrodinger framework, it is then assumed that both

Y and E can be expanded in terms of an "order" parameter X,


Y= o +U1 + 2 + ... 2 (4.4a)


and


E = 0o + XE1 + X2 + .... (4.4b)


By substituting (4.4) into (4.1) and equating the various "orders" of X,

one obtains the (Y.3 as solutions of the inhomogeneous equations

n-2
(e Ho)@ = (V E1)n E (4.5)
o o n 1 n

with the perturbation energies [(k) defined as


Ek+l =

Recalling the definitions


I = I0 ) ( 0)-j1(@I, P = 1 &, (4.7)


and

-1
R = (% -PH P) -p, (4.8)
0 0

then using R 0 = 0, (2.54), and the intermediate normalization


ePk = k0 (4.9)


(4.5) may be conveniently written as






n-2
= R (V E1 ) n-- Z R 6 oEn (4.10)


If the expansions (4.4) are to yield a valid solution, they must con-

verge to the exact eigenfunction and eigenvalue of the Hamiltonian, H;

but this may not be fulfilled for many choices of the splitting (4.2).

Moreover, it is seldom possible to obtain solutions to (4.10) without re-

sorting to some approximate method, a complication which tends to further

obscure the validity of the conventional expansions. To increase the

flexibility of the problem, we prefer to consider the more general finite

expansions

(m) = + + Y 2, + + Xm-l' + X m(4.11a)
o 1 22 m-- m

and

(m) = + XE + 2E + ... + + E1' (4.lib)


for


) = ) (4.12)
k k k

The factors (c(m)), which may be dependent on the truncation order m, are

at our disposal. From the definitions (4.6) and (4.12), the modified

perturbation energies are seen to be


Ek+1 = V ol ) = ckk+l. (4.13)

In the event that the (c(m)) for all i and m are chosen to be unity,

standard perturbation theory is regained leading to 1(m) = Y and &(m) = E

at infinite order. In other cases, though, we have an additional facility

that permits us to insure that (4.11) constitutes a valid expansion that




34

will allow *(m) and &(m) to approach and E as m increases even though the

ordinary linear perturbation series shows no convergence.

The Brillouin-Wigner quantities with

-1
T = (e PH P)-1 P, (4.14)


are defined as


k+= T )Vk (4.15)

and


ek+1 (= PIVIPk)' (4.16)


where


8 = (9o1V + VT(F)Vlo)>. (4.17)


With the previous modification, we similarly obtain

= a(m) (4.18)


and


= 0jolV ) = akm) k+1 (4.19)

Besides the conventional choice of c. or a. as unity for all i, we

have the option of determining the coefficients from the variational prin-

ciple, or more generally the Lippmann-Schwinger extremum principle [34],

where the functions (y.5 or (cpi) are chosen as the basis vectors. The

variational principle has the important advantage of assuring that the

modified energy expansion (4.4b) gives a converging upper bound in any

order. Another possibility is to determine the coefficients by a Pade'

summation of the perturbation energies given by.(4.6) or (4.16). This




35

last alternative can yield numerous different solutions of varying degrees

of desirability. In particular, the BW energy expansion, (4.17), for a

positive definite perturbation and for & less than the first excited un-

perturbed state, is a series of Stieltjes. From this it may be concluded

that determining the a(m) from the [N,N] or [N,N-1] approximants

results in the resultant t(m) for m 1 yielding respectively upper and

lower bounds to E [33]. (See III.2.) This last feature can also be re-

lated to the Lippmann-Schwinger principle [35].


IV.2. A Model Hamiltonian for a Finite Basis

As it is our intention to study energies and other molecular proper-

ties as well as is possible within the n-particle basis if) we are

necessarily limited to results that could be obtained from the full CI

solution. As has been discussed, this in itself is a worthy objective

sincee essentially all of molecular quantum mechanics makes use of some

=sis set, and,as such, the whole field can rigorously do no better than

a full CI within the selected basis.

Moreover, in the vast majority of practical applications of perturba-

tion theory, especially if higher order corrections than p, are desired,

the functions defined by (4.10) or (4.15) must be calculated in an approxi-

mate manner, usually by introducing a basis. Once the perturbation quanti-

ties are so approximated, another question arises, namely will the proper-

ties of these approximate solutions be those of the exact ones, and further,

will the approximate ones converge to the exact ones as the basis tends to

completeness.



-The basis [ff)= If f ...f ) in general contains all the configura-
tions of the correct symmetry wnich can be constructed from a suitably
chosen Slater orbital or some other type of basis set.






In order to suitably dispense with these inconveniences from the on-

set, we shall explicitly introduce the orthonormal basis Iff)by defining

a model Hamiltonian as the outer projection of the correct Hamiltonian for

the system, then


C = fHf = jifF) (ff|Hff)(ff (4.20)


for


Of = jf)

From the splitting (4.2), we also have


3C = 1H Hf (4.22)
o fof (4.22)

and


ff= V&' (4.23)


If we now partition the basis IE) into a reference function 0c and n ele-

ments required to be in the orthogonal complement to cpo,


If)= cp0o,h *. (4.24)


With this partitioning, a representation for the projector of the orthogo-

nal complement to CP is


P = 1 o o><(pol = Ih>

A useful relationship between the functions Ih) and an outer projected

operator nf = f'fff is that

(. (4.26)


*The partitioning need not be into one function and the remaining n,
but may also involve a linear transformation among the elements, Iff). The
subsequent discussion and particularly (4.26) will still follow.





37
By using the result of (11.5), it immediately follows that the model

Hamiltonian (4.20) is exactly soluble, i.e.,


EY = Ei 1 (4.27)


for


Sfi i = ( o1 C2"'Tn), (4.28)

with T the matrix of eigenvectors to the Hamiltonian matrix defined by

If). In addition, (4.22) defines anX which always has a spectrum of

exact unperturbed solutions whether or not H itself has a set of eigen-

functions. This last feature eliminates the usual perturbation theory

tyranny of limiting one to a very few possible forms for H since 3C can

be used for any form of unperturbed Hamiltonian which appears to be desir-

able. In the event that one actually has an H that yields a spectrum of

eigenfunctions,


Hok 0 k k0 (4.29)


then by choosing Iff)= 1[e o...9 ), it follows from the definitions that


Koek = k (4.30)

also. Thus, the quantity KX can be viewed as another generalization of

the ordinary theory.

Relative to the finite basis set and the XC of (4.22), the previous

expressions for the resolvents R and T become exactly [24]

R = Ih)<(hi OC oj)1(l = h>R <0(j (4.31a)

To = lh><)(lh osi h)'1( = Ah)To <(hj, (4.31b)

where the relationships P lh) = [b>)and(4.26) were used. If the functions




38

SIh) are eigenfunctions of X then the inverse matrix is diagonal giving

the more customary forms,

n Ihk)(hkl
R = E (4.32a)
o k=l o-

and

n Ihk)(hkl
T = Z 0 (4.32b)
o k=l E-

Then, subject to the intermediate normalization, the RS expressions are


n = h)n (4.33a)

n-2
a = R [(IhV -El6 jh)_ z E n- ] (4.33b)


and

En = (oVh>) n; (4.33c)


and for the BW,


Cn = >1n, (4.34a)


n = To(IhIVI h)> n-l (4.34b)

and

Sn- = (o I VI h)n (4.34c)


These expressions (4.33) and (4.34) which follow from the model Hamilton-

ian (4.20) are identical, of course, to those obtained from the Hylleraas,

Scherr, and Knight (HSK) method which yields approximate perturbation quan-

tities for the exact Hamiltonian H = H + V [22,24,25]. That is, instead
0
of solving the problem (4.1) approximately, we are defining a Hamiltonian

C = KC +7/ whose eigenfunctions and eigenvalues can be exactly determined.
o




39

Our results, however, are still slightly more general, since our 3C con-
o
tains the HSK H as a special case.

From (4.33) and (4.34) the elements occurring in perturbation theory

can be precisely calculated, and the coefficients of the general expan-

sions (4.11) and (4.18) can be chosen to guarantee convergence to the ex-

act result for 3, ensuring that the orthogonality properties of the eigen-

function to an Hermitian operator are obeyed. Furthermore, as If ) be-

comes complete, the solutions of the model Hamiltonian must approach the

correct eigenfunctions and eigenvalues of the problem, and by the varia-

tional theorem always from above. Similarly, by a selective choice of the

coefficients in (4.11) and (4.18), the energy to any order of solution can

also be made an upper bound -- and sometimes a lower bound -- which, con-

trary to much of modern perturbation theory, allows one to always know the

relation between the computed result and experiment.


IV.3. The Fundamental Equations for the
Reduction Process

In the previous section we defined a perfectly general K as a multi-

dimensional outer projection. We shall have occasion to use a form simi-

lar to this in the discussion of excited states, but for the present,

another form arising from an additional outer projection relative to co'


V= 'o o = ocp (cpj, (4.35)


is found to be an especially convenient choice. Obviously, this V0 sat-

isfies (4.3) as well as allowing us to introduce any approximate wave-

function as the reference function in a perturbation formulation. From the

partitioning (4.24), the definition (4.22), and


*This particular form for W has been previously discussed by Musher
[36] and Epstein and Karl [37]. 0




40

P I> = Ih), (4.36)


it follows that


Jcp)(o I fo f Jf)(f oo0)(p = )e ol (4.37)

for e the expectation value of the arbitrary reference function p With

this o we have the definition


V = 3C % (4.38)


The disadvantage of this particular V( is that it leads to divergent ex-
o
pensions [36]. Thus in order to obtain meaningful results, its use neces-

sitates a non-linear summation procedure to obtain the arbitrary coeffi-

cients in (4.12) or (4.18).

Another feature of the ( of (4.35) is that it only has a single non-
o
vanishing eigenvalue. This results in the R of (4.8) and T of (4.14)
o o
assuming the simple forms


Ro = P/& ; T = P/&. (4.39)


Restricting ourselves for the present to the simpler BW expressions,

in the following, we shall deduce a reduction procedure by appealing to

the properties of a moment series expansion and the use of Paddf approxi-

mants. The final closed expression can be identified as an inner projec-

tion and, as such, guarantees an upper bound in all orders. From the de-

fining equations of BW perturbation theory, (2.37) and (2.40), we have


S= (

and


k (TV)kl-1 = (ToV) ko). (4.41)






By using (4.38) and (4.39), with PY0 = 0, we obtain


S= + + I/j( pf CPeCcp) + l/ 2(co JC(P3C)2 o >+ ... (4.42)


and


qk = l/&k(P3C)kPo>). (4.43)

Then, with the definitions


S= (PC)klo) =-lh)ihjHJIh)k-l IhHIcp0> (4.44)
k-I
S h) H lhh b k > 1,


and


k- = oHk) = k -hh Ib, (4.45)


the 1/8 moment expansion (4.42) can be written as


P = + + // + l/ 2 a + ... (4.46)
o 8 o 8 1

where the previously defined BW energies are k = 0k-2/ Consequently,

given the Hamiltonian matrix in the complementary space, Hhh' to con-

struct the quantities (4.44) or (4.45), one only needs to perform various

matrix multiplications.

In principle it should be possible to solve (4.46) as an implicit

function of e, and if the convergence is sufficiently rapid, obtain a good

answer for the eigenvalue. However, this is impossible for two important

reasons. First, it is expected that the series (4.46) should be strongly

divergent. For example, for the series to converge the eigenvalues {Ci}

of Hhh should be such that -1 < i./8 s +1 for each C, [38]. These condi-

tions are unlikely to be satisfied for molecular problems. A second com-

plication is that for any truncation of (4.46) there is necessarily a




42

dependence on the energy zero-point. However, since Padd approximants are

known to have better convergence and existence properties than a linear

series, by using such a summation technique to determine the coefficients

in the elements of the modified energy expansion given by (4.19), it may

be possible to improve the situation. In particular, by applying the

[M,M-1] Pade"approximant for M N, (4.46) can be written as

e(m) = g + e l/(m) [M,M-1], (4.47)


where


[M,M-1] = (E 1/ (m) E (4.48)


and


S= [.i i+1...om i = 0,1,2,...,m (4.49)


Ek= k+l ... k+m-1] k = 0,1. (4.50)

With some slight rearrangement we have


o(m) = + C (m) E E )-I (4.51)
0 0 0t 0

This is the fundamental formula for the "reduction" process, but as

yet it is not obvious if the two previous objections are eliminated. Con-

sequently, let us now consider the functions 1i) defined in (4.44). If we

augment the first m of these functions by cp and then assume that (cpo j)

is a reasonable basis for an approximate solution of (4.1), by using the

fact that (cp |ok = 6ok, we obtain the partitioned form for the secular

equation equivalent to (4.1) as

(m) = + H (g(m)f 1 H (4.52)
0 ocp cp p ( po

where the quantities are defined as




43

A = (m)l(m), (4.53a)


S = (m)Hl(m), (4.53b)


and &(m) is the eigenvalue obtained as a function of the M basis functions

1(m)0). From (4.44) and (4.45) it also follows that


pcp = 1 02 ... m (4.54)
G2 3



m +m1 ... C2m-l

and


A = I ... m-I (4.55)
fx o0 rm-1
1 02


m-1 .. 2m-2 ,


which with m)= (m)) allows the identification with (4.51). In addi-

tion, the intermediately normalized Mth order wavefunctions are given by

1r(m) = + (m)& (m)o 1) -1 (4.56)


Therefore, the summation of (4.46) by means of the [M,M-1] Pade' approxi-

mant is identical to solving the M1411 x M+-1 secular equation equivalent to

(4.52). As such, besides elucidating the nature of the improved result

given by the Pade summation, (m) for any m is guaranteed to be an upper

bound to the full solution; and the result obtained from a variational

calculation must be independent of the energy zero-point. In the event

that M = N, we have simply carried out a linear transformation of the

original problem which is necessarily equivalent to the full CI solution.






The expressions (4.51) and (4.56) are also seen to be consistent with

the definitions (4.18) and (4.19) where the arbitrary coefficients are

obtained via the variational principle. The degree of success of the re-

duced partitioning procedure, however, depends on the [M,M-1] Pade sum

approximating the [N,N-1] result even though M << N, since then ,(m)

should be very close to 8(n) = E, and hopefully, to a desired numerical

accuracy. If so, this should effectively negate the necessity of solving

enormous secular equations that would arise in the conventional approach,

as well as leading to tractable solutions for further use.

The equivalence of the [M,M-1] Pade approximant and the variational

solution in the basis Iop Z( ) could have been concluded in a slightly

different fashion from the deduction of (111.2). This result shows that

a moment expansion of which (4.46) is an example can be related to an

inner projection which in turn is identified as an [M,M-1] Pade'approxi-

mant. The fact that it is an upper bound follows from T being a nega-

tive definite operator.

Having given the basic equations within the framework of BW theory,

it may be asked what happens in the RS case. Recalling the recursion

expressions (4.33) and (4.34), and by using (4.26), (4.36), (4.38), and

(4.39), we have

k-2
k = lh)k '; k /E Chh k- 1 k- (4.57)
o 0=1

and


'k = (>hk ; lk = hh'ok-I (4.58)

where (4.58) is slightly modified from (4.34) to be consistent with (4.44).

From these two relations, it can be seen that.




45

VW = I/ (4.59)
0

02 = 1/ e 2 Mhh 1 1/ 2 12
o 0
W3 = 1/eg3 1hhF2 E2/ A P
o 0




This can be written


A = B T (4.60)


for A = ( 2 ... ), = (1 2 m); and, where it is also ob-

served that T is of upper triangular form. The latter fact ensures that

for any order m, the variational solution relative to either basis set is

the same; that is,

=(m) = + I (m) ) (m) o (Im) a (m) (4.61)


and


e(m) = (m) iC(m) (4.62)

Although the variational solutions in the BW and RS bases are equiv-

alent, a slightly different result could have been obtained in the latter

case if the coefficients (c m)) were determined by applying a [M,M-1]

Pade approximant to the series (4.4b). This result is usually only slight-

ly different from the variational solution [24].


IV.4. The Method of Steepest Descent and the
Geometric Sumrule

In ordinary perturbation theory, the form of the inhomogeneous equa-

tions for the perturbation functions is dictated by the requirement that

(4.4) or (4.40) is convergence. In the case of a divergence expansion, it






is not necessarily obvious why the set of perturbation functions should

still be one which rapidly exhausts the space of interest, as is neces-

sary for the reduction process to be effective. To attempt some justifi-
*
cation of this point, we appeal to the method of "steepest-descent" as

employed by McWeeny [40] in connection with SCF studies.

Given an arbitray trial function, cp = cp + 8cp, where it is assumed

that (cpo 6p) = 0 and eo =0 (cpojHjo), the energy increment due to the

first order variation of the wavefunction is

() + (<6pH eo0 ) (.
68 = (CP)+ &P >4.63)



If 1 h) constitutes a basis in the orthogonal complement to cp we can

assume an expansion for 85p of the form


&p) = Ilh>) (4.64)


for e1 the expansion coefficients. If we neglect the second order term,

(69cjH 6&p), and ask for the direction of the vector in the linear

space corresponding to the maximum possible change in 8g, we obtain


6(68) = 0 = 8a 1 + [(lh|H Jlh) 68l] (4.65)


or, that t 1 is proportional to (IhJHcp ) = b as defined previously. Con-

sequently, the basic functional form for the first order correction should

be


1= lh)lb (4.66)


The optimum amount of cp1 to mix with cp o, that is the step size T, can be


*Although philosophically similar, the terminology "steepest descent"
usually pertains to a different application originally discussed by Rie-
mann and Debye [39].






determined by including the ignored second order term and requiring that


S=0 0 (4.67)


for


P(T) = (/< +Tcpl +To + )>. (4.68)

Then we have the result that


= ())Go 1 )- 10 (4.69)

where (4.66) and the definitions (4.45) were used. This result gives the

first order equations obtained via the steepest descent procedure as

g(1) = + o (e(1) 0 1) (4.70)


and


(1 = o+ 1 (1~o 1 (4.71)

These two equations can be identified as those corresponding to (4.51) and

(4.56) for n = 1, or the [1,0] Padedapproximant solutions.

We can similarly justify the higher order expressions by considering

the function (4.71) as a new reference function and repeating the proce-

dure where now (cpo 6*) = 0, but (4(1) 6>) # 0. If so, by again excluding

the second order term, we have

6 1 )H( )1) |(1) H (1) (1)


6[<6e[ >) + (<6 5 ) + ()]}. (4.72)


By assuming 6* = Ih) 2 we find that




48

C 2 ~ hhb + pb (4.73)


where the constant p indicates a mixing of some amount of b Thus, the

next function which should be employed to augment cp1 is exactly the c2 of

(4.44). By a continuation of this argument, the remaining (p's can be

similarly justified, thereby providing an alternative explanation from a

source external to perturbation theory for the special forms of (4.44).

The optimum step size subject to po' ,cp and cp2 is obtained from a form
(2) = po + 1 + I2, where l and T2 are variationally determined.

This, of course, is the [2,1] Pad~eapproximant of (4.51) and (4.56).

Another approximation could have been obtained by assuming the ) of

(4.71) to be a new reference function andaskingfor the steepest change in

the first order energy relative to this unperturbed solution. This type

of procedure has been employed in a RS framework by Dalgarno and Stewart

[41] and Hirschfelder [42,43]. For the BW case, compare also Goldhammer

and Feenberg [44] and Young, et al. [45].

Besides being directly derivable from the steepest descent procedure,

the first order solutions (4.70) and (4.71) have some other properties

which make them particularly significant. For example, their use corre-

sponds to the application of a geometric sum rule to a series, in this

case (4.46), and hence is often referred to as the "geometric" approxima-

tion. Despite the fact that such an approximation is seldom warranted by

the actual terms in the series, it has been used with a great deal of suc-

cess in a variety of applications [46,47,48,49]. This approximation --

recognized as the [1,0] approximant -- has been previously justified on a

variational basis [50], as in the present case, where it rigorously corre-

sponds to a variational solution corrected through the first order in the

wavefunction.




49

As should be expected from the connection with the method of steep-

est descent, this first order solution gives a substantial improvement

over the unperturbed result, yielding, in the caseswe have studied, about

sixty to seventy per cent of the possible energy improvement that may be

gained from the given basis set. Furthermore, the geometric solution

also has an important simplifying feature when the reference function is

chosen to be the two most common types, namely the Hartree-Fock-Roothaan

(HFR) or the CI variety. Let us consider the first case where Co is

chosen to be an HFR wavefunction and assume the remainder of the configu-

rations are generated by the substitution of a "virtual" orbital. If we

partition h) such that the single, double, etc. excitations are explic-

itly introduced,


i) = b (1) (2)... (n), (4.74)


then from Brillouin's theorem it follows that


0o = (PoHlI0 (2)h (2) IHjcp (4.75)


and


01 = (y IHl h(2) )h(2) H h (2)) (2)h. (4.76)


Consequently, the [1,0] solutions (4.70) and (4.71), relative to an HFR

function, depend solely on the double excitations. The CI reference

function will be discussed in the next chapter where excited states are

considered.













CHAPTER V. EXCITED STATES


The subject of excited states is an important one in quantum chem-

istry, since it is an area where a great deal of information, which is

not readily accessible to experiment, can be obtained from theoretical

consideration. Compared to ground states, however, the difficulties

associated with an adequate determination of excited states are propor-

tionately compounded. In particular, it is expected that the higher ex-

citations that one usually neglects in a CI wavefunction will be of more

significance for excited states, from which it would appear that a reduc-

tion process should be of even more importance. In the following, a

super-matrix formulation for excited states, which is a generalization of

the previous theory, will be presented. From a consideration of these

equations, a modification is made which allows a selective treatment of

the specific state of interest. This last result will also suggest a con-

ceptually appealing approximate treatment. As in the Hartree-Fock case,

the first order solution for excited states is found to have a computa-

tionally convenient property.


V.1. The Super-Matrix Formulation

In order to progress to excited states, it is necessary to consider

a multi-dimensional reference space f) = XXI'..X p), which has the

characteristic projector


-The multi-dimensional partitioning technique has been employed in
lower bound studies by Choi [51] and Wilson [52], where the reference
space consisted of a degenerate set of functions.







Y = y .
X

Within the space spanned by If), for


If) = lj h),


&f is the identity operator, thus


P = & = I|h)<(IhI.


Consider an eigenvalue equation,


10 = Y.E.


for some state Y. with eigenvalue E.. Using the property that


x= O< = C-8,f = '-YCf = e-fHf,


(5.4) can be rewritten as


C(0 + P)Y. = Y.E. .


From (5.6) the two equations,


SY. Ti x i E XPY i


51


(5.1)





(5.2)





(5.3)





(5.4)


(5.5)


(5.6)





(5.7a)


and


P xC i. = PE.Y. P3CPY., (5.7b)


are obtained by left multiplication of 0' and P, respectively. Solving
x
(5.7b) for PT., it follows that

PYi =TE 3X. i (5.8)


where TEi has the usual definition







TE = (E. PXCP)1 P. (5.9)
E i

By substituting the expression (5.8) for PYi., (5.7a) becomes


e& ( + XKTEI )eXYi = (5.10)


or that Q Y. is an eigenfunction to the operator 8' (K + 3CTE 3C)8 with
Xl X i X
eigenvalue E.. (See Lowdin [15(PTIV)] .) Thus, (5.10) is exactly equiva-

lent to (5.4), but a consideration of (5.10) leads us to a perturbation

treatment.

With the definition (5.1), the fact that Y. = Iff)C and the par-

titioning (5.2), (5.10) in matrix form is


yX jX + XT EiX he ff) C = |><)(Vif)C.E,. (5.11)


For [j) linearly independent and C= ( Xi), this requires that
Chi


XX C = E. (5.12)


This gives an alternative operator derivation of (2.10), which was pre-

viously obtained from a direct partitioning of the eigenvalue equation.

Obviously, the essential difference between the basic equation for the

multi-dimensional case, (5.12), and the earlier example for p = 0 is that.

now we cannot remove C to get a direct expression for the energy E..
Xi
Instead, we must solve a secular equation for an effective Hamiltonian.

This consequence ensures that the resultant multiple state solutions will

satisfy the requisite orthogonality and noninteracting properties. Since

the expression (5.12) is an implicit function of E., in practice we must

resort to the more general form


XH (P) t (P) = X (8)f (5.13)
xX\ x X






where for eachchoice of 8, an H (8) may be constructed, from which
XX
diagonalization yields p + 1 roots, ({ -...p ). The eigenvalues (E.i

are given when P = F. = E. for some i.

With the principal definition,

a () = < X |+ XT r3 >, (5.14)


we can proceed as previously by expanding (5.14) via the relation T =

T + T XT, to obtain
o o

XX (8) = IX + (IKCT 0 + CT 0XT + ..."'). (5.15)


With the definition


I k(&) = (I3C(To0C)k+l 4), k 2 0, (5.16)


(5.15) can be written as an expansion in terms of matrices


xx () = xx () + 1() + ( (+ ) + 2(8) + .... (5.17)


Now let us assume the set |y) are the solutions to an unperturbed pro-

blem, that is,


C y) = ) to, (5.18)


for t a diagonal matrix consisting of the unperturbed eigenvalues,

0o ..o 3. Then the expression (5.17) becomes the multi-dimensional
o 1 p
equivalent of the usual BW perturbation expansions which, by taking to

a certain order and then solving (5.12), yields the perturbation correc-

tions for each of the p + 1 roots. This type of treatment for degenerate

RS perturbation theory was originally given by Van Vleck [53]. (See also

Hirschfelder [54].) For a general XC we could apply various treatments
0






to (5.17) to obtain series of non-linear expressions for IXX but in

the interest of brevi\y, we shall first specify a particular unperturbed

Hamiltonian.

Analogous to thi previous single partitioning example, we shall

select for the unpert ibed Hamiltonian


3C C e-
0


(5.19)


which is easily seen Lo satisfy (5.18). Using the spectral expansion and

(5.3) we have, just as before,


T = P/
o


Then from (5.16),


!J-.+1 = 1/ k+l (y IC(PPC)k+lIl),


(5.20)





(5.21)


where the ([k) are

imant to (5.17) we

reduced, effective H:r
()
(m)
XX


e,-ndent of &. By applying the [M,M-1] Pade'approx-

in to order m the super-matrix expression for the

1 tonian matrix,


(5.22)


The super-matrices U and 14 are defined by

2 = ytt 12 m-
'r' =71 2 + 2" am-


r- U
LA+1






l-m-l


"Al' -AF2m-l
'A-2m-1


A = 0,1,...


a = 0,1,...


and


(5.23a)


(5.23b)


fk


o + (efo a )-i .o
0 0 0 1 0






Inspection of (5.22) indicates that the super-matrix product results in

a p+- x p+1 matrix. A somewhat similar kind of treatment has been given

in [15(PTXIV)].

To further illustrate the connection with the previous theory, we

can define the perturbation corrections relative to each unperturbed

state,

(i) = k- H k-k
i) = (PX)kiXi = j khhl Hxi) = 'l) h bi (5.24)


which should be compared to (4.44). Also, we have the analogous sigma

quantities


i) I (i) b I k lb (5.25)
k YXiIcPk i hh i

and for i # j,


i, j) = (x IJpP) = (xi KJ3C pi) 11h (5.26)


which are generalizations of (4.45). The terms (i are required to

describe the interactions arising from the different states |^). With

these definitions, the elements of the k matrices are seen to be

-(o)- (1) (p)
C= k k "k ] (5.27)

where the sigmas are grouped into column matrices,

(0,i)
S(i) k



(p, i)


As before, we can identify the basic reduction equation for multiple

states, (5.22), as the partitional form of an eigenvalue equation, but








this time relative to the basis jyp() () ...(p)>, for

(i) (i) (i) (i)(5.29)
= 1 9l 2 "'' m ) (5.29)

When p = 0 and i = 0, the fundamental expression, (5.22), reduces to the

single dimension case.


V.2. The Selective Reduction Procedure

The advantage of a reduction process based on (5.22) is that all

states are treated equivalently, giving a set of p + 1 improved states at

any order m. This is also a disadvantage, though, since in order to

effect this result one must use (m + l)(p + 1) functions in the computa-

tion. In a typical CI calculation, even though we expect the order m to

be low, p may be a rather large number. To avoid this difficulty, a imod-

ification of this method will be presented, which, requiring only m + p + i

functions, permits a higher order treatment for one specific state.

Although it is possible to obtain the desired result directly from

a consideration of the super-matrix equations, for illustrative purposes

it is convenient to approach the problem from a different perspective.

Therefore, let us consider the possibility of obtaining a wavefunction for

each excited state from a perturbation treatment of its corresponding

unperturbed solution. Then, as in the single partitioning example, we

assume a trial function of the form


S= (1 + TP)Xi, (5.30)


consisting of a single element in the reference space |^), and a term

from the orthogonal complement, jIh). Contrary to a rigorous single parti-

tioning, ei cannot become the correct solution Y. when 8 = = E., as

long as i contains any contributions from the other elements of jy).







However, in the usual application of the theory, the set I ) will be

eigenvectors of a relatively small CI problem and will be necessarily

mutually orthogonal and noninteracting. The only mixing allowed is

found in the Hamiltonian matrix elements involving the remaining (Xj}

and the terms from the orthogonal complement. Consequently, it could be

expected that the contribution to .i from the additional (xj} is small.

In any event, we can obtain the approximate expression


E. (Xil3j Si = (XiC+KTTC IXi (5.31)


from which the perturbation functions (5.24) are obtained, and the mt

order reduction equations analogous to (4.51) and (4.56) are derived;

(m) -(m) =o (i) (i) (i)-1 (i)
S+ 0 ( E 0 E (5.32a)
1 1 1 o o 1 o

and

(m) (m) + (i)i) (i) (5.32b)
Xi + (SE E ) o. (5.32b)


One way to retain the rigorous upper bound properties of the p + 1

approximate solutions, (5.32), is to construct the Hamiltonian and over-

lap matrices relative to ~Tol...Tp3 and to solve the secular problem.

Then one regains the noninteracting property required by the separation

theorem [18,19]. This variation is not very flexible, though, since the

relationship between the [x and ([P i) is already fixed. A superior

alternative can be deduced by proceeding somewhat differently. Within the

space of interest, and subject to (5.19), if the expression(5.14) is writ-

ten in more detail, we have


*The solution corresponding to Xo, the ground state, is still ensured
to be an upper bound although it cannot quite converge to the best solution
in the given space If)= I| ,Ih) since some of the Xj are excluded.










XX o oh ho ohq h oh hp

khQ() "ho (5.33)


Ph ho ph() hp
-1
for Q() = (1,hh hh -1" From this exact expression, (5.32) is seen

to correspond to a reduction process applied exclusively to the diagonal

elements. Consequently, assuming the perturbation functions so obtained,

( ...) are a good set to use in a selective reduction treat-

ment, it follows that we can construct several reduced Hamiltonian ma-

trices, one for each p(i), similar to (5.33), by simply taking the off-

diagonal elements into account. By so doing, we get to order m

S(i,m) +F(i) (i) (i) (i)
M) + 1P (e) eM )MP (5.34)
XX o o o 1 o

where

(i) = (o,i) (l,i) (p,i)], (5.35)
Pk k k k

and

ti,) = (i,j) (i,j) .0(i1,j) (5.36)
k k+ k+m-1"

This form corresponds to a partitioned eigenvalue problem in the basis

IXpi)), thereby allowing complete flexibility in the determination of

the variation solution to yield the best possible result. At the same

time, the proper noninteracting properties are also guaranteed, ensuring

upper bounds for all roots. Since one may construct p + 1 effective Ham-

iltonian matrices of the form (5.34), diagonalization of each one yields

(p + 1) eigenvalues and eigenvectors. From the previous discussion of

steepest descent, however, it follows that by choosing i as the particular







state of interest, the i root should be selectively reduced for m < n.

When m = n, (5.34) for any i becomes exactly equivalent to (5.33) giving

p + 1 identical secular equations. Besides the rigorous solutions, re-

sults obtained from the approximate expressions (5.32) will also be found

to be useful.


V.3. The Geometric Formula for a Multi-
Dimensional Reference Space

In a practical application of the multi-dimensional reduction pro-

cedure, one is interested in a CI problem in the space jfE)partitioned

into two subspaces Is) and Itt). Typically, Is) is composed of a selec-

tion of n configurations, neglecting the remaining nt configurations

which are grouped into Itt). The dimension of Is) is usually chosen to

be much larger than the number of roots desired. In such a case, the

first order solution, a type of geometric formula, has a convenient

simplifying feature.

Let us consider


If> = Is t) (5.37)


for jff)an orthonormal set of configurations. The equation we would like

to solve is (5.4), or equivalently,


HI (f = f E. (5.38)


In practice, though, we usually must resort to solving a truncated prob-

lem within the chosen space, IsY. By so doing, from the definitions

(4.20) and (5.37), we have


(s<13s) = (sIHIs) = ss (5.39)
Ss


and








SssD =Ds 0 (5.40)
SS S S O

for the o of (5.18) and the ns possible eigenvectors,


)s = d o .. n -1] (5.41)
s

Since we are only interested in the first p + 1 solutions, let us define

them as


S= Is) i, i = 0,l,...pSns. (5.42)

These wavefunctions will constitute a multi-dimensional unperturbed ref-

erence space to implement a perturbation solution of (5.38). Now we may

partition Iff) such that


Iff) = 1g,v, tt) = I,lh>, (5.43)


where |-) consists of the remaining ns (p + 1) functions, which may be

obtained from the basis js). The I ) are assumed to be orthogonal to

Iy), and an adequate realization would be the other eigenvectors of (5.41),

which are necessarily noninteracting with the unperturbed solutions. The

following is a consequence of this latter property.

The first order solution to the super-matrix equation, (5.22), is

-(1) + -1
T( = t + it (e yi) 1i" (5.44)


From the relation (5.27), the important components are the { (i'j)] de-

fined in (5.26) where k = 0 and k = 1. With the definition of K and by

using the eigenfunction properties of the jt), it follows that


0(ij) = (iHIJtt)(ttHIxj> (5.45)


and








(ij) = (, (5.46)


for any i,j = 0,1,...,p. This has the convenient consequence that in a

truncated CI calculation the first order improvement to the p + 1 lowest

states can be obtained by ignoring the ns (p + 1) combinations, I|),

and solely introducing the excluded configurations, Itt), into the calcu-

lation. Since the selective reduction equations (5.34) and the approxi-

mate solutions (5.32) are sub-classes of (5.22), this feature is also

present for these methods. Although we employed the n (p + 1) remain-
s
ing eigenvectors of (5.41) in this derivation, in the interest of going

to higher orders, it may be mentioned that this is not required since

only the orthogonality of the l ) to the unperturbed solutions, IY), is

sufficient to ensure the noninteracting property.

When one selects a set of configurations for a CI calculation, it is

assumed that those remaining are relatively insignificant. By determining

the first order improvement to the initial choice Is) by means of (5.45)

and (5.46), this supposition can be placed on a more quantitative basis.

Its assistance in choosing important configurations is also apparent.

Furthermore, since this solution typically accounts for more than fifty

per cent of the obtainable energy improvement, it seems that it would be

a logical extension of any standard CI computation.














CHAPTER VI. NUMERICAL APPLICATIONS OF THE
REDUCED PARTITIONING PROCEDURE


In order to undertake anumerical assessment of the reduced partition-

ing theory, we made some numerical applications within the configuration

interaction framework. Since one of the objectives of the theory is to

minimize the difficulties encountered in configuration selection, we were

interested in limiting these initial computations to molecules where the

full CI wavefunction for a basis of several functions would be easily

tractable, since the full CI, being the best answer in the chosen basis,

will provide the ultimate result to use in our comparisons. Practically,

this limited us to the two-electron systems H2 and HeH+, as the number of

initial basis functions allowed by our computer program (18) would lead

to a few thousand configurations of the proper symmetry for even LiH.

The calculations that we have performed fall essentially into two

categories: ground state studies relative to a single unperturbed solu-

tion,and a simultaneous treatment of several states by employing a multi-

dimensional reference manifold. The first of these will be illustrated

with equilibrium studies of H2 relative to a simple Coulson molecular

orbital wavefunction as the unperturbed state as well as with a Hartree-

Fock-Roothaan reference function. In addition, a complete potential curve

for the lowest + state of HeH subject to a HFR unperturbed solution

will be obtained. As an example of the multi-dimensional case, the selec-

tive reduction of the EC+ excited states for HeH will also be considered,

including some numerical justification of the approximate treatment pre-

viously described. Although this discussion is self-contained in the

62






63

sense that the objective is a theoretical solution, due to the consider-

able experimental interest in the reactions of metastable He [55,56],

these calculations also may be of some interest in connection with this

problem.


VI.1. Summary of the Computational Procedure

The computer program implemented for this problem couples the in-

gredients of a configuration interaction calculation with the perturba-

tion approach described in the previous chapters. The initial basis

functions used are the complex form of the Slater type orbitals (STO's)

located on each center in the molecule. The complex form ensures that

each orbital is an angular momentum eigenstate and thus assists one in

constructing n-particle functions of the proper symmetry. From the

initial basis of STO's, a secondary basis consisting of an orthogonal set

of orbitals is generated. In the calculations presented here, these

are either the Schmidt orthogonalized combinations of the STO's or the

Hartree-Fock-Roothaan eigenvectors one obtains from solving a preliminary

single determinant problem. By associating a or $ spin with each member

of this secondary basis, a set of one-particle pure spin orbitals is ob-

tained. Due to the symmetry of the STO's, the resultant spin orbitals

are specified by the usual 0, n, 6, etc. classification. From the spin

orbitals, n-particle anti-symmetrized products are constructed, where the

symmetry of the state E, 1, A, etc. is easily achieved from the spin orbi-

tal specifications. For homonuclear diatomics the gerade and ungerade

symmetry for the orbitals is also taken into account. By combining these

anti-s :nillltrized products into the proper combinations, spin symmetry is

incorporated into the elements of the basis set, as is the plus and minus

s:,mctry arising from reflection in the plane of the molecule for sigma







states. A configuration.for a diatomic molecule is thus defined as a

pure symmetry function in all respects, as this gives the minimum possi-

ble number that must be considered for the problem of interest.

The molecular integrals are computed from a program written by H. H.

Michels and altered by J. Kouba to give integrals accurate to at least

eight figures. Additional routines from J. Kouba's CI program [57] have

been modified by the author for inclusion into the completed program.

When Hartree-Fock wavefunctions are required, they are obtained from a

program written by H. H. Michels, but with the more accurate integral

routines substituted.

It may be worthwhile to briefly summarize the essential steps in the

remainder of the program, which is used to effect the perturbation calcu-

lations. This section is arranged to calculate CI wavefunctions relative

to a given set of configurations, and then make two sub-partitionings in

each single run. With each partitioned result constituting one or several

unperturbed states, these solutions and the remaining elements of the

orthogonal complement are used to construct the perturbation quantities

by the theory previously described. In this respect, routines have been

written which allow us to work either within the RS or BW framework.

Since a variational solution relative to the perturbation basis is desired,

this can be accomplished by generating the respective Hamiltonian and

overlap matrices from which the eigenvalue equation may be solved. In the

BW case, this is easily accomplished solely from the perturbation energies,

since from the basic definitions, (2.40), it follows that


(CplV 1 q) = ( pl V(To V) o) = (PpqlV 'F= p+ql (6.1a)

and


(plpq) = P~-q (6.1b)





65

In the RS case, the overlap matrix elements must be calculated from the

perturbation corrections (2.52), but from these overlap elements and

the perturbation energies (2.53), the Hamiltonian matrix elements are

efficiently constructed via a relationship derived by L'wdin [15(PTTX)]

giving

p q
pVl) = EP l + kE E= E -k-l. (6.2)

Within BW theory, an iterative procedure for obtaining the [M,M-1]

and [M,M] Pade' approximants by means of the solution of the implicit

equation, 8 = f(s), has also been implemented. This routine incorporates

the technique of Newton-Raphson iterations [15(PTI)], a second order pro-

cess, to increase the rate of convergence. For example, for the [M,M-1]

approximant with a single partitioning


S= () = E + E+) To (6.3)
0 0 0 1 o

from which one readily obtains the quantity


f'(t) = ( E E E -1 (6.4)


which is always negative. From a first order approximation


(1) =f(() (6.5)


the improved value,

(o) (1) _(o)) f,(0(o)
= (O + [((1) )/(1 f( )], (6.6)


follows from an application of the Newton-Raphson formula to


F(8) = P f(s) = 0. (6.7)


Furthermore, the fact that f'(&) < 0 guarantees that for (o) >e,






(1) *
G) < ; but e which can be related to the variational principle

[15(PTI)], must always be greater than 9, ensuring ccr'.verglce from

above. This process is continued until two successive &'s differ by no-

more than 10-0. Since this method requires repeated inversions of

(E 0- E 1) for the various values of 8, although it may be more desirable

in some respects, it is a more time consuming process than the equivalent

secular equation solution.

Despite the particular approach used, one encounters computational

problems with near linear dependency in the perturbation basis set. In

the implicit function process, this complication is manifested in the

inversion of the ill-conditioned matrix (-EE o- E), and, in the eigenvalue

equation, by the removal of the overlap matrix. In the program, this

possibility is investigated at each order of calculation by multiplying

the matrix with its inverse, or checking the transformed overlap matrix.

As the perturbation functions rapidly exhaust the space of interest, a

resulting near linear dependency is indicative of having essentially

achieved the desired solution, but in the process it is often of interest

to gain another one or two figures of accuracy. Consequently, to obtain

slightly higher orders of solution than would otherwise be possible, a

multi-precision extension written by C. E. Reid [58] for the IBM 360/65

has been incorporated into the program. With this package, multi-

precision routines for the Pade' approximant analysis, including a matrix

inversion subroutine, have been written. The use of these routines de-

pends on the assumption that the pertinent quantities need to be calcu-

lated only in double precision (16 figures). Then with these values, the

remaining analysis involving the more sensitive steps is accomplished

with four words (37 figures) of precision. This process can be trusted

up to the point where the linear dependency once .ig:in manifests itself,







or when the ultimate effect depends on differences beyond the range of

the low precision of the initially calculated quantities. Either of

these features becomes apparent in the course of the calculation.

In order to treat multiple states with the selective reduction pro-

cedure, the program uses essentially the same computational framework

as in the single partitioning case. To illustrate this, let us consider

the secular equation approach which requires constructing the "reduced"

Hamiltonian and overlap matrices,

(i) XX X(i)

H Ri) X= (6.8)
H(i) (i)

and
A Ax (i)
AR (i) -x(i) (6.9)

cPX

The partitionings are defined as


S = (6.10a)
XX o

(i) = (pi) W (i,0) (i,l).. (i,p), (6.10b)
XT= 1 2 m

HM(i) (i)jXlV(i)) = (i)
1

and

A = (J~y = 1 (6.11a)
XX X

A(i= (il( ) = (6.11b)
Xcp

A(i) = (i) (i) (i) (6.11c)

for the reference functions, I ), and the sigma terms defined in (4.49),

(4.50), and (5.26).








Since we have t from the unperturbed solution, the quantities

needed are Hi i ,and A To obtain these matrices, it is con-
X~P cpcp
venient to begin by calculating the approximate solution, (5.32), for

each of the p + 1 states. As this is a single partitioning approach, this

is achieved with the basic program structure. In particular, the

quantities


bi =

and the perturbation functions


() = lh)0(i) (6.13)


are generated for each i. From these, one also produces the matrices
(i) (i)
iH and A which defines the secular equation for the approximate

solution, (5.32). By solving this, the approximate energy, &, and

wavefunction, '., are given as the lowest energy solution. In the pro-

cess of obtaining each of these approximate results, by simply storing

the L Ib., H ) and A\ externally, all the basic information

necessary for the rigorous multiple states solutions is available.

The remaining elements required are the terms (o' ij) which arise

from the interaction of the unperturbed states, and are thus unique to

the multi-dimensional case. These present no problem, though, since from

the definition


C{k-) = (XjICcp' ), k 1, (6.14)


one immediately obtains


(ij) lb ( (6.15)
Following this strategy, the pk-rogram efficiently constructs the p + 1

Following this strategy, the program efficiently constructs the p + 1





69

possible reduced Hamiltonian and overlap matrices, from which the secular
th
solution gives p + 1 roots, of which the i one is selectively reduced.

As a bonus the approximate solutions are given, and as our computations

will indicate, these are likely to be sufficiently close to constitute

a satisfactory treatment for many applications.

To place the computational benefits of the reduced partitioning

theory into perspective, it should be noted that in any truncated CI cal-

culation with a basis set, the calculation of the molecular integrals is

the predominant time consuming operation, with a considerable amount of

time also required for their transformation to a secondary basis. Sine.

both of these steps are necessary despite the number of configurations ;-

used in the calculation, it seems logical to get maximum benefit from the

chosen basis by using many more configurations than is customary, or even

all that are possible. To do this in the standard approach would involve

solving an N x N secular equation for several states, which is, at pres-

ent, an impossibility for very large N. On the other hand, the reduction

process does not require such a diagonalization, but, as will be shown,

gives a result in essential agreement with the full solution by only

solving low order secular equations. It is still generally necessary to

construct all the Hamiltonian matrix elements, but even though this is

admittedly a problem for very large N, since these elements are obtainable

solely from the transformed molecular integrals and the configurations,

the extension is reasonable.


VI.2. Results and Discussion: The Single Partitioning Case

As a first example of the reduced partitioning procedure (RPP), we

considered the h.drogen molecule at the equilibrium internuclear separation


*All the calculations reported herein were done on the IBM 360/65
computer of the University of Florida Computer Center.





70

R = 1.40 a.u. The n-particle basis, Ipo h), was generated from an ini-

tial set of nine STO's on each atom, ls(1.20), Is'(1.00), 2s(1.16),

2po(1.71), 2p +(1.71), 3do(2.20), and 3d (2.00), which, using Schmidt

orthogonalized combinations, led to 34 possible configurations of 1e
g
symmetry. As this is the full CI basis, it gives the best answer obtain-

able for the chosen space. The full CI result is E = -1.1712787 a.u.

The principal contribution to the CI wavefunction is the Coulson
**
function configuration


To =)(1s o(i) is g(2)) (6.16)


which constitutes 98% of the full solution and gives E = -1.1280771 a.u.

by itself. Choosing this configuration as the reference function, the

{ck) and the ({.i were calculated from (4.44) and (4.45), and from these

quantities the reduction energies were obtained from (4.51) for each m

from 1 to 13. These numbers are listed in Table 1.

For comparison, the difference between the unperturbed energy and

the full CI result will be defined as


AE(i) = E. (6.17)
1 1

where i indicates the state of interest. If 8. is a HF energy, then AE

may be referred to as E the correlation energy obtainable within the
corr

space if), by analogy with the usual definition of correlation energy.

The most dramatic feature illustrated in Table 1 is the anticipated

extreme divergence of the sigma series which, lacking a non-linear summa-

tion technique, would be useless. However, by applying the [M,M-1] Pade

*The orbital exponents for n = 2 were taken from the optimized basis
set for R = 1.40 a.u. obtained by McLean, et al. [59].
**In fact, Coulson [60]found the best scale factor equal to 1.197
for R = 0.732 A.











TABLE 1

Reduction Energies for H2 with the Coulson Reference Functiona
(R = 1.40)




S (m) = + /( MM-] E Brillouin-Wigner
o 8 corr Sigma Elements


-1.1280771

-1.1582136

-1.1662436

-1.1696479

-1.1707700

-1.1711049

-1.1712244

-1.1712552

-1.1712685

-1.1712741

-1.1712764

-1.1712778

-1.1712782

-1.1712784


33(Full CI)


0.00

69.76

88.35

96.23

98.82

99.60

99.87

99.95

99.98

99.99

99.99

100.00

100.00

100.00


-1.1712787


0 = 0.124303

(0 = 0.279955

02 = 1.594896

03 = 1.25 x 101

04 = 1.18 x 102
43
S= 1.22 x 103
5


23
2 = 2.23 x 10
25


aEnergies and bond distances are always expressed in atomic units.







approximants to the series, it is seen that the reduction energies con-

verge very rapidly toward the full CI solution [61]. From the connection

between the [M,M-1] approximant and the variational treatment, it is more

easily seen why this result is found [61]. This is the same situation

previously observed in an application of BW perturbation theory to the

hydride ion [62], where Pade approximants were also used to sum the series

to give meaningful energies.

As has been mentioned, the fact that the perturbation functions

JcPoc1,...,"c) rapidly exhaust the space of interest causes problems

with near linear dependency. In the examples studied, this usually be-

comes apparent in the ninth or tenth order solution. Consequently, mul-

tiple precision was used to obtain the higher order solutions in Table 1,

permitting another one or two digits of agreement.

Due to the simplification inherent in the first order solution, it

is especially encouraging that this result accounts for almost seventy

per cent of the possible energy improvement. For many problems it may be

difficult to calculate the higher order terms, thus a dramatic energy de-

crease from the "geometric" result is a definite asset.

In order to consider a HFR reference function, the previous STO basis

set was used in a HFR calculation to give a set of one-electron orbitals

as eigenfunctions to the Fock Hamiltonian. From these, another 34-config-

uration CI wavefunction was constructed but with the principal configura-

tion now actually being the HF result. The energy of the HF solution is

-1.1335224 a.u., and, of course, the full CI energy is the same as before.

For a molecule such as H2, where the ls STO combination is a good approx-

imation to the 1r molecular orbital, there cannot really be too much dif-
g
ference between the Coulson function and the HF solution. This is reflect-

ed in a comparison of Table 1 and Table 2. The .elements of the sigma











TABLE 2

Reduction Energies for H2 with a Hartree-Fock Reference Function
(R = 1.40)




M (m) = + l/ (m)[M,M-] % Ef Brillouin-Wigner
o + corr Sigma Elements


-1.1335224

-1.1590861

-1.1668000

-1.1696330

-1.1707910

-1.1711357

-1.1712490

-1.1712697

-1.1712763

-1.1712779

-1.1712785

-1.1712786

-1.1712786


0.00

67.71

88.14

95.64

98.71

99.62

99.92

99.98

99.99

100.00

100.00

100.00

100.00


33(Full CI)


00 = 0.106462

01 = 0.243929

02 = 1.509162

03 = 1.26 x 101

04 = 1.22 x 102

05 = 1.28 x 10
5

= 1.27 22
023 = 1.27 x 10
23


-1.1712787







expansion are very similar as are the reduction energies for each order,

although the results with the IF reference function do seem to converge

a little better in higher order. The energy decrease given by the geo-

metric solution is somewhat better for the Coulson function, which is

probably a result of more improvement being possible. It should be ob-

served that better than ninety-five per cent of the possible correlation

energy is given by only a third order or four-function expansion.

For the HeH molecular ion a very good HF calculation, including

orbital exponent optimization, has been done by Peyerimhoff [63]. For

R = 1.455 a.u., the optimized basis she obtained is s He(1.37643),

s'H (3.87107), 2s (1.54335), 2p (2.64576), 2p' (3.24082), 3d
He He o He o He o H
(2.54147), 4fo He(3.73526), lsH(1.00949), 2sH(1.18036), 2s'H(2.56229),

2po H(1.79089), 3do H(2.41228), from which, she reports an energy of EHF

-2.933126 a.u. By repeating this HF calculation, we obtained the slightly

higher result, EHF = -2.933072 a.u., which we expect is due to our more

accurate integral computation. By augmenting this sigma basis with 2p He

(2.868), 2p l H(0.827), and 3d 2 He(2.686), a full CI wavefunction con-

sisting of 82 configurations of 1E symmetry was obtained. With the HF

solutions as the reference function, the series elements and the reduction

energies are as given in Table 3. It appears the sigma series is even

somewhat more divergent than in the H2 case which might be expected from

the convergence criterion given in (IV.3), but may also be dependent on

the adaptibility of the basis set, or the number of configurations. In

keeping with the greater extent of divergence, the convergence of the re-

duction energies is slightly worse than before, even though the third

order solution still accounts for more than ninety per cent of the available

correlation energy. The geometric result gives about fifty-nine per cent.


*This STO basis will be referred to as basis I.










TABLE 3

Reduction Energies for HeH with a Hartree-Fock Reference Function
(R = 1.455)




(m) f Brillouin-Wigner
M ) = 8 + 1/ (m)[M,M-1] % Ef Brillouin-Wigner
o corr Sigma Elements
--------------------------i-------o-


-2.933072

-2.956818

-2.966665

-2.969824

-2.971765

-2.972594

-2.972941

-2.973046

-2.973105

-2.973138

-2.973153

-2.973162

-2.973168


81(Full CI)


0.00

59.21

83.77

91.64

96.48

98.55

99.41

99.68

99.82

99.90

99.94

99.97

99.98


-2.973176


T0 = 0.260175

,1 = 1.723745

2 = 3.01 x 101

3 = 6.58 x 102



330
0 = 1.64 x 10

a = 4.39 x 105
5



0 = 5.60 x 1030
23


4







To obtain a more balanced basis set for potential curve calcula-

tions for HeH basis II was obtained by excluding the T and 6 functions,

as well as the 4f (He) and 2s'(H) functions, which are least important

to the HF solution. These were replaced by 2p He(3.00),. 2p~i H(2.00),

3d 2 He(3.00), and 3d2 H(2.00). This basis results in 61 configurations

of 1 E symmetry. The HF energy at R = 1.460 a.u. is -2.93302 a.u. In

Figure 1, the HF potential curve and the full CI curve for the X E

ground state can be compared. Since He is a closed shell atom, this is

one of the few special cases when the HF potential separates properly,

giving a HF atom and a proton. The main feature to be observed is the

relative constancy of the correlation correction. In Table 4 the reduc-

tion energies through the eighth order are given as a function of R.

From the percentages of the correlation energy listed, the convergence

with only nine functions is seen to be quite satisfactory, being poorest

at R = 2.50 a.u. and excellent at large internuclear separations. The

first order solution accounts for better than seventy per cent for larger

R values. Figure 2 illustrates this as well as the additional improvement

given by the second order solution. It should be noted also that the cor-

relation energy is not actually constant but shows a rise at R = 1.52 a.u.

and decreases slightly as R increases.

The question of the shapes of the potential curves for the reduction

energies may also be of importance if one is interested in the vibrational

spectra. Thus, to more adequately assess the differences between the

curves in Figure 2, a Dunham analysis was applied to each of these approx-

imations using a program written by Beebe [64]. These results are pre-

sented in Table 5, along with some comparison values obtained in other

calculations by Peyerimhoff [63], Anex [65], Michels [66], and Wolniewicz

[67]. Since it is not necessarily obvious just how to treat the separated









TABLE 4

Reduction Energies for the X + State of HeH
as a Function of Internuclear Separationa
(Single partitioning with a HF reference function)


R 1.00 1.40 1.46 1.52
g(m)


(1)


(2)


(3)


(4)
8


(5)


g(6)


(7)


(8)


2.86018


2.88522


2.89392


2.89705


2.89789


2.89807


2.89814


2.89817


2.89818


2.93249


2.95771


2.96703


2.97071


2.97179


2.97205


2.97218


2.97223


2.97226


2.93302


2.95820


2.96757


2.97130


2.97243


2.97270


2.97283


2.97289


2.97292


2.93240


2.95753


2.96693


2.97070


2.97188


2.97217


2.97231


2.97237


2.97240


E (Full CI) 2.89818 2.97228 2.97295 2.97243
o


AE 0.03800 0.03979 0.03993 0.04003

%AE(G() 65.89 63.38 63.06 62.78

%AE((8) 100.00 99.95 99.92 99.93


aAll energies are nri.;,aive.








TABLE 4 (continued)


R 2.00 2.50 3.50 5.00
(m)


&0


g(1)


(2)


e(3)


(4)
8


8(5)


e(6)


&(7)


5(8)



Eo(Full CI)
0


2.91057


2.93533


2.94456


2.94835


2.94964


2.94994


2.95011


2.95021


2.95026



2.95033


2.88776


2.91243


2.92112


2.92454


2.92557


2.92578


2.92592


2.92600


2.92604



2.92612


2.86746


2.89242


2.90039


2.90304


2.90371


2.90387


2.90393


2.90395


2.90396



2.90398


2.86218


2.88739


2.89507


2.89752


2.89808


2.89819


2.89822


2.89822


2.89823



2.89823


AE 0.03976 0.03836 0.03652 0.03605

%AE(1() 62.27 64.31 68.35 69.93

%AE(e(8) 99.82 99.79 99.95 100.00








TABLE 4 (continued)


R 6.00 8.00 10.00
g(m)


e0

5(1)


(2)


(3)


8
e(4)


(5)


e(6)


(7)


(8)



Eo(Full CI)


2.86169


2.88694


2.89458


2.89702


2.89756


2.89767


2.89769


2.89769


2.89770



2.89770


2.86147


2.88673


2.89435


2.89680


2.89733


2.89743


2.89746


2.89746


2.89746



2.89746


2.86142


2.88668


2.89430


2.89678


2.89727


2.89738


2.89740


2.89741


2.89741



2.89741


AE 0.03601 0.03599 0.03599

%AE((1)) 70.12 70.19 70.19

%AE(e(8) 100.00 100.00 100.00












































- 4
Ca



4-J-

*HQ
0


4-I 0


4J










0
Scd























Cd
to
















0
S-
rl




0
o










0


00



bO

0)
C!
r-4
aFK1



o ,0
1-1 %-.,









































































c\i c \


10

co


U)1
c'Y3
,I- --


03


-o


LUl


S



































a4
q.-
0

4J

r 0 >.
+ 4 o
CdB
4-i ) tiS)



00
4-) 0 -A
n M *i c o
4J 4-I 0 l
d L41 4J
CN 0 W 0 0
44 to ; '

M UO 3 C




o o H
0 *to 0 ,-"


o 0 a)
S ) wl p *

0 > Q)0 0 o


0 4 P 0
;0 0 H







!-i







+



8










C0+
C0
I -
I
I
I













I 0
I I
I
I I

I -
I
I I























SCO c C\j CN1 Q0
I 0 Q 0
I
0










"- "-- c
-














LLI






84

atom limit for the &( and (2) curves, the values calculated at 10 a.u.

were assumed to be a sufficiently close estimate to use in the deter-

mination of the dissociation energies. Anex and Wolniewicz used the

exact value for the He atom, E = -2.90372 a.u., while in Michels' study,

the calculated value of E = -2.87574 a.u. was the reference point. The

values for 10 a.u. should be very close to the calculated atom limit,

but even though our dissociation energies seem to be somewhat greater

than the comparison values, if any error is present, the magnitude should

be slightly underestimated. This is not unexpected, however, since bind-

ing energies defined in this manner are not governed by the variational

theorem. The lowest total energy found is that of Wolniewicz where E =

-2.97867 at the minimum given in Table 5. Anex's result is E = -2.97424,

while Michels reports a value of E = -2.94373 a.u.

From the previous spectroscopic parameters in Table 5, it is indi-

cated that despite the dramatic energy decrease, the actual shapes of the

potential curves for the first and second order reduction energies appear

to be somewhat further from the "best" results than is the HF curve. This

effect of some energy improvement destroying agreement with other proper-

ties is not uncommon in varieties of perturbation and CI studies. In addi-

tion to the shape of the potential curves, another manifestation of this

lies in the accuracy of the density distribution which is usually apparent

from predictions of the dipole moment or other one-electron properties.

Although the HF wavefunction is known to have certain stability fac-

tors [8], for more general cases these questions have not been adequately

pursued. However, one can typically expect better agreement for proper-

ties more sensitive than teenarwhen additional flexibility is built into

the trial wavefunction. Since these low order reduction solutions have



























+a


44
0


4-J



0U

0l (3



uo






0 0
o 4-1






03









O




rt H
o C
0 (


















0 0C
0 1
Oi 4-1



o f
0







CO


0

4-1




I-




















S I
a)





u
[I
















0




',-







0
o*


>
S3






r>
'6


a-I C' ,-4 0
'A ON 10 N
10 r- 0o o
0-I 0N C 0














0 10 10 C-
d) \D L-i 1-4
















,0 r r-o rL
0 0 00o co
0 '< 10 Ca C

M CM N







o o co ai
Co CM 10 10







co a-i 0 a-I

i-i '.o r
'0 4- co L
1o0 10 10


vo Vn V) o
0 10 -- -.- 0-
- 1- 1-1 toj









T-- a-I a-I a-I
r0" r 04

- F Co
* *I -


'.0 -
-4 -


C --i '%D 1n

IN O 1 0o 0

a--I a-I NM


0 0
o o*
r u 0 *-I
o rc o



-i4 o A r-
H4 0 00


0 I
o i
* I







I I







0; i
F- I
N I






0 I
* I
In I
en)


C- I 10)
.0 I .--
a-I I 10


0 0 0 u



141 4.4 4-44-4
w ( 0 0
pag a ex
(3 U 1





86

been obtained with only one and two variable parameters, the variational

principle apparently overly prejudices the calculation toward an energy

reduction with a consequent sacrifice in the accuracy of the potential

curves. To avoid these problems, somewhat higher orders must be included.

As these are incorporated, the near equality of the eighth order solution

and the full CI necessitates that the spectroscopic parameters be essen-

tially identical.

In all of the examples presented in this section,a dramatic energy

decrease was given by the low order solutions and essential agreement

in the higher orders. In each case, the reduction objective was met with

8 to 12 functions, even though the number of configurations varied from

34 to 82. From the nature of the procedure, it is expected that a similar

number of functions should be adequate even with many more configurations.

Since we can have only single and double excitations in the systems

studied, and the first order solution relative to a IIF reference function

depends only on the latter, it may be argued that the energy reduction

obtained from the geometric formula may not be as pronounced for larger

molecules. However, since it is generally conceded that from an energy

criterion the double excitations are the most significant, we still an-

ticipate a substantial improvement over the HFR result even for other

molecules. To obtain reliable spectroscopic properties, though, it is

important to obtain slightly higher order solutions. This feature should

also apply to observables that are determined by averaging over the

electron density.







VI.3. Results and Discussion: The Selective Reduction
Procedure for Multiple States

In order to examine the selective reduction treatment for excited

states, the previous 61 configuration, basis II calculation was parti-

tioned to give a ten-function reference space composed of the HF solution

and the nine possible singly excited configurations. By solving a CI

problem with these ten functions, a set of unperturbed solutions, I) ,

were obtained. From these reference functions, the selective process of

(V.2) was used to calculate reduction energies for the ground state and

the four lowest excited states of 1 symmetry.

The selectivity of the reduction technique is demonstrated in Table

6. Consistent with the steepest descent argument, the percentages of

AE(i) indicate that for each choice of unperturbed solution only the cor-

responding root is substantially affected. The most efficient reduction

is found in the ground state, although the reduction process for the low-

est three excited states is also effective. In each of these cases, the

first order solution is still seen to be significant.

The ground state perturbation solution obtained with the multi-

dimensional reference space is interesting, since,by Brillouin's theorem,

the single excitations cannot mix with the HFR ground state, audleave one

with exactly the same unperturbed energy as in the single partitioning

case. Since more functions are explicitly employed in the multi-dimensional

approach, the increased flexibility must of necessity yield some improve-

ment over the single partitioning case for each order. However, the energy

decrease is found to be negligibly limited to the fifth decimal place, thus

the results in Tables 4 and 5 are the same for all practical purposes.

This consequence attests to the fact that the extent of interaction :nonig

the elements of the reference space and the contribution from the







TABLE 6

Illustration of Selective Reduction for Multiple States of HeHa
(Ten-function partitioning; HF and single excitations; R = 1.460)


Unperturbed o
Energy

Reference
Function
o (1))

%AE(i)

8(8)

%AE (i)


(1)

%AE(i)
(8)

%E (i)


8(1)

%AE(i)

8(8)

%AE (i)


8(1)

%AE(i)
8(8)

%AE(i)


2.93302 1.91869 1.41766 1.10141 0.61147


2.95830

63.31

2.97294

99.97


2.93445

03.58

2.95028

43.22


2.93651

08.74

2.93887

14.65


2.93630

08.21

2.95176

46.93


1.91938

00.83

1.96910

60.95


1.95689

46.19

2.00131

99.89


1.91977

01.31

1.92574

08.52


1.92222

04.24

1.93000

13.67


1.41900

02.13

1.41954

02.99


1.41845

01.26

1.42300

08.50


1.44910

50.06

1.48031

99.76


1.41778

00.19

1.43357

25.33


1.10398

04.79

1.12804

49.62


1.10753

11.40

1.11040

16.75


1.10166

00.47

1.11245

20.57


1.12206

38.48

1.15461

99.12


0.61191

00.27

0.69951

53.52


0.61159

00.07

0.73366

74.28


0.61164

00.10

0.62182

06.29


0.61147

00.00

0.62356

07.35


aAll energies are negative.








TABLE 6 (continued)


i 0 1 2 3 4


0 (1) 2.93392 1.91880 1.41779 1.10141 0.64363
4
%7E(i) 02.25 00.13 00.21 00.00 19.55

(8) 2.94321 1.96909 1.42581 1.11437 0.73923

%AE(i) 25.52 60.94 12.98 24.15 77.67



Full CI (E.) 2.97295 2.00140 1.48046 1.15508 0.77597
Solution i






orthogonal complement is insignificant for this state compared to the

mixing of the ground state unperturbed solution, i.e., the HFR result,

and the perturbation functions derived from it.

Since basis II is very prejuduced toward the ground state, of the

four excited states, only the first, the A + state, behaves correctly

at large internuclear separation, where it converges reasonably well to a
+ 1+
He (1s) + H(ls) separated atom limit. The relationship between the X I and

A curves is illustrated in Figure 3. Although it is not apparent in

the figure, from the full CI results listed in Table 7, the first excited

state actually shows a slight minimum in the vicinity of R = 6.5 a.u.

This feature has been previously observed by Michels [66] who more cor-

rectly positions the minimum at 5.65 a.u.

In Figure 4 the first and second order reduction energies are com-

pared with the unperturbed result and the full CI solution. As before,

the extent of reduction given by the low order solutions is very satis-

factory. In fact, (2)(R) apparently converges to the same separated

atom limit as the full CI, although (1)(R) does not seem to be able to

do so. The eighth order result is observed to be essentially identical to

the full solution.

As has been mentioned, the approximate quantities defined in (5.32)

are obtained as an intermediate in the computational procedure. In order

for these conceptually appealing results to yield reasonable estimates,

the interactions arising from the remaining p unperturbed solutions need

to be small as was indicated from the ground state discussion. If so,

neglecting the mixing should not appreciably affect the spectrum of the

results. That this is actually the case in this example is illustrated in


*A correlation diagram for the lower states of Hell has been given
by Michels [66].



















-4 0 .4"











00 0 00
C 0 C4
) .0 -









to 0 t
o01 *
OO4

N N


*t N
CM CM


0
o

4-J


cc
0




01
a) 1

o .c,
44 0






0)



-4
4 0
Om
r4-







d c
0 4 1






























0







40 0
o -r4
4-1 U










U 0X
o o







0u






O) bo
4-4 *J
4










H 4-1
Q)
S0



0 .1
S4-






ci)



ffi)


0
0
Oc
O












0
OC

Lr)










0
*
























0
Co
O
































o
*


00









cc N
oC



CO C

01 CO
a n


1-1


r-4
'-4
r<
3
r-1 (d CM( O 3
s-Ae cc
0 ,-c r -s-- d .-4 -] wi 4
(*O cO B-S &l B^ c. s^ M


N
0 \0
O -O

0
-4 CM


00 c0

0M CM

CN N