19700033037.Pdf

METHODS FOR THE DIRECT CALCULATION OF

REDUCED DENSITY MATRICES

. John Peter Simons

A thesis submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

(Chemistry)

at the

University of Wisconsin

1970 METHODS FOR 'SHE DIRECT CALCUTATION OF RF,DUCED >k DENSTTY MATRICES

c John Peter Simons

Under the supervision of Associate Professor John E. Harriman

ABSTRACT

This thesis contains the results of four separate efforts to overcome some of the mathematical difficulties involved in formulating atomic and molecular quantum mechanics in terms of reduced density matrices. First the problem of wnstructing approximately N-representable density matrices which can be used in variational calculations is studied in detail. The effects of approximate N-gepresentability on calculated properties are also analyzed.

Secondly use is made of field theoretical Green's functions to directly determine the sectond-order density matrix of an N-fermion

+ NSF Graduate Fellow * This work was supported in part by the National Aeronautics

and Space Administration Grant NGL 50-002-001, system, This method is applied to the ground state of the helium atom as a test case. Thirdly a new technique for calculating, in a self-consistent fashion, the first-and second-order density matrices of atoms and molecules is put forth. This scheme makes use of a generalized random-phase approximation to obtain equations for various spin components of the density matrices. The results of applying this new method to helium, lithium, and beryllium are presented. Finallyg the first-and second-order density matrices of symmetry-projected single determinants are derived and analyzed. In memory of my friend

David Guzaliak iii

ACKNOWLEDGMENTS

I would like to express my sincere appreciation to my colleagues in the Theoretical Chemistry Institute for their encouragement, help, and friendship. I will always remember fondly these happy and productive years at the University of Wisconsin.

I owe special debts of gratitude to the following people:

Professor John E. Harriman for his wise and patient guidance of my graduate research and for his helpful criticism of my ideas;

Professor Joseph 0. Hirschfelder for providing such pleasant and stimulating surroundings and for his willingness to help the younger scientists in the institute;

My wife Peg for her critical and accurate proofreading of my manuscripts and for not complaining about sharing her eccentric husband with science, his other love;

The staff of the Theoretical Chemistry Institute for their help and friendship iv

TABLE OF CONTENTS

Dedication is Acknowledgments iii

INTRODUCTION 1

Chapter One. CONSTRUCTION OF APPROXINATELY N-REPRESEN!UB%E

DENSITY MATRICES 8

I. Introduction 9

II. Development of Formalism 12 III.Consequenees of Approximate N-Representability 26

IV. Methods of Calculation and Effects of

Truncation 31 V. Discussion of Results 38 Appendix 1. Spin orbital expansion of the spin gemfnals 42 A Appendix 2. The Relationship between and T. 44 Notes and References 46

Chapter Two. DIRECT CALCULATION OF SECOND ORDER DENSITY

MATRICES USING GUEN'S FUNCTIONS 49

I. Introduction 50

II. Second-Order Density Matrix and 'Two-Particle

Green's Function 50 V

III. Time-dependent Perturbation Expansion of

iYY> and Diagrams 56

IV. Evaluation of Two-Particle Green's Function 61

Q. ERPA in a basis 80

Notes and References 85

Chapter Three. DIRECTION CALCULATION OF FIRST-AND SECOND-ORDER

DENSITY MATRICES USING A GENE'KALIZED WDOM-PHASE

APPROXIMATION 86

I. Introduction 87

II The Generalized Random-Phase Approximation 89

IT1 0 Density Matrices in the Occupation Number

Representation 99

IV* Use of the GRPA 102

Q. First-Order Density Matrix 107

VI. Self-consistent Determination of 108 ro,rz,y0 , and yz

VI1 . Application to Helium, Lithium, and Beryllium 110

VIII.Error Bounds 117

IX. Conclusions 125

Notes and References 128

Chapter Four. FIRST-AND SECOND-ORDER DENSITY MATRICES OF

SYMMETRY-PROJECTED SINGLE DETERMINANT

WAVE FUNCTIONS 132 vi

I. Introduction 133

11. Derivation of the Density Matrices

for Point-Group Projection 133

140 111 B Example of Finite Point-Groups IV . The Axial Rotation Group 144 V. Example for Axial Rotation Croup 144

VI. The Totally Symmetrfc Components of

&?Y 146

VI1* Totally Symmetric Components of Pirst-

Order Density Matrices Before and After Projeetion 148

VIII. Discussion of Results 15 1

References 15 3 1

INTRODUCTION

For a quantum mechanical system composed of N identical, pairwise-interacting particles, the Hamiltonian operator can be written as follows:

me one-particle operator f(i) operates only on the space-spin variables of the ith particle, which are represented by the index 1.

Similarly the two-particle operator g( i,j) operates on the space-spin variables of both the ithand jth particles. The system wave function '1E/ (1,2,3,. ..N) is the solueion of the time-dependent Schrgdinger equation

with the initial condition

ai: t=t,.$ (39

If the Hamiltonian does not depend explicitly Qn time, the time-

11 dependent Schrodinger equation allows the separation of variables 2

I1 which leads to the time-independent Schrodinger equation

H Y= EY (5)

The solution of Eqn. (5) describes a stationary state of the system whose energy is equal to the eigenvalue E.

The expectation value of any observable Q for a system described by the wave function is given as

4 where Q is the quantum mechanical operator corresponding to the observable Q, and d"t: implies integration over the space-spin variables of all N particles. If the particleg which comprise the system are fermions (bosons) then the wave function (1,2*3,. ..N) must be antisymmetric (symmetric) under permutation of the space-spin variables of any two particles:

This fact allows the expectation value expressed is Eqn. (6) to be

h written in a more suggestive form. For example, if the operator Q

is a one-particle operator 3 then Eqn. (6) can be written as follows:

Eqn. (9) follows directly from Eqn. (6) by repeated application of the following identity:

The first equality in Eqn. (10) is obtained by relabeling dummy integration variables; the second equality is a consequence of Eqn, (7). Arguments similar to those employed above can ba used to rewrite the expectation value of a two-particle operator in the form given below;

where (:) is the binomial coefficient.

It: is clear from Eqns. (9) and (11) that knowledge of the M-particLe 4 wave function is not necessary for the calculation of expectation values of one-and two-particle operators. All of the necessary information Is contained in tho first-and second-order reduced density matrices defined as follows:

and

Higher order density matrices are defined in an analogous fashion. In terms of these reduced quantities, the expectation values given in

Eqns. (9) and (11) can be written as

and

For an N-particle system the wave function depends on 3N

continuous variables and N spin varhbleg whereas ?he second-m-dew 5 density matrix r is a function of only twelve continuous variables and 4 spin variables,independent of the number of particles. Moreover the second-order density matrix contains a11 of the information needed to calculate the properties of systems composed of pairwise-interacting particles. These facts, together with the knowledge that accurate wave functions are extremely difficult to obtain for any but the simpJest systemsg lead naturally to investigating the possibility of determining directly the first-and second-order density matrices.

By using the definitions of the reduced density matrices, e.g,

Eqns. (12) and (El), and the fact that the wave function obeys the

I1 Schrodinger equation, a system of coupled iategro-differential equattons involving the density matrices can be derived. In statistical mechanic6 this hierarchy of equations is known as the BBGKY (Bogolubov, Born,

Green, Kirkwood, Yvon) equations. The main difficulty with such alp approach is that the equations are coupled; the equation which should be used to determine the first-order density matrix contains the secpnd- order density matrix. Likewise the equation for the second-order density matrix is coupled to the third-order density matrixs and so QV.

There are many decoupling procedures which can be used to approximate the solutions of the first few equations in the hierarchy, but they are not based on sound theoretical foundations.

Another approach to the direct determination of first-anad seappdr arder density matrices would be to use trial density matrices in 8 variational calculation. The difficulty with this approach is that the trial density matrices must be restricted to the class of functions which are derivable from an antisymmetric or symmetric wave functi~na~ in Eqns. (12) or (13). Such density matrices are said to be N-represpf- able. The problem of restricting trial density matrices to be N-reprp- sentable is very difficult and has yet to be solved.

Presented in this thesis are the results of four separate efforts to develop techniques allowing the N-particle wave function to be by- passed in favor of reduced density matrices. En the first chapter the problem of constructing approximately N-representable density matrices which can be usad in variational calculations is studied in detail. The i. effects of approximate N-representability on calculated properties are also analyeed.

In chapter two use is made of field-theoretical Green's functions to directly determine the second-order density matriees of N-fermion systems. As a test case the method is applied to the ground state of the helium atom.

Chapter three contains the derivation of a new technique for calculating, in a self-consistent fashion, the first-and second-order density matrices of atoms and molecules. This new method makes use og a generalized random-phase approximation to obtain equations for various spin components of the density matrices. These equations are solved $+ by an iterative procedure. The results of applying this method to the helium, lithium, and beryllium atoms are also presented.

Finally, in chapter four the first-and second-order density maSx$aes of symmetry-projected single determinants are derived. The totally symnzatric components of the density matrices are also presented and discussed.

It is hoped that the research reported in this thesis will help 9 provide theoreticians with a useful sac of tools for calculating the properties of atomic arid molecular systems. If this 4s too ambitious, perhaps some thoughts for more productive research will arise in the readers ' minds. a

CHAPTER ONE

CONSTRUFTION OF APP&OXIMATEz;Y v-RBPRESENTABLE DENYXTY WRICES 9

f 4 ~~~~DUCTIQ~ 4 The pth- order reduced densicy or pwatrix, for a pure

It haa certain properties as immediate cansequences of this definition: it is hermitian, and if '4' is a nomalieed, antiEjpstric function ll'" ll'" is antisymmetric with respect to permu)tat$orrs pf the grinled or af the unprimed variables, and is of trace 1. MoTJfper, a proposed density matrix having these properties is qot necessarily derivable from an antisymetric, N-particle wave funation, The problem of determining t conditions on a proposed density matr;tx euyh that thgp axdsta at least: one antisymmetric functionc y from whi& Ohg g;iuep 2) pqn be obtahed according to Eqn. (1) is known a8 $ha (pure static) N-tepFeseatability pr ob lem. '-" It: has received much attention in fewnt years but remains uns olvecl . One reason for interest in this prqbretn iq chat it would be easier to do a variational calculation with the 2-mtrix d*nec;tly than with a many-electron wave function. 1os18*19 This is par;c&eylarlytrue when , correlation effects are of interest, 8Sn~ethey can be dealt wieh fairly well in two electron systems. If a tria1,depity rnatrqq is not N-representable, howeverg then the energy cunputRd as the traqe of $he product

of the density matrix adan appropriate reduced harpfl0qnian matrix is

3~0%in genezal an upper bwnd to tha ground st;ate ctne~gyof the system, calculated from io ~o~~~~~e~~ese~tab~~density matrix is rigorously bounded behow only by the fwest eigenvalue of the reduced harnillonian.

is al~oct hwer bound for the ground state energy of

d the system, and may be ~~i~~far below it, Unless ~~~~~~@~~~~~ility i 4 c~st~a~~tsaF@ imposed, a density matr&x cal~u~~t~~is thus of doubt-. ful value, 20-23

KL%hough N-representability conditions for the 1-matrix can be seated entirely in terms of its eigenvalues, those far the 2-magrix necessarily involve not ~nkythe matrix elensants but al.so the gemiaale

in terns of which the matrix is expanded. One of the many problema

associated with direct attacks on the N-representabikey of the 2-matr$.x

has been that the conditions on geminal8 seem to involve the very weakly

occupied geminals to exactly &e same extent as the most strongxy

occupied geminal@e l3 This "is unfortunate since the wsrikly ocoupied

geminaPs have relatively little effect on calculated 'physical properties

1% is a conseqyence of the statement of the N-representab$lity question

Pm "all DF nothing" tams, We are thps lad to consider thh goesibirity

of an approximately N-ripresentable density matrix

Some work has been done in which conditions which are ?mown to i be necessary, but not sufficient, for ~-rep~es@ntabi~i~yare impoead

and a variational calculation carried out. The results have been

encouraging. 10924925 Perhaps if enough necessary conditions are imposed / the resultant density matrices will not be too far from being M-repra-

sentable, Tr would clearly be more satisfactory9 however, if some

J quantitarively estimaLad.

In this chapter we will consider the following question, which is closeEy related TO thac of the N-represe?~rability of a 2-matrfx:

Given a set of M orthonormal, anzlsy-rmetric spin gcninals

sentability problem itself. IZ will be of interest if we can eSt&l.iSh D a measure of the escrenL tu which "' is N-representable, and an estimate of the maximum extent to which an energy calciilated from

D(2) can fall below che ground state energy of the sysiem. Since we hope to do a variational calculation, we want to obtain a density mtrix which has some free parameters in it. We wi-3.2. exmina =lie way in which maximization of N-reprcsoncabilicy intoracts with minimization of tha

L approximate energy (D

As a first scep in investigating the puestion posed above, we introduce in Section 11 a continuous measure of the N-representability of a density matrix, and define a procedure which can in principle he used to obtain the density matrix or natri ces of optimal N-reyresentability fox ;1. p,Evcv ~cI~II~I~Lhasi:> SL'C. Tl;c> iiie::hocf. could cinsi1.y bc uxtantlad to consider a generdl p-macr-ix, hut we will confine OUT discussion to the 12

Z-n?atrix, becaue of irs piiysieal hierest Lor system of pairake

interacting fermiorss. We find that tha exac1; N-representabiliLy problem

can be treated as a special case, bug the solution which then results

is not of great interest, leading in general to &e eqerivalear. of a

complete CL C~~CU~~K~GXI.

in Section LI 1 we consider thc effeccs of Zpp'rGXil:la2J H-r~~r~scncs~iLity

on variational energy calcu2aLions. We imd zhat tht nzrimrm extznt in

expamioils on our results. We also show how the genlinal set can be

systemaricalLy expanded to imprwc M-reprcsentabiliry and decrease i maximum possible errors. We conclude wi~ha C7,iscussi~nizl Secti~i:~xi i of the resulrs which have been obtained, and a view to future efforts

11. DEVELOPMENT OF FORMALISM

In this treatment we will assfrine that we have available 8 fixed

set of oxhonormal, antisyinmetric spin gsminals i $,> a They may be J. expiieitly correlated or given as C3 axpansions in Siaeer geminals.

Any member of the family of E-particle functions vh~se2-snat;rix is

expressible in terms of these spin gerninaPs can be written as determiried by

C4t

of its antisymmetric component:

is t;he N particle anzia~pme~ric: pro3 eccion operacoa where @l ...N

The summation on P extends over all 31 permraclons of the N space- i spin variables, and p is th~partfy of P. Becauso a. '1.. .N is projeccion operator and Y is nomalized we know that 0.C- u[\4!1 -< 1 and that 1-1 = 0 impLies that *N'?=O while 1-1 = 1 implies that

Substituting from Eqn. (3) irlto Eqn. (51, we obtain an 'le. .N WY. 14

espression for the measure of aneisymetry for any ilxeinber of ri:~

desired class of functions:

I-Iere 8 and O3 are the 2 pa-rticie ana &--2 ;lax:icLe J-,2 . . .M antisymmetrrc projection operators respectively and "3J ic the stmi

of all signed cransposirions between particles 1. and 2 and particles 26 3.. .N:

Our method of obtaining .the density matrices whic1.i can be expa~ded

in terms of the given sat of spin gerninals and which are 5s nearly

N-representable as possible wfll be to choose the functions( xi } so as to maximize p[Y] and then LO take d to be given by Eqn. (4; W,% In general we wlll noc be able to attain the resu1.t 1-i = L, which wrxiid

imply exact N-representability . We can, however interpret the niaximim

attainable value of 1-1 as a mea.sure oE the N-represenEability of the

density matrix we have obtained. The lorn of U1 ~n~roducedin Eqn. (8) makes ic clsar r;hzf: -1.N in seeking a maximum value of i~for noriiialiaed '? we necd coaside~

only chose functions ~~(3...N) which are antisymmetric irc the N-2 pucfcles.

The spxi geminals 4 are also antis~ymetxac,by assuznprinn, so z1ic.t ~111: 1 - expression for p can. be written as

If the antisymmetric functions xi are varied 50 3s to nlaic~ 1:

stationary subj ecr, to he noma1 ization constrainz:

a set of equations satisfied by the cprimuii {Xi; Is obrained:

where i\ is a lagrange multiplier introduced CQ a!35ure tha~Eqn, (1.1)

4 will be satisfied. In order to make progress in using Eqn. (12) to determine the

optimum { xi we introduce in kppsndix 1 a set of K (perhaps infinice) orthonormal spin orbirals Labeled by {a1,a2 in cezins of which

all the { 4i )can be expressed. For convenience we denoee the spin orbitals rhemsalves by the indices a Then i'

R i 13)

=1 16

IIere (ilcila2) is an expansion coefficie~rand brackets denote a

normalized Slater determinant

(14)

In general chert-?will. be Slarer dcccri;linanrs [a.a, : izut CtlC 21 XJ number M of spin geminals used in k!~density matrix vFLl be ma11eL^,

possibly much smaller, especially if thc spin gminais axe correlated.

can be formed from the u i’

where the set a3...afl has Seen replaced by LY for brevity .

That this expansion is possible for K~Sopcixum {)I i} can most readily

be verified by examining the dependence of the terms in Eqn, (32) on one

particle, say particle 3, and making ~seof the antisyinriletry of the

functions,

Substitution or‘ Eqn. (15) into Eqn. (10) gives

(1.6) 17

which is a hermitian fO1m in the caefficiencs c C,ia 1 . In Appendix 2 we show that the integrai zppea~ingin Eqn. (16) can be further reduced

by use of the identity

27,28 with

Eqn, (16) can then be replace2 by

The functional u[W] is thus a weighted average af the eigenvalues n of the matrix T whose elements are given in Eqn. (17). The rnaximtm WVJ h value of ;I["] is equal to the Largest eigenvalue of and occurs

h when the f Cia;- are the elements of an eigenvector of associatad

with this largest eigenvalue. Thus we are led to consider the eSgeiwaLJe

equar ion The aonnsiiration condition on ? takes the form

be solved directly, without r~q7.ti~ii2gan icerai:i.ve process Yak

reason alone may pJace the present :nethod closer to being compurarioaally 29 useful than same previous schemes. KOwCvOrs we musf; remczlljE2r r,!zar

A I R ghe dimension of T is pI(N-2) which can be an ex~remc~ly!argc V.uJ number, or even infinite if I.? is infinit;e. In procciice ir .r/ouLci

probably be necessary fz~choose 50me truncaratl set of spin orbfcale Lai, i = 1,. . .B' 1 in terms of which co e:cpresa he ~-2pariricLe ~iater

determinants. Also, we have in mind a sinuation in which the cunbcz:

M of spin geniaalls QL is rnuch less than and probably much less

than (:I ) a These approxkmLions will be discussed in 7nare dekai:

in Sect. IV.

We assume that the degeneracy of rhe eigenvalue is largest: 1a 6 and denote the sec of orthonom.al eigenvectors associated with It

by {C::) a = 1,.. .6) . Any linear combination of these will

provide a set of coefficients optimizing p[Y]:

1 19

in which the Ya are arbitrary except: for ehe nornalization condi'iion

A I will be an eigenvector of T with eigeanvadue X and chub wirh 11 = v.%\ 0 ?,

The coefficients Ya can be considered as variational pavr'aiictexs XI

the density matrix, which is gik7en by Eqzr. (2) with

optimally N-representable 2-matrices. In practice there would remain n problems associated with the construction of '& and che detemLnztiOn

h of the { Gia . Except in simple cases, the large dimension of T std.wb

would probably make the time and efr'orr required LO carry out those

steps prohibitively large. For this reason we will examine Pzrer the n possibility of decreasing rhe dimension of T by truncatioai of zlhe -.pr z set of determinanes [a3...c"fa] used in che exyaasfon of che apt;iimim{ J:,: + If the l$i;- were equivalent to the set of all Slater gemii-izls

[a.a.] , which requires that M --1 then the largest eigenvalue I3 (;I, xo would be equal to 1 and its degeneracy 6 would be (t). This is just: a restatement of the fact that (;I independent N particle 30 Slater determinants can be formed from N spin orbitals. $31 The

present approach then reduces (or expands!) to tha equitlalcnt or' a

conventional full configuration interaction calculation. In gract-ice

we hope to use a number, M, of spin geminals which is much smaller, , h

expansion, and 3q.n. (22) is replacad6 by 21

and the N-representability is acasured by

we find it of interest to r~l.atethis treatment lo the ~X~CCN-ropre-

sentability problem for a given density matrix. \.?e note that Cha

{ Cia) are not the only coefficients wikZ had to the mssrix d, which w** From the properties of we knov that there exists n scpare, hermitian

matrix d such that '*HII 1'2

It then follows from Eqn. (2.8) and

rha t 22

Ho-cJever, it is nor necessarily true chat

substitute this expression for the coefficients ia~oEcp. (1.9) a~idob%c7.l%

lJ[Yl = (32)

where

i,j = I,. e .M

we obtain the set of equations 23

The F~~ are Lagrange multipliers associared with che consrraints Of

Eqn. (33). Since we initially know neither the elments of the optiniua

V nor rhe Lagrange multipliers E, Eqra, (34) would have to be solvcd * vw* iteratively, if at al.1. If this cait be done, 'c\re &rain

(35)

We conclude thaz: if che trace of the Lagranga rmJ.t.ip1iex msrrix 2 wcwi

is 1, then the given dmsiry matrix 2s exactly N-rcpresene;;hfe. i*fi :A:- trace E .(: 1 , we interpret its va1.m as a fi&dS71%~of hoj~nczrly rthJ N-representable the density mar;rix is. Because of the large dirr'ienslon

of d 1/2TdL/2 , and because we do not know that Eqn. (34) can In fact - 'EQa*

useful. This discussion has been presented oiily to esrahlish ?lie

connection betwean our approach arid rhs exace E-represec~tabxliCy pzc;bPem

for a given density matrix.

There is another formulation of the exact N-represcntab ~i.ity problem for a given density ruacrix which avoids the difficulLies

associated with Eqn. (34), although again the solution providzd is one

in principle rather than one of pracCLcal. utillty, Given 2 hermiziaa, non-negalive matrix d of wit rzase and e set of orthonomd, bJ.... h antisyxmetric spin geminals {$x3 we fixst corrstirucr. the matrix 2: bvrt, and find the 6 different eigenvecrors associated with the largest

eigenvalue A . If X is not: unity, the density matrix cannot be V 0 24

N-representable, so the questrta-i is aiiswcrcd, Fk Xo is equal to It,

we must determine whether Lhe parmctcrs {Y,) of Eqn. (22) can ba

chosen so the Eqn, (24) yieids given values ~'fChe . The the $ij density matrix will be N-representable if and only if such coefficients

can be found. If we daiim

f 36)

and

(37) i

Eqn. (24) can be rewritten as

2 2 This can be thought of as f/! litlear equztiom in the 8 unkmwns

of which oniy 6 zrc independent, That is, if we know 'ab '

within a singli: arbitrary phase factor. As in all. systems of linear

equations, the existence of solutions is governed by the rank of the

1 matrix P and the rank of the augmented rnarrix which 2s fa-mod by

adjoining the "~o1umnvector" & to z!?-

$11 (39) . 25

are determined and how are arbitrary, my, of the unknowns %b many ensity ma%rfxis CQ be N-representable them must be a non-

trivial solucf

representability of a given density matrix. Because cjf cha large

h dimensions of T and P, Emwewar, this approach is noe: useful, an.3 is \FLp Va& perhaps better characterized as a restacemenl; rar;lzer than. 8 sulti~inilrcf the M-representability problem.

Let us review whac we have found in this section. To construct

optimally %-representable 2-matrices from a given set of orthaneinal,

antisymmetr$c spin gemfnals, we must find the eigenvectors associated

A with the largest efgenvalua h of the p1 dimensional fcaacrix T. 0 ( N-2 ) w The required coefficient matrix is then given by Eqn. (249, aid X L', is a measure of haw N-representable the reisultant density msfiix is, with the value 1 corresponding to exact E?-represencability . Additi~iaal A eigenvectors of 3: assoaiaeed with eigenvalues nearly as large a5 X tuI 0 may also be included eo increase the variational freedom, witih some loes of N-representabilicy, This approach can alse, be rclnCad to CFxi exact N-reprosentability problein for a given density rslatrfx bttc the resultant equaeions are not practical to work with. 26

As we remarked earliero one reason for iin~erestin the N-representability problem is the desire CG do variaeional calcnkaEicns , directly with &he reduced density matrix, whf.ch is a potenti.aliy simpler thing than the wave €unction. If the I-ItimrZconian for the system is of tile forin

Then a r~educedIZaniiLtanian may be defined as 32

such that for an antisymmetric wave furicrion Y

whore R is the marrix of % in the c 1 basis: UbR 9,

(443 - If Y is not antisymmetric the two expressions for E are not ,.. equivalent, and E calculated from d is not; an upper bound to ul*e. Eo the ground state energy of a fermion systzm defined by &(. 33 Since is in any case so'defined as to be a non-negative, hermitlap matrix, 27

it will be true Char:

state energy is related TO u, the measure GI? N-repTecentabflaEy

introduced in the previous seceion, and approa~hesZBLO as 11 approac:ias 1. We begin by introducing a ---remainder function Q(1,2$...N) &fine2 for any Y of the type in Eqn. (3) by

With 1-1 determined from Y by Eqn. (5). This EuncLian is antisymrcetric in the first two particles, and if only antisyrrsnerric x's are con-

sidered, it is also antisymmetric in the last N-2 particles:

If the optimum x's are used, corresponding to 1-1 = Xo, then it also

follows from Eqn. that Q is to all of the @ (12) orthogonal i 28

* It is clear also, by the vari.a;ion principle, ihac E > Eo. W:3 will

r\ -. consider the possible difference between E and E defined in terms

of the density matrix. Of course any uthar symxietric 0re-m two-

electron operator can be substiwted for the :i2nFltonian if an

appropriate reduced operator is also defined, but the compa.rison with 1 has an analogy only for operators chat: are bounded balow. Eo We Look first at the expectation valve of % wich respect to 9

I i

! 29 or

(53)

t since

Then using the triangle inequalicy we Ei.nd

u n IuE - p2E1 5 tI-k 2!-t]. (55)

A more useful relationship can be obzaained by usixig bounds for the terms on the right haad side. By the Schwarz inequality

2 It should be noted that 'k the square of a two-electron opayator, is itself a two-electron aperacor. It follows chat

The remaining tern can be bouaded as

where l@lmax is the eigenvalue of maximum absolure value associated with 'ICin the space spanned by the Slater spin geminals [a.a.J. I.3 This follows from the expansion of Eqn. (15). 30

We have seen earlier L'nar. in the ces~sf the optimum ! xi} when

p = Xo, i-2 is orthogonal to all the spin geminsls $i. Xn this case iC follows that

where is the eigenvalue of maxhmi? abscluta value aasoclaixd

with % iil the difference space berxeen rha'c spanned 3y ali chc; Slate.:

geminals and that spanned by the e The dimsnsiun of thio diffe-renze 'i 'i R space is (*) - M. Our bound on the energy difference 4s rhw

and if the optimum xi are used, 1 e I,, can be replaced by 1 B ' Inlax.

w The occurence of g multiplying E is somewhat unifortunate, but not

really serious since presumably p-4. in cases of intorest.

Since both terms on the right hand side of Eqn. (GO) conrain

(14) the bound can be improved by modifying the sph geminal S~L60

increase 1-1 In the limit when 1-I becomes equaL to L the bourid

Ar*

goes to zero and E =i E. For 's somewhat less than 1, if the

optimum have been used and the term involving e'[maX makes x i 1 a significant contribution, the bound can be reduced by a particular

augmentation of the spin geminal basis. We add to the set {@i 1 the

eigenfunctions o€q , within the difference space, which are associated

with large eigenvalues. Because the new differepce space which re~ain::

does not contain these high energy functians, the effect of chis 31

procedure will be to reduce 1s’ I,, and thus to i~proveour Sound,

Such an augmentation will clearly not decrease the ~ptiin~~~~value of p.

Of course we could augment the spin geminal basis liy adding * of the functions j,;l the difference space. We wc-ufd then bava tr, deal R\ with a set of 1 functions equivzlent ta rhe full 542 of Siarcr gerniwil~ (52

Glearly the possible practical utility of the ITA~T,~IO~described

above for the construction of optimally N-representable densi~y

matrices is dependent; on our abiiity to eonstlruct; the %l.*It Et\ I ,-. .,::I-2k dimensional matrix T and LO find the eigenvectors Cl:’ 1 assuciaced %.a%%

with the largest eigenval.ue Ao. Let us first: turn oiir attention to * rz the evaluation of the elements of T. The operator 2: is defined in WN. Eqn. (18) and the matrix elements of interesr arq those of Eqn. (3.71,

Using the well known rules for evaluating matrpx elemeqts of ana-

and two-particle operators between Slater determinants, we obtain ~1-19

h following expressions for contributions to ‘io,jo:

if &-a k=!3-P, 0 o the-wise (6% 32

& (3,4)$.(3,4)dr if a.4 . .I

(63)

The notation a=@ means that xhe rwo sets i a3...%T 'i and f33.. 'eN 2

are identical; = B 6 means that the two secs differ pniy in a-aK - n the unequal indices yc and B,,,' the other indices in the a set. being the same as the other indices in the f3 se?;; and a-%-all = @-@,-B, means that the two sets differ in the two indices 4r et and f,sB, 35 only. If st~ongZyorthogonal geminals are used, aLI rhe iEntzgraLs zxe zero except in Eqn. (61) and the third ease in Eqn. (631, Since the spin geminals {(ji 1 can also be expanded in terns of rhe spin orblcal set. cy. 1 these results can be expressed entirely in terms of the expslnsion coefficiqnts

of Eqn. (13) which are datermined by 33

should be noted Ehac only the coefficicnrs wit11 are raqufred It a L c: "2 in Eqn. (13). It is coiivenient for the expressions we now wane, and cm-

sisterit: with Eqns. (13) and (64) to allow eirher GX~G~with (i /a2aL)

- (iia a 1. tie expressions for r;he integrd~sczm t~ieizbe kr%t.een 12

if a-g=B-B A m

0 0 .the rw is e

if a-%-aR = B-Bm--Bn

otherwise 34

These integrals provide all Zhe Lrdorinarion we need to evaluate the

h h elements of T. Because of the large dimension of WtC'T it will bt? difficult, or at least time consuning, eo find the largest eigewalue and tho eigenvectors associated with ir. Marry af the macrix eZemenil'cs ax6 zero9 but even though the matrix is quit& spars2 it dx:s rivt q 2.;- r"G have 2-,7 block structure which might aid in Ehe diagonalization,

A The trace of & is of some intexsst. Sici1.d zhf: pwir;zve q-Liant:lty p[\y] is expressed in Eqn. (19) as a pozcncialL7 arbi~rcii:y -dtig'nl:ed

h average of the eigenvalues of T, all the ar,ymvaltJes must be nom-aega6iveI WrW c. and 6 thus cannot: exceed zhc trace of z1. This qua~tiry:,an Zse X0 W& evaluated from the diagonal matrix element expressions include6 a%t?irea

It; is found that the expansion coefficiencs occur only in sums rhat

The trace is thus inde5kndent of the expansion coefficients. XG is

h (R-2) 1 trace T I- 2M -- %am-, 24 f (X-X) f

We note that it is simply proportional to M, thc nu~haror" spin gem:nc:Lii - in the sets. If M = (;) , then the trace is (i).WE: expeer that in this case the eigenvalue 1 will occur with degeneracy (;I. If 11 <(;Id h then trace T (N") w We have noted previously that the nwnber R of spin or'biza1.S (d.3 L may be very large OK even, if correlated geminds are used, infiqFte.

It may thus be necessary to Cruncaze the sdt, using only some smaller number R'. In addition we may find it a practical necessity to use only' R' of the possible N-2 particle deze-rminants made up from these 35

spin orbitals Eit;her of these tuuncaeiona will affect our estimation

the elements d which the optimally of p and also sg we ascribe to ?+representable densicy matrix,

As an initial step in che estimation of the consequuncefi of such

truncations, we note that much of what; we have done nbovci is in €86~valid

for any choice of basis sace The particular choice [ai i=1, is 9 merely one which is convsnionr, rand Is cap&le of leadfzig to oprrir;num reeultx. Eqn. (9) defines ~J[‘u]: for any {xi)$ anci so long as these fuactions are antisymmetric, the expression of Eqn. (10) fsllows. If the x, are. ax- panded in terms of ~omeset of Slacer determZnants built; up f.rm CWE;~S%LOXC~~L

spin orb$tals, we can arrive at Eqn. (19) and Lhe opt:itmm expansion 60-

efficients are obtained from Egn. f20>+ The optimum density matrix hw d iwr i given by Eqn. (24). Hone of Chis requires that tho set: of spin o.rbi.t;ala

be complete for the expansion of the spin geminals 9, or chat the sum

over the Ea-2 particle determinants include all possible choices. Tf

these condieions are not: met, we will not be able in. general to ateafn

the trulgf OpCh’lUIU OY’ find the truly best & possible far the given

set tt$& 1- This loss oE complete optimization may be oEEset by gehs

in convenience, however. Wrsl will+ stiiL1 be able GO estimare the conse-

quences of only approximate N-representability and rhus to decide in a

given case if the resulrs are good enough.

In deciding which spin orbitals and which determinants to include

in the truncated sets we must be guidad by the fallowing considerations:

A want; che largest eigenvalue of lo$ EO unity ir;s we vcA*T* to be close and

degeneracy (QP ashe ntatnbor of othaz aigenv~luesnearly deganaraze wiLh it) to be sufficiently large to give good varinClonal freedom, but nof: so

large as to return us effectively to the CL problem. IC is difficu3.k

to ~o~rn~~a~~sposifis cricoria in t;he general caser Tho fact: chat we

are trying to make Y as nearly antisymmetric a8 yoosfble, togather

however, Those spin orbitals which figure significantLy in fiZae i 9%.I 1 must be fncluded, For a general set of spin orbitals [a * 1, wo cain i i. I'

is large for any k Similarly, the determinants which are mwr, :impcrszn"c,

to include &re those containing the greatest numbers of tl-LC;:mwr' Sa~portxis;

spin orbitals *

Let us consider finally the grobkem of using che density rnatrices

resulting from the above prdcedtares to calculate properties of our sgs~em.

We suppose that we have chosen not only the set of spin geminal8 { $i 1,

but also a set of spin orbitals .[ avi,i=l.a .R' 1 and seleclied soma S

A of the N-2 particle detemninants made up from ehem, We suppose zhaa;, T WbL 37

with

When our interest is in clze energy of the system we inzroduce the,

Thq dimension of the marrix & which must be considerad is the sum 02

Che degenexacies of the eigenvalues which 1iz.ve been includad he lowest eigenvalue in Eqa. (73) is our approximation to the groi:-d state energy of the system, Its associated, normalized eigenvector can be substituted into Eqaa. (69) to determine an approximation go the

2-matrix for the system in lies ground etatea,slnd from this, other properties can be ~~~~~irn~~*In the sme way, highax- eigenvalues and their asaociotead

~~&~~a~~~~o~~can &e usad co approximate gropertias of excited states of

have to work with a nattix sf infinite dimension6

For the ground sr&eP where our ~~p~~~irna~~a~should bc rha best,

A N we can calculate 1-1 from Eqn. (27) and bound /E - pE{ by a6ir.g Eqn, {go),

63 Since E is an upper bound to the true ground stare energy, this astablushas

a maximum ox1 the extent; to which our estimate may fa21 below the true value* Of course if ehe {$,I are poorly chosen or we have too few

linear variatiabaal parmeters our value may be far above the true value.

We have propoead haze a niosTriod wheraby, glvan eonie set of spin;

gemiwals, we can find the 3-matrices axpressibLe in term of them which are as nearly N-representable ;as possible, Wa introduce a family of yave functSons, not necessarily aptisymmetric, which lead go 2-matrFqes

involving only the given set 0% spin geminals. The norm of the antisymmerric

component of 6uch a normalized wave function is taken a@ a measure of tha 39

nearly N-representable density matrix I The lave functinfi itself need

sentability and the deizsrty niatrLLx arc dercrm~nodfrom zhe ~ig:~~valucn

A and eigenvectors of D matrix '2. We have givein cxp.i.i.~itcx:Jressiaris

for the elaments of this matrix.

Y-. If the larges2: eigenvalue of "*T is degenerate, then variakiozlal parameters occur in the density matrix. Lhere no degeneracy, i If is ,,. or parameters desired, large eigenvalacs of T may if more are other VM. be

iachdecl, Variacional freedom is then gained at the expense of ftT-repre-

sentzbillty, Of COUXSQ the spin gemiaals hense selves can be varied, 'but:

aacb change isk the spin geminals requires a re-evaluation of E-rzpresenz-

ability. We have also investigated the N-reprasentd~ility of a dansiry

matrix for which both spin geminzls and expansion coeffi.cfenrs are given.

The problem here is more difficult and although 6 iiev restatement of r;ha

exact N-representability problem results, it is doub~fulChar: practiical

utility will be found in this case. The treatment for i: giva density

matrix is &hus of interest Frimarify in relating the preseat treataent

to other artacks on the N-representability problem. This is not a sev~re

d Limitation, however, since we are less interested in resting a given

density matrix than in obtaining density matrices with embedded variational

parameters such that exact: or approximate N-representability Is main-

tained gs th~parameters ELTO adjusred to miniruize the energy of the. systctn.

Because the energy calculated from a non-N-representable density

matrix is nut an upper bound to the true ground scate energy of the system!, it is necessary to estimate the consequences for such a calwlatfon of having only approximate N-representability We have obtained a u bound on the difference 'beeween E, an energy determined from the density h ziatrfx, an$ E, determined. %rim an ancbspmetrfc wave function. Since

A upper bound to the &rw energy, this esZablishas a 1irfd.t on , Y how far E could possibly be below the true energy. The drlffaarence

betwoen these two energies can be reduced in a systematic way by expanding

tha basis set,, and becomes ZBXO aFj exact H-rapresentability is approachad.

The ~~~n~i~~~~~~~~~1~~with QUfG approach lies in the large size

h 0% the matrix Xt caw bsa ~~~~~~~b~~ in siza t;~Gbei full CI marrix for the problem sf i~~e~~~t~for a basis set of a gfven ~ize. WE! hav~

thus considered possibility of the of orbitals che truncating, set spin and tho set: of 63-2 partrcle determinants which are used in estimating M-represeatability and in determining the optimum density matrix. Su&

~r~~cati~nsare possible and the calculation can bo carried through,

a~~~o~~hthe optimw results potentially available for th@ given spin

g@rninalset will therr not be obtained.

Wa have neglecred the consequences ,of symmetry, other than permutationai ! in the density matrix or in the wave functiond Xr is we&l known that

symznetry restrictions an the wave funcclon lead to certain limitations on

N-represencable density matrices, 2336-45 Them testrictions should pro$ably be imposed %fwe'waplt: satisfactory descriptions of the physfcsl

system of interest. We have not included them in the present discussion

beeabase of the addftlfaeal aomplications chay would add to an already diff$su%&prsb omo The consequences of symmetry should be fusth~rinvestigated, 41

evaluated until an. attempt has been made 'io acttially apply 2.k to a

fully reproduce the interesting pares of ehe problem. Wt?: arc cuntinuiag

i N-regresentabiliry is a useful one, and that this investigatian has in- *

We would like to thank Prof. Darwin Smith for helpEul comments

and discussion. !

I 42

APPENDIX E. Spin orbital axpansion aE the spin geminals,

The spin geminals with which we work are orthonormal and an2isyrf:rnerric.

Each such two particle function can be expanded 91% toms of an orthcrnsxlnal sat of spin orbitals as

of the 1-matrix associated with They are pf c$i and the pseudo natcral spin orbitsls of number spin orbitals required, is :,he sf r.,L be infinite,

To choose a spin orbital hasis €or che whola problem we first fom the apace which is the union of the spaces spanned by the spin arbitals. associated with the various spin geminals. Wa chon find nome archonormal basis for this space. It might be cmvonient co start wich tho{ tiJa1 k associated with ($lT which we tam KO be the spin gemfnal WB expect; to be most important in the final density matrix. Any of the 1 associated with the next spin geminal which cannot be expanded in terms r 1 of the {$: 1 are then orthogonalized ro all the I<,, and added TO the 3 set. Any independent 5. 3 are then orthogonalized and added to the .I set, and $his process continued until the ser is sufficient to expand all. the { 5; 1 This final set will be Labeled by { ai , i = 1...Rj ,

C 1B ar Sy 43

f and we expect that: R will be much less Lhail the sum trnlcss strangiy i

which ths {ai 1 l1AW baen cliasen. A4

h APPENDIX 2. The relationship bccween and T

c. We want to show that the operators ( NE-' "~b ani T , ciefinci: liy

The firsc transformation is obtained by relabe1ir.g the Z~i.;uny vsriab3.es

of integration, and rhe second follows fnom the antisymxe:ry of

@k,@i,Xk and xi. There are N-2 rems in the sum for 3 --< j -< N, so P. the second terns of ( 'fs and T are equivaleat. "J &'ne fom P P,, < and have The third terms in are of rj 2,iS j k, matrix elements 45

Use has again been rnade of rclabeling clnd ot rhe ancisjmmetry of ji xc and There are ("3terms i.11 this siim, corresponding to j < k between 3 and N. The final cerw in the two operators ara rhus sleo

equivalent.

! NOTES AND REFERENCES

1, P. 0. Lgwdin, Phys:. Rev. ,‘A, 1474 (1955). 2, R. McWeeny, Rev. Mod. Phys. 32, 335 (1963).

3. T, Ando, Rev. Nod. Phys, j3690 (19631,

4, A collectioii of papers including both original wik and r~.views

will be found in “Ked~~cdL)er;sity Nst:ricct; ~lrii.11 Appi icacions io

Physical and CtieinLca1 Applicattons ’, A. .J. Colctiic~*land 2. M. Erdall?,

eds., Queen’s Papers cn Pure and Applied Mathematics--Mo, II, Queen’s University, Kiags ton, Ontario, 1968.

5” Cf, also the repores of the 1968 aad 1969 densicy matrix COLI- i ferences at Queen’s University. (Unpublished).

6. An extensive bibliography and review of the field as of early

1969 is included in the Ph,D. Thesis of Xxiiskai, ! Me B. University of Wisconsin, Madison, Wisconsin, 1969.

A, Coleman, Rev, Hod. Phys. 668 (1943). ! 7. 5. 2, 8. E. R. Davidson, J, Math. Phys. lo9 725 (19691,

9. K. F. Freed, J. Chem. Phys. 47, 3907 (1967).

10, C. Garrod and J. M. Percus, J. Math. Phys, 5, L756 (1964)-

11,

12, M, B. Ruskai and J. E. Harrirnaa, Ptiys. Rev. m, 101 (1968).

! 13 e M, B, Ruskai, Phys. Rev, 163, 129 (1969).

14 D, W. Smith, 3. Chem. Phys, 43, S258 (1965). i 15 D. W. Smith, Phys. Rev. E,869 (1966). 16, F. Weinhold and E. B. Wilson, Jr,, 5. Chcm. Phys. 46> 2752;

p_47, 2298 (1967). 47

25. Cf, also the wark of Carrod and Rossina reported in ref:;, 4 and 5.

26, This expression can also be wrttten in a more Byinmetipic form 1 N with the lase term replaced by - C’ PLjP2k. 3Lk503 The equivalence of the two forms in the present context is readily

established.

27, Alternatively, one could relate these expressions to the 1 operator ~(14-P13) end seek to minimize j Y*.$I+Pl3) Ydr

However, it: is not clear that the error bounds which we obtain

later can be derived if the theory is formulated in terms of

this operator.

28. F, D, Peat and R, J, C, Brown, Ingernat, J, Quantum Chem. IS,

465 (1967),

29, The method proposed by Smith (ref. 5, 1968) as a way of testing

for M-representability can lie easily generalized to give a

measure of approximate N-representability, ICs imp lenient at ion

requires iterative solutioti of a set of equations comparable in

dimension to ours, The N-representability test proposed by

Ruskai (ref, 13) requires a sequence of matrix diagonalizations 48

wirh at lsast the final. matrix being as iarge as ours.

30, I), W. Smith and E. G. Larson, private corninmicotion,

3 1. J. E. Harriman and M. 3. Rusks;i, unpublished.

3 2. The constants appearing in the reduced tIam-i.3tonian depend on

the normalization of the 2-matrix. Cf. ref. 21,

33, If YJ is totally symmetric the two expressions are Eigair, ,., equivalent but the description is one of bossn states, so E

is still not in general an upper bound Eo Eo’ the xcrixiori

ground state energy.

34. In what fo11ows we will drop tile subscripts 47 is aL**No for brevity,

35. A. C, Hurley, I.,. Lennard-Jones and A. J. “rople, Zroc. 30y. SOC, i - (London) A220, 446 (1953),

3 6. For a review of density matrix symmet-ry properties, see the

paper by W. A. Binge1 and I?. Kutzelnigg in ref. 4.

37-3 We A. Ringel, J. Chem, ’Phys. 32, 1522 (1960); 1066 (196L);

2842 I_36, (1962).

38, C, Garrod, Phys. Rev. 17_2_, 173 (1968). I 3 9. D, J. Klein, Internat. J. Quantum Chem. S3, 675 (1970). 40. W, Kutzelnigg, Z, Naturforsch, 1058 (1963); a,168 (1965). 4 1. W. Kutzelnigg, Chem. Phys, Letters k, 449 (1969). 42, 2. 0. Lzwdin, Rev. Mod, Phys. -35, 629 (1963).

43 I) R. McWeeny and Y. Mizuno, Proc. Roy. SOC. (London) A259, 554 (1961).

44 a R, McWeeny and W. Kutzelnigg, Internat. J. Quantum Chem. 2, 187

(1968) a

45. D, Micha, J, Chem. Phys. 4L, 3648 (1563).

4 49

CHAPTER TWO

DIRECT CAZCULATION OF SECOND-ORDER

DENSITY MATRICES USING GREEN'S Z"NCTI0NS 50

I. Introduction

In this chapter we show how one can obtain directly the second- order density matrix for a system of pairwise interacting fermions.

We employ the Green's function method, which has recently' been used to calculate the first-order density matrix of the helium atom. The important advantage of this technique is that it permits the direct calculation of reduced quantities; it does not require the calculation of the N-particle wave function.

In Sec. I1 we demonstrate the connection between the two-particle

Green's function and the second-order density matrix I'. The usual time-dependent perturbation treatment of the wave functeon and an introduction to the use of diagrams is presented in Sec. 111. In

Sec. IV we use Green's function diagrams to derive an exact integral equation for . Approximations to the so-called irreducible vertex potential are also introduced. We obtain a matrix equation for the Green's function in Sec. V using a specific approximation to the irreducible vertex potential. The second-order density matrix is then obtained from A by performing a contour integration. One contribution to this integral is evaluated analytically, but there remains a contribution which must be done numerically. Finally we discuss the application of these methods to the ground state of the helium atom.

11. Second-Order Density Matrix and Two-Particle Green's Function

The two-particle propogator (Green's function) & corresponding to the state vector I$> is written in the Heisenberg representation 51

I as follows :

where $; and $H are fermion2 field creation and annihilation operators respectively, T is the Wick time ordering operator, and the integers

1, 2, 3, 4, refer to space-spin coordinates. That depends only on the difference t-ts can easily be seen by recalling that we are in the Heisenberg picture (thus the subscript H) and that the haniltsnian is independent of time. The denominator of Eq. (II'el.) is ineluded so that we can obtain a linked-diagram expansion for . This is discussed in See. IV. The one-particle Green's function in the Heisenberg representation is defined as follows

One-and two-particle operators can be written in the second- quantization language as is shown below

and

where f and j are the usual "first-quantized" one-and two-particle OP OP operators respectively. If the operators defined above are to be in a particular representation (Heisenberg, Schrudinger, interaction, etc.) then we must use field creation and annihilation operators which are expressed in this representation.

The first-and second-order density matrices belonging to the state vector I P can be defined by the following relations:

If we use the second-quantized expressions given in Eq. (11.24 for the operators and the second-quantized state vector, the expectation values in Eq. (II.2b) can be written as

Thus the first-and second-order density matrices in the sacond quantization language can be identified as 53

and

respectively. To include the possibility of wave functions which are

not normalized to unity, and in preparation for the devfation of a

linked-diagram expansion of the density matrices, we can generalize

the expressions in Eq. (11.2d) by dividing the right hand sides by

From the definitions of the one-and two-particle Green's functions

given in Eqs. (11.la) and (11.1) respectively, the following relation-

ships between Green's functions and density matrices can easily be

established

r (1,~) I 2') = - & 2I /Ixl1;2'/ t-t') (I1* 3) L'+ t* We have made use of the antisymmetry of I' and & 3, and we take

the limit t' approaches t from above (t'>t).

In the following secti-ons, we will obtain an expression not for

,& (t-t') but rather for its fourier transform ++&(E) defined by 54

With this, the expression for the second-order density matrix given in

Eq. (11.3) becomes

(I1 6 5)

This integral can be evaluated as a contour integral in the upper half

complex E plane (Pig. (I)).

Figure 1. Contour Y I

x X x'

The upper half plane is chosen so that the integral over the arc vanishes due to the factor e -T"mE). We will see later that (E) has

poles both above the real. axis to the left of the imaginary axis, and

below the real axis to the right of the imaginary axis, ae is shown in

Fig. (E). Because the integral about contour 1 is difficult to carry

out, we choose instead the contour shown in Pig. (11). Both contour 1

and contour I1 enclose the same poles of (E) and so they lead

to the same result when substituted into Eq. (11.5). We choose the

Coulson contour because the integral over the arc (E = Re

can be done analytically and the integral 55

Figure 11. Coulson contour7 (Contour 1x1

along the imaginary axis can be handled conveniently by standard numerical techniques.

Hence we have reduced the problem of calculating the second-order density matrix I’ to that of finding an expression for -& (E) and then evaluating the contour integral:

-R (I1 e 6)

The evaluation of these two integrals is treated in Sec. V. 111. Time-dependent Perturbation Expansion of IY!> and Diagrams

We assume that the hamiltonian of our system can be written as

where Ho is a sum of one-particle hamiltonians (perhaps Bartree-Fuck or hydrogenic) and V is a two-partiela tiine-inckpendent perturbation:

h/ 1 v = vc-i/a', (111 .%b) A"

We also assume that we know the eigenfunctions of h , and thus of Ho.

In Eq. (1II.la) a is a positive real constant which we will eventually allow to approach zero.

The time-dependent Schrb'dinger equation is written as follows:

(I11 D 2)

However, the problem is not yet completely specified. We must also stipulate the state vector ]Y > at some particular time. Therefore we decide that at t = - aI) 1" > is given by some eigenfunction of Ha where

The function I@ > will usually be some Slater determinant composed of spin-orbitals which idre eigenfunctions of h.

To write the SchrEdinger equation its. a more useful form, we transform to the interaction representation, This is done by defining new state vectors and operators as follows:

The SchrSdinger equation and its boundary condition in this representation become:

(111.6) 58

This differential equation and boundary condition are equfvalent to the following integral equation A

(111.7) d -08 which can be iterated to yield the usual perturbation expression for

A- A

The Wick time ordering operator 31 arises by recognizing the identity

.-Go

(111.9)

For convenience we write Eq. (111.8) in the form

!PIN)B -" ur ckJ-- 59 where we have made the identification of terms in Eq. (111.8) fn- volving m-V1's with U1(m> . Because we are interested in obtaining

an eigenfunction of H' -+ V, we integrate from - ~0 to o (at t = O, E = H" -I-vj

and take the adiabatic limit (a + 0) of the temis in. Eq. (111.8).

1% may seem that the introduction of the time ordering operator T accomplished nothing, but as can be seen by reading any book on many-

body theory, its presence is essential to the use of Wick's theorem

and the techniques of Feynman diagrams. 495

To illustrate the use of diagrams in expressing the tame given

in Eq. (111.8) we consider the first order contribution pJg: CE)(O,- WjJW.

This can be written in second-quantfzateon notation as - 414, I -# $fhi dl da re VCI,ZI --a0 (111.11)

8 Each field operator is written in the particle-hole picture as

and each of these particle-hole operators is represented by a directed

line segment as in Fig. (111). 60

Figure 111 Particle and Hole Diagrams

I- $*(xt) Xt particle creation

Xt hole annihilation

hole creation

Xt particle annihilation

In these diagrams, time increases in the positive vertical direction, and a dot represents a space-spin-time point x,t. Recalling that in the particle-hole picture the unperturbed vector 1 Q, > becomes the vacuum state represented by I 0 > ,we find by using Wick's theorem that the contributions to Eq. (111.11) can be written diagramatically as in

Fig. (IV) *

Figure IV Contributions to UT(') (0,- m> 1 Q, 61

The wavy Line represents V(l,2), and integration over the space-spfn- time variables is implied. The construction and use of such diagrams is discussed in a number of texts,4 and so we will not pursue it further here. In the next: section we use somewhat different diagrams (Green's function diagram^)^ to derive an integral equation for the exact two- particle Green's function.

IV. Evaluation of Two-Particle Green's Function

The expression given in Eq. (11.3) for the exact two-parkicle

Green's function can be rewritten in a form more amenable to calculation by using the following relations between the Heisenberg and interaction pictures:

(IV .la> and

(LV. Ib)

(1V.lc)

These relations lead to the following expression for A& : where the subscript (I) has been dropped for convenience.

The denominator in Eq. (IV.2) can be expressed as the sum of a91 time-integrated vacuum diagrams ,'some of which are shown in Fig. (V) .

Figure V Vacuum Diagrams

P * .k

The numerator must: be expressed in terms of the so called

Green's function diagrams. These diagrams are obtained by expanding as in Eq. (111.10) eacih of the three U's appearing in the numerator and then using techniques similar to those used in writing the diagrams in Fig. (IV). Some of the Green's function diagrams are shown in

Fig. (VI).

Figure VI Green's Function Diagrams

Y3 Y 3

- t c> I 2 I t

c/ Y 3 Y 3 63

Y 3

I a t 'k

- - t k

t.* Y 3 A directed line segment leading from x,t to x',t' represents the unperturbed one-particle Green's function G0(x' ,x,t'-t) In these - diagrams all time orderings of t with respect to t and t' are implied. Then disconnected diagrams such as in c> and d) can be factored into their constituent: parts, and, for example, diagrams a> and c> can be combined to give

If higher order diagrams'were included in Fig. (VI), we would find that much more factorization of diagrams would occur and that the above 64 result would generalize to

Notice that the first factor is exactly what we wrote in Figo (V) for the denominator. Thfs type of factorization occurs for all disconnected. diagrams, and thus we can omit such diagrams from the list of Green's function diagrams since they only serv@to cancel. the denominator. This is the linked-cluster result for Green's functions.

Diagrams a) and b) represent contributions to the unperturbed two-particle Green's function 0 . Each directed line segment

0 represents an unperturbed one-particle Green's function

and so diagrams a) and b) yield

Diagrams such as e) and f) only serve to convert the unperturbed one-particle Green's function 6: into the exact one-particle

Green 's function . Therefore we can omit diagrams in which the 65 section containing the wavy Iine(s) is connected to only one of the directed line segments if we _now interpret directed lfne segments to represent not but the exact

With these considerations we can write the essential. diagrams which contribute to Some of these are given in Fig. (VIL)

Figure VI1 Essential Contributions to _I

To understand how these diagrams leiad to an integral. equation for we have written their sum in a slightly different form in Fig. (VIII).

Figure VI11

f 66

where

The box represents a generalized potential which is nsn-local in space and time. We recognize that the sum of the diagrams which are attached to the bottom of the box is identical to the sum of the Green's function diagrams given in Fig. (VII) . Therefore we can immediately write an equation which must satisfy. From Fig, (VKIK) we sese that must be a solution of the following integral equation:

where , which is represented by the box in Pig. (VIIL), is commonly

referred to as the irreducible vertex potential. We can think of as

a non-local potential describing the interaction between two particles moving in the rrsea" of the remaining particles.

The irreducible vertex potential can be evaluated to any desired

order by simply writing all essential Green's function diagrams of that - order and then identifying the contributions to V as that whfch multiples the factor

To prove this statement we expand and in perturbation series and then write the various orders of Eq. (IV,4) in symbo%ic n0t:stion: 68 with

etc. Thus we see that we can identify ,''> as that factor which multiplies GG[GGGG] in the first-order diagrams; likewise -(2)V is the factor multiplying GGCGG-GG] in the second-order diagrams. This result can obviously be generalized to higher orders. .As an example, let us wm- - sider the first-order contribution to V:

The first-order diagrams which contribute can be written as follows:

J 3 Y 3 69

The sum of these two diagrams is equal to

Therefore we can identf€y V'l) as

Higher order terms can be calculated in a similar fashion.

In the next section we will develop Eq. (IV.4) in more detail., using only the first-order irreducible vertex potential given by

Eq. (IV.7). This approximation will be referred to as the extended random-phase approximation (ERPA).

V. ERPA in a Basis

Substituting the first-order approximation for the irreducible vertex potential into the exact Eq. (IV.4), we a?zPive at the follawing approximate equation We now assuuie that the faurisr trleansfom of the exxk one- 6 particle Green's function is k,nown and is given in the form:

where the and a?"e complex E' E' a' E: numbers with

(V. 3b)

and the gi are positive real constants. The number Pl and the

Greek and Roman subscripts are used in the summations to distinguish -% between those poles which lie above the rea%axis (Ea) and those whi.ch lie below (E:) The (not necessarily lfxearly ~~~@~~~~~~t~ func tions $i are assuned to be expanded in some chosen basis set of spin-orbitals 1 x, 1 as

(U e 4) and the Cia are assumed to be known,

The time-dependent one-particle Green's fumction can be obtained by fourier transform from Eq. (V.2) This gives the following rssuxt:

where we have defined

and O(t> is the unit step function.

Wfth this, we can rewrite Eq. (V.1) as follows: 72

We can then fourier transform this equation EO yield the more useful

result given below:

where we have defined G(1,2,3,4lE> as the Eourier transform of

G(1,4,t-tr) @(2,3,t-t') The operator permates the spa~e-3pi.n

variables 3 and 4. Let us now examine the permutation symmetry of

factors in Eq. (V.8) s remembering that (%,2,3,4jE) is anti-

symmetric in variables (l,2) and (3,4).

First we consider the integral term: We have rased the fact that G(1,2,3,4/E) is the fourfer transform ai

che product G(l,4,t) G(2,3,t;), and we recognized chat the variables

(5,6) are dummy. Hence the integral term is antfsymmetrfc in variables

(l,2) and (3,4). Therefore the first term in Eq. (Ya8) must also be

antisymmetric in (1,2) and (3,%)e

Thus we can expand Eq. (V.8) in the following basis of antisym-

metric functions:

(V. IO!

This results in the matrix equation given below: and

Thus given any basis { a = I.,.M and. the exact one-particle x,, I Green’s function as in Eq. (V.2) we can calculate the matrix elements CE) (for any E) and Ver,gh ” E¶. (V,ll) is then a simple Gab cd (E). We write thris equation matrix equation which determines ab cd in matrfx form as

from which it follows that

This is the final expression for - (E) Let us now return to the problem of evaluating the second-order

density matrix as discussed in-Sec. 11. Being careful to notice the

l/i in Eq. (V.l2a), we write the previous expression (Eq. (I1,6)) 75 for P in matrix notation as .follows:

-R with the matrix r defined by dw.%

The evaluation of the arc integral. can. be done a,naIyt%callyby notichg (see Eq. (V.lZb>) that for large R

P This defines the matrix i- Therefore rhe arc integral reduces KO

I..--

This result is exact, arld is easily evalue%ledif the erract ane-particle

Green's function is known.

TQ calculate the integral dorig the imagimasy axis we resort

to the use of numerical ineegratfon techniques. However we first

ment. We notice that each term in (iy) contains factors of Gab 9 cd ehe €am

Therefore we can write &(fy) as where

and

-1 We also decompose G (iy) and .&(iy) into real and imaginary uu- uo components as

(V. 214

and

(V. 21b) 78

To determine T(y) and Ufy) we write the real and imaginary parts 40-4 - of the identity

This results in the following matrix equations:

(V. 23%)

and

(V. 23b)

Then knowing T(y) , T.l(y), and V we can writa Eq. (V. 14) as IC *w

which yields the following equations for &("(y) and &'2)(y) : & -

(V. 244

fV. 24b) 79

Because &'" (y) is an even function of y (see Eq. (V.20%)>t and Gc2)(y) is an odd function of ys can easily be seen that % it & ("(y) is even in y and a(2'(y) is odd in y. Therefore (2) 2 (y) will not contribute to the integral along the imaginary axis because we integrate from -R to R. As a result of the above analysis we can now write the imaginary axis integral as follows:

(V. 25) where we have used the fact that a(''(y) is an even function of y. Auu*r This integral is evaluated numerically.

To review our formal discussion of the calculation of density matrices using Green's function techniques, we now outline the method for determining the second-order density matrix r.

We assume that we are given the one-particle Green's

function as in Eq. (V.2) as well as a set of basis

functions [ Xa,a=l,2,. e .M f e The procedure then is as follows:

1) Form the matrix B defined in Eq. ('v.17). The contribution Iva to I' from the arc integral is immediately written as:

2) Calculate the matrix defined in Eq. (V. l2c) waV 3) Set y equal to zero.

Use Eq. (V.20a) and Eq. (V.20b) to evaluate G") (y) and 4) &%a -G'2) (y) for the currant value of ye 5) Use (V.23) and to obtain Eqs. (V.24) - (Y> e 6) Increase y by some small increment, and return to

step 4) after adding the contribution to from this Q%sz.I? iteration.

The results of such a calculation are presented in Lke next section. Hopefully this method will prove to be very usefu3. for calculating second-order density matrices of reasonable accuracy for atomic systems. Certainly it represents a new and interesting method which should be investigated a great deal more in the future.

VI Results and Discussion

We have applied the Green's function method described above to the ground state ('S) of the helium atom. The basis'' of five

S-type Slater orbitals (normalized) which were used in the present calculation are described in Table I.

Table I. Slater Basis

n exp onent

1 1.4191 m 2.5922 2 4.2625

3 3.9979

3 5.4863 81

It was found to be sufficient to take the upper limit (Rmax.) of the numerical integration equal to 1000 atomic units. The total calculati.on took five minutes on a Univac 1108 computer.

Because helium is a two-electron system we can extract the wave function from our calculation of the second-order density matrix. The radial part of the singlet wave function can be expanded in the following set of symmetrized functions:

where the Ti are the five Hartree-Pock orbitals constructed from the above Slater orbitals. In Tabla I1 the expansion coefficients for

the wave function obtained from the Green's function technique are

compared to those for a complete C.I. wave function within the same basis. 82

Table 11. Expansion Coefficients for the Wave Function

i 3 C.I. Green. ' s Function

1 E .99969 .9996% 1 2 -.02327 -.02330

1 3 00699 00699 1 4 - 0033.8 -.00323 1 5 00245 .00250

2 2 .00303 * 00313

2 3 - 0615l -0 00155

2 4 IO0038 e 00084

2 5 .00066 .00091

3 3 .00050 .00060

3 4 -e 00039 -.0004.2 3 5 -.00040 -.00044

4 4 -e 00009 -.00014

4 5 .00024 a 00026

5 5 - 00020 -.00028

The wave function obtained from the Green's function calculation seems to be in excellent agreement with the C.1, wave function. It is not surprising then that the energies reported in Table IC11 are in such close agreement for the two calculations. 83

Table III. Predicted Energy_

Method Energy

Hartree-Fock -2.8619 a.u.

C.I. -2.8790 a.u.

Green ' s Function -2.8980 a,u.

It is important to note that we must compare our results to the complete C.I. calculation within the original basis raisher than to the results of more extensive computations.

Although the results of this calculation are encouraging they do not constitute a proof that the Green's function method will be a useful tool for larger systems. In order to inteLligently evaluate the method, we must have numerical results from many more computations.

Hopefully these calculations will be performed in the near future.

In review then we see that in this chapter we have described in considerable detail the application of Green's function techniques to the calculation of the second-order density matrix for an interacting

N-fermion system. We derived rn integral equation for the two-parkicle

Green ' s function which was reduced, after making the extended random-phase approximation, to a matrix equation. We then obtained an expression for the density matrix I' involving a contour integral of ,8 . The evaluation of this integral was discussed. We have also presented the results of applying this method to the ground state of the helium atom. Although such calculations have never before been carried out, we believe that these techniques will prove to be very useful in atomic and molecular problems. The most appealing aspect of Green's functions is that they provide a means of directly calculating reduced quantities without ever having to obtain an

N-particle wave function, Certainly this reason alone is sufficient to justify further research. 85

References

1. W. P. Reinhardt and J. D. Doll, J. Chem. Phys. -50, 2967 (1969). 2. Because of the fermi statistics, we have the following equal 4- time commutation relations: [$ (x,t) s$(ygt)], = 6(x-y) e [*+(X,t), * (Y,f)l+ = 0 = [*(x,t>, *(Y,t)l+, where the symbol [, ] -s- means anticommutator.

3. r(1,2,1~,2'> a - ~(2,1,1~,2~)= - r(1~2~2y.v)~Similarly for 3 . 4. March, Young, and Sampanthar, The Many-Body Problem in Quantum

Mechanics, Cambridge Univ. Press, 1967.

5. Thouless, The Quantum Mechanics of Many-Body Systems, Academic

Press (1961).

6. See for example: A. S. Layzer, Phys. Rev. -129, 897 (1963).

7. C. A. Coulson and H. C. Longuet-Higgins, Proc. Roy. Soc. (London)

Al91, 39 (1947).

8. Holes are vacancies in the one-particle functions that are

occupied in 15 > ; particles are one-particle functions which

are unoccupied in 13 e

9. Vacuum diagrams are those with no free creation or annihilat$on

Lines e

10 We are grateful to Prof. W. P. Reinhardt for providing us with

his one-particle Green's function results. 86

CHAPTER THREE

CONSTRUCTION OF APPROXIMATELY

N-RBPRESENTABLE DENSITY MATRICES It is well-knm that the sec sit^ matrix is

o determifie all obsarvab%e p~o~e~~~~sfor systems compos ise intetacti 1es. In zadc%i.t% the s~~~~~o~~er~%~~~y matrix is a function of only ~~~~~@ eont%uxreousvariables, whereas .the system wave fuiiction' depends ot"l 3N ~~~~~~ou~var%ab%es These

ions, together with the aw~adgethat accurate wave ~~~~~~Q~~

density matrices e

One cannot merely use trial second-order density matrices in a variational calculation of the energy; there -is BO variational principle for arbitrary density matrices. @xi'srestrict the class of tria density matrice$ ta those which can be evpress

i where the nomafized N-particle wave fuaption (P,%,.,Ns) is

in a91 of the space-spi s ~~p~~se~~e~by the irategars 1,2p,.IN. Such density matrices are said to be ~~~~~~e~@[email protected] problem of d@~e~~ni~gecessary and sufffcbe on$^^^^^^ to gu~r~~~~~ that a proposed second-order density matrfx can be obtained from (I J ynmnetric w~~efunction as .in Eqn~I (I,l,> is 16KI as the pure

ibity ~~~b'~rn~The &ace ~~~~~~o~ of this problem

vstxi 23%ionaL 4

I calculations e Very few numerical applications' of these methods have been carried out.

Another more promising method which has been successfully applied 7 to and molecular problems is the Green's function technique.

This is a potentially exact scheme in which the flrst-order (second-

order) density matrix is obtained as a contour Integral involving the

Fourier transform of the one-particle (two-particle) Green's function.

The evaluation of the contour integral is done numerically on an auto- matic.gomputer, which is a distinct disadvantage of the method.

In this chapter we present a new method for approximating, in a self-

consistent fashion, the first- and.second-order reduced density matrices

for systems of N pairwise-interacting fermions. Unlike the Green's

functiop technique, this method requires no time-consuming numerical

integration.

Within this scheme, one can bound the errors in expectation values obtained by using the resultant density matrices which might not be N-representable.

This test involves formally generating a special wave function whose

reduced density matrices are then compared to the density matrices

obtained using the proposed procedure.

Before developing the formalism of our method, we briefly review in

Sec. I1 the generalized random-phase approximation (GRPA) as presented 8 by Rowe . In Sec. 111 we discuss the occupation number representation

of density matrices and their spin components. In See. IV the GRSA

is used to evaluate certain contributions to the second-order density

matrix. Sec. V contains closed expressions for the spin components

of the first-order density matrix. In Sec. VI we discuss a scheme which

allows the self-consistent determination of the first- and second-order 89 density matrices. Sec. VI1 contains the results of applying the method

to the helimplithiumjand beryllium atoms e We discuss possible error bounds involving the resultant density matrices in Sec, VIII. Sec. IX contains our concluding remarks.

11. The Generalized Random-Phase Approximation

The equations-of-motion method was originally developed by nuclear physicists8 as a technique for directly determining excitation energies and ground state' transition strengths for nuclei. It is expected that these relative quantities will not be as sensitive to correlations within the stationary-state wave functions as, for example, the energies of the individual states. Thus the results of such calculations often imply a higher order of ground-state correlation than would be expected by considering the approximations made within the method. The GRPA occurs as a special case of the equations-of-motion method.

As an approach" to the equations-of-motion technique, let us recall the operator equations for a harmonic oscillator of frequency

U , and Hamiltonian H :

(11.1)

(11.2) where we have chosen to use units in which% = 1. In attempting to approximately solve Eqns. (11.1) and (11.2) we will expand the unknown

) and the Hamiltonian in a basis of known operators.

These equations will than allow us to determine the expansion coefficients and the excitation energies . Before doing this we transform Egns. (11.1) and (11.2) into forms which will be more easily used. 1%we define the usual set of egg d eigenval~esby

the

and thexef om

(11.4)

It is easily seen that these operators ale raising and lmerfng aperators respectively with respec to the Hamiltonim (energy).

To extend &his type of treatment eo more general ~~~l~oni~~~,we \ assume that the spect of our ~~~~0~~~~ 2s h~~~~~~ up %a the mth * ---' olevel, iSe,

where is 0 cons The space ep by the (at.r-l) lowest eigenfunct%onsof H is ~~f~rr~~to as ragiora of ~~lbe~~

harmonic, than. W=B and I

the harmonfc region is -dimens bnal s pac panned by the ground

for arbitrary R e In a similar fashion we obtain an analogous

equation invo1ving

=-04

for arbitrary R. Also it fo11aJs directly from Eqns. (11.8) and

(11.9) that

If the harmonic region is one-dimensional, 1 +) must b P

&round state of the system, In this case Eqn. (II,11) is the ~~~~~~~~ conjugate of Eqn. (II.XO), so we need only coneider one of them. Yf I@) is only an approximation to the ground state wave ~~n~t~o~~~~~~~ (11.10) and (11.11) are not simply Bemipian conjugates of one another.

TO'regain this Hermitian refationship, we generalize Eqn. (IL.$O) in

Ithe following way:

for arbitrary R. The double commutator symbol is ~~~i~~d

I I If I .p) is the exact ground state, Eqn. '(11.13) is identical to

Eqn. (11.10) e ;t. The solutions of Eqn, (11.13) are to be ~~~~r~~~~@~as

excitarion operators wktieh general;@ excited state wave f 93 operating on the ground state I$?):

$. The energy wk associated with okthen represents the excitation energy from the ground state 1 43) to the excited state To derive the generalized random-phase approximation we restrict the excitation operator to be of the following form:

t where the c.i and c; are fermion creation and annihilation operators respectively and the Ch), tmG&)are coefffcients. The index Nv\ is summed over all single-particle functions (spin orbitals) which are unoccupied in the single-determinant approximation to the exact 11 ground state function a Similarly€ is summed over all oecupied.spin- orbitals. Inserting Eqn. (11.16) into Eqn. (11.13) and ushg the fact that Eqn. (11.13) must hold for any R within the space of operators 5. t spanned by tc,G, cecm], we arrive at a set of equations for the coefficientsjm;+) xL$l, which can be written in matrix form:

(PI a 18) 4

given by Rme reduce to

as.

.-a P

To convert

Eqn, (IIef7) by

in matrix form as ferllawsr

with

and i

98 with obvious noezrtfon,15

D (ZZ .30) i Thus to calculate transition values of aed only know the Q

choice will restrict t'n As a matter of we < &he ~~~i~~@~of development to the most frequently used sph ~~~~~~~~s of i I

approximate exeitxd states without r~~ui~i~gknowledge of

,even when a ~~~~~~~e~ set of orbitals is"ursed. This supports our c

e- rn

f , 1

1 r V, First-Order Density Matrix I i a 1 nts of the first-order density ~a~~~xax i

$ wad-order density mat;rix ,

l i e ! I Let us aosume that a eruncated basi orbitals $la& been chos~ta, and that we wish to evaluate the firsr-and second-order density matr

€3 P this basis. If we substitu e the ~x~~~~~~~n~for and given in Eqn, (1rII.8) into Eqn. (Vel), ment, the fallowing closed expressions for

0 rhe above equations axe closeds and they can be wed i P 0 VI Self-Consistent Determination of and

Ha the preceding sections we hav method cast be used to evaluate various component the first-and seco&-order density matrices. Eqa, (V.2) provldes tal! with a means of directly 0 t and y by using the G

(I Eqns. (IIXe$), (TV.3), (~~~~)3and (IV,5) can be us z and if x@

# The groc dum is as dolfows:

Q Set; and

approximations.

, using tha presant valu

(XT1,5) use the cu to

( ets,

Form and s. (fV.3) and (19,4) 0 Use the currerat and ‘ on the right

side of Eqn. CV.2) to obtai - values for 8 O and 2/ B i? In Eqn. (IxIa8)9 us and to es evaluate and ment is satisfactory and if thB 7old

agree well with the

the calculation is complete. Otherwise return to step P (2) using the -new '25' and That this procedure is an iterative method is clear from the above description. ft is also called §~lfaco~s~~~~n~b cause half of the convergence criterion is that the Eirsk-order density matrix calculated in step (5) must agree with the first-order density matrix obtained by reducing f (step (7)) * That is, the f irst-order density matrix musk. be consistent with the second-order density matrix.

One disadvantage of the iterative method is thar matrix element@

'2/EOm connecting occupied and unoccupihd orbitals are difficult to calculate. If one begins the iteration process with a chqrge density matrix having x,", = Q the procedure never produces any nonzero rzbm However if exact Hartree-Fock orbitals are beiqg used,such matrix elements should be quite small due to Brillouin's theorem. Tkt is, if the C. f. expansion of the Fjave function /4) contains no single excitations, then the lowest: order nonmro contributions to

/cLd~64 jl ceP/+) will come from matrix elements of double excitations with triple excitations. Because the C. I. expansion coefficients of triple excitations are usually quite small, the elemenfiea

I Y&-l will, in general, also be very small. To obtain approximate 0 values for which could then be used in the first step of the iterative procedures one can use first-order perturbation theory. This mly requires a knowledge of the Hartree-Pock orbitals and energies,

There 3s no formal proof that an it rativa procedure such as we have proposed will converge to any m~ani~~fu~resul (I Therefore we mwt test 110

the convergence by using the method to carry out numerical calculations on systems of interest. In the next section we report the results of such calculations on the helium, lithium and beryllium atoms.

VII. Application to Helium, Lithium, and Beryllium

There are a number of reasons behind our decision to choose the

.ground state of the helium atom as our first test case. In the first place, helium is the simplest atomic system to which our method is applicable. In addition the second-order density matrix which we obtain can easily be tested for N-representability, because necessary and sufficient conditions are known for the two-electron case. Finally, we want to compare the results of the present method to results which we have previously obtained for helium using Green's function techniques.

We have chosen as a basis five s-type Hartree-Fock orbitals19 , each of which is given as a linear combination of the five Slater orbitals

(normalized) described in Table I.

Table I. Slater Basis for Helium

n exponent

1 1.4191

1 2.5722

2 4.2625

3 3.9979

3 5 4863

The convergence criteripn used was that M

complete calculation 111

taking thirty seconds on a Univac 13.08 computer. Calculation of the necessary one- and two-electron integrals required twenty seconds, so the GRPA calculation only required ten seconds. This is to be

compared to our Green's function calculation20 in which four minutes were used in the numerical integration step. The convergence was not found to be very sensitive to the initial choice of

expectation values of the one- and two-electron operators which OCGUX

in the Hamiltonian, along with the results of other work, are given in

Table 11.

Table 11. Expectation Values for Helium 1) This work: (-20"-B X} r =-3.8822 a.u.

-+0,98766 a,u. (H) =-2,8945 a.u 20 2) Hartree-Fock H> =-2.8617 a.~. 21 3) c. I. =-2.8790 aeue 22 4) Green's function

The fact that our energy is below the energy of the complete C. I.

immediately tells us that the second-order density matrix which we

have calculated is not N-representable. We will defer further comments

concerning this problem until the next section.

The eigenvectors (natural orbitals) and eigenvalues (occupation

numbers) of the charge density matrix are given in Table 111,

Of course, the spin density matrix is identically zero. 112

Table 111. Natural Orbitals and Occupation Numbers- for Helium

0 c cup ation Numbers

1.9974 1s 0000 -0.0004 0 D 0001 0 e 0000 0.0000

2 5025~10-~ 0.0004 0,9286 -0.3688 -0.0419 -0.0007

4.9696~10~~ 0 e 0001. 0.3612 0.8717 0 3311 0 0053

2.1833~10~~ 0 * 0000 -0.0853 -0.3216 0 9386 0 e 0x4

2.2919~10-~ 0.0000 0 a 0066 0 e 0246 -0 I 0880 0 a 9958

Expansion coefficients refer to the Hartree-Fock orbitals a In carrying out this calculation we first approximate the charge density matrix 0 elements Vcrn (see the end of Sec. VI) by using first-order perturbation theory. These approximate values were then used to begin the iterative procedure. This was found to have negligible effect on our results. The diagonal overlaps between our natural orbitals ( and those of Reinhardt and Doll2' (R-Di) are given in Table LV. The two sets are in fairly good agreement.

Table IV. Diagonal. Overlaps

1 1* 00000

2 0.99817

3 0.99719

4 0.99901

5 0.99999

Although our natural orbitals seem to be reasonably aecuxate, the 113

expectation value of the one-electron operators given in Table 11 is too low.25 This is evidence that the occupation numbers of the second through fifth natural orbitals are not large enough.

By imposing the additional constraint (necessary for N-represent- B ability) that the diagonal elements of be non-negatfve, we obcain an energy of -3.8496 9. 0.9837 = -2.8619 a, u, for helium, Ln this case the natural orbitals are essentially unchanged, but the occupation numbers are altered considerably. The new occupation numbers are given in Table V.

Table V. Occupation Numbers with Constraint

2.9854

1 2441~10-~

2.13l9~10~~

6. ~7xlO-~

2 e 5378~10-~

0 With this )( the predicted expectation value of the one-electron operators agrees very well with the correct result. 25 The fact that the total energy is not very good when compared to the C. E. result 0 indicates that the second-order density matrix which our method yields is probably not very accurate in this case,

In addition to the first- and second-order density matrices of the ground state, the GRPA calculation yields an approximate electronic excitation spectrum of the system. The energy differences 114

between the ground state and excited states which are obtained by doing the GRPA calculation are compared EO the C. I. results for singlet states in Table VI.

Table VI, Singlet Excitation Spectrum for Helium

GRPA c. 'I.

1.4243 a.u. 1.4068 a.u.

5.3068 5.3208

19 522 19.451

118e 50 118.34

For the calculations on the ground states of Lithium and 26 beryllium we used bases of six s-type Hartree Fock orbitals

(restricted H.F.) which are expressed in terms of the (normalized)

Slater orbitals described in Table VII.

Table VIE. Slater Bases for Lithium and Beryllium

Lithfum Beryllium

n exponent n exponent

1 2.4803 1 3 4703

1 4.7071 1 6 e 3681

2 0 3500 2 0.7516

2 0.6600 2 0,9084

2 1.0000 2 1.4236

2 1.7350 2 2 e 7616 115

Convergence was realized in each case after four Iterations. Both computations took ninty seconds; sixty seconds for integral evaluation and thirty seconds €or the GWA iterations. Again the convergence was 0 not found to be sensitive to the initial choice of and for lithium the calculation was quite insensitive to the initial spin- density matrix 8 a The natural orbitals and occupation numbers for lithium and beryllium which are obtained by our method are given in

Tables VIII and IX respectively. In both cases the constraint that 8 the diagonal elements of be non-negative was imposed.

Table VIXI. Natural Orbitals and Occupation Numbers €or Lithium

Occupation Numbers Expans ion Coefficients (rows )

1.9979 l.0000 -0.0002 0.0001 -0.0002 0.0003 -0.OOOl

0.9976 0.0002 0.9994 0.0282 -0.0186 0.0003 0.0000

2.9 6 14xl~l-~-0.0001 -0.0088 0.6743 0.5608 0.3341 -0.3451

1.1524~10-~ 0.0002 0.0319 -0.6121 0.7862 -0,0784 0.00&7

3.0561~10-~ -0.0003 0.0067 -0.3946 -0.2232 0.8264 -0.3338

8.5726~10-~ -0.0001 -0.00l0 0.1184 0.1315 0.4463 0.8772

Table IX. Natural Orbitals and Occupation Numbers for Beryllium

0 c cupat ion Numb e rs Expansion Coefficients (rows)

1.9998 0.9339 0.3575 0.0000 0.0000 0,0000 0.0000

1.9968 -0.3575 0.9339 0.0000 0 D 0000 0.0000 0.0000

2.4175~10-~ 0 0 0000 0.0000 0.7697 -0.6304 0.0986 -0.0205

2. %226~lO-~ 0.0000 0.0000 0.5626 0.7329 0.2285 -0.3072

4.2424~10-~ 0.0000 0 0000 -0.2849 -0.2006 0.8635 -0.3551

7.8827~10-~ 0 e 0000 0.0000 0.0991 0.2596 0.4308 0.8827 116

These natural orbitals and occupation numbers are similar to those

obtained by other workers 27’28’29 using somewhat different basis

functions. Notice that the block structure of the natural orbital 0 expansion coefficients implies that the which are more difficult

to obtain by our iterative scheme, arep as we anticipated, quite small

in the three cases considered here. The ground state energies for

lithium and beryllium calculated by using our density matrices are

presented in Table X. For lithium our spin density at the nucleus

(2.8006) agrees fairly well with the correct value (2,9096). 30

Table X. Ground State Energy of Lithium and Beryllium

Method Lithium Beryllium

This work -7.4419 a.u. -14.579 a.u.

Hartree Fock -7.4327 a.u. -14.572 a.u. 32 31 Radial Limit -7.4420 a.~. -14.592 a.u.

In both calculations the expectation values of the one-electron

operators agree very well with the exact results. Almost all of the

error in the calculated energy is due to error in the two-electron

energy. This supports our earlier proposal that the obtained by Q our method can be -inaccurate, whereas the resultant is usually rather good. This is not surprising because? the detailed effects of 0 particle correlation which enter into can not be adequately described by the limited basis sets which we have chosen. On the other hand it is well known that correlation does not appreciably alter the change density, and so the limited bases should not prohibit us from obtaining accurate first-order density matrices. 117

We have also studied the behavior of the resulting densiry matrices for helium as the basis set is expanded. It was observed that the natural orbitals and occupation numbers converged smoothly to the radial limit results reported in Table III. Such calculations were

carried out with two, three, four, and five s-type basis functions.

Although the results of these examples do not consititute a proof

that the proposed iteration scheme will always converge, they do indi-

cate that the method can be a useful tool for determining first- and second-order density matrices of atomic and molecular systems. Even

though the energy which our method predicts is not accurate, the

resulting natural orbitals can be used in C. I. calculations to obtain better expectation values of two-electron operators.

VIII. Error Bounds

We have seen from the results of the helium calculation that our method does not necessarily yield density matrices which are exactly

N-representable. However this does not mean that these density matrices

can not be used for predicting the properties of atomic and molecular

systems. We learned from the calculations reported that our method can

yield first-order density matrices which are reasonably accurate.

However density matrices which are obtained by the GWA method, the

Green's function method, and other "direct calculation" techniques

might not be N-representable, therefore it is important to examine

the consequences of possible approximate N-representability. In a

recent paper' we have shown that the errors introduced in calculating

expectation values with non-N-representable density matrices can be 118

bounded, and that these errors decrease to zero as the density matrices become more nearly N-representable.

In the method which has been presented in this paper, approximations to the first- and second-order density matrices belonging to the unknown wave function 1 are obtained by an iterative procedure.

Hopefully these approximate density matrices are quite close to the true (N-representable) density matrices of I Concentrating on the first-order density matrix , we can define a measure of deviation from the density matrix belonging to 1

where tf is our approximation to the true

(VIIL. 2)

It should be kept in mind that we are trying to bound the differences in expectation values which are calculated using our d and the 'd~ belonging to I#) Neither of these expectation values are necessarily Suppose now that we are interested in calculating the expectation A, exact. IV value of some one-particle operator & e If we define the i=c difference function

(VscII. 3) then 119

is the deviation of the ealculsted expectation value of P from its value for the wave function I Notice that Trace { g&=] is not necessarily the expectation value of F for an exact wave function.

are expanded in some orthonormal (probably finite) basis, then the quantity can be bounded as fo1low.s; M M (VIII. 5) where M is the dimension of the basis and &*are the representatives and &i 2 of f and E within this basis. The bound on s/ is thus written

as a factor depending on the operator F times a factor which depends only

on the difference function E. St is easy to see from Egn. (V1II.l)

that the second term in Eqn. (VIII.5) is identical to what we have

(VIII .6)

The bound on /AF!can then be written in efther of the following forms

M (VI11 .7) ILgl" 4 p i=(22 F7L 2 where (fli is a matrix element of the operator f" which is still a

one-particle operator. The second . inequality in Eqn. (VIII 7) follows from the fact that the basis in which 'd and f~ are expanded is probably not complete, If we can find a means of evaluating the parameter%A , either form of Eqn. (VIII.7) will allow us to bound the 1. We do not mean to imply that the bounds given above are in any sense good bounds; we only wish to show that knowledge o

can lead to error bounds for expectation values. In order to calculate the value of,@ corresponding to a given f , we must somehow obtain at feast formally. This can be done by using the following property of the known excitation operators

(VIIS e 8)

for all excited states I ) Eqn. (VIII.8) is a consequence of the fact that is the GWA ground state wave function which must be f + orthogonal to all of the excited states ok14) e Because the o* and hence the 04are known once the GRPA caLculation has been performed, the above equation can be used to determine 14). For what follows we find it convenient to relate 14) to its single

determinant approximation 0) by the unitary operator given below:

where S is an antihermitian operator which is to be determined by using

Eqn. (VlI1.8). The expression for an element of the second-order

density matrix

By using the definition of the first-order density matrix and Eqns. 121

(Vel) and (VIII.10) we can write M

In the GRPA method we approximate the right hand side of Eqn. (VIII.11) as follows :

where the projection operator P is given by

(VI11 e 13) B -6 and the @A are the known GRPA excitation operators. Thus the deviation of &; (GRPA) from the true cb;)&ican be written formally as

(V911.14) with

=I-P a (VI11 .15) P If the spin-orbital basis used in constructing @A is complete, the 7c operators 0~and04 form a complete set in terms of which any operator f of the form ci can be expanded as follows: Ctd

B N The W.; in, 1 (/b) and v\/,. , are expansion coefficients. By d d. # using the orthogonality properties of the states 0)[+) and the 122

expansion given in Eqn. (VIII.16), it is easily shown that each element

&ivanishes ,, Therefore as the spin-oribtal basis approaches completeness, it is expected that the 4 will approach zero and the approximate d will approach the true fT e To make use of Eqn. (VEI2.9) in evaluating the parameterr we

recall the following identity for exponential operators 33 e0

(VIII. 18)

Before this equation can be simplified by using Eqn. (VIII.l7), we must investigate in more detail the form of the operator S.

In the nuclear literature34235 the wave function ,+}is usually written as

(VIII. 19) where K is a nomalfzation conshant and

(VIII .20)

We are using the subscript convetion of Sec. 11, 123

This form of the wave function is not especially useful because the operator S is not antihemitfan. This means that Eqn. (VXII.18)

can not be used to evaluate if we insist on using the above wave function. To avoid this problem we can express /

(VIIL.9) with the antihemitian operator S given by

Once the coefficients have been determined, Eqns . (VIEX. 17)

and (VIXI.l8) can be used to calculate E*i e We will return to this

calculation shortly.

To obtain an equation for the Coefficients we make use *in$ of Eqn. (VIII.9) and Eqn. (VIII.17') to write Eqn (VIIL.8) in the form

(VIIS e 22) OA I+) ex-pc-41 04 P L S, OAJ+$ LS,GS,OAJ] +*d60)=~

By substituting the explicit expressions for a?and S given in Eqn. (11.6)

and Eqn. (VI1.21) respectively, and carrying out the commutations

indicated above we can equate to zero the coefficients of the various + +-P independent functions 10) , C; C,/O), C At d Ch C'' O> , etc. This leads to the foll equation involving S,, u"p : t $ Y CA) = - ~h,, "P

In our GRPA calcuLations all of the hdCA1 turned out to be quite

small as did most of the 3md fAl e For each value of there was one g,, (A] whose value was near unity. These results are typical of RPA calculations on atomic systems. Based on these observations 124

Eqn. (UIII.23) implies that the magnitude of the will generally be at least as small as the Abd [A) .

If we represent each of the (M-N) N pairs /hp by 8 slngle

Greek indexp ,the set of GEA coefficients ymp14) and a md CAI can be thought of as forming square N(M-N) dimensional matrices, and so Eqn. (ULI.23) is a simple matrix equation which Can be solved by standard matrix inversion techniques to yield the coefficients

(VIP1 .arc>

It should be pointed out that Eqn. (VPII.23) is identical to the equation which would result if the wave function given in Eqn. (VIPI.19) had been used. In other words no additional complications arise when we introduce the antihemitian form for S given in Eqn. (VIII.21).

With the coefficients given by Eqn. (QII-24) we now

return to the evaluation of the (I By using the identity given in

Eqn. (VIII.17) and carrying out the indicated commutations, the right hand side of Eqn. (VIIE.18) can be written as a sum of terms involving varfous powers of the coefficientsS\,& q. Because the are generally quite small the use of powers of a,na for ordering purposes is justified. The sm of all. terms which do not contain any

(UILI. 25) 125

Notice that there are no contributions in the zeroth order to Ehd

This supports our earlier claim that the quantities xbd should be quite small in general. In addition all terns which are first order ins md,~p are found to vanish identically. Thus the factors contribute to ERA only in the second and higher orders. Beeause -3 theswa ,h P are generally smaller in magnitude than 10 , the second order contributions to E*R. will be of the order of loe6 or smaller.

This is a negligible contribution for our purposes, and so we need not obtain explicit expressions for these second order terms. (01 Because the zeroth order contributions E&. given in Eqn.

{VIII.25) are also quite small, it is not surprishg that the expectation values of the one-electron operators which we have calculated are in good agreement with the correct values. The fact that the magnitudes of all the A,,,d Ihlare loF3 or less implies that the value of,& given by Eqn. (VIII.6) with Eqn. (VIIL.25) is of the order of Therefore M A unless the quantity 'L.8 / ' which enbers into Eqn. (VPII .7) is quite large, the devyation should be very small.

It is our opinion that error bounds such as have been discussed in this section are necessary components of any complete and workable method which attempts the direct calculation of reduced quantities.

IX. Conclusions

In this paper we have shown how the GRPA method can be used to approximately determine the first- and second-order density matrices of atomic and molecular systems. In our method there are no numerical 126

integrations and the size of the arrays to be diagonalized increases much less rapidly with the number of particles than in the C. I. technique. Besides these computational advantages there exists the possibility of obtaining error bounds involving the resultant density matrices. These bounds allow us to estimate the deviarions in calculated expectation values caused by using density matrices that might not be exactly N-representable.

From the application of our method to the ground states of the helium, lithium, and beryllium atoms, we learned that the iterative procedure which we proposed can converge to a meaningful result. In these cases convergence was realized after a few iterations, We also found that the results of the method are not very sensitive to the initial approximations for the charge density matrix. In the case of lithium the results were also insensitive to the initial choice of x'. This indicates that there are probably no inherent instabilities in the iterative method.

We observed that by imposing the additional Constraint that the 0 diagonal elements of be non-negative, the calculated natural orbitals were essentially unaltered but the occupation numbers were

significantly changed e 36 With this constraint the resulting charge density matrices gave nearly exact results for the expeetation values of the one-electron operators occurring in the Hamiltonian. From this observation we inferred that nearly all of the error in the 0 predicted energy is caused by error in . These results also indicate that the occupation numbers which we obtain are probably of 129

reasonable accuracy

Although our method might not yield accurate second-order density matrices or ground state energies which are competitive with the best results, it does show promise as a method for obtaining first-order density matrices and natural orbitals which can then be used in a C. I. calculation. Let us recall that the CRPA technique was developed to predict properties which do not depend strongly on the complex correlations within wave functions. Therefore in our use of the GRPA for evaluating certain contributions to the first- and second-order density matrices, we should not expect to be able to accurately describe detailed particle correlation effects. Because such detailed effects contribute significantly to the second-order density matrix, it is not reasonable to think that our method can consistently yield reasonable second-order density matrices. On the other hand, it is well known that particle correlations have relatively little effect on the first-order density matrix. Thus it is not surprising to find that the CRPAmethod is capable of predicting the small corrections to the Hartree-Pock first-order density matrix.

In order to better assess the value of our method as a tool for calculating first- and second-order density matrices, many more numerical calculations are needed. Hopefully such results will become plentiful in the future.

Acknowledgments

We wish to thank Professor William Reinhardt for helpful discussions, and PI-Q~WWXCVincent MeKoy far his comments on this work. 128

Notes and References

1. We restrict the discussion to pure states of N fermions.

2. .J. Simons and J. E. Harriman, University of Wisconsin Theoretical

Chemistry Institute Report WIS-TCI-382 (to be published).

3. D. W. Smith, Report of the Density Matrix Seminar, Queen's

University, Kingston, Ontario, 3.969.

4. F. D. Peat, The Q-Matrix, Report of the Density Matrix Seminar,

Queen's University, Kingston, Ontario, 1969. Also references

contained therein.

5. W. P. Reinhardt and J. D. Doll, J. Chem. Phys. -50, 2767 (1969).

6. J. Simons, University of Wisconsin Theoretical Chemistry Institute

Report WIS-TCE-374. 11 7. J. Linderberg and Ye Ohm, Proc. Roy. Soc. (London), -A285, 445 (1965).

8. D. J. Rowe, Rev. Mod, Phys., pC0, 153 (1968). Rowe refers to the

GRPA as higher RPA.

9. The GRPA, as presented, is only applicable to the ground state.

Therefore we restrict our treatment ot this case. One can

generalize the GRPA method to apply to excited states, but we will

not do so here.

10. Our treatment follows closely that of Rowem

11 e It is assumed that the ground state wave function can be approximately

represented by a single Slater determinant. One can generalize

the treatment to cover a combination of Slater determinants but

the bookkeeping becomes very messy. 129

12. The notation "singlet and tripletboperators" is based upon the If- fact that operating on a singlet yields a singlet and

operating on a singlet yields a tgiplet.

13. We assume that there are no effects which split the degeneracy

of the triplet level.

3.4. Notice that the dimension of the matrix to be dfagonalized is

given by 2 (M - &J where is the number of occupied orbitals and M is the number of basis functions which we use.

The size of this matrix does not increase nearly as rapidly with

the number of particles as, for example, the C. 1, matrix. This

is an important computational advantage of the method.

15. The function/&, ) is not necessarily an eigenfunction 2 of S . The notation only implies thatl-k,S, is obtained by .+ 2 operating on the ground state e S -dependence

has no effect on our problem; we are just using the

as a set of orthonormal functions which are also

orthogonal to 1

16e We use the following normalization : Trace

17. See, for example, R. McWeeny, Rev". Mod. Phys. -32, 335 (3.960).

18. We sometimes use $ to represent a diagonal element of the 0 0 charge density matrix: 2 ~JL This is done for

notational convenience.

19. We are grateful to Professor William Reinhardt for furnishing

us with the Hartree-Fock basis. 130

20. C. C. J. Roothaan, L. Me Sachs, and A. We Weiss, Rev. Mod. Phys. 32. 186 (1962).

21. A complete configuration interaction calculation within the basis

of the same five Hartree-Fock functions which we used, This

calculation was carried out by us.

22 e Our unpublished resultse

23 a C. L. Pekeris, Phys. Rev. -112, 1649 (1958).

24. Private communication from W. P. Reinhardt.

25. E. F. Davidson, J. Chem. Phys. 2, 875 (1963). Davidson gives a

value of -3.8495 a.u. for the expectation value of the one-

electron operators.

26. E. Clementi, Tables of Atomic Functions, IBM Corporation, S.an Jose'*

California (1965).

27. D. Smith and S. J. Fogel, J. Chem. Phys. -43, S91 (1965).

28. G. P. Barnett, Thao. Chim. Acta. -7, 410 (1967).

29. G. Barnett, J. Linderberg, and H. Shull, J. Chem. Phys. -43

S80 (1965).

30. P. Kusch and H. Taub, Phys. Rev. -75, E477 (1949).

31. C. F. Bunge, Phys. Rev. ~ 168, 92 (1968).

32 A. Weiss, Phys. Rev, m, 9826 (1961). This might plot be the exact radial limit

33. E. Merzbacher, Quantum Mechanics (John Wiley and Sons, Inc. 1969).

34 E. A. Sanderson, Physics Letters 2, 941 (1965). 35. M. Baranger, Phys. Rev. 2,957 (1960). 36. Reinhardt and Dol1 find that accurate occupation numbers are 131

difficult to obtain. Private communication.

37. Prom this point on the unknown wave function #"will probably not be exact. However we will often refer to /+)as the exact function in

the sense that /$)=o(see Eqn. (VIII.8). 132

Chapter Four

First- and Seconddrder Density Matrices of Symmetry-Projected Single Determinants I. Introduction

It has been found that the effects of applying a spin projection operator to a determinental wave function involving different orbitals - for different spins can be conveniently summarized by considering the

first and second order reduced density matricesr 'j2 ~n this chapter ye will be concerned with the density matrices of wave functions obtained by applying to a single determinant projection operators of point- 8 or the axial-rotation group.

In section 11 we consider the effects of pointegroup s~@~ry projectbon, and obtain expressions for the elements of the fLrst and second order density matrices of the prajeited function, An example is considered in section 111, In sections'IV and V we treat the axial- rotation group and an example of this group.

It is also of interest to obtain the totally symmetric components of density matrices, since only these compments contribute to the expectation values of symmetric operators. We consider this problem in section VI. In section VIIj we consider the effect of projection' on,the eigenfunctions of the totally symmetric coaponent of the first order density matrix. We find that these e%gen€unctionscan be taken

I to be the same after projection as before, I

, \

Me stazt with a function 0 which we wish to adapt to the symmetry of a particular group: I 1,

t I 4,34 I w~rcro94 is titc! ~-pnrtislonnCil.rymnictxixur, mmo or ~li~i~~~*tn~I e 1 qN spin-orbitals without any particular symmetry properties, and the arguments 1 *.. N refer to the space and spin coordinates of particles 1 through N, respectively, The normalized$' symmetry-adapted wave function can be obtained by projection as

- -1/2 p* - *v where

3 The projection operator (PY can be expressed as

where the sum is over all operations in tha group. The index V refers to that irreducible representation which characterizes The cow efficients C,(R) are defined in terms of the characters XV(R)$ the dimension n,, of irreducible representation V and the order oE the group, g.

Since we are dealing with a many-particle syetsm the group , operqtors R will. be of the form w i

li convenience, we adopt a special nothation for the effect of I I

operation OR a spin-orbital:

is seen to be $

We first evaluate the normalization constant 0Jv rnakin

the facts that ,, is self-adjoint and ~d~~~~~~n~, ~e If-adjoint and has

the sum'of at1 N: signed permutations of PJ- en

where Pi is the index produced from i by& *l The integral fn (9) is a product of one-electron integrals of the form The expression for LOv therefore takes the form

Notice that the determinants, det R, are to involve only the spin- %@I pearing in @ They are ~@~~~~~~~~~~sof N x N mdtrices.

Thus if we know the matrix elements of the group ~~~~~~~~~~ in &he! spin-orbital basis, we can calculate Wy We will assume that t matrix elements are known.

We now proceed to the calculation of the first order density matrix. Given a symmetric, one-particle operator Q, we CEL~

expectation value < >v, respect: VIv ~epa its Q with ts --?

The first expression is the usual definitio of the expectatloa vola%t, and the second is the density matrix Bguivalent, "he prime on thq integral means that Q (1) fixst ocgserattas on the unprimed ~~o~~~~~~@$

only, then the primed coordinates are set equal to the, ~~~~~~~~ and integration ie carried out. Our met$od wild. be to ~~~~~~~t~ 9sZ &v

the ftrst expreaaion, and then #to ~~@~~~~~ y WV wbl.1 correctly reproduce tbe n;.eeond ~~~e~s~~~~ Since the symmetry operators are unitary,

I We can then write

and thus Therefore, we may identify yi as

N N I

forward, 6 i

where rhe orbiteSs are

I that in equation (11) the determinants are over only the

tals appearing in ; so only the pairs of 5 x 5 subraatr of om operator matrices are to be used to calculate WA2 ' We also find it helpful to construct a table of the vario appearing in the expressions for y and Sv (Table I), -V Using this information and equation (22) we obtain

Similarby, by using equation (24Jg we find the% 14

Table E. Combinations of Group Ope~ati~t~

PUsed to Calculate Density Matrices

-1 T S

E E I

@2 @2 x 0 Q -a V V

(3 Vg E

c2 E

(3 v CT V6 ff v E

c2 1 a E -1 V a VS

(3 wp E a '1 a 1 V

% c2 ff E V0 a

where

and

Equivalent results are of course obtainable more directly, %e

advantage of the present ~~~~0~ inclceaapea with

groups Q 3.44

XV. The Axial-Rotation Group

~ To extend the above treatment for the' first order density matrix

to the axba l-rotation 5 projection operator The proper projection. operataz &s

In this expression sais an operator which rotates through angle Q, If we define

then we obtain for the coefficients

2a -ima &J =..rrs,' das m 2n

! The evaluation of the second order density' i difficult and will not: be gresenfred

V. Example for Axial-Rotation Group 1

As an example in this case we will calculate the first order r.

density matrix of the E = component of an atomic ~~~~~~~~ (wa z 1 t

4 2 will not consider the L behavior). 'We take

effect of a rotation about: the Z-axis by angle 6 can be written L%B

From this and equation (2719 &

where

and

1 = i2p IY

But this I can be written as

... .._ ~ I 146

> 'I This result is of course expected because the Lz = 1 component

of is given by

and the first order density matrix above follows di~ectly. For a

more complicated function, however, ,the density matrix method can

be simpler than the direct method. 6 VI. The Totally Symmetric'Components of *Vy and UUVr The totally symmetric compbnent I' of r can be obtained as WV WV

and

We will consider Y first. w,v 147

But

Letting R = QS, and using the orthogonality properties of the *- I C v(R), we obtain

In a similar way we obtain the matxix elements of the symmetric component of the second order density metrix

1 I Equations (33) and (34) are the final expressions for the coefficients

in the expansion of 0 and 0 respectively, Again a knowledge w=PvY L,, of the matrices of the group operations and the appropriate character

table is all that is needed to perform the calculations. Because equations (33) and (34) are slightly simpler than eqtnaeiona (2%) BE$

VI1*

Before and After Projection L It has been noticed in spin-projection calcdlations that the I natural orbitals of the charge density matrix are not changed by prs-

jectiou, In what follows, we will examine the ~o~r@sp~~dingproblem

in the point-group case.

Let the totally symmetric component af the first order density matrix before projection be denoted by Po With y from t w Mnrv equation (331, we have for an element of the co~~~a~5~of

Because both and ‘are by ~~~ini~io~totally Po USrVY symmetric;

they will commute with all symmetry operations and $0 will their are indexed by i,

try-adapted basis, 8 etrig: operators will have

f I e 491' t,s

Therefore, we may identify the matrix elaient in the 8 basis 88

! i 152

I We have also shown how to project from w given deaasi'cy matrix $he , totally symmetric component, This component is all that is needed in

the calculation of expectation values of symmetric ogeretoro, Fbnelly,

! we have shown that the NSB's of the totall$ symmetric ~~~~~~~~ af g

first order dknsity matrix are unchahged b$ p~oje~ti.on~,

These results are of potentfal use in'twoways:

s a practical method of obtaining expectation values sf ~~~~~~~~' I

operators foi the cbmphent of a particular syimetry, ~~~~~~~~~,

I& * from a no~~6~m~t~yadapte single-determinant wave ~~n~t~~~~

(2) As the Eirst step in obtaining P var%~tionally-useful energy

i & 153

References /I

1, J.Eb Harriman, J. Chem. Phys. 3, 2827!(1964).

2. A. Hardisson and J.EI ~~~~~rn~n~$. Ghem, Phye. 3, E. Wigner,

Mechanics of Atomic Spectra (Academic Press Xnc., New York, L 4. P.O. Lgwdin, Rev. Mod. Phys. 39, 259 (1967). 5e . Hamermesh, Problems (Addison-Wesley Publishing Cohpany Xnc., 1964).

I' F

I I i It e -7-