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PERMUTATIONAL SYMMETRY IN ELECTRONIC SYSTEMS

Ph. D. Thes is Submitted to Iowa State University, May 1971

William Irwin Salmon

Ames Laboratory, USAEC

Iowa State University

Ames, Iowa 50010

Date Transmitted: June 197 1

PREPARED FOR THE U. S. ATOMIC ENERGY COMMISSION DIVISION O F RESEARCH UNDER CONTRACT NO. W -7405-eng- 82

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.Permutational symmetry in electronic systems . .

William Irwin ,Salmon

A Dissertation Submitted to the

Graduate Faculty in Partial Fulfillment of

The Requirements for. the Degree of

DOCTOR OF PHILOSOPHY

Major Subject: Physical Chemistry

Approved : n

In Charge of Major Work.

~ e a u o f Major : Department

Dean of Graduate College

Iowa State University Of Science, and Technology

M & G ~ . .

TABLE OF CONTENTS

Page ABSTRACT

PREFACE

GENERAL CONSIDERATIONS

I n d i s t i n g u i s h a b i l i t y o f E l e c t r o n s

Exc lus ion P r i n c i p l e

Spin Eigensunc t ions

Spin-Adapted Antis,ymmetrized Produc ts

ELECTRONIC WAVE FUNCTIONS AS SUPERPOSITIONS OF SPIN-ADAPTED ANTISYMMETRIZED PRODUCTS

Linear Dependence of SAAP1s

Space, Produc ts

Geminally-Adapted Spin Eigenfunc t ions

L i n e a r l y Independent SAAPts . 7

Energy ~ a t r i x Elements between SAAP1s Cons t ruc ted from Orthonormal O r b i t a l s

Der iva t ion o f - t h e General Energy M a t r i x Element Formula

Der iva t ion of t h e Matr ix Element Formula i n . .

S p e c i f i c Cases

GENERATING SPIN EIGENFUNCTIONS WITHOUT USING GROUP ALGEBRA THEORY

Cons t ruc t ion of Spin Eigenfunc t ions by Spin- Coupling Techniques 4 4

Lowdin's P r o j e c t i o n Opera tors 55

Wigner Opera tors 57 * ^2

Serbe r Sp in Func t ions by Diagona l i za t ion of S 59

The Group Algebra for SN; the Reg'ular Represen- tation :

Minimal Left Ideals, Primitive Idempotents; and Matric Bases

Young Idempotents, Young Operators

Spin Diagrams

Deficiencies of Young Operators

Tableau Chains

Chains of Young Idempotents .and Genealogical . Spin Functions : an Heuristic '~rgument

Definitions of Orthogonal Matric. Bases

Discussion

Basic Lemmas

Lemmas Concerning the Matric Bases I

Existence Proofs

Multiplicative Properties

Orthogonal operator' Bases for Every ~rreducible Representation

CONSTRUCTION OFSPACE FUNCTIONS

Generating Dual Space Functions by.Means of the Matric Bases

Simultaneous Eigenfunctions of .t2 and i2 by Matrix Diagonalization

APPENDIX A: NOTATION

APPENDIX B: THE SYMMETRIC GROUP

APPENDIX C: COMPUTER PROGRAM FOR' SERBER SPIN EIGEN- FUNCTIONS 161

APPENDIX D : COMPUTER PROGRAM FOR THE EVALUATION OF COEFFICIENTS IN THE ENERGY MATRIX ELE- MENTS BETWEEN SAAP 'S 167

APPENDIX E: COMPUTER PROGRAM FOR GENERATING SIMUL- TANEOUS :EIGENFUNCTIONS OF SPIN AND OR- BITAL ANGULAR MOMENTA AS LINEAR COMBINA- TIONS OF SAAPss 180

LITERATURE CITED 201

ACKNOWLEDGMENTS 206

Permuta t iona l symmetry i n e l e c t r o n i c systems

Will iam I rwin Salmon

ABSTRACT 0

Thi s work is concerned wi th t h e s i m p l i f i c a t i o n of

quantum-chemical c a l c u l a t i o n s on e l e c t r o n i c systems through

t h e e x p l o i t a t i o n of pe rmuta t iona l symmetry.

U n r e s t r i c t e d c o n f i g u r a t i o n - i n t e r a c t i o n (CI) wave func-

t i o n s a r e c o n s t r u c t e d according t o c e r t a i n convent ions which

s i m p l i f y t h e c a l c u l a t i o n of e x p e c t a t i o n va lues when S is a

good quantum number. General wave f u n c t i o n s a r e expressed i n

t e r m s of pure-spin components of d e t e r m i n a n t a l f u n c t i o n s .

These b u i l d i n g b locks , c a l l e d "spin-adapted ant isymmetr ized

p roduc t s " , o r SAAP's, a r e designed t o t a k e advantage of dou-

b l e occupancy.

I t i s shown t h a t SAAP8s, when c o n s t r u c t e d from or thonor -

m a 1 o r b i t a l s , can be handled i n C I c a l c u l a t i o n s more e a s i l y

than S l a t e r de t e rminan t s . Simple formulas a r e d e r i v e d f o r

ma t r ix e lements of t h e Hamiltonian and i2. A computer program

i s given f o r t h e e v a l u a t i o n of c o e f f i c i e n t s o c c u r r i n g i n t h e

energy m a t r i x e lements .

Two new methods a r e d e s c r i b e d f o r t h e c o n s t r u c t i o n of

s u i t a b l e s p i n e i g e n f u n c t i o n s . The f i r s t of t h e s e i s an a lgo-

r i t hm and computer program f o r g e n e r a t i n g Se rbe r f u n c t i o n s

by diagonalization of the s2-matrix. The other is a direct

procedure for obtaining orthogonal operator bases spanning

Yamanouchi-Kotani and Serber representations of the symmet-

ric group algebra.

A computer program is given for generating simultaneous

^2 A A

eigenfunctions of L , LZ, i2, and S,.

PREFACE

T h i s d i s s e r t a t i o n i s concerned wi th t h e s i m p l i f i c a t i o n

of c a l c u l a t i o n s on e l e c t r o n i c systems through t h e e x p l o i t a -

t i o n of pe rmuta t iona l symmetry.

Accurate t h e o r e t i c a l d e s c r i p t i o n s of chemical phenom-

ena a r e made e a s i e r when s e c u l a r equa t ions can be f a c t o r e d

i n terms of commuting o p e r a t o r s . I t is i m p r a c t i c a l t o i g -

nore t h i s p o s s i b i l i t y i n any b u t t h e s i m p l e s t ca ses . I n most

quantum-chemical c a l c u l a t i o n s , it i s t h e r e f o r e ' d e s i r a b l e t o

c o n s t r u c t wave f u n c t i o n s from ant isymmetr ized space-spin

~2 f u n c t i o n s t h a t are a l s o e i g e n f u n c t i o n s of S and g Z .

Two problems must be so lved . F i r s t of a l l , one must be

a b l e t o g e n e r a t e s p i n e i g e n f u n c t i o n s f o r any d e s i r e d e igen-

va lues S and MS. I n o t h e r words, one must b e a b l e t o f i n d a

b a s i s f o r any g iven i r r e d u c i b l e r e p r e s e n t a t i o n .of t h e symmet-

r i c group. For systems w i t h more than a few e l e c t r o n s , t h i s

i s more d i f f i c u l t than it might seem. The problem h a s re-

ce ived much a t t e n t i o n i n r e c e n t y e a r s , and a survey of t h e

techniques a v a i l a b l e appears i n t h e t h i r d and f o u r t h chap te r s .

The second problem is t o s t r u c t u r e t h e wave f u n c t i o n

i n such a way t h a t e x p e c t a t i o n v a l u e s can be c a l c u l a t e d con-

v e n i e n t l y . I t i s p a r t i c u l a r l y impor tan t t o o b t a i n a s imple

formula f o r t h e energy. Prev ious a t t empt s have y i e l d e d ex-

p r e s s i o n s i nvo lv ing sums over many permutat ion ma t r ix e lements

o r o t h e r complicated c o e f f i c i e n t s . This s u b j e c t is d i s c u s s e d

i n t h e second chap te r .

W e i n t roduce a p a r t i c u l a r cons t ruc t ion f o r u n r e s t r i c t e d

c o n f i g u r a t i o n - i n t e r a c t i o n wave func t ions which s i m p l i f i e s

t h e c a l c u l a t i o n of expec ta t ion va lues . General wave func t ions

a r e expressed i n t e r m s of pure-spin components of determi-

n a n t a l func t ions . The bu i ld ing b locks , c a l l e d "spin-adapted

antisymmetrized products" , o r SAAP I s , a r e designed t o e x p l o i t

double occupancy.

I t is shown t h a t S A A P ' s , when cons t ruc ted from or tho-

normal . o r b i t a l s , can be handled i n c a l c u l a t i o n s more e a s i l y a

than S l a t e r de te rminants . Simple formulas a r e de r ived f o r . ' A2 mat r ix elements of . the Hamiltonian and L . A computer program

i s given f o r t h e eva lua t ion of c o e f f i c i e n t s occur r ing i n t h e

ene rgy 'ma t r ix elements.

Two new methods a r e . d e s c r i b e d f o r t h e . c o n s t r u c t i o n of

s u i t a b l e s p i n e igenfunc t ions . The . f i r s t o f . t h e s e i s an algo- ..

r i t hm f o r gene ra t ing S e r b e r func t ions by d i a g o n a l i z a t i o n of

^2 t h e S -matrix. The o t h e r i s a d i r e c t procedure f o r ob ta in ing

. . orthogonal m a t r i c bases ' spann ing Yamanouchi-Kotani and Serber - . . .

r e p r e s e n t a t i o n s of t h e symmetr ic group a lgebra .

A computer program is given f o r gene ra t ing simultaneous

"2 A A

e igenfunc t ions of L , Lz , and S Z .

I n t h e d i s c u s s i o n t h a t fo l lows , c e r t a i n s p e c i a l symbols

and convent ions a r e used. These a r e expla ined i n ~ p p e n d i c e s

A and B.

1nd i s t i n g u i s h a b i l i t y of E l e c t r o n s

E lec t rons a r e i d e n t i c a l , meaning t h a t no experiment

can t e l l them a p a r t . This impl i e s t h a t expec ta t ion va lues

a r e independent of any e l e c t r o n numbering scheme. Suppose

t h a t Y ( 1 , 2 , . . . , N ) i s t h e e x a c t wave func t ion ( a s o l u t i o n

of t h e Schrodinger equa t ion ) f o r an N-electron system, and

t h a t P is any of t h e N1 of t h e e l e c t r o n s . hen A

f o r any observable ope ra to r 0,

. . Since. permutat ions a r e u n i t a r y o p e r a t o r s ( ~ p p e n d i x B) , it fol lows t h a t

. . . , f o r any wave func t ion . Thus it must be t h a t

every observable ope ra to r i s i n v a r i a n t under s i m i l a r i t y

t r ans fo rma t ions t h a t permute i t s e l e c t r o n l a b e l s . I n o t h e r

words, every observable o p e r a t o r . a f f e c t s e l e c t r o n s symmetri-

c a l l y .

I f it should happen t h a t Y i s permuta t iona l ly symmet-

ric or antisymmetric,

then it is clear that (1) is satisfied. However, (1) does

not imply that the wave function has this property. In - fact, any product function

will satisfy (1).

The behavior of the operators does induce a behavior

in the wave functions. It follows from (2) that observable

operators commute with all electronic permutations, and

group-theoretical arguments then lead to the conclusion

that eigenfunctions of observable operators span represen-

tations of the ,symmetric group.

Suppose that the operator 8 has, for a given eigen-

value, a ~ e t ' ~ , , . . . ~ ~ of m linearly independent,

. . degenerate eigenfunctions. Then (2) guarantees that the

result (POi) of permuting any eigenfunctio~~ in the set

is a new function

which is itself a vector in the space spanned by the Oi. . .

The number [PI ji is the (j,i)-element of the matrix [PI

representing P, and the functions'{$i} are said to form a

basis for the representation.

1f:the symmetric.group SN contains every symmetry A

transformation commuting with 0, then the degenerate func- ,

tions ' {Oi span an irreducible representation of SN

(apart from accidental degeneracies), and each 6-eigen- '

value will be associated with a particular irreducible

representation.

Exclusion Principle

since permutations commute with the Hamiltonian, the

implication of the argument. above is that solutions of

the N-electron Schrodinger equation for a given energy

must span a representation of the symmetric group. Permu-

tations of electrons do not comprise every symmetry trans-

formation commuting with the Hamiltonian, so' there is no

theoretical reason to suppose that such a representation

will be irreducible.

Nevertheless, experiment demands that solutions of

the Sohrodinger equation for fermion systems must span

the one-dimensional (thus irreducible) antisymmetric repre-

sentation of the symmetric group. In other words, for

every P in SN,

where E ( P ) is +1 when P is even and -1 when P is odd.

Here P .is a transformation which permutes the space and 7

spin coordinates .of the fermions . This result is the Pau.li'Exclusion Principle for

fermions .

Spin Eigenfunctions

It happens that SN contains every symmetry transfor-

mation commuting with the total spin operator g2 .. Thus

spin eigenfunctions Ba(NSM), satisfying the equations

g2ea (NSM) = fi2s (S+1) Ba (NSM) ,

are basis functions for irreducible representations of SN.

Here the permutations transform only the spin coordinates

of the electrons. . .

0 Spin eigenfunctlons are important in quantum chemjs-

try because, for many atoms and molecules, the Hamiltonian, A G , very nearly commutes with s2 and SZ. This means that

eigenfunctions of fi can be rhosen to be also eigenfunctions h

of i2 and EZ. Doing so simplifies energy calfulations by factoring the energy matrix: . . if two trial wave functions

$,(NSM) and $ (NSIM1) are spin eigenfunctions for which 8 S1#S or M1#M,

The energy matrix reduces to a,direct sum of blocks within

which S and M are constant. .

Thus the Pauli and Indistinguishability Principles lead

to two conclusions .regarding electronic wave functions:

(i) the wave functions are antisymmetric with respect

to simultaneous permutations of the space and spin

coordinates of the electrons,;

(ii) they can often be chosen to be eigenfunctions of

, implying that they transform according to irreducible representations of the symmetric group permuting

only the spin coordinates of the electrons.

spin-~dapted Antisymmetrized Products

Slater determinants (Slater, 1929, 1931) are antisym-

metric with respect to simultaneous permutations of space A

and spin and are Sz-eigenfunctions, but they are not in gen-

eral eigenfunctions of s2 . An approximate wave function which is to be a spin eigenfunction .is usually constructed as a

linear combination of slatex determinants. In fact, q

antisymmetric wave function can be expanded in S'later deter-

' minants: such determinants span, the con£ iguration space

A SPater determinant f0r.N electrons is 0btained.b~

applying the antisymmetrizer (Appendix B) to the product

of a space.product function 4(N) and a spin product func-

tion 9 (NM) having the sZ-eigenvalue M:

(A has been defined in such a way that it is idempotent,

but (NM) is not normalized. ) In the discussion that fol-

lows, the orbitals of which $(N) is composed will not be

discussed. They may be atomic or molecular orbitals: what

they are in particular does not concern us at this stage.

The pertinent fact is that $(N) is some product of one-

electron orbitals, which we shall for convenience assume

to be orthonormal.

In analogy to the Slater determinant @(NM), we can h

define an antisymmetric eigenfunction of SZ which is also . an eigenfunction of g2 by replacing the spin product function e(NM) with a spin eigenfunction B,(NSM). The new

function,

will be ' an eigenfunction of ;* because the spin operator

commutes with A . Functions like that given in (3) can be

projected out of Slater determinants by suitable operators,

and we shall refer to .them as "spin-adapted antisymme-

trized products", or SAM'S. Since each spin eigenfunction

F, (NSM) i s a 1 inear com-.hjna,ti=an of spin products a SAAP ci

is a linear combination of Slater determinants.

' Spin-adapted antisymmetrized products span the N-

electron configuration space: any antisymmetric wave func-

tion can be expanded in terms of them. Furthermore, SAAPns

possess an advantage over Slater determinants, in that they

are eigenfunctions of g2.. Slater determinants are easy to ,-

handle without the use of group theory, and lead to con-

venient formulas for the matrix elements of observable

operators. We shall show, using group theory, that SAAP's

lead to formulas no less simple, and thus that they are

more efficient building blocks for wave functions when S

is. a good quantum number.

The antisymmetrizer in (3) masks the true relation-

ship between the space and spin components of the SAAP.

Suppose that there are d(NS) spin eigenfunctions B,(NSM)

for a given M. Then these functions span an irreducible

NS matrix representation [PI of SN: for any permutation

P transforming the spin coordinates of 'the electrons

This willbe called, the spin representation of SN. Using

this relation in (3),

where NS g B ( ~ s a ) = Id(NS)/N! I IE ( P I [PI Ba ( ~ $ 1 . i 6 P

Equation (5) shows t h a t t h e SAAP i s a sum of terms, l

each of which i s t h e product of a s p i n eigenfunct ion and

some kind of space funct ion . The space funct ion , a s shown

i n (6), i s p ro jec ted ou t of t h e "pr imi t ive" ,space product

funct ion + by a Wigner' ope ra to r (Wigner, 1931) . As a con-

sequence, these space func t ions form a b a s i s f o r an i r r e -

duc ib le r ep resen ta t ion of SN, c a l l e d t h e space represen-

t a t i o n : i f k=d(NS)/N!,

Comparison of ( 7 ) and ( 4 ) shows t h a t t h e apin func t ions

t ransform under P according t o t h e matr ix [p lNSr while t h c

space func t ions @ (NSa) t ransform according t o t h e t r ans - f3

pose of E ( P ) [p-llNS. Thus t h e r e is a c l o s e r e l a t i o n s h i p

between the spin and space representations:,they are recip-

rocal to each other in such a way that the SAAP 'is anti-

symmetric. These representations are said to be dual

(Kotani - et - al., 1955) . The spin-adapted antisymmetrized products have been

displayed in two equivalent forms. Either form demands a

procedure for obtaining spin eigenfunctions, and one of

them requires dual space functions. We shall see later

how these might be obtained. First, we examine the ukeiul-

ness of SAAP's.

ELECTRONIC WAVE FUNCTIONS AS SUPERPOS.1TIONS

OF SPIN-ADAPTED ANTISYMMETRIZED PRODUCTS

Linear Dependence of SAAP's

Any antisymmetric wave function that is an eigen-

function of s 2 and Gz can be written as a linear combination of SAAP's having N, Sf and M fixed:

If the sum over space products includes contributions from

different configurations, Y is a configuration-interaction

(CI) function. We assume for generality that this is the

In ( 8 ) , the sums run over every space product for the

configurations of interest, and every spin eigenfunction

for the given N, Sf and M. In general, some of the SAAPVs

will then be linearly dependent. In order that the coeffi-

cients c($,a) will be unique and the secular equation will

be soluble, it is essential to remove this dependence. Two

sources of linear dependence are easily identified.

Suppose that Y includes every SAAP containing the space

product $. SAAPDs containing a space product $'=P$ differ-

ing from $ by only a permutation should not be included in

Y. For

Thus any SAAP containing @' is linearly dependent' on SAAPqs I

already included in Y. The additional one'contributes noth-

ing new.

It follows that, in ( 8 ) , it is sufficient to sum over

'just'those space products containing -different orbitals.

Double occupancy is a second source of linear depend-

ence. If a space product @ contains a doubly-occupied

orbital, there exists a transposition t=t-' such that

t@=@. It-follows that

Thus the only way to avoid having all the SAAP's for given

N, St and M linearly dependent is to construct the spin

eigenfunctions B,(NSM) in such a way that

for every transposition t under which 4 is invariant. A

procedure for doing this is introduced in the next two

sections.

u c t s h a l l be chosen. W e have assumed f o r convenience t h a t t h e

space o r b i t a l s a r e or thonormal . I n a d d i t i o n , w e adopt t h e J

fo l lowing convention. -- f o r t h e s t r u c t u r e - of space produc ts :

t hey w i i l have a l l t h e i r doubles l i s t e d f i r s t , w i t h ascending

labels, fol lowed by t h e s i n g l e s , i n t h e o r d e r of ascending

l a b e l s . , F o r example, of. twelve p o s s i b l e space p roduc t s con-

2 t a i n i n g t h e a tomic o r b i t a l s (Is) 2s2po , w e p i ck t h e f u n c t i o n

A s i n t h i s example, space p r o d u c t s ' c o n t a i n i n g n doubles

w i l l be denoted by t h a t s u b s c r i p t : e . g . , $ , $',. Space prod-

u c t s w i t h t h e s u b s c r i p t T a r e i n v a r i a n t under t h e geminal

p e ' m u t a t i o n s belonging t o a,, , where aka.

Geminally-Adapted Spin Eigenfunc t ions

T h e r e ' a r e i n f i n i t e l y many ways t o make s p i n f u n c t i o n s

f o r g iven-N, S t and M, corresponding t o i n f i n i t e l y many

e q u i v a l e n t s p i n r e p r e s e n t a t i o n s of SN . W e choose t h e follow-

i n g convent ion - f o r s p i n f u n c t i o n s :

(i) The s p i n e i g e n f u n c t i o n s w i l l be or thonormal .

(ii) They w i l l be c o n s t r u c t e d by coup l ing t h e

s p i n s of each geminal p a i r of e l e c t r o n s sep-

a r a t e l y , t h e n coupl ing t h e p a i r - s p i n s t o . .

each o t h e r . I f N i s odd, t h e s p i n of t h e

remaining e l e c t r o n w i l l then be coupled

to the resultant spin.

Spin eigenfunctions constructed in this way, using

Clebsch-Gordon coefficients, were first described by Serber

(1934a, 1934b) . They contain a singlet or triplet component . .

for every geminal pair of electrons, and thus are either

symmetric or antisymmetric,with respect to every geminal

transposition in SN. The Serber functions for N=4, S=1. M=O

[a('l) B(2)+B(l)a(2) I [a(3) B(4)-8(3).,(4) 1/2,

la (1)a(2) B (3)'8.(4)-8 (1) B (2'),a(3)o (4) 1/42,

We shal.1 denote a.spin function antisymmetric in the

first IT geminal pairs, but symmetric in the next one, by the

subscript IT. If there are several such functions, they will

be called BIT1 (NSM) , BIT?, (NSM) , etc. Using this ,notation, the

functions in the example above would be labelled BO1(41O),

C302(410), and e11(410). As a result of the notation,

for every. g in SN,

but in particular,

ge,, = -e,, if g belongs to#,, , where nin.

These relations imply that the matrices representing

geminal transpositions in the spin representation spanned

by Serber spin functions are diagonal: since the functions

are orthonormal,

'glnll~ ,aa = <en ,, (NSM) 1 gGna (NSM) > = 2<Bn ,, (NSM) I %a (NSM) >

for every g in SN. 1n particular, if g belongs toan,, where

n'cn or rl<n" , NS

[ g l r ~ ~ ~ ,na = -6(n"B,na).

Since geminal permutations G are products of geminal

transpositions, we have the more general result

NS = 26 (rWf3,na) for every G in SN. [G1n"f3,na

In particular,, if G belongs to a,, , where irn or nl~a", NS

lG1 nl'B ,na = E (G) 6 (n"b,na), (10)

in which E (G) is +1 when G is even and -1 when. G is odd.

This result has .a special consequence that will prove

use£ ul . We write 'GE&~. " to mean.."(; belongs to dnl " . Since ' . cvcry gcminal pcrmutation ic.n product of mutually commuting

geminal transpositions, with a .factor I or (2p-1,2~) from

the pth geminal pair, the order, of is 2': and (10) gives

when IT'<IT or n's~". This result is, as we shall see, a great

aid in simplifying the expressions for expectation values.

Linearly Independent SAAP's

The two conventions we have adopted further simplify the

wave function (8) when the space products contain doubly-

occupied orbitals. We have already reduced the number of

space products required to the bare minimum:'one product for

each choice of orbitals. The conventions reduce the number

of spin eigenfunctions required.

Consider the SAAP A [ $ €3 1, where ,'<IT. The geminal trans- IT +a -

position g=g '=(21;+1,2;+2) , which belongs to bn but not to 4,' , has the properties

- SO, - '9, and g€3,& = +8&.

As a result,

A[+,~,.,I = A [ (g$,)enh] =Ag[$,.*geITb] = -A[$ T e ITC~ , I,

so that the SAAP is zero. In other words, if a SAAP contains

a space product with doubles in geminal positions in which

the associated spin function is not antisymmetric, then that

SAAP vanishes.

This result reduces the sum over spin functions in the

CI wave function: .we have now

where t h e sum over space produc ts @, i n c l u d e s on ly one prod-

u c t f o r each cho ice of o r b i t a l s , and t h e sum over s p i n func-

t i o n s i n c l u d e s on ly some of them.

The wave f u n c t i o n has been reduced t o t h e b a r e minimum:

t h e SAAP's i n ( 1 2 ) a r e a l l l i n e a r l y independent. I n f a c t , w e

now show t h a t t hey a r e a l l o r thogona l .

The o v e r l a p between two SAAP's w i t h t h e same va lues

of N , S , and M i s

Here w e assume t h a t {)n and hp, f o r o the rwi se t h e SRAP's '

would van i sh . The f i r s t i n t e g r a l on t h e r i g h t is t h e (n&,d f l ) -

e l emen t .05 t h e m a t r i x r e p r e s e n t i n g P ' i n t h e Se rbe r s p i n r ep re - . .

s . e n t a t i o n f o r N , S . Thus

No two space produc ts i n . the C I wave f u n c t i o n c o n t a i n t h e

same o r b i t a l s , s o <@,[PO > i s z e r o u n l e s s @,='#P: P

The i n t e g r a l on t h e r i g h t i s zero unless P belongs t o t h e

geminal group &, under which 0, i s i n v a r i a n t :

Using (11), ' , ; f3)*2" /~! A = 6 ($,t0p)6(7Ta

This proves t h a t t h e func t ions

.where IT<IT' and only one space product i s included f o r each

choice of o r b i t a l s , form a complete ~ r ~ h o n o r m a l s e t spanning

t h e space of N-electron antisymmetric wave func t ions having

s p i n eigenvalues S and M . These SAAP ' s a r e t h e r e f o r e e f f i -

c i e n t bu i ld ing blocks f o r C I wave func t ions when S i s a good

quantum number.

Energy Matrix Elements between S A A P D s

Constructed. from Orthonormal O r b i t a l s

General formula

The importance of t h e space and s p i n conventions i n t r o -

duced i n t h e l a s t s e c t i o n s l i e s i n t h e way i n which they s i m -

p l i f y t h e c a l c u l a t i o n of expecta t ion va lues . I t has been

shown t h a t they f a c i l i t a t e t h e removal of l i n e a r dependence

i n t h e wave funct ion . W e now show t h a t they s impl i fy t h e

c a l c u l a t i o n of energy matr ix elements.

It is .assumed that the wave function (12) is constructed . . . . . . .

from orthonormal orbitals, and that the Hamiltonian is, for . .

practical purposes, spin-free. Except for these conditions,

our results will be perfectly general, and applicable to

either atomic or molecular systems.

The immediate result of (12) is that the energy is a

sum of Hamiltonian matrix elements between spin-adapted

antisymmetrized products. The problem is to express such

matrix elements in terms of elementary one-.and two-elqctron

integrals.

~ u s t as SAAPQs are generalizations of ~iater determi-

nants, we shall obtain matrix element formulas which are gen- . .

eralizations of Slater's matrix element rules. Despite the

.fact that the derivations are complicated by"group theory, the . . . "

results are very nearly as simple as those for.determinanta1

functions. Befbre proceeding to the derivation, we define

notation and display the formulas obtained.

We consider the two SAAP ' s A [ + n (N) (NSM) 1. and

A (N) edB (NSM) ] , where the space products are P

Here n and pk are the orbitals occupied by electron k in k

0; and $I . According to convention, O n and 4 contain n and .. P . P .

p doubles , r e s p e c t i v e l y . I t should be noted t h a t . . an o r b i t a l

nk i n $n can occur a l s o i n $ and an o r b i t a l pm i n $ can P P

occur i n 4,. W e w r i t e , f o r example, n (nk ,On) and. n (n $ ) t o k' P

denote t h e occupancies .of nk i n O n a n d $ . P

I t i s assumed t h a t O n and $ d i f f e r by no more than two P

o r b i t a l s , and t h a t n s n ' and p6p1. Otherwise, t h e energy matr ix

element i s zero.

here i s a permutation, 6,, t h a t rear ranges $ s o as t o P

p l ace it i n "maximum coincidence" wi th $,. This means t h a t

(L$p) and O n a r e i d e n t i c a l except poss ib ly f o r t h e o r b i t a l s

occupied by one o r two e l e c t r o n s .

We break down t h e Hamiltonian i n terms of t h e one-

e l e c t r o n H a r n i l t ~ n i a n s h ~ and' t h e e l e c t r o n i c i n t e r a c t i o n s gi j :

-1 where H i j = (N-1) (h i+h. ) + g i j . 3

The genera l formula f ~ r t h e energy matr ix element t u r n s

ou t t o be [when t h e SAAP ' s a r e normalized according t o (13')l

i n which even E (dr) = {I:} when L i s { Odd)

NS [P 1 n'cl, dB = <enk (NSM) I P 1 edB (NSM) > 1

and pr and pS a r e t h e o r b i t a l s occupied i n (L$ ) by elec- P

t r o n s i and j , r e s p e c t i v e l y . .

I f $,=$ , t h e s u m i n (15)' i s over every d i s t i n c t p a i r P

of o r b i t a l s i n t h e space produc t . For example, i f nl=n2 i s a

double , b u t n 3 and n 4 a r e s i n g l e s , t hen t h e sum may I i n c l u d e

t h e p a i r s ( n 1 2 , ( n l , n 3 ) , (n1,n4) , ( n 3 , n 4 ) , i n which c a s e

it does - n o t i n c l u d e ( n 2 , n l ) , ( r2 ,r3) , o r ( n 2 , , n 4 ) , . O r it may

i n c l u d e ( n l , n i ) (n2 ,n3) , ( n 2 , r 4 ) , and ( n 3 ,n4) , i n which case

it does n o t i n c l u d e (n2 , nl) , ( n l , n 3 ) , o r ( n 1 4 ) . I n o t h e r

words, doubles do n o t con t2 ibu te d u p l i c a t e t e r m s t o t h e sum.

When,@n=@p, t h e a l ignment permuta t ion i s L = 1 .

If $n and $ d i f 5 e r by one o r b i t a l , t h e sum is over P . ,

every d i s t i n c t o r b i t a l p a i r i n @ n c o n t a i n i n g t h e d i f f e r i n g

o r b i t a l . For example, suppose t h a t a , b , c , d are o r b i t a l s ,

and +n=n1n2n3n4=aabc whi le g p = p 1 2 3 4 - p p p -abcd. The d i f f e r i n g

o r b i t a l i n $* i s a , and i n $ i s d . Then t h e sum i n (15) may P

i n c l u d e t h e o r b i t a l p a i r s ( n l , n 2 ) = ( a , a ) , ( n l , s 3 ) = ( a , b ) , and

IT ) = ( a , c ) , b u t n o t (IT* , 7 r 3 ) o r ( 7 ~ ~ ~ 7 ~ 4 ) a s w e l l . I n t h i s ('11 4

example, 6 = ( 1 , 2 , 3 , 4 ) ' . : . .

I f $n and $ d i f f e r by two o r b i t a l s , t h e on ly t e r m occur- P

r i n g i n t h e sum i s t h a t f o r which ni and n j

a r e t h e d i f f e r i n g

o r b i t a l s .

The f u l l power of t h e SAAP formalism becomes e v i d e n t when

one e v a l u a t e s t h e m a t r i x e lement i n s p e c i f i c c a s e s , express -

i n g it i n t e r m s of one- and two-elect ron i n t e g r a l s . W e save

t h e d e r i v a t i o n s u n t i l l a t e r , and g i v e h e r e on ly t h e r e s u l t s .

Case when %=$ -- I n t h i s e v e n t , pr=ni , p s = n j , a n d L = 1 . W r i t i n g

n (n i ) = n ( ~ ~ , $ ~ ) = n ( ~ ~ , $ ~ ) , w e have

t h e sums being over d i s t i n c t o r b i t a l s ( i . e . , on ly one from

each d o u b l e ) . H e r e g=(e2/r12) and h i s a one-e lec t ron H a m i l -

t o n i a n . Case when kn and & d i f f e r one o r b i t a l -- -

L e t t h e d i f f e r i n g o r b i t a l be n i n $ and po i n $ . ? Ti

. . P Then

where t h e sum is over d i s t i n c t o r b i t a l s i n On o t h e r ' t han the

o r b i t a l s n and p a . A double makes on ly one c o n t r i b u t i o n . u

Case when & & differ o r b i t . a l s -- P

W e t a k e t h e d i f f e r i n g o r b i t a l s t o be a,,, nv i n 4,. and

P u t P, i n 4 . There a r e no sums i n t h e formula and no one-

P e l e c t r o n i n t e g r a l s a r i s e . The r e s u l t i s , t h e n ,

Discuss ion

These formu1.a~ are ve ry n e a r l y a s s imple as ' S l a t e r ' s

r u l e s f o r m a t r i x e lements between de t e rminan t s ( S l a t e r , 1929) , t h e ' d i f f e r e n c e being t h a t c e r t a i n d e l t a f u n c t i o n s f o r one-

e l e c t r o n s p i n s have b e e n . r e p l a c e d by s p i n r e p r e s e n t a t i o n '

matr ix elements . f o r t h e permuta t ions C and (i, j )L.

Formula (15) was f i r s t obta ined , i n a s l i g h t l y l e s s

simple form, by K . ~ u e d e n b e r g (p r iva te ' communication, Iowa

S t a t e u n i v e r s i t y , Ames, Iowa, 1968) ' . The formulas shown' here ,

a s we l l a s formulas f o r t h e matr ix elements of p-electron

opera to r s and pth-order . reduced d e n s i t y matr ices , w i l l be

rep'orted by Ruedenberg and Poshusta ( 1 9 7 1 ) . There have been previous a t tempts t o o b t a i n formulas

of t h i s type.. Kotani -- e t a l . (1955) .used group theory t o s i m -

p1 i fy the express ions f o r ene'rgy.matrix. elements between

s p i n components of de terminanta l func t ions . Har r i s (1967)

extended t h i s work, and gave closed- and open-shell . formulas

f o r mat r ix elements . of one- and two-electron operat ,ors , with- . .

out assuming t h a t t h e - o r b i t a l s a r e or thogonal . Even with t h i s

assumption, h i s r e s u l t s . were complicated, . involving sums

over many permutations. Karplus e t -- a l . (1958) obtained matr ix

element formulas f o r one-electron opera to r s .

The case when t h e wave funct ion is expressed a s one SAAP

i~ c i m i l a r t o the extended Hartree-Fock dppruximation of

Lowdin (1955b, 1960) . Matrix elements f o r sp in- f ree opera to r s

i n t h i s formalism w e r e 'ob ta ined by Pauncz , de Heer, and . .

Lowdin (1962) f o r appl ' ica t ion t o t h e a l t e r n a n t molecular or-

b i t a l method. The formulas were genera l ized by Pauncz (1962,

1969) . The r e s u l t s involved var ious complicated c o e f f i c i e n t s , t

closed expressions f o r which were found by a 'number of work-

ers (Percus and Rotenberg, 1962; Sasak i and Ohno, 1963; Smith,

1964; Shapi ro , 1965; Smith and Harris, 1967) . Reviews have

been g iven by Harris (1967) and P.auncz .(1967, 1969) .

The formulas p r e s e n t e d , h e r e avoid t h e s e d i f f i c u l t i e s .

The i r c l o s e r e l a t i o n t o S l a t e r ' s r u l e s ' is emphasized by t h e

ease w i t h wh ich . they c a n ' b e reduced t o t hose r u l e s when t h e

SAAP ' s involved happen t o be S l a t e r ' de te rminants . Consider ,

N f o r example, t h e case when and M=S=-: 2 enb=63 =aa* * a . dB Since t h e s e s p i n f u n c t i o n s c o n t a i n no an t i symmetr ic f a c t o r s ,

it must be t h a t n=p=O and n ( n i ) = l f o r every o r b i t a l n i n i

O O . W e h a v e & ( n h , p ' ~ ) = i and

i n (16) . The r e s u l t is. : t h e formula

s i n c e A [$0 (act* * * a ) I = ( a a * * * a ) A [ ~ o I , ( 1 9 ) is the formula

f o r t h e m a t r i x e lement 1; I A ( O ~ ) ? , where O O c o n s i s t s

e n t i r e l y of s ingly-occupied o r b i t a l s . Thus A (00) i s a "space

only" S l a t e r de te rminant , and (19) is analogous t o t h e f ami l -

i a r formula f o r t h e energy of a d e t e r m i n a n t a l wave f u n c t i o n .

Appendix D contain 's a l i s t i n g f o r a F o r t r a n program t o

implement formulas (16) - (18) . I t f i n d s t h e a l ignment . permu-

t a t i o n L, e v a l u a t e s t h e r e p r e s e n t a t i o n ma t r ix e lements f o r

L and (i, j )& from knowledge of t h e s p i n f u n c t i o n s , and cal-

c u l a t e s ' t he c o e f f i c i e n t s of t h e one- and two-elect ron i n t e -

g r a l s o c c u r r i n g i n t h e energy m a t r i x e lement between two

SAAP1s. T h i s program was based on e a r l i e r , ' m o r e compl ica ted ,

formulas t han t h o s e given here': An updated. v e r s i o n is. 'being

w r i t t e n .

The Serber s p i n f u n c t i o n s used wi th this program w i l l

be d i s c u s s e d l n t h e nex t chap te r . A s i s mentioned t h e r e , i f . .

i s found more conven ien t t o g e n e r a t e t h e s p i n f u n c t i o n s and

then o b t a i n t h e . r e p r e s e n t a t i o n m a t r i c e s from them, t h a n t o . . . .

c a l c u l a t e t h e s e m a t r i c e s d i r e c t l y ., . .

Der iva t ion ' of t h e General ..

Energy Mat r ix Element Formula . . . .

W e seek t o e v a l u a t e t h e i n t e g r a l

where T~T ' , , p ~ p ' , and t h e sum runs ,over a l l of S S ince t h e . . . N'

Hamiltonian i s assumed t o c o n t a i n no s p i n o p e r a t o r s , space . .

and s p i n s e p a r a t e :

. . . . .where we have used the fact that H is ~ermitian.

In terms of the one-electron Hamiltonians hi and the

electron, repulsions g ijt the N-electron ~amilt~nian is

In order to simplify the derivation that follows, we shall

write. A

. H = l l H i j , . i<j

in terms of the operators

Thus the Hamiltonian is written in terms of two-electron

operators. From (20) , we. have

the.sums on i and j,being over electron labels, and the sum

on P being over the symmetric group, SN. The rest of the deri-

vation is devoted to the simplification of this equation. . .

Reduction ---- of the sum over.permutations . . . , .

We assume 'that @n and 4 are the following products of P

. . orthonormal one-electron orbitals : . .

We write (4 ) to denote +at part of 4 occupied P ktfir***rm P

by electrons k,L, . . . ,m. For example, (4p) = pk. Then

This integral is zero unless <nkl(p4 P ) k > = 1 for every k

other than i and j,,.

It is clear that not every P in (21) will make a nonzero

contribution to the i , j -term. Suppose that Qi i s a permu- . .

tation aligning 4 with $n i n such a way that (Qij@P)k=nk P

for every k other than i and 1,. Then

Furthermore, for any geminal permutation G in& P ' G4P=0p and

Thus the set of permutations { Q ~ ~ G I G E & ~ } makes nonzero con-

tributions ' to ' the i, j-term in (21) . We will show that other permutations may do this.

The two orbitals 'iram. 0 . t h a t are, occupied in ( 8 . . 0 ) . e 1 3 P by electrons i and j are uniquely determined,.by the condition

that <H. .(I I Q ; .(I > not vanish. Let these ..two. orbitals. be P, 1 3 n 1 1 P

and ps:

This is not meant t,o suggest that r and s are.uniquely deter-

mined by i and j . If pr or ps is a double in,.,$ P ' then there

rn 9 -n UI a

4J '44

J3 : 3

rd, rl

rn 0 w

. a -nu

k -n rn

2 arna

€3- w

- ! -mu

t-i l=

a, - n

a -n rn

4J - -

E-1 a,

(22) . It must be kept in mind that these' permutations may not all be distinct. The sets

each consist of distinct permutations. We now investigate

the conditions under which the sets may overlap.

The two sets share an element if and only if there are ,

two geminal permutations G and G t ingP such that

But then ij -1 ij ,-1 = GG-' -1 (rts) = (Qrs) QrSGG

i j ij* (rts)G} share an element Thus {Q:~G} and (it j)QrSG} = {QrS

only if (r ,s) E,$ : in fact, then they share all their elements. P -

heref fore, the sum in (21) over all re-

duces to a sum over the permutations in the two sets (23), . .

but this sum should be divided by two if r and s are in gemi-

nal positions and (r &.& (1 .e., if pr=ps) . Using this re- P

sult in (21) , we obtain

where 6 ( pr , ps) is the Kronecker delta.

This result can be simplified by noticing that

NS - NS NS (i) for any permutation P, LPG1 - [Pl,;ltp,BIGl because the .matrices representing geminal permuta-

tions are diagonal;

NS (ii) 1 E (GI [GI dB, dB = 2Pt from (11) .

Gr8, /

Thus we obtain

the sums running over .electrons.

Reduction of the sum over electron pairs

Equation (24) contains redundancies. Suppose that ni=nk . , . .

We make the following observations:

(i) It must be that k=ill and (ilk) is a geminal

transposition.

. G. g 3

r-i -4

t h e number of equal i f n i is: , and IT is :

j con t r ibu t ions i n

( 2 4 ) is:

o double same double 1

double d i f f e r e n t double 4

double s i n g l e 2

s i n g l e double 2

s i n g l e s i n g l e . 1 '

I n genera l , t h e number of equal con t r ibu t ions i s

where dij($,) is t h e number of doubles i n $ 71 represented by

the o r b i t a l s ni and n j '

Equation ( 2 4 ) i s s i m p l i f i e d by c o l l e c t i n g toge the r a l l

t he equal t e r m s , summing only over d i s t i n c t con t r ibu t ions .

This is t h e same a s summinq over d i f f e r e n t p a i r s - of o r b i t a l s

i n 4,. Normalizing t h e SAAP's according t o (13) . we have

where p ( i j ; r s = , [(p-n)/21+dij ($ , ) -6 (n i~n ) - ~ ( P ~ I P ~ )

The meaning of t h e sum needs c l a r i f i c a t i o n . I f

t h e sum runs over every d i s t i n c t p a i r of o r b i t a l s . For ex-

ample, i f $,=$ =IT n n n =aabc, t h e sum inc ludes t h e o r b i t a l p 1 2 3 4

. c, o

-rl a,

same e lec t rons . ' The e l e c t r o n s occupying t h e d i f f e r i n g o rb i -

t a l s i n O n a r e unambiguously def ined by t h e convention adopt-

ed f o r i and j . Any 6r w i t h . t h i s behavior w i l l perform t h e d u t i e s of

every pi! i n t h e sum of ( 2 5 ) . Thus we ob ta in a simpler r e s u l t :

The.exponent of two appearing i n t h i s equat ion i s

a number apparent ly not symmetric i n i t s arguments. However,

where ?iij - - ars i s t h e number of doubles i n O n o t h e r than n i and n o r t h e number of doubles i n 4 o the r than PI and ps .

j P Thus

where 1 . ~(itj) = I-d. . ($,) - 6 (nitnj)-J 2 1 7

1 and P(rts) = [zdrs(Op) - 6 (pr,ps) 1.

This can be cast into a form more convenient for pro-,

gramming by noticing that

a result.obtained by considering all possible. cases:

double . same double 1 - - I = - - ' I (4/8) 1/2 2 2 = 1/42

double ' different double 1 - 0 .= 1 (4/1)li2 = 2

double single Y o = 2 (2/1)li2 = h 1 .

single double 1 5 - O = (2/1) 'I2 = 42 2 single single . 0 - O = O (1/1)1/2 = 1

We obtain the final re'sults

NS - [ (i, j)L] Yh,,tB<~ij~i.rr,l b . s P r } .

This is the general' energy matrix element formula quoted in

(15) on page 20. 1t is also, of course, the matrix element

between SAAP's of any operator expressible as a sum of two-

electron operators. , .

The only properties of the spin eigenfunctions that

were used in deriving this equation were those of (10) and

(11). In other words, we have assumed that the spin function

in a SAAP is antisymmetric in every geminal pair .which is a

double in the space product. We have also assumed that the

. . spin functions can be labelled eTb . indicating that the . .

.functions are. antisymmetric in the. first .rr' geminal pairs, . . . . .

and.symmetr,ic in the next one. As we shall see, Serber spin

functions are not the only ones with 'these properties. It

will turn out, though, that Serber.functions are particularly . .

easy to generate.

2 Q). C

t3 . rn rl a

'44 -4

a, C 3

rl I h

Table 1. S i t u a t i o n s occur r ing when 4 and 4 d i f f e r by one TI P

b , Example b TI TI Example Pa p . j j

( i n +TI)a of O n ( i n Q ~ ) ~ of O p

s d ( p j / * ' * j m ) s d . ( a j / * * * j m )

d d ( m j / * . * j m )

d f d ( p j / - * * p j m > s d ( a j / * - * p j m )

d ' # d (mj/* -'p j m )

% o t a t i o n : "dl' means double , " s " means s i n g l e .

b ~ n t h e examples, o r b i t a l s a r e r ep resen ted by t h e i r sub-

s c r i p t s . The o r b i t a l s occupied by e l e c t r o n s p and j a r e

l i s t e d t o t h e l e f t of t h e s l a s h . The d i f f e r i n g o r b i t a l .

is l i s t e d f i r s . t .

It can be seen, from Table 1 t h a t , when IT does n o t equa l n j v

o r p o t it i s a double or s i n g l e i n bo th $n and 4 . Thus P

[ n ( n j t @ , ) n ( n ,(I = 2e(fiumber of doubles o t h e r t han TI j P j n and p U ) U

(#nu t p,) + l a (number of s i n g l e s o t h e r t han

n and p a ) lJ

= number of e l e c t r o n s occupying o r b i t a l s

o t h e r than and p . lJ (3

Because o f . t h i s , t h e p o s s i b l e v a l u e s of % a r e :

n ( i n 4 n ) P, ( i n O lJ P

The r e s u l t i s t h a t n = , ( N - l ) [ n ( ~ l J l ~ , ) n ( ~ , , O P ) 1 'I2, and s o

t h e ma t r ix element i s

This was t h e r e s u l t q u o t e d in (17) on.page 23.

GENERATING SPIN EIGENFUNCTIONS

WITHOUT USING GMUP ALGEBRA THEORY

Cons t ruc t ion of Spin Eigenfunc t ions

by Spin-Coupling ~ e c h n i q u e s

Yamanouchi-Kotani f u n c t i o n s

. .The e n t i r e . s p i n space f o r N e l e c t r o n s i s spanned by t h e

N 2 e lementary s p i n produc t f u n c t i o n s B k ( N M ) :

N Of t h e s e , t h e produc ts {ek (NM) 1 k=1,2.. . . , (,+,)I Span t h e

p a r t of t h e N-spin space t h a t is s p e c i f i c t o Sz-eigenvalue M.

On t h e o t h e r h a n d , , t h i s subspace i s a l s o spanned by s p i n

e igen func t ions B.(NSM), where j and S t a k e on a l l p o s s i b l e 3

v a l u e s . Thus t h e r e i s a t r ans fo rma t ion from t h e e lementary

s p i n produc ts t o t h e s p i n e igen func t ions :

€I. (NSM) = 1 €Ik (NM) yk (NSM) . J

H e r e t h e produc t f u n c t i o n s B k ( N M ) belong t o t h e r e d u c i b l e

' d i r e c t - p r o d u c t s p i n space f o r N fermions . The c o e f f i c i e n t s

'kj must be chosen i n a S p e c i a l way t h a t f o r c e s €I . (NSM) i n t o

. . .I.

a subspace f o r t h e i r r e d u c i b l e s p i n r ep resen ta t ion def ined

by N and S.

This is specia l case of t h e vector-coupling. problem

solved by Wigner (1931). The s o l u t i o n i s given stepwi'se, by

coupling s p i n s one a t a time. One s t a r t s with t h e s p i n of a

s i n g l e e l e c t r o n , couples 'it t o t h e s p i n of ' another , and pro-

ceeds by coupling t h e s p i n of t h e Nth e l e c t r o n t o t h e re-

s u l t a n t s p i n of t h e f i r s t ( N - 1 ) ... A t each s t a g e , t h e r e a r e

two ways i n which one can ob ta in s p i n S f o r N e l e c t r o n s .

P i c t o r i a l l y ,

This s o r t of spin-coupling p i c t u r e i s c a l l e d a' branchinq d i a - - gram, and t h e two r o u t e s shown correspond t o t h e two equat ions

S - M + 1 '

0 . (NSM) = -\j2S+2 1 1 € 3 . ( N - 1 ,S+Z ,M-I) * a ( N )

1 1 : + 0 . ( N - 1 , S-I,M+-) 6 ( N ) . 3 2

The c o e f f i c i e n t s appear ing h e r e a r e examples of Clebsch-

Gordan o r Wigner c o e f f i c i e n t s , which gua ran tee t h a t t h e

B.'(NSM) form an orthonormal b a s i s f o r an i r r e d u c i b l e ' r ep re - 3

s e n t a t i o n of SN.

I n apply ing t h e s e e q u a t i o n s r e c u r s i v e l y f o r g iven .N, S ,

and M , one makes a sp in-coupl ing cho ice a t e a c h ' s t a g e - a ,

cho ice between Equat ions (28a) and (28b) . I n the end, t h e r e

a r e a number of ways i n which N one-e lec t ron s p i n s can be

coupled s o t h a t t h e r e s u l t a n t s p i n i s S. Each of t h e s e " sp in -

coupl ing schemes" is l a b e l l e d by a v a l u e of t h e s u b s c r i p t j

i n ( 2 8 ) . The schemes can be r e p r e s e n t e d p i c t o r i a l l y as r o u t e s

on an N-elect ron branching diagram l i k e t h e one g iven i n Fig-

u r e 1, where w e have g iven a t each i n t e r s e c t i o n t h e number

of s p i n . f u h c t $ o n s r e s u l t i n g f o r . t h e corresponding v a l u e s o f N

and S. This number, which is independent of M , i s

Thus, f o r example, t h e r e a r e t h r e e s p i n e i g e n f u n c t i o n s f o r

N=4, S = l , f o r each va lue of M . -- - - Since each N-elect ron s p i n f u n c t i o n i s d e r i v e d from a

c h a i n of p r e d e c e s s o r s , ' t h i s p rocedure i s o f t e n c a l l e d a "gen-

e a l o g i c a l c o n s t r u c t i o n " . I t w a s i n t roduced by Yamanouchi

(1936, 1937, 1938) , and a f u l l account has been g iven by

Kotani e t -- a l . (1955) . W e s h a l l h e r e a f t e r r e f e r t o s p i n func-

t i o n s c o n s t r u c t e d accord ing t o (28) as Yamanouchi-Kotani ( Y K )

Figure 1. ~ a m a n o u c h i - ~ o t a n i branching diagram

s p i n . f u n c t i o n s , and t o F igu re 1 as. a YK b ranching diagram.

The YK func, t ions a r e a b a s i s f o r a ve ry s p e c i a l or thogo-

n a l r e p r e s e n t a t i o n o f SN. Not on ly are t h e m a t r i c e s r e p r e s e n t - . .

i n g permuta t ions i n SN f u l l y reduced, b u t i t w i l l be observed . .

from ( 2 8 ) t h a t t h e r e p r e s e n t a t i o n .of t h e subgroup SN-l is a l s o

reduced. I n f a c t , t h e r e c u r s i v e n a t u r e of t h e s e equa t ions has '

. the r e s u l t t h a t . t h e r ep r , e sen ta t i ons of t h e subgroups SN-l,

SN-2 ..., S1 are a l l - f u l l y reduced. Th& YK s p i n r ep re sen ta -

t i o n i s . s a i d t o . b e adapted t o t h e sequence

of nes t ed symmetric groups (K le in , C a r l i s l e , and Matsen,

1970) . W e s h a l l . r e t u r n t o t h i s p o i n t l a t e r .

Serber f u n c t i o n s .

I n t h e l a s t c h a p t e r , w e found it u s e f u l t o have 'orthogo- h

n a l e i g e n f u n c t i o n s of g 2 and SZ t h a t w e r e s imul taneous ly

~2 e i g e n f u n c t i o n s o f a l l t h e geminal s p i n o p e r a t o r s S ( 2 ~ - 1 , 2 p ) , where y l a b e l s a geminal p a i r of e l e c t r o n s . Such f u n c t i o n s

were f i r s t ob t a ined by Serber (1934a, 1934b) , u s ing a genea-

l o g i c a l procedure i h which s p i n s were coupled - two a t a t i m e .

Assume f o r t h e moment t h a t N=2n i s even. ,Then, ' de f in ing

geminal s p i n f u n c t i o n s w ( s . , m .) f o r t h e y t h geminal p a i r , I-' I-' lJ

w,, (0 ,O) = [a ( 2 ~ - 1 ) B ( 2 ~ ) -6 ( 2 ~ - 1 ) a ( 2 ~ ) I / J 2 ,

. . it is' p o s s i b l e t o make 2n-e lec t ron s p i n e igenfun 'c t ions from

t h e s e :

Here t h e sum runs o v e r . a l l cho ices of m l , m 2 , . . . ,m such t h a t n

Cm =M. S i n c e each s i s f i x e d , t h e f u n c t i o n s (30) w i l l be IJ I-'

~2 automatically eigenfunctions of S (2~-1,2.p) for each P.. The

subscript " 7 ~ " On

(NSM) indicates that

Thus Bncr(NSM) is antisymmetric under the geminal transposi-

tions of rtJT. Each geminal spin function w (s ,m') belongs to an

lJ . P lJ

irreducible representation r(s ) for two electrons, so lJ

'7C a (NSM) automatically .belongs to the space for the direct-

product representation

The coefficients must be chosen in a special way that forces

8. (NS'M) into the irreducible space defined by N and S. 7Ca As'before, the solution'is given stepwise, in this case

.by coupling spins two at.a time:

€3 (NSM) = 1 W (s' ,sn,S;M-mn,mn,M) x 7Ta m 7Ta

Here Bra(N-2,s ' ,M-mn) is an (N-2) -electron spin function for

spin s ' . Since sn can be 0 or 1, s t can be S+1, S, or S-1.

The numbers W (s ' , sn ,S;M-mn,mn,M) are the Wigner- coeffi- TCl

cients . There are four equations like (31), corresponding to the

four spin-coupling ("branching"). routes shown in the follow-

i n g diagram:

s '=S+l 1, adding s =1

N-2 E N

s '=S = - S adding s n = O

The t e n Wigner c o e f f i c i e n t s involved a r e a v a i l a b l e i n s t and -

a r d r e f e r e n c e s . ( W i g n e r , 1959, p . 193; Condon and S h o r t l e y ,

1951, p. 7 6 ) .

The d i f f e r e n t s u b s c r i p t s TJCX o c c u r r i n g i n (31) c o r r e -

spond t o d i f f e r e n t r o u t e s on a Se rbe r branching diagram l i k e

tfiat i n F igu re 2. A s i n t h e prev ious c a s e , t h e v a l u e s of d(NS)

are shown a t each i n t e r s e c t i o n .

I t fo l lows from (31) t h a t Se rbe r s p i n f u n c t i o n s a r e a

b a s i s f o r a r e p r e s e n t a t i o n of SN t h a t i s adapted t o t h e se-

of n e s t e d symmetric groups. I t a l s o fo l lows from t h i s equa-

t i o n t h a t t h e r e p r e s e n t a t i o n of every geminal two-elect ron

subgroup i s f u l l y reduced. These f a c t s w i l l prove u s e f u l

Figure 2 . Serber branching diagram f o r s t a t e s leading t o

N=10, S=l

l a t e r on.

'"serber-type". f u n c t i o n s f o r odd N can be made by coupl ing

t h e s p i n of t h e Nth e l e c t r o n t o Se rbe r f u n c t i o n s f o r . N ' = N - 1 .

The r e s u l t i n g f u n c t i o n s w i l i t hen have Serber- type behavior

up t o e l e c t r o n N s . . .

comparison --- of YK and Se rbe r . f u n c t i o n s

The d i f f e r e n c e s between YK and Se rbe r s p i n f u n c t i o n s

are n o t made obvious by t h e branching diagrams, F i g u r e s 1 and

2 . The e a s i e s t way t o r e v e a l t h e d i f f e r e n c e s i s t o examine

t h e £unc t ions r e s u l t i n g from both g e n e a l o g i c a l schemes when,

say, N=4, ~ = l . ' , M=O . We .use t h e notation i n t r ~ d u c e d prev ious-

l y , and show wi th each f u n c t i o n i t s branching r o u t e .

The YK f u n c t i o n s t u r n o u t t o be

On t h e o t h e r hand, t h e S e r b e r f u n c t i o n s a r e

The Se rbe r func t ions , a r e symmetric o r an t i symmetr ic i n

each geminal p a i r : t hey are s imul taneous e igen func t ions of

2 ' A2 , 'gZ, S1 , a n d , s2. The YK f u n c t i o n s a r e less s imple . The

f i r s t one happens t o be t h e same a s t h e S e r b e r f u n c t i o n ell

because i t s .branching diagram unambiguously f i x e s t h e s p i n

of t h e f i r s t geminal p a i r t o be zero . S ince t h e t o t a l s p i n . i s one, t h e s p i n of t h e second p a i r must be s*=l. I n t h e

o t h e r two YK f u n c t i o n s , t h e s p i n of t h e f i rs t geminal p a i r i s

unamb~guously sl=l, b u t t h e second p a i r has no d e f i n i t e s p i n .

I n o t h e r words, t h e func t ion ' s e2 and e3 are s imul taneous

A2 A2 A2 e i g e n f u n c t i o n s o f S , iZ, and sl, b u t n o t of s2; This is be-

cause e i t h e r sZ=l o r s2=0 can couple w i t h s -1 t o g i v e S = l . 1-

Rather t han c o n t a i n i n g a pure c o n t r i b u t i o n from sZ=l o r s2=0,

t h e YK f u n c t i o n s e2 and B 3 c o n t a i n mix tures of both .

However, these ' f u n c t i o n s can be l a b e l l e d w i t h t h e sub-

s c r i p t "n" , j u s t . a s t h e Serber f u n c t i o n s were. One merely

def fnes t h e YK f u n c t i on Y ( N S M ) t o be one f o r which the ITa

branching r o u t e has t h e form A f o r t h e f i r s t n geminal

p a i r s , t h e n t u r n s ' upward f o r t h e nex t . I n t h e example.above,

The consequence of . t h i s n o t a t i o n is t h a t t h e YK f u n c t i o n

Yna and t h e Serber f u n c t i o n ens w i l l bo th be an t i symmetr ic I

i n t h e f i r s t n g e m i n a l ' p a i r s , symmetric i n the ' n e x t , t h e n

bear no f i x e d r e l a t i o n i n t h e rest.

A s was po in t ed o u t i n t he ' l a s t c h a p t e r , t h i s i s the only

behavior r e q u i r e d of s p i n f u n c t i o n s i n t h e SAAP formalism.

E i t h e r YK o r S e r b e r f u n c t i o n s can be used , t h e cho ice depend-

i n g on convenience i n g e n e r a t i n g t h e f u n c t i o n s .

P r a c t i c a l i t y - of spin-coupl ing t echn iques

The g e n e a l o g i c a l c o n s t r u c t i o n of s p i n f u n c t i o n s i s i n -

convenien t because it. is r e c u r s i v e . I n o r d e r t o make an N-

e l e c t r o n s p i n f u n c t i o n , one must f i r s t g e n e r a t e every pred- . e c e s s o r i n t h e g e n e a l o g i c a l scheme. I t can be seen from t h e

branching diagrams t h a t t h e complexity of t h e problem i n -

c r e a s e s r a p i d l y w i t h N .

I n ,o rder t o make t h e t h r e e YK f u n c t i o n s f o r N = 4 , S = l ,

M = O , one must g e n e r a t e t h e ' f o l l o w i n g f i f t e e n func t ions :

N S f u n c t i o n s M v a l u e s r e q u i r e d t o t a l f u n c t i o n s f o r each M

1 1 / 2 1. ' + 1 / 2 , -112' 2

1 2 . 0 ' 0 1

The c a l c u l a t i o n s a r e s o si.mple t h a t t h i s i s no problem. But

t h e r e a r e 90 s p i n f u n c t i o n s f o r N=10, S=' l , M=O. I n o r d e r t o

g e t them, one must g e n e r a t e 660 f u n c t i o n s . a l t o g e t h e r , some

con ta in ing as many as 252 produc t f u n c t i o n s .

A t leas t one computer program i s a v a i l a b l e f o r YK func-

t i o n s (Mat the i ss , 1958) , b u t t h e g e n e a l o g i c a l c o n s t r u c t i o n

of s p i n f u n c t i o n s i s p r a c t i c a l only f o r s m a l l N . I n o t h e r

cases, t h e programs r e q u i r e t o o much s t o r a g e .

ow din ' s P r o j e c t i o n Opera tors

By i n v e r t i n g ( 2 7 ) , one can e x p r e s s any e lementary s p i n

produc t f u n c t i o n Bk(NM) i n terms of a l l t h e s p i n e igenfunc-

t i o n s € 3 . (NSM) having t h e s a m e N and M: 3.

ek (NM) = 1 1 A . (NSM) c (NSM) . S j

I t is appa ren t t h a t t h e q u a n t i t y

1 € 3 . (NSM) c (NSM) j 3 jk

i s t h e p r o j e c t i o n of t h e s p i n produc t Bk(NM) on t h e subspace

f o r sp in-e igenva lue S t a subspace spanned by t h e v e c t o r s ,

( € 3 . (NSM) l a11 j ) . Th i s q u a n t i t y is a l s o c a l l e d t h e S-component 3 -

of B k ( N M ) . Equat ion (32) s ays t h a t , i n g e n e r a l , an e lementary

s p i n produc t func t ion .may c o n t a i n components f o r every v a l u e

ow din (1955b, 1960, 1964) has in t roduced. the o p e r a t o r

which, when o p e r a t i n g on B k ( N M ) , s u c c e s s i v e l y a n n i h i l a t e s n

every spin-component excep t t h e one shown i n (33) . Thus OS

p r o j e c t s an. e igen func t ion of c2 from any s p i n p r o d u c t func-

t i o n .

The a p p l i c a t i o n of (34) .is s t r a i g h t f o r w a r d , s i n c e t h e

D i rac i d e n t i t y g i v e s (McWeeny and S u t c l i f f e , 1969)

where I is t h e i d e n t i t y permuta t ion , ( y , ~ ) i s t h e t ranspo-

s i t i o n in t e r chang ing e l e c t r o n s y and v , and

r (NMk) = {!} i f t h e s p i n s of p alJ1d v vv

I f Bs i s a p p l i e d t o a l l t h e s p i n produc ts f o r g lven N

and M , t h e r e s u l t s w i l l be redundant , . bu t enough l i n e a r l y

independent s p i n . . . e i g e n f u n c t i o n s w i l l be gene ra t ed t o span t h e

spin-space f o r N and S. Lowdin (1964) has developed a pro-

cedure f o r choosing s p i n produc ts t h a t l e a d t o independent

e i g e n f u n c t i o n s . A computer program i s a v a i l a b l e (Ro'tenberg,

The r e s u l t i n g f u n c t i o n s can be o r thogona l i zed wi thou t

d i f f i c u l t y . I n o r d e r t o o b t a i n t h e s o r t of s p i n f u n c t i o n s

which a r e u s e f u l i n t h e SAAP formalism, however, one must

t r ans fo rm t h e Lowdin s p i n f u n c t i o n s by d i a g o n a l i z i n g t h e

r e p r e s e n t a t i o n m a t r i c e s f o r geminal t r a n ~ ~ o s i t l o n s . While

t h i s could be done wi th high-speed computers, it would n o t

be a s p r a c t i c a l a s o t h e r methods t o be d i scussed .

Wigner Operators

There 'are ' s e v e r a l g roup- theore t i ca l approaches t o ' s p i n

f u n c t i o n s . The Wigner s h i f t o p e r a t o r s (Wigner, 1931, 1959)

wi th B f i x e d , w i l l gene ra t e from a s p i n product d(NS) s p i n

e igenfunc t ions spanning t h e spin-Space f o r N and S. D i f f e r e n t . .

va lues of B produce d i f f e r e n t bases f o r t h e same rep resen ta -

t ion. . S e t t i n g a=B produces a Wigner p r o j e c t i o n ~ p ~ r a t o r ,

which can be shown t o be idempotent.

I n o rde r t o make s p i n f u n c t i o n s w i t h t h e s e o p e r a t o r s , . . , .

it is necessary t o know t h e N! s p i n r e p r e s e n t a t i o n ma t r i ces . . . .

[plNS, f o r every P i n SN. These can a l l be genera ted from

t h e ( N - 1 ) ma t r i ces r e p r e s e n t j n g t h e elementary . t r a n s p o s i t i o n s

(k-1 ,k ) , where k runs from. 2 t o N .

A sp in-cobpl ing p rocedure f o r e v a l u a t i n g t h e s e ma t r i ces

was given by Yamanouchi (1936, 1937) and d.is.cussed by ~ o t a n i

e t a l . (1955) . The method was extended t o t h e Se rbe r s p i n -- r e p r e s e n t a t i o n by ~ a t t h e l s s (1959) , fo l lowing a scheme sug-

ges t ed by Corson (1951). These procedures a r e r e c u r s i v e , and

s u f f e r from t h e d isadvantages mentioned e a r l i e r .

I t s o happens (Pauncz . . , 1967 ). t h a t t h e ' YK s p i n represen-

t a t i o n i s t h e same a s Young's orth,ogonal r e p r e s e n t a t i o n . .

(Young, ,1932 ; T h r a l l ; 1941) , ob ta ined by nonphysical argu-

ments. Young's a n a l y s i s l e a d s t o u s e f u l r u l e s f o r e v a l u a t i n g

t h e r e p r e s e n t a t i o n matrices f o r t r a n s p o s i t i o n s (Ruther ford ,

1948; ~ o d d a r d , 1967a; Coleman, 1968) . his method is q u i t e

p r a c t i c a l .

I t i s p o s s i b l e t o g e t a long wi thou t t h e r e p r e s e n t a t i o n

ma t r i ce s . S e t t i n g B=a i n . ( 3 ' 5 ) and-summing over a , one o b t a i n s

t h e new o p e r a t o r

where xNS(P) i s t h e c h a r a c t e r of t h e permuta t ion P i n t h e . .

r e p r e s e n t a t i o n g iven by t h e m a t r i c e s [plNS. T h i s o p e r a t o r ,

when a p p l i e d t o a s p i n produc t , does n o t i n g e n e r a l produce

one of t h e s p i n e i g e n f u n c t i o n s B,(NSM), b u t some f u n c t i o n

i n t h e (N,S)-space spanned. by them. Thus (36) i s t h e group-

t h e o r e t i c a l e q u i v a l e n t of Lowdin's o p e r a t o r . . . .

The f l y i n t h e ointment i s t h a t , f o r t e n e l e c t r o n s ,

t h e r e a r e 101 = 3,628,800 t e r m s i n t h e sum of (35) o r ( 3 6 ) .

It would be extremely time-consuming t o g e n e r a t e t h i s many

r e p r e s e n t a t i o n m a t r i c e s from t h e n ine e lementary m a t r i c e s .

Even t o g e t t h e c h a r a c t e r s r e q u i r e d by (36) would be i n e f f i -

c i e n t compared t o Lowdin's method. The o p e r a t o r s (35) and

(36) have been used t o make s p i n f u n c t i o n s f o r ' s m a l l N

(Smith and Harris, 1967; Harris , 1967) , b u t t h e y are n o t

p r a c t i c a l f o r many s y s t e m s of i n t e r e s t .

Young's theory of t h e symmetric group l e a d s t o a more

v i a b l e approach t o p r o j e c t i o n o p e r a t o r s through group thed ry . . .

Only some permuta t ions a r e r e q u i r e d i n p r o j e c t o r s made i n t h i s

way, and t h e c a l c u l a t i o n s do n o t become s o unwieldy. A d i s -

cus s ion and f u r t h e r ex t ens ion of t h i s method is presen ted i n

t h e nex t chap te r .

Se rbe r . Spin Func t ions

fi2 by. D iagona l i za t ion of S

The f i r s t . new method sugges ted . h e r e f o r t h e cons t ruc-

t i o n of S e r b e r s p i n f u n c t i o n s i s l a r g e l y numerical i n charac- . ,

. . . ter .

W e seek t o c o n s t r u c t s p i n e i g e n f u n c t i o n s e ra ' (NSM) h a v i n g

t h e fo l lowing p r o p e r t i e s : . .

ge'A0 (NSM) = '(NSM) f o r every geminal

t r a n s p o s i t i o n g i n ' SN; .

gem (NSM) = -era (NSM) f o r every geminal

t r a n s p o s i t i o n g i n Br. ( 4 0 )

P r o p e r t i e s (39 ) and ( 4 0 ) call be rewordcd~ Bra (NSM) i s to be

an e igen func t ion of every geminal s p i n o p e r a t o r g2 ( 2 ~ - 1 , 2 p ) , and i n p a r t i c u l a r , i t s e igenva lue under such an o p e r a t o r i s

t o be z e r o when U ~ T .

It is n a t u r a l t o t h i n k o f ' s u c h f u n c t i o n s as l i n e a r com-

b i n a t i o n s of p roduc t s , ' no t of one-e lec t ron s p i n f u n c t i o n s a

and B , b u t of t h e geminal s p i n f u n c t i o n s i n t roduced i n (29 ) :

For t h e moment w e c o n s i d e r on ly t h e case when N=2n i s

even. The produc t

where.M=Cm , w e s h a l l , c a l l a geminal s p i n produc t . .Obviously, lJ

each geminal s p i n - p r o d u c t WM i s an e igen func t ion of t h e gem-

h2 i n a l s p i n o p e r a t o r s S 2 - 1 2 and 5^z(2p-1,2p)., f o r every U.

The s p i n e i g e n f u n c t i o n BTa(NSM) , which i s some l i n e a r

combination

' NSM (NSM) = c ({s,,},{mp}) x Tra {s lJ} {rn 1 lJ

where M=Cm i s f i x e d , i s i t s e l f an e igenfunct . ion o f t he ' CI ^ a

o p e r a t o r s S ( 2 ~ - 1 , 2 ~ ) . Thus B,,(NSM) c o n t a i n s on ly t hose gem-

i n a l s p i n produc ts having t h e same geminal pa i r - sp ins :

each l i n e a r combination ( 4 3 ) has { s l , s2 , ..., s f i x e d . - -- n .

W e s a y . t h a t each l i n e a r combination has a c e r t a i n - "pa i r - sp in

combination", o r "PSC". Each geminal p a i r s p i n i s c a l l e d a

"PS". Furthermore, t h e s u b s c r i p t "n" on BITa(NSM) means t h a t

every W i n (43.) has s = O f o r ~ = 1 , 2 , . . . ,IT. This fo l lows from M P

( 4 0 ) .

Now, g iven t h a t t h e l i n e a r combinations ( 4 3 ) , f o r f i x e d

N , S t M , and IT, a r e , s u b j e c t t o t h e t h r e e c o n d i t i o n s

(i) t h e PSC is f i x e d ;

(ii) s = O f o r , IT; P

(iii) Cm =PI; P

only one more c o n d i t i 0 n . i ~ r e q u i r e d t o produce t h e 8 (NSM) : ITa . .

th'e l i n e a r combinations must d i a g o n a l i z e t h e g2-matrix. T h i s , - -- -- of cou r se , f o r c e s t h e l i n e a r combinations t o be e igenfunc-

^2 t i o n s of S . Serbe r s p i n f u n c t i o n s . can be made, t h e n , by t h e ve ry

s imple a lgo r i t hm shown, in F igu re - 3. The a lgo r i t hm i s s o s i m -

p l e t h a t on ly one p a r t of it r e q u i r e s f u r t h e r exp lana t ion - ^2 t h e c a l c u l a t i o n of t h e S -matr ix over geminal s p i n produc ts .

The N-electron o p e r a t o r s S, , g Z , and i2 'are r e l a t e d by

Determine all PSCqs having the properties

(i) s =O for p=1,2,. .. ,IT; IJ

For each of these, do the follow.ing:

construct every possible gem-

inal spin product having Cm =M; 1.I

calculate the i2-matrix be-

tween these geminal spin prod-

diagonalize this g2-matrix;

keep only those.eigenvectors

having .the desired eigenvalue

Figure 3. Algorithm for construction of Serber spin . ,

functions with eigenvalues S, M i for .use with

a space function having IT doubles

A n Writing S, = 1 2, - ( b ) in terms of the geminal. pairs,

The calculation of the g2-matrix is trivial when the operator

is written in this form. The action of the second term on the

geminal spin functions (41) is given by

where n(r), for example, is the number of times that w(1,0)=~

occurs in WM.

Since the geminal spin products (42) are orthogonal,

the s2-matrix elements between such products (with M fixed)

are given by

The contribution in {}-brackets is zero unless WI; = WM. On

the other hand, the last term is zero unless WM and W' differ M in two geminal pairs, say those numbered K and A. In that

case, the last term is [note that all geminal spins s are u the same in any two products Wi and WM in ( 4 3 ) 1

This integral is zero unless m;+mi = m +mA. In fact, of 256 K

elements in the matrix of such integrals, only eight .are non-

zero: see Figure 4,

The algorithm of Figure 3 has been . programmed . in Fortran, . . ,

for the IBM System 360/65, and the listing is given in Appen-

dix C. The speed of this program is.1imited.b~ the matrix

diagonalization procedure. The one listed,;EIGEN, is an.IBM

Jacobi scheme improved by R. C, Raffenetti, D. M. Silver,

and B. F. Sullivan, of the Theoretical Chemistry Group at

Iowa State University, Ames, Iowa. The time required by E.IGEN

to produce double-precision eigenvectors of a matrix goes

up roughly as the cube of the dimension: in this case, as the

cube of the number of geminal spin products for'a given PSC.

E,IGEN will handle a 10x10 case in less than 0.5 second, and

a 2 5 x 2 5 in less than seven seconds.

As a practical matter, one is interested in using the

spin functions to calculate the representation matrices of

the permutations in the symmetric group. Such matrices are

Figure 4. Matrix of elements.

(All elements not given explicitly are zero .)

needed for the evaluation of expectation values in terms of

wave. functions containing the spin functions.

The N! permutations belonging to SN are products of the'

(N-1) elementary transpositions tk= (k-1,k) ; where k runs from

2 to N . In .practical applications, one therefore generates

only the tk-matrices from thespin functions.

A program has been written which generates all Serber

spin functions for given N, S f and M, and then evaluates all

of the tk-matrices from them. Sample running times in single

precision are:

CPU 'time N G M . cpin functions el.em. matrices (see)

6 1 0 9 5 5.1.7

These running times reflect the fact that the complexity of

spin functions depends on [ M I . An application of these techniques is the program to

generate simultaneous eigenfunctions of spin ,and orbital.

angular momentum, listed in Appendix E. Subprograms SSQEIG

and S E I G E N generate Serber spin functions, and FPMAT is used

to evaluate' permutation matrices. The. operation,of this

program i s expla ined i n t h e l a s t chap te r .

The preceding d i s c u s s i o n l e a d s t o t h e fo l lowing conclu-

s i o n s . The most convenient computer - technique f o r o b t a i n i n g

Serber s p i n r e p r e s e n t a t i o n ma t r i ce s i s t o gene ra t e s p i n f m c -

t i i n s f i r s t , and o b t a i n the ' ma t r i ce s from them. This r e q u i r e s

many a r i t h m e t i c a l o p e r a t i o n s , b u t most i nvo lve on ly i n t e g e r

a r i t h m e t i c , and a r e qu ick ly done. Attempts t o o b t a i n ma t r i ce s

d i r e c t l y from, genea log ica l schemes u s u a l l y r e q u i r e a very

l a r g e amount of s t o r a g e whenmore than ' a few e l e c t r o n s are

involved. The except ion is t h a t YK ma t r i ce s may be ob ta ined

conven ien t ly from Young tab leaux . This approach i s u s e f u l

when s p i n f u n c t i o n s a r e ' n o t r e q u i r e d .

CONSTRUCTION OF SPIN EIGENFUNCTIONS

BY GROUP-ALGEBRAIC TECHNIQUES

We have d e s c r i b e d how t h e g r o u p - t h e o r e t i c a l Wigner op-

e r a t o r s can be used t o g e n e r a t e s p i n f u n c t i o n s . It seems

r easonab le t o expec t t h a t group theo ry might l e a d t o s imp le r

exp res s ions f o r such o p e r a t o r s , ones which do. n o t i nvo lve

sums over every group e lement . We p r e s e n t i n t h i s c h a p t e r

a new method t o ' a c c o m p l i s h t h i s , a method by which YK and

Se rbe r s p i n f u n c t i o n s can b e gene ra t ed d i r e c t l y from Young

t ab l eaux wi thou t t h e need t o e v a l u a t e r e p r e s e n t a t i o n ma t r i ce s .

A s a bonus, t h i s approach a l s o g i v e s d i r e c t l y the d u a l space

f u n c t i o n s .

The o p e r a t o r s w e s h a l l d e s c r i b e form m a t r i c bases i n t h e

symmetric group a l g e b r a . The theo ry behind them i s a b s t r a c t

and r e l a t i v e l y u n f a m i l i a r t o chemis t s . For t h i s r ea son , w e

sha l l beg in by o u t l i n i n g t h e a p p l i c a t i o n of group a l g e b r a

theory t o t h e symmetric group. The r e a d e r ~ e c k i n g s morc

complete t r e a t m e n t i s r e f e r r e d e lsewhere (Weyl, 19.31; van

d e r Waerden, 1950; Johnson, 1960; Boerner, 19'63; owd din,

1967; Poshus ta , 1969; Matsen, 1970; Salmon, 1971) .

Desp i t e t h e a b s t r a c t n e s s of t h e t h e o r y , . t h e o p e r a t o r s +

. obta ined t u r n o u t t o be "concep tua l i zab le" and easy t o apply.

The Group Algebra for SN;

the Regular Representation

While the method of the last section dealt with linear

combinations of'spin product functions, we now construct

linear combinations of permutations which, when operating

on a single product function, produce basis functions for

irreducible representations of SN.

Two such operators are.familiar. The antisymmetrizer,

has already been introduced, and there is', similarly, a

These operators are idempotent and are projection operators

for the antisymmetric and symmetric representations, respec-

tively.

For N > 2 , however, there are other irreducible represen-

tations. This chapter is concerned with the construction of

projectors for - all of the 'irreducible representations. '1n

group-theoretical language, we --- seek a way - to completely - re-

duce the regular representation of &. Let us start with the -- - functional approach of thc last chapter , and show how it

leads to a powerful abstract method for this reduction.

Consider an N-electron function f , such that 'the N 1

functions

are'all distinct. It is convenient to label these functions'

with the permutations that generate them from f(lt2,b..,N):

let fI (1.2,. . . ,N) = f (1,2,. . . ,N) and, if

let fp(l,2, .. . ,N) = f (p1,p2,. . . .pN) = Pf (lt2,. -,N), etc.

The set{fp} is a basis for an important representation

of SN: for every P and f there is in the set an fR such Q' , that

P-f = fR, where R = PQ. Q

1n other words,

P-f = 1 rSQ (P) fS with TSP(P) = 6SR' . s

. . It is easy to show that the matrices r(P) multiply like

. . the permutations. They constitute the regular.representation

of SNt which is shown in elementary texts to be reducible

and to contain every distinct (i.e., nonequivalent) irrep

of,SN. It should be noted that the permutations play a dual

role in the regular representation: they 'are,both the trans-

formations and the labels for the basis functions.

he N 1-dimensional linear space .I? (SN) spanned by the fp

i s s a i d t o be t h e c a r r i e r space f o r t h e r e g u l a r r e p r e s e n t a -

t i o n . I t c o n s i s t s of every l i n e a r combination

of t h e b a s i s f u n c t i o n s f p .

S ince t h e r e g u l a r r e p r e s e n t a t i o n i s r e d u c i b l e , i t s ca r -

rier space F(SN) i s decomposable' i n t o t h e d i r e c t sum of sub- ,

spaces i n v a r i a n t and i r r e d u c i b l e w i t h r e s p e c t t o t h e opera-

t i o n s of t h e group. S ince t h e r e g u l a r r e p r e s e n t a t i o n c o n t a i n s

every nonequiva len t i r r e p , FA) c o n t a i n s - a c a r r i e r space f o r

every d i s t i n c t . i - r rep .

The meaning of t h e s e terms can be c l a r i f i e d through an

example. Suppose t h a t f ( 1 ,2 ,3 ) = a(1) b (2),c (3 ) = abc. Then

F(S3) c o n s i s t s of every l i n e a r combination of t h e form

X(1 ,2 ,3) = x abc + x2bac + x cba + x4acb + x cab + x6bca. 1 3 5

I t t u r n s o u t that t h i s l i n e a r space can be decomposed as t h e

d i r e c t sum of t h e fo l lowing f o u r i r r e d u c i b l e subspaces:

subspace 1, spanned by ell= abc+bac+cba+acb+cab+bca:

2abc+2bac-cba-bca-cab-acb, subspace 2 , sp.anned by

22- acb+bca-cab-cba;

- acb-bca+cab-cba ,. subspace 3 , spanned by {:311 3 2- 2abc-2bac+cb&-bca-cab-cb;

subspace 4 , spanned by 6341= abc-bac-cba-acb+cab+bca.'

By "direct. sum" is meant the following:

(i) Every function in F(S3) can be written as a

sum of functions in the subspaces.

(ii) The subspaces share no functions other than

the null'- they are independent. Here, in fact,

their basis functions are all orthogonal.

The subspaces are said to be "invariant" under SN because

the r.esult of operating with a permutation on a vector from

one of the subspaces is again a vector in that s'ubspace. For

example ,

The invariant subspaces.are "minimal" or "irreducible" be-

cause they cannot be decomposed into s'maller invariant sub-

spaces. Here, in fact, two of the subspaces are one-dimension-

a1 . The carrier space F(S ) for the regular representation

of SN is decomposed by finding projection operators for the

various minimal invariant subspaces. To this end, we recast

the linear function space P (SN) in terms oT operators:

an element

is written in the form

From t h i s p o i n t of view, each e lement X(1 ,2 , . .. , N ) i n . F ( S N )

corresponds t o an o p e r a t o r l i k e [Cx(P)P] . The " p r i m i t i v e P

func t ion" f ( 1 , 2 , . . . , N ) i s t h e same i n every c a s e , and is t h u s

supe r f luous . The p r o p e r t i e s of F(SN) can be d i s c u s s e d wi thou t

mentioning t h e p r i m i t i v e f u n c t i o n .

For t h i s r ea son , t h e space F(SN) of f u n c t i o n s can be

r ep l aced by t h e e q u i v a l e n t , l i n e a r space A ( S ~ ) . c o n s i s t i n g

I of a l l o p e r a t o r s of t h e form

The space A(SN) is c a l l e d t h e group a l g e b r a of SN. I t i s t o

be cons idered n o t on ly a s a set of o p e r a t o r s , b u t a l s o as a

l i n e a r v e c t o r space spanned by t h e group e lements .

Like F (SN) , A (SN) i s a c a r r i e r space f o r t h e r e g u l a r

r e p r e s e n t a t i o n of SN. F ind ing o p e r a t o r bases f o r minimal

i n v a r i a n t subspaces of A ( S N ) corresponds t o f i n d i n g b a s i s

f u n c t i o n s f o r minimal i n v a r i a n t subspaces of F ( S N ) , and i n

t h i s s ense is e q u i v a l e n t t o f i n d i n g b a s i s func t ion ' s f o r

i r r e p s of SN.

We s h a l l s e e , f o r example, t h a t a b a s i s f o r a c e r t a i n

minimal i n v a r i a n t subspace of A(S4) c o n s i s t s of t h e o p e r a t o r s

where Si and A are the symmetrizer and qntisym- ,.*.,k it.. . ,k metrizer on the numbers i, . . . ,k, respectively, and

These operators, applied to, the spin primitive

aBaB = a (1) 8 (2) a (3) b (4) , generate the basis functions

Comparison with (9)' shows that these are Serber spin func-

tions for N=4,- S=l, M=O. Either the operators of (45) or the

functions of (46) can be thought of as .a basis for the corre-

sponding irrep of SN.

Just as the group algebra is an abstraction from the

function space F(SN), the regular representation has a more

abstract meaning in terms of A (SN) . Equation (44) defines a matrix representative for each group element when the basis

in A(SN) is chosen to be those same group elements. Again,

the permutations in SN play the dual role of transformations

a, k,' +

Minimal Left Ideals,

Primitive Idempotents , and Matric Bases

Carrier spaces of representations into which the regu-

lararepresentation reduces are subspaces of A(s~) that are

invariant under left-multiplications by group algebra ele-

ments. Given an element U in A(SN), it is easy to see that

the set of elements

L = {XU I XEA (sN) }

is such a subspace. The set L is said to be the - left ideal

generated by U, and U is called its generator. Every sub-

space of A(SN) that is invariant under all left-multiplica-

tions .is a left ideal. Left ideals are thus c,arri,er spaces

for the representations into which the regular representa-

tion reduces.

Corresponding to the reduction of the regular represen-

tation, its carrier space A(SN) decomposes as the direct sum

.of certain left ideals: we write

It may be that a left ideal Li contains left 'ideals of

smaller dimension, in which case Li itself decomposes. By

carrying this process as far as it will go, A(s~) can be

written.as the d&rect sum of certain minimal - left ideals,

each of which is nondecomposable, 'or i r r e d u c i b l e . The - niinimal

l e f t i d e a l s i n t o which A(S,) decomposes a r e c a r r i e r spaces -.

, ( .- .. f o r t h e i r r e d u c i b l e r e p r e s e n t a t i o n s conta ined i n t h e r e q u l a r -- -- \', I .

r e p r e s e n t a t i o n . A s i s well-known, t h e i r r e p "a" occurs da

t i m e s i n t h e r e g u l a r r e p r e s e n t a t i o n i f i t h a s dimension da.

a S i m i l a r l y , da e q u i v a l e n t minimal l e f t i d e a l s ' {L: 1 i=1 ,2 , . . . ,d 1

f o r i r r e p cr occur i n t h e decomposit ion of A(SN). W e w r i t e

ln which t h e sums a r e d i r e c t .

W e wish t o o b t a i n o p e r a t o r bases f o r t h e s e minimal l e f t

i d e a l s , f o r such o p e r a t o r s can be used t o g e n e r a t e b a s i s func-

t i o n s f o r t h e i r r e p s .

I t can be shown t h a t every l e f t i d e a l c o n t a i n s a t l e a s t

one idempotent g e n e r a t o r , e , c a l l e d a g e n e r a t i n g u n i t ; - A gen-

e r a t i n g u n i t f o r a minimal l e f t i d e a l i s c a l l e d a p r i m i t i v e

idempotent. I t t u r n s o u t t h a t an element e is a p r i m i t i v e

idempotent i f and only i f

whcre X is 3 element of A(SN) and h ( X ) is a number t h a t de-

pends on X. Obviously, i f e i s t o be idempotent, it must be

t h a t X ( I ) = l . P rope r ty (47) i s used t o i d e n t i f y g e n e r a t i n g

u n i t s f o r i r r e d u c i b l e c a r r i e r spaces .

Idempotents t h a t g e n e r a t e d i f f e r e n t minimal l e f t i d e a l s

occu r r ing i n t h e decomposition of A(SN) a n n i h i l a t e each o t h e r .

If L: and LB are generated by idempotents eq and eB respec- 9 j '. tively, it can be shown that

These idempotents are the diagonalv elements {eq=eYi} of

a set. a {ea la11 a; i,j=1,2, ..., d 1 ,

. . of operators in A(SN) having the multiplicative property

This property guarantees that the 1 (da) = N1 elements' {ea 1 a i j

are linearly independent. For if

then from (49),

for any. 8, k t and n . Like the permutations, then, - the eyj - - form a basis - for

the whole proup algebra, and there is.a transformation between - the two basis sets:

Because of this and the fact that the eqj multiply like the

"elementary matr icesw 2 , = ( 6 . . ) , they a r e given t h e name 1 3

mat r i c b a s i s .

I t i s important t o note t h a t eSj = e a e ~ : - - i j j j eSje:. This

means t h a t t h e subse t

of t h e ma t r i c b a s i s belongs t o t h e minimal l e f t i d e a l gener- '

a t e d by eq. Since t h e matr ic b a s i s elements a r e l i n e a r l y in -

dependent, B? c o n s t i t u t e s a b a s i s f o r t h e 2 minimal l e f t 3 - -- -

i d e a l - f o r i r r e p a - occurr inq the decomposition -- of t h e 9 3

algebra . From

it follows t h a t t h e c o e f f i c i e n t s i n ( 5 0 ) a r e elements

' of an i r r e d u c i b l e r ep resen ta t ion matr ix f o r P . I t can be seen

, from (51) t h a t t h e s e t s B' and B: , where k#j. span two ca r - j

rier spaces f o r t h e same i r r e p .

Mult iplying ( 5 0 ) by [P-'1 ik , suhning over P I and apply-

ing t h e Orthogonal i ty Theorem f o r i r r e p matr ices , one o b t a i n s

the express ions

These r e l a t i o n s a r e o f t e n used t o f i n d t h e ma t r i c b a s i s ele-

ments.

Now it is poss ib le t o s e e what proper ty of t h e ma t r i c

b a s i s corresponds t o or thogonal i ty i n t h e i r r e p matr ices . W e

d e f i n e f o r each element X = 1 x(P) P i n A ( S * ) P

xi = .I x* (P) P-I = 1 x* ( P - ~ ) P , . .

where x* ( P ) i s t h e complex con jugate of t h e number x(P) . This

de f . in i t ion i s reasonable i n view of i t s a p p l i c a t i o n t o i n t e -

g r a l s over funct ions : i f @ and $ a r e well-behaved func t ions ,

, The a d j o i n t of ey j i s , t h e r e f o r e , !

Comparing t h i s with

w e ' see t .hat t h e property

impl ies t h a t

Thus a mat r ic b a s i s with property (53) spans c a r r i e r spaces

f o r u n i t a r y i r r e p s . I f t h e c o e f f i c i e n t s ' i n t h e matric b a s i s

elements a r e r e a l , then t h e i s r e p matr ices a r e or thogonal ,

I n order t o genera te b a s i s func t ions f o r orthogonal irre-

duc ib le r e p r e s e n t a t i o n s of t h e symmetric groupo t h e r e f o r e , w e

r e q u i r e a m a t r i c b a s i s of o p e r a t o r s wi th t h e m u l t i p l i c a t i v e

proper ty ( 4 9 ) . and t h e a d j o i n t p rope r ty ( 5 3 ) . This m a t r i c b a s i s

i s t o be a s s o c i a t e d wi th t h e i r r e p s of sN by buildin.; it

around p r i m i t i v e idempotents f o r t h e minimal l e f t i de -a l s .occur -

r i n g i n t h e decomposition of t h e group a lgebra . . . .

The p r i m i t i v e idempotents epi=e4 a r e t o be cons t ruc ted t o

have t h e p r o p e r t i e s

I t w i l l fo l low t h a t (see page 137)

Thus t h e idempotent d iagonal elements of t h e m a t r i c b a s i s w i l l

be p r o j e c t i o n ope ra to r s f o r i r r e d u c i b l e c a r r i e r spaces .

Young Idernpotents, Young Operators

Minimal l e f t i d e a l s of A(SN) can be generated us ing a

method developed by Al f red Young (1901, 1902, 1928, 1930 ,

,1932). An account of t h i s method, w i th a complete b i b l i o g - . ,

raphy, has been given by Rutherford (1948) . Weyl (1931) and

Boerner (1963) have descr ibed t h e connect ion between YoungDs

work and group a lgebra theory .

Since t h e r e a r e a s many c l a s s e s of SN a s t h e r e a r e par-

t i t i o n s of N , t h e p a r t i t i o n s of N provide a way of l a b e l l i n g

t h e i r r e p s of SN: f o r N = 4 , t h e l a b e l s a r e .

p a r t i t i o n p i c t o r i a l l a b e l

~ h e s e p i c t o r i a l l a b e l s f o r i r r e p s a r e c a l l e d Young diaqrams

o r p a t t e r n s . I f t h e row leng ths of a Young diagram a r e

P l 1 P 2 t o m - t 'rr (where p l ) p p a p r ) t h e diagram i s named

Tha diagrams a r e used t o make Young tableaux. -- --- A t ab leau

i s a p a r t i c u l a r way of a r ranging t h e numbers 1 , 2 , . . .,N i n t h e \

boxes of t h e diagram. For example, t h e diagram ff f o r N=3

gives r i s e t o t h e fol lowing tableaux:

(We s h a l l o f t e n omit t h e boxes f o r convenience'.)

Each tableau is used to build operators. Given a tableau . .

T , let ' (R = {rj be the set of all permutations which inter-

change only numbers on the same row. This set is.a group - the row group. We.simil.arly define a column group, C . = {c).

these are or" the tableau 3 4

Note that 6L is the direct product of the groups for individ-

ual rows; and that .c is the direct' product of individual column groups,

The row operator is defined to be'a symmetrizer on the - row group :

This i s the product of symmetrizers for the individual rows.

The column operator is defined- to be an antisymmetrizer

on the column group:

where E(C) is +1 when c is even and -1 if c is odd. This is

the product of ipdividual column antisymmetrizers.

The tableau operator is the column operator. followed

by t h e row opera to r : . . II

.. . (Some authors d e f i n e E (T) =CR.) This opera tor i s given t h e . .

s p e c i a l symbol E because -- it i s e s s e n t i a l l y idempotent (idem-

po ten t wi th in 'a ' numerical f a c t o r ) ' and qenera tes , a minimal - l e f t i d e a l . I t is c a l l e d t h e Younq idempotent f o r t ab leau -

' T , and it s a t i s f i e s (47) . young tab leaux and idempotents have t h e fol iowing i m -

p o r t a n t proper.ty: i f T and T I a r e tab leaux belonging t o t h e

same diagram, then E (T) and E ( T I ) genera te minimal i n v a r i a n t

subspaces f o r .equi.valent r ep resen ta t ions ; i f T and T ' belong

t o d i f f e r e n t diagrams, E (TI and E ( T I ) genera te minimal l e f t

i d e a l s f o r nonequivalent r ep resen ta t ions . Since each diagram

l a b e l s a d i s t i n c t i r r e d u c i b l e representa t ion ' , : t h e Young

idempotents can be used t o genera te i r r e d u c i b l e subspaces . .

f o r every d i s t i n c t i r r e p . . . .

One f u r t h e r d e f i n i t i o n is requi red i n o rde r t o c l a r i f y . .

the correspondence between diagrams and i r r e p s . A s tandard

tab leau i s def ined t o be a t ab leau i n which the numbers along . .

each row inc rease t o t h e r i g h t and numbers on' each column

inc rease downward. The'diagrams, s tandard tab leaux, and Young . . . .

idempotents f o r N = 4 a r e shown i n Figure 5 . . ,

I t can be shown , t h a t the.number of standa.rd tab leaux . . . .

. . . . . f o r t h e diagram D = { ~ } = { P ~ , P ~ , . .. ,pr} . i s . '.

. , . . .

Diagram, D D Standard tableaux. Ti

r . I.:

Number of standard

tableaux. dD

D Young idempote.nts , Ei

Figure 5 . Example of N=4

t h a t these numbers s a t i s f y t h e equat ion

and hence t h a t dD i s t h e dimension of t h e i r r e p bf SN c o r r e -

sponding t o t h e diagram D .

The situation i s a s fol lows. Each Young diagram D l a b e l s

a d i s t i n c t i r r e d u c i b i e r ep resen ta t ion rD , t h e aimension of

which i s given by d D , t h e number of s tandard tableaux. This

number i s a l s o t h e number of equ iva len t c a r r i e r spaces f o r

D r occurr ing i n t h e decomposition of t h e group algebra. Thus

t h e r e i s a one-to-one . r e l a t i o n between the s tandard tab leaux

D D { T ~ I 1 , 2 , . . . d f o r diagram D and t h e equ iva len t c a r r i e r

spaces f o r rD occurr ing i n t h e decomposition of A(SN) . Since . .

t h e Young idampotent f o r each s tandard t ab leau genera tes

an i r r e d u c i b l e subspace of A ( S N ) , t h e r e i s a one-to-one

r e l a t i o n between these minimal l e f t i d e a l s and t h e i r r e d u c i -

b l e c a r r i e r spaces .occurr ing i n t h e decompo&ition of t h e

group a lgebra . J u s t wh,at t h i s r e l a t i o n i s w i l l become c l e a r e r . .

a s we proceed.

Suppose t h a t the . s tandard tab leaux f o r diagram D a r e

D r e l a t e d by permutations p i j :

representation. For a,diagram containing two rows at most,

(56) becomes . . . N! (p1-p2+1)

This gives the dimension of the spin representation corre-

sponding to diagram D. Using the example of N=4, we have the

spin representations

ml (dimension 11,

RT] (dimension 3) ,"'

; a :. (dimension 2 ) .

Comparison with the branching. diagrams, Figures. 1 and ' 2, . .

. . reveals the f oilowing correspondence :

B * .S=O.

Indeed, the- general re1,ation between the diagram {p1,p2)

and the spin representation labelled by N and S is given by

o r 1 = ( ~ / 2 ) + S t p 2 = ( N / 2 ) - S .

Let us use t h e techniques descr.ibed .on t h e l a s t s e v e r a l

pages t o de r ive sp in funct ions f o r N = 4 , S=1, M=O. The s tand-

ard tableaux a r e

s o t h a t D Pl1 = 1 1 p2 1 = (3.41, = ( 2 , 3 ) ( 3 , 4 ) . P31

The Young opera to r s f p r T; a r e

These o p e r a t o r s , appl ied t o t h e s p i n product 8=aBaB f o r M = O ,

g ive t h r e e l i n e a r l y independent s p i n funct iohs :

. . . D i r e c t a p p l i c a t i o n of g2 shows t h a t t h e s e func t ions - a r e sp in

e igenfunct ions , wi th eigenvalue S=l .

Def' iciencies of Young Operators

From a p r a c t i c a l p o i n t of view, sp in func t ions .gener-

a t e d by Young opera to r s have two shortcomings: they a r e n o t

or thogonal , and they correspond t o n e i t h e r t h e YK nor t h e

Serber spin-,coupling scheme'. Consequently, t h e s e func t ions

do no t have t h e p r o p e r t i e s demanded by t h e SAAP formalism.

One reason f o r t h i s i s t h a t Young opera to r s do no t com-

pose an orthogonal ma t r i c b a s i s . It can be shown t h a t they

mul t ip ly , no t according t o (49), b u t according t o t h e equa-

t i o n s

E ? . . E ~ ' = g D D ' $ . E~ * ( a number), 1 3 mn j m i n

where 6 i s no t always zero'when j#m. j m . .

I n a d d i t i o n , t h e r e is nothing. about t h e cons t ruc t ion

of Young opera to r s ' t h a t would a s s o c i a t e them with any par-

t i c u l a r spin-coupling scheme. . . ,

Neither do these opera to r s possess t h e a d j o i n t property

of (53)., Since row and column opera to r s a r e s e l f - a d j o i n t , .

(symmetrizers and ,antisy~[uneLl-izez-s are Hermitian) ,

'This l a s t de f i c i ency can be remedied by d e f i n i n g new

D D D D ' opera to r s p . . C .R .C R ? C ? R ~ . such o p e r a t o r s a r e e a s i l y 11 3 I j Or I I j

seen t o s a t i s f y ( 5 3 ) . Their p r o p e r t i e s have been s tud ied by . . "

Gallup (1968, 1969) , who has used them t o genera te pro jec ted

Hartree-product wave funct ions .

According t o ( 4 9 ) , t h e diagonal elements of a ma t r i c

b a s i s mul t ip ly according t o

That i s , t h e s e elements a r e idempotent and they a n n i h i l a t e

each o t h e r from t h e l e f t and r i g h t . I t can be shown t h a t t h e

genera t ing u n i t s f o r t h e minimal l e f t i d e a l s i n t o which t h e

group a lgebra decomposes a l s o have t h i s proper ty , a s w e l l

a s t h e proper ty ( 4 7 ) c h a r a c t e r i s t i c of p r i m i t i v e idempotents.

Young idempotents a r e p r imi t ive . Two Young idempotents

from d i f f e r e n t diagrams a n n i h i l a t e each o t h e r from t h e l e f t

and r i g h t . However, two Young idempotents from t h e same d ia - - gram may n o t do t h i s . In o t h e r words, Young idempotents

"almost" mul t ip ly l i k e t h e d iagonal elements of a m a t r i c

b a s i s (McIntosh, 1960) . ,

Examining t h e s i t u a t i o n more close ' ly , w e may draw t h e

fol lowing conclusion. There occur i n t h e decomposition of

D A(SN) d equ iva len t i r r e d u c i b l e c a r r i e r spaces for t h e i r r e p

l a b e l l e d by diagram D. These c a r r i e r spaces a r e generated by

D D t h e m a t r i c b a s i s idempotents e l , e 2 , ..., e The Young AD

D D u idempotents E l , E 2 , . . ., E~ genera te c a r r i e r spaces f o r t h i s

r e p r e s e n t a t i o n a l s o . Thus t h e r e must be equivalence t r ans -

formations r e l a t i n g t h e Young idempotents and t h e m a t r i c

b a s i s idempotents.

I n cons t ruc t ing from Young idempotents a ma t r i c b a s i s

s u i t e d t o t h e SAAP formalism, w e must, t h e r e f o r e , . . bu i ld '

opera to r s t h a t

. . (i) a r e r e l a t e d t o a spin-coupling scheme;

(ii) mul t ip ly l i k e a ma t r i c b a s i s ;

(iii) have t h e a d j o i n t property (53) . AS w e s h a l l s e e , t h i s can be accomplished by mul t ip ly ing . ..,

Young idempotents from t h e l e f t and r i g h t by c e r t a i n opera-

t o r s ,

. . .Tableau! Chains

I t i s well-known t h a t s tandard tab leaux can be der ived

from a genealogica l scheme s i m i l a r . t o ' t h a t , involved i n spin-

coupling (Jahn and van Wieringen, 1951; Pauncz, 1967; Cole-

man, 1968;,McWeeny and ~ u t c l i f f e ~ ' 1 9 6 9 ; Kleln -- e t a l . , 1970).

Since Young s p i n diagrams {p1,p2 ) l a b e l s p i n r ep resen ta t ions

of SN through t h e r e l a t i o n s pl= ( N / 2 ) + S t p2= (N/2) -St t h e YK -.

branching diagram s a n ' b e given i n t h c form shown i n F igure

6 . I n o t h e r words, t h e Young diagrams can be considered t h e

r e s u l t of a "box-coupling" . . procedure: one s t a r t s wi th '[7

and adds boxes one by one, s u b j e c t t o t h e condi t ion t h a t

P1>.P2'

F igure 6 i s a kind of shorthand f o r t h e :genealogical . .

cons t ruc t ion of s t a n d a i d tab leaux. I f w e ' s t i t wi th t h e

t a b l e a u II) and add, one by one, t h e numbers~2,3 , ..., N i n

1 2 3 4 ..--r- N . .

Figure 6 . YK branching diagram f o r Young d iag rams '

such a way t h a t t h e r e s u l t i n g t ab leaux a r e s t anda rd , w e ob-

t a i n F igure 7.

Each r o u t e i n t h i s f i g u r e r e s u l t s i n a unique s t anda rd

t ab leau . Conversely, each s t anda rd t a b l e a u uniquely d e f i n e s

i t s predecessors a long t h e r o u t e . This fo l lows from t h e f a c t

t h a t removal of t h e h i g h e s t number from a s t anda rd t a b l e a u

f o r N numbers produces a s t anda rd t a b l e a u f o r ( N - 1 ) numbers.

124 in the Thus, f o r example, o n e c a n work backward from,

fo l lowing way:

Figure 7 . YK branching diagram f o r standard Young tableaux

The s i g n i f i c a n c e of t h i s i s t h a t each s t a n d a r d . t a b l e a u

can be uniquely as soc ia ted with a YK branching r o u t e , and

t h e r e f o r e can be uniquely as soc ia ted with a YK s p i n funct ion .

To use t h e example of page 5 2 , N=4, S = l (o r D= ) , . w e have

t h e correspondence

S t a ~ l d a ~ r f Lableau ranching r o u t e

I t w i l l be observed t h a t each number on t h e upper row of a

s tandard t ab leau corresponds t o an upward'movement i n the

assoc ia ted branching r o u t e , and each number on t h e lower row

corresponds: t o a downward movement.

S t r i c t l y speaking, ' it is no t the tab leau i t s e l f t h a t

corresponds t o a YK branching rou te , b u t t h e u:nique "chain1'

of t ab leau predecessors from which it d e r i v e s . For: example,

t h e branching r o u t e i s a shorthand f o r t h e t a b l e a u

cha in

Such a cha in involves the a d d i t i o n of one number a t a t i m e ;

and i s c a l l e d . a 1-chain. - I n g e n e r a l , w e denote by Tr t h e s t anda rd t a b l e a u ob-

t a i n e d from T i by removing i t s k h i g h e s t numbers, v i z . D N , N-l, ..., N-k+l. Thus t h e 1-chain de f ined by Tr is w r i t t e n

Each s t anda rd t a b l e a u i s a l s o a s s o c i a t e d wi th a unique

2-chain, i f N i s even. Removal of two numbers from a s t anda rd - t a b l e a u r e s u l t s i n a sma l l e r t a b l e a u which i s a l s o s tandard .

Thus one can work backward from a given s t anda rd t a b l e a u and

d e f i n e i t s predecessors i n a Serber-type genea log ica l scheme.

For example,

1-1 --P m i , 131 13151

I n g e n e r a l ,

I n o t h e r words, s t anda rd tab leaux can be considered cons t ruc t -

ed according t o the Serber branching diagram of F igure 8.

Figure 8: Serber branching diagram

f o r standard Young . . tableaux

We have ind ica ted i n each case t h e p a i r d o £ numbers being

added, and t h e i r p o s i t i o n s r e l a t i v e t o t h e o r i g i n a l tab leau .

A t each s t a g e i n such a branching diagram', a geminal

p a i r . o f numbers 2 ~ - 1 , 2 ~ is added t o a t a b l e a u conta in ing

p-1 geminal p a i r s . I t w i l l be observed t h a t t h e add i t ion

of * * 12p-11 2p 1 always corresponds t o s =1, and t h e addi- Y . . . .

t i o n o f t w o numbers on t h e s a m e column always has t h e e f f e c t

of adding s = O . There i s an ambiguity, however, when 2p-1 Y

and 2 p a r e on n e i t h e r t h e same row nor t h e same column. One

case must correspond t o t h e a d d i t i o n of s =1 and t h e o t h e r Y

t o s = O . We a r e f r e e . t o make a choice, s o long a s w e a r e Y

c o n s i s t e n t . I n t h e foi lowing pages, we s h a l l . a s soc ia te

* * * pJ . .

rnI with :s = 1 P

and 0 . . -1

ELI with s. = 0 . IJ

.Now it is c l e a r t h a t t h e concept of t ab leau chains pro-

v i d e s the l i n k between Young's theory of t h e symmetric group

and t h e genealogica l c,Gnstruction of s p i n func t ions . However, . . . . .

we have a l ready pointed ou t t h a t Young opera to r s do n o t gen-

e r a t e YK o r Serber s p i n func t ions . C lea r ly , t h i s i s because

they do no t , i n themselves, ca r ry information s p e c i f i c t o

W e begin t o remedy t h i s de f i c i ency by de f in ing chains . .

D'k a r e t h e Young of Young idempotents. Suppose t h a t E: and Er - idempdtents f o r t h e tab leaux T: and T;lk, r e s p e c t i v e l y . Then

t h e m-chain of s t anda rd t ab leaux

i s a s s o c i a t e d wi th t h e m-chain

of Young idempotents. ( W e assume t h a t N i s a m u l t i p l e of m. )

D f k t o be Carrying t h i s one s t e p f u r t h e r , w e d e f i n e Lr

t h e minimal l e f t i d e a l genera ted by t h e Young idempotent

D E:'~. Thus each s t anda rd t a b l e a u Tr d e f i n e s a unique m-chain

of minimal l e f t i d e a l s .

Chains of Young Idempotents

and Genealogical Spin Funct ions:

an H e u r i s t i c Argument

It was rr~entfoned p rev ious ly t h a t t h e YI< s p i n f u n c t i o n s

f o r f i x e d N and S form a b a s i s f o r t h a t s p e c i a l o r thogona l

i r r e p of SN i n which SN-l, SNm2, . . . , S1 a r e a l s o r ep resen ted

by or thogonal , i r r e d u c i b l e m a t r i c e s . The r e p r e s e n t a t i o n is

s a i d t o be adapted t o t h e sequence of groups

We shall say that a representation with this property is

In a similar way, the Serber functions (for even N) are

adapted to the sequence

In addition, every geminal two-electron subgroup of SN is

represented irreducibly. A representation with these two

properties is said to be Serber-adapted.

The adaptation of representations to sequences of - - - - nested symmetric groups -- is the group-theoretical signifi-

cance - - of a genealoqical spin-coupling scheme.

Now suppose that Ly is a subspace of the group algebra,

A (SN) , with the following properties:

(i) Ly is invariant under lef t-multiplications by

elements of SN and transforms according to

D' the minimal leftideal Lr; . .

(ii) the elements of Ly transform among themselves

under left-multiplications by elements of SN-k

D ,k like elements of the minimal left ideal Lr , for k=1 , 2 , . . :, N - 1 .

Property ( = ) means that Ly is a carrier space for an irre-

ducible representation of S N . From property (ii), we see that

Ly is also a carrier space for irreducible representations of

. . . . . .

SN-l f SN-2' .. .. sl. ~ h u s L~ i s a c a r + i e r space f o r a YK-

adapted r e p r e s e n t a t i o n of SN.

I n a s imilar way, a subspace LS of A(SN) i s a carrier

space f o r a Serber-adapted r e p r e s e n t a t i o n of SN i f

(i) LS i s i n v a r i a n t under l e f t l u l t i p l i c a t i o n s by.

D elements of SN and t r ans fo rms l i k e L r ; ,

(ii) the elements of LS t rans form among themselves

under l e f t - m u l t i p l i c a t i o n s by e lements of SN-k

l i k e e lements of L: '~ f o r k=2,4, . . . ,N-2;

(iii) t h e e lements of LS a r e e i t h e r symmetric o r

antisymmetri 'c w i th r e s p e c t t o l e f t - m u l t i p l i c a -

. t i o n s by g e m h a 1 t r a n s p o s i t i o n s . ' . '

Before d e f i h i n g o r thogona l m a t r i c bases f o r genea log i -

c a l r e p r e s e n t a t i o n s , it i s i n s t r u c t i v e t o see' .wha t . p r ed i c -

t i o n s can be made abou t t h e s t r u c t u r e of such o p e r a t o r s by

ex tending t h e p re sen t . a rgumen t . We s h a l l s e e : t h a t ldempotent

g e n e r a t o r s f o r YK- and Serber-adapted c a r r i e r spaces can be

deduced r a t h e r e a s i l y .

The minimal l e f t i d e a l a s s o c i a t e d w i t h t h e s t a n d a r d

D t a b l e a u Tr is d e f i n e d ' t o . . be L~={xE:}, where X sweeps the r .

whole group a lgeb ra . Pt can be shown ( ~ u t h e r f o r d , ,1948,

p. 20) t h a t Young idempotents ha3e the prope r ty

D where eD = ( N l/dD) > 0 does n o t depend on r , and i [XE,] i s

D the c o e f f i c i e n t of t h e i d e n t i t y i n XEr, when it i s expanded

i n terms of t h e group elements. I t follows t h a t

D s o t h a t (XEr) is e s s e n t i a l l y idempotent i f it conta ins t h e D

i d e n t i t y . I n o the r words, - new idempotent genera tors . - of kr

can be made 2 l e f t -mul t ip ly ing E:. --- - Consider, f o r example / t h e element

This opera to r belongs t o L:. To t h e l e f t , it has

D , N - 1 . E: l N - l t which genera tes Lr I

D ,N-2. lN- lED tN-2) , belonging t o Lr : (Er r .

D ,N-3 'D t N - l ~ D t N - 2 ~ D tN-3) , belonging t o Lr (*r r . r I

Thus E y ( D , r ) behaves under l e f t - m u l t i p l i c a t i o n s by elements

Of SN-k (where k=1,2, . . . , N - 1 ) l i k e an element of L:'~. The

Young idempotent E: has been "YK-adapted" by mul t ip ly ing it

from t h e i e f t by the 1-chain

of Young idempotents from which it der ives .

S i m i l a r l y , we. may expect a Serber-adapted idempotent . .

t o take t h e form

where Sr 2k e i t h e r symmetrizes o r antisymmetrizes t h e geminal

D , 2k p a i r (N-2k-1, N-2k). Since t h e opera to r s t o t h e l e f t of Sr

do n o t con ta in t h e e l e c t r o n l a b e l s on which it opera tes , t h e

pair-symmetry opera to r s can a l l be brought o u t t o t h e l e f t :

Thus, when E S ( D , r ) is appl ied t o a p r i m i t i v e funct ion , it

w i l l gene ra te a . funct ion which i s . . e i t h e r symmetric o r a n t i -

symmetric i n each gemina l .pa i r .

Assuming t h a t E ( D , r ) and ES ( D , r ) , when expanded i n y .:. . . .

terms of t h e group elements, conta in t h e i d e n t i t y , they a r e . . .

e s s e n t i a l l y idempotent. However, they a r e n o t Hermitian, . . . . . .

s o they cannot be t h e idempotent d iagonal elements of the . . . .

mat r i c bases we seek.

I t is easy Lo see ' t ha t t h e fol lowing opera to r s &e Her-

mi t ian :

i n which

I t can be shown t h a t t h e s e o p e r a t o r s a r e , i n f a c t , Her-

mi t ian idempotents genera t ing YK- and Serber-adapted c a r r i e r

spaces f o r i r r e d u c i b l e r e p r e s e n t a t i o n s of SN. I t can a l s o

be shown', however, t h a t they do no t mul t ip ly l i k e t h e diago- /

n a l elements of a ma t r i c b a s i s . I t may be t h a t

f o r example. Thus t h e s e opera to r s cannot be used t o genera te

orthogonal b a s i s func t ions .

W e p resen t i n t h e next s e c t i o n m a t r i c bases f o r YK- and

~ e r b e r - a d a p t e d o r thogona l r ep resen ta t ions . I t w i l l be seen

t h a t these matr ic base's a r e symmetry-adapted' i n a way s i m i -

t t .. : l a r t o EYEy and ESES. Their d e f i n i t i o n s d i f f e r only t o t h e

degree . necessary i n order t o o b t a i n , t h e c o r r e c t mul t ip l i ca - . . . . .

t i o n p r o p e r t i e s .

Def in i t ions of

Orthogonal Matr ic Bases

Glossary. of n o t a t i o n ' - B Let T be a s tandard t a b l e a u , f o r a diagram D with N r . .

D boxes, and l e t ar and C' be i t s row and column groups. Le t . r

D D D Rr and Cr be the row a,nd column opera to r s f o r Tr , and le t

. . . .

D D D Er = RrCr be i t s Young idempotent. Then E: has t h e proper ty

D where 8 > O depends only on D , X i s any element of t h e group

D a lgebra , and i [XEr] i s t h e c o e f f i c i e n t of t h e i d e n t i t y , I ,

i n the expansion of xED i n , terms of group elements. I n par- r D t i c u l a r , i [Er]=l (Rutherford, 1948, p.14) , s o t h a t

EDED = e D ~ ; . r r

I t can be shown (Rutherford, 1948, p.65)' t h a t eD = ( ~ ! / d ~ ) ,

where dD i s t h e dimension of t h e r e p r e s e n t a t i o n l a b e l l e d by

diagram D . ' .

The row and column opera to r s a r e . s e l f - a d j o i n t , s o t h a t . .

D" . .

Let t ing oo b e t h e ord&r of t h e row group f o r a n y t ab leau . be-

longing . to diagram D , ' .

s o t h a t . .

D - D D ' D D D D Df = C'.RD#C" = 0, c R c = o a . c E Er Er r r r r r ' r r r r *

We d e f i n e p:s t o be t h e permutation t h a t . r e a r r a n g e s t h e . . .

D numbers i n T: t o form Tr :

D -1 . D . - Thus p;S is: such t h a t prr-I and p:r = (prS) . Furthermore,

t h e t ab leau opera to r s have t h e p r o p e r t i e s

D W e denote by T:lrn t h e s tandard t ab leau obtained from Tr

by removing t h e m h ighes t numbers, i . e . , N , N - 1 , ..., N-m+l .

D T h e n i f m i s a f a c t o r of N , TI de f ines t h e m-chain of s tandard

tab leaux

There corresponds an m-chain

of Young idempotents.

A - matr ic b a s i s - f o r or thogonal YK-adapted - r ep resen ta t ions

The s tandard t ab leau T: de f ines t h e 1-chain

= f o r every D and r. of s tandard tab leaux, where Tr

W e 'def ine f o r t h i s 1-chain a chain of idempotent opera-

t o r s , i n t h e fol lowing manner:

where k: is the number

D D D D kr = Oa. €3 ' P r I

i n which

I t should be notcd t h a t t h e s e opeyaLo~s.are Hemitian.

The idempotents e: a r e used t o c o n s t r u c t t h e m a t r i c

., b a s i s e lements

The d i agona l e lements e:r of t h i s b a s i s a r e i d e n t i c a l t o t h e

idempotents e: de f ined by (57) . For a p p l i c a t i o n t o p r i m i t i v e f u n o t i o n s , it i s mare con-

v e n i e n t t o u se an a l t e r n a t i v e expres s ion f o r t h e m a t r i c b a s i s :

D D , 1 D D D D D D , 1 D D 1 / 2 . . . e = e C R p R C e / (krks) . . r r r r s s s s . ,

- D , l D D D D D D , 1 D D D D 1/2] P C R R C e / [ ~ o ~ ( P , P , ) - e r r s s s s s s

The 1-chain d e f i n e d by T i i s

. f o r which t h e Young idempotents a r e

Neglecting numerical f a c t o r s ,

The e n t i r e m a t r i c r b a s i s , t hen , c o n s i s t s of the opera to r s

The whole m a t r i c b a s i s i s n o t requi red £0; t h e construc-

t i o n of b a s i s func t ions f o r t h e i r r e p . The opera to r s

D . D : {ell , e i l } span a minimal l e f t i d e a l a s s o c i a t e d w i t h T1;

D s i m i l a r l y , {e12, e:2} span a minimal l e f t i d e a l a s s o c i a t e d

D w i t h Ti . E i t h e r of t h e s e s u b s e t s can be used t o g e n e r a t e

b a s i s f u n c t i o n s .

A s an example, w e apply e:2 a n d e:2 t o t h e s p i n produc t

f u n c t i o n 8=aBa. S ince t h e diagram D= corresponds t o S=1/2,

w e should obtain.YK s p i n f u n c t i o n s f o r N=3, S = ~ = 1 / 2 . W e have

D eD 12 = 4 S l 2 = ( 2 , 3 ) * 8 and eZ2 = 1 6 6 , where 6) = A12S13A12.

and e:;e = 1 6 ~ 0 = ' 4 8 j a ~ - ~ a ) a . . .

These a r e , indeed , t h e (unnormalized) YK s p i n f u n c t i o n s ob- . .

t a i n e d f o r -N=3, S=1/2, M=1/2 from t h e spin-coupl ing equa t ions . .

( 2 8 ) . Not ice t h a t e:28 corresponds t o t h e branching r o u t e . .

, w h i l e e:2 cor responds t o N, and t h a t t h e s e f u n c t i o n s . .

a r e or thogona l .

The same f u n c t i o n s , w i t h i n a numerical f a c t o r , are ob-

D t a i n e d by means of t h e m a t r i c b a s i s e lements ?yl and e21.

A matr ic b a s i s f o r or thoaonal S e r b e r - a d a ~ t e d r e m e s e n t a t i o n s

When N i s even, t h e s tandard t ab leau T: def ines t h e 2 -

chain D I N - 4 - ... - D - Tr D , 2 - - Tr ,

D D t N - 2 is e i t h e r ml o r , depending on T r . where Tr

D For T r , a geminal opera tor S: i s def ined i n t e r m s of t h e

p o s i t i o n s ' o f t h e two h ighes t numbers, N - i and N . Denoting t h e

row and column on which a number k appears a s rk and ckf w e

d e f i n e

I n o t h e r words, S: symmetrizes the numbers (N-1) and N i f

D Tr conta ins t h e s e numbers i n t h e p o s i t i o n s -1 o r

. . p=i] m, bu t antisymmetrizes them i f T: con ta ins

Geminal opera to r s S f o r o t h e r tab leaux Tr r i n t h e

2-chain a r e def ined analogously . A s e t of Hermitian idempotents i s d e f i n e d , r e c u r s i v e l y

f o r t h e 2-chain:.

D D D,2 t D D D,2)/k: e: = (ErSrer ) (E S e r r r

where k: i. t h e number D D o D kr = o a 0 e 'r

: i n which

The idempotents e: a r e used t o c o n s t r u c t t h e m a t r i c

b a s i s e lements

D D D,2 t D D D D,2 D D 1/2 = (E S e prs (EsSses / (krks) ers r r r

I t should be noted t h a t a d i a g o n a l e lement eEr i n t h i s b a s i s

i s i d e n t i c a l t o t h e e lement e: de f ined b y (60) . . .

A s i n t h e prev ious c a s e , t h e m a t r i c ' b a s i s e lements can

be g iven i n a s l i g h t l y s imp le r form. The r e s u l t i s

. . . .

A s an example of t h e a p p l i c a t i o n of t h e s e o p e r a t o r s l

w e g e n e r a t e t h e Serber s p i n f u n c t i o n s f o r N=4, S = l , M=O, . .

us ing t h e p r i m i t i v e f u n c t i o n aBaB. The young, .diagram i s

D= , f o r which t h e s t anda rd t ab leaux a r e :, . . . .

s o t h a t . . . .

Pll = I t = (3,4), P 2 1 pY1 = (213) ( 3 , 4 ) , 1

- rl C

..> a -

(62) and n e g l e c t i n g numerical f a c t o r s ]

where " '14~123 A A S 1 4 3 4 12 .

These a r e the ope ra to r s t h a t were d i sp layed i n ( 4 5 ) on

Applying them t o 8=a6apt 'one o b t a i n s t h e Serber func-

t i o n s shown i n (46) on t h a t page. The branching r o u t e s can be

read d i r e c t l y from t h e geminal symmetrizers and antisymme-

t r i z e r s . i n t h e ma t r i c b a s i s elements.

General d e f i n i t i o n of t h e or thogonal ma t r i c bases -- It i s convenient t o t r e a t t h e ' m a t r i c bases f o r 1-chains

and 2-chains t o g e t h e r , under one master formula. , L e t t h e

m-chain def ined by t h e s t anda rd t a b l e a u T: be denoted by

where m is a f a c t o r of N .

For each s t anda rd t a b l e a u Tr D ' J m i n t h i s cha in , an H e r m i -

t i a n o p e r a t o r Mr D t j m i s de f ined i n terms of only t h e h i g h e s t

m numbers, i .e., t h e numbers N - j m , N - j m - 1 , . . . , N- ( j + l ) m + l .

When m = l , t h i s ope ra to r i s taken t o be t h e i d e n t i t y . When

m=2 , it is de f ined t o be a two-electron symmetrizer o r a n t i -

symmetrizer, a s d i scussed p rev ious ly . . .

A set of ~ e r m i t i a n idempotents. i s de f ined r e c u r s i v e l y

i n t e r m s of each m-chain: .

i n which

These idempotents a r e used t o d e f i n e t h e m a t r i c b a s i s

D i n which, it w i l l be no ted , e:r = el.

I t i s convenient t o u se t h e m a t r i c b a s i s e lements i n the . . . . 3

s impler form . .

For u s e i n gene ra t ing b a s i s f u n c t i o n s f o r t h e i r r e p of

SN l a b e l l e d by D', a s u b s e t {e:s 1s f i x e d } of t h e m a t r i c b a s i s

is used. The o p e r a t o r s i n t h i s s u b s e t a l l have t h e form

i s f ixed .

D , m D D D eD = (number). er rs MrPrsps

D D D D D , m Q s = C E M e S S S S

Discussion

W e s h a l l prove i n t h e next s e c t i o n t h a t t h e m a t r i c bases

def ined by (58)- (64) can be used t o genera te b a s i s func t ions

f o r or thogonal r ep resen ta t ions of SN. More p r e c i s e l y , w e w i l l

show t h a t

(i) none of t h e elements e:s vanishes;

(ii) these elements mul t ip ly l i k e a m a t r i c b a s i s ;

D t = eD . (iii) . they possess t h e a d j o i n t property ers sr'

( i v ) they a r e l i n e a r l y independent and span t h e P

group a lgebra , A (SN) ;

(v) t h e diagonal elements e:r a r e p r imi t ive idem-

po ten t s genera t ing t h e minimal l e f t i d e a l s

occurr ing i n t h e decomposition of A (SN) .

That t h e ma t r i c bases a r e YK-adapted (when m = 1) o r

Serber-adapted (when m .= 2) i s e a s i e r t o see . Using ( 6 4 ) , neglec t ing numerical f a c t o r s , and not ing that. . . . M: commutes

D , 2 m , e t c . , wi th er , er . .

D D D D t 2 s D .mCD tmED tmMD CrErPrs (es S S . S s e s lm D I Z " ) ~ :

i s a product of commuting opera to r s . When m = i . G: i s simply

t h e i d e n t i t y . When m=2., it i s a s t r i n g of geiiina'l symme-.

t r i z e r s and ant isymmetr izers .

Comparison of ( 6 6 ) with t h e heur ' i s t ica l ly-der ived oper-

D a t o r s E ~ E : and EBE; of . the previous s e c t i o n shows t h a t ers

i s YK-adapted when m = l and Serber-adapted when m=2.

Orthogonal YK-adapted r e p r e s e n t a t i o n mat r i ces w e r e f i r s t

obtained by Young (1932, p. 218) . This i s t h e r e p r e s e n t a t i o n

known i n t h e l i t e r a t u r e a s ."Young's orthogonal representa-

t i o n " . Pauncz (196.7) has shown t h a t t h i s ' r e p r e s e n t a t i o n i s

i d e n t i c a l t o t h a t obtained by Yamanouchi. A ma t r i c b a s i s f o r . . .

such a r e p r e s e n t a t i o n 'can be obtained from t h e r e l a t i o n s ( 5 4 )

between orthogonal ma t r i c b a s i s elements and permutations.

One ob ta ins t h e so-ca l led "orthogonal u n i t s " (Rutherford,

1948, p . 50)

where t h e sum runs over t h e e n t i r e symmetric group. Goddard

(1967a, .1967b, 1968) has employed t h i s ma t r i c b a s i s , in

quantum-chemical c a l c u l a t i o n s .

I n nuclear theory , Jahn and co-workers (Jahn and

van Wieringen, 1951; E l l i o t t , Hope, and Jahn, 1953; Jahn,

1954) have used m a t r i c bases f o r orthogonal YK- - and Serber-

adapted r e p r e s e n t a t i o n s . The l a t t e r were obtained from t h e

or thogonal u n i t s (67) by f ind ing t h e t ransformat ion between

YK and Serber r e p r e s e n t a t i o n s .

General d i scuss ions of ma t r i c bases , considered accord-

ing t o t h e l r expansions i n permutat ions, have been g iven by

Matsen and co-workers (Matsen, 1964; Klein, C a r l i s l e , and

Matsen, 1970) . . .

I n a l l of these accounts , m a t r i c b a s i s elements were

descr ibed a s l i n e a r combinations of a l l N! permutations i n . .

SN. Thus m a t r i c bases were expressed a s sets of Wigner oper-

a t o r s . he disadvantages of t h i s approach were d iscussed i n

t h e l a s t chapter .

To t h e a u t h o r ' s knowledge, t h e only previous a t tempt

t o ob ta in m a t r i c bases ' d i r e c t l y from t h e s tandard Young tab-

leaux was t h e d e r i v a t i o n by T h r a l l (1941) of "Young's

semi-normal u n i t s " . These have been d iscussed by Rutherford

(1948). The work repor ted i n t h e p r e s e n t chapter i s an ex-

tens ion of T h r a l l ' s approach t o orthoqonal r ep resen ta t ions

u s e f u l i n quantum chemistry.

The formulas given i n t h e previous s e c t i o n would appeitr

t o avoid t h e drawbacks of o t h e r methods f o r obta in ing b a s i s

func t ions . Referr ing t o equat ion ( 6 5 ) , one s e e s t h a t b a s i s

func t ions f o r any i r r e p of SN can be generated by a s e t of

opera to r s cons t ruc ted from symmetrizers, ant isymmetr izers ,

and t h e permutations p:s r e l a t i n g s tandard tableaux. Further-

more, t h e " r i g h t h a l f " of each opera to r , given by (65b) , is

f i x e d throughout t h e c a l c u l a t i o n .

Although the ma t r i c bases presented here are def ined

r e c u r s i v e l y , t h i s does n o t cause s e r i o u s computational d i f f i -

c u l t i e s . The recurs ion g ives r i s e t o a number of row and

column opera to r s which must b,e appl ied i n succession t o a

p r i m i t i v e funct ion . A s can be seen from t h e examples i n t h e

l a s t s e c t i o n , one a p p l i e s a symmetrizer o r antisymmetrizer

t o t h e p r imi t ive , c o l l e c t s terms, and then a p p l i e s another .

The opera to r s a r e a l l "read" d i r e c t l y from t h e s tandard tab-

1eaux:A computer program f o r such a procedure would n o t re-

q u i r e l a r g e amounts of s t o r a g e - the chief drawback of o t h e r

approaches. Such a program would have t o perform very many

permutations and c o l l e c t i o n s of terms, b u t these opera t ions

involve only d a t a t r a n s f e r r a l s and i n t e g e r a r i t h m e t i c , and

can be performed quickly .

A computer program is being w r i t t e n t o genera te Serber

s p i n funct ions by means of t h e m a t r i c b a s i s elements ( 6 2 ) .

Basic Lemmas

Before proceeding t o t h e lemmas and theorems s p e c i f i c

t o or thogonal ma t r i c bases , w e summarize some elementary re -

s u l t s . t h a t w i l l be needed.

The d e f i n i t i o n s ( 5 7 ) - ( 6 4 ) used i n t h e cons t ruc t ion of

mat r ic bases involve numerical f a c t o r s i [ x ] , t h e c o e f f i c i e n t

of t h e i d e n t i t y i n an element x of t h e group algebra. This

funct ion def ined on A ( s ~ ) has two p r o p e r t i e s which w e s h a l l . .

f i n d u s e f u l .

Lemma - 1:

I f 1-1 i s a number and x is an element of A (SN) , then

Proof: I f x = I . ~ ( P ) P , then px = 1 p c ( P ) P , s o t h a t

i L 1 - 1 ~ 1 = 1-1S (I) . But 1-1i[xl = 1-15 ( I ) a l s o .

Lemma 2 : - I f x and y a r e elements of A(SN) , then i [xy] = i [yxl .

Proof: I f x = 1 c ( P ) P and y = 1 n ( P ' ) P ' , then

and i [yxl = 1 n ( P ) S (P-') .

Since t h e sums run over an e n t i r e group, t h e s e exp res s ions

are i d e n t i c a l .

Not ice t h a t Lemma 2 i m p l i e s t h e fo l lowing c y c l i c p rop-

e r t y : i [xyz] = i [ zxyl = i [yzxl ,

f o r any e lements x , y , z of t h e group a l g e b r a .

W e now r e p e a t t h e d e f i n i t i o n of t h e a d j o i n t o p e r a t i o n

and prove two r e s u l t s .

D e f i n i t i o n :

For any e lement x = 1 c ( P ) P i n A (SN) , t h e a d j o i n t

element f s defined to be

x' = 1 E * ( P , P - ~ ,

where * deno te s t h e complex con juga te .

Lemma 3 : - t t t . For any x and y i n A(SN) , ( X Y ) = Y

Proof : Def in ing x and y as b e f o r e ,

Lemma - 4 :

t For any x i n A (SN) o t h e r t han t h e n u l l , i [xx 1 > 0 .

Proof: 1f x = I c ( P ) P , t h e n xt = I E * ( ~ ) ~ - l t Sb that

i f a t l e a s t one c o e f f i c i e n t c ( P ) i s nonzero.

W e s h a l l make f r e q u e n t u se of two p r o p e r t i e s o f t h e

D and Er. These are proved i n Ruther- t a b l e a u o p e r a t o r s R r , C r ,

f o r d ( 1 9 4 8 ) , ' s o they a r e quoted h e r e w i thou t p roof .

Lemma - 5: .

F o r , e v e r y D , r , and s t . .

s o t h a t

Lemma 6: - For every D , D ' , r , s , and every x i n A (.S ) ,

. . !J.

Lemmas Concerning t h e Matric ~ a s . e s

Leitut~iil 7 : - I f t h e numbers N - 1 and N a r e on d i f f e r e n t rows and

d i f f e r e n t columns i n 2 s t a n d a r d t a b l e a u T: c o n t a i n i n g N num-

bers, then E er Dt * does not contain the transposition r

- 1 N : i .e., the coefficient of (N-1,N) in E~ r . eDt r is

Dt does not operate on the numbers N-1 Proof: The element er

D Df were to contain (N-1,~) , Er and N. Therefore, if Er er

would have to contain a permutation of the form (N-1,N)$,

where 6 is a permutation which does not affect N-1 or N. We D shall show that Er can contain no such permutation.

D There are two forms possible for Tr, namely

and . . . .k.. . .N ..

It is sufficient to consider only the former; With T: of the

D form (68) , Er will contain only permutations.~~of the, f orm ... r c c r ~ - l N N N-1 Ct where rN is a row permutation for the row

containing N, etc., and 1, 2 are permutations which do not

operate on N-1 or N.

If r c c r ~ - l N N N-1 E = (~-l,~)g,

where Q does n o t ope ra t e on N - 1 o r N: W e w i l l prove t h a t (70)

i s impossible .

r c c According t o ( 7 0 ) , rN-l N-l must be a permutat'.orL

i n which N i s rep laced by N - 1 , and N - 1 by N . W e know from t h e

form (68) of t h e t a b l e a u , however, t h a t rN- l rN~N~N-l has . t h e

( . .k. . .N-1) ( . . .N) ( . . .k. . .N) ( . . . N - 1 ) , (71)

where t h e d o t s r e p r e s e n t numbers o t h e r than k t N - 1 , o r N .

Now, because of t h e form of t h e t a b l e a u , no --- two of t h e s e

permutat ions can - s h a r e any numbers o t h e r than k t N - 1 , and N . -- --- Thus i f , i n cN = ( . . .k . . .N) , N i s rep laced by a number a

o t h e r than k o r N - 1 , t h e product rN-l r c N%-l w i l l be a

permutation ( . . . N L ) because n e i t h e r rNml nor rN w i l l ope ra t e

on A. Consequently, i f any permutation of t h e form (71) can

s a t i s f y ( 7 0 ) , it w i l l be one i n which t h e numbers r ep resen ted

by d o t s p l ay no p a r t a t a l l . W e may j u s t a s w e l l cons ide r t h e

s impler t a b l e a u

But then

W e have proven t h a t E: can con ta in no permutation of

t h e form ( N - 1 , ~ ) c if T: is of t h e form ( 6 8 ) . The proof f o r

(69) is s i m i l a r .

Lemma 8: -

f o r every D and r , and f o r m = l o r m=2.

Proof: W e d e a l he re w i t h o p e r a t o r s de f ined i n terms of a

s i n g l e s t a n d a r d t a b l e a u and i t s m-chain. W e t h e r e f o r e drop

D D t h e s u p e r s c r i p t s and s u b s c r i p t s , and denote Er by E, Mr by

D , m M , and er by e-.

For a 1-chain, it can be shown (Ruther ford , 1948, p. 28)

t h a t - E =, E + ( t e r m s o p e r a t i n g on N )

Therefore , i [Ee- ] = i [ E - e - ] + i [ + ~ e - 1 . The l a s t t e r m is zero

because e- does n o t o p e r a t e on N , and tN is made up only of . . . . -

permutat ions t h a t o p e r a t e on N . Thus h e cannot c o n t a i n t h e

i d e n t i t y . This proves t h e theorem when m = l .

For a 2-chain, t h e r e a r e t h r e e cases .

D (i) I f N - 1 and N appear on t h e same row of Tr, then

D M ~ R ~ = Rr because R: c o n t a i n s t h e idempotent < and is a r r group sum. Thus

W e have used Lemma 2 and t h e f a c t t h a t M commutes w i t h e-.

(ii) I f N - 1 and N..appear on t h e same column of T~ r t t h e

D D - s o t h a t argument is s i m i l a r : CrMr - C r l

- - i EM^-] = i [ R C M ~ - ] = i [RCe 1 = i [Ee 1 .

(iii) If N - 1 and N occur' on d i f f e r e n t rows and d i f f e r e n t D . . columns i n T r , t hen . .

- The l a s t t e r m con ta ins i [ E * ( N - 1 ,N) me-] = i [ E e ( N - 1 , N ) 1 , which 1s zero u n l e s s ' E e - c o n t a i n s (N-1 I N ) . W e , proved i n

Lemma 7 t h a t t h i s i s impossible . ,

I n a l l t h r e e cases , i [ ~ ~ e - ] = k*i [Ee- I . , where k>O. By

an argument e x a c t l y p a r a l l e l t o t h a t f o r 1-chains, it can be - -

shown t h a t i [ E e - ] = i [ E e I . This proves t h e theorem f o r

Exis tence Proofs

.Our purpose i n t h i s s e c t i o n . i s t o s h o w ' t h a t none of t h e

m a t r i c b a s i s e lements v a n i s h o r blow up. The d e . f i n i t i o n s i n -

D valve f a c t o r s pr i n the denominators. W e begin by proving

t h a t t h e s e q u a n t i t i e s a r e never zero. AS a by-product, w e . . .

a r e a b l e t o show t h a t ' the d iagonal elements of t h e m = t r i c

,basis are idampotent. . , .

Theorem - 1:

For any D and r , and f o r m = l o r m=2 ,

D D D D Dtrn]. > 0; ( a ) . 'p: 2 i [~ : E: M: e:fm] = i [cr R r cr M r er

(c ) e: i s idempotent and s e i f -ad j o i n t .

Proof: The proof is by induct ion . Using t h e n o t a t i o n of

t h e previous lemma, we assume t h a t

- - - -t - i [E-e-] # 0, e e = e , e = e l

then. show t h a t t h e s e p r o p e r t i e s recur : t h a t

and a l s b t h a t i [ C E M ~ - I > 0, i [Me-1 # 0.

This i s shown i n f i v e s t e p s .

(i) W e assume t h a t i[E-e-1 # 0, s o t h a t i EM^-I # 0

by ~emm'a 8. This i s t h e induct ion f o r p a r t (b). Therefore,

d t x = EM^- is n o t t h e n u l l , and i [ x x ] > 0 by Lemma 4.

D (ii) i [ C E M ~ - ] = i [ C R C M ~ - ] = i [CR R C M ~ - I /oa

using Lemmas 1 and 2 . By cons t ruc t ion , M is idempotent and

commutes wi th e-: M e - = Me-M. I n a d d i t i o n , w e assume t h a t - - -

e idernpotent, s o Me- = Me e M I and . .

t - -t - Furthermore, ( R C M ~ - ) = e t ~ t ~ t ~ t . W e assume t h a t e = e ,

t t and M = M , C = C , Rt = R by cons t ruc t ion . Thus

and t D t D i [CEM~-1 =, i [ ( R C M ~ - ) ( R C M ~ - ) 1 /oa = i [xx 1 /oa > 0.

This is t h e induc t ion f o r p a r t (a). W e have y e t t o j u s t i f y

equa t ions (72) . (iii) Since p = . .i [CEM~-] '# 0 by (ii) , t h e q u a n t i t y

i [ E e l = i [EC-MCEM~-] / (Bp ) i s defined . But

= E -8i [ C E M ~ - ] us ing Lemma 2 ,

s o t h a t i [ E e ] = i [EMe-] # 0 by (i). . . . . . . . .

( i v ) Assuming t h a t e- i s idempotent,

(v) et = (e-MCRCM~-) t/ (ep) , , s i n c e 8 and p a r e r e a l . -

W e assume t h a t e-t = e , s o t h a t

This completes t h e induct ion scheme. W e now prove t h a t

t h e - induct ion has a base. D , N - 1 D , N - 1 - D , N - 1 = I

SO t h a t el For a 1-chain, Er . - . .

is idempotent and s e l f - a d j o i n t , and

D I N - 1 DIN-'^ = = I # 0. . i [E, ". D,.N-2 - For 2-chains, Er - er . - - [1 t (1 ,2 ) ] /2 , s o that

is idempotent and s e l f - a d j o i n t , and er

Q.E.D.

Theorem - 2:

None of t h e elements eD i s t h e n u l l . rs.

Proof: . We prove t h a t E;eFSE: 'does n o t vanish. This i s

b y Lemmas 6 and 2 [ t h e argu- The underl ined p a r t is:.' E r e p,

ment i s s i m i l a r t o t h a t i n s t e p (iii) of Theorem 1 1 , s o t h a t

and i [EDgeDt rn ] a r e I t w a s shown i n Theorem 1 t h a t p r , ps , s s s D i s n o t t h e n u l l nonzero. Also, €3 > O by Lemma 6 and prsEs

(Rutherford, 1948, p. 1 6 ) . This completes t h e proof .

W e have now proved that t h e d e f i n i t i o n s (57)-(64) of

t h e or thogonal mat r ic bases y i e l d e x i s t i n g , nonvanishing

opera to r s .

M u l t i p l i c a t i v e P r o p e r t i e s

Theorem 1 has a l ready shown t h a t t h e diagonal elements

D e = e r a r e idempotent. This f a c t , and t h e two lemmas t h a t rr fol low, a r e enough t o . e s t a b l i s h t h e ma t r i c b a s i s mul t ip l i ca -

t i o n r e l a t i o n .

F i r s t w e must show t h a t

I t i s c l e a r t o begin with t h a t

D D , m D , m D D D , m Mr " r . e . . r , M r = M r e r ,

D f m a r e idempotent and commuting. I t remains because M: and e r

only t o show t h a t . .

This is the 'purpose of Lemmas 9 and 10'.

Lemma 9: - D

(For 1-chains) Suppose t h a t two s tandard tab leaux Tr

and T: belong t o t h e same diagram, D , and d i f f e r i n t h e posi-

D;l and TS t i o n of t h e h i g h e s t number, N . Then Tr D r l belong t o

d i f f e r e n t diagrams, and'

f o r e v e r y . x i n t h e group algebra. D D (For 2-chains) L e t two s tandard tab leaux, Tr and Ts,

belonging t o t h e same diagram, D , d i f f e r i n t h e p o s i t i o n of

a t l e a s t one of t h e two h i g h e s t numbers, N - 1 and N . Then

e i t h e r

. . . . .

o r T:'~ and, T:'* belong . . t o d i f f e r e n t diagrams. I n the l a t t e r . . .

case ,

. . f o r every x : i n A (SN) . . .

D ' l and Ts Proof: For 1-chains, it i s obvious t h a t Tr D t l w i l l

belong t o d i f f e r e n t diagrams. The conclusion fol lows from

Lemma 6 .

For 2-chains, t h e :argument i s s i m i l a r e x c e p t when

D T D t 2 and Ts D , 2 w i l l belong = ( N - l , N ) T s . In . such a c a s e , Tr D

t o t h e same diagram, b u t w e have de f ined M: and Ms such t h a t

one w i l l symmetrize N - 1 and N , and t h e o t h e r w i l l antisymme-

t r i z e them. I n t h i s c a s e ,

Lemma 10:

L e t T: and T: be d i f f e r e n t s t anda rd t ab leaux belonging

t o t h e same diagram, D. Then

Proof: I f T: and T: d i f f e r i n th.e p o s i t i o n s , of t h e i r h i g h e s t

one ( f o r 1-chains) o r .,two ( f o r 2-chains) numbers, then Lemma

D , m f o r 9 a p p l i e s d i r e c t l y , and, s i n c e M: commutes wi th et

every t ,

D D D,meD,m D D,meD,mMD = M M Mrer s .s r s r s

D D D,2%D,m D,m = (nuriber) M M e r s r r 'r

where one o r t h e o t h e r of t h e under l ined f a c t o r s i s t h e n u l l .

Otherwise, t h e r e is a number k such t h a t removal of t h e

D D,km h i g h e s t km numbers from: Tr and T: r e s u l t s i n t ab leaux Tr

D f k m d i f f e r i n g i n t h e p o s i t i o n s of t h e i r h i g h e s t m and Ts

numbers. Then r e c u r s i v e s u b s t i t u t i o n . .

g i v e s

D,km D , (k+l)m D , (k+l)%D,km . = (number) M: *-• Mr er es s M:

where t h e under l ined f a c t o r i s t h e n u l l , by t h e argument

given ' above. T h i ~ provcD thc. lcmms.

Lemmas 6 and 1 0 and Theorem l ( c ) p u t us i n p o s i t i o n t o D show how t h e e l emen t s ers m u l t i p l y .

Theorem 3: , - ,

eD e D ' = 6 DD I D rs t u ' s t

where t h e under l ined- f a c t o r vanishes . i f D#Di , by Lemma 6 .

Therefore ,

D eDt DDt D,m D D CDEDMDeD,meD,%DpD CDEDMDeD,m ers tu = ' er M r r s s s s s t t t u u u u u

D -2 D D D D -1/2 (9 (P,P~P~P,)

D D,m By Lemma 10,the underlined factor is GstMses , so &

eD eD =. 'DD ' Df~DPDcDEDMDeDtmpD CDEDMDeDtm rs tu 'ster r r s s s s s s u u u u u

Using the fact that p:,c$; = C D D D E p from~ernma5, S S S U f

eD eDt = 6 DD Dfm'D D D D D Dtm D D D MDeD,m rs tu C E . M e 'st er Mrprs s s s s CsEsPsu u u

, . . . By Lemmas .6 and 2, the ' underlined ' part is

so that

eD eD' = 6 DD eDfmMDpD CDEDpD M D e D f ~ 0- 0 0 1/2] rs st 'st r r rs s s su u u / [e--(prpu)

. . DD ' D D D D D,m D': D .D 1/2

Df%D~Dp C E M ~ / [eJprpU) 1 = ' 'st er r rs su u u u u

DD' Dfm D D CDEDMDeDI~t~ = ' 'ster M r P r u ~ ~ ~ ~

This ' proves the theorem..

Orthogonal Operator Bases

f o r Every I r r e d u c i b l e Representat ion

I t fol lows from Theorem 3 t h a t a ma t r i c b a s i s

{e:! 1 a l l ~ , r ' , s } c o n s i s t s of

l i n e a r l y independent elements. The argument was given on

Thus t h e YK- and Serber-adapted mat r i c bases i n t r o -

duced he re span - t he e n t i r e group a lgebra .

Furthermore, they have been def ined i n such a way t h a t

Thus t h e s e ma t r i c base8 have t h e a d j o i n t proper ty .(53) . . . I t

fol lows that a subse t . .

D Bs = ieD rs 1 s f i x e d , a l l r }

spans a c a r r i e r space f o r an or thogonal r e p r e s e n t a t i o n of SN.

We say t h a t B: i s an opera to r b a s i s f o r an or thogonal repre-

s e n t a t i o n , o r f o r s h o r t i an orthoqonal opera to r b a s i s . . . . .

D e D with s f i x e d , Now B:' c o n s i s t s of elements e:s = ers

D spanning a l e f t i d e a l generated by t h e idempotent ess. A s a

matter. of f a c t , t h i s l e f t i d e a l is minimal,. f o r w e now show

t h a t e:S i s a p r i m i t i v e idempotent.

Theorem - 4:

For any D , D ' , r , .and s t and any element x. i n t h e group

a lgebra , D e: x eD' = err D ' DD ' D s x ess = 6 A (x)ers ,

where X(x) is a number t h a t depends on x.

Proof :

D D ' D D ' -1 x ( e e p r % )

Applying Lemma 6 t o t h e underl ined p o r t i o n ,

. . D D l - D D ' D,%D D D ~ D ~ D ~ D , ~ ; e x e - 6 . e r r s c P r r r s s s s

. D D D D , m Dt%DcDI(eD,-2( D D -1 E M e xe ' [ p s r r r r s s s p r p s )

, . W e have a s a spec i .a l case of t h i s r e s u l t ;

D for . a r b i t r a r y x. Thus ek has, t h e proper ty ( 4 7 ) : t h e d iagonal

elements of t h e ' m a t r i c bases a r e p r i m i t i v e idempotents.

These idempo.tentsl un l ike t h e Young idempotents, gener- . .

a t e t h e minimal l e f t i d e a l s occurr ing i n t h e decomposition

of t h e group a lgebra . This we prove by showing t h a t t h e iden-

t i t y , t h e genera t ing u n i t of the whole group a lgebra , decom-

D poses a s t h e sum of t h e l i n e a r l y independent elements er ,

which genera te minimal l e f t i d e a l s .

Theorem 5: -

Proof: Let T = 11 e:r. Since t h e m a t r i c b a s i s spans A(SN) , D r

an a r b i t r a r y element x can be expanded i n the form

. . . .

I t fol lows t h a t

XT;= 111 L L ef! = 1111 1 ~h(e:~eFI) D r s D ' t D r s D t

= LI11. 1 ~ : ~ ( 6 ~ ~ ~ 6 eD = 111 cESeEs D r s D 1 t . s t r s D r s

f o r a r b i t r a r y x. S i m i l a r l y , Tx = x. I t fol lows t h a t T = I.

I t should be noted t h a t t h i s theorem cannot be proved

with Young idempoten t s '~ : i n p lace of t h e e:. This i s because

D D E E # 6 r s ~ : , i n genera l . Young opera to r s do n o t mul t ip ly r s

l i k e a ma t r i c b a s i s . :

D The d minimal l e f t i d e a l s f o r i r r e p D: t h a t occur i n

t h e decomposition of t h e . group a lgebra a r e those genera ted

. ' D D by t h e idempotents {e r 1 r=l, 2 , . . . ,d } . The minimal l e f t i d e a l s

D D generated by { E ~ I r=1 ,2 , . . . .d 1 can be shown t o d i f f e r from

t h e s e by equiva lence t ransformat ions . .

W e conclude by summarizing t h e u s e f u l p r o p e r t i e s of t h e

D mat r i c b a s i s e lements ers.

Each d i s t i n c t i r r e d u c i b l e r e p r e s e n t a t i o n of t h e symmet-,

r i c group is l a b e l l e d by a Young diagram, D . Spin represen-

t a t i o n s a r e l a b e l l e d by diagrams wi th one o r two rows.

The i r r e p . . l a b e l l e d by D occurs dD t i m e s i n the r e g u l a r

D r e p r e s e n t a t i o n . S i m i l a r l y , d c a r r i e r s p a c e s " f o r , t h a t i r r e p

occur i n t h e decompos'ition of t h e . g r o u p a lgebra . Each of

t h e s e i r r e d u c i b l e c a r r i e r spaces i s a minimal l e f t i d e a l

a s s o c i a t e d wi th a s t anda rd , t a b l e a u T: belonging t o t h e d i a - . . .

gram D..

The minimal l e f t i d e a l a s s o c i a t e d wi th T: is genera ted

D by e = e s and spanned by t h e s u b s e t ss

BE = (.e:sls f i x e d , a l l r )

of t h e matric b a s i s . . .

W e have shown how' to c o n s t r u c t ma t r i c bases f o r orthogo- . .

n a l YK- and Serber-adapted irreps. Basis func t ions f o r t h e s e

i r r e p s a r e genera ted by applying t h e ope ra to r& i n B:, f o r

s u i t a b l e D and a r b i t r a r y s t t o a p r i m i t i v e f u n c t i o n , +. These . .

b a s i s funct ions will be or thogonal , s i n c e

CONSTRUCTION OF SPACE FUNCTIONS

Generating. Dual Space Functions

by Means of the Matric Bases

. . sometimes it is convenient to consider a SAAP, not in

the form Qa(NSM) = A[$(N)ea(NSM) 1, ( 7 4 )

.'but in the alternate form

where $ and 0 are duLl space and spin functions. This sub- B B

ject was discussed on.pages 7-9. The spin functions span an

irreducible representation . ,

of.the symmetric group. The space functions span the dual . . -

representation -1 NS) t _ E (PI . . ([P I

When a SAAP is constructed in the form (75) , there is . .

no sum over N1 permutations, as there is in (7'4). Thus it

may be more.convenient'to construct SAAP's from dual space

and spin functions, if . these . can be generated easily. The

construction of dual functions by means of Wigner operators

has been discussed by Kotani -- ct a1. (1955), Harris (19671,

and Sullivan (1968). Goddard (1967a, 1967b, 1968) has made

extensive use of matric basis elements (young's orthogonal

u n i t s ) f o r t h i s purpose.

We now d i scuss how dua l space and s p i n funct ions a r e re-

l a t e d i n terms of Young diagrams. We have shown t h a t s p i n

r ep resen ta t ions a r e l a b e l l e d by diagrams w i t h one o r two

rows. I t t u r n s out t h a t space func t ions t ransform according

t o i r r e d u c i b l e r ep resen ta t ions a s s o c i a t e d with diagrams hav-

ing one o r two columns.

A diagram obtained from another diagram by in terchanging

rows and columns is s a i d t o be conjugate t o it. For example,

F . , i s conjugate t o

, . EEP

while is s e l f -conjugate. ~ h u s space and, . .spin . diagrams

a r e conjugate . This f a c t seems t o have been f i r s t mentioned

by Weyl (1931) , who, gave t h e proof by tensor . . . methods. . The

proof t h a t fol lows uses m u l t i p l i c a t i o n proper . t ies of Young

idempotents, and is more i n keeping wi th t h e rest of our d i s -

cussion. The proof c o n s i s t s of two theorems.

Theorem - 6 :

D L e t I' . be t h e i r r e d u c i b l e r ep resen ta t ion of SN corres-

A ponding t o the Young diagram D . I n p a r t i c u l a r , l e t I' be t h e

antisymmetric r e p r e s e n t a t i o n , corresponding t o t h e diagram

N {l 1 . Then t h e d i r e c t product rA 63 I'D i s t h e i r r e d u c i b l e re- . ,

p r e s e n t a t i o n corresponding t o t h e diagram con.jugate . . t o D .

A D ;. Proof: Since FA i s one-dimensional, I' @ I' 1s i r r e d u c i b l e ,

- n rn W rn

II - Q

4J a m

II 4J X

rA and rD, r e s p e c t i v e l y . . Then fAfD transforms according t o

rU . Thus it must be t h a t ----

f o r a t l e a s t one value of s . -- - -- W e now eva lua te E:'fAfD d i r e c t l y , making use of t h e f a c t

D ' t h a t f D = X E f, f o r some value of t. I n t h e fol lowing, w e t t D " denote t h e row and column groups f o r a s tandard t ab leau Tv

by dlD1'and c:" , r e spec t ive ly . W e use t h e symbols A(&) and v

$(a) t o mean t h e antisymmetrizer and symmetrizer f o r a group

&. Then

where t h e sums run over re&:' and CEC;'. Thus

Now l e t 6 ' be t h e Young diagram conjugate t o D l , and T D ' 9

be t h e s tandard t ab leau conjugate t o T;'. Then

.., c D ' and -

s o t h a t

The underl ined opera to r is t h e n u l l u n l e s s 6 '=D (Rutherford,

- fAfD = 0 unless D ' = D.

Comparing t h i s r e s u l t with ( 7 6 1 , it i s seen t h a t D ' must be

t h e diagram conjugate t o D . This completes t h e proof.

Theorem - 7:

L e t rD and r D ' be r e a l i r r e d u c i b l e r ep resen ta t ions of . .

A S corresponding t o Young diagrams D and D l . L e t I' be t h e N

antisymmetric represen ' ta t ion , corresponding t o t h e diagram

{ l N } = { l , l , ... , l L

Then t h e d i r e c t product I 'D.@ r D ' con ta ins rA only i f D

and D ' a r e conjugate diagrams. If D and D ' a r e conjugate ,

rD 8 r D ' conta ins rA once only. . .

. . D Proof: The number of t imes t h a t rA i s contained i n r @ rD '

D D where x ( P I i s t h e cha rac te r of t h e permutation P i n . I' . ... A D D However, it was shown i n Theorem 6 t h a t @ = r , -

A D D where 6 is conjugate t o D. Thus x ( P ) x ( P ) = x ( P ) . Using

t h e or thogonal i ty proper ty of r e a l simple c h a r a c t e r s ,

.., a(A,D@Dg) = ( N ! ) - ~ I X D ( ~ ) X D ' ( p )

=' 6 ( 6 , ~ ' ) . Q.E.D.

W e have proved t h a t b a s i s func t ions { $ 1 and (€3 1 f o r B B

i r r e d u c i b l e r ep resen ta t ions of S N c a n be used t o c o n s t r u c t

antisymmetric funct ions of t h e form

only i f t h e i r r e p s spanned b y ' { $ 1 and (€3 1 a r e a s soc ia ted B B with conjugate Young diagrams.

Suppose t h a t 4 i s a space p r i m i t i v e funct ion and 8, a

s p i n p r imi t ive . A s discussed i n t h e l a s t chapter , s p i n func-

t i o n s f o r t h e s p i n diagram D can be generated from €3 by oper-

a t i n g on it with . t h e s e t

B: = {e:s 1 s f ixed , a l l r )

of mat r ic b a s i s elements, f o r a r b i t r a r y s . S imi la r ly , space

funct ions f o r t h e diagram 6 conjugate t o D ,can be generated

from 4 by means of t h e opera to r s

... sD = {efUlu f i x e d , a l l t } , u

for arbitrary u .

A space-spin wave function o f the form

w i l l s a t i s f y the Pauli Pr inc iple . For

and, using (51) ,

But conjugate diagrams correspond t o dual i r r e p s , s o

8. and

W e assume t h a t t h e s p i n p r i m i t i v e , 8, i s chosen t o be

an e i g e n f u n c t i o n of sZ. Then w i l l b e an e i g e n f u n c t i o n of

-2 S and sZ , s i n c e i t s s p i n components are gene ra t ed by an

o p e r a t o r b a s i s f o r a s p i n r e p r e s e n t a t i o n . Thus (77) shows

t h e c o n s t r u c t i o n of a SAAP by means of matric b a s i s e lements .

^2 , Simultaneous E igen func t ions of i2 and S

by Matr ix D iagona l i za t ion

S ince s p i n - f r e e atomic Hamil tonians a r e s p h e r i c a l l y

symmetric, t hey c~mmute wi th t h e o r b i t a l angu la r momentum

o p e r a t o r s i2 and iZ. For t h i s r ea son , it i s u & u a l l y conven-

i e n t i n atomic c a l c u l a t i o n s t o u se a t r i a l wave f u n c t i o n

-2 - which i s an e i g e n f u n c t i o n of L , L Z , g 2 , and g z . It i s e a s y

t o ex tend t h e method of pages 59-67 t o cover d r b i t a l angu la r

momentum.

The g e n e r a l CI wave f u n c t i o n i s of t h e fprm . . ( 1 2 ) : ,

The sum over @, i n c l u d e s on ly space produc ts c o n t a i n i n g . . d i f - . .

f e r e n t o r b i t a l s : no two space p roduc t s are r e l a t e d . . . by a permu-

t a t i o n . The wave f u n c t i o n Y(NSMS) i s g e n e r a l i n t h e s ense

t h a t it may c o n t a i n one c o n f i g u r a t i o n o r many, depending on

t h e SAAP's i nc luded . I t ' i s a l r e a d y an e i g e n f u n c t i o n of i2 and h

and w e assume t h a t t h e enb have been c o n s t r u c t e d by one

of t h e methods desc r ibed e a r l i e r . h

Each space product i s an e igen func t ion of LZ: w e w r i t e

@ , ( N % ) . I f t h e summation i n c l u d e s on ly space products w i th

one va lue of %, t hen Y w i l l be an e igen func t ion of iz wi th

t h a t e igenvalue :

I t i s p o s s i b l e t o choose t h e c o e f f i c i e n t s c ( @ , rb ) i n 1T

such a way t h a t Y i s a l s o an e igen func t ion of c2. The pro-

c e d u r e . ' i s s i m i l a r t o t h a t used f o r s p i n func t ions . O n e c a l -

c u l a t e s trhe t2 -ma t r ix over SAAP'S . .

wi th N , ML, S t and MS f i x e d , then f i n d s t h e l i n e a r combina-

t i o n s of SAAP's t h a t d i a g o n a l i z e t h e matrix. ' . ..

^2 The c a l c u l a t i o n of t h e L -matr ix is more complicated

than t h a t of t h e s p i n ma t r ix , b u t , aga in , t h e computation i s

. . g r e a t l y s i m p l i f i e d by t h e space and s p i n convent ions w e have . . . .

in t roduced .

S ince

The second i n t e g r a l is

by equat ion (13) . This leaves only t h e i n t e g r a l

I n terms of one-electron ladder opera to r s ,

t h e sums running over a11 e l e c t r o n s .

W e d e f i n e A r 1 i f i _ ( i ) L + ( j )+ , conta ins

A, , ( i , j ) = the same o r b i t a l s a s $ P ;

(0 otherwise.

I f A n p (it j ) = 1, def ine P;; t o be permutation which con- A A

v e r t s $ i n t o L ( i ) L + ( j ) 4,. Then P -

o r , sin.ce t h e ma t r i ces r e p r e s e n t i n g geminal permutat ions a r e

d i agona l ,

-1 np N S NS I = ( N 1 ) I I A n p ( i , j ) E ( ~ ? ~ ) [ P . . I

1 3 1 J nh, p'B IG1 66 t PB i j

W e have used equat ion (11) . P u t t i n g ( 7 9 ) and (80) i n t o (78) , w e o b t a i n t h e r e s u l t

Appendix E con ta ins a F o r t r a n l i s t i n g f o r a program t h a t

gene ra t e s s imultaneous e igenfunc t ions of L , iz, i2, and iz f o r any e igcnvalucs . Equation (81) i~ used i n t h i s program -

Subrout ine FLSQME c a l c u l a t e A2 t h e L -matrix . . elements

between SAAP's. I n a l l , t h e program c o n t a i n s s i x subprograms.

The i r i n t e r r e l a t i o n s a r e shown i n F igure 9 .

Sample running t i m e s , t o o b t a i n e igenfunc t ions f o r

every e igenvalue L , a r e shown below:

N S MS % Conf igura t ion SAAP s CPU t i m e (set)

I t should be noted t h a t t h e s e a r e "worst-czise1' t i m e s .

The CPU t i m e s i n c l u d e t h e i n t e r n a l p rocess ing of l a r g e

amounts of t e s t i n g ou tpu t . Also, h ighe r v a l u e s of IblL"L( would

reduce t h e number o f o r b i t a l p roducts r e q u i r e d , and s o lower

t h e running t i m e .

Schae f fe r and H a r r i s (1968) have r e p o r t e d a method f o r

c o n s t r u c t i n g L-S e igen func t ions a s l i n e a r combinations of

S l a t e r de t e rminan t s , us ing ma t r ix d i a g o n a l i z a t i o n . They d e a l

on ly w i t h ML=L, MS=S. Running t i m e s a r e comparable t o t hose .

r epor t ed h e r e . Rotenberg ( 1 9 6 3 ) wrote a machine-langu'age pro-

gram f o r t h e I B M 7090 t o gene ra t e L-S e igen func t ions by means

of Lijwdin p r o j e c t i o n o p e r a t o r s . Running t i m e s ' f o r . t h e exam-

p l e s above were n o t r e p o r t e d . Ne i the r of thes'e ' p rocedures , of

course , g e n e r a t e s wave kunct ions a s l i n e a r combinations of

SAAP s . Al l .p rog rams t h a t gene ra t e L-S e igen func t ions r e q u i r e

a g r e a t d e a l . o f s to rage i f they a r e t o d e a l wi th more than ,

say, e i g h t e l e c t r o n s . I n f a c t , t h i s seems t o be t h e ch ie f

l i m i t a t i o n on t h e i r use. The program given i n Appendix E i s

designed t o handle a maximum of e i g h t e l e c t r o n s . A s i m i l a r

program, b u t with d i f f e r e n t s to rage arrangements, is being

developed t o handle a s many as four teen e l e c t r o n s .

START 0 / Given N , L, ML, S t M

/ S' /I //and t h e con f igu ra t ions / -----,-,,--,-,,,-,----J

I I for each c o n f i g ~ r a t i o n : ~

I I I I

1. G e t a l l space I I ......................... I pr6ducts I I (SSQEIG) I I I I I

I I I For given space prod-

I I 1 uct: . I

I I I t I I I

I g ene ra t e a l l . appro- .. I I I J 1 I I I 2 . G e t a l l s p i n I

I p r i a t e PSC1s; I I I I I I e i gen func t ions -a I I I 10 I g e t . a l l s p i n f u n c t i o n s 1 I . . I I 1 I '--*-----------------C;--

Figure '9.. Organiza t ion of program genera t ing simultaneous e igen func t ions of

I I I I

p------------------- I I I

(FLSQME I I I I I I I I

^ 2 1 1 2 1 B . C a l c u l a t e t h e I I I G e t va lue of L - , ~2 I

I I I L -matrix over f r""--"------"------ 7 I I matr ix element I I (SEIGEN) I

I S W ' s I I I

I between two SAAP ' s 1 15 1 I I I

i2, iz, s2, and iz

L--, ----,---- - - - -A I I I tl4 I I I

.................... I I (FPMAT ) I I I 1 I I I I

; G e t a l l s p i n f u n c t i o n s I I

I I I 16 I I

........................ Get ~ a t r i x element I I I (EIGEN)

I I I I

I I of a given permu- I I I I I I

I I 1 C. Diagonalize t h e &!Matrix , diagona l iza t ion : ; I t a t i o n between 1 I I I I I c2-matr ix I yields e igenvec to r s a s I f given geminal s p i n I I I I I

I I I I w e l l a s e igenva lues I I p roduc ts I I I I I L------------------4 I----------- ----------A

APPENDIX A: NOTATION

The symbol "d" - is used to mean "is defined to be".

"Irrep" means "irreducible representation".

The set X ={xl, x2, ..., xn} is sometimes denoted

or by. X = {xilall i} .

The symbol "E" means "belongs to". For exampie, xi€X

When a summation is written without explicit limits,

it should be understood to run over the entire set

to which the index belongs.

Dirac bra-ket notation is used for integrals: . . . .

APPENDIX B: THE SYMMETRIC GROUP

The symmetric group, SN, c o n s i s t s of t h e N! permutations

of N o b j e c t s . Here we consider permutations of e l e c t r o n l a -

b e l s , a s though e l e c t r o n s could be l a b e l l e d .

Let a , b , and c be t h r e e one-electron o r b i t a l s . By

a (1) b (2) c ( 3 ) , .we denote t h a t space product i n which o r b i t a l

a i s occupied by e l e c t r o n 1, b by 2 , and c by 3. The t r ans -

formation t h a t changes a ( l ) b ( 2 ) c ( 3 ) i n t o the new product

a ( 3 ) b (1) c ( 2 ) , say , i s a permutation of a l l t h r e e e l ec t rons :

it rep laces e l e c t r o n 1 by e l e c t r o n ' 3 , 2 by 1,:. and 3 by 2.

One s tandard n o t a t i o n f o r t h i s permutation would be

This i s t h e so-ca l led "two-row" -- no ta t ion . The same permuta-

t i o n i s sometimes w r i t t e n

A- more ' compact no ta t ion f o r t h e same example would be

The symbol " (1 ,3 ,2) I' i s c y c l i c : it reads , s t a r t i n g a t t h e

l e f t , " e l e c t r o n 1 i s replaced by e l e c t r o n 3, 3. by 2 ,. and; 2 by . .

1". I n t h i s n o t a t i o n , t h e permutation (nl ,n2,.. . . ,n ) is k

c a l l e d a "k-cycle" - . Our example was of a 3-cycle. A 2-cycle

permutation interchanges two o b j e c t s , .and i s c a l l e d a

t r a n s p o s i t i o n . The i d e n t i t y , I , i s a one-cycie,

All permutations can be written as products of trans-

positions. For example, (1,3,2) = (1,2) (1,3) = (1,3) (2,3) =

(2,3) (1,2). If a permutation 1s the product of an even number

of transpositions, it is said to be an - even permutation;

otherwise; it is - odd.

The cycle structure of a permutation is a list of the

cycles occurring in it, given in the order of decreasing cy-

cle length. The notation is similar to that for partitions

of the number N. The following are examples from S 4 :

permutation cycle structure

(1,3,4,2): (4)

(lf3) (2,4): '{2,2} . . .

(I,? ,.3) = (1,2 ,.3)'*I ' {3 ,.I}

(1,3) = (1,3) -1.1 {2,12}

I {141 ..

The cycle structures of permutations can be used to . .

classify 'them: it can b e shown that all permutations with . .

the same cycle structure are equivalent. It is also true that

a permutation and its i:nverse have the same cy:cle structure,

Transpositions are their own inverses.

It is convenient to introduce a shorthand.for manipu-

lating permutations. Our first example could be written

-bca = (1,3,2)abc,

in which the orbitals are listed in the order of the occupy-

ing e l e c t r o n l a b e l s . Operations wi th permutations a r e s impli-

f i e d i f , i n t h i s n o t a t i o n , t h e permutation i s re'ad: "move

t h e o r b i t a l i n t h e f i r s t p o s i t i o n t o t h e t h i r d p o s i t i o n ,

t h a t i n t h e t h i r d p o s i t i o n t o t h e second p o s i t i o n , and t h a t

i n t h e second p o s i t i o n t o the f i r s t pos i t ion" . The r e s u l t is

the same as be fo re , b u t we th ink i n terms of o r b i t a l permu-

t a t i o n s . \

Using t h e c y c l i c permutation n o t a t i o n , t h e symmetric

group f o r t h r e e o b j e c t s , S3, c o n s i s t s of t h e following s i x

pe'rmutations: I , 1 2 , 1 3 ( 2 , 3 ) , ( 1 , 2 , 3 ) , and (1 ,3 t2 . ) , . .

where I i s the . i d e n t i t y .. The m u l t i p l i c a t i o n ' t ab le f o r this

group i s

The antisymmetrizer f o r SN i s def ined to . be

where t h e sum runs over t h e whole group and E ( P ) is +1

when P is even and -1 when P is odd. Since the sum extends

over a complete group, the antisymmetrizer is essentially

invariant under left- and right-multiplications by permuta-

tions :

Similarly, AP' = E(P')A . From this it follows that A is idempotent:

We now find the Hermitian adjoints of permutations and

antisymmetrizers. Consider the N-electron integral < P ~ I @ > , where and $ are well-behaved functions. The Hermitian conju-

t gate of P, P', is defined by <P$ I $> = <$ I P $>. On the other

hand, the integral is a number and is unaffected by a permu-

tation of the dummy variables. Thus

-1 Comparison shows that P' = P . It follows that - the antisymmetrizer -- is se'lf-adjoint:,

<A$ I $> = ( N I ) . - ~ I E ( P ) < P J I I O > = (N!) - ~ I E ( P ) <I/J 1 P-'$> P P . .

~ h u s A' = A . s i n c e = A , it fo l lows t h a t .

W e have merely summarized some important r e s u l t s needed

here. For a complete account of this materia l , the reader is

referred t o the book by Hamermesh (1962) .

APPENDIX C : COMPUTER PROGRAM FOR.

SERBER S P I N EIGENFUNCTIONS

I M P L I C I T R E A L * 8 ( F ) ? INTEGER(A-ETG-Z) D IMFNSIOV PS(4)?C(3314)1SEIGV(1?)~FLEIG(1691~

1 F C ( 1 3 r l 7 9 C c c 8 * * 8 * 8 * * * * * * * * * * 8 8 * * * * * 8 8 * 9 : C C 5SOEIG GENERATES SERBER S P I N EIGENFUNCTIONS FOR USE t TN SAAP'S WITH NDO DOURLY-OCCUPIED ORBIT4LS: I . € * C S P I N FUNCYTONS ANTISYYYETRIC I N THE F I R S T MDO GEMINAL C PAIRS- C c C INPUT - c hJP = C NUMBER @F GEMIN4L PAIRS, OR ONE-YALF THE NUMBER C OF ELECTRONS? WHICH I S ASSUYED EVEN. C NO0 = C YUMRER OF DOUBLY-OCCUPIED ORBS I N THE SAAP. r SKFFP = C TnTPL S QUANTUM NUMRER. r MKEEP = C, TOTAL SZ-EIGENVALUE* C c. * * * 8 8 t * * 8 * * * * t t * * * * 4 8 * * * * 8 * C c,

FACT2 = 7 , 0 7 2 0 6 7 8 1 ~ 8 6 5 4 7 5 0 - 0 I TYP = NP + NP TYNP = 2**NP

5 MAGYT = IARStMKEEP) NPS = 0

C SWEFP DECIMAL REPS O F P S ' S 013 40 DPS'! TTTNP T O = DPS - 1

C CONVERT DEC REP TO PS'S PSSUM = 0 DO PI) P = l ? N P P I = 2**tNP-P) PSP = T D / P I P S ( P 1 = PSP IFfo.GT,NDO) GO TO 9 IF(PSP.NE.0) GO TO 4 0

9 TD = TI3 - PSP*PI 10 PSSUM = PSSUM + PSP

C KEEP ONLY PS COMBINATIONS 4PPROPRIATE TO YKEEP IF(PSSUM. LTq M I G M T ) GO T O 40

NPS = NPS 9 1 GET SSO-$IGENFUNCTTONS CORRESPONDING TO MKEEP AYD GIVEN P S g 5 CA'LL S E I G E N ( N P r P S r MKEEP? S E I G V r FLEIGrNPQODr L I IF(NPROneME;O) GO TO 3 5 NPS = NPS - 1 GO TI7 40 WSF = .o DO 37 l S € F = l r N P R O D N7 = ( I S E F - I I*NPROD IF(SEIGV(ISEFI.NE~SKEEPI GO TO 37 N-SF = NSF + P DO 3@ I PROD=?. r NPROD FC(NSF9 TPRQD) = FCEIG(N~+T PROD! CONTINUE IF(NSF. ,EOoOI GO TO 39

OUTPUT A V 4 I L A R L E AT T H I S POINT - NPS =

TNDEX OF THE PA1Q.-SPIN CnMRINATION (PSC). P t ( P 1 =

S P I N O F THE PTH GEMTNAL P b l R , NSF =

NUMRER V F SPIW FIGENFUNCTIONS HAVING TOVPC SPTN SKFEP AND SZ-EIGENVALUE MKEEP, FOR THE PSC WITH INDEX RIPS.

NPROD = NUMRER OF GEYINAL S P I N PRODUCTS (GEMPRODS) FROM WHICH THE S P I N FUNCTIONS FOR THE PSC LABELLED NPS ARE CONSTRUCTFD.

F C ( I 9 J ) = C l l E F F I C I E N T OF THE JTH GEMPROD I N THE I T H S P I N FUNCTION FOR T H I S PSCI

L ( J P P ) = CODE LABEL FOR THE TWO-ELECTRON S P I N FUNCTI'IN OCCUPIED BY THE GEYINAL P A I R ' P ' I N THE JTH GEMPROD, THE CODE PS AS FOLLOWS:

@ O e MEANS ( A B - R A ) / D S O R T ( 2 ) ' 3 ' MEANS ( A A ) ' 2 O MFANS ( A R + S A l / D S B R T ( 2 ) ' 1 ' YEANS ( R B I ,

1 FINSF, E Q e 0 l , NPS=NPS-1 ,

C?NTI NUE RETIJRN END

SURROUTIME S E I G E M ( N P ~ S F ~ X ~ M T F T X ~ S E ~ G V * ~ L E I C , * N P R O D * L ~ T F P L X C I T R E A L * B ( F ) r TNTEGEP(A-E*G-Z ) REAL*8 OSORT DTMENSION S F I X ( ~ B * L ~ B E L Q ~ ) ~ T S ~ ~ ~ ~ T M ( ~ I ~ S ~ P ~ T ~ I T

1 M ( l . 3 , 4 ) ~ L ( ~ ~ ~ ~ ) ~ F L I N T ( ~ ? ~ ~ S F : T G V ( ~ ~ ~ T I D X ( ~ ~ ) ~ 2 F L E I G ( 1 6 9 )

C C * * * * * * * * * * S * * * *

c S E Z G F N RECEIVES PAIR-SPINS ANI? TCT'AL MS. FROM SSQEIG* C AND F I N D S SSQ-EIGENFUNC.TIONS S A T I S F Y I N G THAT DATA. 'C C I Y Q U T REQUIRED - TOTAL MS ( M T F I X ) * P A I R - S P I N S ( S F ' I X C VEC.TOR) r AND THE NUMBER O F GEMINAL P A I R S ' (NPP* ' . C * t * * * * * * * * * * * * * $ C. T H I S SECTTON PRODUCES NPPnI? GEMINAL S P I N PROOUCTS'QF C THF SPECIF IED"TYPE, THE NTH QNE- H A V I N G THE P A I R - c FUNCTION L A ~ E L S (L(NTI\ TI=I,NP)~ P~IR-SPINS C (S(N,flrI=LeNPle AND P A I R - Y S ' S ( Y Q N T I I , I = B * N P O ~

1'00 WPROD = 0 L L I M P I = 4**NP 00 2 0 0 II=~,LLIMPI. TMT = 0 NMBQ = 11-1 TN = NYRR DO 170 I 2 = l , N P P I = : 4 * * ( N P - 1 2 1 . L A R , E L ( I 3 9 = TRIIPY . . TN = T N - L A B E L ~ I ~ ! * P I . .

T S ( P 2 ) = 1 . . I F ( L A 9 E L ( I 2 ) s E O ~ O ) T S ( . 1 2 1 = 0

I F ( V S ( T Z ) . N E e S F I X ( Y Z H GO TO 2 0 0 T M ( 1 2 ) = T S ( T Z ! $ ( L A B E L ( f 2 ) - 2 )

. 170 TYT = T Y T + T M ( I 2 ) l F ( T M T - M T F I X ) 2 0 0 9 1 8 0 1 2 0 0

1 8 0 NOROD = NPROD 4 f 190 1 2 = I , N P

S ( N P R O D T T ~ P = V S J I 2 0 Y (NPROD. I2 ) = T W ( 1 2 1

100 L ( Y O R O D r 1 Z ) = L 6 R E L ( I ? ) 20'0 CflNTlNUE

I F ( NPROO. NE. 0 ) G O ' , T O 299 RETURN

C I - . * * * * * * * * * * * * * C SSQ-MATRI x RETWFEY GEYPROOS* STORED AS THE M A T R I X C. ' INT"n . . . . . . .

C * * * * * * * * * * * * S

C ?9F CnlJNT = C

DO 560 I 2 = l r N P R O D . . . .

0 0 ' 5 6 0 I S = l r I 2 INT. = Q

.COIJNT = COUNT + 1 YD = O DO 420 I 3 = 1 * N P I F ( L ( I P , I ~ ~ , N E ~ L ( I ~ ~ T ~ . ~ P ND=ND+I,

. 1 2 0 CONTINUE I F f N D * N E e O ) GO TO 46Q

C C . , DIAGONAL ELEMENTS . . C

430 00 450 1 3 = 1 , M P ' 1 LRL = L ( I l r l 3 ) I F ~ ' L R L . E O , C I c.0 T n 450

I F ( L R L - L E * 2 . I hlD=ND+1, 4 5 0 CONTINUE . .

I N ? = YTFIX*(MTFIX+~ 1 + 2*NO ' ,

G 0 TC! 540 C C . . OFF-DIAGONAL E ~ ~ M ~ N T S , C

a60 I F ( N D - 2 1 5 4 0 r 5 1 0 y 5 . 4 0 ': : 510 ' DO 5 2 0 ' 1 3 = 2 ~ h ! p .'

I F ( P A R S ~ M ( I I I I ~ ) - M ( I ~ V ? ~ I l t ,GGT., l . ) GO TD 5 2 0 I 3 M t ' = 13 - 1 . , . . . .

. . OQ 51.8' T 4 = P v T3Y1 ' ,

I F ( S ( I ? , 1 3 ) + S ( I ? , T T 4 1 + ~ ( 1 2 , 1 3 ) + ~ ( i 2 1 1 4 ) . ~ ~ GO TD 5 1 9 Ml.34; = M ( , I l . r I 3 ) 4 M ( I l q 1 4 1 T F ( M ~ ~ ~ ~ N E . M ( I ~ . . , . . I ~ ~ + M ( I ~ ~ I ~ ) ) GO TO 5 i 8 . . ,

I F ( I A B S ( M ~ ~ ~ ) , G T ~ ~ 1 GO TO 5 1 8 I N T = 1NT + 2

51 P CQWTIhOJF '20 C.ONT~NUF 5 4 0 FL TNT(COUNT 1 = . l .NT 560 CQNTTNUE

I F (PIPROD-1 ) 970:*000s 61.0 600 F L = I G ( ? ) = ! "OD0

G O T@ 6 2 C . . C . .

C * * * t * * * : * * * . ' * + * c ? k * * *

D I A G O N A L . 1 Z E T!dC :SSQ-MATRIX, GET SSQ-EIGENFUNCTIONS ,-, * * J C * * * * * * * : * * * * * ' a $ *

, . 610 ~ 4 L L F 1 & E N ( F L T N T , r F L F I G q # ~ ~ @ ~ ~ i r I D X ~ ~ ~ 0 ~ - l 4 ) 6 2 C DO 640 T l = l r N ? R O D

NZ = TI*('91+1 I / ? . . fr) = F L I R I ? ( N ~ ) ..

.APPENDIX D: COMPUTER PROGRAM FOR THE EVALUATION OF

COEFFICIENTS I N THE ENERGY MATRIX ELEMENTS

BETWEEN SAAP ' S

SUPPROGRAY 1.

SURROUTINE Y E P ~ ( N ~ L A B L I M ~ L B L ~ L & ~ ~ R A F ~ L O P ~ R O P ~ A ~ B ~ N G P ~ I F G C t T E S L )

I M P L I C I T R E A L * ~ ( F ~ P ) ~ I N T E G E R ( A - E ~ G - O ~ Q Q Z O DIMENSTON L s L ~ ~ ~ ~ ) ~ M O C C ~ ~ ~ ~ O ) ~ O R B ( ~ ~ ~ O I ~ E L O C C ( ~ ~ ~ O ~ v

1 N 0 ( 2 l l I D E N T ( 1 ~ l ~ ~ D 0 R 8 ( 2 ~ 2 ) ~ D E L ( 2 ~ 2 ) ~ T ( 8 ~ ~ E ( 8 ~ 2 ~ ~ 2 I R P ( ~ ~ ~ ~ ~ I S S ( ~ ~ ~ ~ ~ I R S ( ~ ~ ' ~ ~ ~ I S T ( ~ ~ ~ ~ F J ) ~ I D ( ~ ~ ~ ~ ~ 3 B L A N K ( ~ ~ O ~ ~ N G P ( Z ) ~ F G C ~ ~ ~ ~ O D O T F . S L ( ~ ~ Z O ~ ~ ) ~ 4 F S C ( ~ ~ ~ ~ ~ O ) ~ S C ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ D ~ ~ ~ ~ O ~ ~ P ~ R S S E ( % ~ ~ O ~ ~ O I ~ 5 P R S ( l ~ l I ~ P R R S S D ( 1 ~ 1 1 3 O ) t P R R S S ( 1 v L I ~ P R S R S ( l ~ I I ~ 6 D B L S ( 2 )

COMMON RLANK,EPPlRLt A , R t L O C L I L O C L P ~ N L P R O D ~ N R P R O D ~ N P v l T N P ~ T T N P ~ N C V C ~ F A C T ~ F N O R M ~ F P A B ~ F A C T ~ ~ O P B ~ N O C C ~ F S C ~ S L

C C * t * t t * f * & * t 8 8 * * * * * * * 8 ~ * * * * * * *

C C CALCULATES COEFFICIENTS OF ONE- AND TWO-ELECTRON C INTEGRALS OCCURRING I N AN ENERGY MATRIX ELEMENT C RETWEEN TWO SAAPIS CONSTRUCTED FROM ORTHOQORMAL C ORBITALS AND SERBER SPT N FUNCTIONS, C C VERSION Ad 9/70. CONTAINS TESTING OUTPUT. C C C INPUT - C C N = C NUMBER OF ELECTRONS (ASSUMED EVEN1 C L B L ( S I D E v E L 1 = C NUMERICAL LABEL OF ORBITAL CONTAINING ELECTRON C ' E L ' I N L E F T SAAP ( S I D E = 1) OR R I G H T SAAP C ( S I D E = 2 1 0 C L A R L I M = C THE HIGHEST NUMERICAL ORBITAL LABEL USED. C L A F = C INDEX L A B E L L I N G THE L E F T SAAP. C RAF = C INDEX L A B E L L I N G THE R I G H T SAAPB C A = C INDEX L A B E L L I N G THE L E F T S P I N FUNCTION. C R = C INDEX LARELLING THE R I G H T S P I N FUNCTION. C LOP = C INDEX L A B E L L I N G THE LEFT ORBITAL PROOUCTo C ROP = C INDEX LABELCIYG THE RIGHT ORRITAL PRODUCT* C NGP(S1DE) = C NUMBER OF GEMINAL S P I N PRODUCTS I N S P I N FUNC-

C T I O N ON LEFT ( S I D E = l ) OR RIGHT (SIDE=ZIr c TESL(SIDE,GP,MU) = C NUMERICAL LABEL FOR THE TWO-ELECTRON S P I N FUNC-. C T I O N CONTAINING THE GEMINAL P A I R : ( 2 * M U - l r Z * q U ) C I N THE GEMINAL S P I N PROOUCT ' G P ~ , ' A P P E C \ R . I N G I N C THE SERBER S P I N FUNCTION ASSOC1,ATED WITH 'SI 'DE' a

C FGC(S1DETGP) = '

C C O E F F I C I E N T OF THE GEMINAL S P I N PRODUCT 'GP' C I N T H E SERRER FUNCTION INDICATED BY 'SIDE'o C C C. UPDATED VERSIONS OF T H I S PROGRAM MAY RE OBTAINED c THROUGH THE YHEORETICAL.CHEMISTRY GROUP^ IOWA STATE C UNTVERSITYv AMESp IOWA. C C * * * ~ * * * * d r $ * * * * * * * * * * . * * * * * . * $ $ t $ C

FACT2 = 7 . 0 7 1 0 6 7 8 1 1 8 6 5 - 7 5 0 - 0 1 NP = N / 2 NLPROD = N G P ( 1 ) NRPROD = NGP(20 YNP = N TTNP = 2**NP 00 10 S=1,2 DO 10 L = l t L A B L I M

1C N O C C ( S T L ) = 0 W R I T E ( 3 r l l l

1 1 FoRP+lAT(@IgTa* * * * * * * * * * * * * * * * * * a / / / )

WRTTE(39121 N 12 FORVAT( ' M E 0 1 I N P U T s / / / ' * * * * ' / /

1 @ NUMBER OF ELECTRONS- ' p I P / / / ) 00 1 R S I D E = I T Z NOGP = NGP(ST0E) T F ( S I D E e E Q e 2 1 G3 TO 14 W R I T E ( 3 9 1 3 ) ( L B L ( ~ T E L ) ~ E L = I P N )

13 FORMAT(# L E F T SAAP' / / /SXpe O R B I T A L PRODUCT-'r9Xr 1 8 ( I 2 9 2 X ) P

GO Tn 16 1 4 W R I T E ( 3 9 1 5 1 ~ L B L ( 2 9 E L l v C L ~ l r M ) 1!5 FORMAT(' R IGHT S A A P s / / / 5 X T ' ORBITAL PRODUCT-'99x1

1 8 ( I 2 , 2 X l ! 16 W R I T E 1 3 t l 3 D 17 F O R M A T ( / / / 5 X p a SP!N E IGENFUNCTION-" )

DO 1 8 GP=l,NOGP F S C ( S I D E v l r G p 1 = FGC(SIDE,GP) DO 171 M U = l r N P

271 SL(SIDE,GPTMUI = Y E S L ( S I D E T G P T Y U ~ 18 WRITE(3y I .99 ( F G C ( S I O E ~ G P ) ~ ( T € S L ( S I D E ~ G P ~ M U ~ ~ M U = I ~ N P 1 ~ 19 FORMAT(28XvD25a16~6Xp4Il)

W R I T E ( 3 9 1 9 1 9

. . . . , .

170 . .

. - ' 193, F f - ~ R p l B T ( " l @ ~ ' i b : * * * 8 * * $ * : * . 8 * * 9 ~ " * 1 b g / / /

1 0 I N T E R M E D I A T E * * * * * 8 * , f * * . ' / / / / , 2/' O R B I T A L D A T A B L O C K ' 1

C C C MAKE L I S T S O F L E F T AND R I G H T , O R R I Y A L S ( O R B I T

' C O C C U P A N C I E S (NOCCDv AND HIGHEST-NUMBERED E L E C T R O N C L A B E L S O C C U P Y I N G O R B I T A L S I . E L O C C ) c C

S . = 1 20 NORB = 0

DO 50 E L = l v N L = - L B L ( S T E L ) S W = d

' . I F ( N O R B o E Q o O 1 GO T O 40 . . . DO 30 O = l v N O R R I F ( O R B ( S , O ) ~ N E ~ L ) GO TO 30 SW = 1 . .

36' C O N T I N U E I F ( S W ' e E Q o 0 ) GO T O 40 NOCC( S 9 L ) = NOCC:(.S ,L 1 + 1 E L O C C ( S v L D = E L .. . ,

GO TO 50 40 NORR = N O P R . + 1

O R B ( S T N O R R ) = L . . . . . . . . . . . N O C C ( S q L - ) . = P . . .

E L O C C ( S , L ) = E L ,

5 0 C O N T I N U E .NO( S ) = NORB W R I T E ( 3 9 5 1 ) S v N O ( S 1

5 1 F O R M A T ( / / S X ~ @ F O R S I D E ' o f l r q , T H E R E ARE " T I 2 9 '

1 O R B I T A L S - ' / / / 3 0 X q ORB NOCC , E L O C C ~ / ~ ) D B L S ( S,,) = 0 , .

. , 00 52 I=1 ?NOR% '0 t . O R B ( S T I I I F I N n C C ( S s O ) c % Q . 2 j D B L S ( S I = O B L S ( S 1 9 1

52 W R I T E ( . 3 , 5 3 ) 01 N O C C Q S p O l 9 E L O C C ( S , O I 53 F O R M A T ( 3 1 X , H 2 c l O X q I P 9 8 x 9 1 2 ) . .

I F ( S e E Q e 2 ) . GO T O 60 S = 2 GO TO 20 .

6 0 S = B , F N Q R M = O F L Q A T ( Z * * ( D B L S ( l ) + , D B L S ( 2 0 ) ) . FNORH ' = O S Q R f (FN0R.Y) . . F A C T = ( D F L O A T ( N - l ) 1 * F M B R Y R'L - = C R L S ( 1 )

70 D I F F = 0 SP = 3 - 5 . .

DO 106 L = 1 9 L A B L P M . D = N O C C ( S T L I - N O C C ( S P I L ~ .

I F ( D e L F . 0 ) GO TO 100 D I F F = OIFF+D IF (D IFFeLE .21 GO TO 8 0 IDENT(LOP9ROP) = 3 W R I T E ( 3 7 7 1 1

7 1 FORMAT( / / / / / ' ***** D I F F I S GREATER TH4N 2- ' 1 -

1 'MEDI SETS IDENT(LOPIRDP1 = 3 AND QUITS ****st 1 RETURN

80 EL = E L O C C ( S t L I IF(DeGT.1) GO TO 90 OOPB(S,OIFF) = L DEC(S,DIFF1 = EL GO TO 100

Q O OORR(Sq11 = L DORB(Sv2) = L DEC(S11 ) = EL-1 DEL(S,21 = EL

1 0 0 CONTINUE IF(DTFF.EQ.0) GO TO 1 1 0 IF(S.EQo2) GO 80 1 1 0 s = 2 GO TO 70

110 IDENT(LOP1ROP) = O I F F WQ?TE[391111 O I F F

111 FORMAT(// / /SX, ' ** DXFF = IOENT(L0PqROP) = ' , I 1 1 IF(PIFF.EQeO1 GO TO 1 2 0 W R I T E ( 3 q 1 1 2 1

1 1 2 FOPYAT( / / 30X1 ' D IFFERING ORBITALS ' a I 'ELECTRONS I N D? FFERING ORBITALS1/

DO 114 S=lr2 H R I T E ( 3 , 1 1 3 ) S v ( O O R B ( S ~ ? t ~ I = l r O I F F I

3 1 3 FORMAT(20Xl' S I D E ' p X l p l O X , I 2 9 2 X 1 1 2 1 114 W R I T E ( 3 p l l 5 ) ( D E L ( S I I ) ~ I = I , D I F F ) 1 1 5 F O R M A f ( 0 + o , 7 2 X ~ 1 2 ~ 3 X ~ 1 2 ~ 1 2 0 Df l 130 E L = l r N 130 C I S T ( E L 1 = L B L ( l r E L )

IF(DIFF.EQ.0) GO TO 1 5 0 DO 140 E L = l t D I F F

3 4 0 L I S T ( D E L ( ~ P E L ) ~ = D O R B ( 2 r E L ) C c C CONVENTION - PERMUTATION CONVERTING ROP TO LOP IS C ( E ( l , l ) p E ( l l 2 l ) * * * ( E ( N C Y C 9 l ) r E ( N C Y C ~ Z ) ) o C I. El HIGHEST-NUMRERFD CYCLE OPERATES F I R S T ON RDp. C

1 5 C NCYC = 0 W R I T E ( 3 r l 5 1 )

i51 F O R M A T ( / / / / / s pFRMUTATION B L O C K m / / ) 00 1 7 0 C H K I = I y N DO 1 6 0 CHK2=CHKl,N

IF(LBL(29CHKllcNEeLIST(CHK2)1 GO TO 160 I F(CHKLm EQoCHK2 1 GO TO 170 NCYC = NCYC+X E(NCYCv?.O = C H K I E ( N C Y C v 2 9 = CHK2 SAVE = L I S T ( C Y K 1 ) CTST(CHK11 = L I S T ( C H K 2 1 L I S T ( C H K 2 ) = SAVE GO TO 170

160 CONTINUE 170 CONTINUE

EPP = ( - l ) * * N C Y C 00 I 7 0 1 S = l v 2

1701 WRYTE(3 ,172 ) S I ( L O L ( S v E L I , E L ~ l v N ) 1 7 2 FORMAT(20X, ' ORB PRQO " 9 I l p 4 X v 8 ( 2 X ~ I Z I 1

W R I T E ( 3 v 1 7 2 P 1 NCYC 1 7 2 1 F O R M A T ( / / / 2 0 X , ' NCYC = ' 9 1 1 )

I F ( N C Y C e E Q e O 1 GO TO 1 7 3 1 W R I T E ( 3 v l 7 3 ) ((E(CYCrSIvS=l92)vCYC=1vNCYC)

173 FORMAT( / / / 20X I ' PERMUTATION T O A L I G N R I G H T PRO0 W I T H ' P 1' L E F T PROD- ' , 8 ( ' ( ' 9 2 I l , ' I " 1 )

1731 FPAB = FME(O,E) W R I T E ( 3 v 1 7 4 ) AvRvFPAB

174 FORMAT(~0',21X~I~9'p't11~'-EL€MENT OF S P I N REP Y A T R l ' r P ' X FOR ALIGNMENT PERMUTATION I S F P A B = ' 9 0 2 5 . 1 6 1

W R I T F ( 3 , P 7 5 1 1 7 5 F O R M A T ( / / / / / ' P - C O E F F I C I E N T S AND OTHER DATA REQUIRED ' r

I' R Y C I ' / / ) 0 = D I F F 4 1 GC TO ~ 2 0 0 ~ 3 0 0 ~ 4 0 0 9 v D

C LOP = ROP 2 0 0 DO 2 1 0 O = l t N 2 1 0 I D ( L A F v O 1 = 0

NO1 = N O ( 1 I W R I T E ( 3 9 2 1 1 1 NO1

211 F C l P Y A T ( 2 0 X q q I N S T R U C T I O N BLOCK 200 NO1 = ' 9 1 1 / D

00 2 2 0 O P = f v N U l L P = ORR( l ,OP! LOCLP = ELOCCQPvLPO

C I D ( C P F v R A F P I S LABEL OF O R B I T A L INDEXED "OP'e T H I S C L A B E L TS 4LSO CALLED ' L P ' o

I D ( L A F P O P P = L P W R I T E ( 3 v 2 1 2 1 OPTLP

212 F O R M A T ( 3 0 X v g O P = Q v I l , ' , L P = Q R B ( I v O P I = @ , I 2 9 ' - 9

I ' C A L L F P I ' I C P D I L A F 9 O P 9 IS T H t LOEFF OF INTEGRAL ( L P / H ( l I / L P I

P D Q L A F v C P l = F P 1 ( 0 9 L P v L P q N v E v N O B ) W R I T E Q 3 v 2 1 3 1 C A F , O P ~ ? D ( L A F ~ O P I ~ L A F ~ O P P P O ( L A F ~ ~ )

2 1 3 F O R M A T ( 3 5 Y v ' I D ( ' ~ I l p ' q ' v l f v ' 9 = ' v I 2 / 3 5 X v 1' P D ( ' o I l q O v ' v l l , ' I = ' r D 2 5 e l . 6 0

00 220 O=OP9NO1 L = ORR(P*09 LOCL = ELOCCfP9L) IF(L0CLaNEcLOCLPB GO TO 214 IFQNOCCI19LDoEQcZD LOCL = LOCC-1

C P S R S S E ( L A F I O ~ O P B ~ WHERE O.GE.OP9 IS COEFF OF INTEGRAL C (LTLP/G(992)/LqLP)9 IN DIRAC NQTATIONe

214 P S R S S E ( L A F ~ O I O P B = F P ~ ~ O ~ L ~ L P I C ~ L P ~ O ~ E ~ WRITE(3q22PB L A F ~ O P O P ~ P R R S S E ( L A F ~ O ~ O P O

C PRRSSE(LAF9OPqOO IS COEFF Of ( L ~ L P / G ( ~ V ~ ~ / L P P L ) ~ P R R S S E P L A F I O P ~ O I = FP2(0pLpCPpLoCP9lpEO

220 WRITE(3922fP L A F ~ O P ~ O ~ P R R S S E ( L A F V O P ~ O ) 221 FORYAT(35Y9' P R R S S E ( 0 9 1 P ~ s o d 9 1 1 e Q ~ o p I ] 1 ~ a B = '9025eI6)

PFTURN C C LOP AND POP DIFFER R Y ONE ORBITAL9 VIZe C IRS(LAF9PAF) = L IN LOP9 C IRS(RAF9LAFP = R IN Rope

300 L = DClRBcl~19 IRS(LAFvRAF1 = L R = D O R B ( ~ I L ) IRStRAFvLAF) = R LOCL = DEL(l9lD NO% = NO(1B NO2 = NO(2) 1 = 0 06 33@ Ol=lpNOP LP = ORP(l9OP) DO 310 02=lqN02 IF(LP.aEQ.ORB(2902)8 GO TO 320

310 CONY INUE GO TO 330

320 I = P O P C I S V ~ L A F I R A F ~ I I = 1TH ORBITAL COMMON TO LO? AND ROPI C V I Z a p LPO

I S T I L A F P R A F ~ I ! = LB 336 CONTINUE

NT = I C P+'IS(LAFpRbFP = COEFF OF (L/H(lP/RP.

336 PRSBLAFpRAFP = FP1 Bf 9L9R9N9E7N019 00 340 P=lYNI LP = HST(LAF9RAFqI) LOCLP = ELOCC[P9LP) I F ( LOCLPe NEe LOCL B G O TO 337 PF(NOCC(l9LPOeEQo2) LOCLP=LOCCP-1

C PSRSSDBLAF9RAFoIQ = COEFF OF (L9LP/G(l92l/RpLP)s 3 3 7 P R R S S O ( L A F T R A F ~ B B = FPZ(l9LpLPpRpLP9OiEI

C P R R S S D ( R A ~ P L A F ~ I B = COEFF OF (trLP/G4lpZ)/CPpRBs 340 P R R S S D ( R A F I L A F ~ I ) = FP2(19L,LPqR9LP~lqE)

RETURN

C C L O P AND ROP D I F F E R B Y TWO ORRTTALSI V I Z e r C I R R ( L A F I . R A F ) = L AND I S S ( L A F T R A F I = L P I N L O P 9 C I R R ( R A F q L A F l . = R AND I S S ( R A F q L A F ) = R P I N ROP.

4 0 C t P = D O P 8 ( l p 2 ) I S S ( L A F 9 R A F ) = L P W R P T E ( 3 9 4 f O l L A F T R A F ~ C P

L10 F O R M A T ( 3 5 X p ' I S S ( ' 9 . ? P t ' r ' r I 1 9 ' ) = ' 9 1 2 1 R P = D O R B ( 2 9 2 ) I S S t R A F , L A F D = R P W R I T E ( 3 1 4 1 0 ) R A F v L A F p R P L = D O R R ( Z I I ) I R R ( L A F 9 R A F ) = L W R I T E ( 3 9 4 2 0 D L A F v R A F y L

42.0 F O R M A T ( 3 5 X 1 ° I R R ( ' r H f 9 e 9 ' 9 1 1 9 ' ) = ' 9 1 2 ) R = D O R 8 ( 2 9 1 0 I R R ( R A F 9 L A F ) = R W R I T E ( 3 9 4 2 0 ) R A F v L A F o R L O C L = D E L 6 l r l ) L O C L P = O E L ( 1 9 2 P

C P R R S S ( L A F V R A F ) =. C O E F F OF ( L 9 L ? / G ( I 9 2 ) / . R l R P ) . P R R S S ( L P F 9 R A F ) = F P ~ ( ~ ~ L ~ L B ~ R ~ R P I O ~ E ~ '

W R I T E ( 3 9 4 3 0 ) L A F 9 R A F q P R R S S ( t A F q R A F ! 430 F O R M A T ( 3 5 X p Q D R R S S ( ' p I l l . r \ r 9 r 1 1 9 q = " e 0 2 5 e 1 6 )

C PRSRS(LAF: lRAFI = C O E F F OF ( L p L P / G ( l v Z ) / R P , R ) . p R S R S ( L A F 9 R A F ) = - F P ~ ( ~ , L ~ L P ~ R ~ R P T ~ ' ~ E ~ W R P T E ( 3 ~ & 4 0 9 L A F , 9 R A F 9 P R S R S ( L A F 9 R A F 9

440 F O R M A T ( ~ ~ X , Q P R S R S ( . ' 9 1 1 1 ' 9 ' 9 1 1 9 Q I = 0 r D 2 5 . 1 6 1 R E T U R N

SUBPROGRAM 2 e

DOlJBLE P R E C I S T O N F U N C T I O N F P ~ ( D I F F ~ L ~ R ~ N ~ P T N O B D I M P L I C I T REAL*S(F)TINTEGER(A-E~G-Z) D I M E N S I O N O ~ ~ ( 2 9 3 0 0 9 ~ ~ 8 r 2 ~ ~ ~ ~ ~ ~ ~ 2 9 3 ~ ~ ~ s ' ~ ~ ~ ~ ( 1 2 ~ ~ COMMON B L A N K ~ E P P ~ P L ~ A ~ B ~ L O C L 9 L O C L P ~ N L P R O D p N R P R O D ~ ~ ? ~

I - T N P t T T N P t N C Y C 9 F A C T 1 F ~ O R M , F P A B , ~ ~ ~ ~ 2 9 0 d ~ 9 ~ O ~ C C c CALCULATES COEFF FCIENT OF ONE-ELECTRON INTEGRAL C ( L / H B l P / R ' ) c LOP AND R 3 P D I F F E R 811 OTFF O R B I T A L S . c, . -

. . F = P * O @ O . 0 = L I F ( D P F F . E Q o O ) GO ' T O 25

F = 2eooo O = R

25 FPI = OsODO I F ( N O C C ( ~ ~ L ) ~ P J E I ~ ~ G O TO 30 FP1 = 4rODO * FPI(L,L,O,LqPqO)/F

30 DO 50 1=19NOI LP = ORB(lp1V IF(LPeEQoL9 GO T O 50 '

FD = PaOD0 IF(DIFF.EQeO1 GO TO 40 IF(LPeEQoRD F D = 2 o 0 G O

10 FP1 = FP1 9 F ~ * F P I ( L T L P T ~ T L P P P , O ) 50 CONTINUE 60 FP1 = FP1/FACT

RETURN END

SURPROGRAH 3,

DOUBLE PRECISION FUNCTION F P ~ ( ~ I F F T L T ~ P , R ~ R P ~ S W , P D IMPLICIT REAL*R( F) 9 INTEGERi A-EqG-Zl DIMENSION P ( 8 ~ 2 ) T O R B ( 2 ~ 3 0 9 ~ N O C C ( 2 ~ 3 0 1 ~ B L A N K ( ~ ) COMMON B L A N K ~ E P P T P L ~ A ~ B ~ L O C L ~ L O C L P ~ N L P R O D ~ N R P R O D t N P ~

1' TNPtTTNP*NCYCrFACT,FNORM,FPA8~FACf2rOR0~NOCC C C CALCULATES C O E F F OF INTEGRAL ( L I L P / G ( ~ ~ ~ ) / R ~ R P ) IF C SW=09 OR OF I L ~ L P / G ( P T Z ) / R P ~ R ) IF SW=lo (OP AND ROP C DIFFER BY OIFF ORBITALS. C

FC = 1.0DO C

300 IF(LoEQoLP9 GO TO 3 1 0 IF(RoEQ.RpI GO TO 325 GO T O 350

310 IF(NOCC(P~LteEQe29 G O T O 325 FSTR = OeODO GO TO 375

3 2 5 FC = 2,000 IFISWoEQef O G O T O 375

350 FSTR = FC * F P I ( L 9 L P T R ~ R P , P 9 S W ) / FNORM 175 FP2 = FSTR

RETURN END

SUBPROGRAM 4.

DOUqLE PRECISION FUNCTInN F P I ( L T L P ~ R T R P T P ~ S W I IMPLICIT REAL*8(F)r INTEGER(A-ErG-Z) DIMFNSION N O C C ( ~ ~ ~ O ) ~ P ( ~ ~ ~ ) * O R R ( ~ T ~ O ~ ~ B L A N K ~ ~ ~ O I COMYnN B L A N K ~ E P P * P L ~ A ~ R I L O C L ~ L O C L P ~ N L P R O D T N R P R O D T ~ ' J P T

1 T N P ~ T T N P ~ N C Y C ~ F A C T ~ F N O R Y T F P A B ~ F A C T ~ ~ O R B ~ N O C C C. C CALCULATES QUANTITY C (2**(PRS(C9LPl+PRS(R~RPt+PPS(LOPM))) * EPP * C * FHE(SW*P,A,B) C WHERE C PRS(LTLP) = NO. DIFFERENT DOUBLY-OCC ORBITALS REPRE- C SENTED BY L AND LP (IF C=LPr THIS NUMBER IS ZEROIT C PRS(P*RPI = SIMILAR* C PRS(L0PM) = NO. OF DOURLY-OCC ORBITALS I N LEFT OR3 C PROD AFTER L AND LP ARE REMOVED* C EPP = +1 OR -1 IF P IS A N EVEN OR ODD PERMUTATIONT C FME(SHtPrA,B) = (Ao6)-ELEMENT OF SPIN REP MATRIX FOQ C PERM 'P' IF SW=OT OR FOR PERM (IgJl*P IF S W = I (WHERE C I AND J ARE THE ELECTRONS OCCUPYING L AND LP I N LEFT C ORR PROD!, c

WRITE(391) LqLPrRrRPrSH 1 FORMAT(BOX*' FP~-('~12~413~')-'~

FMATEL 5 FPAB IF(SWoEQo0I GO 70 10 FMATEL = FME(lrP1

10 PWR = PL IF(LeEQrLP9 PWR = PL-P PRRP = NOCC(ZrR1 4 NOCC(2rRP) - 2 TF(SrEQoRP1 PRRP-0 FC = DFLOAT((2**~PWR+PRRPll*EPB) FPI = FC * FMATEL * (OFCOAT((-l)**SW)) WRITE(3*19) PWR~PRRP~SWTEPP,FMATEL

19 kOkMAT(80X,s Z*t('rIZ9' +'rf2*' l * (-f)**'qIlq 1 ' * 'rI2~' * 'rDl3o6) RETURN EN0

DOUBLE PRECISION FUNCT'TON FME ( SWTE I I M P L I C I T R E A L * 8 ( F ) 9 INTEGER(A-EtG-Z)

. DIMENSION ~ ~ C ( 2 ~ 1 t 2 0 ) 9 ~ ( 2 t 2 0 ; 4 I t T ( 4 ) ~ 9 4 ( 4 ) t S L ( 8 ) t 1 S E P ( P b I p F ( 8 t 2 1 ~ F C O E F F ( ~ ~ ~ ~ N O C C , ( ~ ~ ~ O ) , ~ O R B ( ~ ~ ~ ~ ! T 2 R L A N K ( l 2 0 ) . .

C O M M O N . 0 L A N K 9 , E P P ~ P L p A ~ R 9 L O C L t L O C L P ~ N L P R O D ~ N R P R O D t N P o 1 TMPTTTNPpNCYCtFACT9FNOPM~FPABtFACT2tORB9NOCCtFSCtL

C C CALCULATES (AIB)-ELEMENT OF S P I N PEP MATRIX FOR PERM C @Pa I F SW=09 FOR PERM (1141*P I F SW=19 WHERE I AND J C ARE THE ELECTRONS OCCUPYING ORBITALS L AND L P I N ?HE C LEFT ORR PROD. C

FME = 00 OD0 DO 400 LPROD=PoNLPROO DO 4 0 0 RPROD=ltNPPROD FPMAT = OoOOO IF[NCYCeNEeO) GO TO 3 0 5 TF(SWeEQo11 GO T O 3 0 5

C WHEN NCYC=Ot PERMUTATION IS TAKEN TO B E THE IDENTITY, C UF SW=O,

DO 301 T 3 = l o N P I F ( L ( ~ ~ L P R O D ~ I ~ ) ~ N E ~ L ( ~ ~ R P R O D ~ I ~ ~ I GO TO 3 7 0

3 0 1 CONTINUE FPMAT = h*ODO GO TC) 370

C 305 0 0 305 S I D E = P t 2

PROD = LPROD I F ( S I O E o E Q o 1 ) GO TO 306 PROD = RPROD

3 0 6 COUNT = 0 C FOR F I X E D S I D E ( L E F T OR RIGHT 1 AND GEYPRODp SWEEP ALL C SFPRODS AND CUNVERT SUITABLE DECLABELS TO BIMLABECS

00 360 I ~ = ~ T T T N P 13M = I 3 - 3. DO 3 1 0 I 4 = l o N P P I = Z**(NP-14) T ( I 4 ) = I 3 M / P I 4 1 IFQL('SIDEpPRODtI4DoNE.OD GO TO 3 0 7 s = o GO TO 3 0 8

307 S = 9 3 0 8 MQ14) = S * ( L ( S I D E t P R O D p I 4 1 - 2 )

T F ( T ( I 4 0 e E Q . 1 ) GO T O 3 1 0 C S K I P SEPROD LABELS WH A R E NOT ASSOCIATED WITH THE C GIVEN GEMPROD

I F ( M ( I 4 ) e N E e O D GO TO 300 310 13M = I 3 P - T ( I 4 ) * P I + P I

C COUNT SEPRODS ASSOCIATED WITH GIVEN GEHPROD COUNT = COUNT + 1

C FOP EACH SEPROD KEPT, GENERATE THE SINGLE-ELECTRON C S P I N FUNCTION LABELS ( S L ' S ) AND THE COEFFICIENT ( F C )

FC = 1 . O D O DO 230 I 4 = 1 9 N P T I 4 = 2 * I 4 T14M3 = TI4 - 1 IF(M(I4) .NE.O) GO TO 315 I F ( T ( I 4 ) . N E e Z l GO TO 325 S L ( T I 4 M 1 1 = 0 S C ( T I 4 1 = 1 FC = FC*FACT2 fF(L(SPOEtPRODt141eEQo2) GO TO 330 FC = -FC GO T n 330

3 1 5 SL (T ILM1D = 1 XF(L(SIDE~PROD914).EQ.3*) GO TO 320

. S L ( T I 4 M I D = 0 320 S L ( V I 4 1 = S L ( T 1 4 M 1 )

GO TO 3 3 0 3 2 5 SL(TP4M1 P = 1

S L ( T I 4 ) = 0 FC = FC*FACfZ

330 CONTTNUE I F ( S I D E e E Q e l 1 GO TO 3 4 0

C I F SIDE = 2 , PERMUTE THE SL'S IF(NCYCe E Q a O l GO TO 3 3 7 DO 336 K = l t NCVC I = NCYC + 1 - K TEMP = S L ( E ( I 9 2 ) ) S L ( F ( I 9 2 ) ) = S L ( E ( I q 1 ) )

336 S L ( E ( I 9 1 ) i = TEVP 337 IF(SWeEQeO1 GO TO 3 4 0

TEMP = SL(LOCL1 SL (L0CC) = SL(LOCLP1 SL(LOCLP) = TEMP

340 SEPROD = O DO 345 I4=1 ,TNP

3 6 5 SEPRCD = SEPROD 4 S L 4 1 4 1 * ( 1 0 * * ( T N P - I 4 H C I F S ? D E = l ( LEFT17 STORE SEPROD A S SEPBCOUNT), FC AS C FCOEFF(C@UNTI

IF (S fDEeEQ.2 ) GO'TO 350 SEP(COUNT1 = SEPPUD FCOEFF(C0UNTI = FC GO TO 360

15@ CONTINUE DO 355 I 4 = l r N S P L

I F ( S E P R O D 9 N E . S E P ( 1 4 1 ) GO TO 3 5 5 FPMAT = FPMAT 4 F C * F C O E F F ( I a )

355 CONTINUE 360 CONTINUE

C I F S I D E = l T STORE NUMBER OF SEPROOS ASSOCIATED WITH C LEFT GEMPROD

I F ( S I D E . F Q c 2 ) GO TO 365 NSPL = COUNT

365 CONTINUE 37C FME = FME + F S C ( l T A T ~ P ~ 0 ~ D * F S C ( 2 T B ~ R P R 0 D l * F P M A T 400 CONTINUE

RETURN END

C C C SAMPLE OAT A CARDS C C

4 0 1 2 3. 4 2 3

0.57735026918962570 OG 1 3 -0. 5 i7 '?5026918962570 00 2 2

0.5773502691896257D 00 3 P 4 Q 3 1 2 4 2 3

0.5773502691896257D 0 0 1 3 -Ce5773502691896257D 00 2 2

0.5773502691 896257Q 00 3 1

APPENDIX E : COMPUTER PROGRAM FOR' GENERATING

SIMULTANEOUS EIGENFUNCTIONS OF SPIN AND ORBITAL

ANGULAR MOMENTA AS LINEAR COMBINATIONS OF SAAP'S

SURPROGRAY 1..

I M P L I C I T REAL*8(FltINTEGEQ(A-EtG-Z) REAL*A nSQRT DIMENSION N Q N ( 8 ) t L ( 2 0 r 8 ) r M L ( 2 C , 8 ) r L L ( 2 0 , 9 ) r

N S P R O D ( ~ ~ ~ ~ ) ~ S L ( ~ ~ ~ ~ ~ ~ ~ ~ F S C ( ~ O L ~ ~ ~ ~ ~ ~ N S E F ~ ~ ~ ~ ~ I ~ 3 M C ( 2 ~ 1 3 t 4 ) t P R S ( 5 t 2 0 I t N O C C ( l l I O t P C ( 5 t 2 O I * N P S ( ~ ~ ~ 3 FLINT(1275)*NLP(S)rLQN(8!*FLEIG(2500ItBLANK(4I, 4 I D x ( ~ ~ I ~ P S C O O E ( ~ ~ ~ ~ ~ ~ M ( ~ I ~ S L O I S K ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ) ~ 5 F S C O S K ( 5 * 1 6 * 1 3 * 1 3 )

COMMON F S C v S L * M S t M L t L r L L t N S P R O D * N t T T N P v P L C C * * # * * * * * * * * * * * S * * * * * * * * * * * * * *

C C LSE2 C C T H I S PRnGRAq CONSTRUCTS SIMULTANEOUS EIGENFUNCTIOhiS C OF L S Q * L Z t SSQ* AND S Z * THESE EIGENFUNCTIONS BEING C LINEAR COMBINATIONS OF SAAP'S CONTAINING A SPACE PRO- C DUCT AND A S P I N EIGENFUNCTIOk!. THE S P I N FUNCTIONS C SPAN A SERBER-TYPE REPQESENTATION OF THE SYVYETRIC C GROUP. C C INPUT I S THE NUMSEP OF ELECTRONS ( N l r TOTAL S ( S T ! * C TOT4L MS ( M S T l r TOTAL L ( C T ) , TOTAL ML ( M L T ) , HIGHEST C LQN OCCURRING ( H I L l r HIGH'S1 NQN OCCURRING ( H I N l r c nNn THE NUMBER OF CONFIGURATIONS (NCONFI. N IS C ASSUMED TO RE FVFN. C C. FnP E A C H CONFIGIJP\AT!CNv THE PROGSAY NEEDS TYE NUMSER C OF ORRITALS REQUIRE@ TO SPECIFY THAT CONFIGURATION C (NMNP I AND THE L I S T OF NQN'S AND LQN' S s

C C NOTE TO THE USER - T H I S DFCK IS DIYENSIOWED TO C HANDLE MOST CASES OF INTEREST WITY UP T O 8 ELECTRONS. C CERTAIN CASES MAY REQUIRE HIGHER DIMENSIONS. THE C ARRAYS QFSCDSKO AND ' S L D I S K * SHOULD BE PLACED I N EX- C TFRrJAL STORAGE, THEY M A Y THEMSELVES BE STORED I N RUCK C CORE* DP THEIR FUNCTIOY M A Y BE PERFORMED 9 Y TAPE OR C DISK. STATEMENTS I N V O L V I N G THESE ARRAYS A R E I N D I - C CATF9 BY 'CTEMP' MARKERS. c C UPD4TFD VERSIONS OF T H I S PROGRAM MAY BE OBTAINED C THROUGH THE THEORETICAL CHEMISTRY GROUP, IOWA STATE C UNIVERSTTY* AYES* fOWb, C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C

C C FOLLOWTNG I S BLOCK TO SELECT ORBITAL PRODUCTS FOR C GIVEN MLTv INDXVIDUAL L Q N g S q SUBJECT TO CONCITIONS C . . THAT DOUBLES ARE L I S T E D F I R S T 1 THA'T .OOUBCE.S ARE .

C LISTED WITH ASCENDI'NG LABELS* AND THAT SINGLES ARE C L I S T E D WITH ASCENOTNG L.ABELS. c

P . E A D ( l v 9 0 1 ) N * S T ~ M S T P L T T M L T ~ H I L ~ H I N*NCONF .9C1 FORMAT ( 8 1 5 !

W R I T E ( 3 9 0 0 2 1 N ~ H I N ~ H I L ~ ST,*MS.Y?LTTMLT'PNCOMF 9 C 2 F0RM4T( / / / / / / / / / / ' PROBLEM 0,ESCRI P T 1 0 ~ - ' / / 1 0 ~ / ' 1 2 ~

1 ELECTRONS* HIGHEST NQN = q ~ I E r s ~ HIGHEST LQN = ' r

2' 1 1 . / / 2 5 X ~ ' S T = @PYPISXV'MST ' ' q 1 2 / 2 5 X v q L T = " , I 2 1 . .

, 3 &XI 'MLT = ' r I 3 , / / 2 6 X o I f T ' CONFIGURAT?ONS1l ' . ND2 = N / 2

TTNP = 2 ** N02 FACT2 = 7 , 0 7 1 0 6 7 8 1 1 e 6 5 4 7 5 0 - @ I H T L p 1 = H I L + 1 '

YM = 2 * H I L + 1 L Y M 4 X = ( H I L + l ) * * ? . L A B L l M = H I N * (HTN-1 I * ( 2 r H I . N - 1 ) / b + ' L M H A X NPL = 0 . . . .

C - . .

C SWEEP C(3NFIGUP.ATTOhlS-. DO 113 . C L V N C O N ~ :' COWNT = 0

. . . . . N R A F = O . . . . . ' . . NLPRTID = C;

R E A Q ( l v 9 0 0 1 N M N P ~ ( N Q N ( ~ U ) ~ L Q N ( M U I T Y U = I T . N M N P I 900 F O R ~ A T ( ? ~ ~ ~ X T ~ O ~ ? I ~ T Z X ! I . . . .

W R I T F ( 3 - r 9 0 3 t Cr (NQN(MU! v ~ ~ ~ ( r n l ~ 9 9 ~ ~ = l ~ ~ Y ~ ~ ) 9@?, F O R M A T ( I / / / / / / / / / Q l * t 4 * & * * 8 8 9 t * * * * * * * * g * * * * * * e l / ,

1 ' CORIFIGURATION . 'TI~T'- T ~ 4 ( 2 ~ ~ S i . ~ ) ! WRI.TE:( 3v '?04)

Q Q ~ .FORHA:T( / / / / / I N P = N - NMNP .

N P P ~ = NP 4 1 . NPP2 = N P P l 4 P '

NT? => NMNP - YP L TM =' HM ** NMNP . ., . .

c C. SWFEP.DECIMAL L A R E L S FQR SP,ACE PSODUCTS.. KEEPVNG C ONLY THOSE WHIC~H ':SUIT THE. I N P U T DATA FOR THE' G I V E N

. . C CONFIGURATION

DO 4 MU=lrNMNP 6 H.( MU) = -. LQN(MU! . . .

U = l v = 2 : . . . . e NLPROF = NLPRCID 9 -.I . . DO 8 O!?R=l r L A R L I M " , . .

. . . . . . . . .

8 NOCC(0RR) = 0 YSIJM = 0 03 ? @ M U = l r N Y N c EM = M(MU) ML(NLPROD,MUI = EM I F ( I A R S ( E M l e G T a L Q N ( M U ) I GO TO 40 I N f = t I F ( M U m L F s N P ) I N C = 2

10 MSUM = MSUM + ?NC*EM IF(MSUMeNE+YLT! GO TO 4 0 DO 30 MU=Z 9NMNP FN = NQN(MU) EL = LQRI(MU1 C(NLPRODrMU1 = FL LAREL = E N * ( E N - 1 1 * ( 2 * E N - I l l 6 + E L * ( E L + 1 ) + 1 I + YL 1 NLPROO, MU

L L ( N L P R O D I Y U ) = LABEL I F ( Y U e E Q e 1 ) GO Tf' 20 IF(YUaEQ.NPP?) GO TO 2 0 M U M I = MU - 1 I F ( L L ( N L P R 0 0 ~ ~ U 1 ~ L T Q L L ( N C P R O O ~ Y U M l ) I GO TO 40

; r ~ ) IN(, = I T F ( M U * L E * N P ) TNC = 2 NL = NOCC(LAR€LI NL = NL 4 I N C TFQNLaGTe21 GP TO 40 NOCC(LAI3ELD = NL

3 C CCINTINUF GO TO 5 0

40 NLPROD = NLPROD-9 50 l F ( Y ( U ) e E O e l Q N ( U ) l GO TO 501

M ( I J ) = M(U1 4 1 GO TO 6

5 C l U = U + 1 I F ( U e E Q e V D GO TO 502 I F Q M ( U ! e E Q s L Q N ( U I l GP TO 501 W = U GO TO 504

502 PF(MQV).NE,LQN(V)) G f l T O 503 I F (V. EQm NMNP I GO TO 506 V = V + l GO TO 5 0 2

5(?? W = v 5c4 U = 1

W Y I = W - 1 DO 5 0 5 M U = l , W Y ?

5 0 5 Y(MU) = -LON(MU) M ( W ) = M(W1 9 I GO Tr? 6

506 N L P ( C I = NLPROD

TF(NLPROD.NE,O) GO TO 5 2 W R Y T E [ 3 r l 0 1 4 )

1014 F O R M A T ( / / / / ' THERE ARE NO SUITABLE ORBITAL PROOUCTS -' ! 'DATA APE INCONSISTENT' )

GO TO 1 1 3 52 MULIM = NMNP

C C FOR FACH CONFTGUP.ATPON* 'SWEEP ALL SUITABLE ORBITAL.

PRODUCTS* CONVFRT THEM TO STANDARD FORM DO 95 I = l r N L P k O D S H I F T SLOCK NYNP = MULIY NP = N - NMNP NTP = NMNP - MP IF(NPP2.GT.NMNPl GO TO 71 DO 68 YU=NPPZvMUL!M IF (MU*GToNMNP) GO TO 71 MUMI = MU - f I F ( L L ( I * V U O e N E o L L Q I * M U M P I l GO TO 6 8 NMNP = NMNP - 1 S A V E l = L L ( 1 vMUMI 5 4 V E 2 = NP + 1 IF(NP,EQ,OD GO f 9 56 DO 53 NU= lqNP TF(LL( I *NU) .LE.SAVEP) GO TO 53 SAVE2 = NU GO TO 56 CONTINUE X1LPM = MUM1 - 5 A V E 2 I F ( X 1 L I ~ o E Q e O l GO TO 6 2 DO 59 X I = ~ V X I L T M OM = ,MU - X T CL(IoOM1 = LL(!rOY-19 L L ( I SAVE^) = S A V E l I F ( N M N P e L T I M U 1 GO TO 67 00 6 5 XT=MUvNMNP L L ( 1 9 x 1 = L L ( I e X l + 1 P NP=N-NMNP NTP = NMNP - NP CONTINUE P R S ( C v I 1 = NP END S H I F T BLOCK '

GO TO 8 1

'UNPACK MU SUBSCR TPT I F ~ N P ~ E Q ~ O I GO Td 81 00 80, MU=lvNMNP '

MUMI = MU - 1 .".

J = O . . K '= NMNP - M U M I .

O = Y l J Y l - NTP TF(DoGTo0) J=D MEW = N-MUMI-J NEW1 = NEW - 1 LABEL = L L ( 1 t K ) L L ( f q N E W ) = LABEL L L ( T * N E W I ) = LABEL CHYYl = 0 FN = 1 CHK = EN*( EN41 ) * ( 2 *EN+11 /6 TF(CHK*GE*CABELI GO TO 725 CHKMl = CHK EN = EN + 1 GO TO 72 NQN(NEW1 = EN NQY(NEWI.1 = EN LAREL = LAREL - CHKMI EL = 0 CHK = ( E L + l ) * * 2 IF(CHK.GEeLABELl GO TO 735 EL = EL+1 GO TO 73 L ( I 9 N F W ) = E L L ( T * N E W I ) = EL E M = LABEL - EL* (EL+19 - 1 M L ( T * N E W ) = E M M L ( I r N E W 1 1 = EM CONT I NUE END MU EXPANSION BLOCK

GET SPIN EIGENFUNCTTONS TO GO WITH ITH S P A C E PRODUCT FOR CONFIGURATION C CLTM =: C . NPL = NPL + 1 P L ( C 9 I I NPL SW = 0 PF(I.NE.1) GO TO 8 1 IF(CLIMsEQ.3.1 GO TO 88 CLTY = C - 9. 00 87 CC.=lrCLPM' J L I M ' = NLP(CC) IF(C.C*EQ.C) 4.LIM = I-l DO 86 J = X I S L I M IF(SW-EQ.19 GO TO 84 1 F t P R S Q C q . I )eNEePRS(CC,4) ) GO T O 86 P L ( C 9 I ) = P L ( C C 3 d ) NPL = NPL - 1 SW = 2 . . CONF = CC PROD = J

GO TO 8 8 0 34 IF(PRS(CCvJI.GE.PRS(Cv1 I) GO T@ 86

CONF = CC PROD = J GO TO 880

@t CONTINUE e7 CONTINUE

IF(SW.NEoC) GO TO 88 SW = 1 GO T@ 83

8 P CALL S S Q E I G ( N ~ ~ ~ N P ~ S V ~ M S T O C T P ~ M P L ~ N P R S P ~ N S P R O O ~ N S E F T 1 PSCODFvFCETG. FSCDSKv S L O I S U I

N P S ( N P L I = NPPSP GO TG 84

C NPRSP W I L L BE ZERO I F F THEPE ARE NO SUITABLE S P I N C FICENFUNCTIONS Tn GO W I T H THE CURPENT ORB PROD

8Pt7 ORGL = PL(CONF.PROD) P L I = P L ( C . 1 ) NPRSP = NPS(PRGL1 %DO = PRS(C.1) CHV = 2 * * ( N 0 2 - N 0 0 ) COUNT = 0 00 9 A 1 PSC=?,MPRSP IF(SW.EQ.2) GO TO 88GC I F ( PSCODF(PRGC.PSCI. GE* CHK) GO TO 881

8 9 C G COUNT = COUNT + 1 NSF = NSFF(PRGL*QSC! NSP = NSPROD(PPGL. PSCD NSPROD(NPL~CO1JNT 1 = NSP NSEF(NPCvC0UNTI = NSF

CTFPP DO !?A02 I S P = l r N S P Dt l 8801 S E F = l * N S F

8R02 F S C D S K ~ P L I ~ C O U M T ~ S E F ~ I S P ~ = FSCDSK(PRGL~PSCISEFTISP! 90 8802 PR=foNDZ

@802 S L D I S K ( P L 1 v C O U N T o I S P I P R s = SLDISK(PRGC*PSC, ISPoPR) ' C f EMP

8P1 CONTINUE f F ( S W . E O . 2 0 GO T O 8 9 N P S ( N P L I = COUNT

C C FORM LSQ-MATRIX (UPPER TRIANGLE) FOR CURRENT CONF1.G

R 9 RPL = PLfC.1,) NUPS = N P S f R P L ) IF (NEPS. E Q e O ) GO TO 95 DQ 94 RPS=lvNRPS '

NSP = N S P R O D ( P P L ~ R P S ) , . NRSEF = N S E F f R P L v R P S I

00 93 PSEF=fvNRSEF NRkF = NRAF + I.

sw = d CTEYP . . .

DO 'eQ0L T S P = l r N S P F S C ( ~ V R S E F P I S P I = F S C D S K [ R ? L I ~ . P S ~ R S E F ~ I S P I 00 8 9 0 1 P P = l r N D 7

9901 S t t 2 , I S P * P R ) = S L D I S K ( R P L * R P S , I S P * P R ) CTEMP

MU = ~ R i ~ ~ ( 3 ~ 8 9 1 I NRAF ,.

801' FORMAT(/ / / /SX9O SAAP N U M B E R . ' , I 3 / / / / 2 0 X + * S P A C E ' r 1 'PRODUCT1*48X* 'SPIN E1GENFUNCTIONq/ /23X* 'N L f 4 ' 9

2 ~ O X V ' C O E F F I C I E N T ' ~ ~ S - X ~ ~ G E N I N A L S P I N PRODUCT'///.) 892 'IF~MU~GT.NO GO TO 894

EL = L ( I * H U ) E M = M L I I r M U ) E N = NQN(MUB WRTTE(39893) ENqFLvEM

8 9 3 F O R M A T ( ' + ' r 2 2 X v 3 4 1 2 0 ) S W = 1

844 I F ( M U e G T e NSP) GO T O 896. WRITE(^*^^^) FSC,(2rRSEFrMUBt(SL(2rMUrPR)rPR=P,ND2)

945 FORMAT( q + c ,65X~f!19, l t , l6X97T1) SW = 7.

996 I F ( SW. EQ. 0 ) GO 1.0 8 9 8 , . .

SW = 0 WRYTE(3q8Q71

,807 FORMAT( / ) MtJ = MU+f . .

GQ T r 3 8 9 2 . . . . . 9 9 8 W R I T E ( 3 r 8 9 9 1 899 F O R M A T ( / / / [ / )

NLAF = C! . ..

or! 91 ' L~ROD=I o ~ ~ ~ ~ ~ @

L P L = PL(C9LPROD) NLPS = N P S ( L P L ) Y F ( N L P S I EQ.0 I .GO, TO 91. DO 90 L P S = l v N L P S . . NLSP = NSPROD(LPL*LPSB NLSEF ' = PISEF.(LPL*LPSI . ,

DO 9 0 i L S E F = L vNLSEF CTEMP i

DO 8991 I .SP=1 ,NLSP F S C ( P r L S E F r I S P 1 . = FSCDSK(LPLrLPS*LSEF,ISP)

. DO 8991 PR=1* ND2 . '.,8991. S L ( l v T S P * P R ! = S L D X S K ( L P L r L P S 9 P S P r P R I . '

CT'EYP N L 4 F = NLAF + f COWNT . = COWNT + 1.. F C T N T ~ C O W N T I = F L S Q M E ~ C . ~ C P R O D ~ L P S , L S E F , ~ , I.RPS,RSEF,

1 N r M L T r P R S r L A R L I M I .

IF(NLAFaEQ.NRAF) GO TO Q3 90 CONTINUE 91 CONTINUE

C 93 CONTINUE 9L CONTINUE 9 5 CONTINUE

C, ( 9 5 I S END OF I -LOOP) C c C DIAGONALIZF THE LSQ MATRIX

TF(NRAF.GT.1 GO TO 102 F L F I G ( 1 ) = 1 e O D C GO TO 1 0 4

101 CALL E I G E N ( F L I N ~ , F L E I G I N R A F T ~ ~ I D X ~ ~ ~ O D ~ ~ ~ ) 1 C 4 NLEF = 0

W R I T E ( 3 r 1 0 1 2 1 CTLTTMLTI ST-YST 1012 FORMAT('1CONFIGURATION ' v I 1 9 ' e L I S T OF SIMULTAWEOUS',

1 ' EIGENFUh'CTIONS OF LSQ. L Z r SSOT AND SZT W I T H 9 / 2 I 8 X r ' L T = '11197 .X~ ' 'MLT = ' r 1 4 9 7 X ~ ' S T = ' r I Z r 7 X q 3 'MST = ' o I 1 / / / / / 4 X , ' E F N O ~ ' ~ ~ ~ X I ' C O E F F I C I € N T I P 4 15X , 'SAAP ' / / / )

DO 1 2 0 I ! = l r N R A F N l = I L * ( 1 1 + 1 8 / 2 N2 = 11-1 )*NRAF f D = F C I N T ( N t 1 f D = ( ~ S Q R T ( Z ~ 0 0 0 * 4 e O D O * F D ~ ~ I ~ O D O ~ / 2 e O O O L F I G V = FD F D = FO-LEIGV IF(FD.GT.0,5) L E I G V = L E I G V + 1 I F ( L E 1 G V e N E a L T ) GO T O 3.10 NLFF = NCFF + I U R I T F ( 3 ~ i 0 1 3 l N L E F T ( F L E I G ( N ~ + I ~ ) ~ I ~ ~ I ~ = ~ T ~ J R A F )

1013 f~RMAT(//////'0'r4Xr~2~5~(~7x~o23.16,10X~'sAAp('rI~q 1 " 1 ' / 7 X ) !

W R I T E ( 3 r 1 0 1 7 I L E I G V 1017 F ~ ~ M ~ V ( ' + ' V ~ ~ X ~ ' ( C O R R E S P ~ TfY L-EIGENVALUE OF ' 9 1 2 9 ' ) ' l

11 0 CONTINUE IF(NLEF,NE.O) GO TO 113 WRITF (3 .1101 ) CT

1?C1 FORYAT( / / / / / / 'OTHE EIGENVALUE LT = ' 9 1 2 9 1 'DOES NOT OCCUR FOR T H I S CONFIGURATION')

c 113 CONTINUE

C ( 1 1 3 I S END OF C-LOOP) c C

RETURN END

SURPROGRAM 2.

SURROUTINE SSQEIG(NP*NDOPSKEEP(MKEEP*C*ORRPRDIPRGLT 1 NPS*MGP~NSEF~PSCOOEPFLEIG*FSCOSKPSLDTSK)

I M P L I C I T R E A L * 8 ( F I T INTEGER(A-EoG-Zl DIMENSION P S ( ~ ) ~ N G P ( S T ~ ~ ) T C ( ~ T I ~ T ~ I V M ( ~ V ~ ~ V ~ ~ T

? S E I G V ( ? ~ D ~ F L F I G ( ~ ~ O O ~ T N F E F ( ~ ~ ~ ~ ~ T F C ( ~ ~ I ~ T ~ ~ ~ T 2 P S C O D E ( ~ ~ ~ ~ ) T F S C D S K ( ~ T ~ ~ T P ~ * ~ ~ ! ~ S L D ~ S K ( ~ ~ I ~ T ~ ~ V ~ I

COMMON F C T L T M c C C SSQEIG C I N n S EIGENFUNCTIONS OF S**2 AND SE FOR THE C GIVEN P A I R I N G LABEL 'PRGL'o C C

FACT2 = 7.0710678118654759-01 TNP = NP + NP TTND = 2**NP

5 YAGMT = IABS(MKEEP1 NPS = 0

C SWFEF DECIMAL REPS OF PS'S 00 40 D P S t l v T T N P D p S M l = DPS - 1 T 9 = DPSMl

f CONVFRT DEC REP TO PS'S PSSUM = 0 0111 10 P=l. TNP P I = 2 ** ( N P - P l PSP = T D / P I 1FQP.GT.NOO) GO TO 9 I F ( P S P e N E e 0 ) GO TO 40

9 P S ( P ) = PSP TD =, TD - PSP*PI

10 PSSUM = PSSUM + p$P C KEEP ONLY PS COMBINATIONS APPROPRIATE TO MKEEP

IF(PSSUMa LTe MbGMT) GO P O 40 NPS = NPS + 1

C GET S5Q-EIGENFUNCTIONS CORRESPONDING TO SZ-EIGENVPLUE C OMKEEP', AND GTVFN PS'S

CALL S E I G E N ( N P T N P S T P S T Y K E E P T S E I G V ~ F L E I G P N P R O D T ~ R G L ~ C N G P ( P R G L r I 1 I S NOe OF GEMPRODS ASSOCIATED WITH P S c C ' 1 ' AND P A I R I N G LAREL @PRGL'

N t P ( P R G L * N P S I = NPROD PSCODE(PRGL*NPS) = D P S Y l IF(NPROD.NE.0) GO TO 1 5 NPS = NPS - 1 GO TO 40

C NSEF(PRC,L,I) I S NO, T)F SSQ-EIGENFUNCTIONS WITH GIVEN C EIGENVALUE WHICH ARISE FROM TVH PSC FOR PRGL

1 5 NSF = 0' DCI 37 ISEF=I *NPROO N2 = ( I S E F - 1 )*NPROD IF( SEIGV( ISEF).NE. S K E E p I GO TO 37 NSF = NSF + 1 DO 9 0 IPROO=lrNPROD

CTEYP 3c FSCDSKI P R G L ~ N P S *NSF* IPROD) = FLEIG( NZ+IPROD,I

(.TEMP . .37 CONTINUF

NSEF(PRGC*NPS) = NSF IF(NSF.'EQ.O) GO TO 39

CTEMP D9 3 8 IPROD=lrNPROD DO 38 P = l rNP

38 SLDISK(PRGL.NPS,TPROOqP) = L(1* !PRODvP! . .

C.TFMP 39 W R I T E ( 3 r 4 1 0 1 NSF

410 FORMAT(~OOXI' AND NSEF = , ' r T 2 ) T F l NSFs E O . 0 ) YPS=NPS-1

4C CONTINUE . .

RETUP.N . .

END . . . .

, SUBPROGRAM 3.

SUBROUTINE S E I G E N ( N P ~ N P S ~ S ~ I X t M T F I X ~ S E I G V t F L € I G t 1 NPRODvPRGL)

I M P L I C I T R E A L * 8 ( F l * INTEGER(A-EtG-Z) REAL*8 OSQRT DIMENSION S F T X ( ~ ) ~ L A B E L ( ~ ) ~ T S ( ~ ~ ~ T M ( ~ ~ T S ( I ~ ~ ~ ~ *

1 M ( 2 r 4 3 ~ 4 ) ~ L ( 2 ~ 1 3 ~ 4 ) * F L I N T ( 9 1 ~ v S E I G V ( I 3 ~ ~ I D X ( I 3 ) ~ 2 F B L A N K ( 3 3 8 ) r F L E I G ( 2 5 0 0 )

COMMON FBLONKqLqM C C * s s * q * * * * s * * * s * C SEIGEN PECEIVES PAIR-SPINS AND TOTAL MS FROM SSQEIGt C AND FINDS SSQ-ETGENFUNCTIONS S A T I S F Y I N G THAT DATA. C C 1hSPU"PEQIJIRED a TOTAL M S ( M V F 1 X ) r PAIR-SPINS (SPIX C VECTOR)r N / 2 (WPI. c * * * * * * * * * + * * * * + C C T H I S SECTION PRODUCES NPRnD PRODUCT FUNCTIONS OF THE C S P E C I F I E D TYPE* THE NTH ONE HAVING THE PAIR-FUNCTION C LABELS ( L I P R G L ~ N P S * N * I I ~ I = l ~ N P ~ * PAIR-SPINS

C ( S ( N , I l r I = l , N P I , AND P A I R - H S @ S (M(NPS*NI I ) , I=PINP) . 1 0 C NPPOD = 0

L L I M P l = 4**NP DO 2 0 0 I I = ~ V L L I M P I . TMT = 0 NMRR = 11-1 TN = NYBP DO 170 T2= l ,NP P I = 4 * * ( N P - I 2 1 L A R E L ( I 2 ) = T N / P I TV = TN - L A B E L ( I 2 l * P I T S ( I 2 1 = P l F ( L A R F L ( 1 2 ) o E O . O I T S ( I ' ) = C I F ( T S ( I 2 I . N E e S F I X ( I 2 ) ) GO TO 2CC TM( 1 2 ) = T S ( I 2 ) t ( L A B E L ( 1 2 1 - 2 1

1 7 0 T Y T = TMT + f M ( I 2 O I F ( T M T - M T F I X I 200,180,200

180 NPROD = NPROD + f DO 190 ?2=1,NP S ( N P R O D * I 2 1 = T S ( I 2 9 M ~ P I Y P R O D I I ~ ) = TM(128

190 L ( l v N P R O D * I 2 1 - L A R E L ( ? 2 ) 200 CONTINUE

IF (MPR0DeNEeOI GO TO 299 RETURN

C C * * * * * * * * S * * * * C SSB-MATRIX RETWEEN PRODS OF S P E C I F I E D TYPE. STORED A S C THF MATRIX ' I W T ' + C * B * t * * * S f * S * t C

as? COUNT = 0 DO 560 l 2 = 1 * N P R O D DO 5 6 0 I 1 = 1 , 1 2 I N T = 0 COUNT = COUNT + 1 RID = 0 DO 4 2 0 X 3 = l r N P I F ( L ( l r ~ l ~ I 3 ~ e N E . L ( l ~ I 2 , I 3 ) ) ND=MD+f 0

420 CONTINUE IF(ND,NE.O) GO TO 4 6 0

C C DIAGONAL ELEMENTS C

4 3 0 no 4 5 0 1 3 = l , N P L H C = L ( l r P l r f 3 ) I F ( L R L e E Q o O ! GO TO 450 I F ( CRL, LEe 2 I ND=ND+l

4 5 0 CONTINUE I N 1 = M T F I X * ( M T F I X + l ) + 2*ND

- C OFF-DIAGONAL ELEMENTS C

4 6 0 I F ( N D - 2 1 5 4 0 , 5 1 0 ~ 5 4 0 510 DO 5 2 0 1 3 = 2 t N P

. I F ( I A B S ( M ( 1 ~ 1 1 ~ 1 3 ) - M ( l r 1 [ 2 r f 3 ) 1eGTo1 ) GO TO 520 1341 = 13 - 1 DO 5 1 8 I ~ = l t 1 3 M l I F ( S ( I l t T 3 ) + S ( I l r 1 4 ~ + 5 ( Y 2 t I 3 ~ + S ( ? 2 ~ 1 4 1 , N E 4 GO TO 5 1 8 M I 1 4 = Y ( l t I l , I 3 1 + M ( l t I l t l 4 1 I F ( Y ~ ~ ~ ~ M E ~ M ( ~ . I Z ~ I ~ ) + M ( ~ ~ I ~ P I ~ I ) GQ TO 5 1 8 I F ( IARS(M1341 .GT . l ) GO TO 5 1 8 I N T = I N T + 2

53 8 CONTINUE 5 2 0 CONTINUE F 4 0 FL INT(CCUNT) = I N T 5 6 0 CONTINUE

IF (NPP0D-1 ) 9 7 0 r 6 0 C , 6 1 0 6 0 0 F L E I G ( 1 I = 1.000

GO TO 6 2 0 C C $ * * * . * * * * * * * * * I p : f * * C OIAGONALIZE THE SSQ-MATRIX, GET SSB-ETGENFUNCTIONS C * * * * * * * * * * * * * * * * S

6i 0 C4LL E I G E N ( FLTNTvFLEIG, NPRODvl, I D X t 1000-14) 620 00 640 I l = I ,NPROD

N l = 1 1 * ( 1 1 + 1 I l 2 FD = F L I N T ( N 1 ) FD = .(DSQRT( leOD0+4e OT!C*FD)-1.000) /2.0D0 S F T G V ( I I 1 = FO FD = FD - S E I G V ( I 1 ) IF(FD,GT.O,SDOI SEIGV(IZI = SEIGV~III + I

btO CONTINUE R E T U ~ N

9 7 0 STOP END

SUSPROGRAM 4,.

FUNCTION FLSQME(C.;!~I , LPS tLSEF tCJ , J,RPS,R:S.EFrNtMLT,PRSt I. L A B L I M )

I M P L I C I T REAL*^( ~i 1NTEGER.QA-EtG-Z) REAL*8 DSORT DIMENSION P R S ( ~ ~ ~ O ~ ~ M ( ~ O ~ ~ I ~ L O C C ( ~ ~ ~ I P R ~ C C ( ~ P P ~ ~

3 C A B E L ( ~ ) ~ L B L ( ~ O ; ' ~ ) ~ E ( ~ ~ ~ ) ~ B L A N K ( ~ O ~ ~ ~ F S C ( ~ ~ I ~ ~ I ~ ~ ,

2 P L ( 5 ~ 2 0 ! ~ N S P R O D Q 5 ~ 1 6 1 ~ ~ ( 2 0 , 8 ) . '

COMMON F S C P B L A N K ~ M , L ~ L B L , N S P R O D , ? N P ~ T T N P ~ P L . C C FLSQME CALCULATES THE INTEGRAL OVER L**2 BETWEEN C TWO SAAP'S, C

FLSQME = O.ODO NO2 = N f 2 "LC = P L ( C I t I 1 PLR = PL (CJ ,J1 NLPROD = NSpROD(PLL,LPS1 NRPROD = NSPPOD(PLRtRPS? TF(CImNE*CJI GO TO 1C I F ( I o N E o J ) GO YO 1@ I F ( LPS. NE, RPSI GO TO 10 IF(LSEF.NEeRSEF 1 GO TO 10

c C D I AGONAL-TFRM CONTRI BUT I O N

FLSME = HLT*( MLT+?. l 20 00 70 N U = l t N

MNU = M( I ,NUI LNU C(TINU) IF~MNU. EQ*LNU) GO TO 70 DO 6 8 MU=1 P N DO 11 CHKI=L,LARLIM LOCC(CHK1) = 0

21 ROCC(CHK1 l = 0 DO 12 CHKl= l ,N L L = LRL(1,CHKI. I RL = LRL(JtCHK1-1 LAREL (CHK1 l = L L LOCC(LL1 = LOCC(LL1 + 1

1 2 ROCC(RL1 = ROCC(RL1 + 1 MMU = M( IgMU1 LMU = L ( I t M U B IF(MU,NErNUD GO TO 1 5 MMU =. MMU 9 f GO TO 2 0

9 5 IF(MPrllJ.EQo-LMUI GO TO h e C C PPPLY OPERATOR L-QMU)L+(NU) TO LEFT ORBITAL PRODUCT c ( 1 )

LSLMU = LBL(1qMUD LRLNU = L H C ( l r N U 1 LAREL(MU1 = LRLMU - 1 L 4 R E L t N U l = LBLNU 4 1 LOCC(LBLMU1 = LOCC(LBLMUl - 1 LOCC(CRLNU1 = LOCC(LBLNU) - 1 LOCC(LAREt(MUI9 = LOCC(LAREL(MUI9 + 1 LOCC(LABEL(NU)) = LOCC(LABEL(NU!! + 1

C DOES ( L - ( M U ) L + ( N U I * I O CONTAIN THE SAME ORBITALS AS C THE RIGHT ORB PROD ( J ) 3

20 DO 30 C H K l = l . L A B L I M IF(RCICC(CHKl).NE,LOCC(CHKl)) GO TO 68

30 CONTINUF C I F SO. F I N D THE PERMUTATION ( E l THAT CONVERTS C ( L - ( M U ) L + ( N U I * I ) TO THE RIGHT ORB PROD J e THE PER4 C I S FOUND AS A PRODUCT OF TWO-CYCLES.

MCYC = 0 DO 6 0 CHK l= l ,N DO 5 8 CHK2=CHKI.N I F ( L B L ( J . C H K l ) o N E . L A B E L ( C H K Z ) ) GO TO 58 IF(CHKPoEQ.CHK2) GO T C C O NCYC = NCYC 4 1 E(NCYCv1) = CHK l E (NCYC*2 ) = CHK2 SAVE = LAREL(CHK1) L A S E L ( C H K I 1 = LABEL(CHK2) LABEL(CHK2) = SAVE GO TO 60

5 8 CONTINUE 60 CONTINUE

C GET THE CONTRIBUTION TC! FLSDYE FROY THE L - ( M U ) L + ( N U l C TERM

FME = 0.000 DO 62 LPROD=l.NLPROD DO 62 RPP@D=1*MRPROD

6 2 FMF = FME + FSC(PTLSEFTLPRODI * FSC(ZvRSEFvRPROD1 * 1 F P M A T ( N D ~ ~ N C Y C T E , T N P ~ T T N P ~ P L L T L P S T L P R @ D T P L R T R P S T 2 RPRODI

I. 007 COMT I NUE IF(FME.EQoOeODOl GO TO 68 FCMUNU = (LMU-MMU+f)*(LMU+M#Ul*(LNU-MNU)*(LNU+MNU+l) FCMUNU = DSQRT(FCMUNU1 FLSQME = FLSQME + ( ( - f ) * * N C V C ) * FME * FCMUNU

68 CONTINUF C 6 8 I S END OF MU-LOOP

70 CONTINUE C 7 0 I S END OF NU-COOP C C NORMALIZATION

PWR = ( P R S ( C J I J ) - P R S ( C I T I ) ) IF(PWR.GE*Ol GO TO 75 PWR = -PWR FNORM = 1.0DOt ( 2**PWR 1 GO 70 80

75 FNORM = 2 ** PWP S O FLSQMF = FLSQME*OSQRT(FNORM)

RETURN END

FUNCTION F P M A T ( N P , N C Y C S ~ E I T N P ~ T T N P , P L L , L P S ~ L P R O O ~ P L R T 1 RPS,RPRODI

I M P L I C I T REAL*R( F I 9 INTEGER4A-EvG-ZI DIMENSION T(4),Y(4I~SL(8)rL(2~13~4ltSEP(lb)rE(812),

1 F C Q E F F ( P 6 l r F B L A N K ( 3 3 8 ) COMMON FBLANKvL

C C * * * w e * * * * * * * * C, CALCULATES ( L E F T GEMPROD/P/RIGHT GEMPROD), WHERE C GEMPROD DATA I S I N COMMON* AND PERMUTATION CONVENTICN C I S THAT ( 1 2 3 ) MEANSeORBXTAL 1 REPLACES ORRITAL 2 9

C ETCo E. G. ( 1 2 3 )ARC = CAB. c 8 4 $ $ * * * * * * 8 * *

c FACT2 = 7 . 0 7 1 0 6 7 8 1 1 8 6 5 4 7 5 0 - 0 1 FPMAT = OoODO I F ( N C Y C S a M E o 0 I GO TO 3 0 5

C, WHEN NCYCS=O. PERMUTATION I S TAKEN TO BE THE IDENTPTYI C THEN FPMAT I S OVERLAP BETWEEN LEFT AND RIGHT S P I N C GEMPRODSO

DO 301 I 3 = l r N P I F I L ( ~ ~ L P R O D T I ~ ) ~ N E ~ L ( ~ , R P P O D , I ~ ~ I GO TO 370

3 0 1 CONTINUE FPMAT = 1.000 GO TO 3 7 0

C 305 DO 3 6 5 S I D E = 1 * 2

PL = P L L PS = LPS PROD = LPROD I F Q S I D E m E Q a l ) GO TO 306 PL = PCR PS = RPS PROD = RPROCJ

306 COUNT = 0 c FOR F T X E D SIOE AND GEM PROD^ SWEEP ALL SEPREDS AND C CONVERT SUITABLE DECLABELS TO BINCARELSO

DO 3 6 0 1 3 ~ 1 ITTNP 13M = I 3 - 1 DO 310 I4=l*Nf' P I = Z* * (NP-14) T ( I 4 ) = I 3 M / P I + 1 IF(L(SIDEQPRODTI~D.NE.O) GO TO 307 s = o GO T@ 3 0 8

507 S = 1 308 M ( 7 4 1 = S * ~ L ( S I D E P P R O D V 1 4 ) - 2 )

I F I T ( I 4 ) e E Q . l I GO TO 310 CI c SKIP SEPROO LABELS MHICH ARE NOT ASSOCIATED WITH THE C GIVEN GEMPROD*

IF(H1149eME.O) GO TO 3 6 0 3i0 I ~ M = I ~ M - T(I~)*PI + PI

C . COUNT SEPRODS ASSOCIATED WITH GIVEN GEMPROO COUNT = COUNT + 1

C FOR EACH SEPROO KEPT 9 GENERATE THE SINGLE-ELECTRON C S P I N FUNCTION LABELS ( S L ' S ) AND THE COEFFICIENT ( F C )

FC = 1,000 00 330 1 4 = l r N P T I 4 = 2*14 T I 4 M 1 = T I 4 - 1 I F ( M ( I 4 ) m N E c O ) GO TO 315 I F ( T ( 1 4 ) e N E e Z ) GO TO 325 S L ( T I 4 M l l = 0 S L ( T I 4 ) = 1 FC = FC*FACT2 IF(L(SIOE~PRODvI4)cEQ.2) GO TO 330 FC = -FC GO TO 3 3 0 . . . .

5 1 5 S L ( T I 4 M 1 ) = 1 IF(L(SIDEvPROD,I4)mEQe3l GO TO 320 S L ( T P 4 H l ) = 0 .

320 S L ( T I 4 ) = S L ( T I ~ M ~ I . . GO TO 3 3 0

325 S L ( T I 4 M 1 ) = 1 .. S L ( T I 4 k = 0 . . . FC = FC*FACT2

330 CONTINUE IF (S IDE.EQe19 GO TO 740

C I F S IDE = 29 PERMUTE THE SC'S DO 3 3 6 K=I+NCYCS.: I = NCYCS + 1 - .K TEMP = S L ( E ( I v 2 9 B S G ( E ( I v 2 ) b , = S L Q E ( I r l I ) '

336 S L d E I f p l I B = TEMP C GENERATE PROOUCT.,'SEPPODO FROM SL'S

3 4 0 SEPROD = 0 DO 3 4 5 I 4 t f r T N P

345 SEPROD = SEPROO + S L ( I 4 l * ( l O * * ( T N P - I 4 H C I F S l .DE= l r STORE ,SEPROD AS SEP(COUNT1 P FC . A S C. FCOEF,F (COUNT 1

I F ( S I D E a E Q e 2 ) GO 70 350 SEp(C0UNTI = S E P R ~ D FCOEFF(COUNT) = FC GO TO. 3 6 0

350 CONTI'NUE DO 355 1 4 = l * N S p i I F ( SE'PR0O;NEe S E P ( 1 4 l ) GO TO 355

FPMAT. = 'FPMAT 4 FC*FCOEFF( 14) ' 3 5 5 CONT PNUE 360 CONTINUE

C I F S I D E = l . STORE NUMBER OF SEPRODS ASSOC14TED WITH C L E F T GEMPROD . .

TF(S IDEmEQe21 GO TO 365 NSPL = COUNT

365 CONTTNUE 370 RETURN

SURPROGRAM 6 ,

SURROUTTNE EIGEN(A~R,NIMvTIDXTCVG) C C . COMPUTE EIGENVALUES AN0 EIGENFUNCTIONS OF A REAL C SYMMETRIC MATRIX C C DESCRIPTION OF PARAMETERS - C A - O R I G I N A L M A T R I X * OESTROYED I N COHPUTATIONs C RESULTANT ElGENVALUES ARE DEVELOPED I N OIAGO- C NAL OF MATRIX A* C R - RESULTANT MATRIX O F EPGENVECTORS (STORED C COLUMNWfSEo I N SAME SEQUENCE AS EPGENVALUES I C N - ORDER OF MATRICES A AND R C MV - INPUT CODE C 0 COMPUTE EIGENVALUES ONLY (R NEED MOT C BE DIMENSIONED BUT MUST S T I L L APPEAR C I N C A L L I N G SEQUENCE9 C 1 GENERATE R MATRIX---COMPUTE EIGEN- C VALUES ONLY C 1 GENERATE R MATRIX---COMPUTE EIGEN- C VALUES AND EIGENVECTORS AND SORT C -1 SAME AS P EXCEPT R I S INPUT C 2 GENERATE R MATRIX---COMPUTE ETGEN- C VALUES AND EIGEMVECTORS BUT 00 NOT C SORT C -2 SAME A S 2 EXCEPT R I S I N P U T C CVG - CRITERION FOR CONVERGENCE C CVG I S POSLTTVE---FINAL NURM=CVG C CVG I S NEGATIVE---FINAL NORM IS COM- C PUT ED FROM CVG C C c ORIGINAL MATRIX A MUST RE REAL SYMMETRIC (STORAGE C MOOE=19. MATRIX A CANNOT BE I N THE SAME LOCATION AS

C M A T R I X R. A I S C O L U M N W I S E U P P E R T R I A N G U L A R A N D R I S C C O C U M N H I S E S Q U A R E , E A C H S T O R E D I N O N E - D I M E N S I O N A L C ARR AY S. C C C

I M P L I C I T R E A L * 8 4 A - H v O - Z ! D I M E N S I O N P ( l 9 r R ( f ) v P O X ( 1 !

C G E N E R A T E I D E N T I T Y M A T R I X I F ( Y V 0 2 1 r 2 l v 1 0

1@ I J = O 00 2 0 J = l r N D O 20 I = l r N I J = I J + P P ( I 9 1 = O e O D O P F ( T o E Q a J P R Q 1 J D = l a O D 0

2 0 C O N T I N U E 2 1 M X = I A B S ( M V I

I F ( N a E Q o P 1 R E T U R N C, C O M P U T E I N I T I A L A N D F I N A L NORMS ( A N O R M A N D A N O R M X )

25 ANORM=O. OD+OO I D X ( 1 1 - 0 00 35 I = 2 r N J L I M = I - 1 I O X ( I ) = I D X ( J L I M ) + J L I M I A = I D X ( I ) DO 35 J = P , J l I M I A = I A + l

35 A N O R M = A N O R M + A Q I A ) * A ( P A ) I F I A N O R M B 1 6 5 r l 6 5 9 4 0

40 AMORM=Ze 0 0 + 0 0 * 0 S O R f ( ANORM) D I V = 2-000 / D F L Q A V ( 1 A 4 1)

41 ANRMX=CVG I F Q A N R M X l 4 2 9 4 3 ~ 4 3

42 ANRMX = A N O R Y * D I V * D A B S ( A N R M X O 43 I F B A N R M X - G T e ANORMD GO T O 165

C I N I T I A L I Z E I N D I C A T O R S A N D C O M P U T E T H R E S H O L D r T H R T H R = A N O R M

45 T H R = T H R * D I V 1 5 5 I N D = O

DO 1001 L = 2 v N L V O = L - 1 L O = I D X ( L I LL=L+LQ no 1001 M = ~ , L W O - M ( S = T D X ( M I

C C O M P U T E S I N A N D C O S L Y = L Q + M

02 IF(DABS(A(LM)P-THR)10@1r65r65 65 I N D = l

MM=M+MQ X=Oe5D+00*(A(LLI-A(MM)D

6 8 Y=-A( LMI/DSQRT(A(CM!*Af LMl+X*XI IF(X1 7 0 9 7 5 ~ 7 5

70 Y=-Y 7 5 SINX=V/DSQRT( 20 OD+OO*( 1eOD+OO+t OSQRT(l.OD+OO-Y*YI I 1 1

SlNX2=SPNX*SINX 7 8 COSX=DSQRT(l~OD+OO-SfMX2O

COSX2=COsX*COsX SINCS =SINX*COSX

C ROTATE L AND M COLUMNS ILQ=N*(L-19 IMQ=N*(M-1)

\ DO 125 PslrN IQ=IDX(I) IF( I-Ll 8 0 ~ 1 1 5 ~ 8 0

80 IF4 I-MI 8 5 ~ 1 1 5 ~ 9 0 85 IM=I+MQ

GO TO 9 5 90 IM=M+IQ 95 IF(1-CD ~ C O T B O ~ V ~ O ~

1CO ILmItLQ GO TO 110

105 IL=C+PQ 110 X=A(IL)*cOSX-A(IMI*SfNX

A(IM)=A(ILI*SINX+A(IIM9*COSX A(PLI-X

115 I F ( M X I 1 2 0 ~ 1 2 5 ~ 1 2 0 120 TLR=ILQ+I

IMR=IMQ+I X=R(ILR)*COSX-R(IMRP*SINX R(IYR)=R(ILR)*SINX+R(IMRD*COSX R( ILR.I=X

1 2 5 CONTINUE X=A(LPll*(SINCS+SINCS) Y=A(Lh)*COSX2+A ( M M D*SINX2-X x=Ad LL ) * S INX2+A( WM I*COSX2+X A(LMD=O.ODO A(LLD=Y A( MMD*=X

l C O l CONTINUE 150 IF( IND-1) 16C1955r160

C COMPARE THRESHOLD WITH FINAL NORM 160 1 ~ ( ~ ~ k - ~ ~ ~ ~ ~ ) f 6 5 9 1 6 5 9 4 5

C . SORT FIGENVALUES AND EIGENVECTORS 165 IF(MX.NEO11 REV'UF~N

18=0 DO 185 I = ~ T N JLIM=I-1 I Q= I O+N

L L = I + I D X ( I ) JQ=-N 00 1 8 5 J = ~ P J L I M J Q = J Q + N Y M = J + I D X ( J ) I F ( A ( L L 9 - A ( M M D l 1 7 0 * 1 8 5 , 1 8 5

170 X = A ( L L ) A ( L L ) = A ( M M D A ( f l M I = X

175 DO 1 8 0 K = l * N I [LR=IQ+K I M R = $ Q + K X = R ( I L R ) R ( T L R I = R ( I M R I

I R O R ( P M R I = X 1 8 5 C O N T I N U E

RETURN END

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ACKNOWLEDGMENTS

The au thor i s indebted t o P ro fes so r Klaus Ruedenberg f o r

sugges t ing t h i s problem and guiding i t s s o l u t i o n . H e a l s o

wishes t o thank P r o f e s s o r R. D. Poshusta f o r h e l p f u l d i scus-

s i o n s concerning group a lgebras .

S p e c i a l thanks a r e owed t o t h a t unknown person who, i n

1950, l e f t a chemistry textbook i n a t r a s h can, and t h u s

i n a d v e r t e n t l y s t a r t e d t h e a u t h o r ' s i n t e r e s t i n chemistry .

digital.library.unt.edu · PERMUTATIONAL SYMMETRY IN ELECTRONIC SYSTEMS Ph. D. Thesis Submitted to...

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