digital.library.unt.edu · permutational symmetry in electronic systems ph. d. thesis submitted to...
TRANSCRIPT
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
ThU report wacl prepared ar an account of woth sponsored by the United Qwes Govsmment. Neither the United S M a nor tho United State8 Atomic C o m ~ , nar snp of their eolploya, wt #V Of their coa@&om, 8Wbc6nfSaCt~. or their empbyaar, makes.any warn*, e x a m or h p l i ~ d , Or OSJWW legal UnbiliLy ue rw@&blUty fLrr the actlascy. onat pleteness or urefdnew OY m y Mormation, qp*tUS, product or p r o m dsrelowd, or repremat$ that s t r would not infrlnp privately owned rights.
DISCLAIMER
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.
This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that i t s use would not infringe privately owned rights.
Available from: National Technical Information Service Department A Springfield, VA 221 51
Price: Microfiche $0.95
iii
.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
Page
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
Page
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
vii
Permuta t iona l symmetry i n e l e c t r o n i c systems
Will iam I rwin Salmon
Lil
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
h
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 irreduci- ble 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 func- tion 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 combina- tion 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
case;
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
are :
[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,
NS b
'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
and
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 -- -
P -
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
k
O)
A
C
C
rn 4~
a
o a
Q
0
a al
z
zw
o
-4
-4
4
a
l=
l=
k
$4 4J
'"
rn 9 -n UI a
Q
' '
-
3
3
a 0
•
•
."ak
m
-n
0
La -4
8
8 0
•
•
C
l=
k
-4
5
a 4J
-4
0
C
.4
-4
lk
4J '44
4
l=
C
J3 : 3
Wo
vl
+
.f
:g
-n
A
-n
..
0
4
-n
JJ 4
-4
N
0
k4
Ja
l=
l=
rd, rl
X
Q
- $
82
•
V
n
0
OC
JJ
-tJ
0
-
nk
rn
0
P
Q)
-4
a,
-4
- n
rn 0 w
.d
m
Q
k
C5 -
l=
Q
ka
e
F: a
II a,
-4
a
. n
N
'4
a g
I1 -
4k
P
- 4J
'4
4d
\o
4
A
l=
C
(3
I 0
C
I G
- n
-4
0
. CI
CZ
p - n
A
0
rb al
.n
4~ h
. I=
C
.n
d
kk
I=
I=
U1 U)
-4
k
4
-4
- n
.
h
-4
a
E a
,a
o
4J rb
- n
C
4
a .
k
a
o Q
o
Y
a>
a
. .
8 w
5
L-l
0
(d
a)
C
8
0
'44 m
e
. a -nu
4J a
n m
C
a, $
k
-4
II
C
0
tl) N
-4
k -n rn
0
-4
k
II -4
a,
4 3
-0
a,
a h
4
rl
01 .d
k
-nrn
C
r(
4
+ +
k
a 4J
01 w
A
+'WA
-4
g
4J a
-4
k
-
Q
2 arna
-4
a
4~ o
4~ a,
-I4
-4
.4
l=
k=
II a
- n
€3- w
rn
rn '
l=
G+
'
- ! -mu
C
.nm
C
0
rn
hr
n
-4
k
ul 4J
-4
p-
'r
n
o
QO
is
Q
-4
-4
t-i l=
a
-2 A
a,
4~ -n
'4
3.
a
a
a, a
a, - n
.
4~ -4
'4
5
JJ -n
4
A
-4
A
19 a
a a,
a -n rn
$4 a,
-4
JJ
I I
0
-n
U1 X
-n
4J - -
dk
~
aa
. 4
-4
+'
c -
v .
3
4
cm 4
5 a,
Y
a O
-a
,"
*
l=
t= rn
-4
0
-4
'
N
3
0
-n
-
+,
Y
Ew
C
rn
O
a, '
P
2 II
-4
-4
C
4J
X
-
a, a
ao
a
. %
A
l=
aw
E
a,
4J E
G
C
.
7" A
U1
Q
9 4J
,=
a
8
h
0
-4
a
8
- n
04
2 8
4
4
-4
-4
Y
Q
) -n
Q)
0
- 4
l=
l=
0
w
-4
k
- E
3: a
:@
v
a, k
a,
~
LI
rn
a -
a, ll
rn
kO
r
n4
Jr
n
II a,
-4
4J
-4
4
-4
C
2
2 I=
0
I l
=h
rn
a,
a, 3
0
+,
8
h -n
k
8
aa
a,
H
-n
4
4-
C
~C
rn
La U
l4
J
8
k
k
-4
rd
k
-4
Q
) 0
E-1 a,
a, 0
n
0
ld
k
7 4J
h
-4
a,
3
2:
a,
2 0
0
w
HI
3
C
G
3
s.
.Q
U
3
3 k
(22) . It must be kept in mind that these' permutations may not all be distinct. The sets
each consist of distinct permutations. We now investigate
the conditions under which the sets may overlap.
The two sets share an element if and only if there are ,
two geminal permutations G and G t ingP such that
But then ij -1 ij ,-1 = GG-' -1 (rts) = (Qrs) QrSGG
i j ij* (rts)G} share an element Thus {Q:~G} and (it j)QrSG} = {QrS
only if (r ,s) E,$ : 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.
~ -
a
U
B r-i
tn C
a,.
-rl k
m
m
-4
. .m
a
.a
k
tn
k
a,. a
-l.
C
-4
r-i
a, '3
r-i
. 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 ~ )
j
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
U
3
cU w
m
Q
4
.
' a .
W..
. c, o
a
-4
C
A
E.
m
k
k
0
a, l=
c,
8-
c,.
$ %
0
a
W
0
'h
C c,
kO
k
0
-rl a,
Wu
a
a 0
CU
k
og
a
ZE
a,
r
rl
a,
C
+J
ac,
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
and
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.
0
4
rn Q)
U
5 a Q
) k
E
2 Q). C
t3 . rn rl a
4J -4
R
k 0
;
4J h
.Q k
Q)
W
'44 -4
a a
8
2 '.
a l=
8
C
a, C 3
k
c, 0
a,
.
C - 1
l=
v
rl I h
'r
l
I z u
Y
X
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
32
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
h
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
(27 k
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 )
7 3
and
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
n
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:
N-2
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
I
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
i
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
3 jk
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 .
A
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-
ucts;
diagonalize this g2-matrix;
keep only those.eigenvectors
having .the desired eigenvalue
S.
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,
p=l
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
Thus
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.'
72
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
N
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
-4
k
a
. g
a, N
k
-4
- d
ka
a
a, k,' +
' C
a
,w
a
Cf
da
,
Oa
,E
P
k
a,'
0
ax
+'
0.
kd
"I
g
-4
a,
C
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 ,
i j
. . 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 , . .
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 ,
and
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 .
I
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
H . .
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 :
E
-4 "I
8 k
E:
X
a.
o
a,..
'H
a a,
. m
.
4J k'
m
a, -4
N
U
-I+
0
k
representation. For a,diagram containing two rows at most,
(56) becomes . . . N! (p1-p2+1)
dD =
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
where
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
AD U
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
and
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 . ,
. ,
rs
- 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.
Thus
. .
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 ,
Tr
D D t N - 2 is e i t h e r ml o r , depending on T r . where Tr
191
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
UJ a
3
..> a -
m
'a, C
k
U
0
m
4
r-i
(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
page 74. 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: .
where
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
where
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 " ) ~ :
where
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
zero.
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,
then
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
f orm
( . .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..
135
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
page 78. 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
. . . .
D r s
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 ,
. W
- 3
aa"
- n rn W rn
U
WII)
II
w
- cl5
W
- 3
n rn
U
. wrn
II - Q
W
h
n
4J a m
U
Wrn
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
w
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 -
q n
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 )
,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
where
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
I /
/ S' /I //and t h e con f igu ra t ions / -----,-,,--,-,,,-,----J
I
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
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
I I I
V 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
(1)
(ii)
(vii)
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
159
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
C END
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
179
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
END'
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
201
LITERATURE CITED
Boerner, Hermann. 1963. Represen ta t ions o f groups. Amsterdam, The Nether lands , North-Holland Pub. Co.
Coleman, A. J . 1968. The symmetric group made easy. Advan. Quantum Chem. 4, 83.
Condon, E . U . and S h o r t l e y , G . H . 1951. The theo ry of a tomic s p e c t r a . Cambridge, England, Cambridge Univ. P r e s s .
Corson, E . M. 1951. P e r t u r b a t i o n methods i n t h e quantum me- chan lc s o f n -e l ec t ron systems. London, England, B lack ie and Son, Ltd.
E l l i o t t , J. P . , Hope, J . , and Jahn , H. A. 1953. T h e o r e t i c a l s t u d i e s i n n u c l e a r s t r u c t u r e . I V . Wave f u n c t i o n s f o r t h e n u c l e a r p - s h e l l . P a r t B. <p (n ) 1 (n-2) ( 2 ) > f r a c t i o n a l paren- t a g e c o e f f i c i e n t s . Trans . Roy. Soc. (London) ~ 2 4 ' 6 , 2 4 1 .
Gal lup, G . A. 1968. P r o j e c t e d Hartree produc t wavefunct ions . J . Chelil. Phys. 4'8, 1752. . .
Gallup, G. A. 1969'. p r o j e c t e d Hartree produc t wavefunctions. 11. General c o n s i d e r a t i o n s of Young o p e r a t o r s . J. Chem. Phys. 50, 1206.
Goddard, W. A. 1967a. Improved quantum theo ry o f many- A 2 e l e c t r o n systems. I . Cons t ruc t ion of e i g e n f u n c t i o n s of S which s a t i s f y P a u l i ' s P r i n c i p l e . Phys. Rev. 157, 73.
Goddard, W. A. 196 7b. 'improved quantum theo ry ' of many- e l e c t r o n sy'stems. 11. The b a s i c method. Phys. Rev. 157 , 81.
Goddard, W. A. 1968. Improved quantum theory. o.f many-electron systems. 111. The GF method. J. Chem. Phys. 48., 450.
Hamermesh, Morton. 1962'. Group theory . Reading., Mass. , Addison-Wesley Pub. Co., I nc . ,
Harris, Frank E. 1 9 6 7 . ' ~ o l e c u l a r o r b i t a l theory . Advan. Quantum Chem. 3, 61. . .
Jahn , H. A. 1954. Direct e v a l u a t i o n of f r a c t i o n a l pa ren tage c o e f f i c i e . n t s u s ing Young o p e r a t o r s . General t heo ry and <412,2> c o e f f i c i e n t s . ~ h y s , Rev. 96, 989.
Jahn, H. A. and van Wieringen, H. 1951. Theoretical studies in nuclear structure. IV. Wave functions for the nuclear p-shell. Part A. <p (n) 1 p (n-1) p> fractional parentage coef f i- cients. Proc. Roy. Soc. (London) A209, 502.
Johnson, $D. F. 1960. Group theory in solid-state physics. Rept. Progr. Phys. 23, 66.
Karplus, M., Kuppermann, A., and Isaacson, L. M. 1958. Quantum-mechanical calculation of one-electron properties. I. General formulation. J. Chem. Phys. 2, 1240.
Klein, D. J., Carlisle, C. H., and Matsen, F. A. 1970. Symmetry.adaptation to sequences of finite groups. Advan. Quantum Chem. 5, 219.
Kotani, M., Arnemiya, A., Ishiguro, E., and Kimura, T. 1955. Tables of molecular integrals. Tokyo, Japan, Maruzen Co., Ltd.
owd din, P.-0. 1955a. Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 97, 1474.
owd din, P.-0. 1955b. Quantum theory of many-particle systems. 111. Extension of.theHartree-Fock scheme to include degen- erate systems and correlation effects., Phys. Rev. 97, 1509 .'
. .
Lowdin, P.-0. 1960. ~xpansion theorems for the. total wave function and extended Hartree-Fock schemes. Rev. Mod. Phys. 3'2, 328. - Lowdin, P.-0: 1964. Angular momentum wavefunctions construc- ted by 'projection operators. Rev. Mod. Phys. -3, 966.
Lb;wdin, P. -0. 1967. Group algebra, convolutio~ algebra, and applications to quantum mechanics. Rev. Mod. Phys. 39, 259.
McIntosh, H. V. 1960. Symmetry-adapted functions belonging to the symmetric groups. J. Math. Phys. 1, 453.
~ c ~ e e n ~ , R., and Sutcliffe, B. T. 1969. ~ethods of molecular quantum mechanics. New'york, N. Y., Academic Press, Inc.
Matsen, F. A. 196 4. spin-f ree quantum chemistry. Advan. Quantum Chem. L,. 59. . . . .
Matsen, F. A. 1970. Vector spaces and algebras.. for chemistry and physics. New ~ork', ..N. Y., Holt, Rinehart.and Winston'. . .
M a t t h e i s s , L. F . 1958. IBM 704 program f o r g e n e r a t i n g 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 of t h e group of s p i n c o o r d i n a t e permuta t ions . Q u a r t e r l y P rog res s Repor t , S o l i d - S t a t e and Molecular Theory Group, M.I .T . - 30, 43.
Ma t the i s s , L. F. 1959. An i n d u c t i v e method f o r g e n e r a t i n g 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 of t h e group of s p i n coord i - n a t e permuta t ions . Q u a r t e r l y P rog res s Report , S o l i d - S t a t e and Molecular Theory Group, M.I.T. 34, 58. - Pauncz, R. 1962. S t u d i e s on t h e a l t e r n a n t molecular o r b i t a l method. I V . G e n e r a l i z a t i o n of t h e method t o s t a t e s w i t h d i f f e r e n t m u l t i p l i c i t i e s . J . Chem. Phys. - 37, 2739.
Pauncz, R. 1967. A l t e r n a n t molecular o r b i t a l method. P h i l a - d e l p h i a , Penna.,, W . B. Saunders Co.
Pauncz, R. 1969. E l e c t r o n c o r r e l a t i o n i n atoms and molecules . I n Henderson, D . , ed . P h y s i c a l chemis t ry . Vol. 3 . New York, N . Y . , Academic P r e s s , I n c .
Pa'uncz, R . , d e Heer, J:.., and ow din; P.-0. 19'6'2. S t u d i e s on t h e a l t e r n a n t molecular , o r b i t a l method. I . General energy
. exp res s ion f o r an a l t e r n a n t . system wi th c l o s e d - s h e l l s t r u c - t u r e . J . Chem. Phys. 36, 2247. - . .
Percus , J . K . and Rotenberg, A. 1962. Exact e igen func t ions of angu la r momentum by r o t a t i o n a l p r o j e c t i o n . J . Math. Phys. 3, 928. - Poshus t a , R. D . 1969. App l i ca t ions of t h e symmetric group i n t h e o r e t i c a l chemis t ry . Unpublished l e c t u r e ' n o t e s . Arnes, Iowa, Iowa S t a t e U n i v e r s i t y .
2 P r a t t , G . W,., Jr. 1 9 5 3 ; ' ~ i g e n f u n c t i o n s of S by a s p i n ,oper- a t o r method. Phys. ~ e v . 92, 278. -
. .
Rotenberg, A. 1963. c a l c u l a t i o n of e x a c t eigen.£unctions of s p i n and o r b i t a l angu la r momentum us ing t h e p r o j e c t i o n oper- a t o r method'. J . hem. Phys. - 39, 512.
Ruedenberg, K . and Poshus ta , R . D . 1971. ~ a t r i x e lements ' and d e n s i t y m a t r i c e s f o r many-electron s p i n e i g e n s t a t e s b u i l t from orthonormal o r b i t a l s . To be publ i shed . .
. .
Ruther ford , D. E . 1948. , S u b s t i t u t i o n a l a n a l y s i s . E d i n b u g h , Sco t l and , Edinburgh Univ. P r e s s .
Salmon, W . 1. 1971. The theo ry of group a l g e b r a s . UnpubLished l e c t u r e n o t e s . A m e s , Iowa, Iowa S t a t e Univers%ty .
Sasaki, F. and Ohno, K. 1963. Spin-component analysis of single-determinant wavefunctions. J. Math. Phys. 4; 1140.
Schaeffer, H. F. I11 and Harris, Frank E. 1968. Construction and use of atomic L-S eigenfunctions. J. Comp. Phys. 3, 217.
Serber, R. 1934a. Extension of the Dirac vector model to include several configurations. Phys. Rev. 45, 461.
Serber, R. 1934b. The solution of problems involving permu- tation degeneracy. J. Chem. Phys. 2, 697.
Shapiro, J. 1965. Matrix representation of the angular momen- tum projection operator. J. Math. Phys. 5, 1680.
Slater, John C. 1929. The theory of complex spectra. Phys. Rev. 34, 1293.
Slater, John C. 1931. Molecular energy levels and valence bonds. Phys. Rev. 38, . 1109. .
Smith, V. H., Jr. 1964. Construction of exact.spin eigenfunc- tions. J. Chem. Phys. ,Q, 277.
Smith, V. H., Jr. and Harris, F. E. 1967. projection o f exact spin eigenfunctions..Unpublished paper. Preprint No. 182. ~ppsala, Sweden, Quantum Chemistry Group, Unl.versity . . of. .
Uppsala.
Sullivan, J. J. 1968. Antisymmetric projection in the approx- imation of no spin-orbit coupling. J. Math. Phys. 9 , 1369.
. . Thrall,, R. M. 1941. Young's semi-normal representation of the symmetric group. Duke Math. J. 8 , 611.
. . van der Waerden, B. L. 1950. Modern algebra. Vol. 2.. New York, N. Y., F. Ungar Pub. Co.
Weyl, ~e'rmann. 1931. Gruppentheorie.und Quantenmechanik. 2. Auflage. Leipzig,-Germany, S. Hirzel. ~ngiish edition, translated by Robertson, H. P. The theory of groups and quantum mechanics. New' York, N. Y., Dover Publications,
. . . . Inc. 1950.
Wigner , E. P. 1931. Gruppentheorie und ihre ~,nwendun~' auf die Quantenmechanik der Atomspektren. Braunschweig, Germany, F. Vieweg und Sohn ~kt.-~es. . . . .
Wigner, E: P. 1959. Group .theory and its application to the quantum mechanics of atomic spectra. Expanded edition. .Trans- lated by Griffen, J. J. New York, N.Y., Academic Press, Inc.
. . Yamanouchi, Takahiko. 1935. On the calculation of atomic energy levels. roc. Physico-Math. Soc. Japan l7, 274.
Ydmanouchi , Takahiko. 1936. On the. calculation of atomic energy levels. IV. Proc. Physico-Math. Soc. Japan 18, 623.
Yamanouchi, Takahiko. 1937. On the construction of unitary irreducible representations of the symmetric group. Proc. Physico-Math. Soc. Japan 19, 436.
n Yamanouchi, Takahiko. 1938. On atomic energy levels of p p configurations. Proc. Physico-Math. Soc. Japan 2 0 , 547.
Young, A. 1901. On quantitative substitutional analysis. Proc. London Math. Soc. 33, 97. - Young, A. 1902. On quantitative substitutional analysis (second paper). Proc. London Math. Soc. 34, 3 6 1 . - Young, A. 1928. On quantitative substitutional analysis (third paper) . Proc. London Math. Soc. (2), 2 8 , 255.. - Young, A. 1930. On quantitative substitutional analysis (fourth paper). Proc. .London Math. Soc. (2), ..3.1, 253. - Young, A. 1932. On quantitative substitutional analysis (sixth paper). Proc. London Math. Soc. (2), 34, 196. -
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 .