Bonded tableau method for many-electron systems

Journal of Molecular Structure, 198 (1989) 413-425 Elsevier Science Publishers B.V., Amsterdam - Printed

413 in The Netherlands

BONDED TABLEAU METHOD FOR MANY-ELECTRON SYSTEMS*

ZHANG QIANER and LI XIANGZHU Department of Chemistry and Institute of Physical Chemistry, Xiamen University, Xiamen, Fujian (People’s Republic of China) (Received 4 December

1988)

ABSTRACT Bonded tableau functions based on the representation theory of symmetrical groups are proposed for treatment of the problems of many-electron systems. Using the bonded tableau as a basis, the electronic energy matrix elements of a spin-free hamiltonian have been deduced for valence bond and configuration interaction calculations, and the properties of the bonded tableau as well as its applications to some many-electron problems are discussed.

INTRODUCTION

Both valence bond (VB) theory and the configuration interaction (CI) method require an efficient scheme for generating the many-electron bases and for the calculation of hamiltonian matrices. In recent years, several group theoretical approaches, including the symmetric and the unitary group theory, have been applied to many-electron problems of atoms and molecules [l-12]. Based on the symmetrical group algebra, the spin symmetry-adapted n-electron bases are constructed by using the projector of group S,, and the related matrix elements are evaluated by the irreducible representation (IR) matrices of the permutation group. One of the difficulties of this approach in practical calculations is that it is usually difficult to calculate the IR matrix elements of a permutation, namely D(P), although the computation methods for these matrix elements have been improved in some ways. Another approach, based on the unitary group formulation of a modal spin-free hamiltonian, has been extensively investigated. In this formulation, the hamiltonian matrices in terms of Gelfand or other bases are calculated by determining the matrix elements of the generators and their products of the unitary group. However, we note, from the chemical point of view, that the Gelfand basis is not a good physical basis. In other words, it does not possess a simple physical interpretation of *Dedicated

to Professor

0022-2860/89/$03.50

W.J. Orville-Thomas.

0 1989 Elsevier Science Publishers

B.V.

414

chemical interest such as the VB function. If we want to use the Gelfand basis for VB calculation, the VB wavefunction must be expanded in the form of several terms of the Gelfand basis. This is a cumbersome procedure. We develop here a new many-electron basis which describes the classical VB structure; furthermore, the evaluation of matrix elements in terms of these functions can be carried out in an efficient way. This idea was mentioned earlier [ 131, and in this paper we give all the formulae of our method in detail. We outline how the function denoted by bonded tableau can be constructed by using the representation theory of permutation group, and the properties of the bonded tableau are investigated. The expressions of the matrix elements of a spin-free hamiltonian for the bonded tableau states are deduced. We provide some examples of many-electron problems treated by this method, and discuss some of its features. BONDED TABLEAU

First we deduce the symmetry-adapted n-electron functions corresponding to VB states. Although the following description is carried out in VB language, the availability of this kind of function to CI study in the MO method is straightforward if the one-electron orbitals in the bonded tableau are referred to one-electron molecular orbitals. We know that the pairing functions of Heitler-London type take the form [ (a(i),bt_j)]

=2-*[a(i)bCi)+aCj)b(i)]

(afb)

=a(i)aO’)

(u=b)

(1)

and that they are usually used to describe a classical chemical bond or an isolated electron pair. This function is symmetrical with respect to the transposition of two electron indices i and j. One-electron states u,b,... in eqn. (1) are atomic orbitals, but they are chosen here to be Lowdin’s orthogonal orbitals, localized on particular atoms. A VB structure possessing a number of two central bonds and a number of isolated electron pairs may be described by a primitive bonded function which is the product of all pairing functions corresponding to each bond and each electron pair: V,=

to(l),b(2)l-[c(3),d(4)1...

(2)

In this way, different bond structures are related to different primitive bonded functions. The primitive state in eqn. (2) still does not form an irreducible basis for representations of symmetric group S,. The symmetry-adapted VB function of the corresponding bond structure can be obtained by the effect of the standard Young-Yamanouchi projector of group S, on the primitive bonded function

415

(3) where [A] is the Young diagram [2P1q] with p= (n/2) -5 and q= 28, and 0:;) (P) is the standard IR matrix element. It is evident that VA is symmetrical with respect to the transpositions of indices (a$), (c,d),.... If we take the orbitals a,b,c,d ,..., arranged in the order 1,2,3,4,..., then the component in the projector CO/~]must have the same symmetry properties as VA, otherwise the operation of o/i1 on VA will vanish. The component p satisfying these properties is the first Young tableau 1 [A] 1) , which is obtained by assigning indices 1,2,...,n in the boxes of the Young diagram [A] from left to right and from top to bottom in order: 1

2

ip-1

ip

lw)=2p+l

(4)

Equation (3) becomes CO:;1v, ‘sP#13~:~ v,

(5)

This VB wavefunction, CO/:]VA, can be denoted by a bonded tableau, and its properties are discussed below. We know that the permutation operators in projection cause permutations of the electron indices only. The bonded tableau would then consist of linear combinations of primitive bonded functions with identical pairing orbitals, but the pairing square brackets in different terms have different electron indices. It can also be shown that the electron permutation symmetry component or r in w/f1 is trivial in spin-free quantum chemistry. Thus the bonded tableau is a new kind of spin-free VB function and the corresponding bond structure is shown by VA. Using the properties of ( [A] 1), eqn. (5) can now be rewritten as @;“I (A)=

(c,d)...] [ (a$) . . [I : :

=No#

=NAo~:][a(l)b(2)c(3)d(4)

.. . ]

(6)

=N,c#K, In eqn. (6) the primitive bonded function VA is replaced by a tensorial product KA, where the order of orbitals must correspond with that in VA. In this way,

416

different VB structures with the same orbital configuration are described by the related product functions KA of orbitals of different order. We have introduced a tableau of not more than two columns to denote this VB-type function, and in this tableau the one-electron states are listed from left to right and from top to bottom in order of the orbitals that appear in KA. Each row describes a bond (a # b) or an isolated electron pair (a = b ) . The symmetry component r related to electron permutation is omitted, because in the spin-free formulation of many-electron problems the expectation energy is independent of r. For a given configuration of occupied orbitals the primitive tensor product KA can be defined as a primitive permutation RA operating on a given tensor product K,, i.e.

@)!“I(A) =N,o;:~

RAKo

(7)

Compared with the ordinary projection method, the complete set of bonded tableau states of each configuration is now obtained by changing the primitive permutation RA, or equivalently, the primitive state product KA, rather than by changing the projector. The bonded tableau defined in this way has some symmetry properties which provide a simple physical and chemical interpretation for the function. First, it can be shown that the Young tableau 1 [A] 1) has the properties (2i-1,2i)

1 [All)

(2i-1,2j-1)

= ] [all)

1[All)

(8)

= (2i, 2j) 1[All)

where ;,j QP. The first relation is evident. Taking j= i+ 1, the second relation can be verified directly. Then, for any j, the relation can be proved by the mathematical inductive method. Because of these relations, the bonded tableau remains unchanged upon interchange of two pairing orbitals in the same row:

a

.. .. . [ .

b ..=. ... .

I[

a b . .. ... ... . .

The function orbitals:

1

also remains unchanged

(9) upon interchange

of two pairs of pairing

(10)

We can then add all two central bonds and isolated electron pairs to the tableau in an arbitrary order. However, upon interchange of the unpaired orbitals, i.e.

417

the states in the squares of the single-column part of the tableau, tableau function undergoes a change of phase factor - 1, i.e.

the bonded

(11)

it is evident that these properties are necessary for a function which can be used to describe VB structure. Bonded tableaux are constructed from various primitive product functions from a nonorthogonal and over-complete set. The linear dependences of bonded tableaux arise from the following two-element relations:

IL1

a

b

c

;-b; .. .

.. .

a

c

. ..

. ..

(12)

We must define a linearly independent complete set, by which any many-electron functions can be expanded. This set is that consisting of canonical bonded tableaux, which can be defined in several different ways. The first tableau is defined in a similar way as for the Weyl tableau. In this definition all doubly occupied orbitals are first inserted in the same row and located in the upper part of the tableau in decreasing order. Then, all singly occupied orbitals are assigned in the lower part such that state indices increase from left to right along the rows and from top to bottom down the columns. This rule for the arrangement of orbital indices gives a strict and general condition imposed on the canonical states, which will be available for diverse applications such as CI studies, but in VB descriptions, a tableau constructed in this way would probably correspond to a VB structure with cross bonds. For the purpose of VB calculation, we define the functions corresponding to canonical VB resonance structures as the canonical bonded tableaux. By this definition, all isolated electron pairs are also placed in the upper part of the tableau. In the lower part, the singly occupied pairing orbitals are inserted in the same rows of length two, whereas singly occupied unpairing orbitals are inserted in the single-column part, i.e., the tail of the tableau. This shows that the singly occupied portion of the bonded tableau is formally not a Weyl tableau. For both of these kinds of

418

canonical bonded simply given by

tableau

functions,

the normalization

factor

in eqn. (6) is

NA = [n!/frnl 2mA] 1 where mA is the number @l(A).

(13) of doubly occupied

orbitals

in the bonded

tableau

MATRIX ELEMENTS

Consider an n-electron system. In spin-free quantum chemistry, the hamiltonian will be expressed as the sum of one- and two-body operators:

H=F+G = Tfi

(14) +

Cgij i-cj

where fi =

-

IV p - C zic/Ric C

Therefore, hamiltonian matrix elements are the sum of matrix one- and two-body operators and can be written as

elements

of

(@[‘l(A)(H]@l’l(B)) = (2 mA+mB)-t

~~i:l(p)(KA(

(F+G)pI&)

(16)

P

Matrix elements

of

one-body operator

First we consider the matrix elements between two-bonded tableaux for the case of identical configuration of occupied orbitals. Immediately, we find that the left and right tensor products & and KB are identical in part of the doubly occupied orbitals but may differ in the arrangement order of singly occupied orbitals. Let Rc denote the permutation which operates on the electron indices in singly occupied orbitals of KA such that

KB =RBKo = (RBRzl)KA

=RcKA

(17)

(~r”1(A)IFI~~~1(B))=2-“ACd~fD~:‘(g’qgRc1)(KAIFg’qgIKA)

(18)

We have

where g and g’ are elements of the subgroup GA which consists of all permutations KA which remain unchanged (g& - KA). In eqn. (18) we have made a double co-set decomposition of S, with respect to subgroup GA. In the mean time, q is the double co-set generator and d(,, is the repetition frequency. Ev-

419

idently, gG, acts only on the electron indices in doubly occupied orbitals. GAis the set of elements { [e, (12) ] @ [e, (34) ] B...}, the order of which is 2”A. Because g and Rc are commutable and g ) [A] 1) = 1 [A] 1) , we obtain (~[~l(A)lFl~[~l(B))=~I:l(Rc)Cn,f,,

(19)

r

where/L= (rlfl r > is one-electron integral. From eqn. (19) we obtain the nonzero overlap integral between two bonded tableaux (@[‘I (A) 1DLL1(B))

=D;:]

(R,)

(20)

It is evident that two product functions & and KB may have at most one different occupied orbital, otherwise the matrix elements of the one-body operator vanish. We suppose that the configurations of KA and KB correspond to C*:...&?$;‘... c,:...$5:-‘@-‘... where nr= 1,2; n,=O,l.

There is a permutation

such that

TK, = (KA):=...$tO’r)...$sCjs)...

(21)

where ( KA) Fmeans that one of orbitals in KA is replaced by another orbital &. As for deduction of eqn. (19), we make double co-set decomposition C,V GB of S,, where GA and Gn are two subgroups defined as in eqn. (18)) corresponding to KA and KB respectively. Except for the orbitals & and &, the number of doubly occupied orbitals in KA and KB is nc. Then, a subgroup Gc of order 2mc, consisting of the invariant permutations of electron indices in these 2772, orbitals, forms the intersection of GA and Gn. Taking all these results into account, the matrix elements of one-electron operator are finally given by (~r~1(A)JFJ~[~1(B))=2”C[n,(n,+1)]*Cd,1D~:1(q)(KA)Fq)Kc) (22) = [n,(nl+l)]lg!:l~(T)f,, where we have used d (T)=IT-1GATnGBI=)GCI=2mc

(23)

and T is given by eqn. (21) . Matrix elements of two-body operator The matrix elements of the two-body operator are divided into three categories: the matrix elements for configuration of identical occupied orbitals, for configurations with one different orbital and for configurations with two different orbitals. By using approaches similar to those discussed above, we can obtain the following expressions for the matrix elements of a two-body oper-

420

ator. We assume that KA and KB correspond to the identical configuration and, relating these with eqn. (17), we have (~r~1(A)(GI~[~1(B))=2”“Cd,fDI:1(R,qR~1)(K,IG,IK,) Q

where r,s, denote states in the primitive function K. and i,, i,, denote the electron indices in states r,s, respectively, i.e. K,, takes the form

K, =...@r(ir)...qbs(is)...

(25)

If state &is doubly occupied, any of the two-electron indices in &may be taken as i, in eqn. (24)) and we may appoint i, as the odd index. We note that the primitive function KA takes the form

KA =...&.(jr)...qbs(js)...

(26)

Then, in analogy with eqn. (24)) we have another expression of the matrix elements:

(~r~l(A)IGI~[~l(B))=D~:l(R,)[C(n,-l)g,,,, r + ~nrnsJrs]+ r<.S

Cn,n,D1:l[RcO’r,js)l’K,,

T-z.9

(27)

When KA and KB differ in one occupied orbital and are related by eqn. (21)) we obtain

= [nr(nt+l)lf{DIY(T) [&gur,Ut-grr,rtl cl

(28)

It should be noted that n,, n,, n,,... in eqns. (24) and (28) are the occupancies of the corresponding orbitals in KA. If two different occupied orbitals appear in KA and KB, four different situations must be considered. ( 1) When KA and KB correspond to the configurations CA:...@:‘@,“@;‘@:... CB:...~:r-l~sn,-l~;t+l~~+l

where n,.,n, = 1,2; n,,n, = O,l, we have

421

(29) x {D I:]

CT,krs,t, +D !:I liry_is) TI lgrs,ut1

in which T, is defined by T, KH = (&E

(30)

=...~%ci,)..~$,ci,)...

(2 ) When KA and KB correspond

to

c,:...$@:$qU... c,:...Cp;l-‘qq”... where nt,nu=O,l,

we obtain

(~r~1(A)JGI~[~1(B))=[2(n,+l)(n,+l)tD~:1(T,)g,,,,

(31)

in which T2 is defined by (32)

TyKB = (KA)~=...~tCjr)~uCjr+l)... (3 ) When KA and KS correspond

to

c,: ...qiq#I. . .. c,:...~:~-‘~~~-‘~p... where n,,ns= 1,2, we have (~“~1(A)(G(rS’“1(B))=(2n,n,s)iDI:1(T,)g,,,,

(33)

in which T:, is defined by T&s

(34)

= (KA):=...@ttir)$ttis)

(4) When KA and KS correspond

to

C*: ...& .. . c,:...q!q... we obtain (35)

(~‘“‘(A)(GI~‘~‘(B))=DI:‘(Tq)g,,,tt where T4 is defined by T,K,=

(KA)zu =...&(j,)$~~o’~+l)...

The problem of the symmetry

(36)

factor

From the above discussion we have reached an important bonded tableaux are used as a basis, the symmetry factors

result: when the which appear in

422

front of the expressions for the electronic energy matrix elements for one- and two-electron integrals are the irreducible matrix element D ]:I (P) for the first Young tableau 1[All). Therefore D{:](P) plays an important role in the bonded tableau method. We know that the difficulty in the application of symmetric group theory to many-electron problems is the generation of Young’s orthogonal representation matrices, i.e. the general matrix element Ox1 (P), in a sufficiently efficient way. Although the matrix elements for transpositions (i,i+ 1) of neighbor indices can be calculated by hooklength in the Young tableau, and any P can be expressed by the successive products of these transpositions, the evaluation of Djkl (P) for a general permutation P by matrix multiplication may involve rather lengthy and tedious calculus. This procedure was improved by a recursive formula which may be useful in computer implementations (13). However, whether Djil (P) can be calculated by a simple direct formula is still an unsolved difficulty. This difficulty is avoided in our treatment because we need to calculate only Dj:] (P) rather than D&](P). We find that ( [All) has some important properties, such as those given in eqn. (8). These and other aspects of the behaviour of 1 [A] ) can be adapted to develop a simple procedure for calculating D ]:I (P) with considerable simplification. This has been done in our next paper by using the unitary group approach. EXAMPLES

The simplest example is the closed shell of molecules or atoms. The system is described by a bonded tableau wherein all orbitals are doubly occupied. The energy expression can be immediately deduced from eqns. (19) and (24). Because of Dll (i,,i,) = - l/2 for r # s, we obtain the well-known result E = C 2a, + &r,,,,

+ c ar,

+ c 4N?(LL)KS ?
(37)

=~2cu,+;:(25,-;;,) r r,s In a similar way, the energy expression with orbitary configuration can be written E= &a, r

+ C (Q-J,, -ML) r,s

of an open-shell as

molecular

system (33)

where all the coefficients c,, arS, b, can be expressed in terms of Df$] (P). As an example, for an open-shell state described by a bonded tableau in which paired orbitals are inserted in all the rows of length two but the tail of the tableau is assigned by unpaired orbitals, the electronic energy is given as

(39)

423

where for r f s D 1:’ (i,&)

= 1

(the states r and s are located in the same row of the bonded tableau)

E-2 1 (the states r and s are located in different

rows and at

least one of them is the paired orbital). The third example is designed for the computation of matrix elements in VB treatments. Many chemical problems can be well described by VB wavefunctions in which only the singly occupied orbitals need to be considered. A typical example is the r-only treatment of planar conjugated organic molecules. If we do not consider the isolated electron pairs, the VB function can be constructed from a configuration of singly occupied orbitals. From eqns. (19) and (27), the matrix element between two VB states is expressed by (@“‘(A)

]H] @[‘l(B))

CJrsl+ ~D~:‘Pkti,,~sHKs

=D]:I(&)]CCG+ r The diagonal structure

element

(RA =Ra)

T
corresponds

T
to the energy

(40)

of a resonance (41)

D]:‘tir,.U=I

( r and s are bonded)

= - 1 (both r and s are in the tail of the tableau) Z-2 ’ (r and s are unbonded,

and at least one of r and s

is in row of two column ) . The above discussion can easily be extended to the more general multi-configuration and multi-VB-structure cases. We thus find a new approach for evaluating matrix elements, as compared with the classical Rumer’s method [ 151, Lowdin’s determinantal method [ 161, and the diagrammatic Pauling number method [ 17-201. DISCUSSION

Applications of symmetrical group theory to many-electron systems can be carried out using the bonded tableau method. To construct the spin symmetry adapted functions for an n-electron system, the traditional manner is to apply the projector u~$I, corresponding to the Young-Yamanouchi representation of group S,, to a given primitive function K0

424

@$“’ =Nw$lK,

(42)

The complete set of many-electron states is then obtained by taking different indices p, i.e. the intrinsic quantum number of the projector. However, we have introduced here a new way of applying the projector, the main feature of which is that the intrinsic quantum number is fixed as the first Young tableau. The complete set of bases is then generated by the action of a given w$] on different primitive functions corresponding to canonical VB structures rather than the action of co$l with different p on the given primitive function K,,. Why does this yield VB-type wavefunctions? We know that the intrinsic quantum number of a projector of the symmetric group labels the symmetry with respect to the state permutations. When the matrix element D$]P) is the standard representation adapted to the subgroup chain s, 3 s,_11...

2 s,

the symmetry Weyl tableau. chain

label p equivalently corresponds to the Gelfand symbol or the We then obtain the states classified by a canonical subgroup

U(N) 5, U(N-1)

(43)

3 ... 2 U(l)

(44)

Nevertheless, some chemically interested problems require a different basis from the canonical classified one. The VB description needs a basis which must have the symmetry properties given in eqns. (9)- (11)) because it does not matter which bond we write first. We find that 1 [A] 1) possesses these properties. When CI.IJ$’is applied to an orbital product KA, we obtain a function in which the first state in KA is bonded to the second state, the third state is bonded to the fourth, and so on. Therefore, CO::]KA yields a VB-type and spin symmetry-adapted basis, the so-called bonded tableau. When the electronic energy matrix elements for many-electron systems are treated by using the bonded tableau as a basis, another important feature of this method emerges, i.e. only D It1 (P) needs to be calculated. We will show in our next paper that the calculation of II{:] (P) is much simpler than that of D,$l (P). These two features are sufficient to make the bonded tableau useful in handling many-electron problems.

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