operator algebra for the many-body problem in the spin eigenfunction basis
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Operator algebra for the manybody problem in the spin eigenfunction basisWl/odzisl/aw Duch Citation: The Journal of Chemical Physics 91, 2452 (1989); doi: 10.1063/1.457004 View online: http://dx.doi.org/10.1063/1.457004 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/91/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An integrable many-body problem J. Math. Phys. 52, 102702 (2011); 10.1063/1.3638052 Gravitational ManyBody Problem AIP Conf. Proc. 1011, 199 (2008); 10.1063/1.2932288 Locality of the field operators of manybody theory Am. J. Phys. 48, 782 (1980); 10.1119/1.12012 The Many-Body Problem in Quantum Mechanics Am. J. Phys. 37, 116 (1969); 10.1119/1.1975378 ManyBody Problems Phys. Today 17, 92 (1964); 10.1063/1.3051508
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Operator algebra for the many-body problem in the spin eigenfunction basis Wlodzislaw Ouch Zaklad InJormatyki Stosowanej UMK, ul. Grudziqdzka 5, 87-JOO Torun, Poland
(Received 27 January 1989; accepted 17 April 1989)
In the many-body perturbation theory and related methods equations involve sums over orbital indices. Reduction of the many-body equations to such a form is not trivial when spin eigenfunctions are used. A method to achieve such reduction, based on the algebraical properties of the unitary group generators, is described. As an example of application a formula for the second order MBPT with an arbitrary open shell reference state of the RHF type is derived.
I. INTRODUCTION
Many-body perturbation theory (MBPT) is the best known example of a noniterative technique providing an efficient treatment of the correlation problem in atoms and molecules. 1 It is most easily formulated at the level of Nparticle Hamiltonian matrix elements and in this form it has been extended to multireference cases. The greatest advantage ofMBPT is, apart from size consistency, computational efficiency, achieved thanks to the reduction of pertinent formulas from the many-particle to the single-particle level. The energy expressions are given as sums over the orbital indices of integral products divided by appropriate denominators. A voiding explicit construction of many-particle matrix elements is always worthwhile and is the reason for the success of the direct CI method.2 However, it is, in general, not an easy task. Despite all the progress in the group-theoretical approaches to the calculation of matrix elements determinants are still the most commonly employed type of functions. 3 Formulation ofMBPT equations using determinants is straightforward. To handle many types of interactions appearing in higher orders of PT diagrammatic methods of calculation were invented and proved to be very helpful.4 As long as the N-particle functions have little symmetry these methods work very well. Generalization of the theory to the open-shell or multireference cases demands adding at least spin symmetry and that proved to be difficult. Graphical methods of spin algebras were developed to handle spin adaptation of the many-body equations but they never became very popular, although a few very good books exist on the subject. 4
Trying to keep things simple, Pople and co-workers5
suggested the use of unrestricted Hartree-Fock reference functions in the open-shell cases and this approach is the most popular at present. It has two well known disadvantages; spin contamination and the increased computational effort due to the doubling of the number of orbitals. Trying to improve the situation spin-projection techniques were recently proposed by a few authors.6 This does now always solve the problem and leads to rather messy algebra if more than one projector is included. The results are nevertheless very encouraging.
An alternative approach would be to start from the open-shell version of RHF treating all orbitals as singly occupied. In this paper the tools are provided enabling easy derivations of the necessary matrix elements in the general
open shell case and, in perturbative methods, writing them in the form of sums over orbital indices. The most convenient starting point is not the combination of determinants but, from the beginning, spin eigenfunctions with a set of operators acting on these functions. The operators have a number of algebraic properties that simplify the equations and the derivations. After working out these properties derivation of the second order MBPT formulas is presented as an illustration of their application. The techniques developed here should be useful in different formulations of many-body theory.
II. ALGEBRA OF SHIFT OPERATORS
The finite-dimensional space of N-particle, spin-adapted functions is built from properly symmetrized tensor products of N one-particle functions. The spin-independent operators, that map one N-particle function into the other in the many-particle case are conveniently expressed as
N
Eij = I liCk) Xj(k) I· (2.1 ) k~l
Acting on a tensor product Eij operators replace one-particle function [j) with I i), therefore they are known as the "replacement operators" or the "shift operators." On the other hand, commutation relations of these operators
[Eij,Ek/] = OjkEi/ - oi/Ekj (2.2)
are the same as for the generators of the unitary grou p U (n ) , therefore they are also known as the unitary group generators. 7
-9 An arbitrary spin-independent one- or two-particle
operator, defined in the space of one particle functions li(k» is equivalent to a combination of the one- and twoparticle integrals multiplied by these operators. In particular in the perturbation approach to the many-body problem one is using the two-particle perturbation operator
V = ~ I Wlkl)Eij.k, 2 ijkl
= -21 I Wlkl)(EijEk' - OjkEi/) (2.3)
ijkl
to compute elements of the following type:
(OIVIO), (OIVRVIO), (OIVRVRVIO), ... , (2.4)
where R is a resolvent operator introducing appropriate denominators in the matrix element expressions. In practice,
2452 J. Chern. Phys. 91 (4), 15 August 1989 0021-9606/89/162452-05$02.10 © 1989 American Institute of Physics
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W/odzisJaw Ouch: Operator algebra for the spin basis 2453
perturbation operator includes also the one-particle terms but their matrix elements are quite simple. Products of the shift operators that have non-vanishing elements must be of the type
Wljk .. /i=EIjEjk··E/i' (2.5)
These "circular" operators summed over all indices give the Casimir invariants of the U(n) groUp.lO Properties of the traces of such operators over the whole N-particle space were described recently .11 However, in many-body theory it is important to investigate general properties of these operators derived from commutation relations (2.2). For the operators with all indices different one easily finds
15' Iji = Wjlj + Wjj - Wjj = Wjlj + n i - n j , (2.6)
W Ijki = W kljk + 15' Iji - W kjk
= W jklj + W iki - 15'jkj ,
W Ijkli = W jkllj + W ikli - W jklj
= W klljk + W Ijli - W kljk
= W Iljkl + W Ijki - W Ijkl
or in general
(2.7)
(2.8)
W Ij"m"qi = W m"qlj"m + W Ij"qi - W m"qj"m' (2.9)
Some properties are valid only for matrix elements. The shift operators are not Hermitean but diagonal elements of the conjugate circular operator are the same
(01 W Ijk"1i 10) = (01 W i/. 'kji 10). (2.10)
The value of these matrix elements depends on the occupation of the one-particle functions in the state 10). From now on let us introduce a following convention: indices a, b, c for unoccupied (na = nb = nc = 0), s, t, u for singly occupied, and i,j, k for doubly occupied orbitals, while p, q will denote arbitrary indices. Since excitation from unoccupied or to the doubly occupied orbitals is forbidden it is obvious that
(01 W a"'a 10) = (01 W. 'i" 10) = O. (2.11)
Using this result and the identities (2.6)-(2.9) we have
(01 W pap 10) = (01 Wapa + np - na 10)
= (01 W pp 10) = n p '
(01 W iqi 10) = (01 W qiq + n i - nq 10 )
= 2 - nq •
(0115' pabp 10) = (0115' abpa + 15' pbp - W aba 10)
= (0115' pp 10) = n p '
(01 W isti 10) = (01 W stis + W iii - W sIs 10)
= 1- (01 15' sIs 10).
(2.12)
Using Eq. (2.9), one may prove that unoccupied oribtal indices may always be removed
(2.13 )
These simple properties of the circular operators are all that is needed to reduce expressions for their matrix elements to the occupation numbers or to the operators involving singly occupied orbitals only. As one expects, single occupied orbitals lead to some complications. The number of relations
that one may derive for ciruclar operators starting from commutation relations is large. In general diagonal elements of the circular operators with more than two different indices are expressed through the elements of operators with lower number of indices. The proofs of the equalities given below are very simple, usually involving a few commutation relations. Products of operators in which there are two excitations from or to the singly occupied orbitals (e.g., ... EUIEsI or '" ESIEsu) give, of course, zero contribution and may be discarded. Some useful relationships are gathered below:
W sIs = 15' lSI'
[W SIS,W UIU] = W suls - W slus'
(01 [W SIS'W UIU] 10) = 0, (2.14 )
(0115' sIs (W sus + 15' utu) 10) = (01 W sts 10), (01 W SIUS 10) = (0115' sts (1- W,ul)IO),
(01 W SIUS 10) = !(OIWsls + Wsus - W,uIIO).
Many other relations may be deduced in the similar way, but these are already sufficient for some nontrivial applications. We have still to learn how to compute numerical values of the circular operators with singly occupied indices. How does W sIs act on the orbital product I' ·tPStPI··)?
W sIs I' ·tPs· ·tP,··) = ESI I' ·tP,· ·tP,··)
= I' ·tPStPI··) + I' ·tPltPS··) (2.15)
= [1 + (s,t)] I' ·tPstPt··)·
Thus, operator 15' sIs is equivalent to a transposition (s,t) plus one
W SIS = 1 + (s,t) (2.16) acting on the orbital part of the 10) function. If s, 1 are a singlet coupled pair then the orbital part is symmetric for the (s,t) exchange, ifthey are coupled to triplet then the orbital part is antisymmetric. Therefore
(OIW sI 10) = {2 for s,1 si~glet. (2.17) S 0 for s,t tnplet
Although one may directly compute these values it is fruitful to explore the symmetric group approach in connection with the matrix element calculation of circular operators. For matrix elements in the spin-adapted basis
(01 W SIs 10) = (01 W,st 10) = 1 + (S,I) (2.18 )
is expressed through the spin integral (s,t ) corresponding to the transposition of spins number sand t (in the determinantal basis (s,t) is ± 1). If 10) is an open-shell state it may correspond to a number of functions that differ only in the spin couplings. Then (S,1 ) is a matrix of appropriate dimension (representation matrix of the permutation group for the transposition (s,t) , as pointed out already by Kotani 12). Elements of these matrices or spin integrals are easily calculated directly from the branching diagram I3 and in most cases take only a few values that can be tabulated. In a slightly more complicated case we obtain
(01 W slus 10) = (01 W sIs 15' sus - ESIEsuErsEus 10)
= (01 WSls 15' sus 10) = (1 + (s,t»)(1 + (s,u»
= 1 + (s,l) + (s,u) + (s,t,u), (2.19)
J. Chern. Phys., Vol. 91, No.4, 15 August 1989
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2454 WfodzisJaw Ouch: Operator algebra for the spin basis
(01 ~ stuws 10) = (1 + (s,t» (1 + (s,u»( 1 + (s,w»,
where (s,t,u) = (01 (s,t) (t,u) 10) is a diagonal element of a product of matrices corresponding to the transpositions (s,t) and (t,u). If 10) is a single function one may prove, using methods of reference 14 that (s,t,u) is also a product of spin integrals (s,t ) and (t,u). General pattern is clearly visible from the last equation.
It is worthwhile to note the identity for a totally symmetric operator
(2.20)
where P belongs to the permutation group of more than two elements; it comes from the fact, that it is impossible to form spin function antisymmetric in more than two variables. In particular for three spins
(Oil + (s,t) + (s,u) + (u,t) + (s,t)(t,u)
+ (s,u) (t,u) 10) = 0, (2.21)
i.e., diagonal elements of such symmetric combination of permutations are zero. Since (s,t,u) = (s,u,t) the last identity gives
(s,t,u) = - !(011 + (s,t) + (s,u) + (u,t)IO)
= 1- !(Ol~sts + ~sus + ~tutIO). (2.22)
On the other hand, from Eq. (2.21) we have
(s,t,u) = 1 + (Ol~stus - ~sts - ~susIO) (2.23)
which is equivalent to the last identity in Eq. (2.14).
III. APPLICATION TO THE MANY-BODY PERTURBATION THEORY
The properties of the circular operators derived above enable calculation of arbitrary matrix elements in the spin eigenfunction basis, in particular derivation of the formulas for the many-body perturbation theory with the arbitrarily complicated reference state 10). The second order perturbation theory formula is
E2 = L (OIVIK)(KIVIO) K EK -Eo
= L L (OlVlmn--pq) (mn--pqIVIO), (3.1) m<;n p<;q D(m,n,p,q)
where D( m,n,p,q) are appropriate denominators, depending on the partitioning of the Hamiltonian. The part of the perturbation operator (2.3) connecting states differing on two orbitals is 13,15
V2 = 2 -/jmn/jpq(mplnq)EmpEnq
+ (1 - Dmn)( 1 - Dpq )(mqlnp)EmqEnp ' (3.2)
Calculating matrix elements of the shift operators one easily notices that
(OIEmpEnq Imn--pq) (mn--pqIEqnEpm 10)
= (OIEmpEnqEqnEpm 10 ) (3.3 )
provided that all function (differing in the spin couplings) associated with the doubly excited configuration Imn--pq) are taken into account. A few commutations bring Eq. (3.1) to the following form:
E2 = L L {(Ol ~ mpm ~ nqn 10) (mplnq)2 + (01 ~ mqm ~ npn 10) (mqlnp)2
+ 2(01~ mpnqm - ~ mpqmIO)(mplnq)(mqlnp)}ID(m,n,p,q)
+ L L (Ol~ mpm~ mqm - ~ mpqmI0 )(mplmq)2ID(m,m,p,q) m=np<q
+ L L (01 ~ mpm ~ npn + ~ mnpm - ~ mpm 10) (mplnp)2D(m,n,p,p)
+ L L J.- (Ol~ mpm~ mpmI 0 )(mplmp)2ID(m,m,p,p). m=np=q 4
(3.4)
Numerical values of the spin integrals in this expression are immediately evaluated using results of the previous section. It is enough to specify the occupation numbers of the four indices; m = n implies nm = 2,p = q implies np = O. Therefore the last three sums become
E2 = ... L L [(01 ~ qpq 10) + (np - 2)(nq - 1)] Upliq)2IDU,i,p,q) + L L [(01 ~ mnm 10) i p<q m<n a
+ nm (nn - 1)] (malnafID(m,n,a,a) + L L Ualia)2IDU,i,a,a). (3.5 ) i a
Sincenm,nn = 2,1 andnp,nq = 0,1 the four-index sums are separated depending on the occupation numbers into cases with different coefficients multiplying the two-electron integrals. The case with all four indices singly occupied is the most complicated. Fortunately, it does not contribute in the second order if the denominators are antisymmetric in re-
spect to the exchange of pairs of indices s, t and u, w, i.e., D(s,t,u,w) = - D(u,w,s,t} (it is true in most partitionings of the Hamiltonian operator). This comes from the fact that the same integrals are multiplied by the sum of coefficients coming from two intermediate states, 1st -- uw) and luw--st), with denominators of opposite signs. Spin inte-
J. Chern. Phys., Vol. 91, No.4, 15 August 1989
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W/odzisJaw Ouch: Operator algebra for the spin basis 2455
TABLE I. Explicit values of spin integrals for + - + - ... + - + + + + .. spin path. s<t< u order is assumed,
no = N - 2S, where S is the total spin and N the number of singly occupied orbitals in 10). SCP means "singlet-coupled pair."
2( (s,t,u) 2( (s,t,u) 2( (s,t,u) 2(s,t) 1 + (s,t) + (s,t» + (s,u» + (t,u» Conditions
-2 0 0 0 0 no<s -1 1/2 0 0 -1 s<no<t
2 2 1 -2 -2 (s,t) is SCP -2 -2 1 (t,u) isSCP
-1 1/2 1/2 -1/2 -1/2 otherwise
grals for both configurations are the same. First, (01 ff} sus ff} twt 10) = (01 ff} usu ff} wtw 10) and for the second type of spin integral in Eq. (3.4) the difference between the two coefficients is transformed, using Eqs. (2.8) and (2.14) to
(01 ff} swtus - ff} swus - ff} utwsu + ff} utsu 10)
= (01 ff} stws - ff} utwu + ff} utsu - ff} utsu 10) = o. (3.6)
Spin couplings in the 10) state are arbitrary, but the simplest spin integrals are obtained when the "spin up" orbitals are shifted to the last position, i.e., the coupling is up-down-updown" ·up-up-up. All values of the spin integrals are collected in Table I.
Finally, calculating arbitrary matrix elements using the same technique one should remember, that Eij 10) does not give the same functions as those associated with li ..... j) configuration.
IV. SUMMARY
The algebra of circular operators, presented in this paper, gives an elegant method to compute various factors coming from the spin couplings of the S 2 eigenfunctions. It is very similar to the approach of Matsen and Pauncz (cf. Ref. 8, Chap. 9), used by these authors to compute generatorstate matrix elements. To avoid problems with nonortho-
gonality and overcompleteness of the generator states Eij lo} configurations lik-+jl), as defined in Ref. 15 are used here ( each configuration is equivalent to a set of Gelfand states). The same method may be applied to the higher order MBPT and to other many body methods such as propagator techniques. The number of terms in higher orders may be large, 16
therefore it is advisable to employ a symbolic algebra program such as MACSYMA or REDUCE for derivations. One may couple the algebra presented here with the diagrammatic methods used in MBPT or with the graphical methods developed in Ref. 15. The second order formula derived here may be useful in connection with various approaches to the open-shell or multireference perturbation theory. 17
ACKNOWLEDGMENTS
Support of the Institute for Low Temperature Research, Polish Academy of Science, under CPBR 01.12 program, is gratefully acknowledged. I wish to thank Professor Gerald Segal for the great fun we had discussing and developing the open-shell PT program together.
APPENDIX: FORMULAS FOR THE SECOND-ORDER PERTURBATION THEORY
Assuming i <j < k < I the three two-electron integrals with the same indices are denoted
J 1 = (ijlk/), J2 = (il Vk), J3 = (ik VI).
The following order of indices among doubly, singly, and unoccupied orbitals is assumed:
i <j < s < t < u < a < b.
The formula for the second-order energy in the M011erPlesset partitioning is rather long, but it has very clear structure and is ready for coding without any further processing. In the first two lines the usual formula for the closed-shell MP2 is written, then one, two, and three open-shell terms are added.
+ t:t ~b J~c~ :!b+_2;:'~;~3 + t:t O~b ~~~ ~s;~ ~~: + t:t + ~ { J~ + ~~ 1++c~S~);'J~(~~ - J3)
+ Ji + 2(1 + (s~»J~(J2 -J1)}
CS + Co Cj Ct
+ I ~{n (1 + (t,u» + J~ (1 + (s,~) ~ 2( (s,t,u) + (s,u» J 1J2
s<t<u I Cs + Cu Cj ct
n (1 + (t,u» + J~ (1 + (s,u» + 2( (s,t,u) + (s,t) )J1J3 +-------------------------------------------Cs + ct - Cj - Cu
J. Chern. Phys., Vol. 91, No.4, 15 August 1989
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2456 W/odzislaw Ouch: Operator algebra for the spin basis
J; (1 + (s,u» + J~ (1 + (s,t» + 2( (s,t,u) + (t,u) )J2J3 +--------------------------------~--~~
E, + Eu - E; - Es
+ L L {Ji (1 + (s,t» + J~ (1 + (t,u» + 2( (s,t,u) + (s,u) )J)J2
s<'<U 0 E,+Eo-Es-Eu
Ji (1 + (s,t» + J; (1 + (s,u» + 2( (s,t,u) + (t,u) )J)J3 +------------~----------------~--~~
Es + Eo - E, - Eu
J~ (1 + (s,u» + J~ (1 + (t,u» + 2( (s,t,u) + (s,t) )J2J3 } + . Eu + Eo - Es - E,
IS. Wilson, Correlation Energy in Molecules (Clarendon, Oxford, 1982). 28. O. Roos and P. E. M. Siegbahn, in Methods of Electronic Structure Theory, edited by H. F. Schaefer III (Plenum, New York, 1977), p. 277.
3See, for example, A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (McMillian, New York, 1982), or Ref. 1.
4E. EI Baz and B. Castel, Graphical Methods of Spin Algebras (Dekker, New York, 1972); A. P. Iucys, I. B. Levinson, and V. V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum (Gordon and Breach, New York, 1964).
'w. I. Hehre, L. Radom, P. v. R. Schlayer, and I. A. Pople Ab Initio Molecular Orbital Theory (Wiley, New York, 1986).
6H. B. Schlegel, J. Chern. Phys. 84,4530 (1986); P. Knowles and N. C. Handy, Projected Unrestricted MlJller-Plesset Second Order Energies (to be published).
7R. Pauncz, Spin Eigenfunctions: Construction and Use (Plenum, New York, 1979).
SF. A. Matsen and R. Pauncz, The Unitary Group in Quantum Chemistry
, (Elsevier, Amsterdam, 1986).
9J. Paldus and B. Jeziorski, Theoret. Chim. Acta 73, 81 (1988). 10M. D. GouldandG. S. Chandler, Int. J. Quantum Chern. 25, 553 (1984). III. Karwowski, W. Duch, and C. Valdemoro, Phys. Rev. A 33, 2254
(1986). 12M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, Tables of Molecu
lar Integrals (Maruzen, Tokyo, 1955); see also E. Corson, Perturbation Methods in the Quantum Mechanics ofN-electron Systems (Blackie, London, 1951).
13W. Duch and J. Karwowski, Int. J. Quantum Chern. 22, 783 (1982); W. Duch and J. Karwowski, Compo Phys. Rep. 2, 92 (1985).
14W. Duch, Int. J. Quantum Chern. 27, 59 (1985). 15W. Duch, Graphical Representation of Model Spaces. Vol. I. Basics, Lec
ture Notes in Chemistry, Vol. 42 (Springer, Berlin, 1986). 16S. Wilson, Theoret. Chim. Acta 61,343 (1982). 17G. Hose and U. Kaldor, Chern. Phys. 62, 469 (1981); K. Wolinski, H. L.
Sellers, and P. Pulay, Chern. Phys. Lett. 140,225 (1987).
J. Chern. Phys., Vol. 91, No.4, 15 August 1989
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