Direct determination of the cumulants of the reduced density matrices

11 February 2000

Ž .Chemical Physics Letters 317 2000 567–574www.elsevier.nlrlocatercplett

Direct determination of the cumulants of the reduceddensity matrices

Werner Kutzelnigg a,), Debashis Mukherjee a,b

a Lehrstuhl fur Theoretische Chemie, Ruhr-UniÕersitat Bochum, D-44780 Bochum, Germany¨ ¨b Department of Physical Chemistry, Indian Association for the CultiÕation of Science, Calcutta 700032, India

Received 13 September 1999; in final form 16 November 1999

Abstract

Unlike the reduced k-body density matrices, the k-body cumulants generated from them are extensive entities. Weformulate a theory directly in terms of these cumulants, based on stationarity of the energy with respect to variations inducedby k-body excitation operators, that are in generalized normal order – as defined recently – with respect to the exactwavefunction. In terms of this new normal ordering, cumulants arise directly on taking expectation values. The equationsdetermining the cumulants contain explicitly connected terms only, which ensures the extensivity of the theory. Our methodresembles somewhat that of contracted Schrodinger equations, but differs in essential aspects from it. q 2000 Elsevier¨Science B.V. All rights reserved.

1. Introduction

The usual many-electron Hamiltonian only con-sists of one- and two-electron terms, which impliesthat a hierarchy of approximations for the wavefunc-tion C and the energy E should start with one- andtwo-electron terms and improve upon them if neces-sary. Standard methods express, for example, C in

Žterms of a reference function F often a single Slater.determinant and a wave operator U

1p q p q r sCsUF ; UsU qU a q U a q . . . ,0 q p r s p q4

1Ž .

a p sa† a ; a p q sa† a† a a ; . . . , 2Ž .q p q r s p q s r

where a p and a p q are one- and two-particle excita-q r s

) Corresponding author. Fax: q49-234-3214109; e-mail:[email protected]

tions operators, respectively, expressible in terms ofŽ p † .the creation a sa and annihilation operatorsp

Ž .a for a set of orthonormal spin-orbitals c . Thep p

Einstein summation convention over repeated indicesw xis used throughout 1 . The parametrization of C by

Žthe U-coefficients in the sense of CI configuration.interaction is in conflict with the requirement of

multiplicative separability of U and one actuallyrather parametrizes C in terms of the cumulants S p,q

etc., related to the U p asq

S p sU p ; S p q sU p q yU pU q qU pU q , etc. ,q q r s r s r s s r

3Ž .

which amounts to writing the wave operator as anexponential type of operator

1p q p q r sUfexp S ; SsS qS a q S a q . . .Ž . 0 q p r s p q4

4Ž .

0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 99 01410-4

( )W. Kutzelnigg, D. MukherjeerChemical Physics Letters 317 2000 567–574568

in the sense of coupled cluster theory. The clusterŽ .amplitudes S are additively separable extensivek

and allow an expression in terms of connected dia-grams, and are thus well suited for n-electron quan-tum mechanics.

One may wonder, however, why one should basethe theory on a reference function and a cumulantexpansion of the wave operator and whether onecould not use directly a cumulant expansion of thewavefunction C or a quantity more directly relatedto it. One can, indeed construct a cumulant expan-sion of the n-electron density matrix CC ) and thereduced k-particle density matrices. The k-particle

w xdensity cumulants l 2–6 can be defined via akw xgenerating function 5,6 . The l , l , l are related1 2 3

to the corresponding density matrices g , g , g in1 2 3w xthe following way 2–6

l p sg p , 5Ž .q q

l p q sg p q yg pg q qg pg q , 6Ž .r s r s r s s r

Pp qr p qr p qr p q rl sg y y1 g l ydet g g g ,� 4Ž .Ýstu stu s tu s t uP

7Ž .

where P is the parity of the respective permutationof labels and the density matrix elements are definedas

q ² < q < : r s ² < r s < :g s C a C ; g s C a C ; etc. 8Ž .p p p q p q

We refer to the matrix with element g p as g , to theq 1

matrix with elements l p q as l , etc. There is ar s 2

challenge to formulate a theory entirely in terms ofthe cumulants, and this is the topic of the presentLetter.

The method that we are going to propose hassome similarities with that of contracted Schrodiger¨

Ž . w xequations CSEs 6–8,12–20 . This method has tra-ditionally not been expressed in terms of cumulants.

w xOnly quite recently Mazziotti 6–8 has shown that acompact and transparent formulation of the CSEmethod, especially a rationalization of the tricksintroduced in the ‘reconstruction’ of density matricesof higher particle rank from those of lower particlerank, is possible if one introduces the cumulants of

w xthe density matrices. Mazziotti’s analysis 6–8 re-veals that on may look at the CSE method as beingbased on the implicit unconcious use of properties ofthe cumulants.

Our method differs in two important aspects fromthat of the CSEs, even in the reformulation of Mazz-

w xiotti 6–8 . On one hand, we replace the CSEs by theŽ .much simpler generalized Brillouin conditions BCs

whenever this is possible. On the other hand, weŽ .apply the irreducible connected counterparts of the

CSE or BC rather than the original CSE or BC. Thisdoes not only let the cumulants appear directly, usingthe generalized Wick theorem with respect to arbi-

w xtrary wavefunctions introduced by us recently 2–4 .It also leads to a formulation of the stationarityconditions in terms of connected diagrams only,which is not the case for the CSE. It finally leads toa new hierarchy of equations, defined in terms of atruncation at a given particle rank k, such that at anylevel of this hierarchy one-particle, two-particle, . . . ,up to k-particle equations are solved in a self-con-sistent way, and each level uses all the informationobtained at lower levels.

Although a description of an n-electron system interms of g and the l appears – by hindsight –1 k

more natural than that in terms of a reference func-tion F and the cumulants of the waÕe operator, it isnot so difficult to understand why the latter hasdominated for such a long time. The point is that therequirement that the n-electron wavefunction shouldbe antisymmetric, is entirely taken care of by thereference wavefunction F ; application of the waveoperator as a particle-number-conserving Fock spaceoperator does not change this property. There is no

w xn-representability problem 9 on this way.If one formulates the theory in terms of g and1

the l one must, however, explicitly take care ofk

n-representability, i.e. guarantee that g and the l1 k

are derivable from an antisymmetric n-electronwavefunction. This n-representability problem has,to a large extent, inhibited approaches based on thevariation of the energy as a functional of the reduceddensity matrices.

Our approach is not based on the energy as afunctional of the density matrices or their cumulants,we rather start from stationarity of the energy withrespect to variation of the waÕefunction. Only theresulting stationarity conditions are expressed interms of the density matrices or rather their cumu-lants. So we largely circumvent the n-representabiltyproblem. This may only arise, to the extent that the g

or l are not entirely determined by the stationarity

( )W. Kutzelnigg, D. MukherjeerChemical Physics Letters 317 2000 567–574 569

conditions, or as a result of a truncation of thecumulant expansion. We shall come back to thispoint.

In the ‘one-electron approximation’, with whichwe start, n-representabilty is easily checked. Actu-

Ž . w xally g is pure or ensemble n-representable 9 if1

and only if its eigenvalues n lie between 0 and 1pŽ Ž ..for the normalization chosen in 8

0Fn F1 ; n sn 9Ž .Ýp pP

The most important necessary n-representabilityconditions for l are the antisymmetry and the par-2

w xtial trace relations 5 , which take care of the consis-tency between g and l .2

l p q sylq p syl p q slq p , 10Ž .r s r s sr sr

ppr 2l s g yg . 11Ž .Ž . qqr

w xThere are further inequalitites for l 10 , based on2w xthe well-known property 9 that g as well as the2

two-hole matrix h and the particle–hole matrix b2 2

must be non-negative.

2. Stationarity conditions

We write the n-electron Hamiltonian in Fockspace as

1p q p q r sHsh a q g a , 12Ž .q p r s p q2

p ² < < :h s c h c ;q p q

p q ² < y1 < :g s w 1 w 2 r w 1 w 2 , 13Ž . Ž . Ž . Ž . Ž .r s r s 12 p q

Ž .with the excitation operators defined by 2 . Theenergy expectation value is

1p q p q r sEsh g q g g . 14Ž .q p r s p q2

w xSome authors 9,11 – in order to stress the centralrole of g – have rewritten E as2

1 p q r s˜Es H g , 15Ž .r s p q4

1p q p q q p p q q pH̃ s h d qh d yh d yh d� 4r s r s s Õ s r r sny1

p qqg , 16Ž .r s

p q p q q pg sg yg , 17Ž .r s r s r s

Ž .and called the operator given by 16 the reducedHamiltonian.

While it is hopeless to construct g from mini-2Ž .mization of 15 with respect to variation of g , due2

w xto the n-representability problem 9 , one can, ofŽ .course, proceed to minimize 14 with respect to

variation of C . There are two types of stationarityconditions for the energy, the first for stationaritywith respect to arbitrary variations of the wavefunc-

Žtion, the second with respect to unitary norm con-.serving variations

² < < :C Z HyE C s0 18Ž . Ž .² < < : †w xC H ,Z C s0 ; ZsyZ , 19Ž .

Ž .In 18 the Z can be chosen as a set of k-particleexcitation operators

Zs a p ,a p q , a p qr , . . . . 20Ž .� 4q r s stu

Ž . p qIn 19 antihermitean operators like a ya shouldq pŽ .be taken, but one can as well choose the set 20

w x Ž .21 . In 18 E is a Lagrange parameter, that canlater be identified with the energy, while no La-

Ž .grange parameter appears in 19 . Unlike in Eqs.Ž . Ž .18 and 19 , in methods based on reference func-tions and wave operators one varies the wave opera-tor U rather than the wavefunction.

Ž .The set of stationarity conditions 18 is the basisw xof the method of the CSE 6–8,12–19 . For the

choice of all two-particle excitation operators as ZŽ .this set includes the one-particle operators one getscoupled equations that relate g to g and g . In the2 3 4

Ž . w xCSE literature this is referred to as 2, 4 -CSE 8,18 .Of course, g and g are unknowm, but Valdemoro3 4w x16,17 and to some extent Nakatsuji and Yasudaw x14 were able to ‘reconstruct’ approximations to g3

and g in terms of g . One then arrived at a set of4 2

approximate equations for g The relation of this2

‘reconstruction’ of g and g from g to the cumu-3 4 2

lant expansion of density matrices was formulated byw x ŽMazziotti 6–8 . The obvious generalization to n, n

.q2 CSE, for the construcction of g were alsonw xproposed 18

Ž .The set of conditions 19 in terms of the excita-tion operators were earlier studied by one of the

w xpresent authors 22,23 . The first member

p² < < :C H , a C s0 21Ž .q


has long been known as the Brillouin condition ofSCF or MC–SCF theory, and the next memberswere baptized ‘generalized k-particle Brillouin con-ditions’, starting with the two-particle Brillouin con-dition:

p q² < < :C H ,a C s0 . 22Ž .r s

When one evaluates these conditions either in theŽ . Ž .form 18 or 19 one finds that they contain discon-

Ž . Ž .nected terms, e.g. 22 contains contributions of 21multiplied by a factor with disjoint spin-orbital la-

Ž .bels. Something similar arises for the set 18 , i.e. forthe CSE.

Disconnected terms should not be present in anacceptable many-electron theory. Their origin is re-lated to the fact that g is not a separable quantity2Ž .like C and the higher-rank g and is actually notk

the recommended choice for the formulation of amany-electron theory. To express the CSE or BCk k

in terms of the l instead of the g does not help ink k

this respect; the stationarity conditions remain dis-connected.

One can remove the disconnected parts ‘by hand’,but this is extremely tredious. It is much more

Ž .elegant to arrive at a separable connected theory ifŽ .one replaces the excitation operators 20 by their

w xcumulant analogues, characterized by a tilde 4 ,namely by

a p sa p yg p 23Ž .˜q q q

a p q sa p q yg paq yg qa p qg paq qg qa p˜ ˜ ˜ ˜ ˜r s r s r s s r s r r s

yg pg q qg pg q yl p q . 24Ž .r s s r r s

These are nothing but the excitation operators in the‘generalized normal order’ with respect to the exactŽ .correlated C as reference functions. They were

w xintroduced by the present authors 4 , while the veryconcept of generalized normal ordering and of theuse of cumulants was first proposed by one of usw x2,3 . The tilde operators have vanishing expectationvalues with respect to C , and expectation value ofproducts of a operators can be expressed entirely in˜terms of various k-particle cumulants via a general-

w xization of Wick’s theorem 2–4 . The use of the a,˜Ž . Ž .rather than of the a operators in Eqs. 18 and 19

thus leads to expressions involving only extensivequantities.

Ž . Ž .The evaluation of the counterpart of 21 and 22becomes relatively easy if one first writes the Hamil-

Ž .tonian 1 in normal order with respect to C , i.e. as

1p q p q r sHsEq f a q g a , 25Ž .˜ ˜q p r s p q2

with

1 1p p q p q r sEs h q f g q g l , 26Ž .Ž .q q p r s p q2 2

p p pr sf sh qg g , 27Ž .q q qs r

w xand uses then the extended Wick theorem 2–4 toexpress products of a operators as sums of a opera-˜ ˜tors. To the expectation value with respect to C only

Žthe full contractions which are directly expressible.through the cumulants contribute and one gets ex-

plicitly the following irreducible one- and two-par-(ticle Brillouin conditions IBC and IBC , repec-1 2

)tiÕely :

1 1p r p r pr st p r stf g yg f q g l y l g s0 , 28Ž .r q r q st qr st q r2 2

p uq q p q p q t p q t p qf l q f l yl f yl f qg 1yn� Ž .u r s u r s t s r r t s r s p

= 1yn n n y 1yn 1yn n nŽ . Ž . 4Ž .q r s r s p q

1 1p q tu Õw p qq g l 1yn yn y g lŽ .t u r s p q r s Õw2 2

u p q t= 1yn yn q n yn g lŽ . Ž .r s r p t r su

uq p t u p q ty n yn g l y n yn g lŽ . Ž .r q t r su s p t s r u

1uq p t pw tuqq n yn g l q g lŽ .s q t s r u tu r w s2

1 1 1w q tu p Õw p qu Õw p quq g l y g l y g l s0 , 29Ž .t u r w s r u Õ sw u s Õ r w2 2 2

Ž .Eq. 29 has been written in terms of natural spin-Ž . porbitals NSOs , which diagonalize g , i.e. g s1 q

n d pp q

Ž .As an alternative to basing the theory on Eqs. 28Ž . Ž .and 29 , one can evaluate the conditions 18 for the

Ž .counterpart of 19 with the a-operators replaced bythe a operators. An interesting point is that the˜

Ž .Lagrange multiplier E in 18 cancels with the con-Ž .stant part in the Hamiltonian 25 such that the

stationarity conditions become E-independent. Thisleads to the irreducible one- and two-particle CSEs


Ž .ICSE and ICSE , respectively . We get for the1 2Ž .first member, i.e. the counterpart of 28

1r p s r s p tu² < < : ² < < :0s f C a a C q g C a a C˜ ˜ ˜ ˜s q r tu q r s2

1r p s p s r s p tu p tu t pus f l qg h q g l qg l yh l ,� 4 � 4s qr r q tu qr s r qs q r s2

30Ž .

h t sd t yg t . 31Ž .q q q

Ž .There is also a counterpart to 29 . The differenceŽ . Ž . Ž .between 28 and 30 is that in 28 only g and l1 2

Ž .are needed, while in 30 we need g , l , and l .1 2 3Ž .Analogously 29 contains g , l and l , but the1 2 3

Ž .respective two-electron equation not displayed here ,contains in addition l . This makes the Brillouin-type4

Ž . Ž .conditions 28 and 29 simpler and more compact.Ž . Ž .However, Eqs. 28 and 29 cannot by themselves

provide equations determining the diagonal elementsof g and l, since they are identically zero with1

diagonal Z operators like a p or a p q. Thus, we need˜ ˜p p q

to augment them by some additional conditions. Wewill discuss this aspect briefly in Section 3. TheICSE, in contrast, provides equations for both thediagonal and the off-diagonal elements of g and l1

and hence it appears that we do not require anyadditional conditions to determine them. On the otherhand, the ICSEs involve the l of higher ranks ascompared to the IBCs at any given rank of Z-oper-ators used in IBC or ICSE. The two approaches havethus complementary merits, and it is worthwhile tocombine them in a useful theory.

Ž . Ž .While Eqs. 28 and 29 hold for arbitrary en-Ž .semble states, Eq. 30 and its two-electron counter-

part are restricted to pure states or ensembles ofdegenerate states.

All the above equations are most convenientlyexpressed in terms of diagrams. The diagrammaticsas well as a more detailed formulation of our ap-

w xproach will be published elsewhere 10 .

3. Solution of IBC and ICSE

As we have mentioned, the IBC and IBC , i.e.1 2Ž . Ž .Eqs. 28 and 29 , are necessary conditions for the

energy of a pure or ensemble n-particle state to bestationary with respect to arbitrary variations of the

wavefunction, but they are insufficient to character-ize a state entirely. Additional information is needed,e.g. whether we consider the ground state for whichthe energy should have an absolute minimum. Eqs.Ž . Ž .28 and 29 even hold for an ensemble state. The

Žparticle number n and, if necessary the total spin,.etc. must also be specified, since they do not follow

from the sationarity conditions.It is appropriate to define a hierarchy of approxi-

mations. The first member of this, which we refer toas ‘one-particle approximation’ consists in consider-

Ž .ing only 28 and neglecting l . Then one obviously2

has to solve – as is not surprising – the Hartree–Fockequations

pp r p r w xf g yg f s f ,g s0 . 32Ž .qr q r q

This means that the one-particle density matrix g1Ž .commutes with the Fock operator f , defined by 27 .

Ž .We get from 32 the information that f and g have1

common eigenfunctions, but we do not get a condi-tion on the eigenvalues of g , i.e. on the occupation1

numbers n of the NSOs. Fortunately, we know thepw xexact relation 5 that vanishing of l is only consis-2

tent with idempotency of g , hence exactly n NSOs1

will have occupation number 1, the others occupa-tion number 0. For a ground state we shall in view ofthe relation

1 p fEs h q f n 33Ž .Ž .Ý p p p2p

occupy the NSO with lowest energy with n s1.p

This is unique if the highest NSO is nondegenerate.Otherwise also fractional occupations are possible.

2 w xBut only g sg is compatible with l s0 5 , so1 1 2

for fractional n , i.e. for an open-shell state somep

elements of l must be taken into account.2

Let us suppopse that we know the relevant ele-ments of l , e.g. from requirements on the total spin2w x Ž .5 , then we can evoke the partial trace relation 8 .In an NSO basis we only need the diagonal elementsl p q to getp q

n2 yn sl p q , 34Ž .q q p q

which fully determines the NSO occupation numbersŽeven for an open-shell state. The ambiguity of the

solutions of a quadratic equation is related to wether.the nth NSO is strongly or weakly occupied.


Note that for the one-particle approximation theIBC is sufficient and the more complicated ICSE1 1

is not needed. Let us, nevertheless consider what weget from the latter.

Ž .We consider Eq. 30 , where we neglect the l. Inthe natural basis, this leads to

f qn 1yn s0 . 35Ž .Ž .p p q

For psq the above equation is satisfied only if np

is either 0 or 1. This means that a spin-orbital iseither occupied or unoccupied. For the case p/q, itthen follows that f is block-diagonal in the sensethat its elements vanish between occupied and unoc-cupied orbitals, as is familiar from Hartree–Focktheory. So we get the same result as from the IBC ,1

but without the need to evoke the general relationthat vanishing of l is only consistent with idempo-2

tency of g , which may be regarded as a kind of1

n-representability condition.Ž .If we do not neglect l and l and sum Eq. 302 3

over the diagonal elements psq, we confirm, byw xthe way, the partial trace relations 5 both for l2

and l , so these relations can even be obtained from3

the stationarity conditions.The second member of our hierarchy, i.e. the

two-particle approximation consists in consideringŽ . Ž .28 and 29 , and in neglecting l . The higher3

members in the hierarchy are straightforward. Onealways truncates at some l . It is important to notek

here that such a truncation is always physicallymeaningful, at variance with a truncation at some g .k

Ž . Ž . Ž .If we consider both 28 and 29 without l it3

appears straightforward to proceed iteratively, i.e. toŽ .first solve 28 neglecting l , then determine l2 2

Ž . Ž .from 29 , inserting the l into 28 and so forth.2

The simplest approach beyond Hartree–Fock con-Ž .sists in treating 29 perturbatively, regarding f as

the unperturbed hamiltonian and g as the perturba-tion. Then one obtains to first order in perturbationtheory

y1i j i jl s ´ q´ y´ y´ g , 36Ž . Ž .ab i j a b ab

where i and j count spin-orbitals occupied in g and1

a,b spin-orbitals unoccupied in g . This result is1Ž .reminiscent, of course, of the Møller–Plesset MP

perturbation theory. If one does not apply perturba-Ž .tion theory, the system 29 agrees with that of

w xCEPA-0 24 . So there are close relations to standardmethods of quantum chemistry.

What differs from conventional many-body meth-ods is, however, that one should now continue by

Ž .iteration. Going back to 28 in the presence of thel -terms this is no longer the basic equation of2

Hartree–Fock theory but rather of MC–SCF theory.The NSOs now get fractional occupation numbers.

Ž .Consequently in the subsequent solution of 29 onenot only takes care of ‘double excitations’ from theoriginal reference function, but of double excitationfrom an MC–SCF-type wavefunctions. Since themost important ‘triple excitations’ of coupled-clustertheory are double excitations from a double excita-tion, the importance of formal triple excitations isprobably drastically reduced. This means that incases where in coupled cluster theory CCSDT isneeded, l may still be negligible. This needs to be3

studied in more detail. Anyway the following argu-ment is helpful: l first enters to second order in the3

perturbation parameter, it affects l to third order2

and the energy to fourth order. Similarity l enter4

first to third order, if affects l to fourth order, l to3 2

fifth order and E to sixth order, i.e. including l the3

energy is correct to fifth order, and with l to4

seventh order. There is hence an automatic WignerŽ .2nq1 rule. This argument holds for the unupdated

Ž . Ž .g . In an iterative treatment based on 28 and 291

the situation is certainly even better.The above observations indicate that a fruitful

theory to solve for g and l directly will be to use1

the IBC for the nondiagonal elements and to use thediagonal projections of ICSE to determine the diago-nal elements, unless these are not determined bysome generally valid relations, e.g. of the partial-tracetype. This will in a sense combine the merits of bothapproaches. Of course, we have to use the sametruncation schemes for the l in order to be consis-tent.

We should note that, although both IBC and ICSEhave an infinite number of solutions in principle, it ispossible to get the desired one with a good startingpoint. This essentially requires specifying the diago-

Žnal elements of g which indicates which spin-1. Žorbitals are strongly occupied and of l which2

monitors the deviation of the starting g from idem-1.potency . Thus, for a closed-shell singlet ground

state, a good guess will be to take the n to be onep


for the n lowest spin-orbitals for an n-electron prob-lem, and take l to be zero. For ground states of2

nonzero spin, similar choices for the diagonal ele-ments of g can be made, and for simple open shells1

the total spin more or less specifies the elements ofl . For strongly quasi-degenerate cases, an appropri-2

ate MC–SCF calculation has to be done to have agood starting point. The working equations for allthese various possibilities remain the same, only thestarting guesses are different.

4. Outlook

The relations of our new formalism to the CSEw xmethod 8 have been discussed in the text. Let us

point out that in our ‘two-particle approximation’two coupled sets of equations for the determinationof the elements of g and l are solved iteratively,1 2

Ž .while in the 2, 4 CSE method a single set of equa-tions for the g is solved. Essential is, of course, that2

or equations are separable, while the traditional CSEequations are not. There are also close relations tocoupled cluster theory, mainly its unitary variant, tomany-body perturbation theory, as well as to the

w xequations of motion for density matrices 25 . Thesew xwill be analyzed elsewhere 10 in detail.

The approach appears flexible and powerful, inparticular since without any change it can be builtupon MC–SCF theory. The approach is strictly ex-

Ž .tensive size-consistent . In a diagrammatic formula-tion only connected diagrams contribute. Unlike thetraditional pair theories the approach is also exact fora system of dissociating two-electron systems, andeven for non-interacting one-electron systems.

A final comment on n-representability is in order.The ‘one-particle approximation’ is definitly n-rep-resentable, at least for a closed-shell state whereneglect of l at the start is justified. For an n-elec-2

tron system also the n-particle approximation is n-representable, since it creates an antisymmentric pureor ensemble g . Approximations and hence devia-n

tions from n-representability are introduced by ne-glect of the l of higer particle rank and of theirk

indirect effect on l and g . However, these devia-2 1

tions should decrease systematically when one in-creases the particle rank of the approximation. So weessentially rely on the decreasing importance of

higher l . In some cases it may be necessary tok

include some elements of higer l in a selectivek

way. This has to be explored.Let us add a more physical argument. On all

levels of our hierarachy the one-particle density ma-trix g is n-representable, i.e. the one-particle den-1

sity matrix is that of an n-fermion system. Conse-quently the kinetic energy and the exchange energy– which are much more sensitive to the correctparticle statistics than the Coulomb interaction of theelectrons with the nuclear field and with each other– are described correctly. Deviations from n-repre-sentability can only affect the correlation energy.

Details of the approach based on the k-particleirreducible Brillouin conditions and CSEs are beingexplored and will be published in forthcoming pa-

w xpers 10 .

Acknowledgements

We thank DFG and INSA for financial supportvia the INSA–DFG binational programme. We areindebted to D. Mazziotti for pointing out to us thatour work on cumulants may be relevant in the CSEcontext. We finally thank an unknown referee forvaluable comments.

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Direct determination of the cumulants of the reduced density matrices