Chemical Physics 356 (2009) 54–63

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Chemical Physics

journal homepage: www.elsevier .com/ locate/chemphys

Use of a convenient size-extensive normalization in multi-reference coupledcluster (MRCC) theory with incomplete model space: A novel valence universalMRCC formulation

Rahul Maitra a, Dipayan Datta b, Debashis Mukherjee a,b,*

a Raman Center for Atomic, Molecular and Optical Sciences, Indian Association for the Cultivation of Science, 2A & B Raja SC Mullick Road,Jadavpur, Kolkata, West Bengal 700032, Indiab Department of Physical Chemistry, Indian Association for the Cultivation of Science, 2A & B Raja SC Mullick Road, Jadavpur, Kolkata 700032, India

a r t i c l e i n f o

Article history:Received 5 November 2008Accepted 3 December 2008Available online 24 December 2008

Keywords:Valence universal multi-reference coupledcluster theory (VU-MRCC)Incomplete model space (IMS)Intermediate normalization (IN)Excitation operatorClosed operatorEffective hamiltonian

0301-0104/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.chemphys.2008.12.008

* Corresponding author. Address: Raman CenterOptical Sciences, Indian Association for the CultivatioMullick Road, Jadavpur, Kolkata, West Bengal 70004688; fax: +91 33 2483 6561.

E-mail address: pcdm@iacs.res.in (D. Mukherjee).

a b s t r a c t

We present in this paper a size-extensive formulation of a valence universal multi-reference coupledcluster (VU-MRCC) theory which uses a general incomplete model space (IMS). The earlier formulationsby Mukherjee [D. Mukherjee, Chem. Phys. Lett. 125 (1986) 207] led to size-extensive Heff which was bothconnected and ‘closed’, thereby leading to size-extensive energies. However, this necessitated abandon-ing the intermediate normalization (IN) for the valence universal wave-operator X when represented as anormal ordered exponential cluster Ansatz X � fexpðSÞg with S as the cluster operator. The lack of INstemmed from the excitation operator Sq�op which leads to excitations into the complementary modelspace by their action on at least one model function. The powers of Sq�op can in general bring a modelfunction /i back to another model function /j, and this is the reason why X does not respect IN. Sq�op

are all labelled by active orbitals only. To achieve connectivity of Heff , it must be a ‘closed’ operator. Aclosed operator is one which always produces a model function by its action on another model function.Since the decoupling conditions Lq�op ¼ 0, and Lop ¼ 0 for the transformed operator L ¼ X�1HX would bein conflict with Xq�op ¼ 1q�op, the model space projection of X, PXP ¼ P cannot be maintained for the nor-mal ordered Ansatz. This leads to a somewhat awkward expression for Heff . Bera et al. [N. Bera, S. Ghosh,D. Mukherjee, S. Chattopadhyay, J. Phys. Chem. A 109 (2005) 11462] recently tried to simplify the expres-sion for Heff , and accomplished this by introducing suitable counter-terms Xcl in X to enforce Xcl ¼ 1cl . Weshow in this paper that Heff in this formulation leads to a disconnected Heff , though it is equivalent by asimilarity transformation to a connected effective hamiltonian eHeff . Guided by the insight gleaned fromthis demonstration, we have proposed in this paper a new form of the wave-operator which never gen-erates any powers of Sq�op, which is closed. This ‘externally projected’ wave-operator does not need coun-ter-terms Xcl and automatically ensures Xcl ¼ 1cl , thereby yielding directly a closed connected eHeff . Thedesirable features of the traditional normal ordered Ansatz, such as the valence universality, subsystemembedding conditions hierarchical decoupling of the VU-MRCC equations for decreasing valence ranksare all satisfied by this new Ansatz for the wave-operator.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

The coupled cluster (CC) approach [1–7] occupies a pre-eminentposition in providing a compact and systematic description oftreating electron correlation in the theories of molecular electronicstructure in a size-extensive manner [8]. For non-degenerate elec-tronic states, the development uses the wave-operator in an expo-

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nential form: X ¼ expðTÞ, where Ts induce all possible n hole–nparticle (nh–np) excitations from a single determinant /0, where/0 provides a qualitatively correct description of electronic stateof interest. /0 is called the reference function and W ¼ expðTÞ/0

is the generic single-reference CC (SR-CC) Ansatz. The usual choiceof /0 is the Hartree–Fock (HF) determinant for closed shells andunrestricted Hartree–Fock (UHF) or restricted open-shell Har-tree–Fock (ROHF) generalizations for open-shells. With the initialinsight into the nature of the correlation in the SR situations bySinanoglu [9–11] who showed that the pair correlation is the mostdominant component of the correlation energy, the overwhelm-ingly important component in T is the two-body term of viz. T2,which generates all 2h–2p excitations from /0. With HF /0, T1 is

R. Maitra et al. / Chemical Physics 356 (2009) 54–63 55

expected to be small because of Brillouin theorem, although its fullinclusion alongside T2 is computationally quite undemanding. TheAnsatz with the complete inclusion of singles and doubles in anSR-CC theory (SR-CCSD) [12] has been extremely successful forall single-reference situations. From a perturbative point of view,T2 first appears at first order and T1 and T3 first appear at secondorder. The perturbative inclusion of the leading contribution ofthe triples in the non-iterative fashion has generated a spectacu-larly accurate model which is justly famous and is denoted bythe acronym CCSD(T) [13,14]. Somewhat more approximate ver-sion, viz. CCSD[T] [15] and a more accurate iterative version SR-CCSD-Tn [16,17] also exist in the literature.

Prompted by the enormous success of the SR-CC formalisms inthe various truncation strategies, attempts have been made to ex-pand the scope of the CC methodology to encompass situationswhere the reference function is quasi-degenerate with one or morevirtual determinants. Guided by the hindsight of multi-referencegeneralizations of the SR-CC formalism which were developed firstto treat this quasi-degeneracy, and also due to enormous improve-ment in the computational quantum chemistry, SR-CC formalismwas adapted to handle this via inclusion a subset of high-body TScontaining the quasi-degenerate orbitals, leading to schemes withacronyms CCSDt, CCSDtq [18–20] etc which were used to treatbond-breaking where the quasi-degeneracy of anti-bonding orbi-tals with the bonding orbitals becomes progressively important.Another novel development has been the proposal of the methodof moments CC (MM-CC) formalisms which lead to renormalizedand completely renormalized CC formalisms (R-CC and CR-CC)[21,22] which also have been used for both closed-shell andopen-shell single-reference determinants in bond-breaking situa-tions. They are very stable, in particular CR-CCSD(T) providing aviable superior alternative to SR-CCSD(T).

A much more natural starting point for treating quasi-degener-acy in our opinion, however, is to start out with a combination ofdeterminants. This multi-reference (MR) description of the startingfunction is both more flexible and physically appealing wherethere is quasi-degeneracy, as occurs in the description of bondbreaking or where spin symmetry requires use of more than onedeterminants. Historically the generalization of SR-CC to MR-CCwas accomplished using a set of quasi-degenerate functions f/igspanning a complete model space (also called complete activespace). The generalization to the multi-reference situation is nei-ther straight forward nor unique in that different MR-CC formula-tions are tailored to handle different aspects of quasi-degeneracyor spin adaptation. The differences of capturing some specific fac-ets of electron correlation in the various MR-CC approaches is re-flected in the different cluster Ansätze for the wave-operator X.Using the Bloch equation for X [23] and assuming intermediatenormalization (IN) to fix the normalization of the exact function,the effect of the dynamical correlation, viz. the component of cor-relation brought in by the configuration mixing of the virtual func-tions and the non-dynamical correlation due to the quasi-degeneracy of the model function can be elegantly separated[24–28,30,31]. The various MRCC formalisms using complete mod-el space (CMS) generate connected effective hamiltonian Heff [32],the diagonalization of which produces N exact energies for an N-dimensional model space which are all size-extensive due to theuse of CMS. They are also size-consistent in case the various frag-ment asymptotes can be spanned by the CMS chosen. For somecomprehensive discussion on the aspects of size-extensivity andsize-consistency for multi-reference situations, we refer to [28,29].

The two main classes of MR-CC formalisms using CMS addresstwo different physical aspects of dynamical correlation. The valenceuniversal MRCC (VU-MRCC) [24–28] emphasizes computation ofdifferential correlation energy attendant on excitation or ioniza-tion/electron attachment relative to a ground state of predomi-

nantly single-reference in character. This formalism defines avalence universal wave-operator X, designed to generate not onlyexact states fWðnv Þ

k g, (k ¼ 1; . . . ;Nv , where Nv is the number of modelspace functions corresponding to nv active electrons) but also allother eigen-states fWðmv Þ

k gðk ¼ 1; . . . ;Mv Þ where Mvs correspond tothe dimensions of the various subduced CMS, with mv active elec-trons, obtained by deleting active electrons for the nv-valence targetstates, all the way down to the zero-valence ‘core’ problem. The coremodel function is chosen to be a single determinant /0 acting as thereference for the ground state. The qualifier ‘‘valence universal”essentially indicates that a single wave-operator X serves as thewave operator for all mv-valence problems, mv ¼ 0; . . . ;nv . VU-MRCC is thus the natural method of choice for computing spectro-scopic energy differences such as excitation energies, ionizationpotentials, electron affinity, double ionization potentials etc. The in-creased computational demand to treat states of varying degrees ifionization is offset by the ability of the formalism to handle differen-tial correlation energy involving deletion/addition of electrons. Theapplications [33–36], though not as extensive as of SRCC, have estab-lished the power of the formalism. The other variety, called state-universal MRCC (SU-MRCC) [30,31], uses a single wave-operator togenerate a set of functions fWkg from a CMS for a fixed number Nv

of active electrons. SU-MRCC is designed to compute energies ofquasi-degenerate states per se and is suitable for treating severalelectronic states lying close together or even interleaved. Since theworking equations for SU-MRCC are rather complicated and thespin-adaptation of the resulting equations rather involved for non-singlet cases, the applications of the theory have been fewer [37,38].

An elegant reformulation of the VU-MRCC was accomplished byKutzelnigg [39–42] which lucidly brought to the fore the utility ofgenerating a valence universal wave operator X using the powerfuldecoupling conditions of the transformed operator L ¼ X�1HX inthe Fock space itself, rather than realized via the matrix elementsof L : hvðmv Þ

l jLj/ðmv Þk i taken between virtual functions vðmv Þ

l and mod-el functions /ðmv Þ

k equated to zero, 8k ¼ 0; � � � ;Nv . Kutzelnigg wasthe first to emphasize the compactness of the decoupling condi-tions when performed at the operator level in Fock space ratherthan as matrix elements with varying number of valence electrons.The earlier formulations of VU-MRCC by Mukherjee et al. [24,25]and Lindgren [26] and also Lindgren and Mukherjee [27] did notexplicitly use this insightful observation, but instead worked withmatrix elements involving varying number of active electrons. Theterm quantum chemistry in Fock space was also coined by Kutzel-nigg to emphasize this compactness and simplicity when ex-pressed at the operator level.

Despite the obvious attractiveness of the use of CMS, leading toa connected (size-extensive), Heff which automatically yields size-extensive energies, a practical implementation of any MRCC (or forthat matter of any MRPT [43,44]) using effective hamiltonians inCMS if fraught with the danger of encountering the perennial ‘‘in-truder state problem” [32,45,46]. Presence of intruders leads tocomplete instability of the working equations for determiningthe cluster amplitudes. The intruders originate from situationswhere some high lying model functions come close in energy withsome low-lying virtual functions, spoiling the convergence of theMRCC equations and hence of the quality of the all computed rootsof Heff . The ease with which the size-extensivity of an Heff and ofthe associated energies can be monitored in a CMS is severely off-set when there is a reasonable energy spread in the unperturbedenergies of the model functions. More robust solutions to the in-truder problem can be formulated by either transcribing the VU-MRCC equations as dressed MRCI in the union space spanned bythe P-space and the Q-space spanned by the action of S on P[48,49] [eigenvalue-independent partitioning (EIP)] or the similar-ity transformed equation of motion (STEOM) method [50–52].These obviates the numerical instability brought out by intruders,

56 R. Maitra et al. / Chemical Physics 356 (2009) 54–63

since the inordinately large unstable cluster amplitudes are re-placed by the coefficients of normalized eigenvectors spanningthe P and Q space.

If one wishes to stay within the confines of the effective hamil-tonian formulation, it seems eminently reasonable to move theoffending model functions, quasi-degenerate in energy with somevirtual functions, in the virtual space itself, one has then to workwith an MRCC theory with incomplete model space (IMS). As anexample one may imagine that a physically motivated theory forexcitation energies relative to a closed shell ground state warrantsthe use of a model space with hole–particle determinants with re-spect to the ground state vacuum /0, but one has then to analyzecarefully the size-extensivity of the computed energy obtainedby diagonalizing Heff in an IMS. In close analogy with a CI involvingan IMS, even if it were possible to generate a connected Heff in anIMS, its diagonalization would invariably generate size-inextensiveenergies. This is a generic problem of a naive generalization of allMRCCs with IMS. The theoretical constraints required for Heff to en-sure size-extensivity of the energies in an IMS is more subtle andrequires a careful analysis of the essential reasons behind the lackof size-extensivity of eigenvalues of even a connected operator in amatrix space which is IMS.

Alternatives to the use of IMS were developed by Malrieu et al.[47] which they termed as the ‘intermediate hamiltonian formal-ism’. Here one augments an M-dimensional space with anotherbuffer space of dimension ðN �MÞ. Though the first formulationswere determinant based and did not address the aspects of size-extensivity, later formulations were designed to ameliorate this.Thus, Mukhopadhyay et al. [53], Meissner [54–56] and also Kaldoret al. [57,58] formulated other versions of size-extensive interme-diate hamiltonians. As an extreme case of the intermediate formal-ism, one may think of a single-root theory ðM ¼ 1Þ, which aretermed state-specific (SS) theories. SS-MRCC [59–61], SS-MRCEPA[62,63] and dressed-MR-CISD [64,65] formalisms are some repre-sentative size-extensive developments. We will not consider thisalternative strategy in this paper, but instead focus only on theo-ries with IMS.

The problem of generating a connected Heff which acts entirelywithin the IMS for all mv-valence problems 80 6 mv 6 nv , and atthe same time providing size-extensive energies was analyzed firstfor the VU-MRCC formalism by Mukherjee [66–68], and expandedfurther by Kutzelnigg, Mukherjee and Koch [69,70] and also byLindgren and Mukherjee [27]. Mukherjee pointed out that naivegeneralization of the wave operator written in a normal orderedexponential cluster Ansatz, with the various possible cluster oper-ators acting on IMS, leads to situations where the customary andconvenient convention of IN for X becomes incompatible withthe connectivity requirement of both Heff and the associated ener-gies. Only by abandoning IN and adopting an appropriate size-extensive normalization in the normal ordered cluster Ansatz asintroduced by Lindgren for CMS, one can guarantee the size-exten-sivity of an MRCC formalism in an IMS. The lack of IN for X neces-sarily leads to an expression for Heff which involves a ratherawkward inverse of the normal ordered X, and this leads to asomewhat involved formalism, affecting not just the evaluationof Heff itself but also the equations determining the cluster ampli-tudes X, since the Bloch equation for X involves Heff .

For completeness it should be mentioned that Mukhopadhyayand Mukherjee [71] as also Meissner et al. [72] independently for-mulated the analogous SU-MRCC following similar ideas andPahari et al. formulated a state-specific MRCC (SS-MRCC) [73], fol-lowing essentially the same considerations. We are not going todiscuss the various aspects of SU-MRCC and SSMRCC any furtherin this paper.

After the formulation of Mukherjee [66,68], Kutzelnigg,Mukherjee and Koch [69,70] reformulated the problem entirely

in terms of the apparatus of the Fock space quantum chemistryintroduced by Kutzelnigg. They showed that not only the normalordered Ansatz X ¼ fexpðSÞg for X but also a unitary Ansatz leadsto size-extensive Heff . Just as for the original work there are no sim-plifications for Heff analogous to the use of IN for X in CMS. Oneobviously wonders whether a more convenient normalizationand/or a novel cluster Ansatz for X is possible which will lead toan expression for Heff even for IMS which is as simple as in theCMS version of VU-MRCC using IN for X.

In this paper we will propose the use of a new cluster wave-operator which yields just this very compact formulation of asize-extensive VU-MRCC theory with IMS which is analogous tothe one in CMS using IN. In our opinion, the present formulationleads to the simplest formulation of the IMS problem.

Although the formalism suggested in this paper has not beencomputationally tested yet, its computational potentiality for com-puting energy differences of spectroscopic interest cannot be dep-recated. In particular, to treat IP, DIP or high valence excitationsectors for EE, VU-MRCC theories for IMS are natural startingpoints as an alternative to STEOM and the intermediate hamilto-nian formalism. Thus any theoretical simplification in the algebraicstructure of a VU-MRCC theory using IMS is computationallywelcome.

The rest of the paper is organized as follows: in Section 2, wepresent a detailed analysis of the different variants of the wave-operator in IMS, where the various choices of the size-extensivenormalization will be discussed which ensure generation of asize-extensive VU-MRCC formulation in IMS. It is demonstratedin Section 3.1 that, using the normal ordered Ansatz of X wheresame extra cluster operators are added to the usual excitationoperators, it is possible to generate a manifestly disconnectedHeff which is equivalent via a similarity transformation to a con-nected Heff . In Section 3.2, we propose a novel cluster Ansatz forthe wave-operator where only excitation operators appear. Thislatter formulation does not require any counter-terms. Section 4includes the summary of the main conclusions and the futureoutlook.

2. The development of a new VU-MRCC for IMS using aconvenient normalization for

2.1. Aspects of size-extensive normalization of X in an IMS: a briefresume’ of the traditional formulation

Before presenting our new formulation of a VU-MRCC, usingIMS, with a size-extensive normalization for X which is muchsimpler in structure than the original formulation [66,67], weintroduce here certain key concepts pertaining to explicit mainte-nance of size-extensivity of Heff and the associated roots. This willserve the twin purposes of introducing appropriate notations andconcepts as well as of motivating us to the development thatfollows.

In the VU-MRCC, it is customary to define a valence universalcluster wave-operator X which produces exact functionsWðmv Þ

k ¼ XWðmv Þ0k 80 6 mv 6 nv where mv is the valence rank of

the model space spanned by the functions fUðmv Þi g; 8i ¼ 1;Mv .

The ”unperturbed” function Wðmv Þ0k can be written as

Wðmv Þ0k ¼

XMv

i¼1

Uðmv Þi Cðmv Þ

ik ; ð1Þ

where Cðmv Þik are the fully relaxed combining coefficients of the mv-

valence model functions. The Fock space Bloch equation elegantlytakes care of the effect of the dynamical and the non-dynamical cor-relations by way of decoupling equations for the cluster amplitudes(inducing dynamical correlations) and the non-dynamical correla-

R. Maitra et al. / Chemical Physics 356 (2009) 54–63 57

tion which is taken care of via the diagonalization of Heff at the mv-valence stage to produce both the energies Eðmv Þ

k and the coefficients.The equations for cluster amplitudes do not involve Cðmv Þ

ik s, thusobviating the need to know the Cik and the diagonalization of Heff

embeds the full effect of dynamical correlation to get Cðmv Þik s.

In the VU-MRCC formulation one starts from a target IMS for nv-valence problem. Defining the space for this IMS as Pðnv Þ, one canintroduce two complementary spaces both of which incorporatedynamical correlation. The complement of IMS which, togetherwith the IMS spans the CMS for the nv-valence problem, is oneset of virtual function, defined by the projector Rðnv Þ. The clusteroperators inducing the transitions from Pðnv Þ to Rðnv Þ are labelledentirely by active, or valence indices. The second class of virtualfunctions contain at least one inactive index for orbitals. This spaceis characterized by the projector Q ðnv Þ. By construction then, clusteroperators inducing Pðnv Þ to Q ðnv Þ are labelled by at least one inactiveorbital index. The goal of a size-extensive VU-MRCC theory is toproduce Hðmv Þ

eff 8mv ¼ 1;nv which produces size-extensive energies.To motivate towards this formulation as well as for a run-up to

the requirements demanded of a size-extensive theory in an IMS,let us first analyze the theoretical reasons why diagonalization ofeven a connected operator in an IMS produces size-inextensiveenergies. We want to look upon the diagonalization problem lead-ing to every root as an infinite order perturbation theory for the en-ergy whereby we can monitor and identify the generation of bothconnected and disconnected terms (usually represented as dia-grams) and analyzing whether the disconnected terms cancel ateach order of perturbation [74]. If we take one of the model func-tions in the IMS as the vacuum and unperturbed function for gen-erating the perturbation series of one of the target energies andinclude the effect of other model functions of the IMS interactingvia the matrix elements of Heff or H in the Rayleigh–Schrödinger(RS) perturbation theory. Then at every order the RS expansionfor energy contains two distinct types of terms. The first term –to be called ‘‘the direct term” – involves a sum over states involvingtransitions from the starting model function to a sequence of var-ious other model functions, eventually returning to the unper-turbed model function itself. The second term – to be called ‘‘the

*

a

c

Fig. 1. Diagrammatic representation of the normalization term involving double excitaanalysis. (c) Appears naturally from the Rayleigh–Schrodinger perturbation where thedepict either H or Heff depending whether we are analyzing size-extensivity of a CI functioperturbative correction to energy. The complete diagram of (c) is a typical forth order no(c). (a) and (b) has denominators as demanded by the Goldstone rules and the * on the vstarting unperturbed function. (a) and (b) might involve disconnected terms corresponcancelled if the model space is incomplete.

normalization term” – involves a product of a norm correction ofthe perturbed wavefunction and the energy correction with a neg-ative sign. It is the normalization term which is at the root of size-inextensivity for a diagonalization in an IMS. One set of the nor-malization term has no common orbitals between the norm correc-tion and the energy correction. These terms are thereforealgebraically disconnected and hence size-inextensive. By sum-ming together certain sets of similar such terms of the same topol-ogy and by using what is known as the Franz–Mills identity [75],one can rewrite these disconnected terms as a negative of sum overstates akin to the first term Fig. 1 below. The interesting propertyof these sum over states formulae is that, the generation of func-tions obtained by the action of the connected operator whichotherwise yielded the model functions of the IMS reached by theaction of the operator on the unperturbed function, the same oper-ators now act successively on the other model functions of IMSleading to multiply excited functions. For an IMS such successiveexcitations by the effective operator would generally lead to func-tions outside the IMS, in particular to those in the complementarymodel space which together with the IMS spans a CMS. Since thefirst term never involves functions of the complementary modelspace, such disconnected terms from the second term can neverget cancelled. There are of course a set of harmless, connected,entities in the second term where there are common orbitals be-tween the norm correction and the energy shifts which are notdangerous. These are the so-called exclusion principle violating(EPV) terms [76]. If we were to guarantee extensivity while diago-nalizing a connected operator in IMS, then the disconnected nor-malization terms, reorganized as a term with negative of a sumover states should not involve intermediate functions outside theIMS. This implies that no excitation of the effective operator onany function of the model space should lead to excitations out ofthe IMS to the complementary active space. In that case all the dis-connected renormalization terms would have a correspondingcounterpart in the direct term and hence they would cancel eachother at every order of perturbation. Of course, for a CMS such can-cellations are always possible.

*

b

tion out of a single determinant, chosen to monitor extensivity by a perturbativerectangles enclosing the lower vertices indicate energy denominators. The wigglesn or of an Heff . The first term in (c) is the norm correction while the second factor is armalization term. (a) and (b) together, via the Frantz–Mills identity, is equivalent toertices indicate the action of H or Heff on a function reached by a prior excitation ofding to multiple excited states (quadruple in the present case) which may not be

58 R. Maitra et al. / Chemical Physics 356 (2009) 54–63

From the foregoing analysis it is clear that if the effective oper-ator can be confined to only those terms where excitation out ofthe IMS is never possible by its action on any function of IMS, thenthe problem of size-inextensivity in an IMS will never arise. Thisobservation was used by Mukherjee to develop size-extensiveMRCC formalisms in VU-MRCC [66–68]. An operator was definedas ‘closed’ if its action on any function of the IMS keeps it withinthe IMS. An operator was called ‘quasi-open’ if its action on at leastone model function in IMS produces a function in the complemen-tary model space. In addition, since the decoupling conditionsmust also involve null matrix elements between the model func-tions and virtual functions containing virtual orbitals there shouldbe additional cluster operators on X ensuring this decoupling. Suchoperators were called ‘open’. The VU wave-operator X should be sochosen that the transformed hamiltonian L should not contain anyquasi-open or open operator connecting the functions of IMS. Muk-herjee [68] thus proposed the following Ansatz for the size-exten-sive formulation of VU-MRCC in IMS:

X ¼ fexpðSÞg ð2Þ

with

S ¼ Sop þ Sq�op: ð3Þ

In some sense, this Ansatz is minimal in the sense that no closedoperator was included in X. Mukherjee demonstrated that withthe Ansatz (2) the customary and convenient intermediate normal-ization (IN) of X does not generally hold good. IN demands PXP ¼ P.This is however generally not the case with the Ansatz (2), since (a)q-open operators can connect various model functions and (b) pow-ers of q-open operators can be closed. Because of the nature of theAnsatz (2), the derivation of the Bloch equation for IMS must notuse the IN for X. Mukherjee therefore formulated the connectedVU-MRCC equations for IMS without IN [66].

In this formulation the expression for Heff is not as simple as forthe CMS, viz. Heff is not equal to PHXP, but rather a more compli-cated expression having a premultiplication of HX with the inverseof the closed part of X. This made the theory computationally moredemanding since the expression of Heff is more complicated. It isthus worthwhile to look for an alternative normalization wherethe expression for Heff would retain the simplicity of the analogousoperator in CMS. In particular it is possible to envisage an Ansatz

for X for which Xcl ¼ 1cl, such that one expects Heff ¼ fHXz}|{gcl

where z}|{ symbolizes all connected quantities obtained by con-tracting operators of H with those in X. This seems eminently pos-sible since, although Sq�op fixes Xq�op by the decoupling condition,Xcl is still at our disposal – being not related to the decoupling con-dition and Xcl ¼ 1cl is a sensible size-extensive normalizationwhich is expected to yield connected Heff having a structure exactlyanalogous to that in a CMS [81]. Demonstration of the existence ofa wave-operator with precisely such a relation forms the majorcontent of this paper.

As emphasized earlier, it is not enough for an IMS to produce aconnected Heff for guaranteeing size-extensivity. One must ensurethat Heff is also a closed operator, which is equivalent by a similar-ity transformation to a closed connected operator. This similaritytransformation must obviously be defined in terms of a closedoperator, which thus introduces a corresponding normalizationfor the closed part of the wave-operator Xcl. Although the initialformulation by Mukherjee [66,67] and also the later works byKutzelnigg, Mukherjee and Koch [69,70], use the minimalrepresentation of X involving the open and quasi-open clusteroperators of S neither the intermediate normalization norXcl ¼ 1cl is satisfied in general for every mv-valence problem.

We now introduce here the well-known concept of ‘valencerank’ of an operator [27,77], which is equal to the number of h–p

destruction operators of active orbitals appearing in the operator.To be completely general, it is necessary to have two integers asindicating the hole-valence and particle-valence ranks, but weuse a single integer to simplify the notation. In the minimal param-etrization for X the Fock space Bloch equation for the various mv-valence problem is given by

HXPðmv Þ ¼ XPðmv ÞHeff Pðmv Þ 80 6 mv 6 nv : ð4Þ

Taking the closed projection of the equation we have

ðHXÞclPðmv Þ ¼ XclP

ðmv ÞHeff Pðmv Þ 80 6 mv 6 nv : ð5Þ

Using the Ansatz for S and using successively the Fock space Blochequation for all mv , starting from the mv ¼ 0, viz. the core problem,we have

fH expðSÞzfflfflfflfflffl}|fflfflfflfflffl{

g½mv �Pðmv Þ ¼ fexpðSÞHeff

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{g½mv �Pðmv Þ; ð6Þ

where we introduce the notation A½mv � as

A½mv � ¼Xmv

kv¼1

Aðkv Þ: ð7Þ

Aðkv Þ is an operator of valence rank kv . The generalized contractionsymbol z}|{ indicates contraction of all S operators with H or Heff ,excluding contraction from the S operators.

Due to the normal ordering of X, no operator of valence rankgreater than mv can appear for an nv-valence problem. There is thusa hierarchical decoupling of the cluster amplitudes Sðmv Þ and this wastermed as subsystem embedding condition (SEC) by Haque andMukherjee [78], Lindgren and Mukherjee [27] and others. Startingfrom the parent nv-valence problem, if one deletes active orbitaloccupancy successively (for valence holes and/or valence particles)and collects the distinct model functions generated there by, thenone arrives at the various mv lower than nv-valence IMS. This definesthe projector Pðmv Þ as also the complementary active space projectorRðmv Þ and the true virtual space projector Q ðmv Þ. Proceeding hierarchi-cally all the way downwards, one arrives at the zero-valence ‘core’problem, which is just the vacuum used in defining H and X.

It is important to emphasize at this point that any open opera-tor Sðmv Þ

op , acting on an arbitrary kv-valence problem produces a vir-tual function in the space of the Q ðkv Þ if kv is greater than or equal tomv and zero otherwise. In contrast, a quasi-open operator Sðmv Þ

q�op act-ing on a kv-valence model function may or may not produce afunction of the complementary active space for kv P mv ; ratherits action depends on which model function in the kv-valenceIMS it acts upon. However there is at least one mv-valence modelfunction, on which the action of this operator takes it to the com-plementary active space.

At this point it is convenient to introduce a general notation forall excitation operators subsuming open and quasi-open and thisgeneral class would henceforth be denoted as a subscript ‘‘ext”.Starting from the Bloch Eq. (6) and equating the external compo-nents on both sides we arrive at

fH expðSÞzfflfflfflfflffl}|fflfflfflfflffl{

g½mv �ext Pðmv Þ ¼ fexpðSÞHeff

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{g½mv �

ext Pðmv Þ: ð8Þ

Since for an mv-valence problem because of valence universality ofX, the Bloch equation for all the lower valence external operatorsare satisfied

fH expðSÞzfflfflfflfflffl}|fflfflfflfflffl{

g½kv �Pðmv Þ ¼ fexpðSÞHeff

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{g½kv �Pðmv Þ 80 6 kv 6 mv ð9Þ

are satisfied, it follows that:

fH expðSÞzfflfflfflfflffl}|fflfflfflfflffl{

gðkv Þext Pðmv Þ ¼ fexpðSÞHeff

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{gðkv Þ

ext Pðmv Þ 80 6 kv 6 mv : ð10Þ

R. Maitra et al. / Chemical Physics 356 (2009) 54–63 59

In Eqs. (9) and (10) above kv generically denotes a typical valencerank lying between zero and mv .

Similarly by taking the closed projection of (6), we have

fH expðSÞzfflfflfflfflffl}|fflfflfflfflffl{

gðmv Þcl Pðmv Þ ¼ fexpðSÞHeff

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{gðmv Þ

cl Pðmv Þ

¼ fXclHeff

zfflfflffl}|fflfflffl{gðmv Þ

cl Pðmv Þ: ð11Þ

This defines Heff as

fHeff gðmv Þ ¼ fX�1cl fH expðSÞzfflfflfflfflffl}|fflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{

gclgðmv Þ

¼ fexpð�SÞclfHexpðSÞzfflfflfflfflffl}|fflfflfflfflffl{

gcl

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{gðmv Þ: ð12Þ

Eqs. (10) and (12) respectively determine the equations for clusteramplitudes Sext � Sop þ Sq�op and Hðnv Þ

eff . Since with the minimalparametrization of X, Xcl – 1cl it is easy to discern that the tradi-tional formulation of Mukherjee [66] leads to a somewhat involvedexpression for Heff , Eq. (12).

We note at this point that we have the freedom of choosing Xcl

in a manner which leads to the desired simplification of both Heff

and the equations for cluster amplitudes S. In fact Li and Paldusin their recent formulations of SU-MRCC using IMS [79] did tryto use additional closed and quasi-open operators defined withinthe IMS to enforce IN for X. From our analysis delineated above,such a stratagem does not lead to a size-extensive Heff , sinceXq�op must have to be determined by decoupling condition onLq�op, Lq�op ¼ 0 and imposition of any other condition on Xq�op,e.g. enforcing appropriate constraints for IN, would be inconsistentwith the decoupling condition on Lq�op. The only freedom left to usis to choose a suitable normalization condition on Xcl to simplifythe expressions for Heff and the associated Bloch equation. Certaindesirable choices for Xcl guaranteeing size-extensivity of Heff willnow be discussed in the next subsection.

2.2. Imposition of Xcl ¼ 1cl and size-extensivity of Heff

It was pointed out long ago by Chaudhuri et al. [74,80] thatXcl ¼ 1cl can be used in the VU-MRCC for IMS to get a normaliza-tion for X that comes as close as possible to IN. This can be accom-plished by using certain closed counter-terms Xcl which can bedetermined by imposing Xcl ¼ 1cl. Such a formulation in the VU-MRCC context has recently been proposed by Bera et al. [81]. Curi-ously, the use of this counter-term does not necessarily lead to anexplicitly connected Heff and Bera et al. tried to circumvent thisproblem by invoking a rather involved transformation of associ-ated Heff . Since this formulation is instructive to develop the muchsimpler and compact new VU-MRCC formulation we are after, letus digress at this point to present succinctly the approach of Beraet al. [81].

If we introduce the Ansatz

X ¼ fexpðSop þ Sq�op þ XclÞg½nv � ð13Þ

for X, where we impose the normalization condition

fXclgðmv Þ ¼ 1ðmv Þcl ð14Þ

to determine the corresponding Xðmv Þcl then it implies

fexpðSq�op þ XclÞgðmv Þcl ¼ 1ðmv Þ

cl ð15Þ

and

Xðmv Þcl ¼ �fexpðSq�op þ X½mv�1�

cl Þgðmv Þcl : ð16Þ

In Eq. (16) above, the valence rank of X½mv�1�cl on the R.H.S. is neces-

sarily less than mv . Sq�op and Sop components of X are determined bythe decoupling conditions

Lðmv Þop ¼ 0 8mv ð17Þ

and

Lðmv Þq�op ¼ 0 8mv : ð18Þ

Using the Ansatz Eq. (13) for X and proceeding hierarchically fromthe zero-valence problem upwards and using SEC we are led in astraightforward manner to

fHfexpðSþ XÞg½mv �zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{

g½mv �PðmvÞ ¼ ffexpðSþ XÞg½mv �Heff

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{g½mv �PðmvÞ:

ð19Þ

Equating the open and quasi-open components on both sides of Eq.(19) leads to the equations for determining the cluster amplitudeswhich are entirely equivalent to the decoupling conditions for L,Eqs. (17) and (18) above.

One may imagine that any power of Sq�op in Eq. (19) wouldautomatically be cancelled by the counter-term Xcl via Eqs. (15)or (16) but a little reflection indicates that this is unfortunatelynot so in general. In the general situation where Sq�op has both va-lence rank increasing and valence rank decreasing components.Then it can so happen that the counter-term Xcl necessary to cancelthe powers of Sq�op which are closed cannot automatically be elim-inated. Consider for example a case where we have Sq�op operatorsof the type as shown in Fig. 2a and b. Their product may be a closedoperator. Owing to the normal ordering in X, this is necessarily dis-connected. The counter-term Xcl which ensures Xcl ¼ 1cl is alsothen disconnected since it is determined from Eq. (16) whose dia-grammatic depiction is given in Fig. 2c. However the condition(15), as implied by Fig. 2c, leads to a rather awkward and undesir-able situation. Because of the connectivity of all the cluster opera-tors S and X–H or to Heff in Eq. (19), it is imperative that both Xcl andpowers of Sq�op which are closed must all be connected to H or Heff .This in general may not be possible. As an example, the left side ofEq. (19) for a typical case depicted in Fig. 3. In 3a the hatchedsquare are connected composites of H, X and S and the rightmostconnected operator of Sq�op of Fig. 2a is explicitly shown. InFig. 3b we show a similar term containing the operator X ofFig. 2c. It is clear that there is no counter-term containing the prod-uct of two Sq�op of Fig. 2c which can be constructed as a completelyconnected term; the operator of Fig. 2b cannot simply be con-nected from the right to the composite denoted as the hatchedsquare. Thus the typical terms shown in Fig. 3a and b which formthe L.H.S. of Bloch Eq. (19) have disconnected diagrams comingfrom the terms containing Xcl since Xcl by construction is discon-nected. Similarly for the R.H.S. of Bloch Eq. (16), a typical term con-taining Xcl connected to Heff from the left will remain disconnected;there cannot be a counter-term containing powers of Sq�op whereboth Sq�op can be connected from the left to Heff .

From the foregoing analysis of the connectivity of the Blochequation, it should be clear that the mere use of Xcl as counter-terms to enforce Xcl ¼ 1cl while at the same time using the normalordered Ansatz does not necessarily lead to a connected Heff . To getaround this difficulty Bera et al. [81] suggested quite an intricatecombinatoric analysis which in our opinion is really not necessaryto arrive at a connected Heff . There is in fact much simpler andstraightforward way to formulate the problem where the normalordered Ansatz for X is replaced by a new Ansatz and no coun-ter-term is needed at all.

In what follows, we want to demonstrate that this manifestlydisconnected Heff discussed above is in fact equivalent by an oper-ator similarity transformation to a connected Heff . This analysis isimportant in the sense that it indicates how a new Ansatz for Xwithout the use of the closed operator Xcl leads in a straightforward manner to a new Heff which is related by a similarity trans-formation to the one formulated by Bera et al. [81]and at the same

a b c

X

Fig. 2. (a) A typical valence rank increasing Sq�op . (b) A typical valence rank decreasing Sq�op . (c) The normal ordered product of (a) and (b) may be closed e.g. for a hole–particle IMS. The counter-term Xcl to enforce Xcl ¼ 1cl would satisfy the diagrammatic equation shown in (c).

Fig. 3. Diagrammatic depiction of the fact that all X-containing terms may not get cancelled in the VU-MRCC equation (Eq. (22)). (a) and (c) are typical diagrams appearing onthe left and right side of Eq. (22). The counter-terms which could have cancelled (a) and (c) using Fig. 2c cannot appear in Eq. (22) since they are disconnected. The termscontaining X in (a) and (c) therefore remain uncancelled and they are disconnected because of relation shown in (Fig. 2c).

60 R. Maitra et al. / Chemical Physics 356 (2009) 54–63

time ensures Xcl ¼ 1cl. The latter formulation will be presented inSection 3. We will also indicate an alternative avenue closely re-lated to what we present in Section 3.1.

3. A new cluster Ansatz for in VU-MRCC with IMS withoutexplicit use of closed operator in

3.1. Demonstration of the equivalence of Heff of Bera et al. to aconnected eHeff

To demonstrate the equivalence claimed above, we simplify thenotations somewhat. Unless needed, we would not explicitly indi-cate the normal ordered cluster structure of X. Also, we would notexplicitly use the superscript mv to indicate the valence rank. Thevalence universality of X would automatically imply that the equa-tions discussed below are valid for every valence rank mv .

We start out from Eq. (19) for the external projections of Blochequations

fHfexpðSþ XÞgzfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{

gextP ¼ ffexpðSþ XÞgHeff

zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{gextP: ð20Þ

For every connected entity appearing in either in L.H.S. or R.H.S. ofEq. (20), there are always some powers of Sq�op which are closed andwhere all the individual factors of Sq�op as factors of the compositeclosed operators is connected to H or Heff from right and left respec-

tively. The entities containing such closed composites are alwayscancelled by the counter-terms containing the corresponding enti-ties involving Xcl which satisfy Eq. (16). This indicates that we cansimply delete all composite closed powers Sq�op which are all con-nected to either H or Heff . The rest of the closed operators containnon-cancelled Xcl terms and powers of Sq�op where in the latterthere are no closed factors.

To distinguish a normal ordered power of (S + X) which on itsown may be external but which may have factors that are closed,and those powers of S where there are no factors which areclosed, we introduce a component eX from X where there areno X operators and also in any power fSng, we cannot find a fac-tor Sm where m is less than n which is closed. eX thus has thespecial property that in any term in the power series expansion,there cannot appear even one factor with powers of the quasi-open cluster operators that form a closed entity. This is a novelaspect of eX which will be elaborated upon in Section 3 for ournew formalism.

Eq. (20) for external part can be compactly represented as

fHXz}|{gP ¼ fXHeff

zfflffl}|fflffl{gP: ð21Þ

Collecting all the powers of S contracted to H or Heff such that thereare no closed factor of powers of S, as discussed in the definition ofeX, and taking the external projection we can rewrite this as

R. Maitra et al. / Chemical Physics 356 (2009) 54–63 61

fH eXz}|{Xcl

zfflfflfflffl}|fflfflfflffl{gextP ¼ feX XclHeff

zfflfflffl}|fflfflffl{zfflfflfflfflfflffl}|fflfflfflfflfflffl{gextP: ð22Þ

Similarly equating the closed part of Bloch equation on both sides

fðH eXÞcl

zfflfflffl}|fflfflffl{Xcl

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{gP ¼ fXclHeff

zfflfflffl}|fflfflffl{gclP ð23Þ

or equivalently

Heff ¼ fX�1cl ðHildeXÞcl

zfflfflfflfflfflffl}|fflfflfflfflfflffl{Xcl

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{gclP � fX

�1cl ððHXÞcl

zfflfflffl}|fflfflffl{zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{gP: ð24Þ

The middle portion of Eq. (24) implies that Heff is equivalent by a

similarity transformation to a closed operator fH eXq�op

zfflfflfflfflffl}|fflfflfflfflffl{gcl which

contains neither closed powers of S ( in particular those of Sq�op)nor any Xcl. Starting from Eq. (24) we find

eHeff ¼ XclðHeff

zfflfflfflffl}|fflfflfflffl{X�1

cl Þzfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{

: ð25Þ

In terms of eHeff we can write

HildeXzfflfflfflffl}|fflfflfflffl{

Xcl

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{¼ ildeXXclHeff

zfflfflffl}|fflfflffl{zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ð26Þ

as

H eXz}|{ ¼ eX eHeff

zfflffl}|fflffl{: ð27Þ

Hence if the external components of eX are determined from

½H eXz}|{ �ext ¼ eXexteHeff

zfflfflfflfflffl}|fflfflfflfflffl{ð28Þ

and eHeff is computed from

eHeff ¼ ½H eXz}|{ �cl; ð29Þ

then it follows that eHeff is explicitly connected since it is consistentwith the connectivity of Eq. (27). Hence, provided Eq. (28) is used todetermine S, and Heff is computed via Eq. (24) then – despite themanifest disconnected nature of Heff – it is equivalent to the con-nected eHeff – and hence would produce size-extensive energies.For any truncated calculation (i.e. where the rank of S is truncatedand/or all powers of S are not used) one must, however, be carefulnot to approximate X�1

cl in any manner. Otherwise the strict equiv-alence will be destroyed.

The above striking results motivate us to introduce a new sim-pler cluster representation of the wave-operator which directlyleads to an eHeff , which satisfies Xcl ¼ 1cl and does not need theintroduction of Xcl.

3.2. A new cluster Ansatz which exploits the properties demonstratedin Section 3.1

As we discussed in Section 3.1, it is feasible to extract out of avalence universal X, containing both external operators and closedoperator Xcl, such that we can guarantee a connected closed eHeff . Itwould be much more aesthetically pleasing and formally simpler,however, if we can formulate a VU-MRCC theory using IMS whichsatisfies Xcl ¼ 1cl without the intermediary of the operator Xcl. Inthis section we will introduce a new cluster Ansatz for X whichis different from traditional normal ordered representation for X,that possesses just these desirable features stipulated above.

It might appear that, both the generation of a connected set ofequations for the cluster amplitudes of S, without any intermediaryinvolving the closed operator Xcl, and the hierarchical decouplingof the equations for S for different valence ranks are predicatedby the normal ordered exponential structure for X. It may then

come to us as a pleasant surprise that it is necessarily not the case.As we have already gleaned in Section 3.1, one can get a formula-tion generating an effective hamiltonian equivalent to a connectedone using just the factor ~X present in X. It thus strongly indicatesthat a perfectly viable Ansatz for the wave operator satisfyingXcl ¼ 1cl could be written, involving some new cluster operators ~Sas

~X ¼ fexpð~SÞg; ð30Þ

where fexpð. . .Þg symbolizes an ‘externally projected’ normal or-dered exponential which does not contain as closed factors ~Sm

(where m is less than n) in a power f~Sngext . Of course the only classof S which could produce closed powers of S are those containingSq�op and the external projection eliminates such closed powers~Sm

q�op (m is less than n) from f~Snq�opgext . ~X thus contains all possible

terms which an ordinary normal ordered exponential may gener-ate., except those which have powers with subterms as factors withlower powers of ~Sq�op which are closed. For an nv-valence problem,more explicitly

~X½nv � ¼ fexpð~S½nv �op þ ~S½nv �

q�opÞg½nv �: ð31Þ

We should note here that, except the closed powers of ~Sq�op, the firstterm of ~X½nv � is just 1. Before going over to Section 3.2, let us empha-size once again that ~X has the property that in any term in theexpansion, a monomial ~Sn cannot have even one factor with powersof ~S, ~Sm which is closed. There is thus a major difference with a termlike f~Sgn

ext and f~Sgn. As an example a term like f~Sg4ext may have two

factors fð~S2Þclð~S2Þextg since the entire product is external but f~Sg4ext

cannot admit of a factorization that has any closed part in it.It is interesting to observe at this stage that the SEC and the

hierarchical decoupling of the various valence sectors of VU-MRCCequations are also valid for the externally projected Ansatz~X ¼ fexpð~SÞg. Writing ~X is long hand, we have

~X ¼ 1þ f~Sg þ 12!f~S2gext þ

13!f~S3gext þ � � � ; ð32Þ

where 1n!f~Sngext is a compact notation for the externally projected

nth power of ~S As we emphasized at the end of Section 3.1, f~Sngext

is quite different from just the external component of f~Sng, f~Sngext ,where there would in general be some closed factor ~Sm (m<n) inf~Sngext .

~X can hence be written as

~X ¼X

n

1n!f~Sngext: ð33Þ

Defining the effective hamiltonian eHeff corresponding to the choice,Eq. (30) for ~X, the Fock space Bloch equation reads as

H ~XPðmv Þ ¼ ~XHeff Pðmv Þ 80 6 mv 6 nv : ð34Þ

Let us now consider the product H ~X and write it in normal orderusing generalized Wick’s theorem as

H ~X ¼X

n;k6n

1n!

Hð~Sk

zffl}|ffl{~Sn�kÞext

( )n

Ck; ð35Þ

where we have brought all possible contractions of kth power of ~Sjuxtaposed after H. From n ~S operators, we can choose k ~S operatorsfor contraction in nCk ways, hence this factor appears in Eq. (35).

The notation fHð~Sk

zffl}|ffl{~Sn�kÞextg indicates that k ~S operators are con-

nected to H, while (n � k) ~S operators are not connected. The nota-tion ð. . . Þext denotes that there are no closed powers of ~S which arefactors in the product of ~S operators. This expression on the R.H.S.of Eq. (35) is rather important. Eq. (35) can be equivalently writtenas

62 R. Maitra et al. / Chemical Physics 356 (2009) 54–63

H ~X ¼X

k;m¼n�k

1k!m!

Hð~Sk

zffl}|ffl{~SmÞext

( )ð36Þ

and, both k and m can be summed independently of each other, toyield

H ~X ¼ Hð~Xzffl}|ffl{

~XÞext

( )¼ Hðexpð~SÞ

zfflfflfflfflfflffl}|fflfflfflfflfflffl{expð~SÞÞext

( ): ð37Þ

By an entirely similar reasoning, we also can show

~XeHeff ¼ ð~X ~XÞexteHeff

zfflfflfflfflffl}|fflfflfflfflffl{( )¼ ðexpð~SÞ expð~SÞÞext

eHeff

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{( ): ð38Þ

Due to valence-universality of ~X, Eqs. (37) and (38) is valid for eachmv valence sector, viz:

fHðfexpð~SÞgzfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{

expð~SÞÞextg½mv � ¼ fðexpð~SÞ expð~SÞÞext

eHeff

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{g½mv �: ð39Þ

Specifying now the valence rank of mv of individual componentsand proceeding hierarchically upwards in valence rank, exactly asin the VU-MRCC Ansatz with fexpðSÞg, we have

fHfðexpð~SÞgzfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{

expð~SÞÞextgðmv ÞPðmv Þ

¼ fðexpð~SÞ expð~SÞÞexteHeff

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{gðmv ÞPðmv Þ 80 6 mv 6 nv ; ð40Þ

where eHeff is the effective hamiltonian corresponding to the choiceof the wave operator as ~X. Eq. (40) obviously shows explicitly thateHeff at each valence rank is equivalent via the argument given in Eq.(25) of Section 3.1 to a connected, and consequently Eq. (40) is alsoconnected. Compared to the solutions given by Bera et al. [81] theseequations neither necessitate the use of closed disconnected coun-ter-terms Xcl nor produce a disconnected closed Heff that needs to beultimately shown to be equivalent to a connected effective hamilto-nian. The hierarchical decoupling, indicative of SEC and valence uni-versality are also respected.

Since the above formulation produces an eHeff equivalent to asize-extensive one, the size-consistency of the computed energiesis also automatically guaranteed with the choice of a product sep-arable IMS. In particular, the use of a quasi-complete model space(QMS) introduced by Lindgren [82], the product separability of theQMS - by the localizing transformation among the active orbitals ina given group generating certain specific fragment channels – canbe ensured in a straight-forward manner.

We conclude this section by noting that, for a CMS, the Ansatz~X ¼ fexpð~SÞg reduces trivially to ~X ¼ fexpð~SÞg, viz the usual nor-mal ordered Ansatz, since for CMS there are no ~Sq�op, and powersof ~S can never be closed.

4. Summary and future outlook

In this paper, we have taken a fresh look into the aspects of sizeextensivity using the VU-MRCC using IMS. The traditional earlierformulation of Mukherjee [66] used a normal ordered cluster An-satz for the wave-operator X, containing open and quasi-openoperators defined to guarantee size-extensivity of the associatedeffective hamiltonian, and abandoned IN for the wave-operator.This led to a size-extensive effective hamiltonian which is closed,but had a more involved expression containing the explicit inverseof the closed component of the wave-operator. Guided by a pertur-bative analysis, which shows that the size-extensivity of the effec-tive hamiltonian requires the effective hamiltonian to be closed.This necessitates the decoupling of the transformed operatorL ¼ X�1HX for both its open and quasi-open components. Theydetermine the Sop and Sq�op, respectively. The enforcement of theIN for X would require Xq�op components to be zero, which will

be in conflict with the decoupling conditions for Lq�op. The formu-lation which comes closest in the structure for the effective hamil-tonian for CMS with IN for X is one which enforces Xcl ¼ 1cl –rather that PXP = P. We have analyzed the algebraic structure ofan earlier work by Bera et al. [81] on VU-MRCC which ensuredXcl ¼ 1cl via the use of certain disconnected closed cluster operatorXcl. We have demonstrated in this paper that, though the effectivehamiltonian is disconnected, it is nevertheless equivalent via asimilarity transformation to a connected closed effective hamilto-nian, implying size-extensivity of the computed energies. To com-plete this analysis we have also introduced a novel cluster operatorrepresentation Ansatz for a new wave-operator ~X ¼ expð~SÞ where~X excludes all intermediate powers of ~S in any monomial gener-ated by expansion of S, which are closed, even as a factor of anotherwise external operator. We then prove that the hierarchicaldecoupling as is valid for the VU-MRCC for CMS, the size-consiste-ny of the associated eHeff via its equivalence to a connected Heff aswell as the simplicity of the algebraic expression of eHeff can besimultaneously satisfied. There is no need for the use of Xcl typeof operator at all, in our formulation.

The size-extensivity of our formalism will automatically implythe size-consistency of the computed energies, if the IMS is prod-uct-separable into the IMS for the various fragments.

It has not escaped our notice that a very similar idea can be in-voked to generate rather simple working equations for the SU-MRCC and the SS-MRCC as well, which will be the subject of ourforthcoming publications.

Acknowledgements

DM thanks DST (New Delhi) for conferring on him the J.C. BoseNational Fellowship. DD thanks the CSIR (New Delhi) for a seniorresearch fellowship. The authors thank Sarbani Saha for her kindhelp in preparing the manuscript. It is a pleasure to dedicate thispaper in honor of Werner Kutzelnigg on the happy occasion ofhis reaching seventy-five. One of us (DM) cherishes many pleasantmemories of a heartwarming long friendship, and we all wish himmany happy returns of his birthday and many more years of crea-tive life. We thank the referee for his very constructive criticism.

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