Separation of strong (bond-breaking) from weak (dynamical) correlation

Chemical Physics 401 (2012) 119–124

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Separation of strong (bond-breaking) from weak (dynamical) correlation Werner Kutzelnigg Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780 Bochum, Germany

a r t i c l e

i n f o

Article history: Available online 1 December 2011 Keywords: Coupled-cluster (CC) Antisymmetrized product of strongly Orthogonal geminals (APSG) Generalized valence bond (GVB): spinfree Density cumulants Strong and weak correlation Lie algebra

a b s t r a c t A CC (coupled-cluster) ansatz based on a GVB (generalized valence bond) or an APSG (antisymmetrized product of strongly orthogonal geminals) reference function arises naturally if one tries to treat strong correlations exactly (to infinite order), and weak correlations by TCC (traditional coupled cluster) theory. This ansatz is proposed as an alternative to MC-CC (multi-configuration coupled cluster) theory. One uses especially that APSG and GVB are of CC type, but allow to combine separability with the variation principle. The energy and the stationarity conditions are formulated in terms of spinfree density cumulants. The replacement operators corresponding to the APSG ansatz generate a Lie algebra which is a subalgebra of that of all replacement operators. Ó 2011 Elsevier B.V. All rights reserved.

X  l 2 cP ¼ 1

1. Introduction

ð3Þ

P

In a recent series of papers Scuseria et al. [1] have studied the possibility to describe the bond dissociation (and other cases of strong electron correlation) of molecules correctly, but with much less effort than by means of multiconfiguration self-consistentfield (MC-SCF) methods. The concept of symmetry-breaking and symmetry restoring played an essential role in these studies. The details of the chosen approaches and their justification changed gradually between successive papers of this series [1]. However, one paradigm remained unchanged, namely that the parametrization should be of the type of a mean-field theory. This means explicitly that for the description of a 2n-electron molecule, not more than 2n spinfree orbitals should be needed, as many as in unrestricted Hartree–Fock (UHF) and twice as many as in restricted Hartree–Fock (RHF), and that these orbitals should arise in pairs. We feel challenged by this paradigm and look at it from a different point of view, with no reference to symmetry breaking, but with the same aim, namely to make the correct description of bond dissociation easy. If we ask what is the best possible parametrization of the wave function of a 2n-electron molecule in terms of 2n spinfree orbitals, a possible answer is that we should go back to the theory of separated electron pairs [2,3], i.e. to describe the molecular (singlet) state by an antisymmetric product of strongly orthogonal geminals (APSG) [4] (A is the usual antisymmetrizer)

Wð1; 2; . . . ; 2nÞ ¼ A

n Y wl ð2l  1; 2lÞ

ð1Þ

l¼1

wl ðk; lÞ ¼

X l l l  cP /P ðkÞ/P ðlÞ hðk; lÞ P

E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.10.020

ð2Þ

1 hðk; lÞ ¼ pffiffiffi faðkÞbðlÞ  bðkÞaðlÞg 2

ð4Þ

with the strong orthogonality condition

Z

l

3

wP ðk; lÞ wmQ ðk; mÞd rk ¼ 0 for

l–m

ð5Þ

We even allow to restrict this to the ansatz with two configurations for each pair, i.e. with

 l l l l l l  wl ðk; lÞ ¼ cI /I ðkÞ/I ðlÞ  cA /A ðkÞ/A ðlÞ hðk; lÞ;

l

l

cI > 0; cA P 0 ð6Þ

Note that in the two-configuration case, unlike in the general case, we explicitly use that the coefficients of two configurations have opposite sign. This two configuration ansatz [5] is known under the somewhat misleading name generalized valence bond (GVB). (It is all but generalized.) At this point one may wonder why the APSG ansatz (excluding to some extent the special case of GVB) has, after successful pilot studies [6,7], not become popular. The answer is easy. The limitation to separated pairs is very restrictive and is a too crude approximation. Take a CH4molecule and describe it in term a 1s-pair and four CH-bond pairs t 21 ; . . . ; t 24 . Then in APSG one misses the correlation between electrons in different CH bonds (tktl; k – l) [8–10], and also the strong-orthogonality condition is restrictive. In coupled cluster (CC) theory one takes care of all pair correlations (t2k and tktl) at the same level. This looks, at first glance, more balanced, at least as long as one is not too far from the equilibrium   geometry. Near bond dissociation the intrapair correlations t 2k are by far more important, and even of a different type, than the interpair correlations (tktl) and their treatment at the same level is not

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W. Kutzelnigg / Chemical Physics 401 (2012) 119–124

recommended. It was, in a way, the discovery of the inter-pair correlations [8], which disqualified the theory of separated pairs and of privileged orbital pairing. By hindsight, this disqualification was premature. A recent new interest in the APSG ansatz started with the work of Surjan [11], who has also reviewed related studies based on GVB. A generalization of APSG without the strong-orthogonality restriction has been proposed by Cassam-Chenai [12]. It appears too early to comment on it. Single-reference CC cannot describe the bond dissociation correctly, and the usual way out is to build CC upon a multiconfiguration reference function (MC-SCF) i.e. to switch to MC-CC [13,14]. An alternative to MC-CC, that has probably been initiated by Surjan [11] in the context of multireference perturbation theory is to use APSG instead of MC-SCF perturbation theory as a reference and to build upon APSG (or GVB) rather than upon MC-SCF. After all, APSG is a special case of MC-SCF, so this should make sense. The difference of this new strategy, closely related to that of the present study, compared to traditional single-reference CC is that we want to treat intra-pair correlations from a doubly occupied MO at a different level than inter-pair correlations from two different occupied orbitals. In this way we can treat the intra-pair correlation to infinite order, in a variational way, and we have to recur to the less accurate traditional coupled-cluster (TCC) theory only for the inter-pair correlation – and for corrections to the strong orthogonality restrictions, as well as for correlations of higher particle rank. For the nomenclature see Ref. [15]. The acronym TCC was coined in Ref. [16]. TCC is formulated non-variationally in intermediate normalization, based on the method opf momenmts. A modern critical analysis of CC theory (including TCC) is found in Ref. [17]. It is not so much known that the APSG wave function can easily be written in exponential form [18] as known from CC theory, and that it is one of the rare cases where separability and upper-bound property are compatible with a compact formalism [4]. An important observation is that in the frame of CC theory, especially in its unitary form a strict separation of strong and weak correlations is possible. All interesting properties of an n-electron state are determined by its spinfree one particle density matrix C = C1 and its spinfree two-particle cumulant K = K2 [19–22]. We therefore try to characterize any n-electron (exact or approximate) state in terms of C and K. Electron correlation is monitored by K. Vanishing of the two-particle density cumulant K means vanishing of the electron correlation. There are, of course, interesting questions to answer. One of these is related to problems of the possible lack of unitary invariance and to the probable need of a localized representation. Another question is whether there is always a natural pairing of orbitals, or whether there are situations with competitive pairings or even cases of resonance or mesomerism, familiar from the early days of valence bond theory. Maybe APSG will not always be a sufficient reference, but it should be exploited to a maximum extent. The structure of C and K is particularly simple for an APSG function. To prepare this we first have a look at a two-electron bond and its correct dissociation. We then present APSG as a special CC ansatz, and discuss the separation of strong and weak correlation. We continue with an analysis of APSG in terms of density cumulants and with a consideration related to Lie algebra. 2. Spinfree density cumulants The following definitions [23,19,20,22] will be needed. We define electron replacement (excitation) operators in terms of spin orbitals wk in tensor notation as

apq ¼ ayp aq ;

y y apq rs ¼ ap aq as ar ; etc

ð7Þ

where ayp and aq are the usual creation and annihilation operators. Elements of the one- and two-particle density matrices c = c1 and c2, for a state described by the n-electron wave function W are

D   E

 



 pq  cpq ¼ Wapq W ; cpq rs ¼ W ars W

ð8Þ

The corresponding spinfree quantities are (for historical reasons the letter E is used for the spinfree counterpart of a)

EPQ ¼ aPQaa þ aPb Qb EPQ RS

¼

P Q

aPRaaQSaa Pa Qa

C ¼c

þ

þc

ð9Þ

aRPaaQSbb

Pb Qb

a aPbQ RbSa

þ þ D   E  P ¼ WEQ W

PbQ b aRbSb

P aQ b PbQ a PbQb P aQ a CPQ RS ¼ cRaSa þ cRaSb þ cRbSa þ cRbSb

ð10Þ ð11Þ D   E   ¼ WEPQ RS W

ð12Þ

l.c. labels refer to spin-orbitals, cap. labels to spinfree orbitals. The spinfree cumulant K2 = K of the 2-particle density matrix, for short ‘density cumulant’ has the following matrix elements for a singlet state [19,20,22]

1 P Q PQ P Q KPQ RS ¼ CRS  CR CS þ CS CR 2

ð13Þ

The Hamiltonian in spinfree second quantization form, using the Einstein summation convention is

1 P H ¼ hQ EQP þ g PQ ERS 2 RS PQ

ð14Þ

The energy expectation value E can be written as

 1 1 P 1 P RS hQ þ fQP CQP þ g PQ E ¼ hWjHjWi ¼ hQ CQP þ g PQ KRS ð15Þ RS CPQ ¼ 2 2 2 RS PQ 1 PR S P S fQP ¼ hQ þ g PR ð16Þ QS CR  g SQ CR 2 Note that traditionally the letter E is used both for the energy and the spinfree replacement (excitation) operators. The energy consists of a Hartree–Fock-like part (but in terms of a 12 C that is not idempotent) and a correlation energy, which depends on K. C and K cannot be varied independently. They are related by n-representability conditions. These are automatically satisfied if C and K are derived from the same n-electron wave function, but they have to be imposed in a genuine density cumulant functional theory (DCFT), in which the wave function does not show up explicitly [24,25]. 3. Two-electron systems, spinfree The dissociation of a two-electron bond is correctly described by the spinfree two-electron wave function for a singlet state in spinfree NO (natural orbital) form

Uð1; 2Þ ¼ cI uI ð1ÞuI ð2Þ  cA uA ð1ÞuA ð2Þ;

cI > 0; cA P 0

ð17Þ

1 ¼ huI juI i ¼ huA juA i

ð18Þ

0 ¼ huI juA i ¼ huA juI i

ð19Þ

1¼

c2I

þ

c2A

ð20Þ

We have explicitly taken care that the coefficients of the two configurations have opposite sign. The nonvanishing elements of the spinfree one- and two-particle density matrices are (we choose all wave functions and coefficients real)

CIIII ¼ 2c2I ; I I

C ¼ nI ¼

2 CAA AA ¼ 2c A ;

2c2I ;

A A

II CAA II ¼ CAA ¼ 2c I c A

C ¼ nA ¼ 2c2A

ð21Þ

For the non-vanishing elements of the spinfree two-particle density cumulant we get

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W. Kutzelnigg / Chemical Physics 401 (2012) 119–124

1 1 KIIII ¼ CIIII  CII CII ¼ 2c2I  2c4I ¼ 2c2I ð1  c2I Þ ¼ nI nA 2 2 1 A A 1 AA KAA ¼ C  C C ¼ n n I A AA AA 2 A A 2 AA AA II KII ¼ KAA ¼ CII ¼ 2cI cA A I IA 2 2 KAI AI ¼ KIA ¼ CA CI ¼ 4c I c A ¼ nI nA 1 A I 1 AI KIA C C ¼ 2c2I c2A ¼ nI nA AI ¼ KIA ¼ 2 A I 2

ð22Þ ð23Þ ð24Þ

1 AA II KAA K K AA ¼ 2 II AA 1 AA II AA II KIA KIA K K IA ¼ KII KAA ; AI ¼ 2 II AA

½hH ; C ¼ 0; ½f ; C ¼ 0;

ð28Þ ð29Þ

ð30Þ

turns out to be exact for our present example. Relations that are derived from a formulation in terms of wave functions are, of course, sufficient, rather than necessary n-representability conditions. The 2-electron state that we consider, is characterized by two orbitals and by one parameter. In the spirit of wave function theory we would choose the parameter cI (or cA) and derive all other quantities from it. Since we care for a density cumulant functional theory, it is recommended to choose KIIAA as the independent parameter and to identify

k ¼ KIIAA ¼ KAA II

ð31Þ

ð32Þ ð33Þ ð34Þ ð35Þ

The condition for stationarity of E with respect to variation of k is





II j g AA II þ g AA ¼ kðEI  EA Þ g k g AA þ g IIAA k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; g ¼ ¼ II 2 j EI  EA 1þg

ð36Þ ð37Þ

and the stationary energy is

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  II  AA 2 EI  EA EI  EA u t1 þ g AA þ g II E¼ þ 2 2 ðEI  EA Þ2

ð38Þ

1 P ðhH ÞPQ ¼ hQ þ g PI ; 2 QI 1 PI P fQP ¼ hQ þ g PI QI  g IQ 2

ð40Þ

For small k, i.e. in the weak correlation regime we can expand

1 1 nI ¼ 2  k2 þ Oðk4 Þ; nA ¼ k2 þ Oðk4 Þ 2 2  1 2 1  II AA E ¼ EI þ k g AA þ g II þ k ðEA  EI Þ þ Oðk4 Þ 2 4   1 k ¼ g IIAA þ g AA  E Þ þ Oðk3 Þ ðE I A II  II  2 g þ g AA II E ¼ EI þ AA þ Oðk4 Þ 4ðEI  EA Þ

ð41Þ ð42Þ ð43Þ ð44Þ

We get a correlation energy that is dominated by a term quadratic in k. Of course, one does not want to solve the stationarity conditions in first-order perturbation theory, but rather iteratively, which even works far from the low-correlation limit, as it has been known for a long time [26,27]. In the strong correlation limit, i.e. for k ? 1 or j ? 0 we get

nI ¼ 1 þ j;

nA ¼ 1  j;

k¼1

j2 2

þ Oðk4 Þ;

nI nA ¼ 1  j2 ð45Þ

and the energy expanded in powers of j is

E¼

 j 1 j2  AA II  EI þ EA þ g IIAA þ g AA þ ðEI  EA Þ þ g þ g AA þ Oðj4 Þ II 2 2 4 II ð46Þ

The term independent of j is the energy of an open-shell singlet state corresponding to a fully dissociated bond. The optimum j in the regime of bond dissociation is

j¼

Note that KIIAA is usually negative, such that k is positive. We get

k2 ¼ 4c2I c2A ¼ nI nA ¼ nI ð2  nI Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi nI ¼ 1 þ j; nA ¼ 1  j; j ¼ 1  k2  1þj 1j k II E¼ EI þ EA þ g AA II þ g AA 2 2 2 I A EI ¼ 2hI þ g IIII ; EA ¼ 2hA þ g AA AA

ð39Þ

ð27Þ

K monitors the deviation of 12 C from idempotency. Vanishing of K implies that the eigenvalues of C are equal to 2 or 0. nI = nA = 1 is AI AA 1 compatible with KIIII þ KIA IA ¼ KAI þ KAA ¼  2. In density cumulant functional theory [24] (DCFT) it is important to know necessary n-representability conditions as relations between the elements of K, especially between diagonal elements and non-diagonal elements. We find, that a set of relations which was chosen as an approximation in a previous DCFT study [24,25] (where it was formulated at spin orbital level, and where a spin summation was implied)

1 KIIII ¼ KAA KII ; 2 II AA

CIIII ¼ 2

ð26Þ

or specifically

1 2 I 1 ðC ÞI  CII ¼ nI ðnI  2Þ KIIII þ KIA IA ¼ 2 2 1 2 A 1 A AI AA KAI þ KAA ¼ ðC ÞA  CA ¼ nA ðnA  2Þ 2 2

CII ¼ nI ¼ 2;

while all elements of K vanish. The energy expression in this limit is simply EI, the energy of restricted Hartree–Fock theory (RHF), (for a two-electron singlet also that of Hartree theory). Condition for stationarity with respect to variations of /I is either of the two relations

ð25Þ

We have the partial trace relations

1 2 P ðC ÞR  CPR KPQ RQ ¼ 2

Obviously k is a measure of the strength of the correlation. In the limit k ? 0 we have no correlation and the only non-vanishing density matrix elements are

EA  EI þ Oðj3 Þ II g AA II þ g AA

ð47Þ

and the binding energy in the limit of weak binding is

DE ¼ 

ðEA  EI Þ2 þ Oðj3 Þ II 4ðg AA II þ g AA Þ

ð48Þ

The parameter j monitors the deviation of C from idempotency while k monitors the deviation of 12 C from idempotency. While k is a measure of correlation, j is a measure of bonding. For j = 0 the orbitals /I and /A have the same occupation number, for j = 1 the bonding orbital /I is doubly occupied. Increase of j moves charge from the antibonding into the bonding orbital, which is energy-lowering. This energy-lowering linear in j competes with an increase of the electron repulsion which is quadratic in j. In the perturbative treatment of electron correlation in the strong-correlation regime one must not forget that in the limit j ? 0 the eigenvalues nI and nA are degenerate, so that degenerate perturbation theory has to be used. There is an alternative to the NO expansion of a two-electron two-configuration function, namely the so-called uv form.

122

W. Kutzelnigg / Chemical Physics 401 (2012) 119–124 1

wð1; 2Þ ¼ ½2ð1 þ s2 Þ2 ðuð1Þv ð2Þ þ v ð1Þuð2Þ;

s ¼ hujv i

ð49Þ

This formulation is equivalent to (17) but is less convenient since one has to deal with a non-orthogonal set of orbitals. 4. APSG as a special coupled-cluster ansatz

n Y l wI ð2l  1; 2lÞ

ð50Þ

l¼1 l

l

l

wI ðk; lÞ ¼ /I ðkÞ/I ðlÞhðk; lÞ

ð51Þ

with h defined by (4). There is only one I for each l. We apply the exponential operator

eS ;

S¼

X l Al Al X ðlÞ SA E I l I l ¼ S ;

l

l

eS U

ð53Þ

is obviously a (restricted) CC (actually a coupled-pair) ansatz. It is in intermediate normalization, i.e.

hUjeS jUi ¼ 1

ð54Þ

while W as given by (1) is in unitary normalization

hWjWi ¼ 1

ð55Þ

Actually eSU and W differ only in a normalization factor. y

1

y

hUjeS eS jUi ¼

n Y

l

l

hwI j1 þ SðlÞy SðlÞ jwI i ð56Þ

l¼1

This normalization factor can be written as a product of normalization factors for the various geminals [18], the switch between intermediate and unitary normalization is hence trivial. This is a special feature of the APSG ansatz, and is a direct consequence of the strong orthogonality restriction, and is not shared by any other exponential ansatz. For the APSG ansatz separability and a variation formulation are consistent with a finite expansion of the exponential, while for a more general exponential ansatz one must renounce on at least one of the three features: separability, variational behavior, finite expansion length. Note that

S¼

X ðlÞ ðlÞ ðmÞ S ; ½S ; S  ¼ 0;

SðlÞ2 ¼ 0

X ðlÞ ðmÞ X ðlÞ ðmÞ S S ¼2 S S l;m

eS ¼ 1 þ

l
X ðlÞ X S þ S S l

l
þ

X

ð62Þ

This looks, at first glance, like a non-terminating series. However, an exact closed summation is possible [18,17], if the strong orthogonality restriction holds, such that we get back the compact form of the APSG ansatz. It is certainly worth considering a unitary CC ansatz built upon APSG.

SðlÞ SðmÞ SðjÞ þ   

In the intermediate normalization the various Sk operators commute. We can hence write

W ¼ eðTþSÞ U ¼ eT Wint APSG ;

S Wint APSG ¼ e U

ð63Þ

Here S is that part of the cluster operator which is needed for the construction of the APSG function in intermediate normalization, i.e. which takes care of strong correlation including bond dissociation, while T takes care of the remainder, i.e. of weak correlation. If one treats S, so to say, to infinite order, a relatively simple treatment of T may be sufficient. Usually operators contained in S will have to be omitted in T. Of course, if we try to use traditional CC theory for the construction of Wint APSG we don’t gain anything compared with straight TCC for S + T. The point is that TCC is too poor for the strong correlation part, and too poor means here: too far from a variational approach, while TCC is good enough for the weak (dynamic) correlation. Before we come to the appropriate strategy, let us note that there are two possible options. (a) We limit the APSG ansatz to GVB, i.e. take only one A for each l. This should take care of the correct bond dissociation. The supplementary intra-pair correlation energy is treated together with the interpair correlation and possibly correlation of higher particle rank by TCC. (b) We treat the entire intra-pair correlation (including the dynamic one) at APSG level and use then TCC mainly for the inter-pair correlations (and correlations of higher particle rank). It has to be found out, which option is more balanced and hence preferable. A compromise may be to choose GVB as a first step towards APSG. One can hope that the highly sophisticated treatment of S at APSG or GVB level reduces the importance of the remainder T, to the extent, that explicit triple and higher replacements may become obsolete. 6. Cumulant analysis of the APSG function

ð57Þ ð58Þ

CPll ¼ 2 clP

l
X ðlÞ ðmÞ ðjÞ X ðlÞ ðmÞ ðjÞ S3 ¼ S S S ¼6 S S S l;m;j

Since the wave function corresponding to this ansatz is normalized to unity, we need not worry about a normalization factor. The energy expression can be written in terms of a Hausdorff expansion

The cumulants of an APSG function have already been derived at spin-orbital level [20]. We give them here in spinfree form for an APSG wave function constructed from only singlet pairs. For completeness we also give the non-vanishing elements of C1 and C2

l

S2 ¼

ð61Þ

l

ð52Þ

to the reference function U. The spinfree double replacement (excitation) operator EPQ RS has been defined in Eq. (10). We take care of the strong orthogonality, i.e. we require that the l virtual orbitals /A for the lth pair are orthogonal to the orbitals of any different (say the mth) pair. This is a restriction to the E operators. It may be more convenient not to enforce strong orthogonality, but to add instead a penalty term for deviations from strong orthogonality. The function

W ¼ hUjeS eS jUi2 eS U;

 X l Al Al rA EIl Il  EIAllIAll

5. Separation of strong and weak correlation

l

SA ¼ cA =cI

l

l;A

r¼

1 E ¼ hUjH þ ½H; r þ ½½H; r; r þ    jUi 2

Let us start from the Slater determinant wave function

Uð1; 2; . . . ; 2nÞ ¼ A

W ¼ er U;

ð59Þ

P

Pl Pl QlQl

C

Pl Pl Pl Pl

C ð60Þ

l
It is more elegant, but slightly more tedious to construct the APSG wave function from a unitary coupled-cluster ansatz in terms of

Pl Q m Pl Q m

C

 2

¼ nlP

l l

¼ 2cP cQ  l 2 l ¼ 2 cP ¼ nP l m

¼ nP nQ ;

l–m

 1  P P KPll Pll ¼ nlP 2  nlP 2 P P KQll Qll ¼ 2clP clQ ; P – Q ;

ð64Þ ð65Þ ð66Þ ð67Þ ð68Þ ð69Þ

123

W. Kutzelnigg / Chemical Physics 401 (2012) 119–124 P Q

KPll Q ll ¼ nlP nlQ ;

P–Q

ð70Þ

1 P Q KQll Pll ¼ nlP nlQ ; 2

P–Q

ð71Þ

One sees that non-vanishing elements of the density cumulant only occur within one pair, i.e. for l = m, the various pairs for l – m are strictly decoupled. For a GVB function the results simplify to

 2

CIll ¼ 2 clI I

A

CAll

l

¼ nI  l 2 l ¼ 2 c A ¼ nA

C

Il Il Il Il

C

Al Al Al Al

C

Il Il Al Al

C

Il Am Il Am

ð72Þ ð73Þ

l

¼ nI

ð74Þ

¼ nA

¼C

ð75Þ

Al Al Il Il

Al I m Al I m

W ¼ eS U;

S¼

X

s l Nl

ð87Þ

l

with U a separable normalized reference function, e.g. a Slater determinant, let the {Nl} be a basis of particle-number conserving Fock space operators, and the sl expansion coefficients. The energy expectation value is then y

l

¼C

Obviously the operators 2 H1 are a Lie algebra, so are those 2 H2 , operators in different subspaces commute, so the union of the operators in all subspaces constitute a Lie algebra as well. Operators like EPS11TQ11UR11 need not be considered, because they do not act on U. Let us consider the most general formulation of an n electron wave function of CC type

l l

¼ 2cI cA l m

¼ nI nA ;

ð76Þ

l–m

 1  I I KIll Ill ¼ nlI 2  nlI 2  1   1  A A KAll All ¼ nlA 2  nlA ¼ nlI 2  nlI 2 2 I I A A KAll All ¼ KIllIll ¼ 2clI clA   I A A I KIll All ¼ KAll Ill ¼ nlI nlA ¼ nlI 2  nlI  1 1  I A A I KAll Ill ¼ KIllAll ¼ nlI nlA ¼ nlI 2  nlI 2 2

ð88Þ

and the condition for stationarity with respect to variation of the expansion coefficients sl is

ð78Þ

 + * ( Sy )   de   0 ¼ Re U ðH  EÞeS U   dsl

ð89Þ

ð80Þ

If the operators Nl constitute a Lie algebra, we get two alternative expressions for the stationarity condition (89) [29]

ð81Þ

0 ¼ RehUjNyl eS ðH  EÞeS jUi

ð82Þ

In terms of these elements the energy expectation value is easily written down, following the same strategy as for a two-electron system. It is convenient to express all density matrix and cumulant elements in terms of representative cumulants A A

hUjeS HeS jUi y hUjeS eS jUi

ð77Þ

ð79Þ

kl ¼ KIllIll qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l nI ¼ 1 þ 1  ðkl Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l nA ¼ 1  1  ðkl Þ2

E¼

ð83Þ ð84Þ ð85Þ

One must then make the GVB energy expression stationary with respect to variation of the orbitals, subject to the strong-orthogonality constraints, and the coefficients kl, to arrive at conditions that determine the optimized orbitals and coefficients. The knowhow of GVB [5] can also be used.

y

Sy

y

S

0 ¼ RehUje Nl ðH  EÞe jUi

ð90Þ ð91Þ

The second one of these has the form of a contracted Schrödinger equation, which is particularly convenient. Note that this reformulation is not possible for TCC. For the special case that the operator basis is antihermitean, the expectation value and the stationarity condition are

E ¼ hUjer Her jUi

r r de de Her þ er H jU 0 ¼ Uj dsl dsl

ð92Þ ð93Þ

In the case of a Lie algebra this simplifies to either of the two expressions

0 ¼ hUjer ½H; Nl er jUi r

r

0 ¼ hUj½e He ; Nl jUi

ð94Þ ð95Þ

The first of these equations has the form of a two-particle Brillouin theorem [30,20,21].

7. Lie algebraic considerations In CC theory some simplifications arise if the operator basis, into which the operator S is expanded, constitutes a Lie algebra. It is often useful to take advantage of sub-algebras of the full Lie algebra of all replacement (excitation) operators. A special subalgebra is that of all one-particle replacement operators EPQ , which gives rise e.g. to the generalized Brillouin theorem of MC-SCF theory [28]. Another interesting subalgebra is that of all antihermitean operators, as they are needed in unitary CC theory. There are further sub-Lie-algebras related to spin or angular momentum. It has, however, been believed that there is no subalgebra, generated from replacement operators of a particle rank between 1 and the number 2n of electrons in the system. However, we have just found such a sub-Lie algebra. Let us divide the one-particle Hilbert space into n mutually orthogonal subspaces Hl each of which is associated with one of the n pairs of an APSG or a GVB function. For n = 2 (i.e. 2n = 4) we have two subspaces and two corresponding Lie algebras

H1 : EPR11 QS11 ;

EPS11TQ11UR11 ; . . . ;

H2 : EPR22 QS22 ;

EPS22TQ22UR22 ; . . .

ð86Þ

8. Conclusions Coupled cluster (CC) theory involves an exponential wave operator eS, which can be written as a non-terminating power series. This is often forgotten, because one is accustomed to the traditional formulation of TCC in intermediate normalization using the method of moments, where, by a trick, termination at a finite power can be achieved – at the price of loosing variational behavior (i.e. an upper bound for the energy). TCC works nicely if S is small, because then the power series of eS converges rapidly and can be truncated at a low order. This is the case if only dynamic correlation matters. If there is non-dynamic correlation, S becomes large, and the convergence of eS becomes poor. This is not immediately seen in the traditional formulation, but this formulation leads to large errors if S is large [17]. If one realizes that TCC fails for non-dynamic correlation, one must (which is, of course, an old wisdom) try to treat non-dynamic correlation before one applies TCC theory. A good possibility for this is to start with MC-SCF, which is, however, only feasible for relatively small systems.

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There appears to be a relatively simple compromise, namely to use CC for both dynamic and non-dynamic correlation, but to consider first non-dynamic correlation only, summing it to infinite order, which can be done in terms of APSG or even GVB theory, and to add upon this treatment more or less traditional CC for dynamic correlation. One needs to formulate CC theory for a GVB or APSG reference function, which should be possible in terms of generalized normal ordering for arbitrary reference functions [19,20,22]. Probably one has to define a new hierarchy of CC, i.e. not consider replacements (excitations) from one or more occupied spin orbitals, but rather from the same pair or from different occupied pair functions wl(k, l). One must, of course, first check to which extent APSG or GVB is able to describe bond-breaking correctly. It may be that sometimes linear combinations of APSG or GVB functions corresponding to different pairings of orbitals are necessary. One may have to study the classical papers on valence bond theory from the pioneer days of quantum chemistry. The approach advocated here has ingredients both from MO and VB theory. The main message of this paper is the following. A variational (e.g. unitary) formulation of CC theory would be ideal, since it is both separable and variational. It has not yet been used successfully since it requires to truncate an infinite series. The traditional CC theory avoids this truncation, but it is not variational. It is acceptable for the treatment of weak (dynamic) correlation, but it becomes very poor for strong (non-dynamic) correlation. A theory that is both separable and variational and does not require the truncation of a series, is possible if one restricts the operator basis to operators which describe strong correlation, to arrive either at APSG or even GVB. A good compromise is to treat strong correlations in a compact form to infinite order and to use traditional CC only for weak correlations. Acknowledgments This paper is dedicated to Debashis Mukherjee at the occasion of his 65th birthday. It is written in the spirit of our long and fruitful cooperation and friendship. The author is especially grateful to Reinhart Ahlrichs and Volker Staemmler for their constructive comments, because this paper is, to some extent, the continuation of very old joint studies on APSG and related topics.

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