Renormalized CCSD(T) and CCSD(TQ) approaches: Dissociation of the N 2 triple bondKarol Kowalski and Piotr Piecuch Citation: The Journal of Chemical Physics 113, 5644 (2000); doi: 10.1063/1.1290609 View online: http://dx.doi.org/10.1063/1.1290609 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/113/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The antimony-group 11 chemical bond: Dissociation energies of the diatomic molecules CuSb, AgSb, and AuSb J. Chem. Phys. 140, 064305 (2014); 10.1063/1.4864116 On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies J. Chem. Phys. 135, 044102 (2011); 10.1063/1.3613639 Implementation of the locally renormalized CCSD(T) approaches for arbitrary reference function J. Chem. Phys. 123, 014102 (2005); 10.1063/1.1944723 State-resolved unimolecular dissociation of cis-cis HOONO: Product state distributions and action spectrum inthe 2 ν O H band region J. Chem. Phys. 122, 104313 (2005); 10.1063/1.1858437 The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), andCCSDT(Q) approaches J. Chem. Phys. 113, 18 (2000); 10.1063/1.481769

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Renormalized CCSD „T… and CCSD „TQ… approaches: Dissociation of the N 2triple bond

Karol Kowalski and Piotr Piecucha)

Department of Chemistry, Michigan State University, East Lansing, Michigan 48824

~Received 31 May 2000; accepted 14 July 2000!

The recently proposed renormalized and completely renormalized CCSD~T! and CCSD~TQ!methods, which can be viewed as generalizations of the noniterative perturbative CCSD~T! andCCSD(TQf) schemes and which result from the more general method of moments ofcoupled-cluster equations, are applied to the dissociation of the ground-state N2 molecule. It isshown that the renormalized and completely renormalized CCSD~T! and CCSD~TQ! methodsprovide significantly better results for large N–N separations than their unrenormalized CCSD~T!and CCSD(TQf) counterparts. ©2000 American Institute of Physics.@S0021-9606~00!30538-4#

I. INTRODUCTION

One of the most challenging problems in coupled-cluster~CC! theory1–4 is extension of the single-reference CC~SRCC! methods to quasidegenerate electronic states and tostudies of reactive potential energy surfaces involving break-ing of chemical bonds. In order to describe quasidegenerateground states, while retaining the simplicity of the SRCCapproach, one has to incorporate higher-than-doubly excitedclusters in the formalism. Although explicit inclusion of thetriexcited (T3) and quadruply excited (T4) cluster compo-nents is possible, the resulting full CCSDT~CC singles,doubles, and triples!5,6 and CCSDTQ~CC singles, doubles,triples, and quadruples!7–10 methods are very expensive andrather impractical.

A large number of SRCC approximations have been pro-posed that account for the effect ofT3 andT4 clusters usingmany-body perturbation theory~MBPT!. The best examplesof the perturbative CCSDT method are provided by thepopular and widely used noniterative CCSD~T!11 andCCSD@T#12–14 approaches, in which a simple energy correc-tion due to T3 is added to the CCSD~CC singles anddoubles! energy. The noniterative factorized CCSD(TQf) ap-proach, in which bothT3 and T4 effects are approximatedusing the arguments originating from MBPT, has recentlybeen proposed by Kucharski and Bartlett.15 The advantage ofthese methods is their relatively low cost compared to fullCCSDT and CCSDTQ schemes. The problem is that the no-niterative CCSD~T!, CCSD@T#, and CCSD(TQf) methodsand their iterative CCSDT-n and CCSDTQ-n counterparts~cf., e.g., Refs. 14–21! fail to describe bond breaking, whenthe restricted Hartree-Fock~RHF! configuration is used as areference.22–27This failure becomes particularly severe whenmultiple bonds are broken.22–24

An entirely different philosophy has been advocated bythe authors of Refs. 26–29~see, also, Refs. 10 and 25!. In theso-called state-selective~SS! CCSD~T! and CCSD~TQ!approaches10,25,28,29 and their CCSDt and CCSDtq

analogs,26,27 the most importantT3 and T4 components areselected through the use of an active space. The concept ofan active space is also exploited in the reduced multirefer-ence CCSD~RMRCCSD! method of Li and Paldus,30–32 inwhich information about theT3 andT4 clusters is extractedfrom the multireference configuration-interaction~MRCI!wave function. Although all these methods are capable ofproviding a good description of quasidegenerate states andbond breaking~cf., e.g., Refs. 3, 10, 25–27, 30–33!, thecomputer cost of the SSCCSD~T!, SSCCSD~TQ!, CCSDt,CCSDtq, and RMRCCSD calculations and the accuracy ofthe results depend on the size of the active space used toextract information aboutT3 andT4 clusters. Undoubtedly, itwould be desirable to have a robust approach, which com-bines the simplicity of the ‘‘black-box’’ noniterative pertur-bative approaches, such as CCSD~T! or CCSD(TQf), withthe efficiency in which active-space SRCC approaches de-scribe bond breaking.

We have recently introduced an approach which mayhave the desired characteristics.4,34 The main idea of the newtheory, termed themethod of moments of coupled-clusterequations~MMCC!,4,34 is that of the simple, noniterativecorrectiond that, when added to the energy obtained in thestandard approximate SRCC calculations, such as CCSD orCCSDT, recovers the exact~full CI ! result. The correctiondis a functional of the exact electronic wave functionuC& andthe generalized moments of CC equations~the SRCC equa-tions corresponding to projections on the excited configura-tions, whose excitation level exceeds that defining a givenSRCC approximation!. A hierarchy of the MMCC approxi-mations has been introduced, based on different ways of se-lecting the generalized moments of CC equations and differ-ent ways of approximatinguC& in the MMCC energyformula.4,34 The use of simple perturbative approximationsfor uC& allowed us to propose therenormalized and com-pletely renormalized CCSD(T), CCSD(TQ), and CCSDT(Q)methods, which are generalizations of the well-knownCCSD~T!,11 CCSD(TQf),

15 and CCSDT(Qf)15 schemes. Pi-

a!Author to whom correspondence should be addressed; electronic mail:piecuch@cem.msu.edu

JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 14 8 OCTOBER 2000

56440021-9606/2000/113(14)/5644/9/$17.00 © 2000 American Institute of Physics

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lot calculations, using the renormalized and completelyrenormalized CCSD~T! and CCSD~TQ! methods,4,34 indicatethat these methods, unlike their original CCSD~T! andCCSD(TQf) counterparts, are capable of describing thebreaking of a single chemical bond~e.g., the H–F bond inHF! and a simultaneous breaking of two single bonds~e.g.,the O–H bonds in the H2O molecule!. The potential energycurves for these cases of bond breaking obtained with therenormalized and completely renormalized CCSD~T! andCCSD~TQ! methods are in excellent agreement with the ex-act ~full CI ! results over the entire range of nuclear geom-etries. This is illustrated in Fig. 1, where the potential energycurves for the double-zeta~DZ!35 model of HF, obtainedwith the renormalized CCSD~T! @R-CCSD~T!# and com-pletely renormalized CCSD~T! @CR-CCSD~T!# methods, arecompared with the standard CCSD, CCSD~T!, and full CI~FCI! results.4,34 Our CR-CCSD~T! approach completelyeliminates an unphysical hump on the CCSD~T! curve andproduces the potential energy curve, which is virtually iden-tical to the exact~FCI! curve ~see Fig. 1; see, also, Refs. 4and 34!. Even the simply renormalized R-CCSD~T! methodis capable of significantly improving the poor CCSD~T! re-sults at large internuclear separations~cf. Fig. 1!. This find-ing and similar findings for the dissociation of the watermolecule have been intriguing enough that we have decidedto test these methods further. The present paper represents animportant step in this direction.

In this work, we report the results of the pilot calcula-tions for the dissociation of the triple bond in N2 using therenormalized and completely renormalized CCSD~T! andCCSD~TQ! methods. It is well-known that description of thedissociation of multiple chemical bonds represents the mostchallenging test for anyab initio theory. The nitrogen mol-ecule requires sixfold excitations in a CI sense in order to

obtain a correct description of a breaking of N2 into twoground-state N atoms. The standard CCSD, CCSD~T!,CCSDT, and, as we are going to demonstrate in this work,CCSD(TQf) methods completely break down at large N–Nseparations and it is interesting to see how our new renor-malized and completely renormalized CCSD~T! andCCSD~TQ! approaches behave in this challenging situation.

II. THEORY AND COMPUTATIONAL DETAILS

A. The method of moments of coupled-clusterequations

The details of the MMCC formalism have been dis-cussed elsewhere,4,34 so that we restrict our presentation tothe most essential information.

In the SRCC theory, the ground-state wave function ofan N-electron system is written in the exponential formeTuF&, where T is the cluster operator anduF& is anindependent-particle-model reference configuration~usually,the Hartree-Fock determinant!. Let us consider the standardSRCC approach~hereafter referred to as methodA), inwhich the cluster operatorT is approximated as follows:

T'TA5(i 51

mA

Ti , ~1!

whereTi , i 51, . . . ,mA , are the many-body components ofT included in the calculations andmA,N. The equations forcluster amplitudes definingTA are,

Qi~HNeTA!CuF&50 ~ i 51, . . . ,mA!, ~2!

whereHN5H2^FuHuF& is the Hamiltonian in the normal-ordered form,Qi is a projection operator onto the subspacespanned by alli-tuply excited configurations relative touF&,and the subscriptC designates the connected part of the cor-responding operator expression. Once the system of equa-tions, Eq.~2!, is solved forTA , the energy is calculated asfollows:

DE(A)[E(A)2^FuHuF&5^Fu~HNeTA!CuF&. ~3!

When uF& is the Hartree-Fock configuration,DE(A) repre-sents the SRCC correlation energy.

We have recently derived the formula for the nonitera-tive energy correctiond that, when added to the energy ob-tained in approximate SRCC calculations,DE(A), Eq. ~3!,recovers the exact~FCI! energy,DE[E2^FuHuF&. Theresult is4,34

d5DE2DE(A)

5 (n5mA11

N

(j 5mA11

n

^CuQn Cn2 j~mA! M j~mA!uF&/

^CueTAuF&, ~4!

where

M j~mA!5~HNeTA!C, j ~5!

is the j-particle component of (HNeTA)C ,

Cn2 j~mA!5~eTA!n2 j ~6!

FIG. 1. Potential energy curves for the DZ modes of the HF molecule~Refs.4 and 34!. A comparison of the results obtained with the CR-CCSD~T! andR-CCSD~T! methods~designated bym and ., respectively! with the re-sults of the CCSD, CCSD~T!, and FCI calculations~designated by theLsymbol and the short-dashed and dotted lines, respectively!.

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is the (n2 j )-particle component of the SRCC wave operatoreTA defining methodA, anduC& is the exact wave function.The proof of the formula~4! and its generalization to excitedstates, based on the so-called Fundamental Theorem of theFormalism ofb-Nested Equations, has been given in Ref. 4.An alternative derivation of Eq.~4!, independent of the for-malism ofb-nested equations,4,36 has been given in Ref. 34.

Equation~4! represents the basic equation of the MMCCtheory. As explained in the earlier papers,4,34 the M j (mA)components that enter Eq.~4! are the excitation operatorsthat can easily be obtained by considering the projections ofthe SRCC equations, in whichT5TA , onto all possiblej-tuply excited configurations withj .mA . Indeed,

M j~mA!uF&5Qj~HNeTA!CuF&5(J

M J( j )~mA! uFJ

( j )&,

~7!

where

M J( j )~mA!5^FJ

( j )u~HNeTA!CuF& ~8!

are the projections of the SRCC equations, in whichT5TA , onto the individual j-tuply excited configurationsuFJ

( j )&. The quantitiesM J( j )(mA), Eq. ~8!, define the gener-

alized moments of CC equations.37,38 In terms of these mo-ments, the system of equations defining methodA, Eq. ~2!,can be written as

M j~mA!uF&50, ~ j 51, . . . ,mA!, ~9!

and the SRCC energy formula, Eq.~3!, reduces toDE(A)

5^FuM0(mA)uF&5M0(mA). Since the equations definingmethodA are obtained by zeroing the generalized momentsM J

( j )(mA) with j 51, . . . ,mA @see Eq.~9!#, the only mo-ments that enter the correctiond, Eq. ~4!, are the highermoments of methodA, i.e., M J

( j )(mA) with j 5mA

11, . . . ,N.The correctiond, Eq. ~4!, is a functional of the exact

wave functionuC& and, as such, cannot be calculated with-out solving first the FCI problem. We can, however, use asimple guess foruC& ~provided, for example, by some inex-pensive ab initio method! and calculate the approximatevalue of the correctiond. In this way, we can potentiallycorrect the results of approximate SRCC calculations andobtain energies that are much closer to the exact~FCI! en-ergy DE than the approximate SRCC energyDE(A). Theonly requirement that Eq.~4! imposes onuC& is the presenceof some higher-than-mA-tuply excited configurations inuC&.The renormalized and completely renormalized CCSD~T!and CCSD~TQ! methods are based on using the second-order-type MBPT wave functions in defininguC&.

B. The renormalized and completely renormalizedCCSD„T… and CCSD „TQ… approaches

Let us focus on the MMCC approaches, in which wecorrect the results of the standard CCSD calculations~themA52 case!. Let us further restrictuC& in Eq. ~4! to func-tions that have no higher-than-mB-tuply excited (2,mB

,N) components in the corresponding CI expansions. With

these restrictions, the formula for the correctiond that needsto be added to the CCSD energy,DECCSD, to approximatelyrecover the FCI energy reduces to

d5 (n53

mB

(j 53

n

^CuQn Cn2 j~2! M j~2!uF&/^CueT11T2uF&,

~10!

where M j (2) are the generalized moments of the CCSDequations,Cn2 j (2) are the (n2 j )-body components of theCCSD wave operatoreT11T2, andT1 andT2 are the singlyand doubly excited components of the CCSD cluster opera-tor. The value ofj in Eq. ~10! cannot exceed 6, since thegeneralized moments of the CCSD equations correspondingto higher-than-hextuply excited configurations vanish.

Two special cases of Eq.~10! are particularly important:mB53 and mB54. In the former case, defining theMMCC~2,3! approximation, we calculate the energy correc-tion d ~designated here asdMMCC(2,3)) as follows:4,34

dMMCC(2,3)5^CuQ3 M3~2!uF&/^CueT11T2uF&, ~11!

where

M3~2!uF&5Q3~HNeT11T2!CuF&

5 (i , j ,ka,b,c

M i jkabc~2!uF i jk

abc&, ~12!

with

M i jkabc~2!5^F i jk

abcu~HNeT11T2!CuF& ~13!

representing the CCSD equations projected on triexcitedconfigurations uF i jk

abc&. In the mB54 case, defining theMMCC~2,4! approximation, we calculate the correspondingenergy correctiondMMCC(2,4) using the formula4,34

dMMCC(2,4)5^Cu$Q3 M3~2!1Q4 @M4~2!

1T1M3~2!#%uF&/^CueT11T2uF&, ~14!

whereM3(2) is given by Eq.~12! and

M4~2!uF&5Q4~HNeT11T2!CuF&

5 (i , j ,k, l

a,b,c,d

M i jklabcd~2!uF i jkl

abcd&, ~15!

with

M i jklabcd~2!5^F i jkl

abcdu~HNeT11T2!CuF& ~16!

representing the CCSD equations projected on quadruply ex-cited configurationsuF i jkl

abcd&.The completely renormalized CCSD~T! @CR-CCSD~T!#

approach is obtained by replacing the wave functionuC& inthe MMCC~2,3! formula for the correctiond, Eq.~11!, by4,34

uCCCSD(T)&5~11T11T21R0(3)~VNT2!C

1R0(3)VNT1!uF&, ~17!

whereT1 andT2 are obtained in the CCSD calculations,R0(3)

is the three-body part of the MBPT reduced resolvent, andVN is the two-body part ofHN . Essentially, the

5646 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 K. Kowalski and P. Piecuch

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R0(3)~VNT2!CuF&5T3

[2] uF& ~18!

term in Eq. ~17! is the lowest-order~second-order! MBPTcontribution toT3uF& ~precisely, the CCSD analog of thesecond-order contribution toT3uF&). The energy expressiondefining the CR-CCSD~T! method can be given the follow-ing form:4,34

DECR-CCSD(T)5DECCSD1^CCCSD(T)uQ3 M3~2!uF&/

^CCCSD(T)ueT11T2uF&. ~19!

If we further approximateM3(2) in Eq. ~19! by the lowest-order term (VNT2)C,3 , we obtain the renormalized CCSD~T!@R-CCSD~T!# method. Thus, the R-CCSD~T! energy is cal-culated using the formula,4,34

DER-CCSD(T)5DECCSD1^CCCSD(T)uQ3 ~VNT2!CuF&/

^CCCSD(T)ueT11T2uF&. ~20!

Three different variants~labeled by letters a, b, and c! ofthe renormalized and completely renormalized CCSD~TQ!methods are considered in this work. The basic completelyrenormalized CCSD~TQ! approach@the CR-CCSD~TQ! ap-proach introduced in Refs. 4, 34; hereafter referred to as theCR-CCSD~TQ!, a method# is obtained by replacinguC& inthe MMCC~2,4! formula, Eq.~14!, by4,34

uCCCSD(TQ),a&5uCCCSD(T)&1 12T2T2

(1)uF&, ~21!

whereT2(1) is the first-order MBPT contribution toT2 . The

CR-CCSD~TQ!,a energy is obtained using the formula4,34

DECR-CCSD(TQ),a5DECCSD1^CCCSD(TQ),au$Q3 M3~2!

1Q4 @M4~2!1T1M3~2!#%uF&/

^CCCSD(TQ),aueT11T2uF&. ~22!

In variant ‘‘a’’ of the renormalized CCSD~TQ! method@theR-CCSD~TQ!-1,a approach; designated in Ref. 34 as theR-CCSD~TQ!-1 method; cf., also, Ref. 4#, we neglect theT1M3(2) term in Eq.~22! @in the Hartree-Fock case, thisterm is at least a fourth-order term, whereasM4(2) is at leasta third-order term#, replace M3(2) by its lowest-order(VNT2)C,3 contribution, and replaceM4(2) by4,34

M4~2!@CCSD~TQf!#5@VN~ 12T2

21T3[2] !#C,4 , ~23!

whereT3[2] is defined by Eq.~18!. In another variant of the

renormalized CCSD~TQ! method, referred to as theR-CCSD~TQ!-2,a approach@R-CCSD~TQ!-2 in Ref. 34; cf.,also, Ref. 4#, we replaceM3(2) in Eq. ~22! by

M3~2!@CCSD~TQ!#5@VN~T21 12T2

2!#C,3 , ~24!

andM4(2) by

M4~2!@CCSD~TQ!#5 12~VNT2

2!C,4 , ~25!

and again neglect theT1M3(2) term in Eq.~22!.4,34 As ex-plained in Ref. 34, both R-CCSD~TQ! schemes describedabove have a similar physical content.

Variant ‘‘b’’ of the CR-CCSD~TQ!, R-CCSD~TQ!-1,and R-CCSD~TQ!-2 approaches is obtained by modifyingthe formula foruCCCSD(TQ),a&, Eq. ~21!. The first-order esti-mate forT2 in the (1/2)T2T2

(1) term is replaced by the CCSD

value of T2 , so that the functionuC& used in the CR-CCSD~TQ!,b, R-CCSD~TQ!-1,b, and R-CCSD~TQ!-2,b ap-proaches has the form,

uCCCSD(TQ),b&5uCCCSD(T)&1 12T2

2uF&. ~26!

All other details of the CR-CCSD~TQ!,b, R-CCSD~TQ!-1,b,and R-CCSD~TQ!-2,b schemes are the same as in the CR-CCSD~TQ!,a, R-CCSD~TQ!-1,a, and R-CCSD~TQ!-2,a pro-cedures. As we are going to see, the replacement of the first-order estimate forT2 in Eq. ~21! by the CCSD value ofT2

plays a significant role in improving the CR-CCSD~TQ!,R-CCSD~TQ!-1, and R-CCSD~TQ!-2 results for large N–Nseparations in N2 .

Finally, in variant ‘‘c’’ of the CR-CCSD~TQ!,R-CCSD~TQ!-1, and R-CCSD~TQ!-2 methods, we includein uC& the disconnected triexcited termsT1T2 and thelowest-order estimate for the connected tetraexcited clustersgiven by the well-known formula15,20

T4[3] uF&5R0

(4)@VN~ 12T2

21T3[2] !#CuF&, ~27!

whereT3[2] is defined by Eq.~18! andR0

(4) is the four-bodycomponent of the MBPT reduced resolvent. TheT1T2 andT4 components appear for the first time in the third-orderMBPT wave function. The purpose of including these termsin the wave functionuC& that enters the CR-CCSD~TQ!,R-CCSD~TQ!-1, and R-CCSD~TQ!-2 formulas is the inves-tigation if these higher-order terms~particularly, the lowest-order estimates ofT4 clusters! play a significant role in de-scribing the bond breaking in N2 . The formula for the wavefunction uC& used in the CR-CCSD~TQ!,c, R-CCSD~TQ!-1,c, and R-CCSD~TQ!-2,c methods is

uCCCSD(TQ),c&5uCCCSD(T)&1~T1T21 12T2

21T4[3] !uF&,

~28!

where T4[3] is given by Eq.~27!. All other details of the

CR-CCSD~TQ!,c, R-CCSD~TQ!-1,c, and R-CCSD~TQ!-2,cschemes are the same as in the case of methods CR-CCSD~TQ!,a, R-CCSD~TQ!-1,a, and R-CCSD~TQ!-2,a.

The reason for calling the CR-CCSD~T!, R-CCSD~T!,CR-CCSD~TQ!, and R-CCSD~TQ!-1,2 approaches ‘‘renor-malized’’ is the fact that these approaches can be viewed asextensions of the CCSD~T!11 and CCSD(TQf)

15 approaches.For example, if we replace the denominator^CCCSD(T)ueT11T2uF& in the R-CCSD~T! formula, Eq.~20!,by 1 @^CCCSD(T)ueT11T2uF& equals 1, if we neglect thesecond- and higher-order terms4,34#, we obtain the familiarexpression for the CCSD~T! energy,4,34 i.e., DECCSD

1^FuT2†VNR0

(3)VNT2uF&1^FuT1†VNR0

(3)VNT2uF&. Simi-larly, replacing ^CCCSD(TQ),aueT11T2uF& in theR-CCSD~TQ!-1,a formula by 1 leads to the expression de-fining the CCSD(TQf) approach.34 Thus, the main differencebetween the CR-CCSD~T!, R-CCSD~T!, CR-CCSD~TQ!,and R-CCSD~TQ!-1,2 energy expressions and those definingthe CCSD~T! and CCSD(TQf) approaches is the presence ofthe ^CueT11T2uF& denominators in Eqs.~19!, ~20!, and~22!.These denominators play an essential role in improving poorCCSD~T! and CCSD(TQf) results for N2 at large internu-clear separations.

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C. Computational details

In order to test the performance of the renormalized andcompletely renormalized CCSD~T! and CCSD~TQ! ap-proaches in calculations involving multiple bond breaking,we performed a series of calculations for the N2 molecule, asdescribed by the double-zeta~DZ! @4s2p/2s# basis set.35 Weused the DZ basis set, since the purpose of the present paperis to compare our results with the exact~FCI! potential en-ergy curve for N2 , which we obtained using theGAMESS39

package. The following N–N separationsR were included inour calculations: the experimental equilibrium bond lengthRe52.068 bohr and 1.551 (0.75Re), 1.85, 2.35, 2.585(1.25Re), 2.85, 3.102 (1.5Re), 3.35, 3.619 (1.75Re), 3.9,4.136 (2Re), 4.35, and 4.653 bohr (2.25Re).

All CR-CCSD~T!, R-CCSD~T!, CR-CCSD~TQ!, andR-CCSD~TQ!-1,2 calculations and the related CCSD~T! andCCSD(TQf) calculations were performed using the pilotcomputer programs described in Ref. 34. The CCSD calcu-lations needed to obtain theT1 and T2 cluster amplitudeswere performed using the program described in Ref. 40,based on the orthogonally spin-adapted formulation of theCCSD method.41 In all calculations, the ground-state RHFdeterminant was used as a reference. The RHF calculationsand the integral transformation from the atomic to molecularorbital basis were performed withGAMESS.39

III. RESULTS AND DISCUSSION

The N2 molecule is known to be characterized by largeT3 andT4 effects. At the equilibrium geometry (R5Re) andfor a DZ basis set used in this work, the effect ofT3 clusters,as estimated by forming a difference of the full CCSDT andCCSD energies, is -6.182 milihartree. The effect ofT4 clus-

ters, as estimated by forming the difference of the fullCCSDTQ and CCSDT energies is21.912 milihartree. TheCCSDTQ method is virtually exact in this case. According tothe most recent study by Musiałet al.,42 the T5 clusters areresponsible for most of the remaining 0.195 milihartree dif-ference between the full CCSDTQ and FCI energies. For thegeometries near the equilibrium, theT3 and T4 effects areaccurately described by the perturbative CCSD~T! andCCSD(TQf! approaches. Indeed, atR5Re , the differencebetween the CCSD~T! and CCSD energies is26.133 mili-hartree~cf. Table I!, which should be compared to26.182milihartree obtained by subtracting the CCSD energy fromthe CCSDT energy. The difference between the CCSD(TQf!and CCSD~T! energies atR5Re is 21.833 milihartree~cf.Table I!, which is almost identical to the difference betweenthe CCSDTQ and CCSDT energies~21.912 milihartree!.

The situation gets more complicated when the N–Nbond is stretched. First of all, the combined effect of higher-than-doubly excited clusters~as estimated by forming thedifference between the FCI and CCSD energies! dramati-cally increases, from 8.289 milihartree atR5Re to 33.545milihartree atR51.5Re ~see Table I!. At R'1.75Re , theCCSD potential energy curve has an unphysical hump andfor R.3.74 bohr the CCSD potential energy curve is locatedsignificantly below the FCI curve~see Fig. 2; the absolutevalue of the difference between the CCSD and FCI energiesincreases to 120.836 milihartree atR52.25Re ; cf. Table I!.It is very difficult to converge the CCSD equations forR.2.25Re and it is quite likely that somewhere in the regionof R.2.25Re , the solution of the CCSD equations becomessingular.24 The complete failure of the CCSD theory at largeinternuclear distances is related to the absence of importantT3 and T4 clusters. BothT3 and T4 clusters have to be in-

TABLE I. A comparison of various CC ground-state energies with the corresponding FCI results obtained for selected geometries of the N2 molecule witha DZ basis set. The FCI total energiesE @reported as2(E1108)# are given in hartree. The CC energies and their renormalized and completely renormalizedCCSD~T! and CCSD~TQ! analogs are reported in milihartree relative to the corresponding FCI energy values. The equilibrium N–N bond lengthRe equals2.068 bohr.

Method 0.75Re Re 1.25Re 1.5Re 1.75Re 2Re 2.25Re

FCI 0.549 027 1.105 115 1.054 626 0.950 728 0.889 906 0.868 239 0.862 125CCSD 3.132 8.289 19.061 33.545 17.714 269.917 2120.836CCSDTa 0.580 2.107 6.064 10.158 222.468 2109.767 2155.656

CCSD~T! 0.742 2.156 4.971 4.880 251.869 2246.405 2387.448CCSD~TQf) 0.226 0.323 0.221 22.279 214.243 92.981 334.985

R-CCSD~T! 0.868 2.714 7.113 12.536 214.781 2112.745 2169.410CR-CCSD~T! 1.078 3.452 9.230 17.509 22.347 286.184 2133.313

R-CCSD~TQ!-1,a 0.382 1.072 3.242 7.988 4.032 238.370 262.089R-CCSD~TQ!-2,a 0.370 0.734 1.158 1.999 21.331 235.194 257.321CR-CCSD~TQ!,a 0.448 1.106 2.474 5.341 1.498 240.784 269.259

R-CCSD~TQ!-1,b 0.393 1.227 3.939 9.452 17.108 10.682 213.832R-CCSD~TQ!-2,b 0.375 0.969 2.572 5.844 12.099 6.375 218.187CR-CCSD~TQ!,b 0.451 1.302 3.617 8.011 13.517 25.069 14.796

R-CCSD~TQ!-1,c 0.336 1.421 5.090 11.536 13.030 214.667 247.503R-CCSD~TQ!-2,c 0.351 1.154 3.628 8.448 12.537 211.270 242.748CR-CCSD~TQ!,c 0.433 1.518 4.848 11.081 13.927 8.590 26.708

aThe values forR.Re were taken from Ref. 43.

5648 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 K. Kowalski and P. Piecuch

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cluded in the RHF-based SRCC calculations. As demon-strated in Ref. 23~see Table I and Fig. 2; cf., also, Refs. 43and 44!, the full CCSDT method alone is not good enough.This can be seen by looking at the differences between theCCSDT and FCI energies at larger distancesR. For example,the relatively small, 2.107 milihartree, difference betweenthe CCSDT and FCI energies atR5Re increases to 10.158milihartree atR51.5Re ~see Table I!. Moreover, the fullCCSDT curve has a hump for the intermediate values ofR.For R.3.41 bohr, the full CCSDT potential energy curve islocated significantly below the FCI and CCSD curves~seeFig. 2; the absolute value of the difference between theCCSDT and FCI energies increases to 155.656 milihartree atR52.25Re; cf. Table I!, which is a clear sign of the break-down of the CCSDT approach at large internuclear separa-tions R due to the neglect of importantT4 clusters.44 Al-though there is no direct evidence, it is also possible thathigher-than-quadruply excited clusters are non-negligiblewhenR becomes large, since triple bond breaking in N2 re-quires at least some hextuple excitations.3,32 For example,the difference between the SRCISDTQ~SRCI singles,doubles, triples, and quadruples! and FCI energies atR52Re is almost 40 milihartree.32

The inclusion ofT3 and T4 clusters via perturbativeCCSD~T! and CCSD(TQf! approximations leads to disas-trous results at large internuclear separations. The small,2.156 milihartree, difference between the CCSD~T! and FCIenergies atR5Re increases~in absolute value! to 51.869milihartree atR51.75Re, 246.405 milihartree atR52Re,and 387.448 milihartree atR52.25Re. The theoretically bet-ter CCSD(TQf! method, which works extremely well up toR51.25Re ~giving errors relative to FCI less than;0.3 mi-lihartree!, completely fails at large internuclear separations.The difference between the CCSD(TQf! and FCI energiesincreases to 92.981 milihartree atR52Re and 334.985 mili-

hartree atR52.25Re ~see Table I!. The CCSD~T! andCCSD(TQf! potential energy curves are completely unphysi-cal at large internuclear distances: The CCSD~T! curve islocated significantly below the FCI curve and is character-ized by an unphysical hump atR'3.4 bohr; the CCSD(TQf!curve is located significantly above the FCI curve~seeFig. 2!.

As mentioned in Sec. I, the failure of CCSD~T! andCCSD(TQf! approaches is a consequence of the divergentnature of the MBPT series at large internuclear separations.The second factor that contributes to the breakdown of theCCSD~T! and CCSD(TQf! approximations is the unphysicalbehavior of the CCSD method in the region of largeR val-ues. The CCSD~T! and CCSD(TQf! approaches use theCCSD values of theT1 andT2 cluster amplitudes to estimatethe energy corrections due to connected triples and qua-druples. The CCSD method is not a good source of informa-tion about theT1 and T2 clusters for large values ofR andthis must affect the quality of the CCSD~T! and CCSD(TQf!results in this region. The renormalized and completelyrenormalized CCSD~T! and CCSD~TQ! approaches also usethe CCSD values of theT1 andT2 cluster amplitudes. It is,therefore, most interesting to compare the results of therenormalized and completely renormalized CCSD~T! andCCSD~TQ! calculations with the CCSD~T! and CCSD(TQf!results.

The R-CCSD~T!, CR-CCSD~T!, R-CCSD~TQ!-1,2, andCR-CCSD~TQ! results are listed in Table I. The R-CCSD~T!and CR-CCSD~T! potential energy curves are shown in Fig.2, the R-CCSD~TQ!-1,a-c potential energy curves are shownin Fig. 3, and the CR-CCSD~TQ!,a-c potentials are shown inFig. 4 @we do not show the R-CCSD~TQ!-2,a-c curves infigures, since they are similar to the R-CCSD~TQ!-1,a-ccurves#.

It is remarkable to observe the significant improvement

FIG. 2. Potential energy curves for the DZ model of the N2 molecule. Acomparison of the results obtained with the CR-CCSD~T! and R-CCSD~T!methods~designated by, and n, respectively! with the results of theCCSD, CCSD~T!, CCSDT, CCSD(TQf!, and FCI calculations~designatedby the solid line,h, *, s, and the dotted line, respectively!.

FIG. 3. Potential energy curves for the DZ model of the N2 molecule. Acomparison of the results obtained with the R-CCSD~TQ!-1,a-c methods~designated byn, ,, andL, respectively! with the results of the CCSD,CCSD~T!, CCSDT, CCSD(TQf!, and FCI calculations~designated by thesolid line, h, *, s, and the dotted line, respectively!.

5649J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Renormalized CCSD(TQ)

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of the CCSD~T! results, when the method is renormalizedaccording to our MMCC energy formula. Both the simplyrenormalized R-CCSD~T! potential and the CR-CCSD~T!potential energy curve are much better than the potential en-ergy curve obtained in the standard CCSD~T! calculations~see Fig. 2 and Table I!. A significant part of the unphysicalbehavior of the CCSD~T! method at large internuclear sepa-rations is eliminated by our renormalization procedure. The51.869 milihartree error in the CCSD~T! result relative toFCI obtained atR51.75Re reduces~in absolute value! to14.781 milihartree, when the R-CCSD~T! method is em-ployed and to 2.347 milihartree when the CR-CCSD~T! ap-proach is used. The huge 246.405 milihartree error in theCCSD~T! result relative to FCI obtained atR52Re reduces~in absolute value! to 86.184 milihartree, when the CR-CCSD~T! method is employed. None of the renormalizedCCSD~T! methods gives the correct shape of the potentialenergy curve for largeR values, which is a consequence ofthe absence ofT4 ~and other higher-order! contributions inthe R-CCSD~T! and CR-CCSD~T! methods, but the im-provement of the CCSD~T! results is remarkable. It is inter-esting to observe that the CR-CCSD~T! results for largervalues ofR are better than those obtained with the signifi-cantly more expensive full CCSDT method@in fact, theR-CCSD~T! results are not much worse#. At the same time,the R-CCSD~T! and CR-CCSD~T! results for R'Re arepractically identical to the corresponding CCSD~T! andCCSDT results~see Table I!.

The results obtained with the renormalized and com-pletely renormalized CCSD~TQ! approaches, although notperfect, are even more remarkable and intriguing. Alreadythe simplest renormalization procedure offered by theR-CCSD~TQ!-1,a-c and R-CCSD~TQ!-2,a-c approachesleads to significant improvements in the results for large val-ues of R. All three R-CCSD~TQ!-1 curves and all three

R-CCSD~TQ!-2 curves are located between the CCSD andFCI curves for largeR values and the overall description ofthe N–N bond breaking in N2 by the R-CCSD~TQ!-1,a-c andR-CCSD~TQ!-2,a-c methods is much better than that offeredby the CCSD~T!, R-CCSD~T!, CR-CCSD~T!, full CCSDT,and CCSD(TQf! approaches~see Fig. 3 and Table I!. Forexample, the 92.981 milihartree error relative to FCI, ob-tained atR52Re with the CCSD(TQf! method, reduces to10.682 and 6.375 milihartree, when the R-CCSD~TQ!-1,band R-CCSD~TQ!-2,b approaches, respectively, are em-ployed ~the full CCSDT approach gives a large, 109.767milihartree, error at this value ofR). Variant ‘‘a’’ of theR-CCSD~TQ!-1 and R-CCSD~TQ!-2 approaches providesworse results than variant ‘‘b,’’ since variant ‘‘a’’ uses thefirst-order estimate ofT2 in the definition of uC& @cf. Eq.~21!#, which in our view is not a good idea at larger internu-clear separations. The fact that it is important to replaceT2

(1)

in the wave functionuC& entering the energy expressionscharacterizing the renormalized and completely renormalizedCCSD~TQ! methods by the CCSD value ofT2 becomes clearwhen we compare the CR-CCSD~TQ!,a and CR-CCSD~TQ!,b results for large values ofR, such asR>2Re

~see Fig. 4 and Table I!. The CR-CCSD~TQ!,b potential en-ergy curve, which remains reasonably close to and above theFCI curve, is undoubtedly better~in the overall shape! thanthe CR-CCSD~TQ!,a curve.

In general, the use of the complete moments in the CR-CCSD~TQ!,a-c calculations provides improvements in theoverall shapes of the corresponding potential energy curves.This is particularly true for the CR-CCSD~TQ!,b and CR-CCSD~TQ!,c calculations. The humps on the CR-CCSD~TQ!,b and CR-CCSD~TQ!,c potential energy curvesare much less pronounced than those characterizing the cor-responding R-CCSD~TQ!-1,b~2,b! and R-CCSD~TQ!-1,c~2,c! curves~cf. Figs. 3 and 4!. For example, the differ-ence between the R-CCSD~TQ!-1,b energy, at the maximumcorresponding to the hump on the R-CCSD~TQ!-1,b curve,and the R-CCSD~TQ!-1,b energy atR52.25Re equals;21.0 milihartree. A similar difference for the CR-CCSD~TQ!,b curve is only;4.9 milihartree. The inclusionof the lowest-order estimates for the connected tetraexcitedclusters in the wave functionuC& defining the R-CCSD~TQ!-1,c~2,c! and CR-CCSD~TQ!,c methods@see the formula~28!for uCCCSD(TQ),c&# does not lead to significant improvementsin the renormalized CCSD~TQ! results. Variant ‘‘b,’’ whichuses a simple estimate foruC&, Eq. ~26!, that essentially isequivalent to the second-order MBPT wave function, pro-vides a remarkably well-balanced description of the potentialenergy curve for N2 @particularly, considering the simplicityof the R-CCSD~TQ!-1,b~2,b! and CR-CCSD~TQ!,b meth-ods#, although there is room for improvement in the regionof intermediateR values. We should emphasize the fact thatthe R-CCSD~TQ!-1,b~2,b! and CR-CCSD~TQ!,b resultswere obtained by rather straightforward modifications in theformulas that define the original CCSD(TQf! approach, usingthe prescription provided by the general MMCC formalism.The most essential characteristics of the CCSD(TQf! ap-proach, such as the use of the easily available CCSD valuesof T1 andT2 clusters in estimating higher-order effects and

FIG. 4. Potential energy curves for the DZ model of the N2 molecule. Acomparison of the results obtained with the CR-CCSD~TQ!,a-c methods~designated byn, ,, andL, respectively! with the results of the CCSD,CCSD~T!, CCSDT, CCSD(TQf!, and FCI calculations~designated by thesolid line, h, *, s, and the dotted line, respectively!.

5650 J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 K. Kowalski and P. Piecuch

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the noniterative character of the energy correction due totriples and quadruples, are preserved in our R-CCSD~TQ!-1,b~2,b! and CR-CCSD~TQ!,b approaches and their ‘‘a’’ and‘‘c’’ counterparts. At the same time, the overall descriptionof the triple bond breaking in N2 provided by theR-CCSD~TQ!-1,b~2,b! and CR-CCSD~TQ!,b methods@infact, by all renormalized and completely renormalizedCCSD~TQ! approaches# is much better than that obtainedwith the conventional CCSD(TQf! scheme. The fact that allour renormalized and completely renormalized CCSD~TQ!approaches, which are simple, noniterative methods of ac-counting for the effect of tri- and tetraexcited cluster compo-nents, provide better description of the dissociation of N2

than the expensive full CCSDT method is worth noticinghere, too.

As explained in our earlier studies of the bond breakingin HF and H2O,4,34 the primary reason for the observed im-provements in the renormalized and completely renormal-ized CCSD~T! and CCSD~TQ! results at large internuclearseparations is the fact that the^CueT11T2uF& denominatorsentering the R-CCSD~T!, CR-CCSD~T!, R-CCSD~TQ!-1,2,and CR-CCSD~TQ! energy expressions@cf., e.g., Eqs.~19!,~20!, and~22!# increase their values withR ~see Fig. 5!. The^CueT11T2uF& denominator characterizing the R-CCSD~T!and CR-CCSD~T! methods increases its value from;1.0 atR5Re to ;5.5 atR52.25Re . The ^CueT11T2uF& denomi-nators characterizing variants ‘‘a,’’ ‘‘b,’’ and ‘‘c’’ of theR-CCSD~TQ!-1,2 and CR-CCSD~TQ! methods increasetheir values from;1.0 atR5Re to 7.8, 12.9, and 11.7, re-spectively, atR52.25Re . Thus, the CueT11T2uF& denomi-nator plays a role of a natural damping factor, which forlarge internuclear separationsR damps large and unphysicalvalues of perturbative triples and quadruples corrections. No

such denominators are present in the conventional CCSD~T!and CCSD(TQf! energy formulas, and, in consequence, theCCSD~T! and CCSD(TQf! corrections due to triples andquadruples grossly overestimate theT3 andT4 effects, pro-ducing completely unphysical potential energy curves forlarger values ofR. The ^CueT11T2uF& denominators areclose to 1.0 in the vicinity of the equilibrium geometry,where the CCSD~T! and CCSD(TQf! approaches work fine.There is no need to damp the CCSD~T! and CCSD(TQf!corrections due toT3 andT4 in this region and our MMCC-based renormalized CCSD~T! and CCSD~TQ! methods rec-ognize this.

The above results show that the renormalized and com-pletely renormalized CCSD~T! and CCSD~TQ! methods andthe more general MMCC approach provide the frameworkthat may allow us to remove the major drawback of the per-turbative CC schemes, which is their inability to describequasidegenerate states and bond breaking. At the same time,the renormalized and completely renormalized CCSD~T! andCCSD~TQ! methods and the MMCC theory preserve themost essential features of the standard CCSD~T! andCCSD(TQf! approaches, including the simplicity and the no-niterative character of the corrections describing higher-than-doubly excited clusters, the ‘‘black-box’’ nature that madethe CCSD~T! method very popular in chemistry, and theability of the CCSD~T! and CCSD(TQf! methods to describeT3 andT4 effects in nondegenerate situations.

Clearly, the above findings need further studies. It isremarkable to observe the significant improvements that therenormalized and completely renormalized CCSD~T! andCCSD~TQ! methods provide at large internuclear separa-tions, even in a complicated case of the multiple bond break-ing in N2 , but we need to work further on improving certainaspects of the new theory. The behavior of the renormalizedand completely renormalized CCSD~T! and CCSD~TQ!methods in the region of intermediateR distances for mol-ecules having triple bonds, such as N2, needs to be some-what improved @for simpler cases of bond breaking, therenormalized and completely renormalized CCSD~T! andCCSD~TQ! methods work fine over the entire region of in-ternuclear separations4,34#. In the renormalized and com-pletely renormalized CCSD~T! and CCSD~TQ! methods, westill neglect the generalized moments corresponding to pro-jections of the CCSD equations onto pentuply and hextuplyexcited configurations, i.e.,M5(2) andM6(2) @let us recallthat the CCSD momentsM j (2) with j .6 trivially vanish#.These moments may be important for describing the disso-ciation of the triple bond in N2 and their role should beinvestigated. We should also test the renormalized and com-pletely renormalized CCSDT~Q! method,4,34 which can beviewed as an MMCC-based extension of the factorizedCCSDT(Qf) approach15 and which can be used to improvethe results of the full CCSDT calculations by utilizing thegeneralized moments of the CCSDT theory. These variousaspects of the new MMCC formalism are currently investi-gated in our group and the results will be reported as soon asthey become available.

FIG. 5. The dependence of the^CCCSD(T)ueT11T2uF& (h),^CCCSD(TQ),aueT11T2uF& (s), ^CCCSD(TQ),bueT11T2uF& (n), and^CCCSD(TQ),cueT11T2uF& (,) denominators, corresponding to the wavefunctions uCCCSD(T)&, Eq. ~17!, uCCCSD(TQ),a&, Eq. ~21!, uCCCSD(TQ),b&, Eq.~26!, anduCCCSD(TQ),c&, Eq.~28!, which are used to define the CR-CCSD~T!and CR-CCSD~TQ!,a-c approaches, respectively, on the N–N separationRN-N ~in bohr! for the DZ model of the N2 molecule.

5651J. Chem. Phys., Vol. 113, No. 14, 8 October 2000 Renormalized CCSD(TQ)

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ACKNOWLEDGMENTS

This work has been supported by the startup funds pro-vided to one of us~P.P.! by Michigan State University~MSU! and, in part, by the MSU Intramural Research GrantProgram~the New Faculty, Science and Engineering Awardreceived by P.P.!, which allowed us to upgrade the computersystem used in many of the calculations reported in thiswork. We would like to thank Professor Stanisław A. Ku-charski for providing us with the full CCSDT results forlarger N–N distances.

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