Explicitly correlated combined coupled-cluster and perturbation methodsToru Shiozaki, Edward F. Valeev, and So Hirata Citation: J. Chem. Phys. 131, 044118 (2009); doi: 10.1063/1.3193463 View online: http://dx.doi.org/10.1063/1.3193463 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v131/i4 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Explicitly correlated combined coupled-cluster and perturbation methodsToru Shiozaki,1,2 Edward F. Valeev,3,a� and So Hirata1,b�

1Quantum Theory Project and the Center for Macromolecular Science and Engineering,Department of Chemistry and Department of Physics, University of Florida, Gainesville,Florida 32611-8435, USA2Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo,Tokyo 113-8656, Japan3Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061-0002, USA

�Received 12 May 2009; accepted 10 July 2009; published online 27 July 2009�

Coupled-cluster singles and doubles �CCSD� or coupled-cluster singles, doubles, and triples�CCSDT� with noniterative, perturbation corrections for higher-order excitations have beenextended to include the basis functions that explicitly depend on interelectronic distances �r12� in thewave function expansions with the aim of dramatically accelerating the basis-set convergence ofcorrelation energies. The extension has been based on the so-called R12 �or F12� scheme andapplied to a second-order triples correction to CCSD �CCSD�2�T-R12�, a second-order triples andquadruples correction to CCSD �CCSD�2�TQ-R12�, a third-order triples correction to CCSD�CCSD�3�T-R12�, and a second-order quadruples correction to CCSDT �CCSDT�2�Q-R12�. Asimplified R12 treatment suggested by Fliegl et al. �J. Chem. Phys. 122, 084107 �2005�� has beencombined with some of these methods, introducing CCSD�2�T�R12� and CCSD�2�TQ�R12�. TheCCSD�T�-R12 method has also been developed as an approximation to CCSD�2�T-R12. Thesemethods have been applied to dissociation of hydrogen fluoride and double dissociation of water.For the molecules at their equilibrium geometries, molecular properties predicted by these methodsconverge extremely rapidly toward the complete-correlation, complete-basis-set limits with respectto the cluster excitation rank, perturbation order, and basis-set size. Although the R12 schemeemployed in this work does not improve the basis-set convergence of connected triples orquadruples corrections, the basis-set truncation errors in these contributions have roughly the samemagnitude as small residual basis-set truncation errors in the connected singles and doublescontributions even in the dissociation of hydrogen fluoride. In the double dissociation of water, thebasis-set truncation errors in the connected triples contribution can be a few times as great as thosein the connected singles and doubles contributions. © 2009 American Institute of Physics.�DOI: 10.1063/1.3193463�

I. INTRODUCTION

Combinations of coupled-cluster �CC� theory and nonit-erative, perturbation corrections1 constitute the most accurateand efficient class of electronic-structure methods for quan-titative characterization of molecules whose wave functionsare well approximated by single Slater determinants. CCsingles and doubles �CCSD� with noniterative triples�CCSD�T��,2,3 in conjunction with systematic basis setseries,4 has enabled routine, quantitative interpretation andprediction of the energies, structures, spectra, etc. of smallmolecules in the gas phase and even solids.5 Further exten-sions of this class of methods have been vigorouslypursued.6–21 One such extension is to include higher-orderperturbation corrections and/or higher-rank cluster excita-tions in a systematic fashion. A hierarchy of combined CCand perturbation methods was proposed,15 approaching thefull configuration interaction �FCI� limit as the cluster exci-tation rank and perturbation order are individually or simul-taneously raised. Several of the most useful members of the

hierarchy were implemented into efficient computer codes.They are CCSD with a second-order triples correction�CCSD�2�T�,17 with a second-order triples and quadruplescorrection �CCSD�2�TQ�,17 with a third-order triples correc-tion �CCSD�3�T�,19 and with a third-order triples and qua-druples correction �CCSD�3�TQ� �Ref. 19� as well as CCsingles, doubles, and triples �CCSDT� with a second-orderquadruples correction �CCSDT�2�Q�.17 Unlike CCSD�T�with a restricted Hartree–Fock �HF� reference, which cannotdescribe the potential energy curves along single-bond disso-ciation even qualitatively correctly, CCSD�2�T can handlesuch situations by virtue of retaining all diagrammatic con-tributions that are second order and evaluated at the O�n7�arithmetic operations or less �n is the number of orbitals�and, therefore, at a similar computational cost as CCSD�T�.

Another extension is to introduce basis functions of ex-plicit interelectronic distances �r12� in the combined CC andperturbation methods with the aim of addressing the notori-ously slow convergence of electron-correlation energies withrespect to the one-electron basis-set size.22,23 Noga andco-workers24–28 pioneered explicitly correlated CCSD andCCSD�T� methods employing the so-called R12 scheme in-

a�Electronic mail: evaleev@vt.edu.b�Electronic mail: hirata@qtp.ufl.edu.

THE JOURNAL OF CHEMICAL PHYSICS 131, 044118 �2009�

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troduced by Kutzelnigg and Klopper29,30 and refined byothers,31–41 which serves as a basis on which a linear �R12�or nonlinear �F12� basis function of r12 �correlation factors�can be incorporated into virtually any electron-correlationmethod. The methods of Noga et al., however, involved anapproximation �the standard approximation �SA� �Ref. 30��that led to the elimination of a large number of diagrammaticcontributions and simplified formalisms and computer imple-mentations, but at the cost of requiring a large �uncontracted�basis set to be effective. This approximation was lifted insubsequent studies,42–54 but the initial implementations of theexplicitly correlated CC �CC-R12� methods based on com-plete diagrammatic equations55 were possible only with theaid of a computer algebra automating the exceedingly com-plex formula derivation and code synthesis. With thisscheme, CCSD-R12,56 CCSDT-R12,57 and CCSDTQ-R12�Ref. 57� were developed by the authors. A subsequentCCSD-R12 implementation of Köhn et al.58 was also basedon a runtime computer algebra �“string-based algorithm”�similar to that of Kállay and Surján.59

The objective of this work is to merge these twoorthogonal extensions and arrive at explicitly correlatedanalogs of the most important members of the hierarchy ofcombined CC and perturbation methods,15,17,19 namely,CCSD�2�T-R12, CCSD�2�TQ-R12, CCSD�3�T-R12, andCCSDT�2�Q-R12 as well as CCSD�T�-R12. These methodsare implemented in a manner that avoids introducing addi-tional ad hoc approximations �such as the SA� whose effectson accuracy can be severe or unknown. This allows us torigorously quantify the impact of the perturbation and R12treatments by comparing the results of our methods withthose of CC and CC-R12 of various ranks. As a by-productof these developments, we obtain implementations of the so-called � equations of CC analytical gradient theory forCCSD-R12 and CCSDT-R12 because their solutions�� vectors� are an integral part of our perturbation theory.Fliegl et al.,42 Werner and co-workers,46,54 and Valeev andco-workers47–49 proposed useful iterative and noniterativeapproximations of the R12 corrections to CC. We will clarifythe relationship of our methods to these approximations andimplement the CCSD�2�T-R12 and CCSD�2�TQ-R12 methodsusing the iterative approximation of Fliegl et al.,42 which arecalled CCSD�2�T�R12� and CCSD�2�TQ�R12�. These meth-ods require the � vectors for CCSD�R12�, which are alsoobtained in this work. All our implemented methods can uselinear and nonlinear correlation factors and the terms “R12”and “F12” are used interchangeably in this article.

We apply these methods to the severe test cases of bonddissociation in which wave functions have substantialquasidegenerate character and hence the perturbation theoryis expected to break down. Despite this expectation, the com-bined CC and perturbation methods with or without the R12extensions are found to resist such a breakdown well intodissociative region of the potential energy surfaces. Althoughthe basis-set convergence of second-order triples correctionsin CCSD�2�T-R12 is hardly improved by the present R12method, the basis-set truncation errors in these contributionsare as small as the remaining basis-set truncation errors inthe connected singles and doubles contributions, even in the

near dissociation limit of hydrogen fluoride. The basis-settruncation errors in the connected triples contributions in thedouble dissociation of water can, however, be a few times asgreat as those in the connected singles and doubles contribu-tions that are minimized by the R12 scheme. Our numericalresults suggest that CCSD�2�TQ-R12/aug-cc-pVTZ repro-duces the complete-correlation, complete-basis-set �CBS�potential energy curve �excluding the HF energy and corecorrelation effects� of hydrogen fluoride in the ground stateapproximately within 5 mEh in a broad domain of the bondlength up to four times the equilibrium value.

II. THEORY

A. CC-R12

The perturbation corrections to CC-R12 energies andwave functions25 proposed here are based on an applicationof the Rayleigh–Schrödinger perturbation theory to the CC-

R12 effective or similarity-transformed Hamiltonian H,15

H = exp�− T�H exp�T� = �H exp�T��C, �1�

where H= f + v �the zero of energy is taken as the energy of

the single-determinant reference�, f and v are normal-ordered second-quantized Fock and antisymmetrized two-electron integral operator, respectively, and subscript C indi-cates that the operators in the bracket are diagrammatically

connected. The cluster excitation operator T of rank-n CC-R12 �n=2 for CCSD-R12, n=3 for CCSDT-R12, etc.� is thesum of usual one- through n-electron cluster excitation op-

erators �Ti� plus the so-called geminal excitation operator T2,

T = �i=1

n

Ti + T2. �2�

The geminal excitation operator T2 is formally a two-electroncluster excitation operator, promoting two electrons from oc-cupied orbitals into virtual orbitals spanned by an infinitebasis and is written as

T2 = Q12 ����

�i�j

tij����†�†ji� , �3�

where Q12 is the projector onto the space of doubly excited

determinants that cannot be reached by T2. The followingnotation is used: i and j �as well as k, l, m, and n in thefollowing� label occupied orbitals spanned by a finite basisset, � and � �and also �� virtual orbitals spanned by aninfinite, complete set of basis functions, and �¯ � rearrangesthe operators in the normal order. The excitation amplitudes�tij

���, which form an infinite set, are then parametrized asproducts of antisymmetrized integrals Fkl

�� of correlation fac-tor f�r12� and a small number of unknown coefficients tij

kl,60

tij�� = �

k�l

Fkl��tij

kl. �4�

The amplitudes of Ti and the geminal amplitudes �tijkl� are

determined24 by solving what correspond to the usual CCamplitude equations,55–57

044118-2 Shiozaki, Valeev, and Hirata J. Chem. Phys. 131, 044118 �2009�

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��iaH�0 = 0, �5�

��ijabH�0 = 0, �6�

etc., where �0, �ia, and �ij

ab are the reference, singly,and doubly excited determinants, respectively �a and b labelvirtual orbitals spanned by a finite one-electron basis�, andthe geminal amplitude equation,

��ijklH�0 = 0, �7�

where we have introduced the geminal displacement con-figurations,

�ijkl = Q12 �

���

Fkl����†�†ji��0 . �8�

The correlation energy of CC-R12 can be obtained by evalu-ating

E0 = ��0H�0 . �9�

Once these equations are solved for the amplitudes �seeour previous articles55–57 for further details of the formalismsand algorithms�, the single-determinant reference �0 be-

comes the right eigenvector of H in the space of up to

n-tuply excited determinants and ��ijkl�. Let P be a projector

onto this space,

P = �i=0

n

Pi + P2, �10�

where

P0 = �0��0 , �11�

P1 = �a

�i

�ia��i

a , �12�

P2 = �a�b

�i�j

�ijab��ij

ab , �13�

and

P2 = �k�l

�m�n

�i�j

�ijkl�X−1�kl

mn��ijmn . �14�

Here X is the metric,

�X�mnkl = ��ij

kl�ijmn = �

���

Fkl���Fmn

��, �15�

which arises because ��ijkl� are not orthonormal. For the

purpose of this discussion, the complete determinantal spaceis defined as

1 = �i=0

N

Pi + P2 = P + Q , �16�

where N is the number of electrons and Q is the projector

onto the complementary space. Using P, Eq. �9� and Eqs.�5�–�7� are collectively expressed as

PH�0 = PE0�0 . �17�

B. �-CC-R12

Since H is not a Hermitian operator, it has left eigenvec-tors distinct from the corresponding right eigenvectors.1,61

Both right-hand-side and left-hand-side eigenvectors are nec-essary in defining our perturbation corrections to energiesand wave functions.15 The left eigenvector for the groundstate can be written as

��0�1 + �� = ��0�1 + �i=l

n

�i + L2� , �18�

where �i is an i-electron deexcitation operator and

L2 = ����

�i�j

���ij �i†j†���Q12, �19�

= ����

�i�j

�k�l

Fkl����kl

ij�i†j†���Q12, �20�

whose coefficients can be determined by solving the so-called � equations,

��0�1 + ��HP = ��0�1 + ��E0P , �21�

or, equivalently,

��0HP + ��0��H�LP = 0, �22�

where subscript L means that the operators are diagrammati-cally linked. Alternatively, Eqs. �17� and �21� can be derivedby the differentiation of the CC-R12 energy functional E0

= ��0�1+ ��H�0 with respect to the amplitudes of � or T,respectively.

C. CCSD„R12…

The CCSD�R12� method of Fliegl et al.42 retains only

leading-order diagrammatic terms involving T2 in the CCSD-R12 equations. The CCSD�R12� amplitude equations are,therefore, a diagrammatic subset of the CCSD-R12 ampli-tude equations �Eqs. �5�–�7�� and are written as

��iaHCCSD + �HCCSDT2�C�0 = 0, �23�

��ijabHCCSD + �HCCSDT2�C�0 = 0, �24�

��ijklHCCSD + � fT2�C�0 = 0, �25�

where HCCSD is the CCSD similarity-transformed Hamil-tonian. The corresponding correlation energy is

ECCSD�R12� = ��0HCCSD + �HCCSDT2�C�0 . �26�

Defining the CCSD�R12� similarity-transformed Hamil-tonian by

H�R12� = �1 − P2��HCCSD + �HCCSDT2�C��1 − P2�

+ P2�HCCSD + � fT2�C��1 − P2�

+ �1 − P2��HCCSD + � fT2�C�P2 + P2 fP2, �27�

one can consolidate Eqs. �23�–�26� into a single equation

044118-3 Explicitly correlated coupled-cluster methods J. Chem. Phys. 131, 044118 �2009�

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PH�R12��0 = PECCSD�R12��0 , �28�

where P= P0+ P1+ P2+ P2. This form of H�R12� has been cho-sen so that the same sets of amplitude and � equations canbe derived either by considering the left and right eigenvec-

tors of H�R12� or by the differentiation of the CCSD�R12�energy functional �see below� with respect to the amplitudes

of � or T. Note that in this work, the generalized Brillouincondition is imposed in deriving Eqs. �23�–�25� in accor-dance with the original definition of the method by Flieglet al.42

D. �-CCSD„R12…

The corresponding left eigenvectors are obtained bysolving

��0�1 + ��H�R12�P = ��0�1 + ��ECCSD�R12�P �29�

for �= �1+ �2+ L2 or

��0H�R12�P + ��0��H�R12��LP = 0. �30�

Expanding the last equation under the generalized Brillouincondition, one obtains

��0�HCCSD + HCCSDT2�C + ���1 + �2��HCCSD + HCCSDT2�C�L + �L2HCCSD�L�ia = 0, �31�

��0�HCCSD + HCCSDT2�C + ���1 + �2��HCCSD + HCCSDT2�C�L + �L2HCCSD�L�ijab = 0, �32�

��0HCCSD + ���1 + �2�HCCSD�L + �L2 f�L�ijkl = 0, �33�

from which the terms that have vanishing contributions areremoved. Note that these equations, like Eqs. �23�–�25�, also

retain only the leading-order contributions that involve T2

and/or L2. The identical set of equations is obtained by dif-ferentiating the CCSD�R12� energy functional ECCSD�R12�

= ��0�1+ ��H�R12��0 with respect to the amplitudes of T1,

T2, or T2 and has been considered by Neiss and Hättig62 inrelation to nonlinear optical properties by CCSD�R12�. Suchconsistency is not guaranteed because the CCSD�R12� en-

ergy functional does not have exponential dependence on T2

�the same issue exists in the CCSDT-1, CCSDT-2, andCCSDT-3 methods1�. The consistency is ensured by the par-

ticular definition of H�R12� given in Eq. �27� and the use ofthe generalized Brillouin condition. Without the latter,

��0�L2� fT2�C�L�ia enters Eq. �31� in the eigenvector for-

mulation, while it does not in the energy functional formula-tion. The CCSD�R12� equations and the corresponding �equations can be solved at a much lower cost than those ofCCSD-R12.

E. CCSD„2…T-R12 and CCSD„2…TQ-R12

The second-order and third-order perturbation correc-tions to the CCSD-R12 energies are obtained by applying theRayleigh–Schrödinger perturbation theory to the CCSD-R12similarity-transformed Hamiltonian partitioned into a zeroth-

order part �H�0�� and a perturbation �H�1��,

H�0� = PHP + Q� f �0� + E0�Q , �34�

H�1� = H − H�0�, �35�

where P= P0+ P1+ P2+ P2, Q=1− P, and H is defined with

T= T1+ T2+ T2. The normal-ordered Fock operator is divided

into the primary f �0� and secondary f �1� components as fol-lows:

f �0� = f − f �1�, �36�

f �1� = �i,a

�f ia�a†i� + fa

i �i†a�� . �37�

After straightforward algebra,15 one obtains the zeroth-�E�0��, first- �E�1��, and second-order �E�2�� energy contribu-tions that read

E�0� = E0, �38�

E�1� = 0, �39�

E�2� = ET�2� + EQ

�2�, �40�

where ET�2� and EQ

�2� are contributions from the triple and qua-druple excitation manifolds, respectively, and are written as

ET�2� = �

i�j�k�

a�b�c

��0�H�1��ijkabc��ijk

abcH�1��0�i + � j + �k − �a − �b − �c

, �41�

EQ�2� = �

i�j�k�l�

a�b�c�d

��0�H�1��ijklabcd��ijkl

abcdH�1��0�i + � j + �k + �l − �a − �b − �c − �d

.

�42�

The canonical HF reference wave function is assumed and �i

is the HF orbital energy of ith occupied orbital and �a is the

044118-4 Shiozaki, Valeev, and Hirata J. Chem. Phys. 131, 044118 �2009�

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energy of ath virtual orbital. The CCSD�2�T-R12 andCCSD�2�TQ-R12 methods are defined by their correlation en-ergies E�0�+E�1�+ET

�2� and E�0�+E�1�+E�2�, respectively. Thesecond-order correction involves only up to quadruply ex-

cited determinants because the perturbation H�1� can deexciteonly up to two electrons �while it can excite up to six elec-trons�.

F. CCSD„T…-R12

The noniterative correction in CCSD�T�-R12 �Ref. 43� isa diagrammatic subset of the CCSD�2�T-R12 correction. Re-placing the factors in the numerator of Eq. �41� by some oftheir leading-order diagrammatic contributions, one obtains

ET�2� E�T� = �

i�j�k�

a�b�c

��0T†H�ijkabc��ijk

abc�HT�C�0�i + � j + �k − �a − �b − �c

, �43�

= �i�j�k

�a�b�c

��0�T1† + T2

† + T2†�H�ijk

abc��ijkabc�HT2�C + �HT2�C�0

�i + � j + �k − �a − �b − �c. �44�

The correlation energy of CCSD�T�-R12 equals to E0+E�T�.The last expression makes it clear that the noniterative con-

nected triples correction contains terms involving T2, whichcan, in a sense, be interpreted as the basis-set truncation errorcorrection. However, as shown by Köhn63 and confirmed bythe authors �see below�, these extra terms in E�T� �and similarterms in all the other methods considered in this study�hardly improve the convergence of the connected triples cor-rection with respect to the basis-set size.

G. CCSD„3…T-R12

The third-order correction consists of seven terms eachof which sums contributions from excited determinants ofdifferent ranks. Among these, the contribution that involvesonly triply excited determinants is considered in this work.19

The CCSD�3�T-R12 method is, therefore, defined by the cor-relation energy that sums E�0�, E�1�, E�2�, and the following:

ET�3� = �

l�m�n�

d�e�f�

i�j�k�

a�b�c

��0�H�1��lmndef ��lmn

def H�1��ijkabc��ijk

abcH�1��0��l + �m + �n − �d − �e − � f���i + � j + �k − �a − �b − �c�

,

�45�

= �l�m�n

�d�e�f

��0�H�1��lmndef ��lmn

def H�1�T�1��0�l + �m + �n − �d − �e − � f

, �46�

where

T�1� = �i�j�k

�a�b�c

tijkabc�a†b†c†kji� , �47�

ti jkabc =

��ijkabcH�1��0

�i + � j + �k − �a − �b − �c. �48�

Equations �46�–�48� suggest how the 12-fold summation inEq. �45� can be carried out as two much less expensive steps.

H. CCSDT„2…Q-R12

The first nonvanishing correction to the CCSDT-R12 en-ergy also occurs at the second order and it consists of thequadruples and quintuples contributions. In this study, onlythe quadruples correction is considered. It is formally given

by the same expression as Eq. �42�, but P= P0+ P1+ P2+ P3

+ P2 and T= T1+ T2+ T3+ T2 are implied in the definition

of H�0� and H�1�. The � vector obtained from the�-CCSDT-R12 equations is employed.

I. CCSD„2…T„R12… and CCSD„2…TQ„R12…

The correlation energies of CCSD�2�T�R12� andCCSD�2�TQ�R12� are given by ECCSD�R12�+E�1�+ET

�2� andECCSD�R12�+E�1�+E�2�, respectively, in analogy toCCSD�2�T-R12 and CCSD�2�TQ-R12, where E�1�, E�2�, ET

�2�,and EQ

�2� are defined by Eqs. �39�–�42�. It must be understood

that � in these definitions is the solution of the

�-CCSD�R12� equations �Eqs. �31�–�33��. Likewise, H�1�

= H�R12�− H�0� and

H�0� = PH�R12�P + Q� f �0� + ECCSD�R12��Q , �49�

where P= P0+ P1+ P2+ P2.

J. CCSD„2…T,R12

While the CCSD�2�T,R12 method proposed by Valeev andco-workers47–49 is not among the methods implemented inthis study, it is instructive to clarify its relationship with theaforementioned methods. The CCSD�2�T,R12 method is an-other application of the combined CC and perturbation

theory and is derived by setting P= P0+ P1+ P2 and Q= P2

+ P3+ ¯+PN and by using the CCSD similarity-transformed

Hamiltonian HCCSD as H. The first-order correction to theCCSD correlation energy is null and the second-order triplesand R12 correction is

044118-5 Explicitly correlated coupled-cluster methods J. Chem. Phys. 131, 044118 �2009�

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E�2� = ET�2� + ER12

�2� , �50�

where ET�2� is defined formally by Eq. �41� with H= HCCSD

and, therefore, is identical to the non-R12 CCSD�2�T correc-tion. The CCSD�2�T,R12 method introduces another correc-tion ER12

�2� for the basis-set truncation errors, which reads

ER12�2� = − �

i�j�k�l

�m�n

��0�1 + ��H�1��ijkl

�B�ij�−1 �mn

kl ��ijmnH�1��0 , �51�

where

�B�ij��mnkl = ��ij

kl f �ijmn = �B�mn

kl − ��i + � j��X�mnkl , �52�

with B and X �Eq. �15�� being the special intermediates.55

The elements of B are

�B�mnkl = �

�,�,�Fkl

���f��Fmn

�� . �53�

Valeev and Crawford49 implemented a variant ofCCSD�2�T,R12 called CCSD�T�R12, in which ET

�2� was ap-proximated by E�T� of CCSD�T� and only the leading-orderterms were kept in Eq. �51�.

K. MP2-R12

Setting T=0 and P= P0, our combined CC and perturba-tion theory reduces to the Møller–Plesset �MP� perturbationtheory. The second-order correction to the correlation energyconsists of the doubles contribution and the term accountingfor the basis-set truncation errors,

E�2� = ED�2� + ER12

�2� , �54�

where

ED�2� = �

i�j�a�b

��0v�ijab��ij

abv�0�i + � j − �a − �b

, �55�

ER12�2� = − �

i�j�k�l

�m�n

�V��klij�B�ij��−1�mn

kl �V�†�ijmn. �56�

Equation �56� takes the couplings between the conventionaland geminal doubles amplitudes into consideration. The ele-

ments of V� and B� are

�V��klij = ��0v�ij

kl + �a�b

��0v�ijab��ij

ab f �ijkl

�i + � j − �a − �b, �57�

�B�ij�� �klmn = �B�ij��kl

mn + �a�b

��ijmn f �ij

ab��ijab f �ij

kl�i + � j − �a − �b

. �58�

Here a canonical HF reference is assumed and the Fock ma-

trix elements ��ijab f �ij

mn in the numerator of Eqs. �57� and�58� �the so-called A intermediate� are given by

��ijab f �ij

kl =1

2��

�f�aFkl

�b + f�bFkl

a�� . �59�

These equations define the correlation energy of MP2-R12.

III. COMPUTER IMPLEMENTATION

The CCSD�2�T-R12, CCSD�2�TQ-R12, CCSD�3�T-R12,CCSD�2�T�R12�, CCSD�2�TQ�R12�, and CCSDT�2�Q-R12methods as well as the CCSD�T�-R12 method have beenimplemented. The solver for the � equations for CCSD-R12and CCSDT-R12 �see Ref. 55 for the diagrammatic equa-tions� as well as CCSD�R12� have also been developed. Allthe methods implemented in this study including the � equa-tions are size extensive. Efficient computer codes that exploitspin �within spin orbital formalisms�, spatial �real Abelianpoint group�, and index-permutation symmetries have beenwritten with the aid of the symbolic algebra code SMITH.55

The computational costs of these perturbation correctionshave their respective optimal scaling of noniterative O�n7�for CCSD�2�T-R12, CCSD�2�T�R12�, and CCSD�T�-R12,noniterative O�n8� for CCSD�3�T-R12, noniterative O�n9�for CCSD�2�TQ-R12, and noniterative O�n10� forCCSDT�2�Q-R12, where n is the number of orbitals. Notethat the cost scaling of the CCSDT�2�Q method was reportederroneously as O�n9� in the original paper.17 The character-istics of the various methods discussed in this article aresummarized in Table I.

The computer generation of formalisms and codes em-ploys the combination of strategies and techniques developedoriginally for CC,64,65 the � equations,66 combined CC andperturbation methods,17,19 and CC-R12.55–57 The loop fusionalgorithm17 has been employed to avoid the storage of six-index tensors �having the sizes of triple excitation ampli-tudes� in CCSD�2�T-R12, CCSD�2�TQ-R12, CCSD�2�T�R12�,CCSD�2�TQ�R12�, and CCSD�T�-R12 and eight-index ten-sors �having the sizes of quadruple excitation amplitudes� inCCSD�2�TQ-R12, CCSD�2�TQ�R12�, and CCSDT�2�Q-R12.In CCSD�3�T-R12, one six-index tensor of Eq. �48� needs tobe stored.19 The index-permutation symmetries of a varietyof intermediate quantities of these calculations and how theycan be exploited are elucidated by SMITH on a case-by-casebasis at the time of code generation. Like our prior CCSD-R12, CCSDT-R12, and CCSDTQ-R12 implementations, thepresent implementations and underlying formalisms do notdepend on drastic approximations such as the SA or lessdrastic ones such as the generalized or extended Brillouinconditions.37 However, the CCSD�R12� and related methodsemploy the generalized Brillouin condition in accordancewith the original definition of the method.42

All molecular integrals and some special intermediates�the X and B intermediates mentioned above as well as Vand P given in Ref. 55� have been evaluated with the MPQC

program suite.67 The synthesized codes have been a part ofthe NWCHEM program suite68 and fitted with an interface tothe externally supplied integrals and special intermediates.As the correlation factor, a Slater geminal,36 exp�−�r12� with�=1.5a0

−1 approximated as a linear combination of sixGaussian geminals, has been adopted. See Refs. 56 and 57for further details of the implementation.

IV. RESULTS AND DISCUSSION

As severe cases of quasidegenerate wave functions totest the validity of perturbation corrections, the F–H bond

044118-6 Shiozaki, Valeev, and Hirata J. Chem. Phys. 131, 044118 �2009�

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dissociation of the hydrogen fluoride molecule and thedouble O–H bond dissociation of the water molecule havebeen studied by our methods. The aug-cc-pVXZ basis sets4,69

�X=D, T, Q, and 5� have been used as an orbital basis setand Klopper’s 19s14p10d8f6g4h2i /9s6p4d3f2g basisset31,70 as a complementary auxiliary basis set.34 Only va-lence electrons are correlated. The spectroscopic constants ofhydrogen fluoride have been obtained by solving the one-dimensional vibrational Schrödinger equation with a finite-difference method, neglecting the rotational-vibrational cou-plings.

A. The dissociation of hydrogen fluoride

The potential energy curves of the single-bond dissocia-tion of hydrogen fluoride obtained by the methods that in-clude up to connected triples �either perturbatively or itera-tively� as well as CCSD-R12 are drawn in Fig. 1�a�. Thecorresponding figure for the methods including up to con-nected quadruples are depicted in Fig. 1�b�. Since theCCSD�3�T-R12 curve is indistinguishable from theCCSD�2�T-R12 curve, the former is not shown. Likewise, theCCSDT�2�Q-R12 and CCSD�2�TQ-R12 curves are virtuallyidentical and only the latter is included in the figure. Thecorrelation energies of all the methods considered in thiswork are reproduced in Table II at four selected bond lengthsup to four times the equilibrium value �Re=1.7328 a.u.�. Forsingle-bond breaking, the quasidegeneracy of twodeterminants—the HF determinant and the doubly exciteddeterminant in which the lowest-lying antibonding orbital isoccupied—occurs as the bond is stretched. The inclusion ofconnected triples in CC is sufficient to describe such wavefunctions and CCSDT, therefore, serves as an accuratebenchmark.

As expected from our previous study on the non-R12combined CC and perturbation methods,17 CCSD�2�T-R12and all higher-order methods can describe the single-bond

breaking quantitatively correctly. For instance,CCSD�2�T-R12 and CCSD�2�TQ-R12 can capture 93% and99.5% of the connected triples contribution in CCSDT-R12and the connected triples and quadruples contribution inCCSDTQ-R12, respectively, at the equilibrium bond length

TABLE I. Explicitly correlated combined CC and perturbation methods as low-order Rayleigh–Schrödinger perturbation corrections to CC methods based on

the CC effective �similarity-transformed� Hamiltonian �H� partitioned into the zeroth-order part and perturbation. The partitioning is according to Eqs. �34�and �35�, where the projector on the primary space P is used.

Method Primary space �P�a Effective Hamiltonian �H� Costb Annotation

CCSD�2�T-R12 P0+ P1+ P2+ P2 �H exp�T1+ T2+ T2��C O�n7� Second-order triples correction

CCSD�2�TQ-R12 P0+ P1+ P2+ P2 �H exp�T1+ T2+ T2��C O�n9� Second-order triples and quadruples correction

CCSD�T�-R12 P0+ P1+ P2+ P2 �H exp�T1+ T2+ T2��C O�n7� Noniterative triples correction

CCSD�3�T-R12 P0+ P1+ P2+ P2 �H exp�T1+ T2+ T2��C O�n8� Third-order triples correction

CCSDT�2�Q-R12 P0+ P1+ P2+ P3+ P2 �H exp�T1+ T2+ T3+ T2��C O�n10� Second-order quadruples correction

CCSD�2�T�R12� P0+ P1+ P2+ P2 H�R12� O�n7� Second-order triples correction

CCSD�2�TQ�R12� P0+ P1+ P2+ P2 H�R12� O�n9� Second-order triples and quadruples correction

CCSD�2�T,R12c P0+ P1+ P2 �H exp�T1+ T2��C O�n7� Second-order triples and R12 correction

MP2-R12 P0 H O�n6�d Second-order doubles and R12 correction

aIn all cases, 1= P0+ P1+ ¯+PN+ P2, where N is the number of electrons.bAsymptotic polynomial dependence on n �the number of orbitals� of the overall computational cost. The current implementation involves the construction of

B� �see Eq. �58�� at an O�no6nv

2� cost and its inversion at an O�no8� cost, where no is the number of occupied orbitals and nv is the number of virtual orbitals.

This cost is, however, usually negligible in practice and hence is not included in the overall cost.cNot implemented in this study. See Refs. 47–49 for implementations as well as the original formulas.dThe use of the diagonal fixed-amplitude approximation �Ref. 35� �not considered in this work� can reduce this scaling to O�n5�. A further reduction is possiblevia the use of density fitting �Ref. 32�.

-100.4

-100.3

-100.2

-100.1

-100.0

-99.9

-99.8

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Energy/Eh

R /Re

CCSD-R12CCSD(T)-R12CCSD(2)T-R12CCSDT-R12CCSDT

(a)

-100.4

-100.3

-100.2

-100.1

-100.0

-99.9

-99.8

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Energy/Eh

R /Re

CCSD-R12CCSD(2)TQ-R12CCSDTQ-R12CCSDTQ

(b)

FIG. 1. The potential energy curves of hydrogen fluoride obtained with theaug-cc-pVDZ basis set as a function of the bond length �R� measured in theunit of the equilibrium value Re.

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�1.0Re�. The corresponding ratios at 4.0Re drop only slightlyto 91% and 97%. These ratios pertain to the recovery ofcorrelation energies from higher-rank excited determinants ata given basis set and should not be confused with the recov-ery of CBS correlation energies at a given method. In theentire domain of bond lengths considered, CCSD�2�T-R12,CCSD�2�TQ-R12, CCSDT-R12, and CCSDTQ-R12 arewithin a few mEh of one another. This is in contrast toCCSD-R12 and CCSD�T�-R12, whose predicted dissociationlimits deviate from the CCSDT-R12 or CCSDTQ-R12 valuesby more than 20 and 50 mEh. As is well known, theCCSD�T� curve has a nonphysical barrier toward the disso-ciation and the CCSD�T�-R12 curve displays the same short-coming.

The proportion �93%� of the connected triples ofCCSDT-R12 reproduced by CCSD�3�T-R12 is no greaterthan CCSD�2�T-R12. This is a manifestation of the staircaseconvergence of perturbation theory for electron correlation.CCSDT�2�Q-R12 is within 0.19 mEh of CCSDTQ-R12 byvirtue of the full inclusion of connected triples, but it doesnot constitute a significant improvement overCCSD�2�T-R12 or CCSD�2�TQ-R12, which are much less ex-pensive. The results obtained with CCSD�2�T�R12� andCCSD�2�TQ�R12� agree accurately with those ofCCSD�2�T-R12 and CCSD�2�TQ-R12, respectively. In allcases studied, the errors introduced by the “�R12�” approxi-mation are less than 0.3 mEh, which is smaller than the effectof higher-rank cluster operators. Since the �R12� approxima-tion is known to make correlation energies slightly toonegative,56 the more accurate agreement betweenCCSD�2�T�R12� or CCSD�2�TQ�R12� with CCSDT-R12 orCCSDTQ-R12 than CCSD�2�T-R12 or CCSD�2�TQ-R12 isfortuitous.

We now turn to the question of the basis-set convergenceof correlation energies. Table III compiles a breakdown ofcorrelation energies of hydrogen fluoride into connectedsingles and doubles, connected triples, and connectedquadruples contributions as a function of the basis set.They are obtained by CCSD-R12, CCSD�2�T-R12, and

CCSD�2�TQ-R12 at four different bond lengths. At any bondlength, the CCSD-R12 correlation energies converge rapidlywith respect to basis-set size. The proportion of the CCSD-R12/aug-cc-pV5Z correlation energies recovered by CCSD-R12/aug-cc-pVDZ is invariably 97%–98%. In contrast, thebasis-set convergence of connected triples and quadruples isslow and the convergence rate varies greatly with the bondlength. At the equilibrium bond length �1.0Re�, only 46% and69% of connected triples and quadruples of the aug-cc-pV5Zbasis set are recovered by the aug-cc-pVDZ basis set. Theratios increase considerably to 69% �triples� and 86% �qua-druples� near dissociation �4.0Re�, but they are still muchsmaller than the ratios achieved by CCSD-R12 for connectedsingles and doubles contributions.

This result confirms our previous observation57 that theR12 scheme, as implemented in this work, hardly improvesthe basis-set convergence of correlation contributions fromthe connected triples and higher-ranked cluster operators. Ina recent work,63,72 Köhn has shown that this can be remediedby adopting more flexible parametrization of the wave func-tion albeit at an increased computational cost. However, thebasis-set truncation errors in the connected triples contribu-tions are a few mEh at 1.0Re and 10 mEh at 4.0Re and areon the same order of magnitude as the remaining basis-settruncation errors in the connected singles and doubles con-tributions in CCSD-R12. The basis-set truncation errors inthe connected quadruples contributions are negligible ascompared to these errors. Therefore, from a practical view-point, reducing the basis-set truncation errors in the con-nected triples and quadruples contributions at an increasedcomputational cost may not be meaningful without furtherimproving the basis-set convergence of CCSD-R12. In otherwords, the present combined CC-R12 and perturbation meth-ods are well balanced in that they minimize the errors fromvarious sources uniformly while maintaining a relatively low

TABLE II. The correlation energies �in mEh� of hydrogen fluoride at fourdifferent bond lengths �Re=1.7328 a.u.� obtained with the aug-cc-pVDZbasis set.

Method 1.0Re 1.5Re 2.0Re 4.0Re

CCSD 225.98 249.16 275.32 391.27CCSD-R12 305.39 324.49 347.55 456.36

CCSDT 230.32 256.36 287.46 418.95CCSD�T�-R12 309.33 330.96 359.23 538.05CCSD�2�T�R12� 309.15 330.73 357.53 480.30CCSD�2�T-R12 309.05 330.50 357.31 480.18CCSD�3�T-R12 309.07 330.40 357.12 480.80CCSDT-R12 309.33 331.07 358.55 482.54

CCSDTQ 230.64 257.01 288.57 419.98CCSD�2�TQ�R12� 309.74 331.82 359.32 482.99CCSD�2�TQ-R12 309.63 331.59 359.09 482.87CCSDT�2�Q-R12 309.61 331.61 359.44 483.50CCSDTQ-R12 309.65 331.66 359.53 483.69

TABLE III. The basis-set convergence of the CCSD-R12 correlation ener-gies �E0�, the second-order triples corrections �ET

�2�� in CCSD�2�T-R12, andthe second-order quadruples corrections �EQ

�2�� in CCSD�2�TQ-R12 of hydro-gen fluoride at four different bond lengths �Re=1.7328 a.u.�. The energiesare in units of mEh.

Basis set 1.0Re 1.5Re 2.0Re 4.0Re

E0 in CCSD-R12aug-cc-pVDZ 305.39 324.49 347.55 456.36aug-cc-pVTZ 310.70 330.65 354.33 463.08aug-cc-pVQZ 312.98 332.97 356.61 465.12aug-cc-pV5Z 313.59 333.66 357.34 465.85

ET�2� in CCSD�2�T-R12

aug-cc-pVDZ 3.66 6.01 9.76 23.82aug-cc-pVTZ 6.79 9.73 14.18 31.07aug-cc-pVQZ 7.61 10.71 15.39 33.38aug-cc-pV5Z 7.94 11.11 15.88 34.32

EQ�2� in CCSD�2�TQ-R12

aug-cc-pVDZ 0.58 1.09 1.78 2.69aug-cc-pVTZ 0.65 1.20 1.97 3.05aug-cc-pVQZ 0.66 1.21 1.99 3.10aug-cc-pV5Z 0.66 1.21 1.99 3.12

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overall computational cost even in the severe case ofquasidegenerate wave functions near single bond dissocia-tion.

Our methods may still fail when the magnitude of theconnected triples and quadruples contributions becomes toolarge. However, there is an indication that the basis-set trun-cation errors become relatively smaller with increasing mag-nitudes of connected triples and quadruples �see above�. Thiscan be explained by assuming that the connected triples andquadruples play a dual role of capturing dynamical and non-dynamical correlation. The dynamical correlation requires alarge basis set for a quantitative characterization, whereasnondynamical correlation can be described by a small basisset. If we define, for the purpose of this discussion, the dy-namical correlation as the portion of the connected triplescontribution that increases by a basis-set extension and non-dynamical correlation as that captured already by aug-cc-pVDZ, less than half of the connected triples contribution at1.0Re is nondynamical correlation. When the bond isstretched to 4.0Re, a greater proportion � 70%� of the con-nected triples contribution is nondynamical �and is insensi-tive to the basis-set size� and the basis-set truncation errors inthe connected triples contribution are also relatively less.

In fact, the spectroscopic properties of hydrogen fluorideon which the effects of connected triples are noticeable arepredicted accurately by the combined CC-R12 and perturba-tion methods �Table IV�. The conventional CC methods withthe aug-cc-pVDZ basis set provide the frequencies of thetransition v=1←0 too small. For instance, the frequencypredicted by CCSDTQ is 3881 cm−1, which is smaller thanthe experimental value �3961 cm−1� �Ref. 71� by 80 cm−1.This discrepancy is to a large extent ascribed to the basis-settruncation error as CCSDTQ-R12 reproduces the frequency�3953 cm−1� within 10 cm−1 of the observed �the accurate

agreement is partly fortuitous because the basis-set errors inthe HF energies are considerable�. Various noniterative ap-proximations to CCSDT-R12 yield the computed frequenciesthat are within 5 cm−1 of the CCSDT-R12 value �2 cm−1 ofthe observed� at a small fraction of the computational cost ofCCSDT-R12. Likewise, CCSD�2�TQ�R12�, CCSD�2�TQ-R12,and CCSDT�2�Q-R12 reproduce the CCSDTQ-R12 valuewithin 1 cm−1 �the observed within 9 cm−1�. When the aug-cc-pVQZ basis set is used, roughly equal accuracy isachieved at the CCSD�2�TQ-R12 level �within 3 cm−1 of theexperimental value�. Note that these calculations do not in-clude core-correlation effects, basis-set extension effects inthe HF energies, relativistic effects, or non-Born–Oppenheimer effects, and their intrinsic accuracy may not beas high as what the differences between the computed andobserved frequencies suggest. Nevertheless, the significantimprovement brought to by the R12 scheme with a relativelysmall basis set is not fortuitous. The same observation can bemade by comparing the rotational constants.

It may be concluded that CCSD�2�T-R12 andCCSD�2�T�R12� are the most useful for chemical propertiesof stable molecules at their equilibrium geometries and trans-formations involving single-bond stretch or dissociation.They constitute a qualitative improvement over CCSD�T� inthe latter situations. CCSD�2�T-R12 and CCSD�2�T�R12� candescribe the potential energy curves quantitatively correctlyall the way to near dissociation limits, whereas CCSD�T� isqualitatively wrong. With CCSD�2�T-R12 andCCSD�2�T�R12�, the aug-cc-pVDZ basis set suffices, whileCCSD�T� needs a series of calculations with at least up toaug-cc-pVQZ basis set followed by an empirical extrapola-tion. The CCSD�2�T-R12/aug-cc-pVDZ potential curve ofhydrogen fluoride is estimated to be within 25 mEh of thecomplete-correlation CBS curve �excluding HF and core-correlation energies� up to 4.0Re, assuming that the results ofCCSD�2�TQ-R12/aug-cc-pV5Z are within a few mEh fromthe complete-correlation CBS limit. Under the same assump-tion, the accuracy of the CCSD�2�TQ-R12/aug-cc-pVTZcurve is 5 mEh �again, excluding HF and core-correlationenergies�.

B. The double dissociation of water

The equilibrium geometry of water has been taken fromthe FCI study of Olsen et al.:73 Re=1.843 45 a.u. and HOHangle=110.6°. Figure 2 plots the total energies of water ob-tained by various methods as a function of the length �R� ofthe O–H bonds in the isosceles �C2v� structure. The corre-sponding correlation energies are given in Table V at fourselected bond lengths. They have been obtained with theaug-cc-pVDZ basis set.

When the two O–H bonds are simultaneously stretchedconsiderably, the determinant in which four electrons arepromoted to the two lowest-lying antibonding orbitals as-sumes a substantial weight in the total wave function. As aresult, even CCSDT-R12 cannot describe the potential en-ergy curve qualitatively correctly. Hence, there is a notice-able difference between CCSDT-R12 and CCSDTQ-R12 notonly in the dissociation limits by as much as 40 mEh but

TABLE IV. The anharmonic vibrational frequency ��1←0� �in cm−1� androtational constant B0 �in cm−1� of the 1H19F molecule in the ground statecalculated with the aug-cc-pVDZ basis set unless otherwise specified.

Method ��1←0� B0

CCSD 3929 20.30CCSD-R12 3996 20.53

CCSDT 3886 20.21CCSD�T�-R12 3960 20.45CCSD�2�T�R12� 3959 20.44CCSD�2�T-R12 3962 20.45CCSD�3�T-R12 3963 20.46CCSDT-R12 3958 20.44

CCSDTQ 3881 20.19CCSD�2�TQ�R12� 3952 20.42CCSD�2�TQ-R12 3954 20.43CCSDT�2�Q-R12 3954 20.43CCSDTQ-R12 3953 20.43

CCSD-R12/aug-cc-pVQZ 4017 20.67CCSD�2�T-R12/aug-cc-pVQZ 3972 20.55CCSD�2�TQ-R12/aug-cc-pVQZ 3964 20.53Experimenta 3961 20.56

aReference 71.

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also in the qualitative shape of the potential around R=2.5Re. A nonphysical barrier can be seen in Fig. 2�a�, to avaried extent, in the potentials obtained by CCSDT-R12 andits various approximations. The inclusion of connected qua-druples by CCSDTQ-R12 leads to a barrierless potentialwhich serves as a benchmark potential for our purpose.

Figure 2 and Table V attest to the fact that our perturba-tion treatments of connected triples and quadruples accu-rately reproduce the CCSDT-R12 and CCSDTQ-R12 corre-lation energies, respectively, at all bond lengths studied. TheCCSD�2�TQ-R12 results are no less accurate than theCCSDT�2�Q-R12 results in comparison to CCSDTQ-R12,indicating the rapid convergence of the perturbation seriesirrespective of the choice of a reference CC-R12 wave func-tion. Again, CCSD�T�-R12 deviates from CCSDT-R12 by asmuch as 100 mEh at 4.0Re and is a distinctly poorer ap-proximation to CCSDT-R12 than CCSD�2�T�R12� orCCSD�2�T-R12 that have the same O�n7� asymptotic scalingof the computational costs.

The basis-set convergence of connected singles anddoubles as well as connected triples contributions is exam-ined in Table VI. The CCSD-R12/aug-cc-pVDZ method canrecover 97%–99% of the extrapolated CBS correlation ener-gies at four selected bond lengths. The basis-set truncationerrors in the connected singles and doubles contributions areonly 7–10 mEh. As in the dissociation of hydrogen fluoride,the R12 scheme does not improve the basis-set convergenceof the connected triples contributions. The basis-set conver-gence of ET

�2� obtained by CCSD�2�T-R12 is, in fact, slightlyslower than that of CCSD�2�T. The ratios of ET

�2� obtained byCCSD�2�T-R12/aug-cc-pVDZ to the CBS values are only50%, 60%, 67%, and 72%, respectively, at 1.0, 1.5, 2.0, and

-76.4

-76.3

-76.2

-76.1

-76.0

-75.9

-75.8

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Energy/Eh

R /Re

CCSD-R12CCSD(T)-R12CCSD(2)T-R12CCSDT-R12CCSDT

(a)

-76.4

-76.3

-76.2

-76.1

-76.0

-75.9

-75.8

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Energy/Eh

R /Re

CCSD-R12CCSD(2)TQ-R12CCSDTQ-R12CCSDTQ

(b)

FIG. 2. The potential energy curves of water in the C2v structure obtainedwith the aug-cc-pVDZ basis set as a function of the bond length �R� mea-sured in the unit of the equilibrium value Re. The HOH angle is held fixed atthe equilibrium value and the two O–H bonds are simultaneously stretched.

TABLE V. The correlation energies �in mEh� of water in the C2v structure atfour different bond lengths �Re=1.843 45 a.u.� obtained with the aug-cc-pVDZ basis set. The HOH angle is held fixed at the equilibrium value andthe two O–H bonds are simultaneously stretched.

Method 1.0Re 1.5Re 2.0Re 4.0Re

CCSD 228.35 276.68 349.34 645.59CCSD-R12 290.74 330.82 395.96 679.24

CCSDT 233.90 288.33 374.52 704.09CCSD�T�-R12 295.87 341.57 420.63 830.21CCSD�2�T�R12� 295.88 340.90 414.91 731.23CCSD�2�T-R12 295.39 340.16 414.24 730.78CCSD�3�T-R12 295.61 340.62 415.68 733.85CCSDT-R12 295.96 341.67 418.73 736.16

CCSDTQ 234.38 289.83 374.68 661.93CCSD�2�TQ�R12� 296.62 343.12 419.18 693.47CCSD�2�TQ-R12 296.13 342.38 418.50 692.93CCSDT�2�Q-R12 296.34 342.85 419.61 685.55CCSDTQ-R12 296.39 343.04 419.63 696.25

TABLE VI. The basis-set convergence of the CCSD and CCSD-R12 corre-lation energies �E0� and the noniterative triples corrections �E�T�� inCCSD�T�, CCSD�T�-R12, CCSD�2�T, and CCSD�2�T-R12 of water in theC2v structure at four different bond lengths �Re=1.843 45 a.u.�. The ener-gies are in units of mEh. The HOH angle is held fixed at the equilibriumvalue and the two O–H bonds are simultaneously stretched.

Basis set 1.0Re 1.5Re 2.0Re 4.0Re

E0 in CCSDaug-cc-pVDZ 228.35 276.68 349.34 645.59aug-cc-pVTZ 274.01 317.59 385.79 671.50aug-cc-pVQZ 289.11 331.19 397.49 680.12aug-cc-pV5Z 294.22 335.85 401.68 683.32CBSa 299.57 340.73 406.07 686.67

E0 in CCSD-R12aug-cc-pVDZ 290.74 330.82 395.96 679.24aug-cc-pVTZ 296.27 337.43 402.84 684.12aug-cc-pVQZ 298.11 339.36 404.60 685.46

ET�2� in CCSD�2�T

aug-cc-pVDZ 4.91 9.90 19.75 52.39aug-cc-pVTZ 8.03 13.84 24.86 64.84aug-cc-pVQZ 8.74 14.78 26.13 68.38aug-cc-pV5Z 9.01 15.14 26.64 69.74CBSa 9.30 15.51 27.18 71.17

ET�2� in CCSD�2�T-R12

aug-cc-pVDZ 4.65 9.34 18.28 51.55aug-cc-pVTZ 7.83 13.49 24.14 64.32aug-cc-pVQZ 8.65 14.62 25.82 68.18

aExtrapolated CBS values based on the assumed X−3 dependence of corre-lation energies �Ref. 74� �X is the cardinal number of the aug-cc-pVXZ basisset�.

044118-10 Shiozaki, Valeev, and Hirata J. Chem. Phys. 131, 044118 �2009�

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4.0Re. Although these ratios are similar to those observed forhydrogen fluoride, the basis-set truncation error in ET

�2� at4.0Re is greater � 20 mEh� in the double dissociation of wa-ter because the absolute value of ET

�2� has increased signifi-cantly. This error is three times as large as the basis-set trun-cation error in the connected singles and doublescontributions in CCSD-R12 � 7 mEh�. Hence, unlike in hy-drogen fluoride, an effort to accelerate the basis-set conver-gence of connected triples contribution will be helpful inreducing the overall basis-set truncation errors in water nearthe double dissociation limit. While the connected quadruplecontributions have the opposite sign to the connected triplescontributions and hence E�2�� ET

�2�, this conclusion islikely to remain valid for E�2�. One immediate remedy is touse a composite approach that combines CCSD-R12 with asmall basis set and CCSD�2�T with a larger basis set. Long-term solutions should, however, be either to adopt multiref-erence methods that minimize higher-than-doubles contribu-tions in conjunction with the existing R12 scheme75,76 or toimprove upon the ansatz of R12 for connected triples, asdemonstrated recently by Köhn.63

The computational cost �in wall clock time� ofCCSD�2�TQ-R12/aug-cc-pVDZ for the water molecule�excluding that for integral evaluation� is approximately30 times smaller than that of CCSD�2�TQ/aug-cc-pVQZwhile yielding similar correlation energies. The ratio of thecomputational cost of CCSD�2�T/aug-cc-pVQZ to that ofCCSD�2�T�R12�/aug-cc-pVDZ for the same molecule is�20.

V. CONCLUSION

Various combinations of CC-R12 methods and nonitera-tive, low-order perturbation corrections have been imple-mented. These methods form a series of systematic approxi-mations that are extremely rapidly convergent toward FCI inthe CBS limits with respect to all of the cluster excitationrank, perturbation order, and basis-set size. In theorder of increasing accuracy and computational cost,they are MP2-R12�CCSD-R12 CCSD�R12��CCSD�2�T

-R12 CCSD�2�T �R12� � CCSD�3�T-R12 � CCSDT-R12�CCSD�2�TQ-R12 CCSD�2�TQ�R12�� CCSDT�2�Q-R12�CCSDTQ-R12. The CCSD�2�T,R12 method of Valeev andco-workers,47–49 which is not implemented in this work, canalso be considered as an application of the general perturba-tion theory15 spawning all these approximations.

For molecules whose wave functions are approximatedwell by single determinants, lower-rank members of this se-ries such as CCSD�2�T-R12 and CCSD�2�T�R12� are themost useful for quantitative predictions of energies, struc-tures, and properties. They constitute a considerable im-provement over popular CCSD�T� in that they can describesingle-bond breaking quantitatively, while CCSD�T� cannoteven qualitatively, and capture over 95% of CBS correlationenergies with the aug-cc-pVDZ basis set. This latter advan-tage and the fact that these methods are size extensive makethem appealing for solid-state applications in the future. Torealize these methods, the � equations for CCSD-R12,CCSD�R12�, and CCSDT-R12 have also been implemented

without a diagrammatic truncation. Note that these equationsare the key to the CC-R12 analytical gradient and responseproperty theories.

The R12 scheme, as adopted in this work, is highly ef-fective in eradicating the basis-set truncation errors in theconnected singles and doubles contributions in correlationenergies. However, it does not improve the basis-set conver-gence of the connected triples and higher-rank cluster contri-butions. Nonetheless, our numerical tests have revealed thatthe present R12 methods are applicable to severe cases ofquasidegenerate wave functions near single-bond dissocia-tion limits in which connected triples and quadruples contri-butions are substantial. In the double dissociation of water,the connected triples contributions �and quadruples also� areeven greater in magnitude and the basis-set truncationerrors in these contributions begin to dominate the overallbasis-set truncation errors. In such situations, the basis-settruncation errors in the connected triples contributions inCCSD�2�T-R12/aug-cc-pVDZ can be several times as largeas those in connected singles and doubles in CCSD-R12/aug-cc-pVDZ. It is hence meaningful then to improve upon theR12 ansatz63 to accelerate the convergence of connectedtriples contributions or to reduce such contributions alto-gether by switching to a multireference description of wavefunctions in conjunction with R12 in an effort to minimizethe overall basis-set truncation errors.

The proposed combined CC-R12 and perturbation meth-ods are, therefore, well balanced for a wide variety of appli-cations including most molecules at their equilibrium geom-etries and single-bond dissociation, but they are less so fordouble bond dissociation and may break down for more se-vere quasidegenerate situations. Generally, these R12 meth-ods can achieve the same accuracy at much lower computa-tional costs than their non-R12 counterparts. The density-fitting approximations for the two-electron integrals thatenter the R12 intermediates have been implemented recently,improving the performance of these methods even further.Efficient implementations of these methods based on theseapproximations will soon be released in the freely availableopen-source MPQC program.

ACKNOWLEDGMENTS

T.S. thanks the Japan Society for the Promotion of Sci-ence Research Fellowship for Young Scientist. E.F.V. thanksthe Donors of the American Chemical Society Petroleum Re-search Fund �Grant No. 46811-G6� and the U.S. NationalScience Foundation CAREER Award �Grant No. CHE-0847295�. E.F.V. is an Alfred P. Sloan Research Fellow. S.H.thanks U.S. Department of Energy �Grant No. DE-FG02-04ER15621�, U.S. National Science Foundation CAREERAward �Grant No. CHE-0844448�, and the Donors of theAmerican Chemical Society Petroleum Research Fund�Grant No. 48440-AC6�. S.H. is a Camille Dreyfus Teacher-Scholar.

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