Local treatment of electron correlation in coupled cluster theory

Local treatment of electron correlation in coupled cluster theoryClaudia Hampel and HansJoachim Werner Citation: J. Chem. Phys. 104, 6286 (1996); doi: 10.1063/1.471289 View online: http://dx.doi.org/10.1063/1.471289 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v104/i16 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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Local treatment of electron correlation in coupled cluster theoryClaudia Hampel and Hans-Joachim WernerInstitut fur Theoretische Chemie, Universita¨t Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany

~Received 12 September 1995; accepted 4 January 1996!

The closed-shell coupled cluster theory restricted to single and double excitation operators~CCSD!is formulated in a basis of nonorthogonal local correlation functions. Excitations are madefrom localized molecular orbitals into subspaces~domains! of the local basis, which stronglyreduces the number of amplitudes to be optimized. Furthermore, the correlation of distant electronscan be treated in a simplified way~e.g., by MP2! or entirely neglected. It is demonstrated for20 molecules that the local correlation treatment recovers 98%–99% of the correlation energyobtained in the corresponding full CCSD calculation. Singles-doubles configuration interac-tion ~CISD!, quadratic configuration interaction~QCISD!, and Mo” ller–Plesset perturbation theory@MP2, MP3, MP4~SDQ!# are treated as special cases. ©1996 American Institute of Physics.@S0021-9606~96!01214-5#

I. INTRODUCTION

During the last two decades numerous high qualityabinitio calculations have shown that the treatment of electroncorrelation is essential for an accurate description of molecu-lar energetics, structures, and properties. However, since thecomputational effort of conventional electron correlationmethods increases very steeply with molecular size—at leastby the fifth power of the number of electrons—accurate cal-culations have been restricted to relatively small molecules.One of the main reasons for this steep scaling is the fact thatusually calculations are performed using a basis of canonicalmolecular orbitals, which are generally delocalized over thewhole system. This not only prevents the omission of smallcorrelation effects of distant electrons, but also leads to anunphysically steep increase in the number of virtual orbitalsneeded for the correlation of each particular electron pair.

It is intuitively clear that a localized description of elec-tron correlation is needed to avoid these problems. The firstlocal correlation methods have been already proposed in1964 by Sinanoglu1 and Nesbet,2 and many other variantswere suggested later.3–40 A short review of these methodscan be found in Ref. 40. Most of the methods used a basis oforthogonal virtual orbitals, but since these cannot be local-ized very satisfactorily, the success of such treatments wasquite limited. The problem of localizing a set of orthogonalvirtual orbitals is avoided in the local correlation method ofPulay and Saebo”,35–40who use nonorthogonal atomic orbit-als as correlation space. For each electron pair a differentsubspace~domain! of this basis is selected, which is com-prised of all functions that are spacially close to the corre-sponding localized SCF orbitals. In most cases, the basisfunctions of only two to four atoms contribute to a domain,and thus the size of the correlation space for a given electronpair is independent of the molecular size. Pulay and Saebo”

have convincingly demonstrated for MP2–MP4~SDQ! ~sec-ond to fourth-order Mo” ller–Plesset perturbation theory!,CISD ~singles–doubles configuration interaction!, and CEPA~coupled electron pair approximation41! that their local cor-relation method recovers more than 98% of the correlation

energy obtained in a corresponding full nonlocal calculation.In fact, this fraction increases by improving the basis quality,indicating that part of the loss is due to reduction of basis setsuperposition errors~BSSE!.42,43 Furthermore, they haveshown that the weak correlation of spacially distant electronscan in a good approximation be neglected or treated at alower level of theory. The combined effect of reducing thevirtual space for each pair and approximating weak correla-tions avoids the steep increase of the computational effortwith the molecular size and leads to significant savings ofCPU time.

In general, restricting excitations into domains and ne-glecting distant pairs corresponds to a physically motivatedconfiguration selection scheme. The advantages over con-ventional configuration selection methods are twofold: on theone hand, the pair basis can be fixed as a function of geom-etry, and thus smooth potentials are obtained. Second, thematrix structure of the formalism stays intact, and any logicin the innermost loops can be avoided. In fact, matrix mul-tiplications, which are the fastest operations possible on bothmodern workstations and vector computers, dominate thecalculation, and thus optimum efficiency can be achieved.

In this work we generalize the method of Pulay andSaebo” to the case of singles and doubles coupled cluster~CCSD! theory,44–51 which has become very popular in re-cent years. This method has the advantage of being size con-sistent, and is therefore particularly well suited for the accu-rate treatment of electron correlation of larger molecules.Lower-order correlation methods or approximations to thefull CCSD theory like MP2–MP4~SDQ!, QCISD~quadraticconfiguration interaction52!, and CISD or CEPA are includedin our treatment as special cases. The paper is organized asfollows: in Sec. II we describe the construction of the localbasis and the domains. In Sec. III we present the explicitform of the CCSD equations in the nonorthogonal local basisand describe how these equations are solved. Some method-ological details of our implementation are discussed in Sec.IV. Finally, in Sec. V we present local MP2, CCSD, andQCISD calculations for about 20 molecules. By comparisonwith the corresponding conventional calculations, it is dem-

6286 J. Chem. Phys. 104 (16), 22 April 1996 0021-9606/96/104(16)/6286/12/$10.00 © 1996 American Institute of Physics


onstrated that in all cases 98%–99% of the correlation en-ergy is recovered and that the computational effort is signifi-cantly reduced.

II. DEFINITION OF THE LOCAL CONFIGURATIONSPACE

The construction of our local basis and configurationspaces follows the ideas of Pulay and Saebo”.35–40Since thecomputed energies depend slightly on the details of thismethod, we summarize our procedure in this section.

We assume a closed-shell reference determinantu0&[F0constructed fromm orthonormal localized molecular orbitalsfi , which are linear combinations ofN basis functionsxm

uf i&5 (m51

N

uxm&Rm i . ~1!

For most calculations reported in this paper, we use the lo-calization scheme of Pipek and Mezey,26 which minimizesthe number of atoms at which the orbitals are located. Asdiscussed in detail by Boughton and Pulay,53 this procedurehas some important advantages for the local correlation treat-ment, namely,s-p symmetry is not destroyed, and for con-jugated systems like octatetraene, the localizedp orbitalscorresponding to symmetry equivalent bonds are in fact sym-metry equivalent. In contrast, the popular Boys localizationscheme54 usually destroyss-p separation and often leads tounsymmetrical solutions, that may not even be unique. Forsaturated systems, both localization methods yield very simi-lar results.

The complementaryvirtual or externalspace consists ofatomic orbitals that are orthogonalized on the occupied space

uxm&5S 12(i51

m

uf i&^f i u D uxm&5 (r51

N

uxr&Rrm . ~2!

The expansion coefficients of the projected functionsxm inthe AO basis$xm% are given by

R512D–S, ~3!

where

Dmn5(i51

m

Rm iRn i ~4!

is ~half! the closed-shell density matrix andS is the overlapmatrix in the AO basis. Here and in the following we denoteall quantities in the projected AO basis by a tilde. Projectedfunctions whose norm is smaller than 1026 are deleted fromthe basis; this can happen for generally contracted basisfunctions describing inner shells. For numerical conve-nience, the remaining projected functions are renormalized.

By construction, the external functions are optimally lo-calized at individual atoms but are nonorthonormal with met-ric

S5R†SR. ~5!

Due to the projection against the occupied space, the basis$xm% is linearly dependent andShasm zero eigenvalues. It is

advantageous to eliminate the corresponding redundant func-tions for individual pair domains at a later stage~cf., Sec.II B !.

A. Orbital domains

Excitations are made from the localized orbitals intosubspaces~domains! of the functions$xm%. To each occupiedorbitalfi anorbital domain[ i ] is assigned, which comprisesall projected functionsxm that are spacially close to thecharge density of orbitalfi . We assume that all basis func-tions xm are centered at atoms. The projected functionsxm

are assigned to the same centers as the corresponding origi-nal basis functionsxm , even though some of them are some-what delocalized due to the projection. An orbital domain [i ]consists of all projected functionsxm at a certain subset ofatoms. In order to select the atoms for an orbital domain,they are ranked according to decreasing Mulliken orbitalcharges, and all atoms are included into the domain that con-tribute significantly to the charge~typically until the sum ofthe atomic charges exceeds 1.8!. Then the completenesscheck proposed by Boughton and Pulay53 is carried out bycomputing the functional

f ~R8!5minF E ~f i2f i8!2 dt G , ~6!

wheref i8 is an approximate occupied orbital represented byorbital domain [i ],

uf i8&5 (rP@ i #

uxr&Rr i8 . ~7!

The unknown coefficientsRr i8 are obtained, separately foreach orbitalfi , by solving the set of linear equations

(nP@ i #

Smn Rn i8 5 (r51

N

Smr Rr i for all mP@ i #. ~8!

The minimum value of the functional~6! is then given by

f ~R8!512 (mP@ i #

(n51

N

Rm i8 Smn Rn i . ~9!

If the valuef ~R8! is larger than a threshold of typically 0.02,the basis functions of further atoms are added to domain [i ]until the criterium is satisfied. We confirm the finding ofBoughton and Pulay53 that this simple procedure works quitesatisfactorily for a wide range of different molecules. How-ever, in some cases with large and diffuse basis sets, wefound it necessary to reduce the threshold to 0.01. Typically,for a lone pair only basis functions at the same atom areincluded. For abicentric bonding orbital, the basis functionsof the two connected atoms are selected. Forconjugated sys-temsor other multicenter bonds, complete localization is notpossible and three to four atoms may be selected, sometimeseven more. Our program is fully automatic and open-endedin this respect.

Singly excited local configurations are obtained by ex-citing electrons from the occupied orbitalsfi into the pro-jected AOsxm

6287C. Hampel and H.-J. Werner: Electron correlation in coupled cluster theory

J. Chem. Phys., Vol. 104, No. 16, 22 April 1996


F im5Em i u0&, ~10!

where the spin-coupled excitation operatorsEm i are definedas

Em i5ux ma&^f i

au1ux mb&^f i

bu. ~11!

Due to the nonorthonormality of the external functions, theseconfigurations are not orthonormal

^F imuF j

n &52d i j Smn . ~12!

As will be seen later, this does not lead to major complica-tions in the formulation of the coupled-cluster equations. Thecontributions of all single excitations from a given orbitalfi

are described by the functions

C i5(m

t mi F i

m , ~13!

where t mi are the amplitudes determined in the CISD or

CCSD procedure.

B. Pair domains

Double excitations can be defined in an analogous man-ner

F i jm n5Em i E n j u0&. ~14!

In local CISD and CCSD expansions, only excitations intothe subspaces [i ] and [j ] are included. On first view it mayappear natural to restrict the excitations tomP[ i ] andnP[ j ]. However, as already pointed out by Saebo” andPulay,39 this usually leads to unsatisfactory results. In orderto recover a high percentage of electron correlation energy,excitations must be made from bothfi and fj into pairdomains[ i j ], which are the union of spaces [i ] and [j ].Then pair functions can be defined as

C i j5 (mP@ i j #

(nP@ i j #

T mni j F i j

m n ~ i> j !, ~15!

where theT mni j are the amplitudes optimized in the CCSD or

CISD procedure. This ensures that both singlet~p51! andtriplet ~p521! configuration state functionsF i j ,p

m n 5 12(F i j

m n

1 pF i jn m! are effectively included in theN-electron basis.

For each pair domain [i j ], redundant basis functions areeliminated separately. For this purpose the overlap matrixS[ i j ] of the function subspace [i j ] is constructed and diago-nalized. For each small or zero eigenvalue~for all calcula-tions reported in this paper we use a threshold of 1026! onefunction must be deleted. The number of redundant functionsin a pair domain [i j ] is usually smaller thanm and oftenzero. We have implemented and tested various possibilitiesto select the redundant functions. One possibility is to deleteindividual basis functions, that correspond to the largest co-efficients in the eigenvectors of small eigenvalues. The dis-advantage of this method is that eigenvectors correspondingto zero eigenvalues can mix arbitrarily, and therefore theselection may not be unique. An alternative method, whichavoids this problem, is to delete those functions which havethe smallest diagonal coefficientsRmm ~before normaliza-

tion!. The third method which we have tested does not deleteindividual functions, but eliminates the eigenvectors ofS[ i j ]

which correspond to the small eigenvalues. In this case thecontributions of the redundant vectors to the amplitudesT mn

i j

are projected out at an intermediate stage during the updateprocedure~cf. Sec. III B!. For all calculations reported in thispaper, we use this latter method, which we found most sat-isfactory since any arbitrariness~except for fixing the thresh-old! is avoided. It should be pointed out, however, that theelimination of individual functions would simplify the calcu-lation of analytical energy gradients, since the basis set foreach pair function is then independent of the geometry.Many test calculations that we performed showed that thecomputed energies were very insensitive to the selectionmethod used.

C. Weak and distant pairs

The electron pair correlation energiesei j of the pairfunctionsCi j decrease quickly~'1/r 6! with the distancer ofthe localized orbitals39 fi andfj . An important advantage ofthe local correlation method is that electron pair functionsdescribing the weak correlation of distant electrons may ei-ther be dropped entirely from the calculation or treated ap-proximately in a simplified way39 ~e.g., by MP2!. In ourprogram we distinguish three kinds of pairs:strong pairs,which are fully included,weak pairs, which are treated ap-proximately, anddistant pairs, which are neglected. Weakand distant pairs are found automatically according to thefollowing criteria.

Weak pairs: All atoms contributing to orbital domains[ i ] are separated from those of [j ] by a certain minimumdistancer w . For all calculations reported in this paper weusedr w51 bohr, which simply implies that orbital domains[ i ] and [j ] have no atom in common. Alternatively, weakpairs could be selected on the basis of their MP2 energycontributions, but this has not been done in the present cal-culations.

Distant pairs: In a similar way, a larger distancer d canbe used to determine distant pairs. For instance, a value ofr d54 bohr would imply that the orbital domains [i ] and [j ]are separated by at least one atom which is not part of [i ] or[ j ]. The energy contribution of such pairs is typically lessthan 2%. Since the neglect of distant pairs does not lead tosignificant savings for CISD, QCISD, or CCSD calculations,we will assumer d5` throughout this paper. For MP2 cal-culations, the neglect of distant pairs may lead to some sav-ings, since the effort for the integral transformation can bereduced.

The advantages of neglecting~or simplifying! weakpairs are obvious: The number ofstrongpairs increases onlylinearly with the molecular size. Accordingly, the computa-tional effort can be dramatically reduced without much lossof accuracy, as will be demonstrated in Sec. V.

III. THE CCSD EQUATIONS IN THE LOCAL BASIS

The local CCSD wave function for closed-shell systemsis defined by the following exponential expression:

6288 C. Hampel and H.-J. Werner: Electron correlation in coupled cluster theory



C5exp~ T!C0 , ~16!

where the cluster operatorT is restricted to one and two-particle excitation operators:

T5(i

(mP@ i #

t mi Em i

11

2 (i j

(mnP@ i j #

T mni j Em i E n j with T mn

i j 5T nmj i .

~17!

The amplitudest mi and T mn

i j are the solutions of the CCSDequations, which are obtained by projecting the Schro¨dingerequation from the left withC0, F i

m , andFi jm n :

E5^0uHuC&, ~18!

v mi 5^F i

muH2EuC&50, ~19!

V mni j 5^Fi j

m n uH2EuC&50 ~ i> j !. ~20!

The most efficient explicit formulation for Eq.~20! is ob-tained by projecting with the contravariant functions48,51

Fi jm n5 1

6@2F i jm n2F j i

m n #, ~21!

which in the nonorthogonal external basis satisfy

^Fi jm n uC&5@ST i j S#mn . ~22!

A. Explicit formulation of the residual vectors

The following formulation of the CCSD equations canbe viewed as a generalization of the self-consistent electronpair ~SCEP! theory of Meyer,55 to which it reduces for thecase of singles and doubles configuration interaction~CISD!.The corresponding CCD equations have been given by Pu-lay, Saebo”, and Meyer.48

In order to obtain simple explicit equations that are wellsuited for an efficient computer implementation, we collectthe CCSD amplitudes and partially transformed two-electronintegrals into vectors and matrices. In general, we denotedifferent vectors and matrices by superscripts, and their ele-ments by subscripts. Vectors and matrices are represented bylower-case and upper-case letters, respectively.

The coefficient matricesT mni j have only nonzero ele-

ments if m,nP[ i j ] and therefore possess a sparse blockstructure. Similarly, the vectorst m

i are only nonzero formP[ i i ]. Obviously, it is sufficient to compute the corre-sponding blocks of the residualsV i j andv i . In the followingequations we do not explicitly consider any of these blockstructures and assume that all matrices and vectors extendover the whole projected basis set. However, in our actualimplementation the blocking is fully taken into account. Aswill be discussed and demonstrated in Sec. IV and V, thisleads to significant savings of memory and CPU time.

The two-electron integrals with two occupied orbitalsk,lare collected ininternal coulomb and exchange matrices:

~ Jkl!mn5~klumn !, ~23!

~K kl!mn5~mku l n !, ~24!

~ L kl!mn52~mku l n !2~m l ukn !. ~25!

Integrals with three occupied orbitalsi ,k,l are representedby column vectors

@ kkli #m5~mku l i !, ~26!

@ lkli #m52~mku l i !2~m l uki !. ~27!

Similarly, the internal–external part of the Fock matrix isrepresented by vectors

@ f k#m5@ F#mk5hmk1(i

@2~mku i i !2~m i u ik !#. ~28!

For convenience in later expressions, we define the followingmodified coefficient matrices:

~Ci j !rs5T rsi j 1 t p

i t sj , ~29!

~Ci j !rs5C rsi j 2 1

2T rsi j , ~30!

~Ei j !ks5dkit sj . ~31!

All contributions of two-electron integrals with no internalindices are accounted for by theexternalexchange matrices:

K ~Ci j !mn5(rs

C rsi j ~mr u sn !. ~32!

The contributions of integrals with only one occupied orbitalenter via similar Coulomb and exchange matrices:

K ~Ci j !m~k!5(

rsC rs

i j ~mr u s k!, ~33!

J~Ei j !mn5(ks

E ksi j ~mn uks !5(

s~mn u i s ! t s

j , ~34!

K ~Ei j !mn5(ks

E ksi j ~mkuns !5(

s~m i usn ! t s

j . ~35!

Furthermore, it is convenient to define the operators

K ~Di j !5K ~Ci j !1K ~Ei j !1K ~Ej i !†. ~36!

As will be discussed in Sec. IV B, all of these operators canbe computed either from the two-electron integrals in the AObasis or from reduced sets of transformed integrals, and thusno full integral transformation is necessary. This is an impor-tant aspect forintegral-direct implementations.

In terms of these quantities the CCSD correlation energyis given by

Ecorr5(i> j

~22d i j !tr@ Li j Ci j #12(

if i†t i , ~37!

and the expressions for the residual vectors and matrices takethe compact form:

vi5 s i1SF(k

~2T ik2Tki! r k2bki tkG , ~38!

V i j5K i j1K ~Di j !1Gi j1Gj i †

1SF(kl

~a i j ,kl2d j lbki2dkib l j !CklG S. ~39!




The coefficientsa, b are given by

a i j ,kl5Ki jkl1tr~Ci j K lk!1~ t i†kkl j1 t j†k lki !, ~40!

bki5Fki1 f k† t i1(l

@ tr~ L klCl i !1 t l†l lki #, ~41!

and the matricesGi j are defined as

Gi j5SH t i sj†2(kt k@K ~C j i !~k!1 k j ikk1K jk t i1 J ik t j #†

1T i j X1(k

F ~2T ik2Tki!Yk j21

2TkiZk j2Tk jZkiG J .

~42!

S is the overlap matrix of the local basis as defined in Eq.~5!. The intermediate quantities are vectors

r k5 f k1(lL kl t l , ~43!

si5 f i1~ F2A†! t i2S(kl

T lk l kli1(k

@2K ~Cik!~k!

2K ~Cki!~k!1~2K ik2 Jik! t k#, ~44!

and matrices

A5F(kl

L klT lkG S, ~45!

X5F2A1(k

@2J~Ekk!2K ~Ekk!#2F(kr k t k†G S,

~46!

Yk j5S K k j21

2Jk j D 1F K ~Ek j!2

1

2J~Ek j!G

1H 12 (l

@ L kl~2Cl j2Cj l !2 l kl j t l†#1 f k t j†J S,~47!

Zk j5 Jk j1 J~Ek j!2H(l

@K lkCj l1 k lk j t l†#J S. ~48!

These expressions are entirely equivalent to those given inour previous work,51 except that additional multiplicationswith S occur. As will be demonstrated in Sec. V, the extraeffort for these multiplications is by far overcompensated bysavings due to the sparse structure of the coefficient andresidual matrices.

The corresponding equations for singles–doubles con-figuration interaction~CISD! and quadratic configuration in-teraction ~QCISD! can be obtained by leaving out certainquadratic, cubic and quartic terms in the above equations.Also Mo” ller–Plesset perturbation theory through fourth-order ~with no triples! @MP2, MP3, MP4~SDQ!# can beviewed as special cases and have been implemented in ourprogram. It should be noted, however, that the first- and

second-order wave functions in the Mo” ller–Plesset treatmentmust be obtained iteratively, since the Fock matrix is notdiagonal in the local basis.37

B. Iterative solution of the CCSD equations

The nonlinear CCSD equations are solved iteratively.Convergence is reached if the energy change and all ele-ments of the residual vectorsv i and matricesV i j are smallerthan prescribed thresholds.

In order to obtain a perturbational update, the residualvectors v i and matricesV i j must be transformed to an or-thogonal basis:

v i5W@ i i #† v i , ~49!

V i j5W@ i j #†V i jW@ i j #, ~50!

whereW[ i j ] are the solution of the Fock equations projectedto the pair basis@i j #:

F@ i j #W@ i j #5S@ i j #W@ i j #L@ i j #. ~51!

Here F[ i j ]5R[ i j ]†FR[ i j ] and S[ i j ]5R[ i j ]†SR[ i j ] are the Fockand overlap matrices, respectively, in the basis@i j #, andLab[ i j ]5dabea

i j is a diagonal matrix of~positive! orbital ener-gies.

As mentioned before, the function space@i j # of dimen-sion Li j may be linearly dependent. In order to eliminateredundant functions, the overlap matrixS[ i j ] is diagonalized

@X@ i j #†S@ i j #X@ i j ##ab5dabsa , ~52!

and the eigenvectors are renormalized according to

Xab@ i j #5HXab

@ i j #/Asb if sb.1026,

Xab@ i j #50 if sb<1026.

~53!

Vectors corresponding to small eigenvalues are thus elimi-nated. The remaining vectors form an orthonormal basis intowhich the Fock matrix is transformed

F@ i j #5X@ i j #†F@ i j #X@ i j #. ~54!

Diagonalization ofF[ i j ] yields the orbital energieseai j and a

unitary transformationU[ i j ]

U@ i j #†F@ i j #U@ i j #5L@ i j #, ~55!

which is finally used to obtain

W@ i j #5X@ i j #U@ i j #. ~56!

Since the matricesW[ i j ] are needed in each iteration to per-form the transformations~49! and ~50!, they are computedonce for each pair and stored on disk.

The updates are obtained in the orthogonal basis by first-order perturbation theory

D t ai 5

2vai

eaii2 f i i

, ~57!

DTabi j 5

2Vabi j

eai j1eb

i j2 f i i2 f j j, ~58!




where f i i5^f i uFuf i& are the diagonal elements of the Fockmatrix in the basis of the localized occupied orbitals.

Finally, theDT i j are transformed back to the~redundant!nonorthogonal local basis

Dt i5W@ i i #Dt i , ~59!

DT i j5W@ i j #DT i jW@ i j #†. ~60!

Convergence is accelerated by DIIS56,49,51and found to be asfast as for a nonlocal calculation in the canonical MO basis.

C. Treatment of single excitations

In previous implementations of local correlation meth-ods by Pulay and Saebo”,38,39 the single excitations were ei-ther omitted or not restricted to domains, i.e., excitationswere allowed into the full external basis. This does not havemuch influence on the efficiency of CISD or related linearcorrelation treatments like CEPA~coupled electron pair ap-proximation!. However, this is no longer true for the CCSDcase. For an efficient formulation and implementation of theCCSD equations, it is necessary to use the modified coeffi-cient matrices as defined in Eqs.~29!. To take full advantageof the sparsity of the coefficient matrices in the local basis,the matricesCmn

i j should have the same block structure as thematricesTmn

i j , which have only nonzero elements form, nP@i j #. Including nonlocal single excitations would destroy

this block structure. The local vectorst mi as used in our

method have only non-zero elements formP@i i #. Generally,the domains@i i # and @j j # are also spanned by the pair basis@i j #, but if redundant functions are explicitly deleted~cf.,Sec. II B!, the three spaces must not necessarily be identical.In order to construct the matricesCi j it is therefore necessaryto transform the vectorst i and t j from their original bases@i i # and @j j #, respectively, to the pair basis@i j #. This isreadily accomplished using the transformation matricesW[ i j ] :

t i@ i j #5W@ i j #•W@ i j #†

•S@ i j ,i i #• t i@ i i #. ~61!

These transformations have no effect on the final CCSD en-ergy.

In order to investigate the effect of neglecting nonlocalsingle excitations we have performed test calculations inwhich only the double excitations are treated locally. Asshown in Table I for a number of molecules, the typical lossof correlation energy by treating the singles locally is 0.05%or less and thus negligible.

IV. COMPUTATIONAL ASPECTS

A. Block structure and computational strategies

Due to the restriction of excitations into local domains,the matricesT i j , Ci j , V i j , andGi j have a sparse block struc-

TABLE I. Comparison of local correlation treatments.

Correlation energya

Molecule ESCF MP2~full ! MP2~local! CCSD~full ! CCSD~local! CCSD~local!b CCSD/MP2~local!c

VDZ basis:d

Propane 2118.270 461 20.446 239 20.436 124~97.73! 20.496 129 20.485 978~97.95! 20.486 150~97.99! 20.489 501~98.66!Butane 2157.306 738 20.589 737 20.575 103~97.52! 20.652 361 20.637 837~97.77! 20.638 109~97.82! 20.643 409~98.63!Pentane 2196.342 887 20.733 296 20.714 074~97.38! 20.808 656 20.789 709~97.66! 20.790 081~97.70! 20.797 321~98.60!VTZ basis:e

Water 276.057 402 20.261 173 20.259 802~99.48! 20.267 114 20.265 915~99.55! 20.265 937~99.56! 20.265 915~99.55!Ethylene 278.064 365 20.334 718 20.331 664~99.09! 20.359 209 20.356 464~99.24! 20.356 500~99.25! 20.357 129~99.42!Butadiene 2154.965 520 20.655 828 20.647 707~98.76! 20.691 058 20.683 797~98.95! 20.684 084~98.99! 20.689 982~99.84!Hexatriene 2231.867 599 20.969 445 20.957 613~98.78! 21.015 790 21.005 052~98.94! 21.005 468~98.98! 21.013 171~99.74!6-31G** basis:f

Methylamine 295.220 883 20.328 791 20.323 990~98.54! 20.357 233 20.352 306~98.62! 20.352 418~98.65! 20.353 615~98.99!Hydrazine 2111.178 636 20.353 429 20.349 100~98.78! 20.377 127 20.372 596~98.80! 20.372 720~98.83! 20.373 846~99.13!Oxirane 2152.873 067 20.462 131 20.456 472~98.78! 20.486 758 20.480 952~98.81! 20.481 191~98.86! 20.482 874~99.20!Dimethyl ether 2154.073 239 20.478 361 20.472 294~98.73! 20.513 875 20.507 531~98.77! 20.507 795~98.82! 20.509 348~99.12!Ethanol 2154.089 375 20.479 476 20.471 697~98.38! 20.514 170 20.506 298~98.47! 20.506 537~98.52! 20.508 774~98.95!Furan 2228.631 429 20.710 505 20.703 019~98.95! 20.736 066 20.728 231~98.94! 20.728 828~99.02! 20.731 738~99.41!Alanine 2321.886 368 20.945 978 20.929 648~98.27! 20.986 624 20.970 747~98.39! 20.971 195~98.44! 20.976 383~98.96!Oxalic acid 2376.359 761 20.978 806 20.971 477~99.25! 20.987 286 20.979 868~99.25! 20.980 204~99.28! 20.982 462~99.51!6-311G** basis:g

Benzene 2230.754 026 20.822 767 20.811 696~98.65! 20.857 552 20.846 606~98.72! 20.847 015~98.77! 20.852 189~99.37!6-31111G** basis:h

Imidazole 2224.874 677 20.761 783 20.751 673~98.67! 20.780 504 20.770 649~98.74! 20.771 086~98.79! 20.775 598~99.37!

aPercentage relative to nonlocal calculation in parentheses.bOnly double excitations treated locally.cWeak and distant pairs treated by MP2~see text!.dVDZ: cc-pVDZ [3s2p1d/2s1p] basis of Dunning~Ref. 60!.eVTZ: cc-pVTZ [4s3p2d1 f /3s2p1d] basis of Dunning~Ref. 60!.f6-31G** , Ref. 61.g6-311G** , Ref. 62.h6-31111G** , Ref. 63.




ture. In our program only these nonzero blocks are stored,and pointer lists for each pair are used to address the indi-vidual blocks. In order to compute the residuals, these sparsematrices must be multiplied with integral and overlap matri-ces, which are stored in the full basis. It is therefore neces-sary to gather the relevant blocks before performing the ma-trix multiplication. We found that this causes only a smalloverhead. In Eqs.~42!, ~47!, and~48! one might expect thatthe minimum number of operations is obtained if one multi-plies with Safter performing the summations overk,l . How-ever, as will be illustrated by the following example, this isnot always the case. Consider for instance the terms

Gi j5...S•(k51

Tki•Zk j. ~62!

For simplicity we assume that the dimension of all domains@i j # and @ki# is L, and the full basis dimension isN. Thus,Gi j andTki areL3L matrices, whileZk j areN3N matricesfrom which for each value ofk the relevant block must beextracted. For a sufficiently large ratioN/L it may be advan-tageous to move the multiplication withS into the summa-tion overk

Gi j5...(k51

m

~S•Tki!•Zk j. ~63!

For a given (i j ), the first algorithm@Eq. ~62!# requires(N/L1m)L3 operations, the latter one@Eq. ~63!# 2mL3 op-erations. The optimum choice depends on the ratioN/L andthe number of interacting pairs. If weak pairs are neglected~cf., Sec. II C!, only few terms of the sum overk contribute,and then the second algorithm may become favorable even ifN/L>m. A further advantage of the latter method is that thestorage of intermediate quantities can be avoided. Similarconsiderations apply to all other cases. For some of the im-portant terms we have implemented alternative algorithms,which are chosen automatically depending on the problemsizes.

B. The external Coulomb and exchange operators

The only terms in a CISD, QCISD, or CCSD calculationwhich involve two-electron integrals over four external or-bitals are the operatorsK ~Ci j !. These operators can be com-puted directly from the integrals in the AO basis by the fol-lowing sequence of transformations

Ci j5RCi j R†, ~64!

K ~Ci j !mn5(rs

Crsi j ~mrusn!, ~65!

K ~Ci j !5R†K ~Ci j !R. ~66!

In conventional calculations this requires aboutnpair(N

414N3) operations, wherenpair<m(m11)/2 is thenumber of correlated electron pairs~i j ! andN the number ofbasis functions. Similarly, by transforming the single excita-tion amplitudest i and the matricesEi j to the AO basis51

tsi 5(

mRsm t m

i , ~67!

Ersi j 5Rr i ts

j , ~68!

the operatorsJ~Ei j ! and K ~Ei j ! can be evaluated from theintegrals in the AO basis by a generalized partial integraltransformation

J~Ei j !mn5(rs

Ers~mnurs!5(r

Rr i(s

tsj ~mnurs!,

~69!

K ~Ei j !mn5(rs

Ers~mrusn!5(r

Rr i(s

tsj ~mrusn!,

~70!

and subsequently transformed back to the projected AO basissimilar to Eq.~66!. Furthermore, one can compute the opera-tors K ~Di j ! @cf. Eq. ~36!# directly in the AO basis by usingthe matricesDi j5Ci j1Ei j1Ej i † in Eq. ~65! instead of theCi j . Using the relation

K ~Di j !a~k!5K ~Ci j !a

~k!1@K ik t j1 Jjk t i #a ~71!

it is then possible to eliminate for CISD and QCISD casesthe operatorsJ~Ei j !, K ~Ei j ! andK ~Ci j ! entirely, and a singleset of operatorsK ~Di j ! is sufficient. For the CCSD case oneneeds either theK ~Ci j ! or theK ~Di j ! as well as the operatorsJ˜~Ei j ! and K ~Ei j !.

Unfortunately, the transformation of the matricesCi j orDi j into the AO basis@Eq. ~64!# destroys the local blockstructure, and therefore the locality cannot easily be used.We have somewhat experimented with using sparsity in thematricesDi j . To maximize the number of small elements, wecomputed in each iteration the change of the exchange op-eratorsK ~DDi j ! and then replaced small numbers by zeros.Initially, the sparsity of theDDi j was about 50%, and in-creased to about 80% in later iterations. Nevertheless, nosignificant speedup was achieved. The reason is that Eq.~65!can be evaluated very efficiently in terms of matrix multipli-cations, which run at very high speed~>200 MFLOPS perCPU! on our computer~SGI power-challenge!. Checking forzeros in the matrix multiplication requires a different algo-rithm, which is faster only if the sparsity exceeds about 75%.

Alternatively, one could transform the two-electron inte-grals into the projected AO basis and compute the operators

K ~Ci j ! according to Eq.~32!. This would formally reduce theeffort to approximatelynpairL

4, whereL is the average do-main size, and the transformation is not counted. Since forlarger molecules typicallyN/L'10, the number of opera-tions would be reduced by four orders of magnitude. In orderto take advantage of this, the transformed integrals must bestored on disk. However, it would be sufficient to store atmostnpairL

4/8 integrals; even if allm(m11)/2 pairs are in-cluded, this number is usually much smaller thanN4/8. As-suming againm530, L540, N5300 the number of trans-formed integrals would be 7 times smaller than the numberof AO integrals; this factor would be much larger if onlystrong pairs were considered. Most importantly, the number




of transformed integrals will be independent of the total basissize, and if only strong pairs are correlated, it will increaseonly linearly with molecular size. We are presently develop-ing an integral transformation for this case, but this is notoperational yet. Therefore, in all calculations reported in thispaper the evaluation of the external exchange operators pro-ceeds according to Eqs.~64!–~66! and strongly dominatesthe computational effort. A significant savings is onlyachieved by neglecting weak pairs~cf., Sec. V!.

The operatorsK ~Ci j !(k), J~Ei j !, andK ~Ei j ! require inte-grals over one occupied and three projected external orbitals.Since these operators have no local block structure, oneneeds to store all themN3/2 transformed integrals (mnur i ).If the external exchange operators are computed according toEqs. ~64!–~66!, their internal–external parts are obtainedsimply by an additional transformation

K ~Ci j !m~k!5@R†K ~Ci j !R#mk , ~72!

and therefore require only very little extra computationalwork. The calculation of the operatorsJ~Ei j ! and K ~Ei j !from the integrals (mn ur i ) requires32mLN2 operations. Al-ternatively, as shown in Eqs.~69! and ~70!, these operatorscould be computed directly from the AO integrals by a gen-eralized integral transformation. The effort for this transfor-mation is quite insignificant in a conventional CCSD calcu-lation ~10%–15% of the iteration time!.51 However, in alocal calculation this part may become more dominant, andwe found it much more efficient to compute them from thetransformed integrals. As will be demonstrated in Sec. V, theadditional time for the~partial! integral transformation israther small.

V. SAMPLE CALCULATIONS

The method described in the previous sections has beenimplemented as part of theMOLPRO96 suite of ab initioprograms.57 The new program has been carefully tested bycomparing with a version that is identical to our conventional

CCSD/QCISD/CISD program51 except for the coefficient up-dates. For this purpose the residual matricesV i j were trans-formed from the canonical virtual molecular orbital basis tothe projected local basis

U5R†SR, ~73!

V i j5U†V i jU. ~74!

After performing the update in the local basis as described inSec. III B, the coefficient matrices were transformed back tothe canonical basis according to

DT i j5UDT i jU†. ~75!

Similar transformations apply to the residual and coefficientvectors for the single excitations. It should be noted that anyexisting direct CISD or CCSD program can be modified inthis simple way to simulate the local correlation method, butof course no CPU time is saved in this case.

Tables I and II show a comparison of local and nonlocalMP2, CCSD, and QCISD calculations for a number of dif-ferent molecules with various basis sets, which included upto 292 contracted functions~hexatriene with full VTZ basis!.The set of molecules covers simple saturated systems likealkanes, molecules with conjugatedp bonds like alkenes,and aromatic systems. The calculations with the 6-31G**basis correspond to those published by Boughton andPulay.53 The geometries for these calculations were kindlyprovided by J. Boughton to us, but those for hydrazine, me-thylamine, and oxalic acid appear to be slightly differentthan those used in Ref. 53; in these cases the MP2 energiesare not quite identical. The geometries used in the presentcalculations are available from the authors on request. In allcases we used the Pipek–Mezey localization26 and the pro-cedures and thresholds described in Sec. II and III. Only thevalence orbitals were localized, in order to avoid any arti-facts due to mixing with the core orbitals, which were notcorrelated. Our local MP2 correlation energies are slightlylarger than those of Boughton and Pulay, which is probably

TABLE II. QCISD correlation energies for conventional and local calculations and MP2 energies for weakpairs at the VDZ level.a

Molecule ESCF

Correlation energyb

QCISD~full ! QCISD~local! QCISD/MP2~local!c MP2~weak pairs!d

Propane 2118.270 461420.496 506 75 20.486 326 07~97.95! 20.489 792 74~98.65! 20.019 764 73Butane 2157.306 737520.652 876 83 20.638 293 47~97.77! 20.643 779 64~98.61! 20.030 681 85Pentane 2196.342 887020.809 310 87 20.790 273 09~97.65! 20.797 770 60~98.57! 20.041 588 22Hexane 2235.379 034820.965 745 82 20.942 261 05~97.57! 20.951 770 03~98.55! 20.052 507 67Heptane 2274.415 172121.122 175 87 21.094 248 91~97.51! 21.105 768 44~98.54! 20.063 429 19Octane 2313.451 305721.278 606 11 21.246 238 10~97.47! 21.259 767 90~98.50! 20.074 352 06

Isobutene 2156.123 940120.614 965 04 20.602 929 76~98.04! 20.608 470 32~98.94! 20.028 692 15

Butadiene 2154.923 301020.589 525 70 20.579 417 88~98.28! 20.584 306 10~99.11! 20.025 425 68Hexatriene 2231.807 021020.861 853 10 20.848 262 93~98.42! 20.855 077 47~99.21! 20.034 066 94Octatetraene2308.691 147021.139 740 32 21.121 410 65~98.39! 21.130 753 67~99.21! 20.046 832 08

aCorrelation consistent cc-pVDZ basis from Ref. 60.bPercentage relative to conventional calculation in parenthesis.cApproximate MP2 treatment of weak and distant pairs.dMP2 correlation energies for weak and distant pairs.




due to smaller thresholds for deleting redundant functions. Inall cases about 98% or more of the correlation energy isrecovered in the local calculations. This fraction is slightlylarger for CCSD than for MP2, but almost identical forCCSD and QCISD~cf. the alkanes in Tables I and II!. Theabsolute differences of the correlation energies obtained inthe full and local calculations amount to 10–20 mH~6–12kcal/mol!. While errors of this size might appear significanton an absolute scale, it is to be expected that the effect onenergy differences and the shape of potential energy func-tions is much smaller. Moreover, as will be demonstratedbelow, a considerable fraction of this energy difference isdue to a reduction of the basis set superposition error, andthus local calculations may even be more reliable. Also, itshould be noted that the basis set trunction errors are usuallyabout one order of magnitude larger, and therefore we be-lieve that any errors introduced by the local approximationare negligible in practical applications.

Treating the singles nonlocally~second to last column ofTable I!, which is possible using the test program mentionedabove, increases the correlation energy by only 0.03%–0.05%, and thus the local approximation can also be safelyused for this configuration space. The last columns of TablesI and II show the correlation energies obtained when weakpairs are treated by MP2 only and neglected in the CCSDcalculation. It is found that MP2 overestimates the long-range dispersion type correlations quite significantly~up toabout 30%!, but since the contribution of the weak pairsamounts to only about 5%~cf. Table II!, the absolute error inthe total correlation energy is quite small. In order to reducethese errors, Saebo” and Pulay have suggested to take part ofthe couplings of the weak and strong pairs into account,39 butthis is not further investigated in the present paper.

Table III shows the convergence of the correlation en-ergy with increasing domain size for pentane using the VDZbasis. The first calculation is the same as in Table I, in whichby default for each bond the orbital domain comprises theprojected basis functions of two atoms. In the second calcu-lation we added to the domains of the CC-bond orbitals thebasis functions of all six atoms which are directly bound tothe two C atoms. In the third calculation we did the same for

the CH bonds, i.e., added the basis functions of the threeatoms bound to the C atom. As expected, the correlationenergy converges quickly to the limit obtained when the fullbasis is used for all domains. Of course, in the latter case thesame energy as in a nonlocal calculation is obtained.

A comparison of local calculations with different basissets for propane and butadiene is presented in Table IV. Itcan be observed that the fraction of correlation energy recov-ered in the local calculation increases with increasing basisset. Consider for instance the VDZ [3s2p1d/2s1p],VTZ~D/P! [4s3p2d/3s2p], and VQZ~D/P! [5s4p3d/4s3p], calculations for butadiene, in which thes,p,d func-tion space is systematically increased. The fraction of corre-lation energy increases slightly from 98.49% to 98.63%. Iff -type functions are added at the carbon atoms, a larger ef-fect is observed; the fraction increases up to 99.29% forVQZ~F/P! [5s4p3d2f /4s3p] !. Also the absolute errors de-crease with increasing basis set. While for butadiene with theVDZ basis the absolute error amounts to 8.8 mH, it is only5.0 mH for the VQZ~F/P! basis. We believe that this is dueto a significant reduction of the basis set superposition error~BSSE!.42,43On the other hand, the addition ofd functions atthe hydrogen atoms slightly reduces the ratio, which is at-tributed to an increased BSSE for C. Similar results arefound for propane. These findings demonstrate that for largebasis sets the local correlation methods recovers over 99% ofthe correlation energy; for smaller basis sets most of theenergy loss is likely due to a reduction of the BSSE.

The reduction of the BSSE makes the local correlationmethod particularly useful for calculations on weakly boundsystems, as has already been pointed out by Meyer andFrommhold20 and Pulayet al.58 In order to demonstrate thiseffect, we have computed the CCSD interaction energies for

TABLE III. Convergence of correlation energy with domain size forpentane.a

Orbital domainsNumber ofamplitudes

Correlation energyb

MP2~local! CCSD~local!

22centersc 192 309 20.714 074~97.38! 20.789 709~97.66!1CC-neighborsd 473 661 20.723 241~98.73! 20.798 735~98.77!1CH-neighborse 881 574 20.731 539~99.76! 20.806 906~99.78!Full 1523 385 20.733 296~100.0! 20.808 656~100.0!

aBasis VDZ, all pairs included.bPercentage relative to full calculation in parenthesis.cEach orbital domain contains basis functions of 2 atoms.dAs ~c!, but the domains for CC bonds contain in addition the basis func-tions of the six adjacent atoms.eAs ~d!, but the domains for CH bonds contain in addition the basis func-tions of the three adjacent atoms.

TABLE IV. Basis set effect in local correlation treatments.

Basis ESCF

Correlation energya

QCISD~full ! QCISD~local!

Propane:VDZ 2118.270 461 20.496 507 20.486 326~97.95!VTZ~D/P!b 2118.302 352 20.544 994 20.534 434~98.06!VTZ~F/P! 2118.304 253 20.576 963 20.570 951~98.96!VTZ~F/D! 2118.305 097 20.586 980 20.580 376~98.87!6-31G* 2118.260 171 20.440 589 20.432 825~98.24!6-31G** 2118.273 301 20.493 452 20.483 871~98.06!6-311G** 2118.293 205 20.518 938 20.508 203~97.93!

Butadiene:VDZ 2154.922 921 20.584 030 20.575 213~98.49!VTZ~D/P! 2154.958 404 20.639 229 20.630 209~98.59!VQZ~D/P! 2154.969 072 20.659 194 20.650 162~98.63!VTZ~F/P! 2154.964 025 20.681 796 20.675 359~99.06!VTZ~F/D! 2154.964 781 20.689 290 20.682 210~98.97!VQZ~F/P! 2154.973 659 20.701 843 20.696 873~99.29!6-31G* 2154.905 643 20.546 277 20.539 691~98.79!6-31G** 2154.916 784 20.583 761 20.576 591~98.77!6-311G** 2154.946 815 20.609 364 20.600 651~98.57!

aPercentage relative to nonlocal calculation in parentheses.bVTZ~D/P! denotes correlation consistent cc-pVTZ basis of Dunning~Ref.60! with up tod functions on C andp functions on H.




the argon dimer as a function of the interatomic distanceR.The calculations were performed for full and local CCSDwave functions~without weak pair approximation! using theaugmented correlation consistent triple-zeta~aug-cc-pVTZ!basis set of Kendall, Dunning, and Harrison.59 In the localcalculations, the orbital domains were kept constant for alldistances and correspond to the atomic valence orbitals ofthe two atoms. Thus, intraatomic double excitations~i , j lo-cated at one atom! use the local basis [i j ] of the correspond-ing atom, while interatomic double excitations~i and j lo-cated at different atoms! use the projected basis functions ofboth atoms. Asymptotically the full and local CCSD calcu-lations for Ar2 yield exactly the same energies. The BSSEwas estimated by the counter-poise correction~CPC! of Boysand Bernardi,43 and this correction was applied to all com-puted interaction energies. The results in Table V demon-strate that the CPC is dramatically reduced in the local cal-culations. The counter-poise corrected interaction energies of

the full and local calculations are in close agreement; thedeviations are less than about 1% for all values ofR.

A timing analysis for CISD, QCISD, and CCSD calcu-lations for pentane is presented in Table VI. Since the sameset of operatorsK ~Di j ! is needed in all cases and these domi-nate the computational expense~cf. Sec. IV B and below!,the total times for the local calculations are quite similar forall three methods. As compared to the CCSD, the QCISDapproximation has the advantage that the operatorsJ~Ei j !andK ~Ei j ! are not needed, which not only reduces the itera-tion time, but also the cost for the integral transformation. Inthe CISD and QCISD only transformed integrals with atmost two external orbitals are needed, while for the CCSDcalculation also integrals with three external orbitals are re-quired for the computation of the operatorsJ~Ei j ! andK ~Ei j !. It is particularly obvious that in the local calculationsthe evaluation of the matricesGi j takes only negligible time,while it requires significant time~25%–50%! in the conven-tional calculations.

In order to demonstrate the scaling of the computationaleffort with the molecular size, Table VII shows some timingsfor a series of alkanes for nonlocal~first line for each mol-ecule! and local ~second lines! QCISD calculations. Alsoshown are results for local calculations with MP2 treatmentof weak pairs~third lines!. All calculations were performedin C1 symmetry. In the conventional calculations the numberof amplitudes increases from 671 061 for butane by a factorof 13 to 8 931 651 for octane. The total CPU time increasesby a factor of 44. On the other hand, in the local calculationswith approximate treatment of weak pairs, the number ofamplitudes increases only by a factor of about 2~only count-ing the strong pairs!. For octane, the number of amplitudes isreduced by a factor of 17 in the full local calculation and bya factor of 157 if the weak pairs are neglected. In the lattercase it can be seen that the number of strong pairs and the

TABLE V. Interaction energies and counterpoise corrections for Ar2 usinglocal and full CCSD wave functions.a

R CPC~full !b CPC~local!b DE~full !c DE~local!c

10.0 5.2 0.6 214.6 214.78.0 13.9 1.3 248.4 249.37.5 19.1 1.5 255.3 256.67.0 28.6 3.9 238.4 240.16.0 60.8 7.8 389.9 389.65.5 85.1 8.8 1346.9 1355.65.0 128.2 11.4 3902.2 3947.5

aAugmented polarized correlation consistent aug-cc-pVTZ [6s5p3d2 f ] ba-sis set of Kendallet al. ~Ref. 59!; distancesR in bohr.bCounter-poise corrections~Ref. 43! for full CCSD and local CCSD incm21.cCounter-poise corrected interaction energies for full CCSD and local CCSDin cm21.

TABLE VI. Timing analysisa of CISD, QCISD, and CCSD calculations for pentane.b

MethodIntegral

transformationc

Quantity–Time/IterationTotaltime/it. EcorrK ~Di j ! Gi j d J˜~Ei j !, K ~Ei j !

Conventional:CISD 54.0 521.1 97.5 ••• 640.9 20.660 805QCISD 55.0 517.5 194.6 ••• 774.2 20.809 311CCSD 182.6 516.3 197.8 7.6 800.7 20.808 656

Local:CISD 53.8 523.9 12.6 ••• 546.5 20.647 126QCISD 55.1 521.4 43.1 ••• 589.3 20.790 273CCSD 213.8 518.8 45.0 16.9 613.7 20.789 709

Local/MP2~local!:e

CISD 53.3 126.9 2.8 ••• 132.1 20.664 264QCISD 54.2 127.6 5.7 ••• 136.4 20.797 771CCSD 212.8 125.7 6.5 11.2 151.4 20.797 321

aAll times for SGR R8000/75 MHZ~power challenge! in seconds per iteration. The times depend slightly on the machine load and therefore small~62%!variations are seen for identical parts@transformation,K ~Di j !# in different calculations.bCorrelation consistent cc-pVDZ basis from Ref. 60,C1 symmetry.cTransformation of integrals with two and three external orbitals.dThis includes the evaluation of the intermediate quantitiesYk j and Zk j.eWith approximate MP2 treatment of weak and distant pairs.




corresponding number of amplitudes increases onlylinearlywith the molecular size. For each additional CH2 group, ninepairs and 7548 amplitudes are added. Nevertheless, the smallfraction of amplitudes is sufficient to recover about 97.5% ofthe correlation energy, almost independent of the size~cf.Table II!. Unfortunately, the CPU time is not reduced by thesame amount. The reason is that the evaluation of the exter-nal exchange operators strongly dominates the calculation;while in the nonlocal calculations about 65% of the iterationtime is spent in this part, it amounts up to 97% in the localcalculations. As discussed in Sec. IV B, in our present imple-mentation the locality is not yet used in this part, and there-fore the only savings result from the neglect of weak pairs.Nevertheless, the total time for octane is reduced by a factorof 7.4. As pointed out in Sec. IV B, it should be possible toelimininate the present bottleneck by transforming the two-electron integrals into the local basis for each pair; the totalnumber of transformed integrals needed should be much lessthan the number of integrals in the AO basis. Work in thisdirection is presently in progress in our laboratory. We notethat in the local correlation method of Pulay and Saebo”, theexternal exchange operators also caused a bottleneck, whichto our knowledge has not yet been eliminated.

The potential of the local correlation method for reduc-ing the computational effort is much more obvious in theremaining parts of the calculation. As seen in the second tolast column of Table VII, the time for octane is reduced by afactor of 93 for a local QCISD calculation with weak-pairapproximation. In this case the CPU time increases approxi-mately with the third power of the numberNel of correlatedelectrons. This is exactly as expected, since the dominant

terms are the evaluation of the matricesYk j and Zk j @Eqs.~47! and ~48!# and their contributions to theGi j @Eq. ~42!#,which in our implementation scales proportional to4m2(NL21L3) ~the number of correlated orbitalsm and thenumber of virtual orbitalsN are linear in the number ofelectrons, while the domain sizeL is independent of the mo-lecular size!. In the full calculation, the number of multipli-cations in the dominant terms is 4m3N3, i.e., the CPU timeincreases withNel

6 . The savings are less pronounced for thelocal calculations without weak pair approximation, in whichcase the scaling is proportional to 4m3(NL21L3), i.e.,Nel

4 .In general, the savings are somewhat less than expected bythe ratio (N/L). This is mainly due to the smaller matrixdimensions, which make the MFLOP rate lower than in theconventional calculation.

VI. SUMMARY

In this work we have generalized the local correlationmethod of Pulay and Saebo” to the case of singles anddoubles coupled cluster theory. The singles and double con-figuration interaction and quadratic configuration interactionmethods are treated as special cases. A similar efficiency hasbeen achieved for all three methods; in fact, the difference incomputational effort is smaller than in conventional calcula-tions.

In particular, the local QCISD takes hardly more timethan the CISD. It has been demonstrated for a considerablenumber of molecules that the local approximation only leadsto a negligible loss of correlation energy. Part of this loss isdue to the reduction of basis set superposition errors. At

TABLE VII. CPU-timesa of QCISD calculations forn-alkanes.

Molecule

Number of Time per iterationdTotaltimeeAOsb Pairsc Amplitudesc K ~Di j ! other

Butane 106 91 671 061 152.8 79.0 2012.991 118 683 152.7 28.6 1676.937 26 775 43.6 5.0 483.9

Pentane 130 136 1523 385 517.5 256.7 7038.2136 192 309 521.4 67.9 5419.346 34 323 127.6 8.8 1343.0

Hexane 154 190 3007 378 1356.1 680.9 18 503.7190 283 683 1358.4 137.9 13 713.155 41 871 322.0 14.9 3280.3

Heptane 178 253 5377 560 3075.6 1575.8 42 207.4253 392 805 3041.4 247.5 30 107.364 49 419 635.4 22.0 6437.3

Octane 202 325 8931 651 6459.6 3345.6 89 066.4325 519 675 6396.7 451.1 62 726.273 56 967 1192.1 36.0 12 088.5

aCPU times for SGI R8000/75 MHZ~power-challenge! in seconds. For each molecule, data are given forconventional QCISD~first line!, local QCISD with all pairs included~second line!, and local QCISD withweak and distant pairs treated by MP2~third line!.bcc-pVDZ basis sets from Ref. 60.cPairs and amplitudes of weak pairs are not counted.dThe sum ofK ~Di j ! andother is the total time per iteration.eIncluding integral transformation and MP2, nine iterations.




present, the bottleneck of local calculations is the evaluationof the so-called external exchange operators, that take intoaccount the contributions of two-electron integrals over fourvirtual orbitals. In our implementation these operators arecomputed directly from the two-electron integrals in the AObasis, which avoids a full integral transformation but doesnot allow to make efficient use of the local correlation spacefor each electron pair function. This bottleneck can probablybe avoided by transforming the two-electron integrals intothe local basis. Since only a relatively small subset of thetransformed integrals is needed, it is not expected that thiswill require extensive disk space. Work in this direction is inprogress, but at present it is not yet clear how efficient thetransformation will be. All other parts of the local CCSDcalculations are significantly speeded up relative to conven-tional calculations. In particular, if week correlations of dis-tant electron pairs are omitted from the CCSD treatment,very large savings~1–2 orders of magnitude! are possible.We believe that the local correlation method has the potentialto become a most useful tool for treating electron correlationin larger molecules. Future work in our laboratory will there-fore be devoted to the development of an integral-directimplementation that avoids the storage of all two-electronintegrals, a local treatment of perturbative triple excitations,and to the evaluation of energy gradients for local correlationmethods.

ACKNOWLEDGMENTS

We thank Professor Peter Pulay for his hospitality andsupport during a visit of one of us~C.H.! in his group and formany helpful discussions. This work has been supported bythe Deutsche Forschungsgemeinschaft and the GermanFonds der Chemischen Industrie.

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Local treatment of electron correlation in coupled cluster theory