A general-order local coupled-cluster method based on the cluster-in-molecule approachZoltán Rolik and Mihály Kállay Citation: J. Chem. Phys. 135, 104111 (2011); doi: 10.1063/1.3632085 View online: http://dx.doi.org/10.1063/1.3632085 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i10 Published by the AIP Publishing LLC. 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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THE JOURNAL OF CHEMICAL PHYSICS 135, 104111 (2011)

A general-order local coupled-cluster method basedon the cluster-in-molecule approach

Zoltán Rolik and Mihály Kállaya)

Department of Physical Chemistry and Materials Science, Budapest University of Technology and Economics,P.O. Box 91, H-1521 Budapest, Hungary

(Received 22 March 2011; accepted 12 August 2011; published online 12 September 2011)

A general-order local coupled-cluster (CC) method is presented which has the potential to provideaccurate correlation energies for extended systems. Our method combines the cluster-in-moleculeapproach of Li and co-workers [J. Chem. Phys. 131, 114109 (2009)] with the frozen natural orbital(NO) techniques widely used for the cost reduction of correlation methods. The occupied molecularorbitals (MOs) are localized, and for each occupied MO a local subspace of occupied and virtualorbitals is constructed using approximate Møller–Plesset NOs. The CC equations are solved and thecorrelation energies are calculated in the local subspace for each occupied MO, while the total corre-lation energy is evaluated as the sum of the individual contributions. The size of the local subspacesand the accuracy of the results can be controlled by varying only one parameter, the threshold forthe occupation number of NOs which are included in the subspaces. Though our local CC methodin its present form scales as the fifth power of the system size, our benchmark calculations showthat it is still competitive for the CC singles and doubles (CCSD) and the CCSD with perturbativetriples [CCSD(T)] approaches. For higher order CC methods, the reduction in computation time ismore pronounced, and the new method enables calculations for considerably bigger molecules thanbefore with a reasonable loss in accuracy. We also demonstrate that the independent calculation ofthe correlation contributions allows for a higher order description of the chemically more importantsegments of the molecule and a lower level treatment of the rest delivering further significant savingsin computer time. © 2011 American Institute of Physics. [doi:10.1063/1.3632085]

I. INTRODUCTION

The use of quantum chemical methods has becomewidely accepted to solve practical chemical problems; how-ever, their applicability is strongly limited by the computa-tional expenses growing rapidly with the system size. Thisis especially true for the ab initio electron correlation ap-proaches, such as the coupled-cluster (CC) method. Formoderate-sized molecules which can be qualitatively de-scribed at the Hartree-Fock (HF) level, the CC with singleand double excitations1 (CCSD) and the CCSD with pertur-bative triples correction2 methods [CCSD(T)] should be usedif chemical, i.e., 1 kcal/mol accuracy is aimed at. For thesemethods, the scaling of the computational price is n2

on4v and

n3on

4v , where no and nv is the number of occupied and the num-

ber of virtual orbitals, respectively. For even higher accuracy,the application of higher order CC approaches, such as the CCsingles, doubles, triples3 (CCSDT) as well as CCSDT withperturbative quadruples4, 5 [CCSDT(Q)], methods is required,which are even more expensive. The strong dependence ofthe computation time on the system size makes it impossibleto apply the CC methods to large molecules.

To tackle with this problem, several local correlationmethods have been developed. These approaches utilize thatthe weak interactions of electrons localized far from eachother give negligible contributions to the correlation energy.As a consequence of the local nature of the electron correla-

a)Electronic mail: kallay@mail.bme.hu.

tion, the integrals, intermediates, and wave function param-eters connecting orbitals localized far from each other givenegligible contributions and can be neglected.6–11

One of the pioneers of the local correlation methods isPulay,6 who used an orthogonal set of localized orbitals12–14

as the occupied one-particle subspace and a non-orthogonalredundant set of atomic orbitals (AOs) projected onto thecomplement of the occupied space [projected atomic orbitals(PAOs)] to describe the virtual space. To reduce the dimen-sion of the interacting configuration space and, at the sametime, the computation cost, for all the occupied orbital pairsa restricted set of virtual orbitals were chosen to define theallowed configurations. In the simplest case, those virtual or-bitals were considered which belong to the same atoms wherethe occupied pairs are localized. Further computational sav-ings could be achieved by completely neglecting two-electronintegrals with two occupied orbitals localized very far fromeach other. Pulay and Saebø applied these techniques to thesecond15 and higher order Møller-Plesset (MP) perturbationtheories,16 coupled electron pair (CEPA) models,6, 17 and con-figuration interaction methods.17

A significant progress along these lines has been madeby Werner18–21 and co-workers and Schütz,22–24 who devel-oped local correlation methods including local CCSD andCCSD(T). To approach the linear scaling for these meth-ods, further approximations were introduced. Occupied or-bital pairs were classified according to their distance, in-tegrals and cluster amplitudes containing very distant pairs

0021-9606/2011/135(10)/104111/18/$30.00 © 2011 American Institute of Physics135, 104111-1

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104111-2 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

were completely neglected, while integrals with occupied or-bitals whose distance exceeded a certain limit were only ap-proximately evaluated.

The prescreening of the partly transformed integrals isapplied to obtain reduced or linear scaling approaches in theLaplace-MP2 method published by Ayala and Scuseria,25 inwhich, following the idea of Almlöf and Häser,26 the denom-inators from the second-order Møller–Plesset (MP2) energyformula are eliminated using Laplace-transformation and theMP2 correlation energy is expressed in the AO basis. Ayalaand Scuseria also published an iterative MP formalism and aCC approach27 where the MP and CC equations are solved inthe AO basis and a prescreening of the excitation amplitudesguarantees the linear scaling behavior for large systems. Atthis point, it is worth mentioning the local MP2 (LMP2) the-ory published by Maslen and Head-Gordon28, 29 which is, un-like Pulay’s LMP2 theory, an iteration-free approach usingnon-orthogonal occupied and virtual orbitals.

A large class of the local correlation methods decom-poses the system into fragments of manageable size, and thecorrelation energy is calculated as the sum of the contri-butions of the individual fragments and the corrections forthe interactions of the fragments. This general strategy isfollowed in the old papers of Förner and co-workers,7, 8 inthe incremental method proposed by Stoll and studied bymany others,30–35 in the CC method based on the fragmentmolecular-orbital method,36 in particular, in the divide-and-conquer methods such as the one published by Li and Li,37

and more recently by Ziółkowski et al.38

In other local correlation methods, the correlation en-ergy is expressed as a simple sum of the local contributionswithout the need for the explicit treatment of interactions ofthe fragments. The divide-and-conquer CC method recentlypublished by Kobayashi and Nakai39 applies a buffer regionfor each fragment to incorporate the effect of the fragment-fragment interactions at the calculation of fragment energies.In the natural linear scaling (NLS) CC approach advocated byFlocke and Bartlett,40 the CC energy is decomposed as a sumof occupied orbital contributions. In the NLS CC method, thelocalized natural bond orbitals41 are applied, instead of thelocalized occupied HF orbitals, which allows the straightfor-ward definition of a linear scaling CC method.

From our point of view especially important is thecluster-in-molecule CC (CIM) approach42–44 proposed re-cently by Li and co-workers45–48 and the pair natural orbital(PNO) CC method of Neese and co-workers.49, 50 In the CIMframework, the energy expression of various electron correla-tion methods are expressed as the sum of the energy contri-butions of occupied localized molecular orbitals (LMOs) oroccupied molecular orbital domains. For each localized oc-cupied domain, a restricted set of interacting occupied andvirtual orbitals is defined. Unlike in many local correlationmethods, both the occupied and virtual orbitals form an or-thogonal basis sets. The strongly interacting occupied LMOsare collected into the occupied domains and the energy is ex-pressed as a sum of local domain contributions. The occupiedLMOs are considered strongly interacting if the correspond-ing elements of the Fock matrix is large. The virtual orbitalsare comprised as linear combinations of PAOs. Those virtual

orbitals are assigned to an occupied domain which are local-ized to the same or the neighboring atoms where the occupiedorbitals are located. Before the correlation calculation, the vir-tual functions are orthogonalized, and the redundancies areeliminated. Piecuch and co-workers applied this frameworkto CCSD and conventional and completely renormalized (CR)(Ref. 51–53) perturbative triples calculations.

Our work is also influenced by Neese and co-workers,49, 50 who published a local approximation for theCEPA and CCSD methods using PNOs (Refs. 54–57) asso-ciated to occupied electron pairs. The PNOs assigned to oc-cupied orbital pair i, j are the eigenvectors of the matrix ob-tained as the i, j contributions to the virtual block of the MP2density matrix. For a pair of occupied orbitals, the residualequations (of the CEPA or CCSD theory) are transformedto the form where the virtual space is expressed in the ba-sis of the corresponding PNOs. The definition of PNOs, sim-ilar to the definition of natural orbitals, allows the interpre-tation of the eigenvalues of the density matrix componentsas population-like quantities and justifies the elimination ofthe PNOs with eigenvalues below a given threshold. For largesystems, one can expect that a large portion of PNOs can bedropped without entering significant error, and it might lead toremarkable improvement in performance. Very recently a re-lated local MP2 method has been published by Yang et al.58 Inthe latter approach, termed the orbital-specific virtual (OSV)approximation, a restricted-sized optimized virtual space isconstructed for each localized occupied orbital by performingsingular value decomposition for the MP2 amplitudes diago-nal in the occupied spatial indices, and the restricted virtualspace is utilized to achieve a computationally advantageousrepresentation of the first-order MP wave function.

In this paper, we propose a variant of the CIM approachwhich differs basically from the original ansatz in the con-struction of the local orbital domains using an algorithm rem-iniscent of the PNO and the OSV methods. The importantdifference of our method with respect to the PNO approachof Neese et al. is that in the latter approach the interactingsubspaces are defined for electron pairs, while in our localCC approach the interacting subspaces are constructed for in-dividual electrons. Our method to define the local interactingsubspace for a single occupied orbital also differs from thatapplied in the OSV method since we use the eigenvectors ofan approximate density matrix as the basis of the local in-teracting subspace instead of the singular vectors for certaindouble amplitudes. Since our algorithm for the construction oflocal interacting subspaces uses approximate natural orbitals,we expect that higher accuracy can be achieved with the samenumber of correlated orbitals.

Our local CC method at the present stage is not a lin-ear scaling method since the definition of the local interactingsubspaces does not contain any distance dependent cutoff pa-rameters similar to those applied in Werner’s algorithm dis-cussed above or in the CIM CC approach of Li and Piecuch.Our local CC algorithm scales as fifth power of the systemsize, and the scaling is dominated by the integral transforma-tion from the AO to the molecular orbital (MO) basis for anylevel of the CC theory. This means that the presented approachoffers a good alternative for the ordinary CC methods for large

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104111-3 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

systems even at the CCSD level. The general implementationof our approach also enables local CC calculations includinghigher excitations.

The organization of this paper is as follows. InSec. II A, we describe the theoretical background of our localCC theory. Section II B gives a brief summary of implementa-tion and a detailed analysis of the scaling properties. In Sec. IIC, we compare our approach to the related local correlationmethods. Numerical test calculations are presented in Sec. IIIand concluding remarks are given in Sec. IV.

II. THEORY

A. The local CC ansatz

Our principal aim is to reformulate the equations of theCC theory in a way that the locality of the electron correlationbe exploited. To that end, we follow Li et al.45 and recognizethat the CC energy formula,

ECC =∑

ia

fai tai + 1

4

∑

abij

〈ij ||ab〉τ abij , (1)

is invariant to the separate unitary transformation of the oc-cupied and virtual spaces, where i, j , . . . are general oc-cupied orbitals and a, b, . . . are general virtual orbitals,fai and 〈ij ||ab〉 are the Fock-matrix elements and the anti-symmetrized two-electron integrals, respectively, finally τ ab

ij

= tabij + tai tbj − taj tbi is the effective double-excitation ampli-

tude composed of the tai and tabij single- and double-excitation

amplitudes. Utilizing the invariance for the occupied-occupied rotations, we can suppose that the occupied orbitalsare localized. The energy can be written as a sum of the con-tributions for individual localized occupied orbitals,

ECC =∑

i

δEi, (2)

where

δEi =∑

a

fai tai + 1

4

∑

abj

〈ij ||ab〉τ abij (3)

is still invariant to the virtual-virtual and those occupied-occupied rotations where i is kept fixed. This flexibility al-lows us to choose an individual optimized one-particle basisset for each occupied MO i, suitable to introduce approxi-mations to reduce the calculation cost of δEi . To obtain amethod which allows the elimination of weak interactions be-tween orbitals located far from each other, the calculation ofδEi should be approximated by restricting the summations inEq. (3) to a subset of orbitals denoted by Pi which, in somesense, strongly interacts with orbital i, i.e.,

δEi =∑

a∈Pi

fai tai + 1

4

∑

abj∈Pi

〈ij ||ab〉τ abij . (4)

We will refer to Pi as the local interacting subspace of orbitali and we will call orbital i as the central orbital of subset Pi .Since the cluster amplitudes in the above equation depend onthe indices of the orbitals in the Pi space, they can be de-termined by solving the CC equations in the Pi space sepa-rately for each occupied orbital i. In this manner, considering

the steep scaling of CC methods, significant speedup can begained for extended systems, where the local interacting sub-spaces are expected to be small with respect to the originalMO basis.

To complete the definition of the local CC method, wehave to define the local interacting subspaces. To incorporatecorrelation information to the definition of the local interact-ing subspace Pi , we explore an approach which is related tothe PNO method of Neese and co-workers49, 50 and the defini-tion of OSV orbitals of Yang et al.58

Generally, in the electron correlation problem, lookingfor the important part of the one-particle space, a goodguess is to choose the subspace expanded by the naturalorbitals (NOs) of some correlated wave function with highpopulation.56, 57, 59, 60 To obtain the NOs and their popula-tions the corresponding density matrix has to be diagonal-ized. In practice the first-order Møller-Plesset wave func-tion, denoted here by �(1), is a plausible choice for the wavefunction.54, 61–63 The virtual-virtual block of this density ma-trix, when it is expressed in the HF canonical orbital basis,

Dab = 〈�(1)|a+b−|�(1)〉

= 1

2

∑

cij

〈ac||ij 〉〈bc||ij 〉(εi + εj − εa − εc)(εi + εj − εb − εc)

, (5)

contains the εa, . . . , εj , . . . diagonal elements of the Fockmatrix. Similarly, one can also define a hole density matrix,whose occupied-occupied block reads as

Djk = 〈�(1)|j−k+|�(1)〉

= 1

2

∑

abi

〈ab||ij 〉〈ab||ik〉(εi + εj − εa − εb)(εi + εk − εa − εb)

. (6)

Of course, these expressions are strictly valid for canonicalHF orbitals and only approximations to the MP2 density ma-trix if localized orbitals are used.

To utilize the simplicity of the above density matrices andto incorporate the demand for the locality, we define the D(i)

fragment of the density matrix as

D(i)ab = 1

2

∑

cj

〈ac||ij 〉〈bc||ij 〉(εi + εj − εa − εc)(εi + εj − εb − εc)

(7)

and thus

Dab =∑

i

D(i)ab. (8)

Similar density matrix fragments were used by Neese and co-workers49, 50 to define the local PNOs in the coupled-electronpair and CC framework. We choose the eigenvectors of D(i)

with sufficiently large eigenvalues as the virtual basis set forthe local interacting subspace of central orbital i. If λ

(i)a , the

eigenvalue belonging to eigenvector a of matrix D(i), is large,we suppose that orbital i strongly interacts with orbital a. λ

(i)a

can be regarded as the approximation for the contribution oforbital i to the population of orbital a and it will be calledthe local particle population of orbital a. The virtual part ofthe local interacting subspace Pi is composed of a orbitals

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104111-4 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

with λia eigenvalues greater than a threshold hereafter denoted

by ε.Similarly, for the central orbital i a local hole density ma-

trix D(i)jk can be defined as

D(i)jk = 1

2

∑

ab

〈ab||ij 〉〈ab||ik〉(εi + εj − εa − εb)(εi + εk − εa − εb)

, (9)

where obviously

Djk =∑

i

D(i)jk. (10)

It suggests that the local interacting occupied orbitals shouldbe determined similar to the virtual ones. However, when di-agonalizing matrix D

(i)jk , orbital i would also be transformed.

Since the partial energy δEi is defined for orbital i, it is de-sired to keep the central orbital i intact. To avoid the mix-

ing of orbital i, matrix D(i)jk is slightly modified, the interac-

tion of orbital i and the other orbitals is eliminated, that is,D

(i)ik = D

(i)ki = δik is imposed. This modified matrix will be

called the local hole density matrix. The eigenvectors j of the

modified D(i)jk matrix define the local interacting occupied or-

bitals for the central orbital i. If the λ(i)j

eigenvalue of the local

hole density matrix is large, i.e., λ(i)j

> ε, we suppose that or-

bital i strongly interacts with orbital j and it is included inthe local interacting subspace Pi . At the calculation of δEi ,the local interacting occupied orbitals j with small local holepopulation, λ

(i)j

< ε, are kept frozen.The definition of the local interacting subspace as dis-

cussed above—similar to the PNO (Ref. 49) and OSVmethods58—is based on formulas derived for canonical MOs.In a localized basis, these expressions are only rough approx-imations for the first-order MP wave function and the den-sity matrices calculated therefrom. The use of the canonicalformulas can be justified by two facts. First, numerical ex-perience shows that in most cases the Fock matrix is stilldiagonally dominant in the localized basis, thus the termsneglected due to this approximation are expected to be no-ticeably smaller than the retained ones, and the magnitude ofthe coefficients for various determinants in the first-order MPwave function is expected to be correct. Second, the validityof the approximation is also affirmed by the encouraging nu-merical results published by Neese and co-workers,49, 50 Yanget al.,58 and in the present paper.

After defining the local interacting subspaces using thesingle parameter ε, it is practical to transform the orbitals ofthese subspaces to the local canonical basis,45 where the Fockmatrix for each subspace is diagonal, and to use these basissets to compute the partial CC energies δEi in Eq. (4). Theapplication of the local canonical orbitals on the one hand hasthe advantage that the formulas of the perturbative CC meth-ods [CCSD(T), CCSDT(Q), etc.] are simple in the canonicalbasis, i.e., these energy contributions can be calculated non-iteratively. On the other hand, the application of the canoni-cal orbitals also improves the convergence of the iterative CCcalculations.45 Since the local interacting subspaces are sup-posed to be small, the cost of the integral transformations tothe canonical basis is also small. However, the canonicaliza-

tion in the Pi subspace necessarily mixes the central orbital i

with the other occupied orbitals. Hence the energy expression,Eq. (4), in the basis of the canonical orbitals deserves someattention. Exploiting the unitary invariance of the expression,we can rewrite Eq. (3) as45

δEi =∑

a′i ′j ′∈Pi

Uii ′Uij ′fa′i ′ t

a′j ′

+ 1

4

∑

a′b′i ′j ′k′∈Pi

Uii ′Uik′ 〈i ′j ′||a′b′〉τ a′b′k′j ′ , (11)

where matrix U transforms canonical orbitals i ′ to the centralorbital i,

|i〉 =∑

i ′Uii

′ |i ′ 〉. (12)

According to the above formula, it is useful to transform oneoccupied canonical index of the calculated ta

′i ′ and ta

′b′i ′j ′ cluster

amplitudes, the Fock matrix, and the integral list back to cen-tral orbital i, then to calculate the local energy contributionsusing the

δEi =∑

a′∈Pi

fa′i ta′i + 1

4

∑

a′b′j ′∈Pi

〈ij ′||a′b′〉τ a′b′ij ′ (13)

expression, where orbitals a′, b′, . . . and i ′, j ′, . . . are the vir-tual and occupied canonical orbitals in the local interactingsubspace of central orbital i. It is important to emphasize that,if ε = 0 is used, the local interacting subspaces contain thewhole one-particle basis set, and the sum of the partial energycontributions gives the exact CC energy.

For the perturbative CC approaches, the general struc-ture of the energy expression usually differs from the one inEq. (4), but fortunately the above recipe can be adapted to anymany-body energy formula which contains summations overthe occupied orbitals. For example, the form of a generallydefined perturbative triples correction,5, 45

ET =∑

abcijk

Labcijk T abc

ijk , (14)

can be easily decomposed as a sum of partial energies of oc-cupied localized orbitals,

δETi =

∑

abcjk

Labcijk T abc

ijk , (15)

where the Labcijk and T abc

ijk quantities are again calculated in thelocal interacting subspace. Since the calculation of the pertur-bative CC corrections are more straightforward and less ex-pensive in the canonical basis, the practical form of the partialenergies is

δETi =

∑

a′b′c′i ′j ′k′l′Uii ′Uil′L

a′b′c′i ′j ′k′ T

a′b′c′l′j ′k′

=∑

a′b′c′j ′k′La′b′c′

ij ′k′ T a′b′c′ij ′k′ . (16)

Similar expressions hold for the higher order perturbative CCapproaches, such as CCSDT(Q).

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104111-5 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

TABLE I. Scaling of each step for the local CC algorithm compared to ordinary CC calculations. nb , no, nv ,nl , nlo, nlv , and L refer to the number of basis functions, the number of occupied orbitals, the number of virtualorbitals, the number of local interacting orbitals, the number of local occupied orbitals, the number of local virtualorbitals, and the system size, respectively.

HF-SCF n4b L4

Boys localization n3o L3

Integral transformation from the AO to the MO basis n5b L5

Calculation of the density matrices n2on

3v L5

Integral transformation from the MO basis to the local subspaces nonln4b L5

Canonicalization in the local subspaces non5l L

Calculation of local CCSD correlation energy non2lon

4lv L

Calculation of local CCSD(T) correlation energy non3lon

4lv L

Calculation of local CCSDT correlation energy non3lon

5lv L

Calculation of local CCSDT(Q) correlation energy non4lon

5lv L

Any local CC calculation L5

Conventional CCSD calculation n2on

4v L6

Conventional CCSD(T) calculation n3on

4v L7

Conventional CCSDT calculation n3on

5v L8

Conventional CCSDT(Q) calculation n4on

5v L9

The outlined local CC method, which basically differsfrom the previous variants of the CIM approach in the con-struction of the local interacting spaces, will be referred to asthe local natural orbital cluster-in-molecule (LNO-CIM) CCmethod.

B. Implementation

In our current implementation of the LNO-CIM scheme,a calculation starts with the evaluation of the AO integrals andthe solution of the HF self-consistent field (HF-SCF) equa-tions. Applying the traditional approaches, these tasks scaleas n4

b, where nb is the number of basis functions. In the sec-ond step, the occupied MOs are subjected to localization, forwhich we applied the procedure of Foster and Boys,64 scalingas n3

o. At the present stage, we perform an integral transfor-mation from the AO to the MO basis. The integral transfor-mation for the whole basis set scales as n5

b. Currently, thisis the most time-consuming part of the algorithm. For eachlocalized occupied orbital, the local particle and local holedensity matrices are constructed according to Eqs. (7) and (9).The computation cost for the more expensive local particledensity matrices is proportional to n2

on3v . After the diagonal-

ization of the local density matrices, which scales as non3v ,

the occupied MOs with local hole population smaller than theε threshold are frozen and the virtual MOs with local particlepopulation below the ε threshold are dropped. In the next step,for each occupied orbital i a second integral transformation isperformed in which the original MO integrals are transformedto the low-dimensional local interacting subspaces. The totalcomputational cost of the integral transformations scales asnonln

4b, where nl is the average dimension of the local ba-

sis sets. Supposing that the dimension of the local basis setsdoes not scale with the system size and the number of ba-sis functions are significantly larger than the number of occu-pied orbitals, the cost of the second integral transformation ismuch smaller than the first one. For the calculation of the lo-cal energy contributions, Eqs. (13) and (16), a canonical basis

is used. The transformation of the orbitals of the local inter-acting subspace to the local canonical basis is cheap since thedimension of the subspaces is considered to be small. Thecost of the integral transformation to the local canonical ba-sis scales as non

5l which is a linear function of the system

size. Finally, the solution of the CC equations is performed inthe canonicalized local interacting subspaces. Since for largesystems the dimension of the interacting subspaces does notdepend on the molecular size, the computation cost of thecalculation of the partial CC energies is proportional to thenumber of electrons, thus it is a linear function of the systemsize. We also note that the computational expenses can be sig-nificantly reduced for symmetric molecules in a natural waysince it is enough to determine the δEi quantity for one of thesymmetry-related localized MOs.

The summary of the scaling properties discussed abovefor the various steps of a local CC calculation can be foundin Table I. As the table shows the present implementation ofour local CC method scales as L5 with L as the size of themolecule for any excitation level, while the frequently appliedordinary CCSD, CCSD(T), CCSDT, and CCSDT(Q) methodsscale as L6, L7, L8, and L9, respectively.

The LNO-CIM CC method has been implemented inthe MRCC suite of quantum chemistry programs65 interfacedto the MOLPRO (Ref. 66) package. Routines for the cal-culation of local density matrices and four-index integraltransformations have been developed, and the iterative67 andperturbative5, 68 CC codes have been modified to calculateEqs. (13) and (16). In the current implementation, the in-tegral calculation, HF-SCF, and the localization steps areperformed by MOLPRO, and AO integrals and MO coeffi-cients are passed over to MRCC. The local CC method isin principle available for any single-reference CC method,such as CCSD, CCSDT, and CCSDTQ, . . . , as well asfor the corresponding perturbative approaches, CCSD(T),CCSDT(Q), . . . , though we should mention that local CCcalculations including higher than quadruple excitations arecurrently hardly feasible because of the enormous costs of themethods.

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104111-6 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

C. Comparison to other local correlation methods

In this paragraph, we briefly compare the LNO-CIMmethod to the closely related local correlation approaches. Wedo not consider local CC methods such as those of Werneret al.,18–20 Schütz,22–24 or Ayala and Scuseria27 which signifi-cantly differ from our approach.

The CIM method proposed by Li45–48 and co-workersand the NLS CC method introduced by Flocke and Bartlett40

also apply Eq. (2) as the starting point of the definition oftheir local CC approaches, and the δEi partial energies arecalculated in an orthogonal basis. While Flocke and Bartlettguarantee the local nature of the NLS CC theory by theapplication of localized natural bond orbitals, in both theCIM CC and the LNO-CIM CC approaches localized occu-pied orbitals are used, and the calculation costs are reducedusing low-dimensional domains assigned to the localizedoccupied orbitals. In the NLS CC framework, the HF-SCFcalculations are performed for small fragments, while in theCIM and LNO-CIM approaches the HF-SCF is performedfor the whole molecule. The most obvious difference amongthe NLS, CIM, and LNO-CIM CC schemes is the definitionof the local interacting subspaces. In NLS CC, the localinteracting subspaces are chosen by chemical intuition. TheCIM CC approach of Li and Piecuch applies a complicatedalgorithm based on Mulliken orbital charges, the occupiedblock of the Fock matrix, the spatial distance of the orbitals,and PAOs for the definition of virtual interacting subspaces,while the LNO-CIM approach contains a simple correlationargument which is inspired by Neese and co-workers’ CCmethod based on PNOs. In our method, the accuracy of theapproximation can be tuned by one parameter the thresholdon the local populations of the MOs.

In the local CC method of Neese et al. the virtual orbitalspace interacting with a given occupied LMO pair is truncatedby the application of PNOs. The PNOs are defined for eachoccupied orbital pair i, j as the eigenvectors of a matrix de-rived from the virtual block of the one-particle density matrix,Eq. (5), by applying localized orbitals instead of the canonicalones and restricting the occupied summations to i and j . Afterdefining the PNOs the virtual indices of the residual equationsindexed by i and j are transformed to the PNO basis and theCC equations are solved. It means that in Neese’s method,the correlation contributions of the electron pairs are approx-imated. Unlike in Neese’s method in the LNO-CIM approachthe contributions of individual occupied orbitals are approxi-mated, thus in our case the definition of the local virtual den-sity matrix in Eq. (7) contains a summation for an occupiedindex. The CC equations are solved separately for each occu-pied MO, but the equations are not transformed. Neese’s PNOmethod, similar to ours, scales with the system size L as L5.

Though the OSV approach of Yang et al.58 has so far beenapplied only to MP2 wave functions, some remarks about thismethod are in order because among all the procedures forthe construction of the local orbital domains the one used bythese authors is probably the closest to our algorithm. In theOSV approach, for each localized occupied MO the first-orderMP2 doubles amplitudes corresponding to the simultaneousexcitation of both electrons from that particular spatial orbital

are subjected to singular value decomposition. The resultingorbital-specific virtual space is truncated on the basis of thesingular values. The first-order wave function is parameter-ized in terms of factorized amplitudes expanded in the OSVspaces, and the amplitudes are determined by minimizing theHylleraas functional. The common feature of the algorithmof Yang et al. and the present one is that the local orbitalspaces are constructed for each occupied MO separately, notfor orbital pairs or a group of orbitals, using a correlation ar-gument. The two methods basically differ in the latter, thatis, Yang and co-workers decompose the diagonal MP doublesamplitudes, while in the LNO-CIM scheme the one-particledensity matrices are diagonalized. Another important differ-ence is that we also construct an occupied subspace for eachlocalized MO. The OSV approach could in principle be em-ployed in the CIM framework as well for the definition of thelocal virtual space, however, we prefer our frozen NO schemefor two reasons. First, as it is well known, the use of NOsguarantees the most compact representation of the wave func-tion. Second, the density matrix-based approach also enablesus to identify the strongly interacting occupied orbitals, whichis not inherent in the OSV scheme. Nevertheless, we shouldalso note that the OSV approximation is, of course, the rightchoice for the MP2 method because our approach would re-quire the construction of MP2 density matrices, which is asexpensive as the MP2 calculation itself.

An important difference with respect to any previous lo-cal CC implementations is that our approach is general and,in practice, allows for treating up to quadruple excitations.

III. TEST CALCULATIONS

To assess the performance of the LNO-CIM CC method,numerous test calculations were carried out for linear alkanechains, other quasi-two- and three-dimensional hydrocarbons,and water clusters at the CCSD, CCSD(T), CCSDT, andCCSDT(Q) levels of theory. The energies and the requiredcomputation times are compared to the ordinary CC results.The calculations were performed with Dunning’s correlationconsistent polarized valence double- (cc-pVDZ) and triple-ζ (cc-pVTZ) basis sets.69 For the hydrocarbons standardizedgeometries were chosen with C–C and C–H bond distances of1.55 and 1.09 Å, respectively, and tetrahedral bond angles forall the molecules but dodecahedrane, for which symmetric do-decahedral structure was used. The geometry for the (H2O)n(n = 10, 12, 14, 16, 18, 20, and 22) water clusters were takenfrom the paper of Li et al.45 For each water cluster, the lowest-energy structure was considered. The core electrons were keptfrozen, and the CC energies were converged to 10−8 Eh. Thecalculations were carried out on computers with Intel XeonE3110 3.00GHz CPUs and 8 Gb main memory.

As it is discussed in Sec. II B the scaling behavior of thelocal CC calculations is strongly influenced by the dimensionof the local interacting subspaces: the number of correlatedorbitals and electrons. To investigate these two parameters asa function of system size and basis set size, the dimension ofthe largest interacting subspace and the maximum number ofcorrelated electrons are shown in Fig. 1 as a function of thenumber of carbon atoms in the CnH2n+2 linear alkane chains

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104111-7 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

20

40

60

80

2 4 6 8 10 12 14 16 18 20

ε=10-5

0

40

80

120

2 4 6 8 10 12 14 16 18 20

ε=10-6

0

100

200

2 4 6 8 10 12 14 16 18 20

n

ε=10-7

Max. number of electrons, cc-pVDZ Max. number of orbitals, cc-pVDZ

Max. number of electrons, cc-pVTZ Max. number of orbitals, cc-pVTZ

FIG. 1. The maximum number of orbitals and the maximum number of electrons correlated in the local interacting subspaces for the CnH2n+2 alkane moleculeswith the cc-pVDZ and cc-pVTZ basis sets using ε = 10−5, 10−6, and 10−7 thresholds in the local CC calculations.

with the cc-pVDZ and cc-pVTZ basis sets. For the test calcu-lations, ε = 10−5, 10−6, and 10−7 were used as the thresholdfor the local populations. As Fig. 1 shows, for short alkanesthe dimension of the subspace and the number of electronsare increasing with increasing system size. For larger alkanes,these parameters are saturated. The size of the hydrocarbonchain where the saturation occurs strongly depends onto theε value but is not or is weakly influenced by the size of thebasis set. For ε = 10−5, the number of electrons and the num-ber of basis functions saturate at about n = 6 irrespective ofthe applied basis set. For ε = 10−6 and ε = 10−7, the satura-tion takes place at n = 11 and n = 14, respectively, for the cc-pVDZ basis set. With the cc-pVTZ basis the calculations havebeen performed up to ten carbon atoms, but the tendency ofthe curve for ε = 10−6 shows that the saturation occurs againat around n = 11. It is interesting to note that the number ofcorrelated electrons does not depend on the applied basis setbut the number of orbitals for the local interacting subspaceof maximal dimension is slightly larger for the triple-ζ basisset. The maximum numbers of correlated electrons after thesaturation are 38, 58, and 82 for ε = 10−5, 10−6, and 10−7, re-spectively. For ε = 10−5, the maximum number of basis func-tions in the local interacting subspaces is 59 for the cc-pVDZand 72 for the cc-pVTZ basis set. These numbers are 104 and142 for ε = 10−6 as well as 173 and around 240 for ε = 10−7,respectively.

The various values of ε lead to approximations withrather different electron numbers and local basis set dimen-sions. Investigation of the performance of the method with

these thresholds can give some insight into the applicabilityof the theory. We have performed local CC calculations upto n = 20 for the alkane series at the CCSD level of theorywith the cc-pVDZ basis set and up to n = 10 with the cc-pVTZ basis set. The relative errors and CPU times are shownin Figs. 2 and 3. Looking at the relative error curves, one cansee that these are also saturated for large molecules, and thesaturation is slower for smaller ε values. For these examples,the magnitude of the relative errors seems to be a linear func-tion of ε, i.e., if ε is an order of magnitude smaller, the error ofthe approximation is also reduced by an order of magnitude.Using the cc-pVDZ basis for ε = 10−5, 10−6, and 10−7, therelative errors at n = 20 are roughly 2%, 0.25%, and 0.023%.These values are slightly larger for the cc-pVTZ basis set. Toshow the magnitude of the absolute errors, we note that theCCSD correlation energy is a linear function of the number ofcarbon atoms with a very good approximation, and the CCSDcorrelation energies for the C5H12 and C10H22 molecules are−0.79 and −1.56 Eh, respectively, with the double-ζ basis setand −0.93 and −1.83 Eh, respectively, with the triple-ζ basisset. According to these results, the application of ε = 10−6

has an error of a couple of milli-hartrees for molecules of thissize, while the errors for ε = 10−5 and ε = 10−7 are an orderof magnitude larger and smaller, respectively.

For the investigated ε values, the computation timecurves show rather different pictures. As it is discussed inSec. II B, the most time-consuming parts of the present im-plementation are the integral transformation from the AO ba-sis to the MO basis, the construction of the local density

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104111-8 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

0

1

2

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

) ε=10-5

0

20

40

60

80

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.) ε=10-5

0.1

0.2

0.3

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

) ε=10-6

200

400

600

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.) ε=10-6

0.005

0.010

0.015

0.020

0.025

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

) ε=10-7

0

5000

10000

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.)

n

ε=10-7

relative errorordinary CCSD

integral transformations

CCSD iterationtotal local CCSD

FIG. 2. The relative errors of local CCSD correlation energies as well as timings for the local and ordinary CCSD calculations with the cc-pVDZ basis setusing ε = 10−5, 10−6, and 10−7.

matrices, the integral transformations from the MO basis tothe local subspaces, and the solution of the CC equations.Since the scaling of the first three steps is similar but signif-icantly different from that for the last one, in the followingwe do not analyze separately the computation time demandof the first three steps, but we compare their aggregate com-putation time to that of the CC calculations. For the sake of

brevity, the first three steps together will be called the integraltransformations. To highlight the dominant part of the com-putational costs and to emphasize the different scaling of theintegral transformations and the solution of the CC equations,the computation times for these steps are plotted in the fig-ures. The timings of the ordinary CCSD calculations are alsodisplayed in Figs. 2 and 3.

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104111-9 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

0

1

2

3

1 2 3 4 5 6 7 8 9 10R

el. e

rr.(

%)

ε=10-5

0

150

300

450

1 2 3 4 5 6 7 8 9 10

CP

U(m

in.) ε=10-5

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 10

Rel

. err

.(%

)

ε=10-6

1200

2400

3600

4900

1 2 3 4 5 6 7 8 9 10

CP

U(m

in.) ε=10-6

0

0.010

0.020

0.030

1 2 3 4 5 6 7 8 9 10

Rel

. err

.(%

)

ε=10-7

0

6000

12000

18000

1 2 3 4 5 6 7 8 9 10

CP

U(m

in.)

n

ε=10-7

relative errorordinary CCSD

integral transformation

CCSD iterationtotal local CCSD

FIG. 3. The relative errors of local CCSD correlation energies as well as timings for the local and ordinary CCSD calculations with the cc-pVTZ basis set usingε = 10−5, 10−6, and 10−7.

At ε = 10−5 for both basis sets, the computation cost oflocal CC calculations is dominated by the integral transfor-mations, and the cost of the solution of the CC equations is alinear function of the system size as it is expected. The com-putational cost of the ordinary CCSD calculations is largerthan that of the local CCSD from n = 6 and n = 3 with the

cc-pVDZ and cc-pVTZ basis sets, respectively, due to the L6

scaling of the ordinary CCSD and the L5 scaling of the lo-cal CCSD. For ε = 10−6, it is still true that the local CC ismore cost efficient than the ordinary CCSD calculation start-ing from a relatively small n, especially for the triple-ζ basisset. The crossing points are at n = 9 and n = 4 with the cc-

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104111-10 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

0 5 10 15 20 25

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

)

ε=10-5

0

60

120

180

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.) ε=10-5

0

2

4

6

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

)

ε=10-6

0

600

1200

1800

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.) ε=10-6

0

0.5

1

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

) ε=10-7

0

3000

6000

9000

12000

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.)

n

ε=10-7

relative errorordinary CCSD(T)

integral transformation

local (T) corr. (without CCSD)total local CCSD(T)

FIG. 4. The relative errors of local perturbative triples corrections as well as timings for the local and ordinary CCSD(T) calculations with the cc-pVDZ basisset using ε = 10−5, 10−6, and 10−7.

pVDZ and cc-pVTZ basis sets, respectively. For the investi-gated molecules at ε = 10−6, the costs of the integral transfor-mations and those for the solution of the CC equations are ofthe same magnitude for both basis sets, while for ε = 10−7 thecosts of the integral transformations became very small withrespect to those for the whole calculation. Note that for the cc-pVDZ basis the computation time of the ordinary CCSD be-comes larger than that for the local CCSD calculation only at

n = 15, while in the cc-pVTZ basis a significant speedup canbe observed already at n = 5. The speedups we have foundfor n = 20 with the double-ζ basis set are about 40, 20, and 3for ε = 10−5, 10−6, and 10−7, respectively.

The relative errors and the CPU times for the localCCSD(T) calculations can be found in Figs. 4 and 5. The(T) corrections are two orders of magnitude smaller thanthe CCSD correlation energies. For the C5H12 and C10H22

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104111-11 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

0

10

20

30

1 2 3 4 5 6 7 8 9 10R

el. e

rr.(

%)

ε=10-5

0

150

300

450

1 2 3 4 5 6 7 8 9 10

CP

U(m

in.) ε=10-5

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10

Rel

. err

.(%

)

ε=10-6

1200

2400

3600

1 2 3 4 5 6 7 8 9 10

CP

U(m

in.) ε=10-6

0

0.5

1.0

1.5

2.0

1 2 3 4 5 6 7 8 9 10

Rel

. err

.(%

)

ε=10-7

0

6000

12000

18000

1 2 3 4 5 6 7 8 9 10

CP

U(m

in.)

n

ε=10-7

relative errorordinary CCSD(T)

integral transformations

local (T) corr. (without CCSD)total local CCSD(T)

FIG. 5. The relative errors of local perturbative triples corrections as well as timings for the local and ordinary CCSD(T) calculations with the cc-pVTZ basisset using ε = 10−5, 10−6, and 10−7.

molecules, the triples corrections are −0.017 and −0.035 Eh

with the cc-pVDZ, −0.027 and −0.054 Eh with the cc-pVTZbasis set, respectively. The maximal relative errors of the lo-cal (T) contributions with ε = 10−5, 10−6, and 10−7 are about27 (28)%, 6 (8)%, and 1 (2)%, respectively, for the cc-pVDZ(cc-pVTZ) basis set. The use of ε = 10−6 seems to be a goodcompromise which offers a relatively small error in the order

of milli-hartree with a considerable speedup especially withthe triple-ζ basis set. The application of the local CCSD(T)with ε = 10−7 is still economical. With the triple-ζ basis thelocal CCSD(T) is substantially faster from n = 7 with respectto the ordinary CCSD(T), while for the double-ζ basis thecrossover point is at n = 15. For ε = 10−5 and 10−6, a signif-icant part of the computational costs comes from the integral

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104111-12 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

0

5

10

15

20

25

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

)

ε=10-5

0

1500

3000

4500

6000

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.)

ε=10-5

relative errorordinary CCSDT

integral transformations

local CCSDT iterationstotal local CCSDT

FIG. 6. The relative errors of the triples contributions calculated as the difference of local CCSDT and local CCSD energies as well as timings for the local andordinary CCSDT calculations with the cc-pVDZ basis set using ε = 10−5.

transformations, but for ε = 10−7 the costs of the CC iter-ations become dominant, at least for the investigated set ofmolecules.

To gain some insight into the performance of the LNO-CIM CC approach for higher order CC methods we have alsoperformed CCSDT and CCSDT(Q) calculations for the lin-

ear alkanes with the cc-pVDZ basis set and ε = 10−5. TheCCSDT results can be found in Fig. 6. The relative errors forthe triples contributions—defined as the difference of CCSDTand CCSD energies—are less than 20% for n ≤ 6, where thereference values are available. On the basis of the shape ofthe curve and the behavior of the CCSD correlation energies,

0

5

10

15

20

2 4 6 8 10 12 14 16 18 20

Rel

. err

.(%

)

ε=10-5

0

6000

12000

18000

2 4 6 8 10 12 14 16 18 20

CP

U(m

in.)

ε=10-5

relative errorordinary CCSDT(Q)

integral transformations

local (Q) corr. (without CCSDT)total local CCSDT(Q)

FIG. 7. The relative errors of local perturbative quadruples corrections as well as timings for the local and ordinary CCSDT(Q) calculations with the cc-pVDZbasis set using ε = 10−5.

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104111-13 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

it is expected that the error remains below 20% for largermolecules. Thus the relative error of the triples contributionis smaller than 25% that we have found for the perturbativetriples calculations at the same ε value. As a consequenceof the different accuracy for the perturbative and full triplescontributions, the difference of the local CCSDT and localCCSD(T) energies is far from the exact value. As the lowerpanel of Fig. 6 shows, the computation time of the integraltransformations is significantly smaller than that of the localCCSDT iterations, and the full calculation time is roughly alinear function of the system size for molecules of this size.All in all, the speedup with respect to the ordinary CCSDTmethod is dramatic.

The results for the perturbative quadruples contributionsare shown in Fig. 7. The computation time is dominated by thecalculation of the quadruples correction, which scales linearlywith the system size, and the costs of the integral transforma-tions are negligible. It is hard to judge the accuracy of theresult since the (Q) contribution calculated without approx-imations is only available up to n = 4. For normal butane,the relative error is 15.7%. Considering the relative errors ofthe local CCSD, CCSD(T), and CCSDT approaches, we donot expect the relative error to be larger than 20% for largerhydrocarbons. The quadruples corrections are rather small,e.g., 2.1 and 4.2 mEh for decane and icosane, respectively.Thus, considering the small magnitude of the quadruples cor-rections and the numerical experience that the CCSDT(Q)method generally overestimates the contribution of quadrupleexcitations70 by 10%–20%, while the present local approachunderestimates the (Q) correction, a relative error of 20% isacceptable.

The CCSD and CCSD(T) energies for some quasi-two-and three-dimensional hydrocarbons were also calculatedwith double- and triple-ζ basis sets to investigate how the di-mension of the local interacting subspaces and the accuracy ofthe approximations are influenced. The results for cyclohex-ane, adamantane, decaline, diamantane, and dodecahedraneare compiled in Table II and compared to the correspondingvalues for alkanes with the same number of carbon atoms. Asto the number of electrons in the most populated subspace,one can find that for the two- and three-dimensional hydro-carbons, as expected, a larger portion of the electrons aretreated explicitly than for the normal alkanes. It is also truethat for these molecules the maximum number of orbitals inthe local interacting subspaces is considerably larger than thatfor the linear alkanes. The relative errors of the calculationsare in the same range for the one and the higher dimensionalmolecules at each level of approximation, although, the errorsfor the two- and three-dimensional compounds are slightlylarger. Similar to the normal alkanes, the errors in the triple-ζbasis set are larger than with the double-ζ basis set. The num-ber of correlated electrons in the local interacting subspaceseems to be basis set independent, while significantly moreorbitals are required in the larger basis set.

To assess the performance of the LNO-CIM CC methodfor molecules other than saturated hydrocarbons, we alsocarried out test calculations for clusters of water molecules,which are frequently used as test systems for local correla-tion approaches. The results of the test calculations at CCSD

and CCSD(T) levels of theory with the cc-pVDZ basis setare collected in Table III. Considering the number of elec-trons and the number of orbitals in the local interacting sub-spaces as a function of the system size, one can find that thesedimensions are saturated for ε = 10−5 and 10−6 values forthe investigated molecules and tend to saturate for ε = 10−7.The relative errors for a given ε are similar for the variouswater clusters, slightly larger for larger systems, but seemto saturate with the system size. For the water clusters, therange of the relative errors in the case of the CCSD calcu-lations are 1.7%–1.9%, 0.3%–0.4%, and 0.05%–0.06% forε = 10−5, 10−6, and 10−7, respectively, while the same valuesfor the two- and three-dimensional hydrocarbons are 1.8%–2.8%, 0.2%–0.35%, and 0.01%–0.02%. That is, comparingthe relative errors of the alkane and the water cluster calcula-tions, we can see that the relative errors are similar and thissimilarity also holds for the errors of the CCSD(T) results.

Since, from the practical point of view, the calculation ofenergy differences is more important than that for the totalenergies discussed above, we computed reaction energies forsome reactions of the considered hydrocarbons using our lo-cal CC methods. The reactions, the contributions to the reac-tion energies at the CCSD and CCSD(T) levels with cc-pVDZand cc-pVTZ basis sets, and the relative errors of the localCC results with ε = 10−5, 10−6, and 10−7 can be found inTables IV and V. The CCSD correlation contributions to thereaction energies are two orders of magnitude smaller thanthose for the total energies, thus it is not surprising that theirrelative errors are significantly larger. The local CCSD ap-proach with ε = 10−5, which provides the correlation energywith a relative error of around 2%, approximates the CCSDcorrelation contribution for the reaction energies with errorslarger than 30% for most of the reactions. The use of ε = 10−6

offers more reliable reaction energies with relative errors ofaround 10%, while for ε = 10−7 the errors are below 3%.Thus, the errors are reduced roughly by a factor of three whendecreasing the threshold on the local populations by an or-der of magnitude. For the linear alkane chains, the relativeerrors for the reaction energies, just as for the total energies,converge with increasing system size. The errors for the reac-tions involving quasi-two- and three-dimensional compoundsare not larger than errors for the reactions which containsolely normal alkanes. One can observe again that the rela-tive errors for the triple-ζ basis set calculations are somewhatlarger. Concerning the absolute errors, our results suggest thatε = 10−6 is enough for 1 kcal/mol accuracy for a single bondbreaking, while results obtained with ε = 10−7 are accurateto at least 1 kJ/mol.

The value of the perturbative triples contributionsamounts to a couple of kJ/mol for the reactions involvingexclusively normal alkanes and it is in the 10–30 kJ/mol rangefor the other reactions. These corrections are even smallerthan the CCSD contributions and their relative errors arehigher. Both the reduction of the relative errors with decreas-ing ε values and their convergence with increasing system sizeare somewhat slower for the (T) contribution in comparison tothe CCSD correlation contribution. For ε = 10−5 the relativeerrors are sometimes larger than 60% and for ε = 10−6 theerrors are still in the 15%–35% range. Although the relative

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104111-14 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

TABLE II. The maximum number of electrons and orbitals in the local interacting subspaces as well as the relative errors of CCSD correlation energies and(T) corrections for quasi-two- and three-dimensional hydrocarbons compared to the corresponding normal alkane compounds with the same number of carbonatoms at ε = 10−5, 10−6, and 10−7 with the cc-pVDZ and cc-pVTZ basis sets.

Rel. err. of CCSD Rel. err. of (T)No. of electrons No. of orbitals (%) (%)

Molecule nb 10−5 10−6 10−7 10−5 10−6 10−7 10−5 10−6 10−7 10−5 10−6 10−7

cc-pVDZ basisC6H14 154 38 38 38 58 92 129 1.64 0.19 0.011 26.24 5.30 0.78C10H22 250 38 58 62 59 104 160 1.79 0.23 0.017 26.69 5.99 0.98C14H30 346 38 58 82 59 104 173 1.86 0.24 0.019 26.99 6.26 1.08C20H42 490 38 58 82 59 104 173 1.91 0.26 0.023 27.21 6.47 1.15Cyclohexane (C6H12) 144 34 36 36 55 88 118 1.77 0.19 0.009 26.29 5.79 0.88Adamantane (C10H16) 220 50 56 56 68 115 160 2.14 0.25 0.014 27.77 6.91 1.29Decalin (C10H18) 230 52 58 58 72 123 174 2.11 0.26 0.015 27.37 7.06 1.43Diamantane (C14H20) 296 60 76 76 79 141 203 2.42 0.31 0.017 28.58 7.80 1.67Dodecahedrane (C20H20) 380 62 94 100 80 153 235 2.70 0.35 0.021 29.22 7.70 1.43cc-pVTZ basisC6H14 376 38 38 38 72 128 205 2.46 0.32 0.025 27.00 7.69 1.81C10H22 608 38 58 62 72 142 236 2.62 0.36 0.031 27.68 8.20 2.02Cyclohexane (C6H12) 348 34 36 36 68 122 193 2.77 0.35 0.026 28.20 8.17 2.03Adamantane (C10H16) 524 50 56 56 79 151 242 3.33 0.42 0.031 30.21 9.12 2.54

TABLE III. The maximum number of electrons and orbitals in the local interacting subspaces as well as the relative errors of CCSD correlation energies and(T) corrections for (H2O)n water clusters at ε = 10−5, 10−6, and 10−7 with the cc-pVDZ basis set.

Rel. err. of CCSD Rel. err. of (T)No. of electrons No. of orbitals (%) (%)

n nb 10−5 10−6 10−7 10−5 10−6 10−7 10−5 10−6 10−7 10−5 10−6 10−7

10 240 20 52 74 32 66 110 1.68 0.30 0.045 29.05 7.00 1.8212 288 24 66 90 35 78 136 1.80 0.36 0.048 30.18 7.77 2.0814 336 26 68 104 36 78 143 1.80 0.35 0.054 29.98 7.71 2.1016 384 24 66 106 35 79 144 1.85 0.38 0.056 30.56 8.10 2.2618 432 26 70 116 36 78 149 1.86 0.37 0.060 a a a

20 480 26 80 144 36 87 176 1.85 0.37 0.059 a a a

22 528 26 74 132 36 82 165 1.89 0.39 0.062 a a a

aOrdinary CCSD(T) calculations are not feasible.

TABLE IV. The CCSD correlation contributions to the reaction energies and the relative errors of the local CCSD results at ε = 10−5, 10−6, and 10−6 withthe cc-pVDZ and cc-pVTZ basis sets.

Relative errorCCSD contribution (%)

Reaction (kJ/mol) 10−5 10−6 10−7

cc-pVDZ basisC2H6+H2 −→ 2CH4 –11.8 9.9 3.9 0.0C5H12+H2 −→ C2H6+C3H8 –20.2 32.8 6.8 0.6C10H22+H2 −→ 2C5H12 –20.9 34.1 10.7 1.6C15H32+H2 −→ C8H18+C7H16 –20.8 38.2 11.0 2.0C20H42+H2 −→ 2C10H22 –20.8 38.3 11.0 2.4Cyclohexane + C4H10 −→ Adamantane + 3H2 –61.0 25.1 4.5 0.4Adamantane + C4H10 −→ Diamantane + 3H2 –63.5 33.6 6.5 0.4Cyclohexane + Diamantane −→ Dodecahedrane + 6H2 –75.6 33.6 6.2 0.6cc-pVTZ basisC2H6+H2 −→ 2CH4 –9.6 –15.9 11.4 1.0C5H12+H2 −→ C2H6+C3H8 –19.0 34.7 8.6 0.9C10H22+H2 −→ 2C5H12 –20.2 50.1 12.1 2.0Cyclohexane + C4H10 −→ Adamantane + 3H2 –56.1 51.6 6.2 0.6

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104111-15 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

TABLE V. The (T) contributions to the reaction energies and the relative errors of the local (T) results at ε

= 10−5, 10−6, and 10−7 with the cc-pVDZ and cc-pVTZ basis sets.

(T) contribution Relative error (%)

Reaction (kJ/mol) 10−5 10−6 10−7

cc-pVDZ basisC2H6+H2 −→ 2CH4 –2.4 30.6 15.2 0.0C5H12+H2 −→ C2H6+C3H8 –4.3 51.7 25.2 6.7C10H22+H2 −→ 2C5H12 –4.6 49.7 32.0 8.5C15H32+H2 −→ C8H18+C7H16 –4.7 55.0 32.3 9.6C20H42+H2 −→ 2C10H22 –4.7 55.0 32.1 10.2Cyclohexane + C4H10 −→ Adamantane + 3H2 –14.7 39.0 21.5 7.2Adamantane + C4H10 −→ Diamantane + 3H2 –15.9 48.5 23.9 7.2Cyclohexane + Diamantane −→ Dodecahedrane + 6H2 –27.7 42.0 12.2 1.2cc-pVTZ basisC2H6+H2 −→ 2CH4 –2.5 –3.4 38.7 7.3C5H12+H2 −→ C2H6+C3H8 –4.7 48.7 31.2 11.6C10H22+H2 −→ 2C5H12 –5.1 66.6 35.5 14.2Cyclohexane + C4H10 −→ Adamantane + 3H2 –14.9 66.0 26.6 11.6

errors for ε = 10−6 are significant, the absolute errors are bet-ter than 1 kcal/mol. For ε = 10−7 we have relative errors oflarger than 10% for some reactions, but the absolute errors ofreaction energies are usually below 1 kJ/mol. Again, with thetriple-ζ basis set the errors of triples corrections are slightlylarger.

As the above examples show, the calculation of energydifferences is more challenging than the evaluation of totalenergies. Furthermore, the calculation of accurate results re-quires the application of the more expensive, higher order ap-proximations. For large molecules, the electrons which arelocalized to the neighborhood of a chemical bond involvedin a reaction are more important than the electrons locatedfar from the active center, thus it would be practical to de-scribe the regions far from the active center with a lower or-der method, and to treat only the neighborhood of the activecenter using a higher order theory.44, 71 As Li and Piecuch46

demonstrated the separation of the energy as a sum of thecontributions of the occupied orbitals ensures a rather simpleway to treat the different localized occupied orbitals at differ-ent levels of approximation. Using the local CC method de-fined in Sec. II A, we also tested the applicability of the aboveidea to the reaction C20H42 + H2 −→ 2C10H22. Around thebreaking C–C bond in the middle of the C20H42 chain, we de-fined nine shells containing more and more chemical bondsas it is shown in Fig. 8. The first shell contains the breakingC–C bond and the four C–H bonds of the central carbonatoms. The second shell contains the electrons in the first shelland the neighboring C–C bonds, in the third shell all the eightcentral C–H bonds are incorporated and so on. We calcu-lated the contribution of the CCSD correlation energy and thatfor the perturbative triples corrections to the reaction energywhen the calculation of these contributions was restricted tothe local interacting subspace of the chemical bonds in a givenshell of the C20H42 molecule as well as to the correspondingbonds of the C10H22 and the hydrogen molecules. The restof the molecules was treated at the HF level. The results forε = 10−5, 10−6, and 10−7 can be found in Table VI.

The restricted local CCSD and (T) contributions showfast convergence with respect to the unrestricted local CC re-sults. The relative errors of the restricted local CCSD and (T)contributions compared to the original local CCSD and (T)results are less than 10% for shell 4 and larger shells for eachε, where shell 4 contains the four central C atoms with all oftheir bonds. If shell 6 is incorporated into the correlation cal-culation, the errors with respect to the unrestricted local CCcalculations are in the 0.1 kJ/mol range.

From the practical point of view, it is more important tocompare the restricted local CC results to the contributionscalculated with the ordinary CC methods. For 10−5, the rel-ative errors of the local CCSD contributions to the reactionenergies are large, over 38%. These results give the correctmagnitudes of the CCSD contribution to the reaction energy,but the absolute errors are about 2 kcal/mol. Applying 10−6

or 10−7 as the threshold and restricting the local CCSD cal-culations to shell 4 allows to keep the error of the CCSD con-tribution below the 1 kcal/mol limit. For 10−7, the error canbe even smaller, less than 1 kJ/mol, if the next C atoms withtheir bonds are also included, i.e., the local CCSD calcula-tions are performed for shell 6. The (T) contribution, usingthe cc-pVDZ basis without further approximations, is onlyslightly larger than 1 kcal/mol. Hence, if the calculation ofthe triples corrections is also desired, the error of the localCCSD contribution should be kept below the magnitude of the(T) contribution, i.e., at least the use of ε = 10−7 is needed,and the local CCSD calculations should be performed forshell 6.

The local (T) contribution with the 10−5 threshold haveat least 55% relative error with respect to the ordinary calcula-tion. For ε = 10−6, the relative error of the (T) contributions isstill large, around 30%, and the absolute errors are larger than1.5 kJ/mol. Using ε = 10−7 the errors of the (T) correctionscan be kept below 1 kJ/mol with a local CCSD(T) calculationrestricted to shell 4, where the relative error is about 17%.

As a conclusion, we can say that to obtain the CCSD and(T) contributions to the reaction energy separately with an er-

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104111-16 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

H

C C

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

HH

C

H

H

H H H

H H

C

1

3

4

5

62

7

8

9

FIG. 8. The numbering of the shells for the C20H42 molecule. These shells are used to investigate the convergence of the reaction energy for the C20H42 + H2−→ 2 C10H22 reaction when the number of bonds described with a given correlation method is gradually increased. The C–C bond in shell 1 is broken duringthe reaction.

ror less than 1 kcal/mol the local CCSD calculations should beperformed at least for shell 4 with ε = 10−6, while the localCCSD(T) calculations can be performed using even smallershells with ε = 10−5. Since the error of the reaction energy atthe CCSD(T) level is the sum of the errors of CCSD and (T)contributions, to keep the error of the reaction energy below1 kcal/mol the error of the CCSD and the CCSD(T) correc-tions separately should be even smaller than 1 kcal/mol. Toreach the 1 kcal/mol error limit at the CCSD(T) level, theapplication of ε = 10−6 is required and the correlation cal-culations should be performed for the electrons in shell 6for both contributions, i.e., six carbon atoms and the corre-sponding hydrogens have to be described at the CCSD(T)level. To achieve the 1 kJ/mol accuracy in the reaction en-ergy, ε = 10−7 is required for both contributions, and shells9 and 7 should be treated at the CCSD and CCSD(T) levels,respectively.

Finally, we compared the LNO-CIM CC approach to theCIM CC method of Li et al.45 To that end we performed calcu-

lations for normal alkane chains, CnH2n+2, n = 12, 16, 20, 24,and 28, using the HF-optimized geometry and the 6-31G(d)basis set as in Ref. 45. The CCSD energies were convergedto 10−6 Eh, and the ε threshold was set to 4 × 10−7. The lat-ter value was chosen to match the accuracy, 0.1%, of “level2” CIM calculations of Li et al.45 The results are presented inTable VII.

Though a rigorous comparison is not possible becauseof the significantly different hardware used in the calcula-tions, our results are instructive. The solution of the CC equa-tions, concerning either the entire molecule or the largest localinteracting space, is less expensive with the present local CCapproach from n = 16, which is probably a consequence ofthe more efficient algorithm for the construction of the lo-cal orbital spaces. However, if the costs of the integral trans-formations are taken into account, the method proposed byLi et al. is superior. On the other hand, the evaluation ofthe triples corrections is roughly by an order of magnitudecheaper using our method, furthermore, our results are some-

TABLE VI. Convergence of the contributions of the CCSD correlation energies and (T) corrections (in kJ/mol) to the reaction energy for the C20H42

+ H2 −→ 2 C10H22 reaction when the number of bonds described at the CCSD and CCSD(T) levels is gradually increased. The relative errors with respect tothe ordinary CC results are shown in parentheses. The calculations have been performed with the local CC method using ε = 10−5, 10−6, and 10−7, and thecc-pVDZ basis set. For the definition of the shells, see Fig. 8.

Unrestricted OrdinaryShell 1 2 3 4 5 6 7 8 9 local CC CC

CCSD –20.810−5 –4.9 –8.3 –10.5 –11.8 –12.1 –12.6 –12.7 –12.7 –12.8 –12.9

(76.3) (60.1) (49.5) (43.5) (42.0) (39.6) (39.1) (38.9) (38.7) (38.3)10−6 –7.6 –11.9 –15.7 –17.2 –17.8 –18.1 –18.3 –18.4 –18.5 –18.6

(63.7) (43.1) (24.9) (17.5) (14.6) (13.2) (12.1) (11.8) (11.4) (11.0)10−7 –8.2 –12.9 –16.9 –18.6 –19.4 –19.8 –20.1 –20.2 –20.3 –20.4

(60.7) (38.3) (19.2) (10.8) (7.0) (5.0) (3.8) (3.2) (2.8) (2.4)(T) –4.710−5 –1.0 –1.7 –1.8 –2.0 –2.0 –2.1 –2.1 –2.1 –2.1 –2.1

(78.2) (64.3) (61.1) (56.8) (57.0) (55.1) (55.1) (55.1) (55.1) (55.0)10−6 –1.5 –2.3 –2.8 –3.0 –3.1 –3.1 –3.2 –3.2 –3.2 –3.2

(69.0) (51.6) (41.0) (35.9) (34.4) (33.5) (32.5) (32.4) (32.3) (32.1)10−7 –1.9 –3.0 –3.6 –3.9 –4.0 –4.1 –4.2 –4.2 –4.2 –4.2

(59.3) (36.7) (24.2) (17.2) (14.1) (12.0) (11.0) (10.4) (10.3) (10.2)

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104111-17 A general-order local CC method J. Chem. Phys. 135, 104111 (2011)

TABLE VII. Comparison of the LNO-CIM CC method and the CIM CC approach of Li et al. for normal alkanes CnH2n+2.

Canonical CC LNO-CIM CIM CCa

CPU time Relative error CPU time Relative error CPU time(min)bc (%) (min)b (%) (min)d

CCSD correl. Triples CCSD correl. Triples Integral CCSD correl. Triples CCSD correl. Triples CCSD correl. Triplesn energy corr.e energy corr.e transf. energyf corr.ef energy corr.g energyf corr.fgh

12 232 784 0.08 2.41 60 223 (52) 220 (49) 0.08 3.45 215 (39) 1381 (240)16 961 4273 0.09 2.55 194 349 (52) 345 (49) 0.10 4.93 482 (65) 2787 (470)20 3437 17306 0.10 2.65 613 476 (52) 469 (49) 0.10 ... 1460 (88) 7330 (413)24 8946 51581 0.10 2.71 1985 603 (52) 594 (49) 0.11 ... 1718 (92) 8680 (301)28 21511 142553 0.10 2.75 5460 730 (52) 720 (49) 0.11 ... 2312 (58) 12646 (360)

a“Level 2” CIM results from Ref. 45.bMeasured on Intel Xeon E3110 3.00GHz CPUs.cCanonical CC calculations for n= 20, 24, and 28 were not feasible with the above computer hardware. The corresponding calculations were carried out on machines equippedwith two quad-core Intel Xeon E5620 2.40GHz CPUs and 32 GB of main memory, and, for comparison, the CPU times were scaled based on the CPU times measured on the twoarchitectures for C16H34.dMeasured on Altix 3700Bx2 system of SGI equipped with 1.6 GHz Itanium2 CPUs.eCCSD(T) triples correction.fNumbers in parentheses are the CPU times required for the largest local interacting subspace.gCR-CC(2,3) triples correction (Ref. 45).hCPU times were scaled by a factor of 0.5 because the evaluation of the CR-CC(2,3) triples correction is by a factor of two more expensive than that for the CCSD(T) triplescorrection (Ref. 45).

what more accurate. Even if we consider the costs of the inte-gral transformations and the solution of the CCSD equations,we can conclude that our approach, at least for the currenttest systems, is significantly less expensive for the calcula-tion of the CCSD(T) energy than the method of Li and co-workers. An even higher speedup is expected for higher orderCC methods because the costs associated with the solution ofthe CC equations in a local interacting subspace depend onthe eight or higher power of the size of the subspace.

IV. CONCLUSIONS

We have presented a local CC approach, termed LNO-CIM CC, relying on the CIM idea of Li, which has been ex-tensively studied by Piecuch and co-workers. In our method,the CC equations are solved in the low-dimensional local in-teracting subspaces, thus the costs of the correlation calcu-lations scale as a strictly linear function of the system size.On the other hand, the construction of the local interactingsubspaces is based on a correlation argument which requiresthe computation of local density matrices from a first-orderperturbative wave function ansatz as well as a full integraltransformation and integral transformations from the full MObasis to the local interacting subspaces. Due to these compo-nents of the algorithm, the local CC method is at the presentstage not linearly scaling since its costs are dominated by theintegral transformations with an L5 dependence on the sys-tem size. Comparing the L5 scaling of the local CC methodwith that of the ordinary CC approaches, which is L6 for thecheapest case, for large molecules the application of the lo-cal CC method becomes economical. Although the integraltransformation has an L5 scaling, the cost of this step canbe small or even negligible compared to the solution of theCC equations, especially for high-order CC calculations or

for accurate local CC calculations with large local interactingsubspaces.

In the framework of the LNO-CIM CC method for eachlocalized occupied orbital, its local interacting subspace isconstructed using a single parameter ε, the threshold for thelocal populations. We have tested the performance of themethod for various ε’s calculating approximate CC correla-tion contributions to total energies and reaction heats. Calcu-lations with the cc-pVDZ and cc-pVTZ basis sets show thatthe number of electrons in the local interacting subspaces doesnot depend on the basis set size, but the number of virtualorbitals increases with the size of the basis set. The numberof electrons and the number of orbitals in the local interact-ing subspaces saturate with increasing system size. We havedemonstrated that the CCSD correlation energy and the (T)correction can be cost-efficiently approximated with the lo-cal CC method for larger compounds. The above contribu-tions can be calculated with relative errors of better than 3%and 30%, respectively, even with a loose ε threshold of 10−5,and the relative errors are reduced by an order of magnitudewhen decreasing ε by a factor of ten. We have also calculatedhigher-order CC corrections using ε = 10−5 and found thatthe speedup with respect to the parent methods is dramaticwhile the loss in accuracy is acceptable. Our local CC methodhas been successfully applied to calculate reaction energiesand we have found that the relative errors of the approximatereaction energies are larger than those for the total energies,i.e., to obtain accurate energy differences smaller ε’s shouldbe used. Nevertheless, an ε value of 10−6 seems to be suffi-cient to get reaction energies with 1 kcal/mol, while absoluteerrors of better than 1 kJ/mol are expected with 10−7. We havealso showed that the approach allows for the separate descrip-tion of the electrons localized close to a chemically importantfragment of the molecule using more sophisticated correlationmethods, while the remaining part of the molecule can be de-

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104111-18 Z. Rolik and M. Kállay J. Chem. Phys. 135, 104111 (2011)

scribed with a less accurate correlation method or even at theHF level.

The promising results obtained with the present imple-mentation motivate the further development of the method toachieve (near) linear scaling with the system size while keep-ing the essential parts of the approach. To eliminate the L5

scaling, the local density matrices should be constructed di-rectly in the AO basis using a local algorithm which makesuse of the spatial distance of the AOs. This would allow foravoiding the full integral transformation. To reduce the costof the second integral transformation, the transformation fromthe AO basis to the local interacting subspaces, the implemen-tation of local integral transformation is desirable. Finally, thescaling of the integral calculation and HF-SCF steps shouldalso be reduced. The research along these lines are in progressin our laboratory.

ACKNOWLEDGMENTS

Encouragement by and discussions with Professors PéterPulay, Wilfried Meyer, and Frank Neese are gratefully ac-knowledged. Financial support has been provided by theEuropean Research Council (ERC) under the European Com-munity’s Seventh Framework Programme (FP7/2007-2013),ERC Grant Agreement No. 200639, and by the HungarianScientific Research Fund (OTKA), Grant No. NF72194.

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