General implementation of the relativistic coupled-cluster methodHuliyar S. Nataraj, Mihály Kállay, and Lucas Visscher Citation: J. Chem. Phys. 133, 234109 (2010); doi: 10.1063/1.3518712 View online: http://dx.doi.org/10.1063/1.3518712 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v133/i23 Published by the American Institute of Physics. Related ArticlesEffect of microhydration on the guanidiniumbenzene interaction J. Chem. Phys. 135, 214301 (2011) Dispersion interactions in density-functional theory: An adiabatic-connection analysis J. Chem. Phys. 135, 194109 (2011) Basis set convergence of the coupled-cluster correction, MP2CCSD(T): Best practices for benchmarking non-covalent interactions and the attendant revision of the S22, NBC10, HBC6, and HSG databases J. Chem. Phys. 135, 194102 (2011) The electronic spectrum of the previously unknown HAsO transient molecule J. Chem. Phys. 135, 184308 (2011) Accurate ab initio ro-vibronic spectroscopy of the 2 CCN radical using explicitly correlated methods J. Chem. Phys. 135, 144309 (2011) 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 133, 234109 (2010)

General implementation of the relativistic coupled-cluster methodHuliyar S. Nataraj,1 Mihály Kállay,1,a) and Lucas Visscher2

1Department of Physical Chemistry and Materials Science, Budapest University of Technology and Economics,Budapest P. O. Box 91, H-1521 Hungary2Amsterdam Center for Multiscale Modeling, VU University Amsterdam, De Boelelaan 1083,1081 HV Amsterdam, The Netherlands

(Received 9 September 2010; accepted 2 November 2010; published online 20 December 2010)

We report the development of a general order relativistic coupled-cluster (CC) code. Our implemen-tation is based on Kramers-paired molecular spinors, utilizes double group symmetry, and is appli-cable with the full Dirac–Coulomb and several approximate relativistic Hamiltonians. The availablemethods include iterative and perturbative single-reference CC approaches with arbitrary excitationsas well as a state-selective multi-reference CC ansatz. To illustrate the performance of the new code,benchmark calculations have been performed for the total energies, bond lengths, and vibrationalfrequencies of the monoxides of Group IVa elements. The trends due to the simultaneous inclusionof relativity as well as higher-order electron correlation effects are analyzed. The newly developedcode significantly widens the scope of the ab initio relativistic calculations, for both molecules andatoms alike, surpassing the accuracy and reliability of the currently available implementations in theliterature. © 2010 American Institute of Physics. [doi:10.1063/1.3518712]

I. INTRODUCTION

The development of relativistic theories to molecularsystems has been an important thrust in the field of quan-tum chemistry over the years (for a recent review see, e.g.,Ref. 1). A fully relativistic formalism would be symmet-ric with respect to both positive (electronic) and negative(positronic) energy states and intrinsically requires a four-component description of the wave function. Since applica-tions in physics and chemistry below the sub-MeV energyscales hardly involve the positronic degrees of freedom, how-ever, it usually suffices to make the so-called no-pair approx-imation and exclude those degrees of freedom at some stagein the calculations. This is often done prior to the determina-tion of molecular orbitals and combined with a neglect of pic-ture change in the two-electron interaction to reduce computa-tional costs. A more rigorous but also more costly alternativeis to first determine four-component molecular orbitals andonly invoke the no-pair approximation when treating electroncorrelation. In both cases one finally obtains an effective two-component description of the wave function in which only theelectronic degrees of freedom are accounted for. With severalways2–9 to transform the four-component Dirac Hamiltonianinto two-component Hamiltonians and the possibility to treatthe spin-free (scalar relativistic) and the spin-dependent (spin-orbit coupled) relativistic effects separately,10 a large numberof both two- and four-component relativistic approximationshave been proposed and implemented. Nevertheless, practicalapplications of two- or four-component calculations that in-clude spin-orbit coupling from the start are still rather scarce,particularly for large molecules, as this leads to algorithmsthat are an order of magnitude more costly than the standardalgorithms of quantum chemistry.

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

Although, the Dirac–Hartree–Fock Hamiltonian aug-mented with the two-electron Coulomb and Breit or Gaunt in-teraction terms takes into account the relativistic effects quitecomprehensively, considering the electron correlation effectstogether with the relativistic effects on an equal footing isindispensable for reliable description of the electronic struc-tures and spectroscopic properties of the molecules contain-ing heavier atoms. However, due to the computational costs oftreating both effects many of the published calculations treateither the relativistic effects or the correlation effects, quiteoften both, only approximately.

The relativistic correlation methods reported in the liter-ature include the Kramers-restricted closed shell CC theorywith single and double excitations (CCSD),11 Kramers-unrestricted open-shell CCSD theory with partial triple ex-citations [CCSD(T)],12 and the multi-reference Fock-spaceCCSD(T) theory13; all three in conjunction with the two-and four-component Dirac–Coulomb–(Gaunt) reference wavefunctions by Visscher et al., the two-component CCSD andCCSD(T) implementation using effective core potentials byLee et al.,14 the valence universal Fock-space CC methods de-veloped for atoms by Eliav et al.,15, 16 by Chaudhuri et al.,17, 18

the configuration interaction (CI) method with a general-ized active space concept built over the Kramers-restrictedmulti-configuration self-consistent-field (KR-MCSCF) refer-ence wave function by Fleig et al.,19–22 the second-orderMøller–Plesset perturbation theory,23–25 the generalizedmulti-configurational quasi-degenerate perturbation theory(MCQPT) by Miyajima et al.,26 the complete active-spacesecond-order perturbation theory with the Dirac–CoulombHamiltonian (DC-CASPT2) by Abe et al.,27 relativistic den-sity functional theory (DFT) by Liu et al.,28 independently byHirao et al.,29 Saue and Helgaker,30 and Quiney et al.,31–34

the time-dependent DFT for excitation energies developedby Liu and co-workers35–38 and Saue and co-workers,39 the

0021-9606/2010/133(23)/234109/13/$30.00 © 2010 American Institute of Physics133, 234109-1

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234109-2 Nataraj, Kállay, and Visscher J. Chem. Phys. 133, 234109 (2010)

relativistic quantum Monte Carlo (QMC) method using thezeroth-order regular approximation (ZORA) by Nakatsukaet al.40 Very recently a two-component closed-shell CCSD(T)approach using the relativistic effective core potentials withspin-orbit coupling included in the post-Hartree–Fock step41

and the application of direct perturbation theory (DPT) tocompute the relativistic corrections to some electrical proper-ties of third and fourth row molecules42 have been reported byGauss and co-workers. Thus, a variety of post-HF/DF meth-ods are developed over either quasi- or full-relativistic Hamil-tonians. So far, however, not much attention has been paid tothe treatment of higher-order correlation effects.

The comprehensive treatment of the electron correla-tion effects requires methods beyond those generally applied:DFT, finite-order MBPT, restricted CI, truncated CC meth-ods. Most of the molecular applications such as equilibriumgeometries, vibrational frequencies, transition properties, etc.,however, demand the inclusion of higher-order excitations,in particular, the quadruple excitations43–45 if high accuracyis needed. Nonetheless, the prohibitive scaling of the fullCC/CI methods and the associated high computational costsmake them impractical for the application to heavier sys-tems. Therefore, the development of more efficient approx-imate many-body methods based on the CC formalism arevery desirable and timely.

The successful attempts made in the past in this direc-tion include the automated string-based techniques combinedwith the diagrammatic many-body perturbation theory devel-oped by one of us in the last decade,45–50 which solve forthe arbitrary high excitations in CC and CI methods. Thegeneral implementation of CC programs has also been de-veloped by Hirata and co-workers51–54 using computerizedsymbolic algebra called tensor contraction engine and string-based automated program generation techniques by Olsen andco-workers.55 The relativistic extensions of the general orderrelativistic CC codes, known to our knowledge, are those re-ported recently by Hirata et al.56 available in the UTChempackage57 and by Fleig et al.58 Although, the implementationin Ref. 56 offers a wide range of correlation methods to beused with the relativistic reference wave function, their rela-tivistic treatment has several intrinsic limitations as it includescorrections only due to relativistic effective core potentials(RECPs) and spin-orbit couplings. Thus, their inclusion ofrelativistic effects is not as rigorous and complete as in thepresent work. The implementation in Ref. 58 highlights onthe state-specific multireference CC implementation general-ized to four-component relativistic formalism. However, thecomputational scaling of their method is nn+2

o nn+2v as against

the conventional CC scaling of nno nn+2

v , where n is the highestexcitation and no and nv are the number of occupied and vir-tual orbitals, respectively. The expensive scaling, therefore,limits the efficiency of their program while handling morethan 12 correlated electrons and basis sets of size larger thantriple-zeta quality. In contrast, our single- and state-specificmulti-reference relativistic CC implementation has an opti-mal scaling and offers flexibility in the choice of the avail-able methods which include not only the iterative CC and CImethods generalized to arbitrary levels of excitations but alsogeneral order perturbative CC methods.

In the current paper, we discuss the modifications re-quired for the conversion of the general order nonrelativis-tic MRCC program developed by Kállay and co-workers59 tohandle the two- and four-component relativistic Hamiltoni-ans. We also would like to announce the newly developedinterface which couples the relativistic version of the MRCC

suite59 with the local version of the relativistic quantum chem-istry program suite DIRAC.60 These two program packagestogether have a great potential in handling the relativistic ef-fects, both scalar relativistic and spin-orbit effects, togetherwith the correlation effects to arbitrary levels of higher-orderexcitations. This, therefore, widens the scope of the high pre-cision relativistic correlation calculations and provides thehighest levels of accuracy for both molecules and atoms alike,in the future.

In order to demonstrate the capabilities of the newlydeveloped relativistic CC code, we have chosen to studythe relativistic contributions and the convergence trends ofthe correlation effects in the diatomic oxides of Group IVaelements. Although, the considered monoxides have longbeen the subjects of relativistic and correlation studies, notmuch work has been done on the spectroscopic propertiesof these molecules. Various theoretical investigations haveonly looked in to the electric dipole moment of the heaviermolecules. The dipole moment of PbO has been calculatedusing pseudo-potentials in conjunction with core-polarizationpotentials and spin-orbit terms.61 The dipole moments of ox-ides and sulphides of Pb and Sn have been computed us-ing the nonrelativistic Hamiltonian with the mass velocityand Darwin terms (MVD)62 and also using the Douglas–Kroll approximation63 by Kellö et al. The scalar Douglas–Kroll approximation up to fifth order in the external potentialhas been employed in the study of various spectroscopic pa-rameters of SnO.64 Geometries and dipole moments of theGroup IVa monoxides such as GeO, SnO, and PbO havebeen studied by Dyall using various relativistic and nonrel-ativistic Hamiltonians65 at the uncorrelated level of the the-ory. The essence of most of these and similar calculations oninterhalogens,66 dihalogens,67 hydrogen halides68 is that thescalar relativistic methods in which the spin-orbit correctionsare ignored can be quite inadequate, and also the simultane-ous treatment of both relativistic and correlation effects arenecessary in order to produce accurate results. Thus, the re-liability of the results of most of these approximate meth-ods needs necessarily be questioned even if they agree withthe experimental results. We, therefore, have systematicallystudied the convergence patterns of the results with respectto the correlation as well as the relativistic effects. The com-parative study of the relativistic and nonrelativistic results forthe entire series of light through heavy molecules providescrucial insights for understanding the progressive importanceof the relativistic effects with respect to the increase in theatomic number especially when high accuracies are aimedfor. These are the first calculations to the best of our knowl-edge where the full relativistic effects through the Dirac–Coulomb Hamiltonian at the SCF level and the higher-ordercorrelation effects including excitations up to quadruples inthe post-SCF calculations are considered for these set ofmolecules.

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234109-3 General implementation of relativistic CC J. Chem. Phys. 133, 234109 (2010)

II. THEORY AND IMPLEMENTATION

The starting point of our general order relativistic CC im-plementation is the string-based many-body code developedby Kállay and co-workers45, 47, 59, 69 as well as the relativisticCCSD and CCSD(T) methods of Visscher et al.11, 12 as imple-mented in the DIRAC suite of quantum chemistry programs.60

The specialty of the string-based technique is the useof strings of spin-orbital indices rather than the indices foraddressing of the wave function parameters, integrals, andintermediates.45 In the nonrelativistic case strings are orderedsets of spin-orbital indices written as

P = p1 p2 p3 · · · (p1 < p2 < p3 < · · ·). (1)

Relying on this definition cluster amplitudes can be expressedas two index quantities for arbitrary excitation levels and writ-ten as tAI where A and I are the strings of virtual and occupiedspin-orbitals, respectively. Similarly, molecular orbital (MO)integrals and intermediates are treated as four-index tensors inthe form of W CA

KI where A and I are the strings of virtual andoccupied fixed spin-orbitals (i.e., orbital labels determined bythe projecting determinants in the equations), and C and Kare the strings of virtual and occupied free labels (i.e., indiceswhich are summed over when calculating the correspondingmatrix elements). For the manipulation of quantities storedin terms of strings as well as for the derivation of the work-ing equations automated tools have been elaborated, whichhighly facilitate the implementation of many-body methods ina general way, independently of the excitation rank of the de-terminants included in the wave function. In the following wediscuss the modifications required to enable relativistic calcu-lations with our code.

Our relativistic implementation follows the lines put for-ward by Visscher11, 12, 70 for the CCSD and CCSD(T) meth-ods. We invoke the no-pair approximation to simplify ourHamiltonian which intrinsically includes the spin-orbit cou-pling. We also presume that the MOs are Kramers-paired andtransform according to the irreducible representations (irreps)of the corresponding double groups. However, we do not im-pose any time-reversal restriction on the cluster amplitudes.Consequently we should refer to this approach as Kramers-unrestricted relativistic CC method, though this terminologyis somewhat misleading since for a closed-shell system thewave function has the correct time-reversal symmetry. TheKramers adaptation of the CC method for the general case(see the recent paper by Fleig71), which is closely related tothe spin-adaptation problem of the nonrelativistic theory, isnot a trivial task and not considered in this publication.

In the relativistic case the spin-orbitals are replaced byspinors, but the basic ideas of the string-based technique areobviously valid for the strings of spinor indices as well. Con-sequently the fundamental structure of the code is not af-fected, only some minor modifications are necessary. First,one requires the transformed MO integrals of the relativisticHamiltonian employed and in order to meet that purpose ourcode has been interfaced to the DIRAC package,60 which al-lows us to use a wide variety of relativistic Hamiltonians. Sec-ond, one should take into account the decreased permutationalsymmetry of the integrals. In the nonrelativistic quantum

chemistry—provided that no external magnetic field isapplied—the MOs are real and thus the two-electron inte-grals have eight-fold permutational symmetry. In the relativis-tic case the orbitals are complex, and even if the integrals canbe made real, two symmetries will be lost, and hence only afourfold symmetry can be utilized. Since, in our code, the for-mulas are derived and evaluated in terms of antisymmetrizedtwo-electron integrals, which only have the fourfold permu-tational symmetry by construction, this change do not seri-ously affect our implementation. In practice the CC code doesnot need to be changed, only the integral sort algorithm con-structing the integral lists of the normal-ordered Hamiltonian(see, e.g., Ref. 45) requires some modifications. Third, oneshould consider the lack of spin integration. For the nonrela-tivistic MOs the spatial and spin functions can be separated,and hence during the calculation of the corresponding inte-grals the spatial and spin integrations can be performed inde-pendently. Consequently only certain integrals with a definitenumber of alpha and beta indices survive. For Hamiltonianscontaining the spin-orbit interaction this favorable property islost, and in the general case all the combinations of spinor in-dices are allowed. Thanks to our flexible tools the appearanceof the new integral lists only implies the modification of theintegral sort code again, but the CC codes remain intact sincethe corresponding new terms in the equations are automati-cally generated and no modification to the CC code is nec-essary. At last, an important difference in the relativistic caseis that the symmetry group of the Hamiltonian is not a pointgroup like in the nonrelativistic case but its double group,72

hence the use of double group theory is needed if the costs ofthe calculations are intended to be decreased. While the im-plementation of the former three points mentioned above israther technical but straightforward and does not require ex-tensive changes to our CC code, however the double-groupsymmetry adaptation deserves somewhat more attention.

The notable difference between the double groups andthe conventional point groups used in the nonrelativistic quan-tum chemistry is the existence of a new symmetry operation,the rotation by 2π .72 As a consequence new types of repre-sentation, the so-called fermion irreps appear, and the spinorsalways transform according to these irreps. The double-groupadaptation of our code has been carried out relying on theideas of Visscher.12, 70 We use spinors that are symmetry func-tions of the largest Abelian subgroup of the double group ofthe molecule. As it was demonstrated70 these groups havethe favorable property that the integrals and consequently,all other quantities in a correlation calculation can be madereal, which results in a factor of 4 reduction in the scalingof the method. The aforementioned double groups have one-dimensional fermion irreps which are related by complex con-jugation. If the spinors are Kramers-paired, the time-reversedconjugate of a function belonging to an irrep transforms ac-cording to its complex conjugate irrep.

On the one hand, the transition from conventional pointgroups to double groups necessitates the replacement of thegroup multiplication tables used in nonrelativistic codes bythose for the double groups. On the other hand, special at-tention must be paid to the complex valued irreps of doublegroups. Because of the complex irreps—in contrast to the real

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234109-4 Nataraj, Kállay, and Visscher J. Chem. Phys. 133, 234109 (2010)

irreps of the nonrelativistic theory—particular attention hasto be paid on whether a spinor is used in a bra- or a ket-state.Thus, when determining the symmetry of any quantity, thecomplex conjugate of the irreps for the spinors used in the bra-function must be considered. For instance, a two electron inte-gral 〈pq|rs〉 belongs to the irrep �∗

p ⊗ �∗q ⊗ �r ⊗ �s , where

p, q, r , and s denote spinors, �p is the irrep for spinor p, andthe asterisk refers to complex conjugation. As cluster ampli-tudes are the matrix elements of the cluster operator, here thevirtual indices of the amplitudes stem from the bra state, thecorresponding irreps have to be conjugated. In general, for allindices associated with the virtual quasi-creation operators aswell as the occupied quasi-annihilation operators the complexconjugate irreps must be considered.

In our string-based technique all the quantities are ex-pressed in terms of strings, and the symmetry of a quantityis given by the direct product of the irreps of the strings. Theirrep for a string defined by Eq. (1) can be calculated as

�P = �p1 ⊗ �p2 ⊗ �p3 ⊗ · · · . (2)

According to the above rules the tAI cluster amplitudes haveto satisfy the following condition to be nonzero:

�∗A ⊗ �I = �1, (3)

where �1 is the totally symmetric irrep of the double group.For an intermediate W CA

KI the following restriction applies:

�C ⊗ �∗K ⊗ �∗

A ⊗ �I = �1. (4)

In order to satisfy Eqs. (3) and (4) two important changes havebeen made. First, in our code, the cluster amplitudes and in-termediates are stored in a symmetry-blocked structure, i.e.,the elements of the tensors are grouped into blocks accord-ing to the irreps of the indexing strings. The addressing ofthese quantities has been reorganized, and the routines cal-culating the block addresses have been rewritten to conformto the above equations. Second, all the routines that manipu-late cluster amplitudes and intermediates, that is, perform thetransposition or the contraction thereof have been modified.As discussed in Ref. 45, in our algorithms each loop runningover strings is preceded by another loop over the correspond-ing irrep, and the latter are restricted to ensure that only non-vanishing elements are treated. This loop structure has beenmodified to satisfy the new criterion, viz. Eqs. (3) and (4).

The aforementioned changes have been implementedfor both iterative and perturbative CC approaches. Currentlythe following methods are available with relativistic Hamil-tonians: iterative single-reference CC approaches includingarbitrary excitations (i.e., CCSD, CCSDT, CCSDTQ, . . . );iterative multireference CC approaches for arbitrary com-plete active spaces and excitation levels using the state-selective ansatz of Adamowicz and co-workers;47, 73, 74 severalperturbative single-reference CC approximations for arbi-trary excitation levels proposed in Refs. 69 and 75 (in-cluding CCSD[T], CCSDT[Q], CCSDTQ[P], . . . ; CCSD(T),CCSDT(Q), CCSDTQ(P), . . . ; CCSDT-1a, CCSDTQ-1a,CCSDTQP-1a, . . . ; CCSDT-1b, CCSDTQ-1b, CCSDTQP-1b, . . . ; CCSDT-3, CCSDTQ-3, CCSDTQP-3, . . . ). It ispertinent to mention that the latter perturbative approxi-mations require a special treatment for open-shell systems

if restricted orbitals are used. As it was pointed out inthe nonrelativistic case75–77 the conventional restricted open-shell Hartree–Fock (ROHF) orbitals do not diagonalize theFock-matrix, and thus make the choice of the zeroth-orderHamiltonian ambiguous. To remedy this problem the useof semi-canonical orbitals was suggested, which diagonal-ize the occupied–occupied and virtual–virtual block of theFock-matrix and enable a consistent perturbation treatment.Translating this to the relativistic language would implythe transformation of Kramers pairs to an unrestricted ba-sis and destroy several favorable symmetry properties of theKramers-restricted formalism. Therefore, in line with Viss-cher’s relativistic CCSD(T) method12 we propose to go withKramers-paired orbitals and employ the formulas derived forthe semi-canonical ones. Although we know that this approx-imation is not entirely satisfactory from the perturbation the-oretical point of view, it yields however an error which isacceptable.

Concerning the treatment of relativity our program workswith several relativistic Hamiltonians implemented in theDIRAC suite60 including the Dirac–Coulomb, ZORA,78–80 theDouglas–Kroll,81 and the exact two-component (X2C)82, 83

Hamiltonians.It is worth mentioning that besides various CC methods

listed above, their corresponding single- and multi-referenceconfiguration interaction (CI) methods are also available foruse in our codes. We also remark that all the implementedmethods have been parallelized utilizing both shared- anddistributed-memory parallelism to speed up the executiontimes.

III. BENCHMARK CALCULATIONS

As a first application of the new relativistic code, wehave performed benchmark calculations for the total ener-gies, equilibrium bond lengths, and vibrational frequenciesof the monoxides of Group IVa, such as, CO, SiO, GeO,SnO, and PbO. Since relativistic CCSD and CCSD(T) codeswere previously available11, 12 and the effect of relativity onthe correlation contributions of up to perturbative triple ex-citations were also studied,66–68 we have focused on the it-erative triples and the quadruples increments [i.e., CCSDT-CCSD(T), CCSDT(Q)-CCSDT, and CCSDTQ-CCSDT(Q)]in the current work. These contributions were found to beessential for light atoms and molecules for high-accuracycalculations.84–90

We, therefore, have evaluated the contributions of iter-ative triple and quadruple excitations using the double- andtriple-zeta quality basis sets as the standard practice followedin high-accuracy calculations. The basis sets used in our cal-culations include the correlation consistent polarized core-valence double zeta (cc-pCVDZ) and triple zeta (cc-pCVTZ)basis sets of Dunning and co-workers91, 92 for the light atomssuch as C, O, and Si, and Dyall basis sets93, 94 of similar qual-ity for heavier atoms such as Ge, Sn, and Pb. All these basissets are available in the recent version of the DIRAC suite it-self. The relativistic basis sets given by Dyall are uncontractedwhereas the nonrelativistic Dunning basis sets are contracted.

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234109-5 General implementation of relativistic CC J. Chem. Phys. 133, 234109 (2010)

In order to be consistent, we have uncontracted the Dunningbasis sets in our calculations, for both relativistic and nonrel-ativistic cases alike. The use of uncontracted basis sets espe-cially for the diatomic molecules involving Sn and Pb in thetriple-zeta calculations require large computational resourcesand long execution times. In addition, the number of smallcomponent basis functions generated using the kinetic bal-ance condition will become too large, and considering the twoelectron integrals involving them becomes a burden. Hence,we have used an approximation proposed by Visscher95 inwhich the two electron integral contributions from the small–small (SS) components are neglected in the coupled clustercalculation. To obtain reliable total energies we did, how-ever, also run Hartree–Fock calculations in which the SS in-tegrals are included, yielding a correction that is includedin the reported coupled cluster energy. This simple scheme(similar to the 4DCG∗ scheme discussed by Sikkema et al.96)reduces the computational time and the memory require-ments of the calculations significantly with negligible loss ofaccuracy.

Guided by the observations reported in Ref. 97, we havefrozen the noble gas core and the highest-lying d shell elec-trons in the DZ basis sets, that is, we have only correlated the2s and 2p electrons of oxygen and the ns and np electronsof the Group IVa elements, altogether 10 electrons, in orderto reduce the computational costs further. In the TZ basis setthe last occupied d shell electrons of the heavy atoms werealso correlated in the post-SCF calculations, which amountsto correlating 20 electrons for GeO, SnO, and PbO. For theCO and SiO molecules, to keep the number of correlated elec-trons more or less constant and to perform as complete cal-culations as possible, no electrons were frozen for CO andonly the 1s electrons of O and Si were frozen for SiO. Thisresults in 14 and 18 correlated electrons for CO and SiO, re-spectively. During the core freezing the identification of the dshell orbitals of the heavy atoms and the 2s orbitals of oxy-gen does not pose any challenge for molecules up to SnO asthey are well separated and no orbital mixing is seen. How-ever, for PbO the selection of the core orbitals for freezingis less straightforward as the oxygen 2s orbitals mix signifi-cantly with the 5d orbitals of Pb, and hence, there are a fewmolecular orbitals with significant 2s character. In this par-ticular case we have frozen the ones with the lower orbitalenergy. Further, we have also set a maximum energy thresh-old for the high-lying virtuals in DZ and TZ calculations to be5 Eh and 10 Eh , respectively. Adhering to these limitations, wehave performed CC calculations up to CCSDTQ and CCSDTin DZ and TZ basis, respectively.

A series of test calculations have been performed for SiOto assess the accuracy of the results in view of the limita-tions considered above viz. freezing of the inner core elec-trons and truncation of higher virtual orbitals in the DZ ba-sis. The CCSD energy with the full basis set is obtained tobe −365.09 Eh where as freezing the noble gas cores of bothSi and O, that is, freezing n = 2 shell for Si and n = 1 shellfor O yields a CCSD energy of −364.77 Eh . Hence, the frozennoble gas core approximation overestimates the CCSD energyby 0.32 Eh . The truncation of higher virtuals above the thresh-old energy of 5 Eh results in an overestimation of 0.02 Eh .

Therefore, the total deviation of the energy from its originalvalue is 0.34 Eh for SiO in the DZ basis at the CCSD level ofthe theory due to the approximations considered above. Thiswould be even more significant for larger molecules and forlarger basis sets, however, our main aim in the current workis to compute the higher-order correlation contributions to thetotal energies and other spectroscopic properties and not theabsolute values themselves, and also much of the error causedby the above constraints is supposed to cancel in the higherorder calculations.

A Gaussian charge distribution for the nucleus has beenconsidered in the relativistic calculations, while, a pointcharge distribution has been used in the nonrelativistic cal-culations. However, the basis sets, geometries, symmetrygroups used in both calculations are exactly the same. Theexponents used for the Gaussian distribution of nuclearcharge are 680775029.29, 586314366.55, 434677488.23,252356133.99, 190677181.54, and 137688400.81 for C, O,Si, Ge, Sn, and Pb, respectively. We would like to remark thatthe nonrelativistic test calculations using a Gaussian chargedistribution instead of a point charge distribution showed onlynegligible change in the results of bond lengths and vibra-tional frequencies.

The total energies for the diatomic molecules consid-ered in this work have been calculated at the respective ex-perimental equilibrium bond lengths (re) taken from Ref. 98,which (in Å) are 1.128323, 1.509739, 1.624648, 1.832505,and 1.921813 for CO, SiO, GeO, SnO, and PbO, respectively.In order to calculate spectroscopic parameters such as theequilibrium geometry and the vibrational frequencies, fouradditional energy calculations have been carried out at fourpoints around re with the separation (r − re) of ± 0.02 Å and± 0.04 Å, and second-order polynomials have been fitted tothe resulting points on the potential energy curves. The har-monic frequencies are calculated for the most abundant iso-topes of the elements considered viz. C12, O16, Si28, Ge74,Sn120, and Pb208, whose nuclear masses (in amu) are, 12.0,15.99491, 27.97693, 73.92117, 119.90219, and 207.97665,respectively, which are taken from Ref. 99.

The relativistic calculations have been performed usingthe recently developed version of the MRCC program59 andits interface to the local version of the DIRAC08 program,60

whereas the nonrelativistic calculations have been performedusing the CFOUR program100 and its interface to MRCC pro-gram developed earlier. These two seamless interfaces enableus to perform higher-order relativistic and nonrelativistic cor-relation calculations, respectively.

We should emphasize at this point that the results, espe-cially those for quadruple excitations should be treated withsome caution because of the small number of correlated elec-trons and the small basis sets. The DZ basis set itself is rathersmall, and the quality of the one-electron basis is further wors-ened by the truncation of the virtual space. Furthermore, sincethe oxygen atom bears a partial negative charge in the heav-ier molecules, the use of augmented basis sets for oxygenwould be desirable for a quantitative description. Neverthe-less, our intention was to gain some insight into the behaviorof higher-order correlation contributions for heavier element,and CC calculations including up to quadruple excitations

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234109-6 Nataraj, Kállay, and Visscher J. Chem. Phys. 133, 234109 (2010)

are currently hardly possible with a larger number of basisfunctions. Thus we think that our conclusions are not defini-tive, but still instructive for high-accuracy calculations.

IV. RESULTS AND DISCUSSION

The calculated results of the total energies, equilibriumbond lengths, and harmonic vibrational frequencies togetherwith the relativistic and higher-order correlation contributionsfor the Group IVa monoxides are presented in Tables I–III, re-spectively. In columns four and five, the relativistic and non-relativistic quantities are presented, respectively, at differentlevels of the CC theory, both in DZ and TZ basis sets. Therelativistic contribution, i.e., the difference between columnsfour and five is shown in column six for each level of thetheory for all the quantities of interest. The behavior of thesingle-reference CC methods used in these calculations canbe studied with the aid of columns seven and eight in whichthe difference between the given and the preceding level ofthe theory is looked at in relativistic and nonrelativistic cases,respectively. The column nine in the tables gives the signifi-cance of relativistic effects in the hierarchy of CC methods.

We would like to emphasize that the correlation contribu-tions higher than the perturbative triples [i.e., CCSD(T)] arehighlighted in the current paper. The contributions of the iter-ative triples [i.e., CCSDT] have been studied in the TZ basisand the quadruples have been studied in the DZ basis. Beforediscussing the observed trends, we introduce a terminologyto be used hereafter. The correlation contribution due to par-tial triples [i.e., CCSD(T)-CCSD] will henceforth be referredto as (T)-contribution, [CCSDT-CCSD(T)] will be referredto as T-contribution, and the similar terminology follows for(Q)- and Q-contributions. The total correlation contributionsof triples and quadruples (i.e., [(T) + T] and [(Q) + Q]) arepresented in columns 10 and 11, respectively, and the rela-tive contribution of the perturbative approximations over thetotal triples and quadruples contributions for both relativisticand nonrelativistic cases are shown in the last two columns ofTables I–III.

In this work besides observing that the total energiesin the relativistic case are much lower compared to theirrespective nonrelativistic energies, as is well known in theliterature, we observe the contraction of bond lengths andthe reduction in the vibrational frequencies in the diatomicmolecules studied due to the inclusion of the relativistic ef-fects. The magnitude of the relativistic contribution to thetotal energy monotonically increases from 0.1 Eh for CO to1389.8 Eh for PbO. The contraction of bond length variesfrom 0.0002 Å for CO to 0.0306 Å for PbO, similarly, thechange in the vibrational frequencies falls in the range of2 cm−1 for CO to 81 cm−1 for PbO due to the influence ofrelativistic effects.

In the following three subsections we will discuss thetrends observed in the calculations of total energies, equi-librium molecular geometries, and harmonic vibrational fre-quencies separately due to the inclusion of relativistic andhigher-order correlation effects in detail. In the last sub-section, we will summarize our findings for each molecule

considered in this work individually and compare our resultswith the available calculations.

A. Total energies

The total energies for the diatomic molecules are pre-sented in Table I. From the table, we observe that the en-tire quadruples contribution is an order of magnitude smallerthan the entire triples contribution in the DZ basis for bothrelativistic and nonrelativistic cases alike. This factor is inconsistent with the previously reported observations in thenonrelativistic calculations by Kállay et al.45 Furthermore, theratio of [(Q)/Q] contributions is 10 for CO and five for SiOboth in relativistic and nonrelativistic cases, in line with theprevious nonrelativistic calculations.69 The difference in thehigher-order correlation contributions between the relativisticand nonrelativistic cases in the DZ basis are very small forall the molecules but PbO (for which the SO-coupling effectslead to qualitative changes in the bonding between the atoms).The ability of the (T) approximation to describe the effect oftriple excitations does show some dependence on the basisset. For the DZ basis we see that for the lighter molecules the(T) approximation recovers about 95% of the full triples con-tribution, both in the relativistic and the nonrelativistic case.For PbO the T-contribution is unimportant in the nonrelativis-tic calculation, while in the relativistic case the T-contributionis clearly larger than for the lighter elements. In the TZ-basissuch differences between the nonrelativistic and relativisticcorrelation contributions are less prominent and the relativeimportance of the T-contribution is also smaller.

In the heavier molecules, SnO and PbO the convergencewith excitation level is less quick in both relativistic and non-relativistic calculations. This is probably caused by the close-lying occupied and virtual orbitals, which make the moleculessomewhat multireference in nature. Another explanation ispossibly the limited amount of the higher angular momen-tum functions in the DZ basis of Sn and Pb, which makes thecorrelation treatment vulnerable to basis set incompletenesserrors.

Looking at the magnitudes of the relativistic contribu-tions to T-contributions we conclude that up to the third rowof the periodic table the nonrelativistic calculations will suf-fice for the iterative triples, while from fourth row onwardthe inclusion of relativistic effects may be necessary if oneis seeking an accuracy of ∼1 kJ/mol. For SnO and PbO theQ-contribution is larger than the T-contribution, particularly,in the nonrelativistic case, indicating the importance of higherexcitations for heavy elements. It is interesting to note that theCCSD(T) result in the nonrelativistic case of PbO is closer tothe CCSDT result and does not exhibit the erratic behaviorof the perturbative quadruples approach. The noticeable ob-servation that the relativistic effects have negligible influenceon the (Q)- and Q-contributions for molecules up to SnO sug-gests that one can safely perform nonrelativistic calculationsfor quadruples up to fourth row, however, fifth row elementsneed to be treated relativistically both for the perturbativeand full quadruples contributions. Nevertheless, from the thirdrow the use of the CCSDT(Q) method is not recommended,

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234109-7 General implementation of relativistic CC J. Chem. Phys. 133, 234109 (2010)

TA

BL

EI.

Tota

lene

rgie

san

dco

rrel

atio

nco

ntri

butio

nsto

tota

lene

rgie

sfo

rth

eox

ides

ofG

roup

IVa

(in

Eh).

Bas

isTo

tale

nerg

yC

orre

latio

nco

ntri

butio

na[(

T)+

T]

and

[(Q

)+Q

]co

ntri

butio

n%

cove

red

bype

rt.t

heor

y

Mol

ecul

ese

tM

etho

dR

el.

Non

rel.

Dif

f.R

el.

Non

rel.

Dif

f.R

el.

Non

rel.

Rel

.N

onre

l

CO

DZ

CC

SD–1

13.1

1689

2–1

13.0

4539

6–0

.071

497

––

–C

CSD

(T)

–113

.127

776

–113

.056

271

–0.0

7150

6–0

.010

884

–0.0

1087

5–0

.000

009

CC

SDT

–113

.128

305

–113

.056

800

–0.0

7150

5–0

.000

529

–0.0

0053

00.

0000

01–0

.011

413

–0.0

1140

595

95C

CSD

T(Q

)–1

13.1

2932

0–1

13.0

5781

2–0

.071

507

–0.0

0101

4–0

.001

012

–0.0

0000

2C

CSD

TQ

–113

.129

221

–113

.057

714

–0.0

7150

70.

0000

980.

0000

980.

0000

00–0

.000

916

–0.0

0091

411

111

1T

ZC

CSD

–113

.234

331

–113

.162

680

–0.0

7165

1–

––

CC

SD(T

)–1

13.2

5142

1–1

13.1

7975

5–0

.071

667

–0.0

1709

0–0

.017

075

–0.0

0001

5C

CSD

T–1

13.2

5164

8–1

13.1

7998

4–0

.071

665

–0.0

0022

7–0

.000

229

0.00

0002

–0.0

1731

7–0

.017

304

9999

SiO

DZ

CC

SD–3

64.7

4831

9–3

64.0

6779

3–0

.680

526

––

–67

3C

CSD

(T)

–364

.759

823

–364

.079

278

–0.6

8054

5–0

.011

504

–0.0

1148

5–0

.000

019

CC

SDT

–364

.760

658

–364

.080

114

–0.6

8054

5–0

.000

835

–0.0

0083

60.

0000

00–0

.012

339

–0.0

1232

093

93C

CSD

T(Q

)–3

64.7

6213

6–3

64.0

8158

8–0

.680

548

–0.0

0147

7–0

.001

475

–0.0

0000

3C

CSD

TQ

–364

.761

862

–364

.081

315

–0.6

8054

70.

0002

740.

0002

730.

0000

01–0

.001

203

–0.0

0120

112

312

3T

ZC

CSD

–365

.006

032

–364

.324

667

–0.6

8136

5–

––

CC

SD(T

)–3

65.0

2519

2–3

64.3

4379

6–0

.681

396

–0.0

1916

0–0

.019

129

–0.0

0003

1C

CSD

T–3

65.0

2548

6–3

64.3

4409

4–0

.681

391

–0.0

0029

4–0

.000

298

0.00

0005

–0.0

1945

4–0

.019

427

9898

GeO

DZ

CC

SD–2

172.

6924

27–2

150.

5180

30–2

2.17

4396

––

–C

CSD

(T)

–217

2.70

6184

–215

0.53

1709

–22.

1744

74–0

.013

757

–0.0

1367

9–0

.000

078

CC

SDT

–217

2.70

7037

–215

0.53

2564

–22.

1744

73–0

.000

854

–0.0

0085

40.

0000

01–0

.014

610

–0.0

1453

394

94C

CSD

T(Q

)–2

172.

7090

15–2

150.

5345

33–2

2.17

4481

–0.0

0197

8–0

.001

970

–0.0

0000

8C

CSD

TQ

–217

2.70

8548

–215

0.53

4072

–22.

1744

760.

0004

670.

0004

610.

0000

06–0

.001

511

–0.0

0150

913

113

1T

ZC

CSD

–217

3.13

5076

–215

0.94

6760

–22.

1883

16–

––

CC

SD(T

)–2

173.

1611

89–2

150.

9725

28–2

2.18

8661

–0.0

2611

3–0

.025

768

–0.0

0034

5C

CSD

T–2

173.

1609

52–2

150.

9723

30–2

2.18

8622

0.00

0237

0.00

0197

0.00

0040

–0.0

2587

6–0

.025

571

101

101

SnO

DZ

CC

SD–6

251.

3079

90–6

098.

0502

45–1

53.2

5774

5–

––

CC

SD(T

)–6

251.

3244

15–6

098.

0665

77–1

53.2

5783

8–0

.016

425

–0.0

1633

2–0

.000

093

CC

SDT

–625

1.32

5177

–609

8.06

7348

–153

.257

829

–0.0

0076

2–0

.000

771

0.00

0009

–0.0

1718

7–0

.017

103

9695

CC

SDT

(Q)

–625

1.32

8121

–609

8.07

0297

–153

.257

824

–0.0

0294

5–0

.002

949

0.00

0004

CC

SDT

Q–6

251.

3271

65–6

098.

0693

61–1

53.2

5780

50.

0009

560.

0009

360.

0000

20–0

.001

989

–0.0

0201

314

814

7T

ZC

CSD

–625

1.81

7765

–609

8.54

5097

–153

.272

669

––

–C

CSD

(T)

–625

1.85

2469

–609

8.57

9002

–153

.273

467

–0.0

3470

4–0

.033

905

–0.0

0079

8C

CSD

T–6

251.

8515

61–6

098.

5782

26–1

53.2

7333

60.

0009

070.

0007

760.

0001

31–0

.033

796

–0.0

3312

910

310

2Pb

OD

ZC

CSD

–209

88.8

4713

3–1

9599

.038

825

–138

9.80

8308

––

–C

CSD

(T)

–209

88.8

6216

3–1

9599

.055

232

–138

9.80

6931

–0.0

1503

0–0

.016

407

0.00

1377

CC

SDT

–209

88.8

6373

4–1

9599

.055

486

–138

9.80

8248

–0.0

0157

1–0

.000

254

–0.0

0131

8–0

.016

601

–0.0

1666

191

98C

CSD

T(Q

)–2

0988

.866

576

–195

99.0

5837

1–1

389.

8082

05–0

.002

841

–0.0

0288

40.

0000

43C

CSD

TQ

–209

88.8

6531

7–1

9599

.057

377

–138

9.80

7940

0.00

1259

0.00

0994

0.00

0265

–0.0

0158

2–0

.001

890

180

153

TZ

CC

SD–2

0989

.357

710

–195

99.5

5855

9–1

389.

7991

51–

––

CC

SD(T

)–2

0989

.393

339

–195

99.5

9352

6–1

389.

7998

12–0

.035

629

–0.0

3496

8–0

.000

661

CC

SDT

–209

89.3

9254

1–1

9599

.592

633

–138

9.79

9908

0.00

0798

0.00

0894

–0.0

0009

6–0

.034

831

–0.0

3407

410

210

3

a Dif

fere

nce

betw

een

the

tota

lene

rgie

sob

tain

edw

ithth

egi

ven

met

hod

and

the

prec

edin

gm

etho

d.

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234109-8 Nataraj, Kállay, and Visscher J. Chem. Phys. 133, 234109 (2010)

and the parent CCSDTQ approach should be employed for theestimation of quadruples effects.

B. Molecular geometries

The relativistic contraction of the bond lengths has beenobserved uniformly for all those molecules studied in thiswork. This is concurrent to the observations made in the caseof dihalogens.67 An unexpected bond elongation seen only inthe DZ basis in SnO is in contradiction with the bond con-traction observed in the TZ basis in the same system. Thisspurious result was suspected to be due to the omission of 4delectrons in the correlation calculations. In order to verify this,we performed a test calculation correlating 10 electrons in theTZ basis and observed indeed a similar bond elongation of0.00579 Å in the CCSD and 0.00737 Å in the CCSD(T) case.A further calculation performed using 20 correlated electrons,instead of 10, in the DZ basis showed a bond contractionof 0.00661 Å and 0.00569 Å for CCSD and CCSD(T), re-spectively. This suggests that inclusion of 4d electrons in thecorrelation calculations are necessary for getting the correctresults for SnO. Unfortunately, however, it is rather difficultto perform the quadruples calculations in the DZ basis with20 correlated electrons within the resources available for us.Nevertheless, the absence of similar observation (i.e., bondelongation) in the results of the DZ basis with 10 correlatedelectrons for PbO is intriguing.

The (T)-contribution in the TZ basis is larger than a fac-tor of 100 over the T-contribution for all molecules down thegroup except PbO. For PbO this factor is a mere 4 in the rela-tivistic case and 26 in the nonrelativistic case. In addition, thesign of the T-contribution in the relativistic case in PbO is op-posite to the sign observed in all other cases. These results areless easy to rationalize than the trends observed above in thecorrelation energy as they result from a combination of rel-ativistic effects on the correlation and the usual bond weak-ening (caused by spin-orbit coupling) and bond contraction(caused by the scalar relativistic effects that shrink the valencep-orbitals) trends. The increasing differences between nonrel-ativistic and relativistic results do, however, illustrate the needto treat relativity and electron correlation simultaneously.

The correlation contributions due to the entire triples inthe DZ basis are at least a factor of 6 larger than those forthe entire quadruples. This trend is observed to be the same inboth relativistic and nonrelativistic cases uniformly across theentire group. The ratio of perturbative and iterative quadru-ples contributions [i.e., (Q)/Q] decreases monotonically downthe group and again the performance of CCSDT(Q) is ratherpoor from the third row onward in both relativistic and non-relativistic cases. The relativistic effects to the triples andquadruples are significant for SnO and PbO only. Further, therelativistic contribution to (Q) in the case of PbO shows anegative sign breaking the general trend observed in all othermolecules of the group.

The underlying conclusion of Table II is that the in-clusion of the relativistic effects is necessary in the hier-archy of higher-order correlation calculations, in particular,full triples, and quadruples for SnO and PbO. However,

for lighter molecules the nonrelativistic methods are recom-mended for the higher-order correlation calculations unlessone is interested in the change in the bond lengths of less than0.00005 Å.

C. Harmonic frequencies

The harmonic frequencies are lower in the relativis-tic case in comparison to the nonrelativistic case for theGroup IVa monoxides. This is in agreement to the trends ob-served for dihalogens67 and interhalogens66 by Visscher et al.,and is mainly due to the spin-orbit coupling that mixes in anti-bonding contributions, thus weakening the formal triple bondof the Group IVa monoxides.65

The correlation contribution to (both perturbative and it-erative) triples appears to be approximately same in both therelativistic and the nonrelativistic cases in the TZ basis forlight molecules up to GeO. However, for SnO and PbO thistrend is quite different. The T-contribution in the relativis-tic case is noticeably smaller than that in the nonrelativisticcase for SnO. The (T)- and T-contributions show oppositesigns in PbO, particularly in the relativistic case, breakingthe trend exhibited by other molecules in the TZ basis. TheT-contribution is unusually large in the relativistic case forPbO making the ratio [(T)/T] to be ∼2 unlike in othermolecules in which this ratio is more than 10 in the TZ basis.In the nonrelativistic case this ratio is ∼4 for PbO. From ourresults it can be inferred that the CCSDT(Q) approach per-forms very poorly in the vibrational frequency calculations,both in relativistic and nonrelativistic cases, in almost all themolecules of current interest.

No particular trend has been displayed in the relativisticcontribution to correlation increments for frequencies shownin column nine in Table III. However, it is certainly clear thatthe relativistic contributions to T-, (Q)-, and Q-incrementsare negligibly small for elements up to fourth row. For PbOthe T-contribution is significantly large with a value of about−8 cm−1 in the TZ basis and −18 cm−1 in the DZ basis. TheQ-contribution in PbO is also large and is comparable to theT-contribution and hence, one needs to consider the relativis-tic effects for full quadruples also.

D. The individual molecules

Below we shall discuss some noteworthy features foreach individual molecule treated in this work.

1. CO

The relativistic contributions for the total energies of COare about 0.07 Eh . A small contraction in the bond length of(∼0.0002 Å) and a decrease in the vibrational frequency of(2 cm−1) has been observed due to the inclusion of relativis-tic effects. It is interesting to observe that the influence of therelativistic effects in shortening the bond length and in de-creasing the vibrational frequency of CO molecule is constantthroughout the hierarchy of correlation methods considered.Further, the correlation contributions appear to be the same

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234109-9 General implementation of relativistic CC J. Chem. Phys. 133, 234109 (2010)

TA

BL

EII

.E

quili

briu

mbo

ndle

ngth

san

dth

eco

rrel

atio

nco

ntri

butio

nsto

the

equi

libri

umbo

ndle

ngth

sfo

rth

eox

ides

ofG

roup

IVa

(in

Å).

Bas

isB

ond

leng

thC

orre

latio

nco

ntri

butio

na[(

T)+

T]

and

[(Q

)+Q

]co

ntri

butio

n%

cove

red

bype

rt.t

heor

y

Mol

ecul

ese

tM

etho

dR

el.

Non

rel.

Dif

f.R

el.

Non

rel.

Dif

f.R

el.

Non

rel.

Rel

.N

onre

l

CO

DZ

CC

SD1.

1340

91.

1342

6–0

.000

17–

––

CC

SD(T

)1.

1397

51.

1399

0–0

.000

160.

0056

60.

0056

40.

0000

2C

CSD

T1.

1400

41.

1402

0–0

.000

160.

0002

90.

0002

90.

0000

00.

0059

50.

0059

495

95C

CSD

T(Q

)1.

1407

51.

1409

0–0

.000

150.

0007

10.

0007

10.

0000

0C

CSD

TQ

1.14

060

1.14

076

–0.0

0016

–0.0

0015

–0.0

0015

0.00

000

0.00

056

0.00

056

127

126

TZ

CC

SD1.

1279

61.

1281

6–0

.000

20–

––

CC

SD(T

)1.

1347

81.

1349

6–0

.000

180.

0068

20.

0068

00.

0000

2C

CSD

T1.

1347

51.

1349

3–0

.000

18–0

.000

04–0

.000

030.

0000

00.

0067

90.

0067

710

110

0Si

OD

ZC

CSD

1.53

537

1.53

565

–0.0

0029

––

–C

CSD

(T)

1.54

553

1.54

575

–0.0

0023

0.01

016

0.01

010

0.00

006

CC

SDT

1.54

609

1.54

631

–0.0

0022

0.00

056

0.00

056

0.00

001

0.01

073

0.01

066

9595

CC

SDT

(Q)

1.54

824

1.54

846

–0.0

0021

0.00

215

0.00

214

0.00

000

CC

SDT

Q1.

5474

61.

5476

8–0

.000

23–0

.000

78–0

.000

77–0

.000

010.

0013

60.

0013

715

715

6T

ZC

CSD

1.50

929

1.50

962

–0.0

0033

––

–C

CSD

(T)

1.52

010

1.52

036

–0.0

0026

0.01

081

0.01

075

0.00

006

CC

SDT

1.52

000

1.52

027

–0.0

0027

–0.0

0010

–0.0

0010

–0.0

0001

0.01

071

0.01

065

101

101

GeO

DZ

CC

SD1.

6261

71.

6277

3–0

.001

55–

––

CC

SD(T

)1.

6404

01.

6416

1–0

.001

210.

0142

20.

0138

80.

0003

4C

CSD

T1.

6410

41.

6421

9–0

.001

160.

0006

40.

0005

90.

0000

50.

0148

60.

0144

796

96C

CSD

T(Q

)1.

6443

31.

6454

4–0

.001

110.

0032

90.

0032

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377

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353

206

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8147

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8223

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8356

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300.

0001

50.

0208

30.

0196

010

110

2Pb

OD

ZC

CSD

1.91

373

1.92

827

–0.0

1454

––

–C

CSD

(T)

1.92

364

1.95

427

–0.0

3063

0.00

991

0.02

600

–0.0

1609

CC

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1.93

219

1.95

186

–0.0

1967

0.00

855

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0241

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096

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845

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359

5411

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T(Q

)1.

9409

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110.

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0092

3–0

.000

44C

CSD

TQ

1.93

494

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627

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2133

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0604

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0481

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0123

0.00

275

0.00

442

319

209

TZ

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9026

51.

9184

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.015

80–

––

CC

SD(T

)1.

9193

31.

9413

6–0

.022

030.

0166

80.

0229

1–0

.006

23C

CSD

T1.

9236

71.

9404

6–0

.016

790.

0043

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.000

900.

0052

40.

0210

20.

0220

179

104

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234109-10 Nataraj, Kállay, and Visscher J. Chem. Phys. 133, 234109 (2010)

TA

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2010

–2–1

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–16

–16

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9419

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–14

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9519

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310

3Si

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1107

1110

–2–

––

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)10

9310

96–2

–14

–14

0C

CSD

T10

9310

95–2

00

0–1

4–1

499

99C

CSD

T(Q

)10

8910

92–2

–3–4

0C

CSD

TQ

1091

1094

–22

20

–2–2

211

195

TZ

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9410

96–2

––

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(T)

1082

1085

–2–1

2–1

20

CC

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1083

1085

–21

00

–11

–11

106

104

GeO

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984

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832

–7–1

3–1

30

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833

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10

–13

–12

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106

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828

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–50

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810

820

–10

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798

808

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–12

–12

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980

9–1

01

10

–11

–11

106

106

SnO

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372

5–2

2–

––

CC

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)69

171

1–2

0–1

3–1

42

CC

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692

714

–21

13

–1–1

1–1

111

212

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T(Q

)68

470

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1–8

–80

CC

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971

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16

60

–3–3

307

287

TZ

CC

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368

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0–

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)65

267

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9–1

2–1

31

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653

673

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12

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111

7Pb

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647

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630

–33

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635

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–18

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–12

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141

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–75

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234109-11 General implementation of relativistic CC J. Chem. Phys. 133, 234109 (2010)

for both relativistic and nonrelativistic cases at each level ofthe correlation theory. For a light neutral molecule like COone can therefore safely ignore the relativistic effects in thehigher-order correlation contributions.

2. SiO

SiO being heavier than CO, shows an energy decrease ofabout 0.68 Eh while going from the nonrelativistic to the rel-ativistic case which is an order of magnitude larger than therelativistic effect observed in CO. The influence of the rel-ativistic effects lead to a decrease in the bond length in therange ∼0.0002–0.0003 Å, and a decrease in the frequencyof about 2 cm−1. The relativistic contributions to the higher-order correlation effects are insignificant for SiO and theycan be omitted unless one is looking for completeness in thecalculations.

3. GeO

The relativistic effects begin to appreciably influence thetotal energy of the diatomic molecules starting from GeO inthe monoxide series considered in this work with a contribu-tion of ∼ 22 Eh which, however, is a mere ∼1% of the totalvalue. The bond length contraction of 0.001–0.002 Å in theDZ basis and 0.003 Å in the TZ basis is consistently observedfor GeO at different levels of correlation theory. A bond con-traction of 0.003 Å has also been observed by Dyall65 in anuncorrelated DHF calculation. The relativistic decrease in thevibrational frequency varies from ∼7 cm−1 in DZ basis to∼10 cm−1 in TZ basis. The observed change in the frequencyin the DHF calculation by Dyall65 is 2(3) times smaller thanthe one observed by us in the DZ(TZ) basis, which may bedue to the inclusion of electron correlation in our calcula-tions, or due to the difference in basis set. We would like torecall that we have used optimized relativistic basis sets ofDyall for Ge unlike the case for CO and SiO for which wehave used Dunning basis sets. Since GeO stands in the borderline between the relativistic and the nonrelativistic domains,we recommend to treat the higher-order correlation effects to-gether with the relativistic effects in GeO, if a high accuracyis needed.

4. SnO

The relativistic effects get more pronounced for heaviermolecules such as SnO with contributions of 153.3 Eh whichis ∼2.5% to its total energy. SnO shows a bond elongationof about 0.002–0.004 Å in the DZ basis in contrast to thebond contraction observed for all other molecules studied inthis work. However, in the TZ basis we see a strong bondcontraction of about 0.006–0.008 Å as anticipated. The latterbond length contraction compares well with both the uncor-related DHF results (0.007 Å) of Dyall65 and the correlatedscalar relativistic DKH5 results (0.004 Å) of Wolf et al.64 Thereasons for the observed discrepancy in the change in bondlength between the DZ and TZ calculations are discussed inSec. IV B.

The relativistic contribution to the harmonic frequenciesvaries from −20 to −22 cm−1 at different levels of corre-lation theory in the DZ basis and falls in the range −19 to−20 cm−1 in the TZ basis. In comparison with the above-mentioned calculations of Dyall and Wolf et al. the decreaseof the frequency is about twice as large, which can be un-derstood from the lack of correlation in the calculations byDyall and the lack of spin-orbit coupling in the calculationsby Wolf et al., both of which will weaken the bond by mix-ing in antibonding orbitals. The inclusion of relativistic ef-fects in the higher-order correlation calculations is certainlyrecommended.

5. PbO

The relativistic contribution to the total energy of PbO is∼1390 Eh , which is ∼7% of its total energy. A bond contrac-tion of 0.01–0.03 Å has been observed for PbO at differentcorrelation levels of the theory. The relativistic bond contrac-tion being larger at the CCSD(T) level than that at the CCSDlevel sets an unusual trend for PbO. The decrease in the bonddistance for PbO observed earlier by Iliaš et al.82 with theDC and Barysz–Sadlej–Snijders (BSS) Hamiltonians in con-junction with the CCSD(T) approach (0.003 Å), as well asby Lenthe et al.101 using the ZORA approach (0.002 Å) areapproximately an order of magnitude smaller than that ob-served in this work, which can be ascribed as due to the lackof spin-orbit coupling terms. The SO corrected CCSD(T) re-sult of Metz et al.,102 as quoted in Ref. 101 is 0.037 Å, whichis more or less in agreement with our result. The DHF resultsof Dyall65 show a bond length contraction of 0.0146 Å.

The relativistic decrease in the harmonic vibrational fre-quency is 33–81 cm−1 in the DZ basis and 46–75 cm−1 inthe TZ basis. In the aforementioned calculations by Iliaš et al.and by Lenthe et al. approximately 10 cm−1, while around44 cm−1 by Metz et al., and 88 cm−1 by Dyall has been ob-served as the change in the frequency due to relativistic ef-fects. The latter two results are more or less in agreementwith the present calculations. The decrease in the frequencyat the CCSD level is observed to be unusually large whencompared to that at the higher levels of correlation theory. Onthe other hand, the harmonic frequency seems to be overesti-mated with the CCSD(T) approach, in particular, in the rela-tivistic case and at the higher levels of the correlation theory itdecreases slowly. It is also observed that the ratio of the rela-tivistic contributions to CCSD and CCSD(T) results in the DZbasis in PbO is ∼3 as against a ratio of ∼1 observed in othermolecules. Despite the trends being different, it is undoubt-edly clear that the relativistic effects have strong influence onthe higher-order correlation effects in SnO and PbO.

V. CONCLUDING REMARKS

The development of a general order relativistic CCcode has been reported based on Kramers-paired molecularspinors, double group symmetry, and the full Dirac–Coulombas well as several approximate relativistic Hamiltonians. Thenew program is useful for benchmarking lower-level relativis-tic correlation methods. Further, it also unfolds the way for

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234109-12 Nataraj, Kállay, and Visscher J. Chem. Phys. 133, 234109 (2010)

high precision calculations for systems where the relativisticeffects, in particular, the spin-orbit coupling are strong, and arigorous treatment of the relativity is required.

Benchmark calculations have been performed for the to-tal energies, bond lengths, and vibrational frequencies of theoxides of Group IVa. The behavior of the relativistic contri-butions with increasing level of correlation and atomic num-ber has been monitored with special regard to iterative triplesas well as quadruples contributions. Our results suggest thatup to the third (fourth) row of the periodic table the itera-tive triples contributions for total energies and bond lengths(harmonic frequencies) can be calculated using the nonrela-tivistic CC methods, and the explicit inclusion of relativityis only required from the fourth (fifth) row. The contributionof quadruple excitations in the case of total energies and ge-ometries can be evaluated by nonrelativistic approaches up tofourth- and third-row molecules, respectively, and a full rela-tivistic treatment is only necessary for heavier elements. Forharmonic frequencies the inclusion of relativity for the cal-culation of quadruples corrections is only necessary for thefifth row. The performance of the CCSDT(Q) method doesnot seem to be satisfactory from the third row onwards forany of the considered properties, and thus, the use of the fullCCSDTQ method is recommended. We note again that thereexists some caveat concerning our conclusions especially forquadruple excitations because of the small basis sets used,nevertheless these calculations can be considered as state-of-the-art, and larger calculations are not feasible in the nearfuture.

We would like to remark that in the current calculations,the relativistic correction to the Coulomb interaction calledBreit interaction is not included. Since these contributionsmay be important in the high accuracy calculations, we willconsider including them in the future. In addition, in orderto effectively handle open-shell systems, we plan to considerimplementing either a fully unrestricted spinor optimizationfollowed by MO transformation or the semi-canonical orbitaloption in the future work.

ACKNOWLEDGMENTS

The authors are greatly indebted to Professor DebashisMukherjee (Kolkata, India) for useful discussions. Financialsupport to M.K. has been provided by the European Re-search Council (ERC) under the European Community’s Sev-enth Framework Programme (FP7/2007-2013), ERC GrantAgreement No. 200639, and by the Hungarian Scientific Re-search Fund (OTKA), Grant No. NF72194. M.K. acknowl-edges the Indo-Hungarian (IND 04/2006) project and theBolyai Research Scholarship of the Hungarian Academy ofSciences. L.V. has been supported by NWO through the VICIprogramme.

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