Efficient computer implementation of the renormalized coupled-cluster methods: The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches

Computer Physics Communications 149 (2002) 71–96

www.elsevier.com/locate/cpc

Efficient computer implementation of the renormalizedcoupled-cluster methods: The R-CCSD[T], R-CCSD(T),

CR-CCSD[T], and CR-CCSD(T) approaches

Piotr Piecucha,∗,1, Stanisław A. Kucharskib, Karol Kowalskia, Monika Musiałb

a Department of Chemistry, Michigan State University, East Lansing, MI 48824, USAb Institute of Chemistry, University of Silesia, Szkolna 9, 40-006 Katowice, Poland

Received 15 May 2002

Abstract

The recently proposed renormalized (R) and completely renormalized (CR) coupled-cluster (CC) methods of the CCSD[T]and CCSD(T) types have been implemented using recursively generated intermediates and fast matrix multiplication routines.The details of this implementation, including the complete set of equations that have been used in writing efficient computercodes, memory requirements, and typical CPU timings, are discussed. The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) computer codes and similar codes for the standard CC methods, including the LCCD, CCD, CCSD, CCSD[T], andCCSD(T) approaches, have been incorporated into theGAMESSpackage. Information about the main features of this new setof CC programs is provided. 2002 Elsevier Science B.V. All rights reserved.

PACS: 02.70.-c; 31.10.+z; 31.15.Ar; 31.15.Dv; 31.25.-v; 31.50.Bc

Keywords: Coupled-cluster theory; Renormalized coupled-cluster methods; Noniterative coupled-cluster approaches; Method of moments ofcoupled-cluster equations; Electronic structure programs

1. Introduction

The standard coupled-cluster (CC) methods [1–3], such as CCSD [4] (CC approach with singles and doubles),or the noniterative CC methods that account for the effect of triples or triples and quadruples using argumentsoriginating from the many-body perturbation theory (MBPT), such as CCSD[T] [5], CCSD(T) [6], and CCSD(TQf)[7], in either the spin-orbital [4–6] and spin-free [7–10] or orthogonally spin-adapted [11–13] forms, are nowadayswidely used in accurate quantum-chemical calculations [14–17]. Unfortunately, it is not possible to apply the

* Corresponding author.E-mail address: [email protected] (P. Piecuch).URL address: http://www.cem.msu.edu/~piecuch/group_web (P. Piecuch).

1 Selected as an Alfred P. Sloan Research Fellow (in 2002).

0010-4655/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0010-4655(02)00598-2

72 P. Piecuch et al. / Computer Physics Communications 149 (2002) 71–96

CCSD[T], CCSD(T), CCSD(TQf), and similar methods to potential energy surfaces (PESs) involving bondbreaking, if the spin-adapted restricted Hartree–Fock (RHF) configuration is used as a reference (cf., e.g.,Refs. [18–29] and references therein). The CCSD method itself, on which the noniterative CCSD[T], CCSD(T),and CCSD(TQf) approaches are based, is inadequate for the description of bond breaking, as it neglects theimportant triply and quadruply excited clusters. In consequence, the errors in the RHF-based CCSD results atlarge internuclear separations are large, as it is often the case for PESs involving single bond breaking (cf.,e.g., the PES of F2; the RHF-based CCSD approach produces a potential well which is almost twice as deepas the exact well [18,26]), and the CCSD potentials are plagued by unphysical humps and errors larger than onehundred millihartree, when multiple bond breaking is involved [25,27]. The triples and quadruples correctionsof the CCSD[T], CCSD(T), and CCSD(TQf) methods make the situation even worse, since the standard MBPTarguments, on which the noniterative CC approximations are based, fail due to the divergent behavior of the MBPTseries at larger internuclear separations. As a result, the PESs produced by the CCSD[T], CCSD(T), CCSD(TQf),and other noniterative CC approaches are completely unphysical [18–29].

A few approaches have been suggested in recent years with an intention of removing the pervasive failing ofthe RHF-based single-reference CC approximations at larger internuclear separations. The representative examplesinclude the reduced multi-reference CCSD (RMRCCSD) method [16,30–35], the active-space single-reference CCapproaches [19,21,22,26,36–47], the orbital-optimized CC methods [48,49], the noniterative approaches basedon the partitioning of the similarity-transformed Hamiltonian [50–53], and the renormalized and completelyrenormalized CC approaches [23–29,54]. The latter approaches are based on the more general formalism of themethod of moments of CC equations (MMCC) [23–25,29,55–58], which can be applied to ground- and excited-state PESs.

All of the above methods focus on improving the description of bond breaking, while retaining the simplicity ofthe single-reference description based on the spin- and symmetry-adapted reference of the RHF type. At the riskof introducing spin contamination, one can also improve the description of bond breaking (often, significantly) byemploying the unrestricted Hartree–Fock (UHF) (see, e.g., Ref. [18]) or restricted but “spin-flipped” [59] referenceconfigurations. The problem with the approaches of this type is that the spin contamination of electronic statesmay cause problems in some applications. This is certainly true for the UHF-based CC approaches, which alsointroduce a nonanalytic behavior of PES in the region of transition between the triplet stable and triplet unstablesolutions of the Hartree–Fock equations [18,60]. Although the results for PESs involving single bond breakingand the results for molecular systems having biradical character obtained with the so-called Spin-Flip (SF) CCSDmodel are considerably better than the results of the UHF-based CC calculations [59,61], methods that do not breakthe spin symmetry represent, in our view, the preferred theoretical models. Furthermore, the use of the spin-orbitalformalism in the SFCC and UHF-based CC methods does not allow for a number of simplifications which arenormally possible when the spin symmetry is present in molecular systems. For all these various reasons, in thispaper we concentrate on the single-reference approaches that are based on using the spin- and symmetry-adaptedreference configurations of the RHF type, in which problems related to spin contamination do not exist.

In this work, we focus on the promising renormalized and completely renormalized CC approaches, specifically,on the renormalized (R) and completely renormalized (CR) CCSD[T] and CCSD(T) methods, which can be usedto study PESs involving a breaking or making of single chemical bonds [23,24,26,28,29,54]. If a single bond isbroken, we can use these methods to obtain the results of nearly spectroscopic accuracy, including highly excitedvibrational states near dissociation, which cannot be described by the standard CCSD[T] and CCSD(T) methods atall [28,29]. The higher-level renormalized and completely renormalized CCSD(TQ) and CCSDT(Q) approaches,which can be used to break multiple chemical bonds [25,27,29], are not considered in this paper.

In our earlier papers [23,24,26,28,29,54], we focused on the general formalism behind the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods and on testing these methods using pilot codes. The detailedformulas for the noniterative energy corrections defining the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approximations in terms of molecular integrals and cluster amplitudes have never been presented. Oneof the goals of this paper is to provide such information.

P. Piecuch et al. / Computer Physics Communications 149 (2002) 71–96 73

Recently, we have developed highly efficient computer codes for the R-CCSD[T], R-CCSD(T), CR-CCSD[T],and CR-CCSD(T) methods, in which the renormalized and completely renormalized triples ([T] and (T))corrections are calculated using a relatively small set of recursively generated intermediates and fast matrixmultiplication routines. In defining the relevant intermediates, we have used the general programming strategydescribed in Refs. [62,63]. The details of our implementation of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], andCR-CCSD(T) methods, including the complete set of equations that have been used in writing efficient computercodes, are discussed in this paper. We also address important technical issues, such as memory requirements andtypical CPU timings.

The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) computer codes constitute part of a largerCC package, which enables us to perform the basic sequence of the standard CC calculations, including theLCCD [64] (linearized CC doubles approach), CCD [1–3,65,66] (CC doubles approach), CCSD [4], CCSD[T][5], and CCSD(T) [6] calculations, as well as the aforementioned R-CCSD[T], R-CCSD(T), CR-CCSD[T], andCR-CCSD(T) calculations. These new codes have been incorporated in the popularGAMESS package [67] andwill soon be available to all users ofGAMESS. Information about the capabilities of this new set of CC programs,illustrated by a few numerical examples, is provided.

2. Theory

2.1. The standard single-reference CC theory

In the single-reference CC theory, the ground-state wave function|Ψ 〉 of anN -electron system is representedas follows:

|Ψ 〉 = eT |Φ〉, (1)

where T is the cluster operator and|Φ〉 is the independent-particle-model reference configuration (e.g., theHartree–Fock determinant). Typically, we truncate the many-body expansion ofT at a conveniently chosenexcitation level. This leads to the well-known hierarchy of the standard CC approximations: CCSD [4], whenT is restricted to singly and doubly excited clusters (T � T1 + T2); CCSDT [68,69], whenT is restricted tosingly, doubly, and triply excited clusters (T � T1 + T2 + T3), CCSDTQ [40,62,63,70], whenT is truncated atquadruply excited clusters (T � T1 + T2 + T3 + T4), CCSDTQP [71], whenT is truncated at pentuply excitedclusters (T � T1 + T2 + T3 + T4 + T5), etc.

The explicit equations for cluster amplitudes, which defineT , are usually obtained by projecting the connected-cluster form of the electronic Schrödinger equation [1,2,14–16,23], i.e.(

HNeT)C|Φ〉 = E|Φ〉, (2)

onto the excited configurations generated byT . Here,HN = H − 〈Φ|H |Φ〉 is the electronic Hamiltonian in thenormal-product form,E = E−〈Φ|H |Φ〉 is the energy relative to reference energy〈Φ|H |Φ〉 (correlation energywhen|Φ〉 is the Hartree–Fock state), and subscriptC indicates the connected part of a given operator expression.For example, the CCSD amplitude equations are obtained by projecting Eq. (2), whereT = T1 +T2, onto all singlyand doubly excited configurations,|Φa

i 〉 and|Φabij 〉, respectively,

⟨Φa

i

∣∣(HNeT1+T2)C|Φ〉 = 0, (3)

⟨Φab

ij

∣∣(HNeT1+T2)C|Φ〉 = 0. (4)

Once the system of nonlinear, energy-independent equations for cluster amplitudes is solved, we calculate theenergyE in the following way:

E = 〈Φ|(HNeT1+T2)C|Φ〉. (5)

Eq. (5) is obtained by projecting Eq. (2) onto the reference configuration|Φ〉.


In this paper, we are mainly interested in using theclosed-shell (restricted) reference|Φ〉 and thespin-freeor non-orthogonally spin-adapted formulation of the CC theory, based on drawing the Goldstone, rather thanHugenholtz, diagrams (which means, for example, that we must associate an additional factor of 2l , wherel isthe number of closed loops, to each of the resulting diagrams [72]). Thus, here and elsewhere in the present paper,lettersi, j, k, l, . . . designate thespatial orbitals that are occupied in|Φ〉 anda, b, c, d, . . . are the unoccupiedspatial orbitals. Generic (occupied and unoccupied) orbitals are designated by lettersp,q, r, s, . . . . The excitedconfigurations|Φa

i 〉, |Φabij 〉, etc. are obtained with the help of theorbital (rather than spin-orbital) unitary group

generators [16],

Eai = XaαXiα +XaβXiβ, (6)

whereXaσ andXiσ are the usual creation and annihilation operators, respectively, andσ = α or β are the spin-up(α) or spin-down (β) spin eigenfunctions for a particle with spin12. For example,

∣∣Φai

⟩ = Eai |Φ〉, (7)

∣∣Φabij

⟩ = Eabij |Φ〉, (8)

where

Eabij = Ea

i Ebj . (9)

The orbital excitation operators,Eai , Eab

ij , etc., allow us to define the spin-free form of cluster operator components.For theT1 andT2 components, relevant to this paper, we can write

T1 =∑i,a

tai Eai , (10)

T2 = 12

∑ij,ab

tabij Eabij , (11)

wheretai = 〈a|t1|i〉 andtabij = 〈ab|t2|ij 〉 are the spin-free singly and doubly excited cluster amplitudes. The doublyexcited cluster amplitudes have the following symmetry property with respect to interchanges of occupied andunoccupied orbital labels:

tabij = tbaji . (12)

Thus, the followingtabij amplitudes are linearly independent:tabij with i < j and alla andb; tabii with a � b.

In the spin-free CCSD approach, we determine amplitudestai andtabij by solving Eqs. (3) and (4). The compact

form of the CCSD equations in terms of cluster amplitudestai andtabij and one- and two-electron molecular integrals

fpq = 〈p|f |q〉 andvpqrs = 〈pq|v|rs〉, corresponding to one- and two-body parts of the HamiltonianHN , FN and

VN , respectively, is described in Section 3.1.

2.2. Renormalized and completely renormalized CCSD[T] and CCSD(T) approaches

The ground-state MMCC theory and the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methodsthat result from it are based on an idea of improving the results of the standard CC (e.g., CCSD) calculationsby adding thea posteriori noniterative correctionsδ to CC energies [23–25,29,58] (for excited-state and multi-reference extensions of the MMCC theory, see Refs. [29,55–57]). In the exact MMCC theory, correctionsδ recoverthe full configuration interaction (full CI) energies. In the approximate MMCC methods, such as CR-CCSD[T]or CR-CCSD(T), the exact expressions for correctionsδ, resulting from the MMCC theory, are replaced by theapproximate expressions in such a way that the resulting energies, obtained by adding these corrections to the


standard CC energies, remain close to the corresponding full CI energies. The MMCC approximations preservethe simplicity and the ease-of-use of the standard noniterative CC approaches, such as CCSD[T] or CCSD(T),while offering us a good control over the accuracy of the results in difficult situations, such as bond breaking,where arguments originating from MBPT, on which the CCSD[T] or CCSD(T) methods are based, fail. This isaccomplished by directly relating correctionsδ to a quantity of interest, i.e.E − ECC, whereECC is the energyobtained in the standard CC calculation andE is the exact (full CI) energy.

The MMCC correctionsδ are expressed in terms of thegeneralized moments of CC equations defining a givenCC approximation, i.e. the CC equations projected on the configurations that are not included in the standard CCcalculations. The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods, in which we correct theresults of the standard CCSD calculations, use the CCSD equations projected on triply excited configurations, i.e.

Mabcijk (2) = ⟨

Φabcijk

∣∣(HNeT1+T2)C|Φ〉, (13)

where|Φabcijk 〉 = Eabc

ijk |Φ〉 (Eabcijk = Ea

i EbjE

ck ). If T1 andT2 are the cluster operators obtained by solving the CCSD

equations, Eqs. (3) and (4), then the energy formulas defining the CR-CCSD[T] and CR-CCSD(T) methods are[23–26,28,29,54,55]

ECR-CCSD[T] = ECCSD+ 〈Ψ CCSD[T]|M3(2)|Φ〉/〈Ψ CCSD[T]|eT1+T2|Φ〉, (14)

ECR-CCSD(T) = ECCSD+ 〈Ψ CCSD(T)|M3(2)|Φ〉/〈Ψ CCSD(T)|eT1+T2|Φ〉, (15)

whereECCSD is the CCSD energy and

M3(2)|Φ〉 = 16

∑ijk,abc

Mabcijk (2)

∣∣Φabcijk

⟩. (16)

Wave functions|Ψ CCSD[T]〉 and|Ψ CCSD(T)〉, entering Eqs. (14) and (15), are defined by the simple, MBPT(2)[SDT]-like, expressions,

|ΨCCSD[T]〉 = (1+ T1 + T2 + T

[2]3

)|Φ〉, (17)

|ΨCCSD(T)〉 = ∣∣Ψ CCSD[T]⟩ +Z3|Φ〉, (18)

where the

T[2]3 |Φ〉 = R

(3)0 (VNT2)C |Φ〉 (19)

term in Eq. (17) is a CCSD analog of the connected triples contribution to the second-order MBPT (MBPT(2))wave function and

Z3|Φ〉 = R(3)0 VNT1|Φ〉 (20)

is the disconnected triples correction needed to distinguish between the [T] and (T) corrections (R(3)0 designates

the three-body component of the MBPT reduced resolvent).The simpler R-CCSD[T] and R-CCSD(T) approaches are obtained by replacing theMabc

ijk (2) moments in theCR-CCSD[T] and CR-CCSD(T) formulas, Eqs. (14) and (15), respectively, by their lowest-order estimates, i.e.〈Φabc

ijk |(VNT2)C |Φ〉. We obtain [23,24],

ER-CCSD[T] = ECCSD+ 〈Ψ CCSD[T]|Q3(VNT2)C |Φ〉/〈Ψ CCSD[T]|eT1+T2|Φ〉 , (21)

ER-CCSD(T) = ECCSD+ 〈Ψ CCSD(T)|Q3(VNT2)C |Φ〉/〈Ψ CCSD(T)|eT1+T2|Φ〉 , (22)

whereQ3 is the projection operator onto the manifold of triexcited configurations. Although studies of entire PESsinvolving single bond breaking require using the CR-CCSD[T] and CR-CCSD(T) rather than the R-CCSD[T]and R-CCSD(T) approaches [23,24,26,28,29,54], the R-CCSD[T] and R-CCSD(T) methods are important to


understand the relationship between the standard and renormalized CC approaches (and this understanding isvery important for coding all of these methods). Moreover, the R-CCSD[T] and R-CCSD(T) approaches providesuperior description of moderately stretched single chemical bonds, when compared to the standard CCSD[T] andCCSD(T) methods [23,24,26,28,29,54].

The R-CCSD[T] and R-CCSD(T) approaches reduce to the standard CCSD[T] and CCSD(T) methods, whenthe〈Ψ CCSD[T]|eT1+T2|Φ〉 and〈Ψ CCSD(T)|eT1+T2|Φ〉 denominators in Eqs. (21) and (22) are replaced by 1 [23,24].Indeed, by replacing the〈Ψ CCSD[T]|eT1+T2|Φ〉 denominator in Eq. (21) by 1, we obtain the well-known formulafor the CCSD[T] energy [5],

ECCSD[T] = ECCSD+E[4]T , (23)

where

E[4]T = 〈Ψ CCSD[T]|Q3(VNT2)C |Φ〉 = 〈Φ|(T [2]

3

)†(VNT2)C |Φ〉 (24)

is the noniterative [T] correction due to triples. Similarly, by replacing the〈Ψ CCSD(T)|eT1+T2|Φ〉 denominator inEq. (22) by 1, we obtain the well-known formula for the CCSD(T) energy [6],

ECCSD(T) = ECCSD+E[4]T +E

[5]ST, (25)

whereE[4]T is defined by Eq. (24) and

E[5]ST = 〈Φ|(Z3)

†(VNT2)C |Φ〉 (26)

is the famous 5th order term that distinguishes the CCSD(T) approach from the older CCSD[T] approximation.Approximation of the〈Ψ CCSD[T]|eT1+T2|Φ〉 and〈Ψ CCSD(T)|eT1+T2|Φ〉 denominators by 1 is a justified step fromthe point of view of MBPT, since both denominators equal 1 plus terms which are at least of the second order inperturbationVN . Indeed, if we define

D[T] = 〈Ψ CCSD[T]|eT1+T2|Φ〉 (27)

and

D(T) = 〈Ψ CCSD(T)|eT1+T2|Φ〉, (28)

we obtain [24],

D[T] = 1+ 〈Φ|(T1)†T1|Φ〉 + 〈Φ|(T2)

†(T2 + 12T

21

)|Φ〉+ 〈Φ|(T [2]

3

)†(T1T2 + 1

6T31

)|Φ〉, (29)

D(T) = D[T] + 〈Φ|(Z3)†(T1T2 + 1

6T31

)|Φ〉, (30)

where theT [2]3 andZ3 operators are defined by Eqs. (19) and (20), respectively.

For the closed-shell molecules at or near their optimized geometries, where the MBPT series usually converges,theD[T] andD(T) denominators are often very close to 1, so that the renormalized and completely renormalizedCCSD[T] and CCSD(T) calculations give the results which are almost identical to those obtained with the standardCCSD[T] and CCSD(T) methods. However, for the molecular geometries far from the equilibrium ones, wherethe MBPT series is manifestly divergent, theD[T] andD(T) denominators can be much larger than 1 [24,25].This is precisely the reason of the excellent performance of the CR-CCSD[T] and CR-CCSD(T) approaches atlarger internuclear separations. TheD[T] andD(T) denominators play a role of natural damping factors, whichdamp the excessively large and, thus, completely unphysical values of the noniterative triples corrections at largerinternuclear separations. No such denominators are present in the CCSD[T] and CCSD(T) energy formulas (cf.Eqs. (23) and (25)), and, in consequence, the CCSD[T] and CCSD(T) methods produce completely unphysicalPESs when chemical bonds are stretched or broken [23–29,54].


The above simple relationship between the renormalized and completely renormalized CCSD[T] and CCSD(T)methods and their standard counterparts implies that costs of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) calculations are essentially identical to the costs of the standard CCSD[T] and CCSD(T) calculations. Inanalogy to the standard CCSD[T] and CCSD(T) methods, the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches are then3

on4u procedures in the noniterative steps involving triples and then2

on4u procedures in

the iterative CCSD steps (here and elsewhere in the present paper,no (nu) is the number of occupied (unoccupied)correlated orbitals). The memory requirements characterizing the R-CCSD[T], R-CCSD(T), CR-CCSD[T], andCR-CCSD(T) methods are essentially identical to those characterizing the standard CCSD[T] and CCSD(T)approaches. These observations and the fact that the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T)equations and their standard CCSD[T] and CCSD(T) analogs are formally similar are reflected in the similaritiesbetween the computer codes performing the standard and renormalized CC calculations. The detailed expressionsbehind our computer codes for the CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods that allow us to understand the relative costs of all of these approaches (both in terms of themost expensive steps and in terms of the memory requirements) are discussed next.

3. Detailed algebraic expressions for the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T)methods

Before presenting the actual algebraic expressions for the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) energies in terms of the spin-free cluster amplitudestai and tabij and one- and two-electron molecular

integralsf pq and v

pqrs , it is useful to organize all of these energy expressions, so that we can fully utilize the

similarities between different methods discussed in the previous section, when writing computer codes. Let usdefine the following “numerator” quantities:

NCR[T] = 〈Φ|(T [2]3

)†M3(2)|Φ〉 (31)

and

NCR(T) = NCR[T] + 〈Φ|(Z3)†M3(2)|Φ〉, (32)

whereT [2]3 andZ3 are defined by Eqs. (19) and (20), respectively, andM3(2)|Φ〉 are the projections of the CCSD

equations on triexcited configurations, Eqs. (13) and (16). By replacing theMabcijk (2) moments in Eqs. (31) and

(32) by their lowest-order〈Φabcijk |(VNT2)C |Φ〉 estimates, we can define two additional “numerator” quantities,

N [T] = 〈Φ|(T [2]3

)†(VNT2)C |Φ〉 (33)

and

N(T) = N [T] + 〈Φ|(Z3)†(VNT2)C |Φ〉. (34)

Clearly,

N [T] = E[4]T (35)

and

N(T) = E[4]T +E

[5]ST, (36)

whereE[4]T andE[5]

ST are the noniterative corrections used to define the standard CCSD[T] and CCSD(T) approaches(cf. Eqs. (23)–(26)).


The above numerator quantities,N [T],N(T),NCR[T], andNCR(T), and the denominator expressions (29) and (30)allow us to write the following equations for the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T)energies:

ER-CCSD[T] = ECCSD+N [T]/D[T], (37)

ER-CCSD(T) = ECCSD+N(T)/D(T), (38)

ECR-CCSD[T] = ECCSD+NCR[T]/D[T], (39)

ECR-CCSD(T) = ECCSD+NCR(T)/D(T). (40)

The standard CCSD[T] and CCSD(T) formulas are obtained by simply ignoring theD[T] andD(T) denominatorsin Eqs. (37) and (38),

ECCSD[T] = ECCSD+N [T], (41)

ECCSD(T) = ECCSD+N(T). (42)

Clearly, in order to calculate all of the above energies, we must first solve the CCSD equations to determinecluster amplitudestai andtabij , which enter the expressions forN [T], N(T), NCR[T], NCR(T), D[T], andD(T), and to

calculate the CCSD energy,ECCSD. Thus, we begin our presentation of the explicit expressions for the R-CCSD[T],R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) energies and their standard CCSD[T] and CCSD(T) counterpartsby discussing the CCSD equations. The expressions for theN [T], N(T), NCR[T], andNCR(T) numerators and theD[T] andD(T) denominators are discussed afterwards.

3.1. The spin-free CCSD equations

The most efficient formulation of the restricted CC theory, employing the closed-shell configuration|Φ〉 as areference, is obtained by employing the spin-adapted rather than the more conventional spin-orbital diagrammaticapproach (see, e.g., Refs. [8–13,16,63,68,69,73–75]). Essentially, all techniques of spin adaptation of CC equationsfall into one of the following two classes: (i) methods employing the orthogonal spin adaptation (in diagrammatic[11–13,73–75] or algebraic [76] form) or (ii) methods employing the spin-free or nonorthogonally spin-adaptedformalism, preferably in a diagrammatic form [72] (see, e.g., Refs. [8–10,63,68,69]). In principle, the procedureof orthogonal spin adaptation [11–13,73–76] is a better procedure. In the orthogonally spin-adapted approaches,the number of independent cluster amplitudes is reduced to an absolute minimum, the resulting cluster amplitudeshave a high permutational symmetry with respect to interchanges of occupied and unoccupied orbital labels [11–13,73–76], and the orthogonally spin-adapted CC equations have the sparsest possible form [77]. However, theadvantages of the orthogonal spin adaptation over the more straightforward spin-free formulation of CC theoryprimarily apply to the iterative CC methods with triples and other higher-than-double excitations [75], where thespin-free formalism yields linear dependencies among higher-than-doubly excited cluster amplitudes [63,75]. Thenumbers of the linearly independent singly and doubly excited cluster amplitudes in the spin-free and orthogonallyspin-adapted methods are identical. For these reasons, in deriving and coding the CCSD equations, we decidedto use the spin-free formulation of CC theory, based on generating the Goldstone orbital diagrams, in which weassociate an additional factor of 2l , wherel is the number of closed loops, to each of the resulting diagrams [72].

The final CCSD equations, resulting from applying the above rules to Eqs. (3) and (4) and programmed by us,can be given the following compact form:

Dai t

ai = f a

i + Iae tei − I ′m

i tam + Ime(2teami − teaim

) + (2vma

ei − vamei)tem

− vmnei

(2teamn − taemn

) + vmaef

(2tefmi − t

ef

im

), (43)

for the equations projected on singly excited configurations, and


Dabij tabij = vabij +P(ia/jb)

[taeij I be − tabimI

mj + 1

2vabef c

efij + 1

2cabmnI

mnij

− taemj Imbie − Ima

ie tebmj + (2teami − teaim

)Imbej + tei I

′abej − tamI

′mbij

], (44)

for the equations projected on doubly excited configurations. As mentioned earlier, we use a notation in whichfpq = 〈p|f |q〉 andvpqrs = 〈pq|v|rs〉 are the one- and two-electron molecular integrals corresponding to the one-

and two-body parts of the HamiltonianHN , FN andVN , respectively (FN is the Fock operator in the normal-product form). We also use (throughout the entire paper) the Einstein summation convention over repeated upperand lower indices. TheDa

i andDabij terms are the standard MBPT denominators; in general,

Da...i... = εi − εa + · · · , (45)

where

εp = fpp (46)

are the diagonal elements of the Fock matrix. Thecabij coefficients are defined as

cabij = tabij + tai tbj , (47)

and operatorP(ia/jb), when acting on some expression, implies a sum of two expressions of the same typediffering by a permutation of the(i, a) and(j, b) pairs; in general,

P(pq/rs)= 1+ (pr)(qs), (48)

where(pq) is a transposition of indicesp andq . The recursively generated intermediatesIab , I ij , I ′ ij , I ia , I ijkl , I

iajb ,

I iabj , I ′abci , I ′ ia

jk that enter Eqs. (43) and (44) and the basic intermediates that are needed to form them are definedin Table 1 (along with all other intermediates that are needed to calculate the CCSD[T], CCSD(T), R-CCSD[T],R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) energies).

Once the system of nonlinear algebraic equations, Eqs. (43) and (44), is solved fortai andtabij , we calculate theCCSD energy using a formula (cf. Eq. (5)),

ECCSD= 〈Φ|H |Φ〉 +ECCSD, (49)

where

ECCSD= 2f ia t

ai + (

2vijab − vijba

)cabij . (50)

The CCSD equations, Eqs. (43) and (44), are solved iteratively. As in all earlier implementations of the spin-freeCCSD method (cf., e.g., Refs. [8–10]), the most expensive steps, related to the construction of thevabef c

ef

ij term for

i < j and all values ofa andb in Eq. (44), scale as12n2on

4u. Information about the algorithm used by us to solve

Eqs. (43) and (44) and the memory requirements defining to the CCSD calculations based on Eqs. (43) and (44)are discussed in Section 4.2.

3.2. The spin-free expressions for N [T], N(T), NCR[T], and NCR(T)

According to Eqs. (37)–(42), calculations of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T)energies and their CCSD[T] and CCSD(T) analogs require an evaluation of theN [T], N(T), NCR[T], andNCR(T)

numerators. The spin-free expressions for these numerators, in terms of molecular integrals and CCSD amplitudestai andtabij , used by us to code the standard and renormalized energy corrections due to triples, are as follows:

N [T] ≡ E[4]T = t

ijkabc(2) t

abcijk (2)D

abcijk , (51)

N(T) ≡ E[4]T +E

[5]ST = N [T] + z

ijkabc t

abcijk (2)D

abcijk , (52)


Table 1Intermediates used in the CCSD, CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) equations

Intermediate Expressiona

I ia f ia + 2vimae t

em − vimea t

em

Iab

(1− δab)f a

b+ 2vam

betem − vma

betem − 2vmn

ebceamn + vmn

beceamn − tamfm

b

I ij I ′ ij + I ie t

ej

I ′ ij (1− δij )f

ij + 2vimje t

em − vimej tem + vmi

ef tefmj − vimef t

efmj

χaibc

χ ′aibc

− 12 t

amvmi

bc

χ ′aibc

vaibc

− 12 t

amvmi

bc

χijka v

ijka + tek v

ijea

Iijkl

vijkl

+ vijefcefkl

+ P (ik/j l)tekvijel

I iajb viajb − 12v

imeb c

eajm − vimjb t

am + viaebt

ej

χ ′ iajb viajb − 1

2vimjb t

am + tej χ

′aibe

χ ′′ iajb

viajb

− vimjb

tam + 12 t

ejχaibe

I iabj

viabj

+ vimbe

teamj

− 12v

iemb

taejm

− 12v

imbe

caemj

+ viabetej

− vimbj

tam

χiabj viabet

ej

χ ′ iabj

viabj

− 12v

imbj

tam + χ ′aieb

tej

χ ′′ iabj

viabj

− vimbj

tam + 12χ

aiebtej

I ′abci

vabci

− vamci

tbm − tamvmbci

I ′′abci vabci + vabce t

ei − χ ′ma

ic tbm − tamχ ′mbci − Imc tabmi + 2χam

ce tbeim − χamce tebim − χam

ec tebmi − teaimχbmec + tbamnχ

mnic

I ′ iajk

viajk

+ viaeftefjk

+ tejχiaek

I ′′ iajk

viajk

− vimjk

tam + χ ′′ iaje

tek

+ tejχ ′′ iaek

+ 2χimje

taekm

− χimje

teakm

− χmije

teamk

− taemj

χmike

+ tefkj

χaief

a Summation over repeated upper and lower indices is assumed. For definitions ofcabij

andP (ik/j l), see Eqs. (47) and (48),

respectively.δpq designates the usual Kronecker delta.

NCR[T] = tijkabc(2)M

abcijk (2), (53)

NCR(T) = NCR[T] + zijkabcM

abcijk (2), (54)

where for any six-index quantityxijkabc we define

xijk

abc = 43x

ijk

abc − 2xijkacb + 23x

ijk

bca, (55)

and where we again use the Einstein summation convention over repeated upper and lower indices.The tabcijk (2) coefficients are the spin-free amplitudes corresponding toT

[2]3 , Eq. (19),

Dabcijk tabcijk (2)= P(ia/jb/kc)

[taeij vbcek − tabimv

mcjk

], (56)

where (cf. Eq. (45))Dabcijk = εi − εa + εj − εb + εk − εc . The symmetrizerP(ia/jb/kc) is defined in a similar way

to P(ia/jb) (cf. Eq. (48)),

P(ia/jb/kc)= 1+ (ij)(ab)+ (ik)(ac)+ (jk)(bc)+ (ijk)(abc)+ (ikj)(acb), (57)

where(pqr) represents a cyclic permutation of indicesp, q , andr. Thet ijkabc(2) coefficients, needed to formt ijkabc(2)according to Eq. (55), are complex conjugates oftabcijk (2),

tijkabc(2) = [

tabcijk (2)]∗. (58)


In general,

ui...a... =(ua...i...

)∗, (59)

for any quantityua...i... . In particular, thezijkabc coefficients, needed to formzijkabc in Eqs. (52) and (54) according toEq. (55), are complex conjugates of thezabcijk amplitudes defining operatorZ3, Eq. (20). We have,

zijk

abc ≡ (zabcijk

)∗ = (t iav

jk

bc + tj

b vikac + tkc v

ij

ab

)/Dabc

ijk , (60)

where, according to our notation,

t ia = (tai

)∗. (61)

The Mabcijk (2) quantities, entering Eqs. (53) and (54) and, thus, the CR-CCSD[T] and CR-CCSD(T) energy

formulas, are the generalized moments of the spin-free CCSD equations corresponding to projections of theseequations on triply excited configurations (cf. Eq. (13)). They can be calculated using the following expression:

Mabcijk (2) = P(ia/jb/kc)

[taeij I ′′bc

ek − tabimI′′mcjk

], (62)

where the recursively generated intermediatesI ′′abci andI ′′ ia

jk and the basic intermediates that are needed to form

I ′′abci andI ′′ ia

jk are defined in Table 1.

The most expensive steps in calculating theN [T],N(T), NCR[T], andNCR(T) numerators are then3on

4u steps. They

appear in the final construction of thetabcijk (2) amplitudes (see Eq. (56)) andMabcijk (2) moments (see Eq. (62)). The

N [T] andN(T) numerators, defining the standard [T] and (T) corrections and their R-CCSD[T] and R-CCSD(T)analogs, require that we only construct thetabcijk (2) amplitudes (see Eqs. (51) and (52)). TheNCR[T] andNCR(T)

numerators, defining the CR-CCSD[T] and CR-CCSD(T) corrections, require that we construct both thetabcijk (2) [or

tijk

abc(2)] amplitudes and momentsMabcijk (2) (see Eqs. (53) and (54)). Thus, the cost of the CR-CCSD[T] and CR-

CCSD(T) calculations, in their part dealing with the noniterative triples corrections, is twice the cost of calculatingthe standard [T] and (T) corrections. A very similar remark applies to memory requirements, although memoryrequirements for the CR-CCSD[T] and CR-CCSD(T) calculations can be further reduced if we switch to the moreeconomical version of our code discussed in Section 4.2. We illustrate the actual costs of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) calculations by showing a numerical example in Section 5.

3.3. The spin-free expressions for D[T] and D(T)

According to Eqs. (37)–(40), calculations of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T)energies require an evaluation of theD[T] andD(T) denominators, Eqs. (29) and (30). These are rather trivialquantities that have very little effect on the CPU timings. The spin-free expressions for denominatorsD[T] andD(T) in terms of the CCSD amplitudestai andtabij can be given the following form (again, the Einstein summationconvention is used):

D[T] = 1+ 2t ia tai + (

2t ijab − tijba

)cabij + t

ijkabc(2) y

abcijk , (63)

D(T) = D[T] + zijkabc y

abcijk , (64)

wherecabij andt ia are defined by Eqs. (47) and (61), respectively, and, according to our notation (cf. Eq. (59)),

tijab = (

tabij)∗. (65)

The t ijkabc(2) andzijkabc quantities have already been introduced in Section 3.2 and

yabcijk = tai tbj t

ck + tai t

bcjk + tbj t

acik + tck t

abij (66)


are the amplitudes defining the triexcited component of the CCSD wave function enteringD[T] andD(T) (cf. the(T1T2 + 1

6T31 ) terms in Eqs. (29) and (30)). It is clear from Eqs. (63) and (64) that the calculation ofD[T] and

D(T), with its n3on

3u scaling, is inexpensive compared to the most expensive CCSD steps and steps involved in the

determination ofN [T], N(T), NCR[T], andNCR(T). No extra memory is required to formD[T] andD(T), since wecan calculate these denominators “on the fly”, when determining thecabij , t ijkabc(2), andzijkabc quantities in other partsof the calculation.

4. Implementation of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods andstandard CC approaches in GAMESS

The CCSD, CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods havebeen implemented and interfaced with theGAMESSpackage [67]. In addition to the above methods, our codes canperform the LCCD and CCD calculations. The CCD equations can easily be obtained from the CCSD equations byignoring the equations corresponding to projections on singly excited configurations, Eq. (43), and by eliminatingthetai -containing terms from the equations corresponding to projections on doubly excited configurations, Eq. (44).The LCCD equations are obtained by eliminating the nonlinear (i.e. bilinear) terms from the CCD equations.

4.1. General organization of the CC program in GAMESS

Two new source files,ccsdt.src and ccaux.src, have been added to theGAMESS distribution. Theccsdt.src file contains the main CC driver,CCDRVR, which is called from the mainGAMESS modulegamess.src [67] (through a new subroutineWFNCC written by Dr. M.W. Schmidt). TheCCDRVR driver calls asequence of subroutines performing various CC calculations. Theccaux.src file contains subroutines that areused to solve the iterative CCSD (or (L)CCD) equations as well as various auxiliary subroutines facilitating thealgebraic operations that must be performed to calculate the noniterative [T] or (T) corrections and intermediateslisted in Table 1. Those include the subroutines that add and transpose matrices, multiply them by a scalar, copy onematrix into another, and interchange designated columns of multi-dimensional arrays, to mention a few examples.

The main computational effort, i.e. the construction of the right-hand sides of the CCSD equations, Eqs. (43)and (44), the construction of momentsMabc

ijk (2), Eq. (62), and amplitudestabcijk (2), Eq. (19), and the construction ofall relevant intermediates listed in Table 1, is tremendously facilitated by usingDGEMM, a fast matrix multiplicationroutine belonging to the Basic Linear Algebra Subprograms (BLAS). In this way, all essential operations relatedto CC calculations are reduced to a number of fully vectorized matrix multiplications. The use of recursivelygenerated intermediates, which enabled us to rewrite all of our equations in a completely linearized form, and theuse ofDGEMM have resulted in a highly efficient computer implementation of the standard and renormalized CCmethods discussed in this paper.

The most important CC modules, all called from the main driverCCDRVR, areDRSRTING (integral sorting forthe CC calculations),DRCCSD (solving the LCCD, CCD, or CCSD equations), and three subroutines calculatingthe noniterative [T] or (T) corrections, i.e.DRT3WT2,DRINTRI, andDRT3WT2N. TheDRINTRI andDRT3WT2Nsubroutines are executed only if we are interested in performing the CR-CCSD[T] and CR-CCSD(T) calculations.The memory requirements for all of these subroutines and some information about their function in our CC codesas well as the input parameters that control CC calculations are discussed next.

4.2. Input parameters controlling CC calculations, main subroutines and their function, and memoryrequirements for CC calculations

The first program called by the main driverCCDRVR is DRSRTING. TheDRSRTING routine calls subroutinesCCFOCK andSRTING (both of them written by Dr. M.W. Schmidt) in order to resort one- and two-electron


molecular integrals,f pq and v

pqrs , respectively, obtained after performing the RHF calculations and after

transforming atomic integrals to molecular orbital basis. The purpose of this resorting is the necessity of having thefpq andvpqrs integrals reordered by arranging them in groups according to the number of occupied orbital indices,

so that they can be accessed by our CC codes in a highly efficient manner. The final product ofDRSRTING is adirect access file, which contains the following types of transformed integrals:

(i) vabij /Dabij (no records numbered byi);

(ii) vabij (no records numbered byi);

(iii) vaibj (no records numbered byi);

(iv) vabci (no records numbered byi);(v) vaijk (nu records numbered bya);

(vi) vij

kl (no records numbered byi);(vii) vabcd (nu records numbered bya);(viii) f i

j , i �= j (one record);(ix) f a

b , a �= b (one record); and(x) f i

a (one record).

The total number of records in this file is 5no + 2nu + 3. The diagonal elements of the Fock matrix (f ii = εi and

f aa = εa ) are kept in memory. The total memory requirement for resorting the integrals for the CC calculations is

∼non3u words.

TheDRCCSD routine calls subroutineCCSD, which constructs and solves the LCCD, CCD, or CCSD equations.The CCSD (or (L)CCD) equations are solved iteratively with the help of the direct inversion of the iterativesubspace (DIIS) method of Pulay [78–80] (for the first application of the DIIS procedure to solving the CCequations, see Ref. [81]). A few input parameters, namely,mxdiis, iconv, maxcc, and irest control the LCCD,CCD, and CCSD calculations. The memory requirements associated with solving the CCSD equations are shownin Table 2.

The algorithm for solving the CCSD (or (L)CCD) equations implemented in our codes can de described asfollows: We initiate the process of solving the CC equations by providing the zero-order guess fortai andtabij . The

default zero-order guess is obtained by using the first-order MBPT estimates oftai andtabij . The subsequent values

of tai andtabij are obtained by applying the Jacobi procedure. Thus, if(n)t represents the vector of cluster amplitudes(n)tai and(n)tabij , obtained in thenth iteration of the iterative procedure, and ifΛa

i [(n)t] andΛabij [(n)t] are the right-

hand sides of the CCSD equations, Eqs. (43) and (44), calculated using amplitudes(n)t , then the values of clusteramplitudes in the(n+ 1)st iteration are

(n+1)tai = Λai

[(n)t

]/Da

i , (67)

(n+1)tabij = Λabij

[(n)t

]/Dab

ij . (68)

Convergence is achieved if the maximum values of|(n+1)tai − (n)tai | and|(n+1)tabij − (n)tabij | become smaller than

the prescribed convergence threshold 10−iconv. The default value of the input variableiconv is 7.It is well known that the Jacobi procedure converges very poorly or diverges, when applied to CC equations.

Thus, we combine the Jacobi algorithm with the DIIS extrapolation method in which, in addition to calculating(n+1)tai and (n+1)tabij via Eqs. (67) and (68), we extrapolate the new values oftai and tabij from the most recent

vector(n+1)t and a few vectors(m)t calculated in the earlier iterations (m � n). The maximum number of vectors(m)t and, at the same time, the dimension of a small system of linear equations that has to be solved in the

84P.P

iecuchetal./C

omputer

Physics

Com

munications

149(2002)

71–96

Table 2Memory requirements (in double precision words) and functions of the most important routines used in the present implementation of the CCSD, CCSD[T], CCSD(T),R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods (see the text for further details)

Routine Method Called by Memory requirement Function

CCSD LCCD, CCD, CCSD, DRCCSD MEM1 + MEM2 + MEM3 + MEM4, Constructs and solves

CCSD[T], CCSD(T), MEM1 = no + nu, MEM2 = n2o + 3nonu + n2

u, the LCCD, CCD, or

R-CCSD[T], R-CCSD(T) MEM3 = n3u, MEM4 = n4

o + n3onu + 4n2

on2u + non

3u CCSD equations

CR-CCSD[T], CR-CCSD(T)

T3WT2 CCSD[T], CCSD(T), DRT3WT2 MEM1 + MEM2 + MEM3 + MEM4, Constructstabcijk

(2), t ijkabc

(2),

R-CCSD[T], R-CCSD(T) MEM1 = no + nu, MEM2 = 2nonu, andzijkabcCR-CCSD[T], CR-CCSD(T) MEM3 = 2n3

u, MEM4 = n3onu + 2n2

on2u + non

3u

INTQUA CR-CCSD[T], CR-CCSD(T) DRINTRI MEM2 + MEM4, Constructs intermediates

MEM2 = nonu, MEM4 = max(n3onu,n

3u)+ n2

on2u + non

3u except forI ′′ab

ciandI ′′ ia

jk

INTRIPLa CR-CCSD[T], CR-CCSD(T) DRINTRI MEM2 + MEM4, ConstructsI ′′abci


3u)+ n2

on2u + 2non3

u

INTRIPa CR-CCSD[T], CR-CCSD(T) DRINTRI MEM2 + MEM4, ConstructsI ′′abci


3u)+ n3

onu + n2on

2u + non

3u with less memory

INTRIH CR-CCSD[T], CR-CCSD(T) DRINTRI MEM2 + MEM4, ConstructsI ′′ iajk


3u)+ 2n3

onu + n2on

2u + non

3u

T3WT2N CR-CCSD[T], CR-CCSD(T) DRT3WT2N MEM1 + MEM2 + MEM3 + MEM4, CalculatesMabcijk (2)

MEM1 = no + nu, MEM2 = nonu,

MEM3 = 2n3u, MEM4 = n3

onu + 2n2on

2u + non

3u

a INTRIP is a variant ofINTRIPL, which uses less memory and which is executed instead ofINTRIPL if memory required byINTRIPL is not available.


DIIS extrapolation procedure are defined by the input variablemxdiis. The default values ofmxdiis are definedas follows:

mxdiis =

5, for 6 � nonu,

3, for 3 � nonu < 6,0, for nonu < 3.

(69)

Thus, in the vast majority of cases, the default value ofmxdiis is 5. However, for very small problems (definedhere by small values ofnonu), when the DIIS expansion subspace leads to singular systems of linear equations, itis necessary to reduce the value ofmxdiis to 2–4 (we chose 3) or switch off DIIS completely (which is the casewhenmxdiis is set to 0). Our choice of the defaultmxdiis values of 5, 3, and 0 is based on a large number of testcalculations that we performed in order to determine the optimum choices formxdiis leading to the overall fastestconvergence.

Our code is very good in converging the CCSD equations, even when we use the default zero-order guess fortaiandtabij . This is related to the fact that we want to be able to calculate entire PESs involving bond breaking, whereit is not unusual to obtain large cluster amplitudes whose absolute values are close to 1. As demonstrated in ourearlier papers [23,24,26,28,29,54], the CR-CCSD[T] and CR-CCSD(T) approaches are capable of eliminating theunphysical humps on the PESs obtained with the standard CCSD[T] and CCSD(T) methods and large errors in theCCSD results by adding the completely renormalized [T] or (T) corrections to (poor) CCSD energies. Thus, wedesigned our CCSD codes to converge in relatively few iterations for significantly stretched nuclear geometries,say, 3–5 times the equilibrium bond length for molecules with single bonds and 2–3 times the equilibrium bondlength for molecules with multiple bonds. We do not recommend applying the CR-CCSD[T] and CR-CCSD(T)approaches to cases of multiple bond breaking (in this case, one should resort to the completely renormalizedCCSD(TQ) and CCSDT(Q) approaches [25,27,29] or to the newer MMCC(2,6) method [82]), but the fact thatwe can, for example, converge the CCSD equations for twice the equilibrium bond length in N2 (in this case, theCCSD potential energy curve is completely unphysical [25]) with the default initial guess andmxdiis = 3 to within10−7 hartree in∼30 iterations shows that we have a rather robust procedure for converging CC equations. This willpay back future dividends, when we will augment the present code with the highly efficient routines calculatingthe completely renormalized CCSD(TQ) corrections. These corrections can be used to obtain very good results formultiple bond breaking, in spite of the complete failure of the CCSD approach [25,27,29]. For molecules near theirequilibrium geometries, our codes converge the CCSD equations to within 10−6–10−7 hartree usually in 10–20iterations.

It may, of course, happen that our procedure for solving the CCSD equations does not converge, in spite ofincreasing the maximum number of iterations (input variablemaxcc; the default value is 30) and in spite of changingthe default value ofmxdiis. In order to facilitate the calculations in all such cases, we included the restart option inour codes. We can restart a CCSD (or (L)CCD) calculation from the restart file created by the earlier calculation.This option is invoked by resetting the input variableirest (the default value is 0, i.e. no restart) to some valuegreater than or equal 3 (in general, the value ofirest is used to set the iteration counter in the restarted calculation).Examples of using the restart option include the following situations:

• The CCSD program did not converge inmaxcc iterations, but there is a chance to converge it if the value ofmaxcc is increased. User restarts the calculation with the increased value ofmaxcc.

• User ran a CCSD calculation, obtaining the converged CCSD energy, but later decided to run a CR-CCSD(T)calculation. Instead of running the entire CCSD→ CR-CCSD(T) sequence again, user restarts the calculationafter changing the value of input variablecctyp, defining the type of calculation, fromcctyp = ccsd tocctyp = cr-cc (for values ofcctyp, cf. the discussion below). The CCSD program starts from the convergedamplitudes, converging in one iteration, and the program immediately proceeds to the calculation of the desiredCR-CCSD(T) correction.


• The CCSD program diverged for some significantly stretched geometry. User performs an extra calculation fora different nuclear geometry, for which it is easier to converge the CCSD equations, and restarts the calculationfrom the restart file generated by an extra calculation. This technique of restarting the CC calculations fromthe cluster amplitudes obtained for a neighboring nuclear geometry is particularly useful for scanning PESsand for calculating energy derivatives by numerical differentiation.

Once the CCSD equations are solved, we can correct the results of the CCSD calculations by addingthe standard, renormalized, or completely renormalized [T] and (T) corrections to the CCSD energies. Thesecorrections are calculated by the following three subroutines:DRT3WT2, DRINTRI, and DRT3WT2N. Asmentioned earlier, theDRINTRI andDRT3WT2N routines are executed only when we are interested in the CR-CCSD[T] and CR-CCSD(T) calculations.

TheDRT3WT2 routine calls theT3WT2 subroutine.T3WT2 calculates thetabcijk (2) amplitudes definingT [2]3 (see

Eqs. (19) and (56)) and thezabcijk amplitudes definingZ3, Eq. (20) (or, actually, their complex conjugateszijkabc

defined by Eq. (60)). From the description provided in Sections 2 and 3, it immediately follows that amplitudestabcijk (2) andzijkabc are the only quantities that are needed to construct theN [T] andN(T) numerators of the CCSD[T],

CCSD(T), R-CCSD[T], and R-CCSD(T) methods. The calculation of the more expensivetabcijk (2) amplitudes,

requiring then3on

4u steps, and the calculation ofzijkabc are performed in a loop overi, j , andk (split into three

different cases:i > j > k, i = j > k, and i > j = k). The memory requirements associated with theT3WT2subroutine are shown in Table 2. If we ignore smaller arrays, the memory requirement forT3WT2, i.e. the memoryrequirement for the CCSD[T], CCSD(T), R-CCSD[T], and R-CCSD(T) calculations, isnon

3u words. Thus, if we

can perform the CCSD calculations, we can also perform the entire set of the standard CCSD[T] and CCSD(T) andrenormalized R-CCSD[T] and R-CCSD(T) calculations with essentially the same memory.

The CR-CCSD[T] and CR-CCSD(T) calculations require that, in addition to thetabcijk (2) andzabcijk amplitudes

(or, more precisely, theirt ijkabc(2) and zijkabc analogs; cf. Section 3), we construct momentsMabcijk (2), Eq. (62). As

in the CCSD[T], CCSD(T), R-CCSD[T], and R-CCSD(T) calculations, thetijk

abc(2) andzijkabc amplitudes, needed toconstruct theNCR[T] andNCR(T) numerators of the CR-CCSD[T] and CR-CCSD(T) approaches, are calculatedby subroutineT3WT2 (called fromDRT3WT2). MomentsMabc

ijk (2) are constructed by subroutinesINTQUA,INTRIPL (or INTRIP), and INTRIH, called fromDRINTRI, and T3WT2N, called fromDRT3WT2N. ThesubroutineINTQUA calculates all intermediates that are needed to construct numeratorsNCR[T] andNCR(T), exceptfor the highest-rank intermediatesI ′′ab

ci andI ′′ iajk , which appear in the final formula forMabc

ijk (2), Eq. (62). These

highest-rank intermediates are calculated byINTRIPL or INTRIP (I ′′abci ) andINTRIH (I ′′ ia

jk ). OnceI ′′abci and

I ′′ iajk are constructed, momentsMabc

ijk (2) are calculated using Eq. (62). This is done by subroutineT3WT2N. As in

the case oftabcijk (2) andzijkabc, the calculation ofMabcijk (2) is performed in a loop overi, j , andk (split into three

different cases:i > j > k, i = j > k, andi > j = k).The difference between subroutinesINTRIPL andINTRIP, which are used to calculate theI ′′ab

ci intermediates,is in memory requirements and performance. The fasterINTRIPL routine calculates intermediatesI ′′ab

ci using∼2non3

u words of memory. The need for∼2non3u words is, in this case, a consequence of having to allocatenon

3u

words forχaibc andI ′′ab

ci in the process of constructingI ′′abci (see Table 1). We can, however, rewrite the formula

for I ′′abci , so that only∼non

3u words of memory are used, but this leads to a slower algorithm. It is useful, however,

to have a more economical version of a routine calculatingI ′′abci , since this may allow the user to perform the CR-

CCSD[T] and CR-CCSD(T) calculations even when the memory of 2non3u words is not available. In the present

implementation of the CR-CCSD[T] and CR-CCSD(T) methods, our code compares the available memory withthe memory needed for theINTRIPL andINTRIP runs and automatically chooses the appropriate routine.

TheINTQUA, INTRIPL (or INTRIP), INTRIH, andT3WT2N subroutines are called sequentially, so that wenever have to allocate more memory than∼2non3

u (or ∼non3u) words, when running the CR-CCSD[T] and CR-


CCSD(T) calculations. The complete description of memory requirements associated with theINTQUA,INTRIPL(orINTRIP),INTRIH, andT3WT2N subroutines is given in Table 2. As mentioned in Section 3, the CR-CCSD[T]and CR-CCSD(T) methods aren3

on4u procedures in the noniterative [T] steps and the cost of computing the

CR-CCSD[T] and CR-CCSD(T) triples corrections is only twice the cost of calculating the standard [T] or (T)corrections.

Finally, let us briefly discuss the input variablecctyp, which defines the type of CC calculation. From the generalinformation provided in Sections 2 and 3 (particularly, from Eqs. (37)–(42)), it immediately follows that it is quitenatural to perform the standard, renormalized, and completely renormalized CCSD[T] and CCSD(T) calculationsusing the following grouping of methods:

• Option 1 (cctyp = ccsd(t)). The standard CCSD[T] and CCSD(T) calculations are performed together. Thismakes sense, since the standard CCSD[T] and CCSD(T) calculations do not require constructing theD[T] andD(T) denominators and the cost of addingE

[5]ST to E

[4]T , required to construct the (T) correction, is very small

compared to the cost of constructing theE[4]T correction, present in both approaches (cf. Eqs. (41) and (42)).

• Option 2 (cctyp = r-cc). The standard and renormalized (R) CCSD[T] and CCSD(T) calculations areperformed together. This makes sense, since the only difference between the R-CCSD[T] and R-CCSD(T)expressions and their standard CCSD[T] and CCSD(T) counterparts lies in theD[T] andD(T) denominators,whose calculation takes very little time (cf. Eqs. (37), (38), (41), and (42)).

• Option 3 (cctyp = cr-cc). The CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) calculations are performed together. Clearly, calculation of the CR-CCSD[T] and CR-CCSD(T)corrections requires computing thetabcijk (2) andzabcijk amplitudes or theirt ijkabc(2) andzijkabc analogs (cf. Section 3).Those quantities are needed to construct the CCSD[T], CCSD(T), R-CCSD[T], and R-CCSD(T) correctionsas well as the CR-CCSD[T] and CR-CCSD(T) corrections. TheD[T] andD(T) denominators appearing inthe R-CCSD[T] and R-CCSD(T) expressions must be calculated as well, when we are interested in the CR-CCSD[T] and CR-CCSD(T) energy values. Thus, it would make no sense to calculate the CR-CCSD[T] andCR-CCSD(T) energies alone, without providing an information about the CCSD[T], CCSD(T), R-CCSD[T],and R-CCSD(T) energies.

The above grouping of methods is reflected in the choices of input variablecctyp that our program offers to theuser. The complete set of method choices through variablecctyp, offered by our program, and all other options thatallow the user to control the CC calculations using the code incorporated inGAMESS, are listed in Table 3.

As already mentioned, the cost of performing the R-CCSD[T] and R-CCSD(T) calculations is practically thesame as the cost of the standard CCSD[T] and CCSD(T) calculations. Thus, the choicecctyp = r-cc gives fivedifferent energies (CCSD, CCSD[T], CCSD(T), R-CCSD[T], and R-CCSD(T)) for the price of three (CCSD,CCSD[T], and CCSD(T)). The R-CCSD[T] and R-CCSD(T) approaches improve the standard CCSD[T] andCCSD(T) results at intermediate internuclear separations, but they fail at larger distances [23,24,26,28,29,54].The CR-CCSD[T] and CR-CCSD(T) methods are much better in this regard, since they provide a very gooddescription of single bond breaking at all internuclear separations [23,24,26,28,29,54]. This includes variouscases of unimolecular dissociations [23,24,26,28,29] and exchange chemical reactions, in which single bondsbreak and form [54]. The cost of calculating the CR-CCSD[T] and CR-CCSD(T) triples corrections is twice thecost of calculating the standard CCSD[T] and CCSD(T) corrections. Thus, by just doubling the CPU time forthe noniterative triples correction and by selectingcctyp = cr-cc, we gain access to all six noniterative triplescorrections (the CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) energies) plus,of course, to the CCSD energy. A few examples illustrating the performance of the CR-CCSD[T] and CR-CCSD(T)methods and related timings are discussed in the next section.

In addition to various CC energies, our program prints out the values of the largest cluster amplitudes andnorms of the amplitude vectors. It also performs the popularT1 diagnostics [83], which may help the user to

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Input group Input variable Function Value Meaning

$CONTRL cctyp Chooses a CC calculation after RHF;user must also selectscftyp = rhf

cctyp = none Do not perform any CC calculation (default)

cctyp = lccd Perform the LCCD calculation

cctyp = ccd Perform the CCD calculation

cctyp = ccsd Perform the CCSD calculation

cctyp = ccsd(t) Perform the CCSD, CCSD[T], and CCSD(T) calculations

cctyp = r-cc Perform the CCSD, CCSD[T], CCSD(T), R-CCSD[T], andR-CCSD(T) calculations

cctyp = cr-cc Perform the CCSD, CCSD[T], CCSD(T), R-CCSD[T],R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) calculations

$CCINP ncore Defines the number of core orbitalsfrozen in CC calculations

ncore = N N lowest-energy orbitals are frozen in CC calculation (thedefault is the number of chemical core orbitals).

nfzv Defines the number of virtual orbitalsdropped from CC calculations

nfzv = N N highest-energy orbitals are dropped from CC calculation (thedefault is 0).

iconv Defines the convergence criterion forcluster amplitudes

iconv = N Program stops if the maximum change in cluster amplitudes is10−N (the default value is 7).

maxcc Defines the maximum number of CCSD(LCCD, CCD) iterations

maxcc =N Program stops if convergence is not achieved afterN iterations(the default is 30).

mxdiis Defines the number of cluster amplitudevectors used for the DIIS extrapolation

mxdiis =N LastN amplitude vectors are used for the DIIS extrapolation.The default values are defined by Eq. (69).

irest Defines the restart option irest = N If N � 3, program restarts from the earlier run (the value ofirestis used to set the iteration counter in the restarted calculation).The default is 0 (no restart).

amptsh Defines a threshold for eliminating smallamplitudes from CC calculations

amptsh = R Amplitudes with absolute values less thanR are set to zero.The default is 0.0d+00 (all amplitudes are retained in thecalculations).

nword Defines a memory to be used by the CCprogram

nword = N N words of memory will be allocated to CC calculation. Thedefault is 0, meaning that all memory allocated to theGAMESS

run can be used by the CC program.


assess the importance of nondynamical correlation in some cases. Some information about the magnitude ofnondynamical correlation is also hidden in the values of theD[T] andD(T) denominators, which we also printout. These denominators are close to 1, when nondynamical correlation is small, and become much larger than 1,when nondynamical correlation effects become large (as is the case for bond breaking) [24,25].

5. Examples of the standard, renormalized, and completely renormalized CCSD[T] and CCSD(T)calculations

The performance of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approachesvs. standardmethods has been discussed in a number of papers [23–26,28,29,54]. Previous tests included potential energycurves of several diatomics (HF [23,24,28,29], BH [26], F2 [26], C2 [29], and N2 [25]), a simultaneous breakingof both O–H bonds in the water molecule [24], vibrational term values of HF, including highly excited states neardissociation [28], and the entire PES for the Be+ HF→ BeF+ H reaction [54]. Thus, in this section we describeonly a few examples and provide some information about the performance of our codes.

In our earlier papers, we have demonstrated that the CR-CCSD[T] and CR-CCSD(T) methods remove thepervasive failing of the RHF-based CCSD[T] and CCSD(T) approaches for ground-state PESs involving thedissociation of a single chemical bond. They also provide very good results for cases, such as a simultaneousbreaking of both O–H bonds in water, but they do not work well for multiple bond breaking (in this case, onehas to use the CR-CCSD(TQ) or CR-CCSDT(Q) methods). The R-CCSD[T] and R-CCSD(T) methods provideimprovements in the results of the standard CCSD[T] and CCSD(T) calculations at intermediate internuclearseparations, but they break down at larger distances.

The CR-CCSD[T] and CR-CCSD(T) methods eliminate unphysical humps that appear on the CCSD[T] andCCSD(T) PESs and reduce large errors in the CCSD[T] and CCSD(T) results at larger internuclear separations to afew millihartree. This is demonstrated in Fig. 1, where we show the results of calculations for a very difficult caseof single bond breaking in F2.

The F2 molecule is a very challenging case for the RHF-based CC approaches (cf. Ref. [18]). For example, theRHF-based CCSD approach performs very poorly, producing a potential well which is almost twice as deep as

Fig. 1. The CCSD, CCSD[T], CCSD(T), CR-CCSD[T], and CR-CCSD(T) potential energy curves for the cc-pVDZ (a) and aug-cc-pVQZ (b)models of F2. The results for the cc-pVDZ basis set, including the potential obtained with the full CCSDT method, are taken from Ref. [26].


that provided by the highly accurate CCSDT approach. Fig. 1(a) shows the results of the standard and completelyrenormalized CCSD[T] and CCSD(T) calculations for the cc-pVDZ [84,85] model of F2, reported in Ref. [26] (weused the Cartesiand functions and froze two lowest-energy core orbitals, so thatno = 7 andnu = 21 in this case).Fig. 1(b) shows similar results obtained with the code described in this work for the much larger aug-cc-pVQZbasis set [85,86] (we used the sphericald , f , andg orbitals and froze two lowest-energy core orbitals, so thatno = 7 andnu = 151 in this case).

As we can see, the well-pronounced humps for the intermediate values of the F–F separationRF–F, produced bythe standard CCSD[T] and CCSD(T) approaches, are gone, when the CR-CCSD[T] and CR-CCSD(T) methods areemployed. We also observe tremendous improvements in the description of various basic characteristics of F2, suchas the calculated dissociation energies. For the cc-pVDZ basis set, the reference CCSDT value of the dissociationenergyDe is 1.22 eV (the experimental value ofDe is 1.66 eV [87,88]). The CCSD calculation gives 2.30 eV, whichis ca. twice the CCSDT value. The CR-CCSD[T] and CR-CCSD(T) methods give 1.32 and 1.34 eV, respectively,in very good agreement with the CCSDT value ofDe. As pointed out in Ref. [26], these results are better thanthe results obtained with the recently proposed and more expensive VOD(2) and OD(2) approximations [50,51],reported in Ref. [51], which include the combined effect ofT3 andT4 clusters through the perturbative expansionfor the similarity-transformed Hamiltonian and which require the orbital optimization at the CC level. Although theCR-CCSD[T] and CR-CCSD(T) values ofDe are also in reasonable agreement with the experimental dissociationenergy, a comparison with the experimental value ofDe is not very meaningful, when a small, cc-pVDZ basisset is employed. Thus, we also performed calculations with the aug-cc-pVQZ basis set consisting of 160 orbitals.In this case, the CCSD value ofDe is 3.18 eV, which is almost twice as much as the experimental value ofDe.The CR-CCSD[T] and CR-CCSD(T) methods give 1.90 and 1.94 eV, respectively, in much better agreement withthe experimental value of 1.66 eV. We must emphasize the fact that the CR-CCSD[T] and CR-CCSD(T) methodsproduce these goodDe values with an ease-of-use of the standard CCSD[T] and CCSD(T) methods, which producecompletely unrealistic potentials. Clearly, it is impossible to define the dissociation energy using the results of thestandard CCSD[T] and CCSD(T) calculations, because of the presence of the big humps on the CCSD[T] andCCSD(T) curves (cf. Fig. 1).

The values of various equilibrium properties, such as the equilibrium bond lengthre , provided by the CR-CCSD[T], CR-CCSD(T), R-CCSD[T], and R-CCSD(T) methods, are similar to those obtained in the standardCCSD[T] and CCSD(T) calculations. The standard CCSD[T], and CCSD(T) methods give there values of 1.4148and 1.4130 Å, respectively, when the aug-cc-pVQZ basis set is employed. The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods give 1.4091, 1.4077, 1.4076, and 1.4063 Å, respectively. The experimentalresult is 1.4119 Å. This shows that the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods areuseful alternatives to the standard CCSD[T] and CCSD(T) methods. They provide the results of the CCSD[T] orCCSD(T) quality in the vicinity of the equilibrium geometry, while considerably improving the description of PESin the bond breaking region.

Another example of the excellent performance of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods is provided in Fig. 2 and Table 4. We used the standard CCSD, CCSD[T], and CCSD(T)approaches and the new R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods to calculate allvibrational term values of HF (for the experimental or RKR [89–91] values, see, e.g., Ref. [92]). Potential energycurves, needed to generate those term values, were calculated with the aug-cc-pVQZ basis set (for a similar studyof the vibrational term values of HF with the smaller aug-cc-pVTZ basis set, see Ref. [28]). We froze the lowest-energy core orbital and used the spherical components of thed , f , andg orbitals, so thatno = 4 andnu = 121 inthis case.

Potential energy curves resulting from our calculations are shown in Fig. 2. As in the F2 case, the CR-CCSD[T]and CR-CCSD(T) methods eliminate the humps on the CCSD[T] and CCSD(T) curves and provide considerableimprovements in the calculated dissociation energiesDe (cf. Refs. [23,24,28,29]). For the aug-cc-pVQZ basis setused here, the CR-CCSD[T] and CR-CCSD(T) methods reduce the 0.84 eV error in the CCSD result forDe to0.22 and 0.11 eV, respectively (the experimental value ofDe is 6.12 eV [87]). Again, it is impossible to define


Fig. 2. The CCSD, CCSD[T], CCSD(T), CR-CCSD[T], and CR-CCSD(T) potential energy curves for the aug-cc-pVQZ model of HF.

Table 4Vibrational energiesG(v) (in cm−1) of the HF molecule.a With an exception of thev = 0 energies, all theoretically computed term values(obtained with the aug-cc-pVQZ basis set [84–86]) are reported relative to the RKR values

v RKRb CCSD CCSD[T] CCSD(T) R-CCSD[T] R-CCSD(T) CR-CCSD[T] CR-CCSD(T)

0 [2050.8] [2076] [2049] [2052] [2052] [2055] [2053] [2056]

1 3961.4 79 −6 5 4 14 8 16

2 7750.8 136 −10 9 8 25 16 30

3 11372.8 198 −15 13 12 37 24 46

4 14831.6 263 −20 17 15 49 33 63

5 18131.0 333 −28 20 19 63 42 80

6 21273.8 407 −37 23 24 78 51 99

7 24262.3 489 −49 25 29 95 63 119

8 27098.1 578 −64 25 35 114 76 142

9 29781.6 679 −84 23 41 134 91 168

10 32312.1 792 −111 16 46 156 110 199

11 34687.6 921 −151 1 49 179 133 233

12 36904.1 1069 −213 −27 47 201 160 273

13 38955.8 1241 −313 −81 38 222 192 317

14 40833.6 1443 18 239 232 367

15 42525.3 1683 −20 249 284 423

16 44013.4 1974 −94 245 354 488

17 45274.9 2332 452 564

18 46277.7 2783 596 654

19 46975.7 3365 814 758

a The numbers between the square brackets represent zero-point energies. All other numbers represent theG(v) values relative to thev = 0state.

b From Ref. [92].


the dissociation energy using the CCSD[T] and CCSD(T) curves because of the presence of the humps on thesecurves.

The standard CCSD[T] and CCSD(T) methods do not support bound vibrational states withv > 13, whereasthe errors in the vibrational term values resulting from the CCSD calculations rapidly increase, from 25 cm−1

for v = 0 to 3365 cm−1 for the highest experimentally observedv = 19 state. The simplest way of renormalizingthe CCSD[T] and CCSD(T) methods, via the R-CCSD[T] and R-CCSD(T) approaches, provides considerableimprovements in the poor CCSD results (particularly, in the [T] case). Vibrational term values provided by theR-CCSD[T] approach differ from the RKR values by less than 50 cm−1 for v � 15. More importantly, we can studyvibrational states withv as high as 16. The CR-CCSD[T] and CR-CCSD(T) methods provide somewhat worseresults, when compared to the R-CCSD[T] and R-CCSD(T) term values forv � 16, but, on the other hand, wecan calculate the entire vibrational spectrum, including thev = 19 state, with the CR-CCSD[T] and CR-CCSD(T)approaches. The CR-CCSD[T] and CR-CCSD(T) methods reduce the 3365 cm−1 error in the CCSD result forthe highest experimentally observedv = 19 level to 814 and 758 cm−1, respectively, and similarly impressivereductions are observed for all other term values. These results are very encouraging, particularly if we take intoaccount the “black-box” character of the CR-CCSD[T] and CR-CCSD(T) calculations and their relatively low cost.We plan to extend these studies to even larger basis sets in order to explore the basis set limits of the calculatedspectra of HF and other molecules.

Finally, let us address an issue of relative computer costs of various methods described in this paper by analyzingthe CPU timings and memory requirements for a typical CC calculation. An example described here involves theAM1 structure of the glycine (H2N–CH2–COOH) isomer GLY12 found by Jensen and Gordon [93]. This isomerhas no symmetry. We performed several CC calculations for this system using the standard 6-31(d,p) basis set [94,95], as implemented inGAMESS[67]. We used the Cartesiand orbitals and the lowest five orbitals were kept frozenin the CC calculations, so thatno = 15 andnu = 80. The CPU times and memory requirements characterizing thecalculations with three different choices of the input variablecctyp (cctyp = ccsd, r-cc, andcr-cc) are shown inTable 5. We do not show the results forcctyp = ccsd(t) (the CCSD, CCSD[T], and CCSD(T) calculations only),since they are (to within a few seconds) identical to the results obtained forcctyp = r-cc (the cost of computingtheD[T] andD(T) denominators, entering the R-CCSD[T] and R-CCSD(T) formulas, is negligible). As we cansee (and as mentioned earlier), the CPU time needed to calculate the CR-CCSD[T] and CR-CCSD(T) triplescorrections is only twice as large as the CPU time needed to calculate the standard or renormalized (R) CCSD[T]and CCSD(T) corrections. If we choose the faster and more memory consuming subroutineINTRIPL to generatetheI ′′ab

ci intermediates, the memory required to calculate the CR-CCSD[T] and CR-CCSD(T) triples corrections isabout 50% larger than the memory required to calculate the standard CCSD[T] and CCSD(T) triples corrections.We can use less memory to calculate the CR-CCSD[T] and CR-CCSD(T) corrections if we chooseINTRIP insteadof INTRIPL to calculate theI ′′ab

ci intermediates.INTRIP uses only half the memory used byINTRIPL, so thatthe calculation of theI ′′ab

ci intermediates is no longer the most memory consuming part of the calculation. WhentheI ′′ab

ci intermediates are calculated byINTRIP, the memory allocated to the calculation of thetabcijk (2) andzabcijk

amplitudes (theT3WT2 subroutine) becomes larger than the memory used byINTRIP. In this case, the mostmemory intensive part of the entire CR-CCSD[T] and CR-CCSD(T) calculation is the CCSD iterative process.

As we can see, the CR-CCSD[T] and CR-CCSD(T) calculations are essentially as easy to perform as thestandard CCSD[T] and CCSD(T) calculations. The entire CCSD, CCSD[T], and CCSD(T) calculation for glycine(including the CCSD iterations) takes∼3100 seconds on a single processor of SGI’s Origin 3400 system with400 MHz R12000 nodes. By waiting a little longer (∼4200 seconds) and by using a little more memory (17megawords instead of 14 megawords), we can obtain the complete set of the CCSD, CCSD[T], CCSD(T),R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) results. The less memory demanding option usingINTRIP is equally attractive in this regard, since the complete set of the standard, renormalized, and completelyrenormalized CC energies is obtained in∼4500 seconds. Very similar patterns are observed in other calculations.

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Table 5CPU times and memory requirements for three different choices of the input variablecctyp in the calculations for the AM1 structure of the glycine isomer GLY12 ofJensen and Gordon [93]. The standard 6-31G(d,p) basis set [94,95], as implemented inGAMESS[67], was employed. The five lowest-energy core orbitals were keptfrozen and the Cartesiand orbitals were employed

cctyp Method(s) CPU time(s)a/sec. Memory requirement(s)b/words Total energy(ies)/hartree

ccsd CCSDc 1835 14 282 850 (CCSD) −283.6643843

r-cc CCSDc 1835 (CCSD)+ 1253 (triples) 14 282 850 (CCSD), 11 856 400 (T3WT2) −283.6643843

CCSD[T] −283.6914997

CCSD(T) −283.6893872

R-CCSD[T] −283.6856507

R-CCSD(T) −283.6839616

cr-cc CCSDc 1835 (CCSD)+ 2397 (triples)d 14 282 850 (CCSD), 17 313 200 (triples,INTRIPL)d −283.6643843

CCSD[T] 1835 (CCSD)+ 2615 (triples)e 14 282 850 (CCSD), 11 856 400 (triples,T3WT2)e −283.6914997

CCSD(T) −283.6893872

R-CCSD[T] −283.6856507

R-CCSD(T) −283.6839616

CR-CCSD[T] −283.6837068

CR-CCSD(T) −283.6822188

a Calculations performed on a 32-CPU Origin 3400 from SGI (400 MHz R12000 nodes, 24 GB RAM). Only one processor was used and the calculations wereperformed in a batch mode, with an average load of 30 disk and memory intensive jobs. The calculations in a dedicated mode (all other jobs paused) lead toslightlybetter CPU timings (1675 sec. for the CCSD part; 2060 sec. for thecctyp = cr-cc triples part usingINTRIPL).

b Excluding theno + nu = 95 words used to keep orbital energies. The most memory consuming subroutine is given in parentheses.c Convergence threshold for cluster amplitudes was set at 10−7 (the default value). The CCSD calculation converged in 20 iterations.d The triples corrections calculated withINTRIPL (a faster and more memory consuming routine for theI ′′ab

ciintermediates).

e The triples corrections calculated withINTRIP (a slower and less memory consuming routine for theI ′′abci

intermediates). The memory requirement forINTRIP was only 9 903 200 words, so that the most memory consuming routine in calculating the triples corrections wasT3WT2.


6. Summary and concluding remarks

The standard, renormalized, and completely renormalized CCSD[T] and CCSD(T) methods (as well as theLCCD, CCD, and CCSD approaches) for the case of the restricted closed-shell reference have been implemented.The efficient, fully vectorized computer codes for all of these methods have been developed and incorporated inGAMESS. The high efficiency of the resulting computer programs has been accomplished by using the idea ofrecursively generated intermediates and fast matrix multiplication routines.

Information about the main features of the CC code incorporated inGAMESS, including program organization,computer costs of the standardvs. renormalized or completely renormalized CC calculations, CPU timings, andmemory requirements, has been provided. Input variables controlling CC calculations and method choices allowedby the CC codes incorporated inGAMESShave been discussed.

It has been demonstrated that the new renormalized and completely renormalized CCSD[T] and CCSD(T)methods are as easy to use as the standard CCSD[T] and CCSD(T) approaches. The costs of the renormalized andcompletely renormalized CCSD[T] and CCSD(T) calculations are essentially identical to the costs of the standardCCSD[T] and CCSD(T) calculations. All of these methods are then2

on4u procedures in the iterative CCSD steps

andn3on

4u procedures in the noniterative steps associated with calculating triples corrections.

Unlike the standard CCSD[T] and CCSD(T) approaches, the completely renormalized CCSD[T] and CCSD(T)methods do not fail at larger internuclear separations when single bonds are broken. As a matter of fact, the newCR-CCSD[T] and CR-CCSD(T) methods provide highly accurate description of entire PESs involving single bondbreaking. This fact, combined with the “black-box” character and the relatively low computer costs of the CR-CCSD[T] and CR-CCSD(T) calculations, makes the completely renormalized CCSD[T] and CCSD(T) methods auseful alternative to the standard CCSD[T] and CCSD(T) approaches. This has been illustrated by a few examplesof actual calculations for dissociating molecules.

We are currently working on an efficient computer implementation of the renormalized and completelyrenormalized CCSD(TQ) approaches [23–26,28,29], which allow us to study ground-state PESs involving multiplebond breaking [25,29]. Moreover, the CR-CCSD(TQ) methods improve the quality of the CR-CCSD[T] and CR-CCSD(T) results for single bond breaking [24,26,28]. Finally, we are also working on fast computer codes for theexcited-state analog of the completely renormalized CCSD[T] and CCSD(T) methods, termed MMCC(2,3) [56,57], in which simple noniterative corrections due to triples are added to the excited-state energies obtained in theequation-of-motion CCSD [96,97] calculations. All of these programs will be interfaced withGAMESS.

Acknowledgements

This work has been supported by the U.S. Department of Energy, Office of Basic Energy Sciences, SciDACComputational Chemistry Program (Grant No. DE_FG02_01ER15228; awarded to P.P.). We would like to thankDr. M.W. Schmidt for writing the integral sorting routines that allowed us to incorporate our CC codes inGAMESS.Several very helpful discussions with Professor M.S. Gordon and Dr. M.W. Schmidt about interfacing our codeswith theGAMESSpackage are gratefully acknowledged. Finally, we would like to thank Dr. V. Špirko for his helpwith calculating the vibrational spectrum of HF presented in this work.

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Documents

Efficient computer implementation of the renormalized coupled-cluster methods: The R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches