Argon pair potential at basis set and excitation limits

Argon pair potential at basis set and excitation limitsKonrad Patkowski and Krzysztof Szalewicz

Citation: J. Chem. Phys. 133, 094304 (2010); doi: 10.1063/1.3478513 View online: http://dx.doi.org/10.1063/1.3478513 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v133/i9 Published by the American Institute of Physics.

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Argon pair potential at basis set and excitation limitsKonrad Patkowski1,a and Krzysztof Szalewicz1,21Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA2Laboratoire dAstrophysique de Grenoble, Universit Joseph Fourier, UMR 5571-CNRS, 38041 GrenobleCedex 09, France

Received 24 May 2010; accepted 21 July 2010; published online 3 September 2010

A new ab initio interaction potential for the electronic ground state of argon dimer has beendeveloped. The potential is a sum of contributions corresponding to various levels of thecoupled-cluster theory up to the full coupled-cluster method with single, double, triple, andquadruple excitations. All contributions have been calculated in larger basis sets than used in thedevelopment of previous Ar2 potentials, including basis sets optimized by us up to theseptuplesextuple-zeta level for the frozen-core all-electron energy. The diffuse augmentationfunctions have also been optimized. The effects of the frozen-core approximation and the relativisticeffects have been computed at the CCSDT level. We show that some basis sets used in literatureto compute these corrections may give qualitatively wrong results. Our calculations also show thatthe effects of high excitations do not necessarily converge significantly faster in absolute values inbasis set size than the effects of lower excitations, as often assumed in literature. Extrapolations tothe complete basis set limits have been used for most terms. Careful examination of the basis setconvergence patterns enabled us to determine uncertainties of the ab initio potential. The interactionenergy at the near-minimum interatomic distance of 3.75 amounts to 99.2910.32 cm1. Theab initio energies were fitted to an analytic potential which predicts a minimum at 3.762 with adepth of 99.351 cm1. Comparisons with literature potentials indicate that the present one is themost accurate representation of the argon-argon interaction to date. 2010 American Institute ofPhysics. doi:10.1063/1.3478513

I. INTRODUCTION

Knowledge of an accurate argon dimer potential is criti-cal for several areas of physics and chemistry, and thereforethis potential has been the subject of several empirical andab initio investigations, see Ref. 1 for a review of this work.Whereas for the smallest rare gas dimer, He2, theoretical po-tentials became more accurate than empirical ones in themid-1990s,24 for the argon dimer, the empirical potentialssuch as the one developed by Aziz in 1993 Ref. 5 aregenerally believed to represent best the true interactions. Un-til very recently,6 the most accurate ab initio argon dimerpotential available was the one developed in Ref. 7 and re-fined in Ref. 1. The calculations presented in these referencesemployed basis sets up to augmented sextuple zeta aug-cc-pV6Z ones of Dunning et al.810 plus a 3s3p2d2f1g set ofmidbond functions we will use short-hand notation 33221for these functions and similar ones appearing later in thetext. The frozen-core FC coupled-cluster method was ap-plied at the single, double, and noniterative triple CCSDTexcitation level. After an extrapolation to the complete basisset CBS limit, the CCSDT/FC interaction energy at anear-minimum interatomic separation R=3.75 was97.48 cm1. The effects beyond the FC approximationwere approximately accounted for at the CCSDT level us-ing the aug-cc-pV5Z+ 33221 set and amounted to1.71 cm1 at the minimum, so that the total interaction en-

ergy at this separation was 99.18 cm1. The minimum ofthe analytic fit to the ab initio points was 99.27 cm1 atR=3.77 . These values are within 0.3 cm1 and 0.01 ofthe minimum of the popular empirical potential of Aziz5 andthe depth agrees with the result extracted from spectroscopicmeasurements11 to all digits of the latter value. The authorsof Ref. 1 argued that their potential is more accurate thanAzizs potential5 both in the highly repulsive region and inthe asymptotic region, based on comparisons with experi-mental high-energy data12,13 and on asymptotic constants.However, in the minimum region it was not possible to de-termine which potential is more accurate.

Whereas the interaction energies were computed in Ref.1 at the all-electron AE CCSDT level, some terms be-yond this level were very approximately estimated in thiswork and in Ref. 7. The authors of the latter work performedCC calculations with up to full triple CCSDT excitationsusing the aug-cc-pVTZ+ 332 basis set and found these ex-citations to contribute 0.6 cm1 beyond the CCSDT levelnear the minimum. The effects of quadruple excitations werenot determined at that time. However, the authors of Ref. 1speculated that these effects together with the relativisticterm discussed below are likely to cancel out to a largeextent the triple-excitation contribution beyond the CCSDTlevel. The latter contribution could not be anyway includedin the argon dimer potential of Ref. 1 since the calculationsof Ref. 7 were performed for only a few grid points. Therelativistic effects on the argon dimer interaction energy nearthe minimum have been computed in Ref. 14 using a corre-aElectronic mail: [email protected] and [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 133, 094304 2010

0021-9606/2010/1339/094304/20/$30.00 2010 American Institute of Physics133, 094304-1


lated, scaled zeroth-order regular approximation ZORA tothe Dirac equation approach within the treatment of the elec-tron correlation in the second order of perturbation theorybased on the MllerPlesset partition of the HamiltonianMP2 and amounted to 0.7 cm1. Clearly, the very goodagreement of the potential curve of Ref. 1 with experimentaland empirical data originates, to some extent, from the can-cellation of the errors discussed above. In fact, some of thecontributions neglected in this potential, including the fulltriple-excitation and relativistic effects, each exceed in mag-nitude the 0.3 cm1 difference with Azizs potential5 at thepotential minimum. Moreover, the contribution beyond thefrozen-core approximation was calculated using theaug-cc-pV5Z+ 33221 basis set, which was optimized8 atthe frozen-core level, and such sets often do not provide anaccurate account of core-core and core-valence correlationeffects.15 Thus, it is hard to assess the intrinsic uncertainty ofthe potential of Ref. 1, but it is certainly much larger than0.3 cm1.

In this work, we will improve the argon dimer potentialof Ref. 1 and determine the uncertainty of the new potential.The interaction energy will be calculated as

Eint = EintCCSDT/FC + Eint

CCSDT/AEFC + EintTT/FC

+ EintQ/FC + Eint

QQ/FC + Eintrel

, 1

where the subscript int stands for thecounterpoise-corrected16 interaction energy contribution, i.e.,Eint

M is the difference between the value of the energy contri-bution at the level M for the dimer and the sum of the analo-gous contributions for the monomers, all obtained using thefull dimer basis set. The first term on the right hand side ofEq. 1 is the CCSDT interaction energy in the FC approxi-mation. The second term represents the error of this approxi-mation, i.e., the difference between the AE and FC interac-tion energies computed at the CCSDT level of theory. Theterms third through fifth are contributions from higher exci-tations originating from all frozen-core CCSDT,CCSDTQ,1719 and CCSDTQ2022 treatments, respectively.Finally, the last term on the right hand side of Eq. 1 is therelativistic effect, which will be calculated using theCCSDT method with the second-order DouglasKrollHess DKH23,24 relativistic Hamiltonian. Thus, our potentialwill include all the terms that can be expected to give con-tributions of the order of 0.1 cm1 or 0.1% in the region ofminimum and therefore should be sufficiently converged inthe level of theory to enable us to surpass the accuracy ofbest empirical potentials. This goal also requires very tightconvergence of each contribution in basis set size. Therefore,for each of the contributions, its basis set dependence will beanalyzed and a proper type and size of basis set will bechosen based on the patterns of convergence for the near-minimum interatomic separation R=3.75 . For most con-tributions, the results will be extrapolated to the CBS limit.The Dunning et al. basis set sequences that we intended touse for some contributions: the standard aug-cc-pVXZ Refs.810 and aug-cc-pVX+dZ Ref. 25 sequences as well asthe polarized-core-and-valence basis sets aug-cc-pCVXZ andaug-cc-pwCVXZ,26 are only available for the cardinal num-

ber X up to 6 aug-cc-pVXZ and aug-cc-pVX+dZ or up to5 aug-cc-pCVXZ and aug-cc-pwCVXZ, and the conver-gence with X is sometimes irregular. Therefore, we have de-veloped our own basis set sequences in addition to usingDunning et al. sets. We expect that the inclusion of newterms and the use of larger basis sets than in previous workwill lead to a potential which will provide a better represen-tation of the interaction between two argon atoms than anypotential to date. In order to be able to evaluate quality of ourand literature potentials, we will estimate the uncertainties ofour potential from the patterns of convergence in basis setsize and in number of excitations. No such estimate has beenattempted for the argon dimer so far.

When this work was nearly completed, a new high-levelab initio study of the argon dimer potential by Jger et al.6appeared. The authors of Ref. 6 calculated the interactionenergy in a way similar to our calculations, i.e., they em-ployed Eq. 1 except that the full-quadruples effectEint

QQ/FCwas neglected and the relativistic correction was

computed at the CowanGriffin approximation.27 Whereasthe calculations of Ref. 6 have addressed all importantsources of uncertainty in the potentials of Refs. 7 and 1, thepresent work uses basis sets with the cardinal number largerby 1 for nearly all contributions to Eq. 1, and, moreover,most of these contributions are extrapolated to CBS limits.We will also be able to provide estimated uncertainties forthe Jger et al. potential.

The structure of the rest of this paper is as follows. InSec. II, we describe the development of our basis set se-quences. Section III contains numerical results obtained fordifferent interaction energy contributions using both Dun-nings et al. basis sets and the basis sets developed in Sec. II.The ab initio results for the whole potential energy curve areexamined in Section IV and fitted with an analytic expres-sion in Sec. V. The resulting potential is compared with lit-erature ones in Sec. VI. Finally, Sec. VII contains conclu-sions.

II. DEVELOPMENT OF BASIS SETS

Most of the CISD and CCSDT calculations presentedin this work were performed with the MOLPRO program.28 Inthe case of the 7Z/FC basis set which involves k functionsnot coded in MOLPRO, the CCSDT calculations were per-formed using the DALTON program29 suitably patched to al-low for CC integral files larger than 16 GB and for a two-electron integral threshold tighter than 1015, and the CISDcalculations using the LUCIA code30 interfaced to the DALTONintegrals and SCF orbitals. The Edisp

20 dispersion energieswere calculated with the SAPT code,31 using integrals andSCF orbitals from MOLPRO and DALTON. The CCSDT ener-gies were obtained with the CFOUR code,32 and theCCSDTQ and CCSDTQ energies were calculated using theMRCC program by Kllay and Surjn33 interfaced to MOLPRO.It has to be stressed that the largest basis sets employed by usare so saturated and so diffuse that near-linear dependenciesand numerical inaccuracies sometimes become a problem: tocontrol this issue, we had to run small- and medium-R cal-culations in bases disp-6Z+1 /AE+ 33221 and disp-6Z

094304-2 K. Patkowski and K. Szalewicz J. Chem. Phys. 133, 094304 2010


+2 /AE+ 33221 described later in this section using DAL-TON instead of MOLPRO. On the other hand, for large R thelarge-basis correlated calculations using DALTON exhibitedsome small residual inaccuracies that did not cancel betweenthe dimer and monomer calculations resulting in slightly in-accurate interaction energies as could be verified by a com-parison to the MOLPRO results and to results calculated inother basis sets. These inaccuracies persisted even when theintegral, SCF, and CCSD convergence thresholds were madeas tight as possible that is, the dimer and monomer SCF andCCSD energies were converged to about 1011 hartree orbetter. As only DALTON could perform calculations in thefull X=7 basis sets, for some large-R calculations of thiskind, we had to revert to using MOLPRO with the k functionsremoved from the basis set.

A. Nonaugmented basis setsOur first goal was to extend the aug-cc-pVXZ

aug-cc-pCVXZ sequence for FC AE calculations to X=7 6. This task was performed as a two-step procedure,first for the nonaugmented part of the basis set and then forthe augmentation part. The optimization of the cc-pVXZ partfollowed Dunnings algorithm,34 i.e., even-tempered expo-nent series for consecutive angular symmetries l were chosento minimize the configuration interaction singles and doublesCISD correlation energy for the argon atom. In the originalworks by Dunning and co-workers,8,10,34,35 only thed , f ,g , . . . expansions were optimized in this way, whereasthe s and p expansions were not optimized, but rather se-lected from a number of previously obtained sets whichminimize the argon atom HartreeFock self-consistent fieldHF-SCF energy, in order to reduce the number of primi-tive Gaussian functions in the basis set. Since the computa-tional complexity at correlated levels is nearly independentof the primitive basis set size it depends only on the con-tracted basis set size in the conventional nondirect calcu-lations performed by us, we decided to follow a slightly dif-ferent approach, as described below.

First, a very accurate representation of the 1s, 2s, 2p, 3s,and 3p orbitals of argon was taken from the work ofPartridge.36 The 20s15p primitive set of Ref. 36 was con-tracted to 3s2p and included in all basis sets regardless ofX. Note that Dunning et al. cc-pVXZ basis sets contain dif-ferent representations of occupied atomic SCF orbitals fordifferent X and that except for the largest X=6 these de-scriptions are not as accurate as the 20s15p set used by us.Specifically, the argon atom ground-state SCF energies cal-culated with Dunning et al. basis sets range from526.799 8653 hartree cc-pVDZ to 526.817 4836 har-tree cc-pV6Z, whereas the basis of Ref. 36 gives526.817 4840 hartree. These numbers should be comparedto the benchmark value of 526.817 51 hartree.37 The sec-ond difference compared to Dunning et al. procedures wasthat the even-tempered expansions for all l, including addi-tional s and p functions, were optimized in an unconstrainedway. Initially, the even-tempered expansions for different lwere added one-by-one, in the order d , f ,g , ,s , p, on topof the 3s2p contracted SCF set, and the added expansion

was optimized for the atomic CISD correlation energy asdone by Dunning et al. for all symmetries but s and p,keeping the already optimized functions frozen. Since theatomic CISD calculations are inexpensive, we decided to fol-low the l-by-l optimizations with a simultaneous optimiza-tion of the even-tempered sequences for all l with the ex-ception of the basis sets containing k functions where only apartial simultaneous optimization was performed, resultingin a further lowering of the CISD correlation energies by upto a couple percent.

We have minimized the frozen-core CISD correlationenergy of the argon atom using the algorithm describedabove to construct a sequence of basis sets of the same com-position and contracted size as Dunning et al. cc-pVXZbases. These sets will be denoted as XZ /FC, with X rangingfrom D to 7.

The optimization of an equivalent of the cc-pCVXZ se-quence departed more substantially from the literatureapproach.26,38 To improve the description of core-core andcore-valence electron correlation effects by the cc-pVXZ ba-sis sets, Dunning and co-workers26,38 supplemented themwith additional compact functions optimized for the core-core and core-valence contributions to the CISD correlationenergy. Two basis set families, differing by the relative im-portance of core-core and core-valence effects, were ob-tained and denoted as cc-pCVXZ and cc-pwCVXZ, with Xranging from D to 5. We decided to follow a different ap-proach and to optimize a complete even-tempered expansionfor each l in exactly the same way as in the case of theXZ /FC basis sets rather than by supplementing each suchset, except that, of course, the all-electron CISD correlationenergy was minimized instead of its frozen-core counterpart.The resulting basis sets, denoted by XZ /AE where X rangesfrom D to 6, are of exactly the same contracted size as theXZ /FC and cc-pVXZ bases for the same X. Thus, these basesare significantly shorter than the bases cc-pwCVXZ withthe same X and can be expected to be less accurate. Conse-quently, we have also optimized basis set sequencesXZ+1 /AE and XZ+2 /AE, which are constructed like theXZ /AE bases but with the length of the even-tempered ex-pansion for each l increased by one and two, respectively.For example, while the contracted composition of the cc-pVTZ, TZ/FC, and TZ/AE basis sets is 5s4p2d1f, thebases TZ+1 /AE and TZ+2 /AE have the structure6s5p3d2f and 7s6p4d3f, respectively. For comparison,the cc-pCVTZ basis set is constructed as 7s6p4d2f, so it isonly slightly smaller than TZ+2 /AE.

The FC and AE correlation energies for the argon atomat the CISD level of theory, calculated with Dunning et al.and our basis sets, are presented in Table I and in Figs. 13.The values in this table should be compared with the bench-mark energies of 0.251 353 hartree FC and 0.664 605hartree AE, obtained with very large basis sets, of the com-position 18s17p15d13f11g9h7i, constructed from bases6Z/FC and 6Z+2 /AE, respectively, as follows. The 3s2pSCF contractions were kept constant. The even-tempered ex-pansions for the correlation exponents were first made twiceas dense, i.e., a basis function with exponent ii+1 wasadded between each pair of consecutive functions with ex-

094304-3 Argon pair potential J. Chem. Phys. 133, 094304 2010


ponents i and i+1. Next, the resulting even-tempered ex-pansions were extended by one AE or three FC additionalfunctions in each direction toward larger exponents and to-ward smaller exponents, assuming for i functions the sameexponent ratio as for h in the FC case.

Table I and Figs. 1 and 2 show first that our goal ofextending the cc-pVXZ and cc-pCVXZ sequences by onecardinal number has been realized. On Fig. 1, the sequenceXZ /FC practically overlaps cc-pVXZ for X=D6 and fol-lows the convergence trend to X=7. The cc-pVX+dZ se-quence gives slightly inferior results for X=D and T, but forlarger X lies on the same curve as the other two sequences.Similarly, on Fig. 2, the XZ+2 /AE sequence lies on the line

formed by the cc-pCVXZ sequence and properly extends toX=6. One has to use the XZ+2 /AE sequence to obtain agood match with the cc-pCVXZ sequence, as theXZ+1 /AE and XZ /AE sequences lie above this line. Thisfact shows that the sequences cc-pCVXZ and XZ+2 /AEhave an optimal number of functions in each angular sym-metry, whereas XZ+1 /AE and XZ /AE have too few suchterms. The sequence cc-pwCVXZ performs worse than theother AE-optimized sequences for small X, becomes progres-sively better as X increases, but extrapolates from the X=Qand 5 basis sets to a significantly too negative value.

Comparison of the results in Figs. 1 and 2 obtained withfunctions optimized for a given type of CISD energy shows

TABLE I. CISD/FC and CISD/AE correlation energies in hartree for the argon atom calculated in Dunning et al. cc-pVXZ, cc-pCVXZ, cc-pwCVXZ, andcc-pVX+dZ basis sets as well as in the XZ /FC, XZ /AE, XZ+1 /AE, and XZ+2 /AE bases optimized in the present work. The benchmark near-CBS-limitvalues see text are 0.251 353 hartree for CISD/FC and 0.664 605 hartree for CISD/AE.

CISD/FC X

Basis Basis size for X=5 /6 /7 D T Q 5 6 7

cc-pVXZ 95/144 0.147 212 0.214 610 0.237 379 0.244 919 0.249 150cc-pVX+dZ 100/149 0.152 881 0.216 268 0.238 608 0.245 665 0.249 494cc-pCVXZ 181 0.151 176 0.218 739 0.240 233 0.246 878cc-pwCVXZ 181 0.157 376 0.220 413 0.241 176 0.247 376XZ /FC 95/144/208 0.148 463 0.215 681 0.238 096 0.245 814 0.249 270 0.250 995XZ /AE 95/144 0.103 607 0.184 279 0.207 816 0.227 071 0.238 732XZ+1 /AE 131/193 0.137 941 0.194 029 0.223 613 0.237 381 0.244 061XZ+2 /AE 167/242 0.158 673 0.210 339 0.233 931 0.242 746 0.246 971

CISD/AE X

Basis Basis size for X=5 /6 D T Q 5 6 7

cc-pVXZ 95/144 0.151 554 0.242 583 0.294 454 0.398 962 0.491 804cc-pVX+dZ 100/149 0.207 978 0.314 361 0.368 615 0.460 260 0.531 520cc-pCVXZ 181 0.356 735 0.539 015 0.614 549 0.646 075cc-pwCVXZ 181 0.299 962 0.508 540 0.601 125 0.640 461XZ /FC 95/144 0.156 299 0.251 674 0.329 986 0.409 329 0.502 381 0.549 066XZ /AE 95/144 0.196 244 0.399 049 0.519 329 0.588 836 0.624 951XZ+1 /AE 131/193 0.360 995 0.505 673 0.585 201 0.623 544 0.643 586XZ+2 /AE 167/242 0.439 037 0.559 927 0.616 103 0.640 615 0.653 296

-0.26

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

0 50 100 150 200 250

correlationen

ergy

[hartree

]

N

cc-pVXZcc-pV(X+d)Z

cc-pCVXZcc-pwCVXZ

XZ/FCXZ/AE

XZ+1/AEXZ+2/AE

FIG. 1. CISD/FC correlation energies of the argon atomcomputed in various basis sets as a function of the num-ber N of contracted basis functions.



that the AE correlation energy is much harder to convergethan its FC counterpart notice the different scales on thevertical axes. The 6Z/AE basis set still gives an error of 6%with respect to the AE benchmark value, whereas 6Z/FC isless than 1% from the FC benchmark. Of course, longereven-tempered expansions for each l saturate the AE energyprogressively better: to within 3.2%, 2.8%, and 1.7% for6Z+1 /AE, cc-pCV5Z, and 6Z+2 /AE, respectively, butcosts of such calculations are also substantially higher. Thelast result provides the best recovery of the CISD/AE energy.In the case of the CISD/FC energy, the 7Z/FC basis set re-produces the benchmark value with the error of only 0.14%,probably comparable to the error of the benchmark itself. Infact, the extrapolated result from this sequence, 0.253 929hartree, is 1.02% below the benchmark. Similarly, the ex-trapolation from the XZ+2 /AE basis sets gives 0.670 715hartree, 0.92% below the AE benchmark.

Our XZ /FC basis sets give slightly lower CISD/FC cor-relation energies than the corresponding cc-pVXZ bases op-timized for the same quantity which is an obvious conse-

quence of the fact that the exponents in our approach wereoptimized in a less constrained way. However, the improve-ments are relatively small and hardly visible in Fig. 1. Thisfact shows that the constraints used in the optimizations ofDunning et al. are relatively inconsequential.

Table I also contains the results of CISD/FC calculationswith basis sets optimized for all electrons and vice versa. Ineach case, results obtained with such basis sets are signifi-cantly inferior to those obtained with basis sets optimizedusing consistent levels of theory. Moreover, the convergenceof the former sequences is somewhat irregular and, in par-ticular, the FC-optimized sequences used in AE calculationsextrapolate to nonsensical values. Also, the largest-X resultsare far from the limits: the cc-pV6Z and 7Z/FC basis setsrecover only 74% and 83%, respectively, of the benchmarkCISD/AE correlation energy. Since the calculation of thecore contribution requires subtraction of the AE and FC en-ergies computed in the same basis set, this effect convergesvery slowly in FC-optimized basis sets. Calculations in AE-optimized bases either fully XZ /AE, XZ+1 /AE,

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0 50 100 150 200 250

correlationen

ergy

[hartree

]

N

cc-pCVXZcc-pwCVXZ

XZ/AEXZ+1/AEXZ+2/AE

FIG. 2. CISD/AE correlation energies of the argonatom computed in various basis sets as a function of thenumber N of contracted basis functions.

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 50 100 150 200 250

core

contrib

ution[hartree

]

N

cc-pCVXZcc-pwCVXZ

XZ/AEXZ+1/AEXZ+2/AE

FIG. 3. Core contributions, i.e., the differences betweenCISD/AE and CISD/FC energies, of the argon atomcomputed in various basis sets as a function of the num-ber N of contracted basis functions.



XZ+2 /AE or in part cc-pCVXZ, cc-pwCVXZ convergebetter, consistently with their relatively better performance inFC calculations than of FC-optimized bases in AE calcula-tions as expected since AE-optimized bases do depend oncorrelation of valence electrons. Notice that AE-optimizedbases used in FC calculations converge actually faster in ab-solute terms than in AE calculations. For example, thechange of the CISD/FC energy for the XZ+2 /AE sequencewhen X changes from 5 to 6 is 0.0042 hartree, whereas it is0.0127 hartree for the CISD/AE energy. Apparently, theseincrements are sufficiently close to each other to providereasonably well-converged values of the core contribution,cf. Fig. 3.

Another approach to calculations of the core contribu-tions would be to extrapolate AE and FC energies separatelyin AE- and FC-optimized sequences, respectively, and sub-tract the extrapolated results. If this procedure is applied withthe 1 /X3 extrapolation to the sequences XZ /FC andXZ+2 /AE, the CBS limit of the core contribution obtainedin this way is 0.4168 hartree, whereas a direct extrapola-tion of this contribution computed in the sequence XZ+2 /AE gives 0.4179 hartree, not much different on thescale of increments seen in Table I. This comparison pro-vides an additional argument for the soundness of calcula-tions of core contributions from AE-optimized basis sets, de-spite the somewhat different rates of convergence in FC/AEcalculations discussed above.

One should be careful when comparing results in basissets with the same cardinal numbers, as due to the strategyused in the construction of such families of basis sets, thebases may have quite different numbers of functions and thisrelation may change with X. For example, compared to thecc-pVTZ basis, the TZ+1 /AE, TZ+2 /AE, and cc-pCVTZsets contain 1s1p1d1f=16, 2s2p2d2f=32, and2s2p2d1f=25 additional functions, respectively. However,for X=5 the additional functions are 1s1p1d1f1g1h=36,2s2p2d2f2g2h=72, and 4s4p4d3f2g1h=86, respec-tively, so that the cc-pCV5Z basis set is by far the largestone.

B. Augmentation of basis sets

The next step in the development of basis sets that areadequate for calculations of interatomic or intermolecular in-teraction energies is an augmentation of basis sets developedin Sec. II A with diffuse functions. Such functions are criticalfor description of monomer static and dynamic polarizabil-ities, and consequently of dispersion energies. For systemsdominated by dispersion interactions, such as the argondimer, one often applies two sets of such functions per eachangular symmetry. Various algorithms have been proposed tooptimize exponents of diffuse functions. The method intro-duced by Kendall et al.35 to develop the aug-cc-pVXZ basissets involves optimization of such exponents for the SCF andCISD/FC energies of an atomic anion. This is, however, notpossible for noble gases, and for argon the diffuse exponentsin the aug-cc-pVXZ basis sets were not optimized, but ex-trapolated from the optimized diffuse exponents for sulfurand chlorine.8 Since the argon dimer is a dispersion-boundsystem, a viable alternative is to optimize diffuse functionsfor atomic polarizabilities39,40 or, in a related approach em-ployed here, for the dispersion energy of two argon atoms atsome separation R.41,42 This optimization is made possible bythe fact that the leading-order dispersion correction, Edisp

20, in

symmetry-adapted perturbation theory SAPT43,44 is a varia-tional quantity provided that the monomer SCF functions areexact.45 It should be noted that Edisp

20, just like any other

SAPT correction, is calculated directly, without any subtrac-tion of dimer and monomer quantities. Thus, it can be ob-tained to sufficiently many significant digits even for large R.Moreover, Edisp

20 describes simultaneously and seamlesslyboth the asymptotic part of the dispersion energy and itsshort-range charge-overlap contribution. Since the main roleof diffuse basis functions is to reproduce the asymptotic dis-persion energy through the saturation of monomer polariz-abilities, we decided to minimize Edisp

20 for a large intermono-mer distance R=12 so that the charge-overlapcontributions do not interfere.

Relative to the nonaugmented basis sets obtained in

TABLE II. Values of the Edisp20 /FC correction in cm1 at R=12.0 , calculated in aug-cc-pVXZ,

aug-cc-pVX+dZ, d-aug-cc-pVX+dZ, disp-XZ /FC, disp+d-XZ /FC, d -disp-XZ /FC, and d -disp-XZ /FCbasis sets no midbond. The benchmark near-CBS-limit value see text is 0.12697 cm1. The numbers inparentheses are basis set sizes for the dimer.

X aug-cc-pVXZ aug-cc-pVX+dZ d-aug-cc-pVX+dZ

D 0.113 62 54 0.115 51 64 0.123 27 82T 0.123 69 100 0.125 16 110 0.127 07 142Q 0.125 99 168 0.126 29 178 0.126 92 2285 0.126 55 262 0.126 67 272 0.126 95 3446 0.126 74 386 0.126 75 396 0.126 94 494

X disp-XZ /FC disp+d-XZ /FC d -disp-XZ /FC d -disp-XZ /FC

D 0.119 89 54 0.121 71 64 0.120 20 72 0.120 89 72T 0.125 69 100 0.126 03 110 0.125 77 132 0.126 18 132Q 0.126 45 168 0.126 52 178 0.126 48 218 0.126 42 2185 0.126 77 262 0.126 86 272 0.126 81 334 0.126 83 3346 0.126 90 386 0.126 94 396 0.126 93 484 0.126 91 4847 0.126 96 544 0.126 98 672 0.126 96 672



Sec. II A, one additional exponent for each l was optimized.The basis set obtained from any basis A upon such augmen-tation will be denoted as disp-A. We have optimized thediffuse functions in bases disp-XZ /FC by minimizing thefrozen-core value of Edisp

20, while for bases disp-XZ /AE,

disp-XZ+1 /AE, and disp-XZ+2 /AE, the all-electron disper-sion energy was minimized. We have observed that the larg-est incremental contribution to Edisp

20arises from the diffuse

function of d symmetry. Therefore, we have also augmentedthe XZ /FC and XZ /AE sets with two dispersion-optimized dfunctions and with one dispersion-optimized function foreach of the other symmetries present in the basis set. Theresulting basis sets will be denoted as disp+d-XZ /FC anddisp+d-XZ /AE. The diffuse exponents of each augmentedbasis were first added one-by-one, in the order dspfg. . ., andoptimized keeping all the previously optimized diffuse expo-nents frozen. After that, a simultaneous reoptimization of alldiffuse exponents has been carried out this second step,however, gave negligible improvements. The only exceptionwas the disp-7Z/FC basis set for which the simultaneous

reoptimization step was not feasible, and the optimizations ofthe pfghi exponents were performed with the k functionsremoved so that the MOLPRO,28 not DALTON,29 code could beused for integrals and SCF orbitals otherwise, the optimiza-tions were hampered by the residual inaccuracies in DALTONdiscussed above.

The values of the Edisp20 SAPT correction for R=12

obtained using dispersion-optimized augmented basis setsare presented in Table II, Fig. 4 FC results, and Table IIIAE results. The corresponding values from theaug-cc-pVXZ, aug-cc-pVX+dZ, d-aug-cc-pVX+dZ, andaug-cc-pCVXZ sets are also included for comparison. Notethat no midbond functions have been included in the basissets; at such a large separation, the improvement the mid-bond functions bring about is negligible even for smallerbases. For example, the addition of the 33221 midbond setto the aug-cc-pVQZ basis lowers the Edisp20 /AE result from0.127 20 to 0.127 23 cm1. We obtained benchmarkvalues of Edisp

20 /FC and Edisp20 /AE by employing a

-0.128

-0.127

-0.126

-0.125

-0.124

-0.123

-0.122

-0.121

-0.12

-0.119

0 100 200 300 400 500 600 700

Dispe

rsionen

ergy

[cm-1]

N

aug-cc-pVXZaug-cc-pV(X+d)Z

d-aug-cc-pV(X+d)Zdisp-XZ/FC

disp+d-XZ/FCd-disp-XZ/FCd-disp-XZ/FC

FIG. 4. Values of the frozen-core Edisp20

correction incm1 for two argon atoms separated by R=12 , com-puted in various augmented basis sets no midbond andpresented as a function of the number N of contractedbasis functions.

TABLE III. Values of the Edisp20/AE correction in cm1 at R=12.0 , calculated in aug-cc-pVXZ,

aug-cc-pCVXZ, aug-cc-pVX+dZ, d-aug-cc-pVX+dZ, disp-XZ /AE, disp-XZ+1 /AE, disp-XZ+2 /AE, anddisp+d-XZ /AE basis sets no midbond. The benchmark near-CBS-limit value see text is 0.12874 cm1.The numbers in parentheses are basis set sizes.

X aug-cc-pVXZ aug-cc-pCVXZ aug-cc-pVX+dZ d-aug-cc-pVX+dZ

D 0.113 98 54 0.114 91 72 0.116 98 64T 0.124 52 100 0.125 93 150 0.126 79 110 0.128 71 142Q 0.127 20 168 0.128 10 268 0.127 97 178 0.128 61 2285 0.128 22 262 0.128 39 434 0.128 39 272 0.128 67 3446 0.128 48 386 0.128 50 396 0.128 69 494

X disp-XZ /AE disp-XZ+1 /AE disp-XZ+2 /AE disp+d-XZ /AE

D 0.119 65 54 0.120 85 72 0.123 34 90 0.121 27 64T 0.123 94 100 0.126 14 132 0.127 33 164 0.126 35 110Q 0.126 30 168 0.127 30 218 0.128 05 268 0.128 23 1785 0.127 35 262 0.128 03 334 0.128 30 406 0.128 45 2726 0.128 06 386 0.128 30 484 0.128 40 582 0.128 56 396



13s12p10d9f8g7h6i basis set which is a superset of thedisp-6Z+2 /AE basis with two additional functions added foreach l. The exponents of these additional functions are1aug and augaug /1, where 1 and aug are the small-est exponent of the nonaugmented even-tempered expansionand the optimized diffuse exponent, respectively. The result-ing benchmark values are 0.126 97 cm1 for Edisp

20 /FC and0.128 74 cm1 for Edisp

20 /AE.Tables II and III show that the dispersion energy con-

verges relatively fast to its CBS limit. For X=6, all types ofbasis sets give results within 0.5% of the benchmark. Still,the optimization of diffuse functions specifically for Edisp

20

gives the disp basis sets a clear advantage at recoveringthis correction. This is well visible in Fig. 4 where the curvesdisp-XZ /FC and disp+d-XZ /FC lie significantly below thecurves aug-cc-pVXZ and aug-cc-pVX+dZ. The presenceof the extra d function improves the convergence more thanit could be expected just from the increase of the size. If thepathologies of some doubly augmented expansions discussedbelow are disregarded, the disp+d-XZ /FC sequence givesthe most accurate dispersion energies of all basis sets con-sidered for the whole range of basis set sizes. The sequencedisp+d-XZ /AE performs similarly in recovering AE disper-sion energies.

As mentioned above, in calculations for dispersion-dominated dimers, one often uses doubly augmented basissets. Even triply augmented basis sets have been applied inone of the earlier argon dimer studies.46,47 A straightforwardoptimization of two diffuse exponents for each l to constructdoubly augmented bases d-disp-XZ /FC would be very time-consuming for large X. Therefore, we decided to estimate thesecond diffuse exponents from even-tempered expansionsinstead of optimizing them. For the aug-cc-pVX+dZbasis sets, their doubly augmented extensionsd-aug-cc-pVX+dZ that, to our knowledge, have not beenexplicitly specified in the literature can be obtained by add-ing one more diffuse exponent, equal to augaug /1, foreach l. This prescription was most likely used by Jger et al.6to construct the d-aug-cc-pVX+dZ basis sets used in theirwork. A similar algorithm for the disp-XZ /FC bases would,however, lead to additional functions that are too diffuse,since the dispersion-optimized exponents aug are quite dif-fuse already and several times smaller than 1. Therefore, wedecided to add additional exponents to the disp-XZ /FC setsin two ways. The first way, leading to basis sets denoted asd -disp-XZ /FC, adds the second diffuse exponent in be-tween aug and 1, i.e., equal to aug1. In the second al-gorithm, the original diffuse exponent aug is replaced bytwo exponents: 1 aug /12/3 and 1 aug /14/3; the ba-sis sets obtained in this way will be denoted asd -disp-XZ /FC. Note that the three most diffuse expo-nents form an even-tempered series for both algorithms andthat the geometric mean of two extra exponents in the scheme is equal to the omitted exponent aug. The FCdispersion energies obtained from basis setsd-aug-cc-pVX+dZ, d -disp-XZ /FC, andd -disp-XZ /FC defined in this way are included in TableII. The sequence d -disp-XZ /FC produces a curve which issignificantly above that given by the disp-XZ /FC sequence,

showing that the increase of the basis set by the second aug-mentation gives relatively small improvement per basis func-tion added. The sequence d -disp-XZ /FC performs evenworse as for X=Q it gives a value smaller in magnitude thanthat computed in the disp-QZ/FC basis set and the conver-gence is nonuniform for small X. Still worse, the secondaugmentation of the sequence aug-cc-pVX+dZ leads to apathology in the value computed using the d-aug-cc-pVT+dZ basis which is markedly below the benchmark disper-sion energy. The reason that the variational principle doesnot hold is that the monomer SCF description is relatively farfrom exact in this basis set, whereas the virtual space is verylarge. Furthermore, the resulting values of thed-aug-cc-pVX+dZ dispersion energies behave somewhaterratically with X, which may cause problems in CBS ex-trapolations. Thus, we cannot exclude the possibility that theuse of the doubly augmented sequences may lead to artifactsin argon dimer interaction energies. On the other hand, forthe largest X the results given by all three sequences are veryclose to each other and converge smoothly. Therefore, in ourstudy of the argon dimer, we have decided to use large-Xdoubly augmented basis sets at the CCSDT/FC level. Theexponents and contraction coefficients of all the basis setsoptimized in this work are listed in the SupportingInformation.48

III. COMPONENTS OF INTERACTION ENERGYAT THE MINIMUMA. Frozen-core CCSDT interaction energies

The values of the CCSDT interaction energies for theargon dimer at the near-minimum separation of 3.75 , cal-culated using various basis sets, are presented in Table IVfrozen-core results and Table I in the Supporting Informa-tion all-electron results. Furthermore, Table V lists the cor-responding values of the CCSDT core correction, i.e., thedifference between the CCSDT/AE and CCSDT/FC inter-action energies. Two sets of bond functions were used:33221 Ref. 7 with exponents sp: 0.9,0.3,0.1, df:0.6,0.2, and g: 0.35 and 666311 Ref. 49 with expo-nents spd: 1.6,0.8,0.4,0.2,0.1,0.05, f: 0.9,0.3,0.1, and gh:0.3. We have also performed some calculations to comparewith the results of Ref. 6 and used their midbond set 44332with exponents sp: 1.62,0.54,0.18,0.06, df:1.35,0.45,0.15, and g: 0.9,0.3. It should be noted that thepresence of bond functions is absolutely crucial for the ac-curacy of results. Even at X=6, the CCSDT/FC interactionenergy, amounting to 97.02 cm1 in the aug-cc-pV6Z+ 33221 set, would be just 93.62 cm1 in the pureaug-cc-pV6Z basis. Apart from the calculated energies, theCBS-extrapolated values are also presented in Tables IV andV. The extrapolations were performed according to the X3scheme.50 Specifically, the correlation contribution to the in-teraction energy was assumed to approach its CBS limit likeAX3 with the basis set cardinal number X where A is someconstant. As the SCF contribution exhibits a much fasterbasis set convergence, its value was not extrapolated; in-stead, the value from the larger basis set of the two used inthe extrapolation was included in the final result. In fact, at



X=6, the SCF interaction energies are converged to about0.02 cm1 for all basis set families considered here. Notethat for the aug-cc-pVXZ+ 33221 family of bases, the re-sults in Tables IV and V differ slightly from the values givenin Refs. 7 and 1, as these values were recalculated here usingtighter SCF and CCSD convergence thresholds.

The convergence of the CCSDT/FC interaction energyin FC-optimized basis sets from Table IV is also visualizedin Fig. 5 AE-optimized bases will be discussed in Sec.III B. Whereas all X=6 and 7 results, both computed andextrapolated, agree with each other to within 0.5 cm1, theenergies spread over the range of more than 5 cm1 when allX are considered. In particular, when going from X=Q to T,all basis sets exhibit a dramatic lowering of the computedinteraction energy, resulting in TQ-extrapolated values lying

very high. One can see that the effect is more pronounced inbasis sets with larger augmentation. This is probably relatedto an imbalance between the number of compact and diffusefunctions for smaller X. However, as X increases, the relativenumber of compact functions increases as well and the con-vergence in the range X=57 is rather smooth. Notice that ifone were to be restricted to X=6, no such statement could bemade since the smooth convergence pattern is not yet wellestablished at this X.

Figure 5 shows, furthermore, how important it is to usedouble augmentation. Although the singly augmented ex-trapolated values are quite close to the doubly augmentedones, the actual calculated values are not nearly as well-converged even at the highest available X. Thus, if only sin-gly augmented basis sets were used, one would have to as-

TABLE IV. Argon dimer CCSDT/FC interaction energies in cm1 at R=3.75 , calculated and extrapolatedrows marked Extr. using various Dunning et al. basis sets as well as the basis sets developed in the presentwork. The extrapolated results given in the X=Q, X=5, X=6, and X=7 columns have been obtained from baseswith X ,X+1 equal to TQ, Q5, 56, and 67, respectively. See text for details of the extrapolationprocedure.

Ar basis Bond X=T X=Q X=5 X=6 X=7

aug-cc-pVXZ 33221 98.616 96.148 96.729 97.017Extr. 94.460 97.244 97.396

aug-cc-pVX+dZ 33221 98.389 96.039 96.679 96.978Extr. 94.504 97.323 97.382

d-aug-cc-pVX+dZ 33221 99.661 96.556 97.003 97.146Extr. 94.428 97.479 97.353



disp-XZ /FC 33221 96.994 96.104 96.615 96.806 97.063Extr. 95.389 97.050 97.039 97.493

d -disp-XZ /FC 33221 99.532 96.714 97.059 97.202 97.342Extr. 94.588 97.328 97.372 97.573

d -disp-XZ /FC 44332 97.793 96.712 97.213 97.281 97.380Extr. 95.867 97.649 97.343 97.543

d -disp-XZ /FC 666311 98.724 96.883 97.283 97.369 97.406Extr. 95.482 97.611 97.457 97.460

d -disp-XZ /FC 33221 99.838 96.863 97.614 97.284 97.310Extr. 94.627 98.308 96.804 97.344

d -disp-XZ /FC 44332 97.634 96.784 97.033 97.236 97.342Extr. 96.109 97.201 97.486 97.515

d -disp-XZ /FC 666311 98.601 97.074 97.455 97.361 97.486Extr. 95.898 97.764 97.202 97.691

disp+d-XZ /FC 33221 97.196 96.486 96.599 96.803Extr. 95.904 96.612 97.053

aug-cc-pCVXZ 33221 97.073 95.966 96.644Extr. 95.221 97.333

aug-cc-pwCVXZ 33221 96.848 95.918 96.637Extr. 95.287 97.366

disp-XZ /AE 33221 99.961 96.281 95.878 95.869Extr. 93.406 95.406 95.834

disp-XZ+1 /AE 33221 96.438 95.278 95.546 96.002Extr. 94.401 95.804 96.622

disp-XZ+2 /AE 33221 95.814 95.162 95.800 96.252Extr. 94.670 96.457 96.864disp+d-XZ /AE 33221 98.523 95.912 95.909 95.916Extr. 93.823 95.869 95.910



sume large uncertainties of the final CCSDT/FC results.The doubly augmented bases developed in this work providequite well-converged interaction energies in the range X=57, and the doubly augmented Dunning et al. basesd-aug-cc-pVX+dZ, while not available for X=7, provideresults of similar quality for smaller X. All results computedin doubly augmented basis sets with X=6 and 7 are within0.35 cm1 of each other. The largest bases employed at theCCSDT/FC level are the 767-function setsd -disp-7Z /FC+ 666311 andd -disp-7Z /FC+ 666311, which give the interaction ener-gies of 97.486 and 97.406 cm1, respectively, or97.691 and 97.460 cm1, respectively, when the CBS ex-trapolation is performed. Note that the difference betweenthe calculated results in the two doubly augmented septuple-zeta basis sets is significantly amplified by extrapolation. It issomewhat disappointing that two such large bases, differingonly in the algorithm of second augmentation, do not agreeto better than 0.25% after the CBS extrapolation. At this R,the results from other basis set families at largest X stronglysuggest that the d -disp-XZ /FC+ 666311 extrapolated re-sult is reliable, whereas for the d -disp-XZ /FC+ 666311 sequence, even though the X=7 results might bemore accurate, the extrapolation overshoots. Clearly, whenan accuracy better than 0.25 cm1 is required, the choice ofthe recommended value of the CCSDT/FC interaction en-ergy is a nontrivial task. Moreover, while at R=3.75 , thesequence d -disp-XZ /FC+ 666311 could be identified asinferior to other large-basis families, the situation stronglyvaries with R and no basis set family that we have tested

performs uniformly well across the whole range of distances.Therefore, we decided to base our final estimate of theCCSDT/FC term and its uncertainty on the results fromseveral different basis set sequences instead of just one.

We have calculated the CCSDT/FC interaction energyfor all 33 interatomic distances using four different basis setsequences: d -disp-XZ /FC+ 666311, d -disp-XZ /FC+ 666311, disp-XZ /FC+ 33221 all extrapolated from X=6,7, and aug-cc-pVX+dZ+ 33221 extrapolated fromX=5,6. The complete set of results is presented in Table IIin the Supporting Information. Out of the four extrapolatedresults available for each R, we have identified and discardedthe one that deviated the most from the other three. It turnedout that each of the four basis set families resulted in thelargest deviation and had to be discarded for some R. Thearithmetic mean of the three remaining extrapolated resultswas chosen as the recommended value of the CCSDT/FCinteraction energy. Its uncertainty CCSDT/FC was deter-mined by the condition that all three extrapolated results notincluding the discarded one and at least one computed resultout of the three largest-X values from different basis setfamilies are within CCSDT/FC of the recommended value.In this way, one takes advantage of all the available resultsand chooses the computed value that is best converged. Onthe other hand, not including any computed result in thedetermination of CCSDT/FC would lead to an underestimateduncertainty when the extrapolated values happen to be acci-dentally close to each other. The final estimate of theCCSDT/FC interaction energy at R=3.75 obtained inthis way is 97.4450.063 cm1, where, as expected, thed -disp-XZ /FC+ 666311 extrapolated result ends up be-ing discarded. The behaviors of the extrapolatedCCSDT/FC results relative to the recommended value, aswell as the relative uncertainty of the latter, are presented inFig. 6 as functions of R.

B. Core correction

As expected, the results in Table IV show that the AE-optimized basis sets perform significantly worse than FC-optimized ones in calculations of CCSDT/FC interactionenergies. The very good Q5-extrapolated results in theaug-cc-pCVXZ+ 33221 and aug-cc-pwCVXZ+ 33221bases are likely fortuitous as the nonextrapolated results arenot very well-converged yet. The AE-optimized basis sets areincluded in Table IV mainly to provide the needed referencevalues for calculations of the core contributions.

The values of the core correction at the CCSDT levelare shown in Table V. The corresponding CCSDT/AE in-teraction energies can be found in Table I in the SupportingInformation. Note that the CBS-extrapolated values of thecore correction are the same when obtained by a direct ex-trapolation or by subtraction of the extrapolated interactionenergies. The FC-optimized basis sets, in contrast to the verygood performance on the CCSDT/FC interaction energiesdiscussed in Sec. III A, now converge very slowly and evenat X=6 are about 0.3 cm1 from the limit value ofEint

CCSDT/AEFC discussed below. As it could be expectedfrom the behavior of the CISD correlation energies discussed

TABLE V. Values of the core correction in cm1, at the CCSDT level oftheory for the argon dimer at R=3.75 , calculated and extrapolated rowsmarked Extr. using various Dunning et al. basis sets as well as the basissets developed in the present work. The same 33221 midbond set has beenadded to all bases. The extrapolated results given in the X=Q, X=5, andX=6 columns have been obtained from bases with X equal to TQ, Q5,and 56, respectively. See text for details of the extrapolation procedure.

Basis+ 33221 X=T X=Q X=5 X=6

aug-cc-pVXZ 2.091 1.082 1.701 1.095Extr. 0.345 2.351 0.262aug-cc-pVX+dZ 2.540 1.340 1.590 1.039Extr. 0.464 1.853 0.281disp-XZ /FC 2.191 2.065 2.259 1.050Extr. 1.973 2.462 0.611disp+d-XZ /FC 2.518 10.772 1.754 1.065Extr. 16.795 7.708 0.120

aug-cc-pCVXZ 0.986 0.919 0.864Extr. 0.870 0.807

aug-cc-pwCVXZ 0.960 0.886 0.847Extr. 0.831 0.807

disp-XZ /AE 2.892 1.162 0.966 0.883Extr. 0.101 0.760 0.770disp-XZ+1 /AE 1.152 0.980 0.894 0.857Extr. 0.854 0.804 0.805disp-XZ+2 /AE 0.960 0.893 0.859 0.841Extr. 0.845 0.823 0.817disp+d-XZ /AE 1.626 1.125 0.963 0.880Extr. 0.759 0.794 0.765



in Sec. II A, the extrapolated core corrections converge evenworse. In particular, Table V shows that the value of the corecorrection in Dunnings et al. aug-cc-pV5Z+ 33221 basisset 1.701 cm1, taken as the final measure of this quantityin the potentials of Refs. 7 and 1, is too large in magnitudeby about 0.9 cm1 compared to the limit value discussedbelow. A warning sign could have been the convergence ofthe CCSDT/AE results in the aug-cc-pVXZ+ 33221 se-quence, X=T,Q,5 ,6, which is quite irregular and, in fact,similar irregularities are present in all FC-optimized basissets. Also an earlier work by van Mourik et al.51 presentedsingle-point calculations in AE-optimized bases which gaveresults smaller in magnitude, close to the values recom-mended by us.

The AE-optimized sequences converge distinctly slowerin calculations of CCSDT/AE interaction energies than theFC-optimized series in the CCSDT/FC calculations al-though much better than FC-optimized ones in AE calcula-tions. However, the difference of the two types of interac-tion energies, i.e., the core contribution, convergesreasonably well. The Eint

CCSDT/AEFCcontributions in Table

V computed in all AE-optimized basis set families at thelargest available X 5 or 6 show a spread of only0.042 cm1. Apparently, the slow convergence of CCSDTstems mainly from the valence part and similarly affects FCand AE calculations. Therefore, basis set truncation errorspartially cancel when calculating the difference. We decidedto compute the core correction from the CBS extrapolationemploying the disp-5Z+2 /AE+ 33221 anddisp-6Z+2 /AE+ 33221 basis sets and to estimate its uncer-tainty as the difference between the extrapolated result andthe one obtained in the disp-6Z+2 /AE+ 33221 basis. AtR=3.75 , the core correction estimated in this wayamounts to 0.8170.024 cm1. Note that these error barsencompass the results extrapolated from all AE-optimizedbasis set families listed in Table V using the highest availableX except for the disp-XZ /AE and disp+d-XZ /AE results.

The calculation of the core contribution by subtraction ofthe limit CCSDT/FC interaction energies Sec. III A fromthe CCSDT/AE ones extrapolated, as above, from thedisp-5Z+2 /AE+ 33221 and disp-6Z+2 /AE+ 33221 ba-sis sets leads to a highly inaccurate result of 0.24 cm1.

-99

-98

-97

-96

-95

-94

100 200 300 400 500 600 700 800

Interactionen

ergy

[cm

-1]

N

aug-cc-pVXZ+(33221)ext-aug-cc-pVXZ+(33221)aug-cc-pV(X+d)Z+(33221)

ext-aug-cc-pV(X+d)Z+(33221)d-aug-cc-pV(X+d)Z+(33221)

ext-d-aug-cc-pV(X+d)Z+(33221)d-aug-cc-pV(X+d)Z+(44332)

ext-d-aug-cc-pV(X+d)Z+(44332)d-aug-cc-pV(X+d)Z+(666311)

ext-d-aug-cc-pV(X+d)Z+(666311)

-99

-98

-97

-96

-95

-94

100 200 300 400 500 600 700 800

Interactionen

ergy

[cm

-1]

N

disp-XZ/FC+(33221)ext-disp-XZ/FC+(33221)disp+d-XZ/FC+(33221)

ext-disp+d-XZ/FC+(33221)d-disp-XZ/FC+(33221)

ext-d-disp-XZ/FC+(33221)d-disp-XZ/FC+(666311)



ext-d-disp-XZ/FC+(666311)

FIG. 5. CCSDT/FC argon dimer interaction energiesin cm1 at R=3.75 computed in FC-optimized ba-sis sets from literature top panel and from this workbottom panel.



The reason is that the CCSDT/AE interaction energy in thesequence disp-XZ+2 /AE+ 33221, unlike the core contribu-tion, is still not well-converged, as indicated by the conver-gence pattern. A better result, 0.70 cm1, is obtained if theaug-cc-pCVXZ+ 33221 sequence is used to calculate theCCSDT/AE energy. We believe this better performance ispartly fortuitous, as discussed above, but partly results fromthe fact that this basis set contains a large number of FC-optimized exponents and therefore reproduces the valencepart of the interaction energy better than the completely AE-optimized disp-XZ+2 /AE+ 33221 family. Apart from theunavailability of the aug-cc-pCVXZ basis for X=6, an addi-tional reason we have decided to use the disp-XZ+2 /AE+ 33221 sequence rather than the aug-cc-pCVXZ+ 33221 one for the whole potential is that the compositionof the latter sequence changes quite dramatically betweenX=Q and X=5 due to a decontraction of the valence orbitals,which potentially may cause problems in CBS extrapola-tions. In any case, if we have chosen the aug-cc-pCVXZ+ 33221 sequence to compute the core correction, the resultnear the minimum would have been different by only0.01 cm1, and the uncertainty would have increased to0.057 cm1.

C. Explicitly correlated calculations of the CCSDTcontributions

The most accurate calculations of the helium dimer po-tential used explicitly correlated functions at the CCSDlevel52,53 in the first-quantized Gaussian geminalapproach.5456 Whereas this approach is too time-consumingto be applied to Ar2, a related so-called CCSDT-F12approach5759 developed recently is applicable to very largemolecules. To check our orbital calculations and possiblyobtain even better estimates of limit interaction energies thandescribed above, we have performed calculations using theversions CCSDT-F12A and CCSDT-F12B Ref. 58 ofthis method, which differ by the type of approximations in-volved. The largest basis set that can presently be applied insuch calculations is aug-cc-pV5Z+ 3322. The 3322 mid-

bond set, extensively used in SAPTDFT calculations,60 dif-fers from the 33221 one by the omission of the g function.Larger basis sets are not possible since the CCSDT-F12approach of Ref. 58 relies inherently on density fitting andrequires auxiliary basis sets.61,62 Such an auxiliary basis foran X=6 main set would have to include at least k functions,and therefore MOLPRO calculations have to be limited to X=5. Our calculations at the CCSDT-F12A/FC level gavethe interaction energy of 97.720 cm1 and the value ex-trapolated from this basis set and from the aug-cc-pVQZ+ 3322 one equals 97.592 cm1. These values are close tothe limit of 97.4450.063 cm1 obtained from orbital ba-sis sets but outside its error bars. The CCSDT-F12A corecorrection obtained with the X=5 basis is very poor in fact,positive. The CCSDT-F12B results are significantly worsein both cases. We have also performed calculations using theVXZ-F12 basis sets specifically designed for F12 calcula-tions with X up to Q,63 but the results had similar uncer-tainties to those discussed above. Thus, the CCSDT-F12method cannot yet compete with the orbital basis calcula-tions and extrapolations such as those presented in Tables IVand V. Certainly, the competitiveness of the CCSDT-F12method will increase when X5 calculations become pos-sible and perhaps when an exact CCSDT-F12 method, de-veloped very recently,59 is applied. However, such an ap-proach may also become very expensive.

D. Contributions beyond CCSDTThe values of the higher-level CC contributions from Eq.

1 computed at R=3.75 are listed in Table VIEint

TT/FC, Table VII EintQ/FC, and Table VIII

EintQQ/FC. All these contributions were calculated with the

FC approximation, as indicated by the symbols. The high-level CC effects are progressively more costly to calculate,and the largest basis sets that we were able to use contained294, 178, and 76 contracted functions, respectively, com-pared to 767 functions at the Eint

CCSDT/FC level. As one cansee from Tables VI and VII, the triple-excitation contributionbeyond the perturbative CCSDT approximation and the

1

0.5

0

0.5

1

2 3 4 5 6 7 8 9 10 12 14 16 18 20

diffe

renc

e(%

)

R []

ddispXZ/FC+(666311)ddispXZ/FC+(666311)

dispXZ/FC+(33221)augccpV(X+d)Z+(33221)

daugccpV(X+d)Z+(44332)

FIG. 6. Percentage differences of extrapolatedCCSDT/FC interaction energies computed using fivebasis set sequences from the final CCSDT/FC result.The latter was chosen based on the results in the firstfour sequences by discarding the most outlying valueand taking an average of the remaining three see textfor details. The solid black lines represent the assumeduncertainty of the final result, estimated as described inthe text. The extrapolations used X=6 and 7 for thebasis sets developed in this work and X=5 and 6 forDunning et al. bases. Note the logarithmic scale on thehorizontal axis.



perturbative quadruple-excitation contribution are quite sub-stantial, each larger in magnitude than 1 cm1, and both arequite difficult to converge with respect to the basis set, re-quiring calculations at the edge of current computationalpossibilities. Fortunately, those contributions cancel out to alarge extent as predicted in Ref. 1, which makes theCCSDT-level potentials of Refs. 7 and 1 reasonably accu-rate.

In the calculations of the EintTT/FC term, we were able

to use the sequences aug-cc-pVXZ+ 332,aug-cc-pVX+dZ+ 332, disp-XZ /FC+ 332, and disp+d-XZ /FC+ 332 up to X=5 and the sequenced-aug-cc-pVXZ+ 332 up to X=Q. We have also used thesame sequences without midbond functions and, for selectedX, with the 3321 and 33221 midbond sets. The resultscollected in Table VI show that the convergence, althoughslow, is fairly systematic starting from X=T. All the se-quences ending at X=5 and containing midbond converge toa value close to 1.29 cm1 and after extrapolation give about1.44 cm1. The d-aug-cc-pVXZ+ 332 sequence terminatedat X=Q is consistent with other sequences at the same car-dinal number. The effects of second augmentation and of theadditional d functions are almost completely negligible. Theeffects of bond functions are not negligible, but muchsmaller than at the CCSDT level. As our recommendedvalue of Eint

TT/FC, we have chosen the aug-cc-pVXZ

+ 332 Q5-extrapolated result of 1.444 cm1. The uncer-

TABLE VI. The full triples contribution EintTT/FC to the argon dimer in-

teraction energy in cm1 at R=3.75 , calculated using various Dunninget al. basis sets as well as with some basis sets developed in the presentwork. The column marked Extr. shows results extrapolated from a givenbasis set and the set from the same sequence and with the same midbondfunctions, but with the cardinal number X smaller by one.

Basis Size EintTT/FC Extr.

aug-cc-pVDZ 54 0.054aug-cc-pVDZ+ 332 76 0.532aug-cc-pVTZ 100 0.824 1.194aug-cc-pVTZ+ 332 122 0.785 1.339aug-cc-pVQZ 168 1.052 1.218aug-cc-pVQZ+ 332 190 1.111 1.349aug-cc-pVQZ+ 33221 213 1.143aug-cc-pV5Z 262 1.200 1.357aug-cc-pV5Z+ 332 284 1.273 1.444aug-cc-pVD+dZ 64 0.012aug-cc-pVD+dZ+ 332 86 0.430aug-cc-pVT+dZ 110 0.885 1.253aug-cc-pVT+dZ+ 332 132 0.860 1.404aug-cc-pVQ+dZ 178 1.104 1.264aug-cc-pVQ+dZ+ 332 200 1.162 1.382aug-cc-pVQ+dZ+ 33221 223 1.194aug-cc-pV5+dZ 272 1.226 1.353aug-cc-pV5+dZ+ 332 294 1.294 1.432d-aug-cc-pVTZ 132 0.697d-aug-cc-pVTZ+ 332 154 0.760d-aug-cc-pVTZ+ 3321 161 0.779d-aug-cc-pVQZ 218 1.028 1.269d-aug-cc-pVQZ+ 332 240 1.112 1.369d-aug-cc-pVQZ+ 3321 247 1.133 1.392d-aug-cc-pVQZ+ 33221 263 1.146

disp-DZ/FC 54 0.252disp-DZ /FC+ 332 76 0.588disp-TZ/FC 100 0.622 0.990disp-TZ /FC+ 332 122 0.688 1.225disp-QZ/FC 168 1.059 1.378disp-QZ /FC+ 332 190 1.141 1.472disp-QZ /FC+ 33221 213 1.169disp-5Z/FC 262 1.232 1.413disp-5Z /FC+ 332 284 1.290 1.446disp+d-DZ /FC 64 0.211disp+d-DZ /FC+ 332 86 0.503disp+d-TZ /FC 110 0.639 0.998disp+d-TZ /FC+ 332 132 0.698 1.204disp+d-QZ /FC 178 1.060 1.368disp+d-QZ /FC+ 332 200 1.141 1.464disp+d-QZ /FC+ 33221 223 1.172disp+d-5Z /FC 272 1.233 1.413disp+d-5Z /FC+ 332 294 1.290 1.446

TABLE VII. The perturbative quadruple-excitation contribution EintQ/FC to

the argon dimer interaction energy in cm1 at R=3.75 , calculated usingvarious Dunning et al. basis sets as well as with some basis sets developedin the present work. The column marked Extr. shows results extrapolatedfrom a given basis set and the set from the same sequence and with the samemidbond functions, but with the cardinal number X smaller by one.

Basis Size EintQ/FC Extr.

aug-cc-pVDZ 54 0.547aug-cc-pVDZ+ 332 76 0.776aug-cc-pVDZ+ 3321 83 0.775aug-cc-pVTZ 100 1.206 1.483aug-cc-pVTZ+ 332 122 1.497 1.801aug-cc-pVTZ+ 3321 129 1.535 1.855aug-cc-pVQZ 168 1.612 1.908aug-cc-pVD+dZ 64 0.858aug-cc-pVD+dZ+ 332 86 1.190aug-cc-pVT+dZ 110 1.379 1.598aug-cc-pVT+dZ+ 332 132 1.690 1.901aug-cc-pVQ+dZ 178 1.706 1.945

disp-DZ/FC 54 0.776disp-DZ /FC+ 332 76 1.011disp-TZ/FC 100 1.428 1.703disp-TZ /FC+ 332 122 1.591 1.836disp-QZ/FC 168 1.660 1.829disp+d-DZ /FC 64 0.781disp+d-DZ /FC+ 332 86 0.982disp+d-TZ /FC 110 1.428 1.701disp+d-TZ /FC+ 332 132 1.587 1.841

TABLE VIII. The full quadruples contribution EintQQ/FC to the argon dimer

interaction energy in cm1 at R=3.75 , calculated using various basissets.

Basis Size EintQQ/FC

aug-cc-pVDZ 54 0.104aug-cc-pVDZ+ 332 76 0.086

disp-DZ/FC 54 0.135disp-DZ /FC+ 332 76 0.109



tainty for EintTT/FC

, estimated as the difference between theextrapolated result and the result calculated in the X=5 basisset, amounts to 0.171 cm1. Such error bars encompass vir-tually all computed and extrapolated results in the largestbasis sets in Table VI and therefore are expected to be reli-able.

In the calculations of the EintQ/FC term presented in

Table VII, the largest basis set that we could use wasaug-cc-pVQ+dZ. As in the case of full triple excitations,the convergence is slow but relatively smooth. The extrapo-lated results with X=Q are all in the range from 1.8 to1.95 cm1. Somewhat surprisingly and unlike the case ofthe Eint

TT/FC term, the presence of an additional set of dfunctions can substantially influence the convergence of theconnected quadruple contribution: the aug-cc-pVX+dZ ba-sis sets perform clearly better than the disp-XZ /FC anddisp+d-XZ /FC ones, which are in turn much better than theaug-cc-pVXZ sets. The presence of midbond functions alsoimproves the convergence, as indicated by the small-X re-sults with midbond being closer to the large-X results thanthe small-X ones without midbond. However, at this level oftheory, the aug-cc-pVQ+dZ calculations were already verytime-consuming and we could not add any midbond func-tions to this basis set. Thus, our final recommended value forthe quantity Eint

Q/FCwill be computed from the best con-

verged family of basis sets, aug-cc-pVX+dZ, by a TQextrapolation. This recommended value at R=3.75 amounts to 1.9450.239 cm1, where the uncertainty wasestimated as the difference between the extrapolated resultand the one calculated in the X=Q basis set.

In the calculations of the small quadruple-excitation ef-fect beyond CCSDTQ included at CCSDTQ level pre-sented in Table VIII, we were able to use only up to theaug-cc-pVDZ+ 332 and disp-DZ /FC+ 332 basis sets.Whereas such basis sets work very poorly for the Eint

TT/FC

term, these bases do recover about 50% of EintQ/FC

. TableVIII also shows that the results given by various basis setsdiffer by up to 60%, whereas the same bases gave order ofmagnitude differences in calculations of Eint

TT/FC. Thus, it

appears that small basis sets give more reasonable results forEint

QQ/FC than for EintTT/FC

. We have arbitrarily selectedthe value of 0.086 cm1 computed in the aug-cc-pVDZ+ 332 basis set as our recommended result for Eint

QQ/FC

and assigned to it a 50% uncertainty, i.e., 0.043 cm1.Our final estimate of the nonrelativistic argon dimer in-

teraction energy at R=3.75 is 98.6760.304 cm1,where we have added the uncertainties quadratically. Notethat we have neglected the effects of the FC approximationon the interaction energy contributions beyond the CCSDTlevel as well as the effects of quintuple and higher CC exci-tations. Both of these effects should be minor and should beabsorbed by the current uncertainty.

E. Relativistic contribution

The relativistic effects at the van der Waals minimum,computed at the CCSDT/AE level with the second-orderDKH Hamiltonian,23,24 amount to 0.624, 0.615, and0.622 cm1 in the completely decontracted aug-cc-pVXZ

+ 33221, X=Q and 5, and disp-5Z /AE+ 33221 basis sets,respectively. We have taken the value of 0.615 cm1 as ourrecommended one. The spread, which can be assumed as anestimate of uncertainty at the DKH level, is only0.009 cm1. We had to use fully decontracted basis setssimilarly to the case of the zinc dimer,64 as the DKH relativ-istic correction obtained from contracted basis sets is notaccurate, especially at large distances. This is because theoccupied orbitals exhibit different basis set requirements forthe nonrelativistic and relativistic cases. With the contractedorbitals from nonrelativistic calculations, the latter require-ments cannot be satisfied. With decontracted sets, the relativ-istic orbitals are well described and the basis set convergenceis very good, as shown above. Note also that there is virtu-ally no difference between the calculations in decontractedFC- and AE-optimized basis sets. The reason is that decon-traction introduces compact s and p functions and makes theFC-optimized aug-cc-pV5Z basis set much more suitable forall-electron calculations even though compact df . . . func-tions are still missing. Our value of the relativistic correc-tion agrees well with the MP2/ZORA result14 amounting to0.7 cm1 and with the CCSDT results65 of 0.48 and0.62 cm1 in two different basis sets calculated by a sub-traction of the values cf. Table V of Ref. 65 obtained usingrelativistic and nonrelativistic effective core potentials.66

It should be noted that for the He2 system, the calcula-tions using the CowanGriffin approach or the second-orderDKH Hamiltonian at the CCSDT level recover only about10% of the relativistic contribution to the interactionenergy.67 The remainder of this contribution stems mainlyfrom the relativistic electron-electron interactions via thetwo-electron Darwin term and the Breit spin-spin and orbit-orbit corrections, not included in the DKH method note thatthe contribution called Breit term in Ref. 67 corresponds tothe orbit-orbit term only in the notation of Ref. 68. We haveverified that the situation is not nearly as bad for the argondimer by calculating all BreitPauli terms at theCCSDT/AE level using the DALTON code.29,68 In theaug-cc-pwCV5Z basis set, the two-electron Darwin, Breitspin-spin, and Breit orbit-orbit corrections to the interactionenergy at R=3.75 amount to 0.012, 0.024, and0.059 cm1, respectively. The sum of the BreitPauli termsmissing at the DKH level is 0.071 cm1 or 11.6% of theDKH relativistic correction. Thus, the DKH approximationfor Ar2 works far better than for He2, but the relativisticelectron-electron terms are still quite sizable. To account forsuch terms, we have assumed a 15% relative uncertainty forthe DKH relativistic correction at each R. Including the DKHcontribution and adding all uncertainties, including the DKHone, quadratically, one arrives at the final estimate of theinteraction energy at R=3.75 equal to99.2910.318 cm1.

F. Comparison with literature calculationsat the potential minimum

We compare in Table IX the interaction energy contribu-tions at R=3.75 obtained by us with the values from Refs.1, 7, and 6. The error estimate of our final result was ob-



tained as square root of the sum of the squares of the errorsof components. Since Jger et al.6 have not included thisdistance in their set of grid points, we have calculated thevalues ourselves using exactly their level of theory. Uponthis calculation, we have discovered that we were unable toreproduce Jger et al. CCSDT/FC results in thed-aug-cc-pV5+dZ+ 44332 basis set on the other hand,we could exactly reproduce their results in the larger,d-aug-cc-pV6+dZ+ 44332 basis as well as their contribu-tions beyond CCSDT/FC. We suspect that the second aug-mentation of the d-aug-cc-pV5+dZ+ 44332 basis setmight have been introduced in some nonstandard way in Ref.6.

Reference 1 extrapolated EintCCSDT/FC from the

aug-cc-pVXZ+ 33221 sequence with X=5 and 6 to get avalue which is only 0.030 cm1 0.049 cm1 when current,tighter convergence thresholds are employed from that rec-ommended by us, well within our error bars. The core cor-rection term Eint

CCSDT/AEFCcomputed in Ref. 7 is, however,

about two times too large in magnitude. As discussed earlier,the reason for this discrepancy is the use of an FC-optimizedbasis set in this calculation. The too large magnitude of thecore correction fortuitously makes up for terms not includedin calculations of Refs. 7 and 1 and the total interactionenergy of Ref. 1 is only 0.11 cm1 from the result obtainedby us.

As mentioned above, Jger et al.6 generally used foreach term basis sets with X smaller by one than applied inour work. Only the Eint

CCSDT/FC term was extrapolated in Ref.6 and this work did not provide any estimate of uncertainty.We have performed extrapolations for the remaining termsincluded in Ref. 6 except for the relativistic correction. Allthese extrapolations used the largest-X set applied for a giventerm in Ref. 6 and the corresponding X1 set. The uncer-tainties were estimated as the differences between the ex-trapolated results and the values in the larger of the two basissets.

The EintCCSDT/FC term was CBS-extrapolated in Ref. 6

from bases d-aug-cc-pVX+dZ+ 44332, X=5 and 6. Thisquantity agrees with our recommended value to within ourerror bars, but its internal error bars are about 2.5 timeslarger. The core correction computed in the aug-cc-pwCV5Zno midbond basis set used by Jger et al. amounts to0.912 cm1, significantly outside our error bars. However,the extrapolation from this value and the aug-cc-pwCVQZresult 1.010 cm1 gives an extrapolated energy of0.8090.103 cm1, in nearly perfect agreement with thevalue recommended by us. The corrections Eint

TT/FCand

EintQ/FC

, computed in Ref. 6 in the d-aug-cc-pVQZ+ 3321and aug-cc-pVTZ+ 3321 basis sets, respectively, are bothabout two uncertainties off our recommended values, whilethe corresponding extrapolated values are within our errorbars. Fortunately, the errors of the Eint

TT/FCand Eint

Q/FC

terms in Ref. 6 relative to our recommended values, althougheach of them is larger than 0.3 cm1, are of opposite signand cancel each other to within 0.1 cm1. The Eint

QQ/FC

correction was neglected in Ref. 6. The relativistic correctioncomputed using the aug-cc-pwCV5Z basis set and theCowanGriffin approximation is only 0.012 cm1 from ourvalue computed using the DKH Hamiltonian. The total inter-action energy of the Jger et al. approach is only 0.056 cm1from our result, however, this excellent agreement stems par-tially from the cancellation of errors between different con-tributions. If the extrapolated corrections are employed in-stead of the nonextrapolated ones, the resulting interactionenergy is 0.014 cm1 from our value, an even closer agree-ment. The partly fortuitous character of this agreement isapparent from the total internal uncertainty of the extrapo-lated result which is as large as 0.468 cm1 we have addedthe uncertainties quadratically including a contribution of0.086 cm1 accounting for the neglect of Eint

QQ/FCand the

contribution of 0.092 cm1 as the uncertainty of the relativ-istic correction, assumed the same as for our potential. Notethat the uncertainty of the actual approach of Ref. 6 is stillhigher due to the nonextrapolated terms. Assuming that eachnonextrapolated quantity has an uncertainty twice as large asthe corresponding extrapolated quantity and adding the con-tributions quadratically, one arrives at an estimate of the totaluncertainty of the approach of Ref. 6 equal to 0.871 cm1,nearly three times our uncertainty.

IV. POTENTIAL ENERGY CURVE

All the contributions to the argon dimer interaction en-ergy, Eq. 1, as well as the total interaction energy Eint andits uncertainty , have been presented in Table X for 33interatomic separations R. One can see that the strong can-cellation between the triple-excitation effects beyondCCSDT and the quadruples effects, observed for the equi-librium separation, holds for a wide range of R.

Another interesting, and somewhat surprising, observa-tion from Table X is that the core correction becomes posi-tive for R larger than 4.2 . This change of sign, displayed inFig. 7 for a wide selection of basis sets, was observed for allsufficiently large AE-optimized bases tested by us includingthe aug-cc-pwCV5Z set employed by Jger et al.6, whereasthe aug-cc-pVXZ+ 33221 bases, employed with X=5 in

TABLE IX. Comparison of interaction energy components at R=3.75 with literature. See text for specifications of basis sets and extrapolations used forvarious terms. All energies are in cm1.

EintCCSDT/FC Eint

CCSDT/AEFC EintTT/FC Eint

Q/FC EintQQ/FC Eint

rel Eint

References 1 and 7 97.475 1.706 99.181Jger et al., Ref. 6a 97.407 0.912 1.133 1.535 0.627 99.347Jger et al. + extrap. 97.4070.15 0.8090.10 1.3920.26 1.8550.32 0.6270.09 99.3050.47This work 97.4450.06 0.8170.02 1.4440.17 1.9450.24 0.0860.04 0.6150.09 99.2910.32aComputed in the present work using the level of theory and basis sets exactly as in Ref. 6.



Refs. 7 and 1, predict negative values of the core correctionfor all distances. Apparently, FC-optimized basis sets even aslarge as aug-cc-pV6Z+ 33221 and aug-cc-pV6+dZ+ 33221 can sometimes give a wrong sign for the core cor-rection. Moreover, medium-sized AE-optimized basis setslike disp-5Z /AE+ 33221 and aug-cc-pCVQZ+ 33221also predict a wrong sign.

As mentioned before, the CCSDT/FC energies in thed -disp-7Z /FC+ 666311, d -disp-7Z /FC+ 666311,and disp-7Z /FC+ 33221 basis sets, calculated with DAL-TON, exhibited numerical inaccuracies for large R. Therefore,for R6.0 d -disp-7Z /FC+ 666311 and disp-7Z /FC+ 33221 and R6.5 d -disp-7Z /FC+ 666311 thefull-basis results were substituted by the values calculatedusing MOLPRO with the basis set that differed from the fullset by the omission of the k functions the complete set of

results after this substitution is given in Table II of the Sup-porting Information. We have checked that although theomission of k functions would have produced a significantdifference for R around the minimum or smaller, this differ-ence becomes nearly negligible around 56 . Because ofnumerical inaccuracies, we were not able to obtain suffi-ciently converged values of the core correction in thedisp-5Z+2 /AE+ 33221 basis set, as well as of the relativ-istic correction, at R=20 , and we set these two values tozero. For this distance, the core and relativistic correctionscontribute very little anyway, as seen in Table X.

V. FIT OF THE ARGON DIMER POTENTIAL

The 33 ab initio argon-argon interaction energies listedin Table X formed the data set for fitting an analytic poten-

TABLE X. Contributions to the ab initio argon dimer interaction potential, the total interaction potential, and its uncertainty. The energy unit is 1 cm1. Thelast digits in the interaction energies and their uncertainties are not meaningful but are given to enable precise comparisons with other calculations and areproduction of our analytic fit.

R EintCCSDT/FC a EintCCSDT/AEFC b EintTT/FC c EintQ/FC d EintQQ/FC e Eintrel. f Eint g

2.00 36 274.12 291.03 50.47 45.26 2.49 272.74 35 718.05 75.762.20 18 201.29 169.30 32.33 29.36 1.53 153.93 17 882.56 43.542.40 8 794.33 95.25 20.96 19.92 0.97 82.08 8 619.02 24.472.60 4 055.01 52.03 13.70 13.90 0.63 42.05 3 961.36 13.532.80 1 743.54 27.67 9.01 9.85 0.42 20.89 1 694.56 7.333.00 659.043 14.294 5.977 7.012 0.288 10.134 633.868 4.0753.20 176.688 7.148 4.009 4.984 0.203 4.826 163.942 2.0573.40 19.704 3.435 2.727 3.534 0.146 2.273 26.074 0.9193.50 63.177 2.336 2.262 2.976 0.125 1.559 67.661 0.6273.60 86.158 1.560 1.884 2.508 0.107 1.071 89.306 0.4643.65 92.325 1.306 1.722 2.303 0.100 0.889 95.001 0.4003.70 95.902 1.020 1.576 2.116 0.093 0.739 98.108 0.3523.75 97.445 0.817 1.444 1.945 0.086 0.615 99.291 0.3183.80 97.409 0.647 1.325 1.788 0.080 0.514 98.953 0.2943.85 96.162 0.508 1.216 1.645 0.075 0.430 97.454 0.2743.90 94.004 0.393 1.118 1.514 0.070 0.361 95.084 0.2574.00 87.970 0.222 0.949 1.286 0.060 0.257 88.726 0.2214.20 72.653 0.035 0.692 0.934 0.046 0.139 73.024 0.1554.40 57.653 0.038 0.514 0.687 0.035 0.083 57.836 0.1204.60 44.939 0.059 0.388 0.512 0.027 0.055 45.031 0.0914.80 34.801 0.060 0.298 0.386 0.021 0.040 34.847 0.0745.00 26.956 0.052 0.232 0.294 0.017 0.031 26.980 0.0595.30 18.530 0.039 0.163 0.200 0.012 0.022 18.538 0.0375.60 12.968 0.029 0.117 0.140 0.009 0.017 12.970 0.0306.00 8.281 0.018 0.078 0.089 0.006 0.011 8.281 0.0206.50 4.938 0.009 0.048 0.054 0.003 0.007 4.938 0.0137.00 3.075 9 0.004 9 0.031 3 0.033 6 0.002 2 0.004 8 3.075 9 0.007 98.00 1.326 3 0.001 9 0.014 2 0.014 6 0.001 0 0.002 2 1.326 0 0.003 8

10.00 0.332 2 0.000 3 0.003 8 0.003 7 0.000 2 0.000 6 0.332 1 0.000 712.00 0.108 88 0.000 13 0.001 23 0.001 22 0.000 09 0.000 16 0.108 81 0.000 2014.00 0.042 59 0.000 03 0.000 48 0.000 48 0.000 04 0.000 09 0.042 67 0.000 1016.00 0.018 97 0.000 01 0.000 21 0.000 21 0.000 01 0.000 04 0.018 98 0.000 0320.00 0.004 92 0.000 02 0.000 04 0.000 05 0.000 00 0.000 00 0.004 91 0.000 01

aComputed using the results extrapolated from bases d -disp-XZ /FC+ 666311, d -disp-XZ /FC+ 666311, disp-XZ /FC+ 33221 X=6,7, andaug-cc-pVX+dZ+ 33221 X=5,6, as described in the text.bExtrapolated from the disp-5Z+2 /AE+ 33221 and disp-6Z+2 /AE+ 33221 basis sets.cExtrapolated from the aug-cc-pVQZ+ 332 and aug-cc-pV5Z+ 332 basis sets.dExtrapolated from the aug-cc-pVT+dZ and aug-cc-pVQ+dZ basis sets.eCalculated in the aug-cc-pVDZ+ 332 basis set.fComputed using the second-order DKH Hamiltonian, the CCSDT/AE level of theory, and the completely decontracted aug-cc-pV5Z+ 33221 basis set.gObtained by adding the partial uncertainties quadratically; see text.



tial. The form of the fitting function was similar to our pre-vious argon dimer study,1

VR = A + BR + C/R + DR2 + ER3eR

n=3

8

f2nbRC2nR2n

, 2

where A ,B ,C ,D ,E , ,b, and the leading asymptotic con-stants C6 and C8 are fit parameters. In Eq. 2, f2nx are theTangToennies damping functions,69

f2nx = 1 exk=0

2nxk

k!. 3

The higher asymptotic constants C10,C12,C14,C16 were thesame as in Ref. 1, i.e., the value C10=50 240 a.u. was takenfrom Ref. 70 and C12C16 were estimated in Ref. 1 bymeans of Thakkars extrapolation71 the small change of theextrapolated constants C12C16 due to a change in C6 and C8relative to Ref. 1 was neglected to simplify fitting.

The fit of Ref. 1 used the value C6=64.691 a.u., de-duced, using the dipole oscillator strength distributionDOSD approach, from experimental data in Ref. 72, whereits accuracy was estimated as 0.04%. However, we foundthat this value is incompatible with our current tight-uncertainty large-R results. The validity of the error bars ofRef. 72 was also questioned in a recent study by Kumar andThakkar,73 who introduced an improved DOSD algorithmand obtained a value C6=64.42 a.u. with an estimated un-certainty of 1%. The ab initio study of the present worksuggests that the latter result, as well as the older value C6=64.30 a.u. obtained by Kumar and Meath,74 is much closerto the exact one, whereas the value C6=63.50 a.u. employedin the HFDID1 potential5 as well as the result C6=63.861 a.u. fitted to the ab initio data by Jger et al.6 aresomewhat underestimated. The behavior of the asymptoticparts of various potentials is illustrated in Fig. 8 where theratio of the undamped expansions to our ab initio computedinteraction energies is plotted. At the largest Rs, the posi-tions of the curves are proportional to the values of the C6

0.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

4 4.5 5 5.5 6 6.5 7 7.5 8

core

corr

ectio

n[c

m

1 ]

R []

augccpV5ZaugccpV6Z

augccpV(5+d)ZaugccpV(6+d)Z

augccpCVQZaugccpCV5Z

augccpwCV5Zdisp5Z/AEdisp6Z/AE

disp5Z+2/AEdisp6Z+2/AE

FIG. 7. Behavior of the core correction at theCCSDT level of theory for 4R8 calculatedusing different basis sets. The 33221 set of midbondfunctions was included in all bases.

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025

6 8 10 12 14 16 18 20

ener

gyra

tio

R []

ab initioPatkowski et al. 2005

AzizSlamanAziz

Jaeger et al.Boyes

fitted C6,C8 this work

FIG. 8. Comparison of the ab initio interaction energiesfor large R with the energies predicted by the undampedasymptotic expansion utilizing various Cn constants.The values plotted are expressed relative to the ab initiointeraction energy at a given R: thus, all the ab initioresults are equal to 1, only their uncertainties vary. Thecurves marked Patkowski et al. 2005, Aziz-Slaman, Aziz, Jaeger et al., and Boyes utilizethe asymptotic constants from Refs. 1, 76, 5, 6, and 75,respectively.



constants and reflect the differences discussed above noticethat for R=20 , the ab initio value becomes less accurate.Clearly, the present theory provides very accurate bounds onthe C6 constant.

Due to the lack of an unambiguous choice of the refer-ence C6 constant, we decided to optimize it ourselves, to-gether with C8 and all the other adjustable parameters A, B,C, D, E, , and b. These parameters were determined bynonlinear least-squares fitting of the 33 ab initio energies,with weights proportional to 1 /nonrel

2, where nonrel listed in

Table III of the Supporting Information48 are uncertaintiesof the nonrelativistic part of the interaction energy, i.e.,nonrel differ from total Table X by the neglect of therelativistic contribution. Note that using total to determineweights would have resulted in a very similar analytic poten-tial: we chose to keep using nonrel as extensive applicationsof our potential were already underway at the time we esti-mated the uncertainty associated with the relativistic contri-bution. The resulting fit parameters are gathered in Table XI.Contrary to the approach of Ref. 6, we chose not to fit C10since it is extremely hard to determine it to good accuracy inthis way. The ab initio value of C10 from Ref. 70 employedby us should be accurate enough for our purposes. The fittedanalytic potential is within the ab initio error bars for all datapoints except for R=20 where it is barely outside them.The fitted value of C6 is 64.2890 a.u., well within the errorbars of Refs. 74 and 73, and the fitted value of C8 is1514.86 a.u. whereas the value computed in Ref. 70 is1644 a.u.

The work of Ref. 1 included a reasonably accurate de-termination of the argon dimer potential in the small-R re-gion, with ab initio interaction energies calculated for Rdown to 0.25 . Even though no particular behavior for R

2.0 was enforced in our fit described above, it behavesreasonably down to about 1.0 and recovers t

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Argon pair potential at basis set and excitation limits