Basis-set convergence of the molecular electric dipole moment

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Basis-set convergence of the molecular electric dipole moment

Asger Halkier

Department of Chemistry, University of Aarhus, DK-8000 A˚ rhus C, Denmark Wim Klopper

Theoretical Chemistry Group, Debye Institute, Utrecht University, Padualaan 14, NL-3584 CH Utrecht, The Netherlands

Trygve Helgaker

Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway Poul Jørgensen

Department of Chemistry, University of Aarhus, DK-8000 A˚ rhus C, Denmark 共Received 15 April 1999; accepted 9 June 1999兲

The electric dipole moments 共␮兲 of BH and HF are computed in conventional calculations employing different correlation-consistent basis sets at the levels of Hartree–Fock theory, second-order perturbation theory, and coupled cluster theory with single and double excitations, and single and double excitations with a perturbative triples correction. The basis-set convergence of␮ is examined by comparison with results obtained with explicitly correlated wave function models.

Inclusion of diffuse functions in the basis set is essential for accurate calculations of␮. They speed up the convergence at the Hartree–Fock level significantly and make the convergence at the correlated levels systematic. Once the outer valence regions important for ␮ are described accurately via the diffuse functions, the convergence at the correlated levels is governed by the interelectronic Coulomb singularity. For the aug-cc-pVXZ basis sets, the correlation contribution to

␮ ^follows ␮X corr⫽␮lim

corr⫹aX^⫺³, which is similar to the form for the correlation energy, and extrapolated values based on this form represent a significant improvement on the ordinary basis-set results. Our estimates of the exact dipole moments ␮e(HF)⫽1.8037⫾0.0007 D and ␮0(BH)

⫽1.3586⫾0.0007 D are in excellent agreement with the experimental values ␮e(HF)⫽1.803

关S0021-9606共99兲31233-2兴

I. INTRODUCTION

Basis-set convergence and extrapolations are among the most important issues of contemporary molecular electronic wave function theory—see for example Refs. 1–14 and ref- erences therein. When examining basis-set convergence, it is first of all mandatory to have a hierarchical sequence of basis sets with systematic improvements from level to level. The correlation-consistent polarized valence and core-valence basis sets, denoted cc-pVXZ and cc-pCVXZ, respectively, and their diffusely augmented counterparts, denoted aug-cc- pVXZ and aug-cc-pCVXZ, of Dunning and co- workers^10,15–20constitute such hierarchies and have been ex- tensively used in our previous studies of basis-set convergence of the correlation energy,^8,11 the Hartree–Fock self- consistent field 共SCF兲 energy,¹⁴ and various first-order one- electron properties.⁹ Second, the basis-set limit needs to be established from a source that is independent of the basis sets under study. For correlated methods, efficient tools for this purpose are the explicitly correlated R12 methods,^21–25 which yield results close to the basis-set limit, because they include terms in the wave function that are linear in the in- terelectronic distance r₁₂, as required by the Coulomb cusp condition.

As the energy 共E兲 is the fundamental property of ab initio quantum chemistry, most recent studies of basis-set

convergence and extrapolations have been concerned with E—and especially with the correlation energy. Less attention has been given to electric properties such as the dipole moment 共␮兲, and in the studies devoted to these properties, ei- ther the basis-set limit has not been established with R12 methods⁹ or the R12 results have not been compared with conventional results obtained with the correlation-consistent basis sets.¹³

In this study, we present results for ␮of the molecules BH and HF obtained in conventional calculations employing the different correlation-consistent basis sets at the following levels of wave function models: SCF, second-order Møller–

Plesset perturbation theory 共MP2兲,²⁶ coupled cluster singles and doubles共CCSD兲,²⁷and CCSD augmented by a perturba- tional correction for connected triple excitations 关CCSD共T兲兴.²⁸ Furthermore, results obtained at the explicitly correlated MP2-R12, CCSD-R12, and CCSD共T兲-R12 levels are presented. With these data, we investigate the basis-set convergence and the accuracy of different extrapolations for

␮, to improve our understanding of these issues and to ex- plore similarities/differences in the basis-set convergence of

␮and E. As a corollary of this investigation, we obtain very accurate theoretical values for ␮of BH and HF, which are compared with experimental values.

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II. THEORY

It is well-known that the convergence of the correlation energy with respect to the size of the basis set is very slow in conventional calculations because the expansion of the two- electron operator r₁₂in one-electron functions is very slowly convergent. Following earlier work by Schwartz,²⁹ Carroll et al.,³⁰and Hill,³¹we recently proposed the expression⁸

E_X^corr⫽E_lim^corr⫹AX^⫺³ 共1兲

for the convergence of the correlation energy E_X^corrobtained with the correlation-consistent basis set with cardinal number X (D:2,T:3...) toward the basis-set limit E_lim^corr. As shown in Ref. 8, this expression exhibits the correct asymptotic behav- ior of the correlation energy for large X compared with the theoretical analysis of the partial-wave expansion of the he- lium atom.³¹ The extrapolation for the correlation energy is thus theoretically motivated and the extrapolated E_lim^corrvalues obtained from Eq.共1兲have accordingly been found to be in good agreement with those obtained from large R12 calculations.^8,11

The Cartesian components of ␮are obtained as energy derivatives

␮␥⫽⫺⳵^E*

⳵^F␥

冏

_F_⫽₀^, ^␥^⫽^{x,y ,z} ^共²^兲

of the energy E* obtained from the perturbed Hamiltonian

H⫽H₀⫺␮–F, 共3兲

where H₀ is the Hamiltonian of the unperturbed molecule, and the last term is the interaction operator between the dipole moment ␮ and a static homogeneous external electric field F. As the dipole interaction operator is a simple one- electron operator, the convergence of the correlation part of E* is governed by the electronic cusp condition originating from H₀. Consequently, the convergence of the correlation part of E* and that of the unperturbed energy should be the same. The asymptotic convergence of the correlation contribution to ␮is therefore expected to be of a form similar to that of the correlation energy

␮X corr⫽␮lim

corr⫹aX^⫺³. 共4兲

However, two comments are appropriate.

First, the dipole interaction operator is linear in the elec- tronic coordinates r_iso, in order to obtain accurate results for

␮, an accurate representation of the wave function is needed in the outer valence region, which is a low-density region and therefore relatively unimportant for the correlation energy of the unperturbed molecule. The convergence of ␮ implied by Eq. 共4兲 may therefore not be observed for basis sets that do not include the diffuse functions needed to de- scribe the outer valence region properly, even though the same basis sets follow Eq. 共1兲 for the convergence of the correlation energy. We shall examine this particular problem by performing calculations both in the cc-pVXZ and cc- pCVXZ basis sets, which do not provide an accurate representation of the outer valence region, and in the aug-cc-

pVXZ and aug-cc-pCVXZ basis sets, which contain the diffuse functions needed for accurate calculations of␮^.⁹

Second, Eq. 共1兲 is a truncated series that contains only the leading term in an inverse power series in X.⁸ Expres- sions that include the next terms of the series are given by

E_X^corr⫽E_lim^corr⫹AX^⫺³⫹BX^⫺⁴⫹¯ 共5兲 for the correlation energy and consequently

␮X corr⫽␮lim

corr⫹aX^⫺³⫹bX^⫺⁴⫹¯ 共6兲

for ␮^corr, where we note the relationships a_␥⫽⫺⳵^A/

⳵^F␥兩F⫽0 and b_␥⫽⫺⳵^B/⳵^F␥兩F⫽0. As it is not possible to predict the dependence of A on F_␥ compared to the depen- dence of B on F_␥a priori, the relative importance of the X^⫺³ and X^⫺⁴ terms in Eq.共6兲 may vary from case to case. Al- though the X^⫺³ term will always dominate the convergence for sufficiently large X, the X^⫺⁴ term may thus dominate the convergence for some—or even all—of the cardinal numbers for which it is possible to perform the calculations. To examine this particular problem, we shall perform extrapolations of the more general form

␮X corr⫽␮lim

corr⫹aX^⫺␣ 共7兲

with different values of␣^.

Finally, for extrapolations of the correlation energy based on Eq. 共1兲, the most accurate extrapolation beyond a given cardinal number X⬘ is obtained by employing only the results for X⬘ ^{and X}⬘⫺1, since reference to results of lower cardinal numbers—which are further away from the region where the truncation of Eq. 共5兲to Eq. 共1兲 is valid—is then avoided.¹¹For␮, we shall therefore consider only two-point extrapolations for consecutive X, which for the general form in Eq. 共7兲 yields the following expression for the extrapolated basis-set limit

␮lim corr⫽␮X

corrX^␣⫺␮X⫺1

corr 共X⫺1兲^␣

X^␣⫺共X⫺1兲^␣ . 共8兲

III. RESULTS AND DISCUSSION A. Computational details

For linear molecules there is only one nontrivial compo- nent of ␮ ⁽␮z), and we thus only report␮⫽兩␮z兩⫽兩␮兩 below.

All calculations have been carried out at the experimen- tal equilibrium geometries of r_e(BH)⫽2.3289 a₀ and r_e(HF)

⫽0.917 Å, respectively.³²

Conventional SCF, MP2, CCSD, and CCSD共T兲calcula- tions have been carried out for the cc-pCVXZ and aug-cc- pCVXZ basis sets (X⫽D-6) with all the electrons correlated and for the cc-pVXZ and aug-cc-pVXZ basis sets (X

⫽D-6) with only the valence electrons correlated. These calculations have been performed with a local version of the Dalton program³³ that contains the coupled cluster code of Koch and co-workers.^9,34–36

The explicitly correlated MP2-R12, CCSD-R12, and CCSD共T兲-R12 calculations have been performed with the

DIRCCR12-95 program,³⁷ employing two different basis sets

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for each molecule: A smaller set 共179 contracted Gaussian functions兲denoted d-aug-cc-pV共Q/5兲Z, and a larger set共285 functions兲denoted d-aug-cc-pV共5/6兲Z. For H and F, the sin- gly augmented aug-cc-pV共Q/5兲Z basis set is described in detail in Ref. 38. It combines the low angular momentum functions of the aug-cc-pV5Z basis set关s and p 共F兲, s共H兲兴with the higher angular momentum functions of the aug-cc-pVQZ basis set. The doubly augmented d-aug-cc-pV共Q/5兲Z basis is constructed by adding to the aug-cc-pV共Q/5兲Z basis one set of diffuse functions to each angular momentum shell, and has been developed for the purpose of computing the electric dipole moment of the HF dimer.³⁹In a similar manner—that is, by combining the d-aug-cc-pV5Z and d-aug-cc-pV6Z basis sets—we have derived the d-aug-cc-pV共5/6兲Z basis set, and the same procedures have been followed for the B atom.

All the results have been obtained from calculations in which finite perturbations with electric field strenghts of F

⫽⫾0.0001 a.u. 关⬅e/(4␲⑀0a₀²)兴 have been applied and the dipole moment is thus obtained from central-differences numerical differentiation of the energy. The conventional SCF, MP2, and CCSD results have furthermore been evaluated analytically to check the accuracy of the numerical differentiation.

Finally, in the comparison with the experimental dipole moments the following conversion factor is used: 1 a.u.

⬅ea₀⫽2.541 77 Debye共D兲.

B. Convergence in Hartree–Fock calculations

The results for BH and HF are collected in Tables I and II, respectively. For diatomics, the SCF basis-set limit may be obtained from numerical Hartree–Fock studies, and the SCF basis-set limit result for␮of HF has been determined in this way by Sundholm et al. to be 0.756 08 a.u.⁴⁰ However, since the bond distance employed in that study is slightly different from the one used here, we have computed ␮ ^of both HF and BH with the numerical SCF program of Kobus and co-workers.^41,42 These new numerical results are the ones labeled ‘‘numerical’’ in Tables I and II.

We first note that the effect of core-correlating functions on ␮ calculated at the SCF level is very small—as expected—and we shall therefore focus solely on the valence basis sets in the remainder of this section. On the other hand, the inclusion of diffuse functions in the basis increases the accuracy of ␮ significantly. The basis-set error for an aug- cc-pVXZ basis set is in general significantly smaller than for the corresponding cc-pVXZ basis set, and for X⭓4 the aug- cc-pVXZ basis-set error is more than an order of magnitude smaller than that of the corresponding cc-pVXZ basis set.

Errors as large as 2.0 milliatomic units 共m.a.u.兲 and 0.5 m.a.u. persist at the cc-pV6Z level for HF and BH, respectively, and already the aug-cc-pVQZ basis set provides re-

TABLE I. Calculated␮ein atomic units for the BH molecule at the experimental re. For SCF the total dipole moment is given, whereas for the correlation models the correlation contribution to the dipole moment (␮model⫺␮SCF) is given.

Basis set SCF ⌬MP2 ⌬CCSD ⌬CCSD共T兲

Valence only

cc-pVDZ 0.657 16 ⫺0.059 04 ⫺0.157 10 ⫺0.162 73

cc-pVTZ 0.671 39 ⫺0.053 09 ⫺0.133 19 ⫺0.140 70

cc-pVQZ 0.679 43 ⫺0.050 25 ⫺0.128 32 ⫺0.136 45

cc-pV5Z 0.682 75 ⫺0.049 03 ⫺0.127 69 ⫺0.136 14

cc-pV6Z 0.684 46 ⫺0.048 32 ⫺0.127 49 ⫺0.136 06

aug-cc-pVDZ^a 0.687 96 ⫺0.057 26 ⫺0.152 74 ⫺0.158 57

aug-cc-pVTZ^a 0.686 49 ⫺0.052 15 ⫺0.133 61 ⫺0.141 49

aug-cc-pVQZ^a 0.684 93 ⫺0.049 60 ⫺0.129 33 ⫺0.137 69

aug-cc-pV5Z 0.684 95 ⫺0.048 20 ⫺0.127 99 ⫺0.136 52

aug-cc-pV6Z 0.684 98 ⫺0.047 49 ⫺0.127 41 ⫺0.136 02

Numerical^b 0.684 96

R12/d-aug-cc-pV共Q/5兲Z 0.684 90 ⫺0.046 12 ⫺0.126 93 ⫺0.135 27

R12/d-aug-cc-pV共5/6兲Z 0.684 95 ⫺0.046 08 ⫺0.126 73 ⫺0.135 25

All electrons

cc-pCVDZ 0.657 30 ⫺0.061 74 ⫺0.155 33 ⫺0.162 40

cc-pCVTZ 0.671 46 ⫺0.054 93 ⫺0.129 71 ⫺0.139 58

cc-pCVQZ 0.679 51 ⫺0.050 51 ⫺0.122 71 ⫺0.133 79

cc-pCV5Z 0.682 74 ⫺0.049 00 ⫺0.121 72 ⫺0.133 28

cc-pCV6Z 0.684 45 ⫺0.048 17 ⫺0.121 41 ⫺0.133 14

aug-cc-pCVDZ^a 0.687 57 ⫺0.060 09 ⫺0.151 06 ⫺0.158 28

aug-cc-pCVTZ^a 0.685 70 ⫺0.054 08 ⫺0.129 79 ⫺0.140 00

aug-cc-pCVQZ^a 0.684 89 ⫺0.049 98 ⫺0.123 77 ⫺0.135 09

aug-cc-pCV5Z^a 0.684 95 ⫺0.048 23 ⫺0.122 05 ⫺0.133 70

aug-cc-pCV6Z 0.684 97 ⫺0.047 36 ⫺0.121 34 ⫺0.133 11

R12/d-aug-cc-pV共Q/5兲Z 0.684 90 ⫺0.047 11 ⫺0.121 85 ⫺0.132 64

R12/d-aug-cc-pV共5/6兲Z 0.684 95 ⫺0.046 07 ⫺0.120 76 ⫺0.132 07

aFrom Ref. 43.

bNumerical Hartree–Fock.

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sults that are significantly more accurate than those obtained with the cc-pV6Z basis set.

The basis-set errors in the aug-cc-pVXZ series are below 0.1 m.a.u. for X⭓5 and X⭓4 for HF and BH, respectively.

The basis-set convergence of ␮ within this series of basis sets is thus quite fast, but unfortunately it is not very systematic. For BH, the error is reduced by a factor of 50 from the triple-zeta to the quadruple-zeta level and from that point ␮ is for most practical purposes converged to the basis-set limit. For HF, the convergence is more smooth but the reduction in the error from level to level varies unsystemati- cally between a factor of 2 and 6. It therefore appears diffi- cult to find an analytical form for the basis-set convergence of ␮at the SCF level.

C. Convergence in correlated calculations

Turning to the results for the correlation contribution to

␮ ⁽␮^corr) in Tables I and II, we first note that the effect of core correlation is rather small, as the difference between the results obtained in a valence basis set and the corresponding core-valence basis set is less than 5%. The basis-set convergence of ␮^corr for the series with all electrons correlated in the core-valence basis sets is thus virtually the same as for the series with the valence electrons correlated in the corresponding valence basis sets. In the following, we therefore investigate the cc-pVXZ and cc-pCVXZ results together

关jointly denoted cc-p共C兲VXZ兴 and the aug-cc-pVXZ and aug-cc-pCVXZ basis sets together 关jointly denoted aug-cc- p共C兲VXZ兴.

The R12 results for␮^corrobtained with the two different R12 basis sets differ on average by 0.34 m.a.u. The accuracy of the R12 results obtained in the d-aug-cc-pV共5/6兲Z basis set, however, is appreciably higher than this difference, and the R12 results are estimated to be within 0.2 m.a.u. of the basis-set limit. The R12 results in the smaller d-aug-cc- pV共Q/5兲Z basis set are in all cases except one closer to the basis-set limit 关as given by the d-aug-cc-pV共5/6兲Z R12 re- sults兴than the conventional aug-cc-p共C兲V6Z results are. Al- though the R12 results in the smaller basis are not sufficiently accurate to serve as reliable basis-set limit estimates, they thus represent a significant improvement on the conventional results obtained in similar—or even larger—basis sets.

The basis-set convergence of ␮^corr within the aug-cc- pVXZ and aug-cc-pCVXZ series of basis sets is monotonic but not particularly fast. From triple to quadruple zeta the error is on average reduced by a factor of 2.45 and from quadruple to quintuple zeta and quintuple to sextuple zeta the similar factor is 1.90 and 1.73, respectively. The X^⫺³ form of Eq. 共4兲implies that the error is reduced from one level to the next by factors of 2.37共T,Q兲, 1.95 (Q,5), and 1.73共5,6兲, respectively. The good agreement between these values and the observed ones is the first indication that the basis-set

TABLE II. Calculated␮ein atomic units for the HF molecule at the experimental re. For SCF the total dipole moment is given, whereas for the correlation models the correlation contribution to the dipole moment (␮model⫺␮SCF) is given.

Basis set SCF ⌬MP2 ⌬CCSD ⌬CCSD共T兲

Valence only

cc-pVDZ 0.767 02 ⫺0.042 82 ⫺0.046 91 ⫺0.051 15

cc-pVTZ 0.763 70 ⫺0.046 82 ⫺0.046 28 ⫺0.052 50

cc-pVQZ 0.760 76 ⫺0.045 20 ⫺0.042 36 ⫺0.048 92

cc-pV5Z 0.760 20 ⫺0.043 62 ⫺0.040 64 ⫺0.047 21

cc-pV6Z 0.758 17 ⫺0.043 74 ⫺0.040 46 ⫺0.047 17

aug-cc-pVDZ^a 0.759 76 ⫺0.050 72 ⫺0.050 33 ⫺0.056 34

aug-cc-pVTZ 0.757 50 ⫺0.048 53 ⫺0.046 07 ⫺0.052 85

aug-cc-pVQZ 0.756 34 ⫺0.045 84 ⫺0.042 40 ⫺0.049 27

aug-cc-pV5Z 0.756 17 ⫺0.044 83 ⫺0.041 35 ⫺0.048 23

aug-cc-pV6Z 0.756 14 ⫺0.044 31 ⫺0.040 85 ⫺0.047 74

R12/d-aug-cc-pV共Q/5兲Z 0.756 20 ⫺0.043 23 ⫺0.040 42 ⫺0.047 22

R12/d-aug-cc-pV共5/6兲Z 0.756 14 ⫺0.043 28 ⫺0.040 26 ⫺0.047 11

All electrons

cc-pCVDZ 0.767 60 ⫺0.042 89 ⫺0.046 70 ⫺0.051 03

cc-pCVTZ 0.763 65 ⫺0.045 43 ⫺0.044 72 ⫺0.051 13

cc-pCVQZ 0.760 72 ⫺0.043 69 ⫺0.040 69 ⫺0.047 44

cc-pCV5Z 0.760 17 ⫺0.042 03 ⫺0.038 89 ⫺0.045 63

cc-pCV6Z 0.758 15 ⫺0.042 12 ⫺0.038 69 ⫺0.045 57

aug-cc-pCVDZ^a 0.760 08 ⫺0.050 68 ⫺0.049 98 ⫺0.056 09

aug-cc-pCVTZ^a 0.757 39 ⫺0.047 11 ⫺0.044 49 ⫺0.051 47

aug-cc-pCVQZ^a 0.756 28 ⫺0.044 30 ⫺0.040 70 ⫺0.047 76

aug-cc-pCV5Z^a 0.756 14 ⫺0.043 22 ⫺0.039 58 ⫺0.046 64

aug-cc-pCV6Z 0.756 13 ⫺0.042 68 ⫺0.039 06 ⫺0.046 14

R12/d-aug-cc-pV共Q/5兲Z 0.756 20 ⫺0.041 69 ⫺0.038 85 ⫺0.045 79

R12/d-aug-cc-pV共5/6兲Z 0.756 14 ⫺0.041 59 ⫺0.038 47 ⫺0.045 47

aFrom Ref. 43.

bNumerical Hartree–Fock.

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convergence of ␮^corr within the aug-cc-pVXZ and aug-cc- pCVXZ series of basis sets follows Eq. 共4兲. We also note that, although the error is reduced from double to triple zeta, this reduction varies considerably from case to case. The basis-set convergence is thus less systematic when double- zeta quality results are included, as seen also for the energy.¹¹

In the left half of Table III, we have listed the mean error, the standard deviation, the mean absolute error, and the maximum absolute error of the␮^corrresults obtained with the cc-p共C兲VXZ basis sets compared to the d-aug-cc-pV共5/

6兲Z R12 results. The same information is also given for the aug-cc-p共C兲VXZ results. Comparing the statistical data for the cc-p共C兲VXZ and aug-cc-p共C兲VXZ basis sets we note—

somewhat surprisingly—that the mean error for a cc- p共C兲VXZ level is in fact lower than that of the corresponding aug-cc-p共C兲VXZ level. However, the standard deviation for the cc-p共C兲VXZ levels are significantly larger than for the corresponding aug-cc-p共C兲VXZ levels, so the results obtained in a basis set without diffuse functions are more scattered in accuracy than those obtained in the corresponding diffusely augmented basis set. Furthermore, the basis-set convergence within the cc-p共C兲VXZ series of basis sets be- comes nonmonotonic six times, incompatible with the X^⫺³ form. The convergence of␮^corrwithin the cc-pVXZ and cc- pCVXZ series of basis sets is thus less systematic than within the aug-cc-pVXZ and aug-cc-pCVXZ series of basis sets. Moreover, the maximum absolute error are larger for the cc-p共C兲VXZ levels than for the corresponding aug-cc- p共C兲VXZ levels, and the average reduction in the error from quintuple to sextuple zeta for the cc-p共C兲VXZ basis sets is only a factor of 1.36, significantly lower than predicted by Eq.共4兲. We thus have several clear indications that the basis- set convergence of␮^corrwithin the cc-pVXZ and cc-pCVXZ series of basis sets does not follow the X^⫺³ form.

D. Extrapolations in correlated calculations

In the right half of Table III, we have listed the mean error, the standard deviation, the mean absolute error, and the maximum absolute error of the two-point extrapolated

␮lim

corrresults obtained with␣⫽3 in Eq. 共8兲compared to the

d-aug-cc-pV共5/6兲Z R12 results. The two-point X^⫺³ extrapolation based on the aug-cc-p共C兲VDZ and aug-cc-p共C兲VTZ results represents a reasonable improvement on the unextrapolated aug-cc-p共C兲VTZ results as the mean absolute and maximum absolute errors are reduced by a factor of about 2.2 and 1.6, respectively. Proceeding to higher cardinal numbers, these factors increase steadily, so the improvement on the unextrapolated results obtained from the extrapolation becomes larger as X increases. This is exactly as expected for the X^⫺³ extrapolation scheme which—as mentioned above—exhibits the correct convergence in the asymptotic region for large X. Furthermore, already the 关aug-cc- p共C兲VTZ, aug-cc-p共C兲VQZ兴 X^⫺³ extrapolated results have mean absolute and maximum absolute errors comparable to those of the unextrapolated aug-cc-p共C兲V6Z results. Finally, except for the关aug-cc-p共C兲VDZ, aug-cc-p共C兲VTZ兴pair, the mean error of the X^⫺³ extrapolated results obtained for the 关aug-cc-p共C兲V(X⫺1)Z,aug-cc-p共C兲VXZ兴 basis sets are, for most practical purposes, converged to the basis-set limit.

These observations further substantiate that the basis-set convergence of ␮^corr within the aug-cc-pVXZ and aug-cc- pCVXZ series of basis sets does indeed follow Eq.共4兲.

For the cc-p共C兲VXZ basis sets the situation is different.

For these basis sets, the two-point X^⫺³ extrapolated results usually represent an improvement on the unextrapolated results, but the improvement is not as large as for the aug-cc- p共C兲VXZ basis sets and in one case the mean absolute error is in fact slightly larger for the extrapolated results than for the unextrapolated ones. Furthermore, the improvement on the unextrapolated results obtained from the extrapolation does not become larger as X increases, contrary to what is expected for the X^⫺³ extrapolation. It thus appears that the basis-set convergence of ␮^corr within the cc-pVXZ and cc- pCVXZ series of basis sets does not follow the X^⫺³ form, which is a consequence of the lack of diffuse functions in these basis sets.

Considering the excellent performance of the two-point X^⫺³ extrapolation for the aug-cc-p共C兲VXZ basis sets, the importance of higher-order terms in the series in Eq. 共6兲 is expected to be small. To investigate this issue in more detail, we have performed extrapolations of the form in Eq.共8兲with

TABLE III. Mean error (⌬¯ ), standard deviation (⌬std), mean absolute error (⌬¯_abs), and maximum absolute error (⌬max) of the unextrapolated cc-p共C兲VXZ and aug-cc-p共C兲VXZ results and of the two-point X^⫺3extrapo- lated basis-set limits based on the 关cc-p共C兲V(X⫺1)Z,cc-p共C兲VXZ兴 results and on the 关aug-cc-p共C兲V(X

⫺1)Z,aug-cc-p共C兲VXZ兴results. All the results are given in m.a.u.

Unextrapolated Extrapolated

⌬

¯ ⌬std ⌬¯_abs ⌬max ⌬¯ ⌬std ⌬¯_abs ⌬max

cc-p共C兲VDZ ⫺14.7 12.7 14.8 34.6

cc-p共C兲VTZ ⫺6.2 1.7 6.2 9.0 ⫺2.7 4.1 4.6 6.0

cc-p共C兲VQZ ⫺2.3 1.0 2.3 4.4 0.6 1.6 1.5 3.2

cc-p共C兲V5Z ⫺1.0 1.0 1.0 3.0 0.4 1.3 1.1 1.7

cc-p共C兲V6Z ⫺0.8 0.7 0.8 2.2 ⫺0.5 0.4 0.5 1.3

aug-cc-p共C兲VDZ ⫺15.8 8.2 15.8 30.3

aug-cc-p共C兲VTZ ⫺6.5 1.2 6.5 9.0 ⫺2.7 2.4 3.0 5.5

aug-cc-p共C兲VQZ ⫺2.7 0.6 2.7 3.9 0.1 0.9 0.7 1.7

aug-cc-p共C兲V5Z ⫺1.5 0.4 1.5 2.2 ⫺0.1 0.3 0.3 0.7

aug-cc-p共C兲V6Z ⫺0.9 0.3 0.9 1.4 0.1 0.2 0.2 0.4

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different␣. The value of␣that minimizes the mean absolute error of the extrapolated␮lim

corrresults compared to the d-aug- cc-pV共5/6兲Z R12 results has been found for the different pairs of关aug-cc-p共C兲V(X⫺1)Z,aug-cc-p共C兲VXZ兴 basis sets (X⭓4), and these ‘‘optimal’’ values are 3.36 共T,Q兲, 2.92 (Q,5), and 2.81共5,6兲, respectively. We note that due to the small uncertainty in the d-aug-cc-pV共5/6兲Z R12 results as representatives of the true basis-set limits, these values have a small uncertainty on the last digit. The main observation to be made here is that the ‘‘optimal’’ values are close to 3 and in particular significantly closer to 3 than to 4. Therefore, the first (X^⫺³) term in Eq.共6兲 appears to be sufficient for extrapolations based on the aug-cc-p共C兲VXZ basis sets and one does not need to include higher-order terms in the extrapolations of ␮^corr^.

E. Comparison with experimental dipole moments From the numerical SCF result and the CCSD共T兲-R12 correlation contribution obtained with all electrons correlated in the d-aug-cc-pV共5/6兲Z basis set in Table I, we obtain an estimate of 0.5529⫾0.0002 a.u. for the CCSD共T兲 basis-set limit result for␮e of BH, where the uncertainty is in accor- dance with the discussion above. This value compares favorably with the previous best estimate of 0.5520⫾0.0015 a.u.

for this CCSD共T兲basis-set limit.⁴³The error in␮eof BH due to the incompleteness of the CCSD共T兲wave function model was investigated in Ref. 43. From comparison of CCSD共T兲 and full configuration interaction共FCI兲results in a variety of diffusely augmented correlation-consistent basis sets it was found that the error of the CCSD共T兲 model was virtually constant with the basis set and had a magnitude of 0.0018

⫾0.0002 a.u. Combining this with the new value of the CCSD共T兲 basis-set limit result, we obtain 0.5511

⫾0.0003 a.u. for the FCI basis-set limit of ␮e(BH) 共we treat the uncertainties as standard deviations兲. Finally, inclusion of the correction for rovibrational averaging of ⫺0.0166 a.u.⁴³—obtained at the CCSD共T兲/aug-cc-pV5Z level—gives the final value ␮0(BH)⫽0.5345⫾0.0003 a.u.⫽1.3586

⫾0.0007 D. This value is consistent with the experimental value␮0(BH)⫽1.27⫾0.21 D,⁴⁴as it is well inside the experimental error bar and has a significantly smaller uncertainty.

Because of the large uncertainty in the experimental

␮0(BH), the high accuracy of the theoretical results is not fully revealed in the comparison with the experimental value.

However, for HF, the experimental value is ␮e(HF)⫽1.803

⫾0.002 D,⁴⁵and the theoretical confirmation of this result is thus a more challenging task which has the potential of re- vealing the high accuracy of the theoretical results. From the numerical SCF result and the CCSD共T兲-R12 correlation contribution obtained with all electrons correlated in the d-aug- cc-pV共5/6兲Z basis set in Table II, we obtain the following estimate of the CCSD共T兲basis-set limit result for␮e of HF of 0.7106⫾0.0002 a.u. This value also compares favorably with the previous best estimate of 0.710⫾0.001 a.u. for the CCSD共T兲 basis-set limit.⁴³ Adding the correction of

⫺0.0010⫾0.0002 a.u. for the incompleteness of the CCSD共T兲 model⁴³—based on the difference between FCI and CCSD共T兲results for␮e(HF)— we arrive at the following

final theoretical value ␮e(HF)⫽0.7096⫾0.0003 a.u.⫽1.8037

⫾0.0007 D, which is in excellent agreement with the experimental value, and clearly confirms the high accuracy of the theoretical results.

IV. CONCLUSION

Results of conventional SCF, MP2, CCSD, and CCSD共T兲calculations within different series of correlation- consistent basis sets for ␮e of BH and HF have been presented. The results have been compared with similar results obtained using the explicitly correlated MP2-R12, CCSD- R12, and CCSD共T兲-R12 wave function models. The inclusion of diffuse functions in the basis set has been shown to be essential for accurate calculations of␮e. Without diffuse functions, errors in SCF calculations of up to 2 m.a.u. persist at the sextuple-zeta level of basis set 共cc-pV6Z兲, whereas diffusely augmented basis sets of quadruple-zeta quality 共aug-cc-pVQZ兲give results that are within 0.3 m.a.u. of the SCF basis-set limit. Furthermore, without diffuse functions, the basis-set convergence of the correlation contribution to

␮eis sometimes nonmonotonic and it becomes very slow for high cardinal numbers. The results are less scattered in accuracy once diffuse functions are included in the basis, and the convergence of the correlation contribution becomes more systematic. When diffuse functions are included in the basis, the basis-set convergence of the SCF part of ␮e is significantly faster than that of the correlation contribution—as also seen for the energy. The convergence of the correlation contribution, however, is, as opposed to that of the SCF part, systematic. This makes reliable extrapolations of the correlation contribution possible, and the slow convergence of the correlation part may thus be remedied.

The basis-set convergence of the correlation contribution to␮e within the aug-cc-pVXZ and aug-cc-pCVXZ series of basis sets follows the form␮X

corr⫽␮lim

corr⫹aX^⫺³. This form is completely analogous to the one for the correlation energy and is a consequence of the electronic cusp condition, which governs the convergence of the correlation contribution to

␮e once the outer valence regions important for the dipole operator are described accurately via the diffuse basis functions. Two-point extrapolated values of ␮lim

corr from the X^⫺³ form represent a significant improvement on the ordinary basis-set results for the aug-cc-pVXZ and aug-cc-pCVXZ basis sets. Extrapolated values based on augmented triple- and quadruple-zeta results are within 2 m.a.u. of the basis-set limit and the corresponding augmented quintuple and sextuple-zeta extrapolated values are within 0.5 m.a.u. of the basis-set limit. The cc-pVXZ and cc-pCVXZ basis sets provide only a crude representation of the outer valence region, and the errors originating from this dominate the basis-set error for␮e for these basis sets. Consequently, the basis-set convergence of the correlation contribution to ␮e does not follow the X^⫺³ form for the cc-pVXZ and cc-pCVXZ basis sets.

As a corollary of our calculations we have obtained highly accurate estimates of the CCSD共T兲basis-set limit values for␮eof BH and HF. Combination of these with corrections for the incompleteness of the CCSD共T兲wave function

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model has lead to reliable estimates of the exact ␮共within the Born–Oppenheimer approximation兲. Our final theoretical values are ␮e(HF)⫽1.8037⫾0.0007 D and ␮0(BH)⫽1.3586

⫾0.0007 D, which both are in perfect agreement with the experimental values ␮e(HF)⫽1.803⫾0.002 D and ␮0(BH)

⫽1.27⫾0.21 D.

Comparison of the corrections for the incompleteness of the CCSD共T兲 model and the uncertainty of the established CCSD共T兲basis-set limit results for␮indicates that for sys- tems for which we may perform large R12 calculations or extrapolations based on the larger aug-cc-p共C兲VXZ basis sets, the dominating error in the resulting estimated CCSD共T兲basis-set limit value for␮will be that originating from the incompleteness of the CCSD共T兲model. For tri- and tetra-atomic molecules, results for wave function models more elaborate than CCSD共T兲is thus needed for highly accurate studies of␮^.

ACKNOWLEDGMENTS

We would like to thank Angela Wilson for providing us with the core-valence functions of the cc-pCV6Z and aug- cc-pCV6Z basis sets. This work has been supported by the Danish Research Council 共Grant No. 9600856兲 and by the Research Council of Norway 共NFR Supercomputing Grant No. NN2694K兲. The research of W.K. has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences.

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