Accurate correlation consistent basis sets for molecular coreâvalence ...

JOURNAL OF CHEMICAL PHYSICS

VOLUME 117, NUMBER 23

15 DECEMBER 2002

Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited Kirk A. Petersona) Department of Chemistry, Washington State University, Pullman, Washington 99164-4630 and the William R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352

Thom H. Dunning, Jr.b) William R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352 and the Joint Institute for Computational Sciences, University of Tennessee, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

共Received 31 May 2002; accepted 18 September 2002兲 Correlation consistent basis sets for accurately describing core–core and core–valence correlation effects in atoms and molecules have been developed for the second row atoms Al–Ar. Two different optimization strategies were investigated, which led to two families of core–valence basis sets when the optimized functions were added to the standard correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence–electron correlated calculations, i.e., for the core–core plus core–valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms 关D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 共1995兲兴. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell 共core–valence兲 correlation energy converges more slowly than the intrashell 共core–core兲 correlation energy. Since the effect of including the core electrons on the calculation of molecular properties tends to be dominated by core–valence correlation effects, a second scheme for determining the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core–valence 共intershell兲 correlation energy, except that a small amount of core–core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core– valence basis sets (cc-pwCVnZ), significantly improve the convergence of many molecular properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster 关CCSD共T兲兴 calculations on a series of second row homonuclear diatomic molecules (Al2 , Si2 , P2 , S2 , and Cl2 ), as well as on selected diatomic molecules involving first row atoms 共CO, SiO, PN, and BCl兲. For the calculation of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1520138兴

I. INTRODUCTION

In correlated ab initio electronic structure calculations the so-called frozen core approximation, in which correlation effects involving the electrons in the low-lying core orbitals are neglected, is motivated both by chemical intuition and computational efficiency. For these reasons the vast majority of calculations employ this approximation, and the errors arising from this source are generally small in comparison to those associated with basis set incompleteness and inadequate treatment of electron correlation. If, however, the goals of a calculation are to obtain chemical accuracy of thermochemical properties, i.e., errors less than 1 kcal/mol, a兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

0021-9606/2002/117(23)/10548/13/$19.00

or molecular structures accurate to within 0.01 Å, then the effects of correlating the electrons in the core orbitals generally must be addressed in the calculations. Unfortunately, nearly all commonly used Gaussian basis sets, including most of the popular correlation consistent and Pople-style basis sets, have been developed only for the description of valence electron correlation. The absence of correlating functions for the core electrons in these sets can lead to unreliable results when all of the electrons are included in the correlation calculation. Thus, any serious work where the core electrons are included in the correlation treatment must consider the augmentation of these valence basis sets with additional functions appropriate for describing core correlation effects. In the past, a number of investigations have focused on the effects of core–valence correlation on molecular ener-

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© 2002 American Institute of Physics

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J. Chem. Phys., Vol. 117, No. 23, 15 December 2002

getics and spectroscopic constants 共see, e.g., Refs. 1–3兲. These include the seminal work of Bauschlicher, Langhoff, and Taylor,2 which discussed the systematic construction of atomic natural orbital 共ANO兲 basis sets for 1s core correlation of first row atoms, as well as differentiating between core–core and core–valence correlation. Outside the scope of the present work is the core polarization potential 共CPP兲 method of Meyer and co-workers,4 where the effects discussed here are recovered by effective operators. In fact, at least for molecules containing only first row atoms, calculations involving CPPs have been shown to closely reproduce accurate all-electron results.5 Within the framework of the correlation consistent family of Gaussian basis sets, previous work for the first row atoms B–Ne involved the optimization of functions for core correlation to be added to the standard cc-pVnZ sets.6 This work led to the cc-pCVnZ 共core–valence兲 basis sets. The exponents of the augmenting functions, which were added to the cc-pVnZ sets in He-atomlike correlation consistent shells, i.e., (1s1p) to the cc-pCVDZ set, (2s2p1d) to the cc-pCVTZ set, and so on, were optimized using the energy difference between all-electron and valence–electron correlated singles and doubles configuration interaction 共CISD兲 calculations. This strategy leads to augmenting sets that systematically describe both core–core 共intrashell, 1s 2 ) and core–valence 共intershell, 1s 2 – 2s 2 2p n ) correlation effects. Subsequent benchmark calculations demonstrated that the resulting cc-pCVnZ basis sets (n⫽D, T, Q, and 5兲 led to systematic convergence of molecular energies and spectroscopic properties 共see, i.e., Refs. 3 and 7兲. The goal of the present work was to develop core– valence basis sets that systematically converge to the complete basis set limit for the second row atoms Al–Ar to facilitate accurate, all-electron correlated, ab initio calculations involving these atoms. Two different strategies were developed in the present work, which led to two different families of core–valence correlation consistent basis sets. The first involved a straightforward extension of the cc-pCVnZ basis sets developed for the first row atoms to the second row, i.e., shells of augmenting functions were optimized for the difference between all-electron and valence–electron CISD energies. The second scheme was motivated by the expectation that the effect of including the core electrons in calculations of molecular properties would be dominated by intershell correlation effects and little affected by the correlation among just the electrons in the core orbitals. This strategy involved biasing the functional optimization toward the description of the core–valence correlation energy, yielding basis sets denoted by cc-pwCVnZ 共correlation consistent polarized weighted core–valence basis sets兲. As shown in subsequent benchmark calculations, the latter scheme results in faster, smoother convergence towards the complete basis set limit for the effects of core correlation on molecular energetic and spectroscopic properties. Given the success of this procedure, analogous basis sets were then derived for the first row atoms with suitable benchmark calculations also being carried out. Section II of this paper briefly outlines the methodology used in this work, while Sec. III details the results of the two

Correlation consistent basis sets

10549

optimization schema for the second row atoms. Section IV presents the results of benchmark calculations on the homonuclear diatomic molecules, Al2 , Si2 , P2 , S2 , and Cl2 , using both families of core–valence basis sets, cc-pCVnZ and cc-pwCVnZ. Section V presents cc-pwCVnZ basis sets for the first row atoms B–Ne and compares their performance against the cc-pCVnZ sets for CO, SiO, PN, and BCl. Finally, the results for both the first and second row atoms are summarized in Sec. VI.

II. COMPUTATIONAL METHODOLOGY

The core–valence basis sets were constructed by adding sets of primitive Gaussian functions to the standard correlation consistent valence basis sets (cc-pVnZ) sets of Dunning and co-workers.2,8 The exponents for the augmenting functions were optimized in singles and doubles configuration interaction 共CISD兲 calculations on the atomic ground states. In general, nearly identical exponents can be obtained by carrying out optimizations at coupled cluster levels of theory, but, as described below, the use of CISD wave functions affords more flexibility. Symmetry equivalencing was imposed on the atomic HF orbitals by state-averaging the degenerate components of the atomic states; however this was not done in the CISD calculations. In each case the exponents were chosen as even tempered expansions, i.e., ␨ i ⫽ ␣␤ i⫺1 and 共␣, ␤兲 were optimized. Only the spherical harmonic components of the d, f , g, etc. basis functions were used; this minimizes problems with linear dependence. For the calculations on the second row atoms Al–Ar, the very low-lying 1s electrons were not included in the correlation treatment. Thus, the resulting core–valence basis sets resulting from this work are not appropriate for describing second row 1s 2 correlation effects. However, the effects of correlating the electrons in the 1s orbital are expected to be negligibly small for most properties of interest to chemists. The MOLPRO suite9 of ab initio programs 共v2000.1兲 was used throughout this work.

III. CORE–CORE AND CORE–VALENCE CORRELATION IN THE SECOND ROW ATOMS, Al–Ar A. Establishing correlation consistency

Following the methodology outlined previously for the development of the valence basis sets for the second row atoms,8 initial optimizations were carried out for the sulfur atom. Basis sets for the other elements of the row were assumed to behave similarly to sulfur. In configuration interaction calculations involving the second row atoms, excitations can arise from three shells of electrons, the K shell (1s 2 ), L shell (2s 2 2 p 6 ), and M shell, i.e., the valence shell (3s 2 3 p n ), electrons. For the purposes of this work, correlation of the K shell electrons is not included. From this point on, the term all-electron will refer to calculations including the electrons in the L and M shell orbitals; the 1s 2 pair will be treated using the frozen core approximation. In this case, correlation of the core electrons can be divided into two classes: intrashell, core–core correlation 共LL兲 and intershell,


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K. A. Peterson and T. H. Dunning, Jr.

core–valence correlation 共LM兲. An all-electron correlated calculation equals the sum of the contributions: LL, LM, and MM. For the first row atoms 共B–Ne兲, it was found that correlation consistent shells of basis functions could be defined that led to a systematic convergence of the K-shell correlation energy.6 For these atoms, the correlating functions were added in H/He-atomlike groupings to the standard cc-pVnZ basis sets, i.e., 1s1 p to the cc-pVDZ set, 2s2 p1d to the cc-pVTZ set, etc. These choices were initially determined in calculations involving only 1s 2 correlation, i.e., KK-only calculations, with the expectation that the basis functions appropriate for describing KL correlation effects would be bracketed by those for describing KK and LL correlation effects, yielding a systematic convergence to the all-electron correlated complete basis set 共CBS兲 limit as both sets are increased in size. However, it was found that better convergence to the CBS limit was obtained if the exponents for these functions were optimized for the difference between all-electron and valence–electron CISD calculations, i.e., for ⌬E corr⫽E CISD共 KK⫹KL⫹LL兲 ⫺E CISD共 LL兲 . This change does not affect the aufbau principle used to construct the cc-pCVnZ basis sets. For the second row atoms the appropriate correlation consistent groupings were obtained analogously to the first row case; CISD calculations were carried out on sulfur whereby only the 2s 2 2p 6 electrons were correlated. The base set for these initial exponent optimizations corresponded to an uncontracted (20s12p) set obtained from the standard cc-pV5Z basis. From one to four d functions were added to this set and the exponents of the functions optimized using the LL 共core–core兲 correlation energy. Then, one to three f functions were added to the (20s12p4d) base set and the exponents optimized, one to two g functions were added to the (20s12p4d3 f ) set and the exponents optimized, and, finally, a single h-type function was added to the (20s12p4d3 f 2g) set and its exponent optimized. The incremental contributions of these functions to the LL correlation energy, 兩 ⌬E corr兩 , are shown in Fig. 1. Just as for valence electron correlation the convergence is monotonic with easily-defined correlation consistent groupings, (1d), (2d1 f ), (3d2 f 1g), and (4d3 f 2g1h), which are the same as the valence correlation groupings found for B–Ne. It can be seen from Fig. 1 that the higher angular momentum functions are somewhat more important, e.g., the first h function is energetically more important than the second g function and the variation within the (4d3 f 2g1h) group is a bit larger than that within the (2d1 f ) group. This is nearly identical, however, to what was observed in the valence basis sets of B–Ne, and the groupings are still distinct from each other.2 The optimizations of the s and p core correlating functions proceeded in a similar fashion. For the s functions the (20s12p3d2 f 1g) set was contracted to 关 3s12p3d2 f 1g 兴 and the exponents of one to four added s functions were varied to minimize the CISD LL correlation energy. The p functions were similarly optimized in a contracted 关 20s2p3d2 f 1g 兴 basis set. The results for the s and p func-

FIG. 1. Contributions of polarization functions, as well as s and p functions 共inset兲, to the correlation energy of the 2s 2 2p 6 electrons in sulfur. The absolute values of the incremental correlation energy lowerings, 兩 ⌬E corr兩 , are plotted in mEh against the number of functions in the expansions for spd f gh functions. The solid lines are exponential fits.

tions are shown as an inset to Fig. 1. The incremental contributions of the s functions fall off rather quickly, while those of the p’s more closely resemble those of the d’s. Thus, just as in the valence sets for B–Ne, the (1s1p) set is paired with the (1d) to form the group of functions to be added to the cc-pVDZ set to form the core–valence ccpCVDZ set, the (2s2 p) set is paired with the (2d1 f ) to form the core–valence cc-pCVTZ set, and so on.

B. cc-pCVn Z basis sets for Al–Ar

As discussed previously for the first row atoms,6 one could just add the additional functions optimized for core– core 共LL兲 correlation to the cc-pVnZ basis sets to form the cc-pCVnZ basis sets. On the other hand, it is generally accepted that intershell correlation effects, e.g., LM correlation, have the largest effects on molecular properties, and LM correlation effects would converge slowly with respect to basis set size in a scheme based only on LL and MM optimizations. Hence the groups of exponents to be added to the cc-pVnZ basis sets to form the cc-pCVnZ sets for the second row atoms are obtained by minimizing the difference in energy between an all-electron and valence–electron CISD calculation, i.e., ⌬E corr共 LL⫹LM兲 ⫽E CISD共 LL⫹LM⫹MM兲 ⫺E CISD共 MM兲 .

共1兲




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TABLE I. The composition and size of the cc-pVnZ and cc-pCVnZ basis sets for the second row atoms Al–Ar. Only the core correlation functions added to the cc-pVnZ sets are noted in the cc-pCVnZ column. N V is the number of functions in the cc-pVnZ set and N CV is the total number of functions in the cc-pCVnZ set. Set DZ TZ QZ 5Z

cc-pVnZ 关 4s3p1d 兴关 5s4p2d1 f 兴关 6s5p3d2 f 1g 兴关 7s6p4d3 f 2g1h 兴

NV

cc-pCVnZ

N CV

18 34 59 95

⫹(1s1p1d) ⫹(2s2p2d1 f ) ⫹(3s3p3d2 f 1g) ⫹(4d3 f 2g1h) a

27 59 109 181

The original 关 7s6p 兴 contraction of the (20s12p) HF set is recontracted to 关 11s10p 兴 . See the text.

a

The augmenting groups were added in the same manner as those obtained from Fig. 1 and the exponents for the new core functions were optimized in the presence of the appropriate valence basis set. The latter allows for coupling between the core and valence functions and is particularly important for the spd core functions.

FIG. 3. Exponents of the optimal core correlating functions in the cc-pCVnZ 共䊏兲 and cc-pwCVnZ 共䊊兲 basis sets for sulfur.

FIG. 2. Convergence of the CISD correlation energy 共in mEh ) of core–core 共LL兲, core–valence 共LM兲, and valence–valence 共MM兲 electron pairs for 共a兲 sulfur and 共b兲 aluminum using the valence-only cc-pVnZ 共䊏兲 codes and core-valence cc-pCVnZ 共䊉兲 basis sets.


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FIG. 4. Convergence of the CISD correlation energy 共in mEh ) of core–core 共LL兲 and core–valence 共LM兲 electron pairs for 共a兲 sulfur and 共b兲 aluminum using the cc-pCVnZ 共䊏兲 and cc-pwCVnZ 共䊉兲 basis sets.

For the augmenting s and p functions, we have chosen to use the exponents determined in the CISD optimizations rather than attempt to use primitive functions selected from the HF sets since close exponent matches could not, in gen-

eral, be found. The one exception is for the large cc-pCV5Z basis sets where strong linear dependencies arose primarily between the optimal p correlating functions and the p functions in the HF sets. In these cases, the cc-pV5Z (20s12p)

TABLE II. Spectroscopic constants for the X 3 ⌸ u state of Al2 by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Basis CVDZ CVTZ CVQZ CV5Z wCVDZ wCVTZ wCVQZ wCV5Z Expt.a

De 共kcal/mol兲

Valence

All

⌬(mEh )

Valence

⫺483.887764 ⫺483.913366 ⫺483.919986 ⫺483.921603 ⫺483.888911 ⫺483.913664 ⫺483.920110 ⫺483.921655

⫺484.278602 ⫺484.483841 ⫺484.549029 ⫺484.572975 ⫺484.147492 ⫺484.422338 ⫺484.523158 ⫺484.562315

⫺390.84 ⫺570.47 ⫺629.04 ⫺651.37 ⫺258.58 ⫺508.67 ⫺603.05 ⫺640.66

28.05 28.37 31.66 31.66 32.64 32.60 32.96 32.90 28.29 28.35 31.71 31.67 32.66 32.61 32.97 32.91 31.7共14兲

All

␻e (cm⫺1 )

re 共Å兲 ⌬ 0.31 0.00 ⫺0.04 ⫺0.06 0.06 ⫺0.04 ⫺0.05 ⫺0.06

Valence

All

2.7586 2.7525 2.7231 2.7162 2.7126 2.7021 2.7115 2.6992 2.7525 2.7496 2.7208 2.7122 2.7122 2.7002 2.7112 2.6982 2.701共2兲

⌬ 共mÅ兲 ⫺6.1 ⫺6.9 ⫺10.5 ⫺12.3 ⫺2.9 ⫺8.6 ⫺12.0 ⫺13.0

Valence

All

277.4 278.6 284.3 283.7 285.6 285.5 286.0 286.0 278.6 277.6 284.7 284.1 285.7 285.7 286.1 286.1 285.8

⌬ 1.2 ⫺0.7 ⫺0.1 0.0 ⫺1.0 ⫺0.6 0.1 0.1

a

Dissociation energy and bond length from Ref. 16 and the harmonic frequency from Ref. 17. Atomic spin–orbit effects have been removed from the experimental D e . The quoted experimental uncertainties in the last digit are given in parentheses.




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TABLE III. Spectroscopic constants for the X 3 ⌺ ⫺ g state Si2 by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Valence

All

⌬(mEh )

⫺577.929147 ⫺577.980245 ⫺577.994183 ⫺577.997630 ⫺577.932541 ⫺577.981170 ⫺577.994495 ⫺577.997811

⫺578.324607 ⫺578.557861 ⫺578.633295 ⫺578.659838 ⫺578.211123 ⫺578.499885 ⫺578.607960 ⫺578.649009

⫺395.46 ⫺577.62 ⫺639.11 ⫺662.21 ⫺278.58 ⫺518.71 ⫺613.46 ⫺651.20

Basis CVDZ CVTZ CVQZ CV5Z wCVDZ wCVTZ wCVQZ wCV5Z Expt.a a

De 共kcal/mol兲 All

⌬

61.82 71.03 74.24 75.21 62.42 71.34 74.33 75.28

0.46 0.20 0.23 0.25 0.18 0.21 0.26 0.26

Valence 61.36 70.83 74.00 74.97 62.24 71.13 74.08 75.01 75.6

␻e (cm⫺1 )

re 共Å兲 Valence

All

2.2935 2.2910 2.2629 2.2580 2.2532 2.2458 2.2517 2.2431 2.2849 2.2846 2.2601 2.2540 2.2529 2.2445 2.2515 2.2424 2.246

⌬ 共mÅ兲 ⫺2.6 ⫺4.9 ⫺7.4 ⫺8.6 ⫺0.3 ⫺6.1 ⫺8.4 ⫺9.1

Valence

All

492.85 494.40 510.49 511.13 515.31 516.86 515.63 517.51 496.55 495.89 511.49 512.45 515.37 517.20 515.80 517.80 511.0

⌬ 1.5 0.6 1.6 1.9 ⫺0.7 1.0 1.8 2.0

Reference 18. Atomic spin–orbit effects have been removed from the experimental D e .

sets were re-contracted to 关 11s10p 兴 , i.e., the nine most diffuse s and p functions were left uncontracted, and the optimal (4d3 f 2g1h) functions were added to form the ccpCV5Z basis sets. The resulting composition and sizes of the cc-pCVnZ basis sets are summarized in Table I and the optimized exponents are explicitly given in Table 1 of the supplementary data.10 The convergence of the individual energy contributions to the CISD all-electron correlation energy, LL, LM, and MM, with respect to the cc-pCVnZ basis set used is shown in Fig. 2 for both the sulfur and aluminum atoms. It is straightforward to separately calculate the LL 共core–core兲 and MM 共valence only兲 CISD correlation energies. But, only the double excitation contributions to the LM correlation energy could be directly calculated with MOLPRO. In this case the LM singles correlation energy was approximated from the difference in singles-only calculations involving allelectron, LL-only, and valence-only, i.e.,

more approximate nature of this singles contribution should not introduce significant errors. Note that since the CISD method is not size extensive, the sum of the individual LL, LM, and MM contributions do not yield the all-electron CISD energy. Another strategy to obtain the LM contribution would be to simply use an analog of Eq. 共2兲 in CISD calculations, i.e., E corr(LM)⫽E corr(all)⫺E corr(LL)⫺E corr(MM). Unfortunately, this tends to strongly underestimate the LM correlation energy due to the lack of size extensivity. For example, for the sulfur atom as shown in Fig. 2共a兲, the cc-pCV5Z basis set yields a CISD LM correlation contribution of 58.0 mEh when calculated as LM(pairs)⫹LM(singles) 关the latter via Eq. 共2兲兴. This can be compared to the result of 30.1 mEh obtained by subtracting E CISD(LL)⫹E CISD(MM) from the all-electron correlation energy. The former, larger LM correlation energy compares much more favorably to the value calculated by Nesbet11 in Bethe–Goldstone calculations of 57.4 mEh , which itself was estimated to be accurate to 1%–2%. Figure 2 clearly demonstrates that the new cc-pCVnZ basis sets result in regular convergence towards the apparent CBS limit for each correlation contribution 共LL, LM, and MM兲. On the other hand, the use of the valence cc-pVnZ basis sets for core correlation is totally inadequate, e.g., the cc-pVQZ basis set recovers only 40% of the cc-pCVQZ LM correlation energy for S and a tiny 14% of the LL correlation

E corr共 LM, singles兲 ⫽E corr共 all-electron, singles兲 ⫺E corr共 LL, singles兲 ⫺E corr共 MM, singles兲 .

共2兲

This result was then added to the LM pair-only correlation energy to yield the total LM correlation contribution. Since the singles contribution is generally small compared to the total magnitude of the LM contribution, i.e., ⬍10%, the

TABLE IV. Spectroscopic constants for the X 1 ⌺ ⫹ g state of P2 by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Valence

All

⌬(mEh )

Valence

⫺681.728515 ⫺681.812108 ⫺681.835720 ⫺681.842267 ⫺681.735738 ⫺681.813967 ⫺681.836331 ⫺681.842624

⫺682.128628 ⫺682.397251 ⫺682.486169 ⫺682.516885 ⫺682.025341 ⫺682.341737 ⫺682.461532 ⫺682.506351

⫺400.11 ⫺585.14 ⫺650.45 ⫺674.62 ⫺289.60 ⫺527.77 ⫺625.20 ⫺663.73

90.20 90.82 105.40 105.95 111.88 112.54 113.99 114.71 92.43 92.74 106.05 106.69 112.05 112.77 114.09 114.84 117.2



␻e (cm⫺1 )

re 共Å兲 ⌬ 0.62 0.55 0.66 0.72 0.32 0.64 0.72 0.75

Valence

All

1.9346 1.9329 1.9095 1.9054 1.9010 1.8952 1.8990 1.8922 1.9252 1.9247 1.9065 1.9015 1.9006 1.8941 1.8988 1.8917 1.8934

⌬ 共mÅ兲 ⫺1.7 ⫺4.2 ⫺5.9 ⫺6.8 ⫺0.5 ⫺5.0 ⫺6.5 ⫺7.1

Valence

All

756.2 759.3 773.5 777.0 783.7 788.4 785.5 790.6 763.8 764.7 773.9 778.1 783.8 788.8 785.8 791.1 780.8

⌬ 3.1 3.6 4.6 5.1 0.9 4.1 5.0 5.3

Reference 18.


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TABLE V. Spectroscopic constants for the X 3 ⌺ ⫺ g state of S2 by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Basis CVDZ CVTZ CVQZ CV5Z wCVDZ wCVTZ wCVQZ wCV5Z Expt.a a

De 共kcal/mol兲

Valence

All

⌬(mEh )

Valence

⫺795.336648 ⫺795.462848 ⫺795.496844 ⫺795.506407 ⫺795.347558 ⫺795.465740 ⫺795.497891 ⫺795.506941

⫺795.740338 ⫺796.055613 ⫺796.157842 ⫺796.192854 ⫺795.638614 ⫺796.001638 ⫺796.133800 ⫺796.182511

⫺403.69 ⫺592.77 ⫺661.00 ⫺686.45 ⫺291.06 ⫺535.90 ⫺635.91 ⫺675.57

82.69 83.19 95.31 95.76 100.18 100.66 101.89 102.39 85.30 85.59 95.85 96.34 100.31 100.82 101.95 102.47 102.9

⌬

All

␻e (cm⫺1 )

re 共Å兲

0.50 0.45 0.48 0.50 0.29 0.49 0.51 0.52

Valence

All

1.9348 1.9331 1.9058 1.9027 1.8957 1.8913 1.8932 1.8880 1.9200 1.9197 1.9027 1.8988 1.8952 1.8903 1.8929 1.8875 1.8892

⌬ 共mÅ兲 ⫺1.7 ⫺3.1 ⫺4.5 ⫺5.1 ⫺0.3 ⫺3.9 ⫺5.0 ⫺5.4

Valence

All

701.4 703.4 718.4 720.5 729.5 732.3 731.5 734.6 709.9 710.4 718.6 721.2 729.6 732.7 731.7 735.0 725.7

⌬ 1.9 2.2 2.9 3.1 0.5 2.6 3.1 3.3


energy. The cc-pVnZ results for Al 关Fig. 2共b兲兴 are even more unreliable for core correlation. In this case the cc-pVTZ basis set actually recovers more LL correlation energy than the cc-pVQZ set. Calculations involving second row atoms that include core correlation with only the standard cc-pVnZ basis sets are strongly discouraged. It can also be observed from Fig. 2 that the CISD allelectron correlation energy for S is dominated by the core– core 共LL兲 correlation energy, 288 mEh compared to 155 mEh for valence-only, and 58 mEh for the core–valence 共LM兲 contribution 共cc-pCV5Z basis set results兲. This is in marked contrast to the first row atoms where nearly 76% of the CISD all-electron correlation energy for the oxygen atom was due to valence-only correlation, 8% was due to core–valence correlation, and only 17% was due to core–core correlation. For the sulfur atom, valence 共MM兲 correlation accounts for only ⬃33% and the core–valence contribution 共LM兲 is just ⬃12%. Since the cc-pCVnZ exponents were optimized for the sum of the LL and LM contributions via Eq. 共1兲, it might be expected that the large energy disparity between core– core and core–valence correlation energies would introduce significant bias into the exponents, particularly for the smaller basis sets. Nonetheless, for the sulfur atom, the ccpCVDZ basis set recovers 61% of E corr(LL) and 48% of E corr(LM) of the values obtained with the cc-pCV5Z set. At the cc-pCVTZ level, the values are 87% for E corr(LL) and 83% for E corr(LM). Hence, somewhat surprisingly, the LL

and LM contributions are converging at very similar rates towards the CBS limit with the cc-pCVnZ basis sets, albeit as expected the core–valence term is converging somewhat more slowly and is particularly undervalued with the ccpCVDZ set. C. Improved convergence for core–valence correlation, cc-pw CVn Z basis sets for Al–Ar

One way to improve the convergence of the intershell correlation energy would be to optimize the augmenting functions for E corr(LM) and completely neglect E corr(LL). However, since typical all-electron calculations also include core–core correlation, such basis sets would not be expected to systematically converge to the total correlation energy. It is possible, however, to optimize functions in calculations that strongly weight core–valence correlation over core–core correlation. In such a scheme the convergence of the LM intershell correlation energy would be improved at the expense of a somewhat slower convergence of the core–core correlation energy, but since the latter would be included in the minimization scheme these basis sets should still exhibit systematic convergence to the all-electron correlated CBS limit. In these optimizations the same groups of functions were added to the cc-pVnZ sets as in the cc-pCVnZ basis sets, but the exponents were optimized with ⌬E CISD共 LM⫹wLL兲 ⫽E CISD共 LM兲 ⫹wE CISD共 LL兲 ,

共3兲

TABLE VI. Spectroscopic constants for the X 1 ⌺ ⫹ g state of Cl2 by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Basis CVDZ CVTZ CVQZ CV5Z wCVDZ wCVTZ wCVQZ wCV5Z Expt.a a

De 共kcal/mol兲

␻e (cm⫺1 )

re 共Å兲

Valence

All

⌬(mEh )

Valence

All

⌬

⫺919.271391 ⫺919.435641 ⫺919.482157 ⫺919.495636 ⫺919.284170 ⫺919.439235 ⫺919.483782 ⫺919.496398

⫺919.678093 ⫺920.037750 ⫺920.154408 ⫺920.194426 ⫺919.574617 ⫺919.986076 ⫺920.131361 ⫺920.184510

⫺406.70 ⫺602.11 ⫺672.25 ⫺698.79 ⫺290.45 ⫺546.84 ⫺647.58 ⫺688.11

42.50 53.42 56.94 58.35 43.94 53.70 57.04 58.39

42.76 53.65 57.13 58.53 44.13 53.91 57.22 58.57

0.27 0.23 0.20 0.18 0.19 0.21 0.18 0.18

59.7

Valence

All

2.0482 2.0465 2.0075 2.0045 1.9953 1.9914 1.9917 1.9874 2.0325 2.0319 2.0057 2.0022 1.9948 1.9906 1.9915 1.9870 1.9879

⌬ 共mÅ兲 ⫺1.7 ⫺3.0 ⫺3.9 ⫺4.3 ⫺0.6 ⫺3.5 ⫺4.2 ⫺4.5

Valence

All

511.6 512.5 549.9 551.8 558.1 560.0 562.1 564.1 517.0 517.6 550.3 552.2 558.5 560.5 562.1 564.2 559.7

⌬ 0.9 1.9 1.9 2.0 0.5 1.9 2.0 2.1





10555

FIG. 5. Effects of correlating the 2s and 2p core electrons on the dissociation energy D e , the equilibrium bond length r e , and the harmonic frequency ␻ e of P 2 at the CCSD共T兲 level of theory.

where w is an arbitrary weighting factor and E CISD(LM) is obtained from a direct calculation of the doubles contribution plus Eq. 共2兲 for the singles. After some experimentation it was found that w⫽0.01 was sufficient to yield exponents that both systematically converge the LL contribution to the correlation energy and speed up convergence of the LM contribution compared to the cc-pCVnZ sets. The resulting basis sets, the structure of which are identical to the cc-pCVnZ sets shown in Table I and the exponents of which are given

explicitly in Table 2 of the supplementary data,10 are denoted as cc-pwCVnZ for correlation consistent polarized weighted core–valence basis sets. The exponent distributions of the core functions for the S atom between the cc-pCVnZ and cc-pwCVnZ basis sets are compared in Fig. 3. While the exponents in the weighted sets are always smaller than those in the cc-pCVnZ sets, this trend is most apparent for the exponent of the highest angular momentum functions in each set, e.g., the tight f function

FIG. 6. A comparison of calculated core correlation contributions to D e , r e , and ␻ e at the CCSD共T兲/cc-pwCV5Z level of theory for the set of homonuclear diatomic molecules containing second row atoms.


10556



FIG. 7. Convergence of the CISD correlation energy 共in mEh ) of core–core 共KK兲 and core–valence 共KL兲 electron pairs for 共a兲 oxygen and 共b兲 boron using the cc-pCVnZ 共䊏兲 and cc-pwCVnZ 共䊉兲 basis sets.

in the TZ set or the tight g function in the QZ set. It can also be seen that as the basis set size increases, the exponents of the two families of core–valence basis sets begin to coalesce. Last, Fig. 4 compares the convergence of CISD

E corr(LL) and E corr(LM) using the cc-pCVnZ and cc-pwCVnZ basis sets for both the Al and S atoms. Both sets lead to regular, systematic convergence of the LL and LM correlation energy contributions, but the cc-pwCVnZ sets

TABLE VII. Spectroscopic constants for the X 1 ⌺ ⫹ state of CO by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Basis CVDZ CVTZ CVQZ CV5Z wCVDZ wCVTZ wCVQZ wCV5Z Expt.a a

De 共kcal/mol兲

Valence

All

⌬(mEh )

Valence

⫺113.059983 ⫺113.161153 ⫺113.190193 ⫺113.199186 ⫺113.064864 ⫺113.163830 ⫺113.190963 ⫺113.199526

⫺113.134711 ⫺113.264151 ⫺113.303351 ⫺113.315390 ⫺113.125345 ⫺113.264427 ⫺113.303357 ⫺113.315344

⫺74.73 ⫺103.00 ⫺113.16 ⫺116.20 ⫺60.48 ⫺100.60 ⫺112.39 ⫺115.82

242.04 242.49 252.28 253.03 256.17 257.04 257.40 258.33 242.37 242.87 252.68 253.46 256.29 257.19 257.46 258.40 259.6

All

␻e (cm⫺1 )

re 共Å兲 ⌬ 0.45 0.75 0.87 0.92 0.50 0.78 0.90 0.94

Valence

All

1.1435 1.1429 1.1343 1.1326 1.1311 1.1289 1.1306 1.1282 1.1425 1.1418 1.1333 1.1314 1.1309 1.1286 1.1305 1.1281 1.1283

⌬ 共mÅ兲 ⫺0.6 ⫺1.7 ⫺2.2 ⫺2.4 ⫺0.7 ⫺2.0 ⫺2.3 ⫺2.4

Valence

All

2143.84 2146.77 2155.77 2163.12 2164.51 2173.84 2165.30 2175.14 2141.46 2145.30 2155.40 2163.68 2164.73 2174.31 2165.35 2175.39 2169.8

⌬ 2.9 7.3 9.3 9.8 3.8 8.3 9.6 10.0





10557

TABLE VIII. Spectroscopic constants for the X 1 ⌺ ⫹ state of SiO by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Valence

All

⌬(mEh )

Valence

⫺364.086835 ⫺364.202482 ⫺364.233359 ⫺364.242925 ⫺364.096350 ⫺364.206221 ⫺364.234367 ⫺364.243324

⫺364.322320 ⫺364.545333 ⫺364.612951 ⫺364.636000 ⫺364.266071 ⫺364.518959 ⫺364.601029 ⫺364.630799

⫺235.49 ⫺342.85 ⫺379.59 ⫺393.08 ⫺169.72 ⫺312.74 ⫺366.66 ⫺387.47

162.24 162.50 182.89 183.31 188.39 189.11 190.26 191.14 165.61 165.70 184.28 184.90 188.68 189.55 190.35 191.30 192.9



␻e (cm⫺1 )

re 共Å兲 ⌬ 0.26 0.42 0.72 0.88 0.09 0.63 0.87 0.95

Valence

All

1.5531 1.5527 1.5200 1.5174 1.5158 1.5117 1.5146 1.5097 1.5377 1.5373 1.5155 1.5118 1.5145 1.5096 1.5143 1.5091 1.5097

⌬ 共mÅ兲 ⫺0.4 ⫺2.6 ⫺4.1 ⫺4.9 ⫺0.5 ⫺3.7 ⫺4.8 ⫺5.3

Valence

All

1152.13 1151.94 1232.53 1235.14 1238.32 1243.82 1240.51 1247.39 1159.25 1158.02 1238.41 1242.94 1239.21 1245.78 1240.66 1247.98 1241.6

⌬ ⫺0.2 2.6 5.5 6.9 ⫺1.2 4.5 6.6 7.3


etry (⫺0.3 a o ⭐r⫺r e ⭐⫹0.5 a o ). Spectroscopic constants were obtained from the polynomial derivatives by the usual Dunham analysis.14 The dissociation energies were obtained relative to atomic calculations that did not employ symmetry equivalencing. For the X 3 ⌸ state of Al2 , however, the orbitals used in the CCSD共T兲 calculations were obtained from state-averaged SCF calculations. For each species, calculations were carried out for both valence-only and all-electron 共neglecting 1s 2 ) correlation. It is important to note that the same basis sets were used in both calculations, since as clearly seen from Fig. 2 the additional core correlating functions in the new core–valence basis sets make non-negligible contributions even in valence-only calculations and this is especially true in molecular calculations. This is primarily due to the importance of tight d functions for the second row atoms. This effect has been systematically investigated for valence-only correlation, which resulted in the development of the cc-pV(n⫹d)Z basis sets.15 In fact the tight d functions used in these latter sets were very similar to some of the ones added in the cc-pwCVnZ basis sets. It should be noted, however, that for studies of core correlation effects, the augmenting functions developed in this work should be used with the cc-pVnZ 共or aug-cc-pVnZ) basis sets and not the cc-pV(n⫹d)Z ones. Tables II–VI contains the results of CCSD共T兲 calculations for Al2 , Si2 , P2 , S2 , and Cl2 , respectively, for both the cc-pCVnZ and cc-pwCVnZ basis sets. The systematic con-

lead to better overall convergence of the LM correlation energy at the expense of a slower convergence of the LL contribution. The faster convergence of the intershell 共LM兲 correlation energy with the cc-pwCVnZ basis sets is particularly evident for the Al atom, where the cc-pwCVTZ and cc-pwCVQZ results are nearly equivalent to the cc-pCVQZ and cc-pCV5Z values, respectively. IV. MOLECULAR CORE–VALENCE CORRELATION EFFECTS IN SECOND ROW ATOMS: Al2 , Si2 , P2 , S2 , AND Cl2

In order to characterize the convergence of molecular core–valence correlation effects with the new basis sets, the spectroscopic constants of the second row homonuclear diatomic molecules Al2 , Si2 , P2 , S2 , and Cl2 were calculated with both series of basis sets, cc-pCVnZ and cc-pwCVnZ. Near-equilibrium potential energy functions were calculated at the singles and doubles coupled cluster level of theory with a perturbative treatment of connected triples, CCSD共T兲.12 The open shell species utilized the CCSD共T兲 variant that employs restricted open shell Hartree–Fock orbitals, but the spin is unrestricted in the solution of the CCSD equations.13 For these molecules the resulting spin contamination was always less than 0.01 and hence negligible. The potential energy functions were obtained as 5thor 6th-degree polynomial fits to 7 ab initio energies unequally distributed about the approximate equilibrium geom-

TABLE IX. Spectroscopic constants for the X 1 ⌺ ⫹ state of PN by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Valence

All

⌬(mEh )

Valence

⫺395.457493 ⫺395.552864 ⫺395.578969 ⫺395.586795 ⫺395.466057 ⫺395.555886 ⫺395.579771 ⫺395.587142

⫺395.694836 ⫺395.896942 ⫺395.960982 ⫺395.982495 ⫺395.641145 ⫺395.870283 ⫺395.948869 ⫺395.977225

⫺237.34 ⫺344.08 ⫺382.01 ⫺395.70 ⫺175.09 ⫺314.40 ⫺369.10 ⫺390.08

115.93 116.35 134.47 135.03 141.06 141.80 143.34 144.19 118.98 119.24 135.54 136.20 141.32 142.15 143.44 144.33 148.6



␻e (cm⫺1 )

re 共Å兲 ⌬ 0.42 0.56 0.74 0.85 0.26 0.67 0.83 0.89

Valence

All

1.5220 1.5213 1.5017 1.4989 1.4953 1.4914 1.4942 1.4897 1.5127 1.5122 1.4980 1.4946 1.4946 1.4903 1.4940 1.4894 1.4909

⌬ 共mÅ兲 ⫺0.6 ⫺2.8 ⫺3.9 ⫺4.5 ⫺0.6 ⫺3.4 ⫺4.3 ⫺4.7

Valence

All

1276.28 1278.30 1329.06 1334.61 1339.04 1346.45 1342.75 1351.13 1291.43 1292.57 1330.45 1336.78 1339.77 1347.71 1342.78 1351.30 1337.2

⌬ 2.0 5.5 7.4 8.4 1.1 6.3 7.9 8.5

Reference 18.


10558



TABLE X. Spectroscopic constants for the X 1 ⌺ ⫹ state of BCl by valence-only and all-electron CCSD共T兲 calculations. Ee (Eh ) Basis CVDZ CVTZ CVQZ CV5Z wCVDZ wCVTZ wCVQZ wCV5Z Expt.a a

De 共kcal/mol兲

Valence

All

⌬(mEh )

Valence

⫺484.372989 ⫺484.463214 ⫺484.489858 ⫺484.497430 ⫺484.380076 ⫺484.465605 ⫺484.490784 ⫺484.497859

⫺484.612334 ⫺484.810576 ⫺484.875928 ⫺484.897904 ⫺484.553435 ⫺484.783254 ⫺484.863945 ⫺484.892776

⫺239.34 ⫺347.36 ⫺386.07 ⫺400.47 ⫺173.36 ⫺317.65 ⫺373.16 ⫺394.92

113.56 114.03 118.80 119.24 121.18 121.64 122.04 122.51 114.25 114.57 119.04 119.48 121.25 121.71 122.08 122.55 123.4共11兲

⌬

All

␻e (cm⫺1 )

re 共Å兲

0.47 0.44 0.46 0.47 0.31 0.44 0.46 0.48

Valence

All

1.7437 1.7414 1.7256 1.7216 1.7225 1.7166 1.7218 1.7154 1.7390 1.7372 1.7235 1.7182 1.7222 1.7158 1.7217 1.7150 1.7153

⌬ 共mÅ兲 ⫺2.2 ⫺4.1 ⫺5.9 ⫺6.5 ⫺1.8 ⫺5.3 ⫺6.3 ⫺6.7

Valence

All

846.20 849.17 840.02 844.09 838.66 844.70 837.56 844.04 850.16 852.68 840.89 846.51 838.90 845.19 837.82 844.34 840.3

⌬ 3.0 4.1 6.0 6.5 2.5 5.6 6.3 6.5

Experimental r e and ␻ e from Refs. 19 and 20, respectively. Experimental dissociation energy from Ref. 21. Atomic spin–orbit effects have been removed from the experimental D e . The quoted experimental uncertainty in the last digit is given in parentheses.

vergence of E e , D e , r e , and ␻ e with respect to increases in the basis set is generally observed in all cases, and the contributions due to core correlation on D e , r e , and ␻ e are fairly well converged even at the QZ level, especially for the cc-pwCVQZ sets. On comparison to the experimental values, a correlation of the core electrons generally improves the agreement between the CCSD共T兲 results and experiment for D e and r e , although the latter will apparently tend to be somewhat shorter than experiment at the all-electron correlated basis set limit. On the other hand, the CCSD共T兲 harmonic frequencies tend to overshoot the experimental value with the addition of core correlation by as much as 10 cm⫺1 . In regards to the dissociation energies, the largest calculations tend to underestimate the experimental results, but extrapolation to the CBS limit, as well as the inclusion of scalar relativistic effects, is expected to remove most of the remaining errors between theory and experiment. Figure 5 compares the convergence of the core correlation contributions to these properties for the cc-pCVnZ and cc-pwCVnZ basis sets for the P2 molecule. Indicative of the better description of core–valence correlation effects, the cc-pwCVnZ basis

sets exhibit a more systematic convergence than the cc-pCVnZ sets. In addition, the results based on the cc-pwCVQZ basis set are nearly of cc-pCV5Z quality for substantially less computational cost. In fact, even the cc-pwCVTZ results recover about 70– 85% of the cc-pwCV5Z results, which are essentially converged. Figure 6 compares the core correlation contributions to D e , r e , and ␻ e obtained at the CCSD共T兲/cc-pwCV5Z level for this series of homonuclear diatomic molecules. In contrast to the analogous first row species, the core correlation contributions to D e are nearly negligible for the early members of this series. Only those for P2 and S2 are appreciable (⫹0.75 and ⫹0.52 kcal/mol, respectively兲. By far the largest effect of core correlation is on the equilibrium bond lengths, and these smoothly decrease in magnitude from ⫺13.0 mÅ for Al2 to ⫺4.5 mÅ for Cl2 . Obviously for the calculation of accurate equilibrium structures for molecules containing second row atoms, core correlation effects must be taken into account. The trend for the harmonic frequencies closely follow those of the dissociation energies with a maximum contribution of just ⫹5.3 cm⫺1 for P2 .

FIG. 8. Effects of correlating the 1s core electrons on the dissociation energy D e , the equilibrium bond length r e , and the harmonic frequency ␻ e of CO at the CCSD共T兲 level of theory.




V. cc-pw CVn Z BASIS SETS FOR THE FIRST ROW ATOMS. B–Ne REVISITED

Given the improved convergence of core correlation effects for the second row atoms with the weighted core– valence basis sets in comparison to the cc-pCVnZ sets, it was of interest to determine if a similar advantage could be achieved for the 1st row atoms. Groups of core correlating functions were added to the valence cc-pVnZ basis sets just as in the previous cc-pCVnZ work,6 but the exponents of the augmenting functions were now optimized via the first row analog of Eq. 共3兲, ⌬E CISD共 KL⫹wKK兲 ⫽E CISD共 KL兲 ⫹wE CISD共 KK兲 ,

共4兲

with w again set equal to 0.01. Optimized exponents to be added to the cc-pVnZ sets to form the cc-pwCVnZ basis sets are given in Table 3 of the supplementary data10 for the elements B–Ne. Contributions to the KK and KL CISD correlation energies for the oxygen and boron atoms are shown in Fig. 7 using both the cc-pCVnZ and new cc-pwCVnZ basis sets. Just as observed in Fig. 4 for sulfur and aluminum, the weighted CV sets lead to faster convergence of the core–valence correlation energies at the expense of slower core–core correlation energy recovery. The improved core– valence correlation recovery is most apparent for the early members of the row, e.g., B or Al. Benchmark calculations on the closed shell ground states of CO, SiO, PN, and BCl were carried out to assess the relative convergence characteristics of the old cc-pCVnZ and cc-pwCVnZ basis sets at the CCSD共T兲 level of theory. Potential energy functions and spectroscopic constants were obtained in an identical manner as detailed in Sec. IV. The results of these calculations are shown in Tables VII–X. The cc-pCVnZ results for CO were taken from our previous benchmark calculations on this molecule.7 As observed and discussed above for the second row homonuclear diatomic molecules, systematic convergence of core correlation effects is observed when using the new cc-pwCVnZ basis sets 共as well as for the cc-pCVnZ sets兲. Figure 8 compares the CCSD共T兲 core correlation contributions to D e , r e , and ␻ e for CO with the two families of core valence basis sets. While the differences in convergence rates between the two basis sets are not nearly as dramatic as observed for P2 in Fig. 5, the cc-pwCVnZ basis sets do represent an improvement over the cc-pCVnZ sets, particularly for the equilibrium bond length. The results for SiO and PN 共Tables VIII and IX兲 are similar to those of CO, however, in these cases there is a clearer advantage in using the cc-pwCVnZ basis sets. For SiO the cc-pwCVQZ core correlation contributions are nearly identical to the cc-pCV5Z values and the cc-pwCVTZ results are almost of cc-pCVQZ quality. VI. SUMMARY

Two families of basis sets, cc-pCVnZ and cc-pwCVnZ, n⫽D, T, Q, and 5, have been developed for describing core correlation effects in molecules containing the second row atoms Al–Ar 共the term all-electron here does not include the two electrons in the K shell兲. The cc-pCVnZ basis sets are based on the procedure described previously for the first row

10559

atoms and involved optimizing the exponents of the augmenting core correlating functions based on the difference between the all-electron and valence–electron correlation energies. These sets systematically converge to the all-electron complete basis set limit, but some irregularities in the convergence of molecular core correlation contributions are observed with the smaller sets of this sequence. The weighted cc-pwCVnZ basis sets are constructed using core correlating functions which were optimized for the atomic core–valence correlation energy with a small 1% weighted contribution from the core–core correlation energy. These sets also systematically converge to the all-electron correlated CBS limit, but yield more rapid convergence of the intershell, core–valence correlation energy at the expense of slower convergence of the core–core correlation energy. It is the former that is most important in the calculation of molecular properties. Benchmark calculations at the CCSD共T兲 level of theory on the homonuclear diatomic molecules involving second row atoms indicate that the cc-pwCVnZ basis sets do lead to faster, more systematic convergence of the core correlation contributions to dissociation energies, equilibrium bond lengths, and harmonic frequencies compared to calculations with the cc-pCVnZ sets. Analogous cc-pwCVnZ basis sets were also determined for the first row atoms B–Ne and compared to the existing cc-pCVnZ basis sets. Diatomic benchmark calculations showed that the weighted core–valence basis sets (cc-pwCVnZ) also lead to a more accurate treatment of core–valence correlation for molecules containing first row atoms. All of the basis sets developed in this work are available from the EMSL Gaussian Basis Set Library.22 ACKNOWLEDGMENTS

This research was performed in the William R. Wiley Environmental Molecular Sciences Laboratory 共EMSL兲 at the Pacific Northwest National Laboratory 共PNNL兲. Operation of the EMSL is funded by the Office of Biological and Environmental Research in the U.S. Department of Energy 共DOE兲. PNNL is operated by Battelle Memorial Institute for the U.S. DOE under Contract No. DE-AC06-76RLO 1830. This work was supported by the Division of Chemical Sciences in the Office of Basis Energy Sciences of the U.S. DOE. W. Meyer and P. Rosmus, J. Chem. Phys. 63, 2356 共1975兲; H.-J. Werner and P. J. Knowles, ibid. 94, 1264 共1991兲; C. W. Bauschlicher, Jr. and H. Partridge, ibid. 100, 4329 共1994兲; J. M. L. Martin, Chem. Phys. Lett. 242, 343 共1995兲; H. Mu¨ller, R. Franke, S. Vogt´ner, R. Jaquet, and W. Kutzelnigg, Theor. Chem. Acc. 100, 85 共1998兲; D. Feller and K. A. Peterson, J. Chem. Phys. 110, 8384 共1999兲. 2 C. W. Bauschlicher, Jr., S. R. Langhoff, and P. R. Taylor, J. Chem. Phys. 88, 2540 共1988兲. 3 K. A. Peterson, A. K. Wilson, D. E. Woon, and T. H. Dunning, Jr., Theor. Chem. Acc. 97, 251 共1997兲. 4 W. Mu¨ller, J. Flesch, and W. Meyer, J. Chem. Phys. 80, 3297 共1984兲. 5 A. Nicklass and K. A. Peterson, Theor. Chem. Acc. 100, 103 共1998兲. 6 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 共1995兲. 7 K. A. Peterson and T. H. Dunning, Jr., J. Mol. Struct. 共Theochem兲 400, 93 共1997兲. 8 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 共1993兲. 9 MOLPRO is a package of ab initio programs written by H.-J. Werner and 1


10560


P. J. Knowles with contributions from J. Almlo¨f, R. D. Amos, A. Bernhardsson et al. MOLPRO2000, 2000. 10 See EPAPS Document No. E-JCPSA6-117-309247 for three tables of supplementary data. A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage 共http://www.aip.org/pubservs/ epaps.html兲 or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information. 11 R. K. Nesbet, Phys. Rev. A 3, 87 共1971兲. 12 K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 共1989兲; C. Hampel, K. A. Peterson, and H.-J. Werner, ibid. 190, 1 共1992兲. 13 M. Rittby and R. J. Bartlett, J. Phys. Chem. 92, 3033 共1988兲; J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 共1993兲; P. J. Knowles, C. Hampel, and H.-J. Werner, ibid. 99, 5219 共1994兲.

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