Use of Mo”Ller-Plesset Perturbation Theory in Molecular Calculations: Spectroscopic Constants of First Row Diatomic Molecules

JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 12 22 MARCH 1998

Use of Mo”ller-Plesset perturbation theory in molecular calculations: Spectroscopic constants of first row diatomic molecules Thom H. Dunning, Jr. Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352 Kirk A. Peterson Department of Chemistry, Washington State University and the Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352 ͑Received 9 December 1997; accepted 17 December 1997͒

The convergence of Mo”ller–Plesset perturbation expansions ͑MP2–MP4/MP5͒ for the spectroscopic constants of a selected set of diatomic molecules ͑BH, CH, HF, N2, CO, and F2͒ has been investigated. It was found that the second-order perturbation contributions to the spectroscopic constants are strongly dependent on basis set, more so for HF and CO than for BH. The MP5 contributions for HF were essentially zero for the cc-pVDZ basis set, but increased signiﬁcantly with basis set illustrating the difﬁculty of using small basis sets as benchmarks for correlated calculations. The convergence behavior of the exact Mo”ller–Plesset perturbation expansions were investigated using estimates of the complete basis set limits obtained using large correlation consistent basis sets. For BH and CH, the perturbation expansions of the spectroscopic constants converge monotonically toward the experimental values, while for HF, N2, CO, and F2, the expansions oscillate about the experimental values. The perturbation expansions are, in general, only slowly converging and, for HF, N2, CO, and F2, appear to be far from convergence at MP4. In fact, for HF, N2, and CO, the errors in the calculated spectroscopic constants for the MP4 method are larger than those for the MP2 method ͑the only exception is De͒. The current study, combined with other recent studies, raises serious doubts about the use of Mo”ller–Plesset perturbation theory to describe electron correlation effects in atomic and molecular calculations. © 1998 American Institute of Physics. ͓S0021-9606͑98͒00912-X͔

I. INTRODUCTION authors found that classical ‘‘single configuration’’ systems such as hydrogen fluoride, with a weight of nearly 95% for The treatment of electron correlation in atomic and mo- the Hartree–Fock wave function in the MPn expansion, can lecular calculations is critical for an accurate prediction of lead to expansions that are either divergent or appear to di- molecular properties. Ignoring correlation effects can lead to verge in low orders of perturbation theory. The reason for errors of 100% or more in bond energies and reaction ener- this behavior has yet to be firmly established. Analyzing getics, 10%–20% in vibrational frequencies, and as much as their perturbation theory calculations on the neon atom, 5% or more in bond lengths. Mo”ller–Plesset perturbation Christiansen et al.11 found that the divergence was due to theory, as implemented in a number of popular quantum avoided crossings between the Hartree–Fock configuration 1 2 3 chemical packages ͑GAUSSIAN, CADPAC, MOLPRO, ACES and states dominated by quintuple and higher excitations. 4 II, etc.͒, especially second- through fourth-order perturba- Cremer and He5 noted that the convergence behavior of the tion ͑MP2, MP3, MP4͒ theory, is the most widely used tech- perturbation expansion changed depending on the magnitude nique for quantifying the effects of electron correlation on and sign of the triple excitation terms. The pathology in the molecular properties. Techniques and applications software perturbation expansion appears to arise when multiple elec- 1 5 for computing MP5 and MP6 energies and wave functions tron pairs, e.g., the bond and lone pairs in HF, are in close are even coming into use, despite the dramatic increase in proximity. computational demands associated with higher orders of per- In this paper we examine the convergence of Mo”ller– turbation theory. Plesset perturbation theory for molecular properties. The per- Mo”ller–Plesset perturbation theory is based on a single turbation theory expansion of a molecular property, Q, can reference, Hartree–Fock wave function and it has long been be written as recognized that this imposes limits on the radius of convergence of the MPn expansions. The numerical implications of Q͑MPϱ͒ϭQ͑HF͒ϩ⌬Q͑MP2͒ϩ⌬Q͑MP3͒ this limitation have been investigated for a number of ‘‘dif- ϩ⌬Q͑MP4͒ϩ⌬Q͑MP5͒ϩ , ͑1͒ ficult’’ small molecules.6–8 More recently, it has become ap- ¯ parent that convergence problems in the MPn expansion are where not limited to molecules whose zero-order wave function is poorly described by a single configuration.5,9,10 These latter ⌬Q͑MPn͒ϭQ͑MPn͒ϪQ͑MPnϪ1͒ ͑2͒

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II. THEORETICAL AND COMPUTATIONAL CONSIDERATIONS The correlation consistent basis sets of Dunning and co-workers,12–17 labeled cc-pVnZ with nϭDϪ6, form a hi- erarchial family of basis sets that become more and more complete as n increases. As has been demonstrated in nu- merous benchmark studies,18–26 calculations with the correlation consistent basis sets exhibit systematic convergence toward their apparent CBS limits. In fact, the basis set convergence can often be modeled by a simple exponential function:27 Ϫ␣͑nϪ2͒ QnϭQϱϩ⌬Q2e ͑3͒

to obtain an estimate of the complete basis set limit, Qϱ ,of property Q. ͓Other extrapolation procedures have been ex- plored by Wilson and Dunning26 and Martin and co-workers FIG. 1. Definition of apparent and intrinsic errors for the computed quantity, Q. Types I and II refer to two distinct behaviors: convergence to a ͑see, e.g., Ref. 28͒.͔ In the present work Eq. ͑3͒ was used to value which lies on the same side of the experimental value, or convergence estimate CBS limits whenever possible. Otherwise, the esti- to a value on the other side of the experimental value. Examples for both mates were taken from calculations with the cc-pV6Z basis types of convergence behavior are observed here. sets, which is very close to the CBS limit. We focus here on valence-electron calculations on BH, CH, HF, N2, CO, and F2. For atoms on the left-hand side of is the incremental change in Q from (nϪ1)th order to nth the periodic chart, core and core-valence correlation effects order perturbation theory and Q(MPn) is the value of Q at are significant and their inclusion is essential to obtaining nth order perturbation theory. For the perturbation theory agreement with the experimental results. For example, for expansion, Eq. ͑1͒, to be a useful approach for describing the BH, all-electron calculations29 show that inclusion of core/ effect of electron correlation on property Q, the series must core-valence correlation effects increases De by 0.2 kcal/ Ϫ1 converge, i.e., limn ϱ⌬Q(MPn)ϭ0, and must converge to mol, decreases re by 0.003 Å, and increases ␻e by 11 cm the exact solution→ of the electronic Schro¨dinger equation, ͑at the MP4 level͒. Thus, the intrinsic errors associated with i.e., limn ϱQ(MPn)ϭQ(exact). If a finite basis set is used, valence-electron calculations include this error in addition to then the→ series must converge to the full configuration inter- any errors arising from the limitations of the electronic struc- action value for Q in the chosen basis set. In this paper, we ture method, as well as those resulting from the breakdown examine the behavior of the perturbation series expansion, of the Born–Oppenheimer approximation and the neglect of Eq. ͑1͒, for the spectroscopic constants (De , re , ␻e , ␻exe , relativistic corrections. However, neglect of core/core- Be , ␣e) of a set of diatomic molecules ͑BH, CH, HF, N2, valence correlation effects has essentially no effect on the 9 CO, and F2͒ chosen to be representative of a wide range of convergence behavior of the Mo”ller–Plesset perturbation ex- molecules. Computational restrictions limit our studies of pansion, which is the subject of interest here. CH, N2, CO, and F2 to the MP2–MP4 methods and of BH To minimize potential sources of experimental errors, and HF to the MP2–MP5 methods. As we shall see, the we have limited the current study to diatomic molecules for convergence behavior of the Mo”ller–Plesset perturbation ex- which accurate spectroscopic constants have been pansion alluded to above gives rise to significant problems in determined.30 For each molecular species, potential energy computing molecular properties. functions were calculated by fitting nine computed energies In characterizing the behavior of the perturbation expan- that covered a range in ⌬r(ϭrϪre)ofϪ0.4a0р⌬rр sions for molecular properties, it is important to distinguish ϩ0.7a0 to seventh- or eighth-order polynomials in ⌬r. between the apparent and intrinsic accuracies of a MPn Spectroscopic constants were then determined from the fitted model. In Eq. ͑1͒, Q(MPn) should be obtained from the polynomial coefficients by a Dunham analysis.31 In the cal- exact solution of the MPn equations. This is the only way to culation of dissociation energies, the dissociated limits were establish the intrinsic accuracy of the MPn method. Past obtained from ROHF–MPn32 calculations on the isolated at- 1 studies of the convergence of the Mo”ller–Plesset expansion oms, except in the case of the MP5 calculations, where an were limited by the use of finite basis sets. In this case, one unrestricted Hartree–Fock ͑UHF͒ based method was used. can only determine the apparent accuracy of a MPn method, Unfortunately, software limitations prevented us from carry- which may be very different than the intrinsic accuracy of ing out UHF-MP5 calculations on the boron and carbon at- the method. The relationship between the intrinsic error and oms, so we are unable to report values of De for these spe- the apparent error is illustrated in Fig. 1 for two common cies at the MP5 level. types of convergence behavior with respect to basis set.In All electronic structure calculations employed the this paper we use the well-known convergence properties of MOLPRO program package of Werner, Knowles and co- the correlation consistent basis sets to obtain reliable esti- workers ͑for RHF-MPn͒,3 the ACES II program package of mates of the complete basis set ͑CBS͒ limits of the spectro- Bartlett and co-workers ͑for ROHF-MPn͒,4 and the GAUSS- scopic constants for the molecules of interest here. IAN 94 package ͑for UHF-MP5͒.

TABLE I. Spectroscopic constants for boron hydride, BH(X 1⌺ϩ), from valence-electron MPn calculations with the correlation-consistent valence basis sets.

Ee De re ␻e ␻exe Be ␣e Method Basis set ͑hartrees͒ ͑kcal/mol͒ ͑Å͒ (cmϪ1) (cmϪ1) (cmϪ1) (cmϪ1)

Expt.a 84.8 1.2323 2366.9 49.4 12.021 0.4120

HF cc-pVDZ Ϫ25.125 337 62.4 1.2360 2492.6 44.1 11.951 0.3656 cc-pVTZ Ϫ25.129 973 64.0 1.2221 2483.0 42.6 12.225 0.3740 cc-pVQZ Ϫ25.131 351 64.3 1.2203 2487.3 42.8 12.261 0.3762 cc-pV5Z Ϫ25.131 615 64.3 1.2200 2488.2 42.9 12.267 0.3753 cc-pV6Z Ϫ25.131 680 64.3 1.2199 2488.6 42.9 12.269 0.3755 Est. CBSb 64.3 1.2199 2489.0 42.9

MP2 cc-pVDZ Ϫ25.185 941 75.5 1.2468 2421.8 44.1 11.746 0.3681 cc-pVTZ Ϫ25.203 457 79.9 1.2268 2431.0 44.8 12.131 0.3923 cc-pVQZ Ϫ25.209 462 81.1 1.2248 2437.1 44.3 12.172 0.3904 cc-pV5Z Ϫ25.211 504 81.5 1.2241 2439.0 44.7 12.185 0.3912 cc-pV6Z Ϫ25.212 390 81.7 1.2241 2438.1 44.5 12.186 0.3907 Est. CBSb 81.9 1.2241 2438.0 44.6

MP3 cc-pVDZ Ϫ25.203 752 77.8 1.2519 2384.7 45.0 11.650 0.3744 cc-pVTZ Ϫ25.219 998 82.0 1.2313 2392.8 45.8 12.043 0.4019 cc-pVQZ Ϫ25.224 555 83.0 1.2293 2399.8 45.4 12.082 0.3997 cc-pV5Z Ϫ25.225 636 83.2 1.2286 2401.4 45.8 12.095 0.4005 cc-pV6Z Ϫ25.225 964 83.3 1.2285 2401.4 45.7 12.098 0.4003 Est. CBSb 83.4 1.2285 2401.0 45.7

MP4 cc-pVDZ Ϫ25.210 097 78.4 1.2550 2357.6 46.4 11.592 0.3835 cc-pVTZ Ϫ25.225 973 82.6 1.2347 2363.4 47.0 11.977 0.4115 cc-pVQZ Ϫ25.230 420 83.6 1.2327 2370.8 46.8 12.016 0.4091 cc-pV5Z Ϫ25.231 539 83.8 1.2320 2372.2 47.1 12.029 0.4100 cc-pV6Z Ϫ25.231 930 83.9 1.2319 2372.1 47.0 12.031 0.4098 Est. CBSb 84.0 1.2319 2372.0 47.0

MP5 cc-pVDZ Ϫ25.212 753 1.2565 2343.2 47.4 11.565 0.3897 cc-pVTZ Ϫ25.228 517 1.2361 2350.3 47.8 11.950 0.4168 cc-pVQZ Ϫ25.232 926 1.2340 2358.2 47.6 11.991 0.4144 Est. CBSb 1.2332 2359.0 47.9 aTaken from Ref. 30. bEstimated CBS limit using Eq. ͑1͒ or from cc-pV6Z calculations.

III. RESULTS in regard to the rate of convergence of the MPn expansions. 1 ϩ 1 ϩ Tables I–III summarize the results of the new MPn Here, we limit our discussion to BH( ⌺ ), HF( ⌺ ), and CO(1⌺ϩ). valence-electron calculations on BH, CH, and F2. The results of valence-electron calculations on HF, N , and CO dis- ؉ 1 2 cussed here were taken from previously reported A. BH„ ⌺ … 17,24,33,34 work. The dependence of the perturbation theory expansions of De , re , and ␻e on basis set is illustrated in Fig. 2 for BH. As IV. DEPENDENCE OF PERTURBATION can be seen, ⌬Q(MP2) is strongly dependent on basis set: CONTRIBUTIONS ON BASIS SET ⌬De increases by 4.3 kcal/mol, ⌬re decreases by 0.0066 Å, and ⌬␻ increases by 20.3 cmϪ1 from the cc-pVDZ set to In this section we investigate the convergence character- e the cc-pV6Z set. The dependence of ⌬Q(MPn) with nϾ2 istics of the individual terms in the perturbation theory ex- on basis set, on the other hand, is rather weak, with the pansion, ⌬Q(MPn), with basis set. As has been shown,35 the changes, in general, decreasing with increasing order of per- cc-pVDZ basis set recovers only 59%–86% of the valence turbation theory. For example, for MP4 theory, ⌬D in- correlation energy of the ﬁrst row atoms ͑Ne–B͒, while the e creases by just 0.04 kcal/mol, ⌬r increases by only 0.0003 cc-pVTZ set recovers 84%–96%. The percentage of correla- e Å, and ⌬␻ decreases by just 2.2 cmϪ1 from nϭ2to6 tion energy recovered steadily increases and the percentage e (cc-pVnZ). range steadily decreases to the point that the cc-pV6Z set recovers 98%–99% of the correlation energy for Ne–B. B. HF X 1⌺؉ Given these differences, it would not be at all surprising to „ … see signiﬁcant variations in the convergence patterns of the For the HF molecule, the dependence of ⌬Q(MPn)on MPn expansions for the various cc-pVnZ sets. This is, in basis set is stronger in HF than in BH; compare Figs. 2 and fact, the case and, as we shall see, MPn calculations with 3. Thus, ⌬De(MP2) increases by 9.4 kcal/mol as the basis small basis sets can provide one with a false sense of security set increases from cc-pVDZ to cc-pV6Z; for ⌬De(MP4) the

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TABLE II. Spectroscopic constants for CH(X 2⌸) from valence-electron MPn calculations with the correlation-consistent valence basis sets.

Ee De re ␻e ␻exe Be ␣e Method Basis set ͑hartrees͒ ͑kcal/mol͒ ͑Å͒ (cmϪ1) (cmϪ1) (cmϪ1) (cmϪ1)

Expt.a 83.94 1.1199 2858.5 63.0 14.457 0.5340 RHF cc-pVDZ Ϫ38.268 782 54.66 1.1179 3048.7 58.8 14.509 0.4815 cc-pVTZ Ϫ38.277 032 56.83 1.1052 3036.7 54.0 14.844 0.4740 cc-pVQZ Ϫ38.279 342 57.21 1.1039 3043.7 54.9 14.880 0.4776 cc-pV5Z Ϫ38.279 917 57.32 1.1037 3047.1 55.1 14.889 0.4771 Est. CBSb 57.4 1.1035 3049.0 55.3

RMP2 cc-pVDZ Ϫ38.352 196 72.15 1.1329 2930.1 59.5 14.127 0.4880 cc-pVTZ Ϫ38.382 930 78.29 1.1140 2943.5 56.0 14.610 0.4930 cc-pVQZ Ϫ38.392 844 80.03 1.1118 2955.3 56.6 14.668 0.4941 cc-pV5Z Ϫ38.396 409 80.66 1.1112 2959.5 57.1 14.684 0.4947 Est. CBSb 81.0 1.1109 2961.0

RMP3 cc-pVDZ Ϫ38.371 260 74.34 1.1183 2896.1 57.9 13.982 0.5032 cc-pVTZ Ϫ38.401 452 80.27 1.1162 2908.5 58.4 14.499 0.5067 cc-pVQZ Ϫ38.409 856 81.81 1.1161 2909.6 58.5 14.554 0.5066 cc-pV5Z Ϫ38.412 285 82.27 1.1155 2912.8 58.8 14.572 0.5073 Est. CBSb 82.5 1.1152 2914.0

RMP4 cc-pVDZ Ϫ38.376 705 74.87 1.1419 2839.3 63.5 13.905 0.5123 cc-pVTZ Ϫ38.407 158 80.96 1.1215 2857.9 60.0 14.415 0.5203 cc-pVQZ Ϫ38.415 584 82.57 1.1195 2870.3 60.5 14.469 0.5200 cc-pV5Z Ϫ38.418 069 83.06 1.1188 2874.2 61.0 14.485 0.5209 Est. CBSb 83.3 1.1184 2872.0 aTaken from Ref. 30. bEstimated CBS limit using Eq. ͑1͒ or from cc-pV6Z calculations.

1 ϩ TABLE III. Spectroscopic constants for diatomic ﬂuorine, F2(X ⌺g ), from valence-electron MPn calculations with the correlation-consistent valence basis sets.

Ee De re ␻e ␻exe Be ␣e Method Basis set ͑hartrees͒ ͑kcal/mol͒ ͑Å͒ (cmϪ1) (cmϪ1) (cmϪ1) (cmϪ1)

Expt.a 39.0 1.4119 916.6 11.2 0.890 19 0.013 8

HF cc-pVDZ Ϫ198.688 937 Ϫ34.38 1.3476 1180.1 5.60 0.977 2 0.008 00 cc-pVTZ Ϫ198.758 042 Ϫ27.50 1.3291 1266.9 6.70 1.004 6 0.008 00 cc-pVQZ Ϫ198.774 464 Ϫ27.26 1.3275 1263.9 6.50 1.007 0 0.008 10 cc-pV5Z Ϫ198.779 058 Ϫ27.16 1.3267 1266.0 6.60 1.008 2 0.008 10 cc-pV6Z Ϫ198.779 645 Ϫ27.05 1.3265 1266.7 6.58 1.008 6 0.008 13 Est. CBSb Ϫ27.0 1.3263 1268.0 6.6

MP2 cc-pVDZ Ϫ199.079 669 29.97 1.4240 933.3 8.06 0.875 2 0.010 14 cc-pVTZ Ϫ199.274 647 39.71 1.3979 1016.6 8.72 0.908 1 0.010 04 cc-pVQZ Ϫ199.337 965 41.88 1.3971 1006.5 8.33 0.909 2 0.010 04 cc-pV5Z Ϫ199.362 468 42.75 1.3959 1008.0 8.41 0.910 7 0.010 05 cc-pV6Z Ϫ199.371 974 43.24 1.3954 1008.8 8.39 0.911 4 0.010 06 Est. CBSb 43.8 1.3949 1010.0 8.4

MP3 cc-pVDZ Ϫ199.081 584 19.16 1.4169 934.2 9.01 0.884 0 0.010 95 cc-pVTZ Ϫ199.273 122 25.95 1.3858 1043.4 9.32 0.924 0 0.010 37 cc-pVQZ Ϫ199.334 329 27.73 1.3829 1040.6 8.88 0.927 9 0.010 34 cc-pV5Z Ϫ199.355 777 28.52 1.3811 1045.3 8.90 0.930 4 0.010 30 cc-pV6Z Ϫ199.363 162 28.95 1.3804 1046.8 8.88 0.931 3 0.010 29 Est. CBSb 29.4 1.3797 1049.0 8.9

MP4 cc-pVDZ Ϫ199.097 770 27.11 1.4506 817.3 11.70 0.843 4 0.013 30 cc-pVTZ Ϫ199.298 779 36.84 1.4172 924.3 11.70 0.883 6 0.012 40 cc-pVQZ Ϫ199.362 465 39.22 1.4166 916.2 11.10 0.884 3 0.012 30 cc-pV5Z Ϫ199.384 859 40.23 1.4155 917.7 11.20 0.885 7 0.012 30 cc-pV6Z Ϫ199.392 555 40.74 1.4149 918.8 11.18 0.886 4 0.012 31 Est. CBSb 41.3 1.4143 920.0 11.2 aTaken from Ref. 30. bEstimated CBS limit using Eq. ͑1͒ or from cc-pV6Z calculations.

1 ϩ FIG. 2. Basis set dependence of the perturbation contributions, ⌬Q(MPn), for De , re , and ␻e for BH(X ⌺ ).

corresponding increase is 2.7 kcal/mol. For re and ␻e the tron correlation on molecular properties. Double zeta sets, changes at the MP4 level are comparable to that at the MP2 even those that include a set of polarization functions, only level: 0.0038 vs 0.0029 Å and 60.9 vs 62.6 cmϪ1, respec- account for a fraction of the correlation energy and can have tively. Note also that the changes in MP2 and MP4 theory difficulty in reproducing the effects of correlation on mo- are in the same direction as are the changes in MP3 and MP5 lecular properties. Much larger basis sets, cc-pVQZ and be- theory, but that the changes in the two sets are opposite to yond, are required to obtain consistently reliable results. 10 one another. In fact, for re and ␻e the variation in ⌬Q(MP2) Olsen et al. found that the convergence behavior of the and ⌬Q(MP4) with basis set are comparable as are the Mo”ller–Plesset perturbation expansions was strongly depen- changes in ⌬Q(MP3) and ⌬Q(MP5). dent on the nature of the basis set. In particular, for HF they For the cc-pVDZ set, ⌬Q(MP5) is essentially zero for found that the series converged for the cc-pVDZ set and a all of the properties considered here. However, in all cases modified cc-pVTZ set, but diverged for the aug-cc-pVDZ set ⌬Q(MP5) increases significantly in magnitude from the cc- ͑the convergence was much slower for the modified cc-pVTZ pVDZ to the cc-pVQZ set ͑the largest set for which it was set͒. In the present case, although there would be a signifi- feasible to carry out MP5 calculations͒. For example, ⌬De cant difference between the spectroscopic constants com- decreases by 1.6 kcal/mol, ⌬re decreases by 0.0026 Å, and puted with the cc-pVDZ and aug-cc-pVDZ sets, the results Ϫ1 ⌬␻e increases by 46.9 cm . These results illustrate the dif- tend to the same value as the basis set increases in size. For ficulty of using small basis sets to assess the impact of elec- example, comparing the spectroscopic constants obtained

1 ϩ FIG. 3. Basis set dependence of the perturbation contributions, ⌬Q(MPn), for De , re , and ␻e for HF(X ⌺ ).

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Ϫ1 1 ϩ 2 1 ϩ 1 ϩ 1 ϩ 1 ϩ TABLE IV. Intrinsic errors in De ͑kcal/mol͒ re ͑Å͒, and ␻e (cm ) for BH(X ⌺ ), CH(X ⌸), HF(X ⌺ ), N2(X ⌺g ), CO(X ⌺ ), and F2(X ⌺g ) from valence-electron MPn calculations with the correlation-consistent valence basis sets.

Species Quantity MP2 MP3 MP4 MP5

BH De Ϫ2.9 Ϫ1.4 Ϫ0.8 re Ϫ0.0082 Ϫ0.0037 Ϫ0.0003 ϩ0.0009 ␻e ϩ71 ϩ34 ϩ5 Ϫ8

CH De Ϫ2.9 Ϫ1.4 Ϫ0.6 a re Ϫ0.0090 Ϫ0.0047 Ϫ0.0015 Ϫ0.0003 a ␻e ϩ102 ϩ55 ϩ13 Ϫ2

HF De ϩ4.2 Ϫ3.1 ϩ1.1 Ϫ1.0 re ϩ0.0013 Ϫ0.0050 ϩ0.0020 Ϫ0.0011 ␻e ϩ2 ϩ100 Ϫ22 ϩ34

N2 De ϩ12.0 Ϫ12.4 ϩ3.7 re ϩ0.0117 Ϫ0.0123 ϩ0.0114 ␻e Ϫ150 ϩ186 Ϫ157

CO De ϩ12.7 Ϫ8.8 ϩ5.0 re ϩ0.0056 Ϫ0.0114 ϩ0.0125 ␻e Ϫ42 ϩ146 Ϫ147

F2 De ϩ4.2 Ϫ10.9 ϩ1.7 re Ϫ0.0166 Ϫ0.0317 ϩ0.0031 ␻e ϩ92 ϩ131 ϩ2 aEstimated ROHF-MP5 intrinsic error from UHF-MP4 and UHF-MP5 calculations.

with the cc-pV6Z and aug-cc-pV5Z sets ͑which are approxi- re , ␻e , ␻exe , etc., due to the neglect of non-Born– mately the same size͒, we ﬁnd that the computed re’s differ Oppenheimer and relativistic corrections. For De , however, Ϫ1 only by 0.0002–0.0003 Å, ␻e’s by 2 – 5 cm , and ␻exe’s relativistic corrections are more signiﬁcant, especially for the Ϫ1 17,24 by 0.1 cm for MP2–MP4 calculations. heavier atoms ͑O,F͒, and we have corrected the experimental

values of De for spin–orbit effects in the atoms. The error ؉ 1 C. CO„X ⌺ … due to neglect of core/core-valence correlation may be sig-

In CO, ⌬De(MP2) and ⌬re(MP2) are both strongly de- nificant, especially for molecules containing the lighter at- pendent on basis set: ⌬De increases by 9.5 kcal/mol and ⌬re oms ͑B–N͒; we will estimate these errors as appropriate. decreases by 0.0050 Å from the cc-pVDZ to the cc-pV6Z Cremer and He5 classified the molecules that they exam- set. The dependence of ⌬␻e on basis set is somewhat less ined into two types. In Class A systems the Moller–Plesset Ϫ1 ” pronounced, ␻e increases by only 18.7 cm over the same perturbation expansion of the energy converged monotoni- basis set range. The dependence of ⌬Q(MP3) and cally; in Class B systems the energy expansion oscillated. ⌬Q(MP4) on basis set can also be quite significant, al- Olsen et al.10 found that the series diverged only for Class B though less so than for Q(MP2), e.g., D (MP3) de- ⌬ ⌬ e systems, especially if larger basis sets were used. We ob- creases by 2.4 kcal/mol, ⌬D (MP4) increases by 2.7 kcal/ e serve the same type of behavior here. Thus, for the exact mol, and ⌬␻ increases by 15.2 cmϪ1 from the cc-pVDZ to e MPn expansions there also appear to be two convergence the cc-pV6Z set. For ⌬re(MP3), ⌬re(MP4), and patterns, Class A and Class B. ⌬␻e(MP3) the changes are significantly smaller than in ؉ 1 ⌬Q(MP2). A. BH„ ⌺ … For BH, the dissociation energy (D ), equilibrium bond V. CONVERGENCE OF PERTURBATION THEORY e EXPANSIONS length (re), and harmonic frequency (␻e), as well as the anharmonicity constant (␻exe), equilibrium rotational con- In this section we use the estimated complete basis set stant (Be), and vibration–rotation coupling constant (␣e), ͑CBS͒ limits for the value of the spectroscopic constants, Q, show convergence behavior typical of a Class A molecule as at the various orders of perturbation theory to investigate the defined by Cremer and He.5 In Fig. 4 the successive pertur- convergence of the Moller–Plesset perturbation expansions. ” bation theory contributions to D , r , and ␻ are plotted. We consider both the dependence of ⌬Q(MPn) on the order e e e of perturbation theory as well as the intrinsic errors associ- For both De and ␻e , the magnitude of ⌬Q(MPn) decreases ated with Q(MPn). For the latter, which are shown in Table monotonically with increasing orders of perturbation theory; IV, we compare Q(MPn) with the experimental values of for re , there is an initial increase, resulting from a significant property Q, rather than to that obtained from the exact solu- underestimation of re by HF theory, followed by the ex- tion of the Schro¨dinger equation. This is expected to intro- pected decrease. However, the perturbation expansion ap- duce only small errors into the estimated intrinsic errors for pears to be converging very slowly, e.g.,

1 ϩ FIG. 4. Incremental changes, ⌬Q(MPn), in the perturbation expansion of De , re , and ␻e for BH(X ⌺ ).

⌬re͑MP5͒/⌬re͑MP4͒ϭ0.38, for the BH molecule, despite the fact that CH is an open- shell (2⌸) system and BH is a closed-shell (1⌺ϩ) system; ⌬␻e͑MP5͒/⌬␻e͑MP4͒ϭ0.45. compare the results in Tables I and II. Only for the second- order corrections, Q(MP2), are differences between CH From the data in Table I, De appears to be converging to- ⌬ ward the experimental value with increasing orders of per- and BH signiﬁcant. This is undoubtedly a reﬂection of the larger nondynamical correlation effects in BH than in CH. turbation theory, but re(MP5) and ␻e(MP5) overshoot and undershoot their respective experimental values. The intrin- The largest differences in the third- and fourth-order contri- sic errors for the valence-electron calculations on BH are butions are for ⌬␻e(MP3) and ⌬␻e(MP4). Cremer and He given in Table IV. As noted previously, inclusion of core/ found that CH2 and CH3 were Class A systems. core-valence correlation effects increases De by 0.1–0.2 ؉ 1 kcal/mol, decreases re by 0.003 Å, and increases ␻e by C. HF„X ⌺ … 10 cmϪ1.29 Taking these corrections into account, at MP4 Unlike BH and CH, the successive perturbation contri- theory De is underestimated by 0.6 kcal/mol, re is underes- Ϫ1 butions to the spectroscopic constants of HF alternate in timated by 0.003 Å, and ␻e is overestimated by 15 cm . sign, see Fig. 5. For De , the magnitudes of the perturbation corrections decrease with increasing order of perturbation B. CH X 2⌸ „ … theory: ϩ45.5 kcal/mol ͑MP2͒, Ϫ7.3 kcal/mol ͑MP3͒, The perturbation theory expansions for the spectroscopic ϩ4.2 kcal/mol ͑MP4͒, and Ϫ2.1 kcal/mol ͑MP5͒. Since the constants of the CH molecule behave very similarly to those terms in the ⌬De(MPn) series alternate in sign with decreas-

1 ϩ FIG. 5. Incremental changes, ⌬Q(MPn), in the perturbation expansion of De , re , and ␻e for HF(X ⌺ ).

Downloaded 22 Feb 2001 to 192.101.100.146. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html 4768 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 T. H. Dunning, Jr. and K. A. Peterson

1 ϩ FIG. 6. De(MPn), re(MPn), and ␻e(MPn) for HF(X ⌺ ). Horizontal lines mark the experimental values of these constants.

ing magnitude, the error in the fifth-order value for De has to ␣e exhibit similar and unusual convergence behavior. These be less than 2.1 kcal/mol—a rather discouraging result given quantities both show a slight ͑0.5%–1.0%͒ decrease from the high computational cost of the calculation. The actual MP2 to MP3 and then a much more significant ͑3.0%–3.5%͒ error in the fifth-order value for De is just 1.0 kcal/mol, increase from MP3 to MP4. There is no semblance of con- which is less than 2.1 kcal/mol, but still shows that fifth- vergence for these quantities. order perturbation theory is unable to predict the bond energy of a simple molecule such as HF to better than 1 kcal/ D. N X 1⌺؉ and CO X 1⌺؉ mol even at the complete basis set limit. 2„ g … „ … For both re and ␻e , the perturbation series does not The dissociation energy of N2 from the valence-electron converge uniformly, e.g., ͉⌬Q(MP4)͉Ͼ͉⌬Q(MP3)͉. This MPn calculations also exhibits the oscillatory behavior char- behavior is consistent with the conventional wisdom that the acteristic of Class B systems.5 The perturbation series ap- effects of electron correlation are over-emphasized in even pears to be only slowly converging to the experimental val- orders of perturbation theory, and that this over-emphasis is ues of the spectroscopic constants; see Fig. 7. For example, corrected in odd orders. The even and odd orders of pertur- the intrinsic errors in De are: ϩ12.0 kcal/mol ͑MP2͒, bation theory, separately, appear to be converging, although Ϫ12.4 kcal/mol ͑MP3͒, and ϩ3.7 kcal/mol ͑MP4͒, while the more terms would be required to conclude this with any corresponding ⌬De(MPn) are: ϩ118.2 kcal/mol ͑MP2͒, certainty. HF was identified as a Class B system by Cremer Ϫ24.4 kcal/mol ͑MP3͒, and ϩ16.1 kcal/mol ͑MP4͒.Asa and He,5 and the behavior observed here is consistent with further indication of problems with the perturbation expan- this designation. sion for N2, note that the intrinsic errors in re and ␻e are As shown in Table IV, the intrinsic errors in De for the essentially the same magnitude in the MP4 calculations as in MPn series are: ϩ4.2 kcal/mol ͑MP2͒, Ϫ3.1 kcal/mol the MP2 calculations: ϩ0.0115 vs ϩ0.0117 Å for re , and Ϫ1 ͑MP3͒, ϩ1.1 kcal/mol ͑MP4͒, and Ϫ1.0 kcal/mol ͑MP5͒. Ϫ157 vs Ϫ150 cm for ␻e . For the anharmonicity, ␻exe , For both re and ␻e the situation is more complex; see Fig. 6. the series is clearly not converging, with intrinsic errors of: For these molecular constants the intrinsic errors in MP4 ϩ4.35 cmϪ1 ͑MP2͒, Ϫ4.71 cmϪ1 ͑MP3͒, and ϩ6.85 cmϪ1 theory are greater than those for MP2 theory: ϩ0.0020 vs (MP4͒. Again, the corrections due to the inclusion of core/ Ϫ1 ϩ0.0013 Å for re , and Ϫ22 vs ϩ2cm for ␻e , respec- core-valence correlation effects are substantially smaller than tively. For odd orders of perturbation theory, on the other those noted above. hand, the situation is reversed with MP5 being more accurate In the current set of molecules, the most erratic behavior than MP3: Ϫ0.0011 vs Ϫ0.0050 Å for re and ϩ34 vs in the MPn expansions of the spectroscopic constants is ob- Ϫ1 ϩ100 cm for ␻e , respectively. Although the differences served for carbon monoxide, although the overall pattern is in the errors for re and ␻e are small, they are outside the very similar to that for N2. As for N2, De(CO) from the estimated errors bounds associated with the CBS limits for valence-electron calculations appears, at best, to be only these molecular constants. The differences are also much slowly converging: larger than the estimated core/core-valence corrections which ⌬De͑MP2͒ϭϩ87.5 kcal/mol, for HF are 0.2 kcal/mol in De , Ϫ0.0006 Å in re , and Ϫ1 36 ϩ5cm in ␻e . Clearly, in HF one does not necessarily ⌬De͑MP3͒ϭϪ21.5 kcal/mol, obtain more accurate results with MP4 theory than with ⌬D MP4͒ϭϩ13.8 kcal/mol. MP2 theory, despite the much greater computational ex- e͑ pense. As we shall see below, HF is not an isolated case. The slow rate of convergence is even more pronounced for Among the remaining spectroscopic constants, ␻exe and re and ␻e :

1 ϩ FIG. 7. De(MPn), re(MPn), and ␻e(MPn) for N2(X ⌺g ). Horizontal lines mark the experimental values of these constants.

؉ 1 ⌬re͑MP2͒ϭϩ0.032 Å, ⌬re͑MP3͒ϭϪ0.017 Å, E. F2„X ⌺g … The final system considered, F , provides further evi- ⌬re͑MP4͒ϭϩ0.024 Å, 2 dence of the often inconclusive nature of Moller–Plesset per- Ϫ1 Ϫ1 ” ⌬␻e͑MP2͒ϭϪ299 cm , ⌬␻e͑MP3͒ϭϩ188 cm , turbation expansions in molecular calculations. Here we find Ϫ1 ⌬␻e͑MP4͒ϭϪ293 cm . that MP4 theory has substantially smaller intrinsic errors than either MP2 or MP3 theory, see Tables III and IV. In This behavior is reflected in the intrinsic errors for r and ␻ e e fact, for r and ␻ the MP4 results are quite accurate, the plotted in Fig. 8 where it is seen that the errors for these two e e intrinsic errors being just 0.003 Å and 1 cmϪ1, respectively. spectroscopic constants, as well as for ␻exe , are substantially greater in MP4 theory than in MP2 theory: However, the plots of ⌬re(MPn) and ⌬␻e(MPn) in Fig. 9 clearly suggest that this agreement may be fortuitous— ϩ0.0125 Å (MP4) vs ϩ0.0056 A (MP2) for re ; Ϫ1 Ϫ1 ⌬r (MP4) and ⌬␻ (MP4) are both far larger in magnitude Ϫ146.8 cm ͑MP4͒ vs Ϫ41.9 cm ͑MP2͒ for ␻e ; and e e Ϫ1 Ϫ1 ϩ5.8 cm ͑MP4͒ vs ϩ0.1 cm ͑MP2͒ for ␻exe . Cremer than the intrinsic errors in re(MP4) and ␻e(MP4). This sus- and He5 classified CO as a Class B system, which is consis- picion is also supported by the fact that the negligible error Ϫ1 tent with the above. in ␻e(1 cm ) seems to be at odds with the much larger

1 ϩ FIG. 8. De(MPn), re(MPn), and ␻e(MPn) for CO(X ⌺ ). Horizontal lines mark the experimental values of these constants.

Downloaded 22 Feb 2001 to 192.101.100.146. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html 4770 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 T. H. Dunning, Jr. and K. A. Peterson

1 ϩ FIG. 9. Incremental changes, ⌬Q(MPn), in the perturbation expansion of De , re , and ␻e for F2(X ⌺g ).

error in re(0.003 Å); typically, re and ␻e are strongly magnitude of the fifth-order contribution to De increased to coupled and an error of 0.003 Å in re would lead to a far 1.6 kcal/mol for the cc-pVQZ set. This illustrates the diffi- Ϫ1 larger error in ␻e than 1 cm . culty of using small basis sets to benchmark the effect of electron correlation on molecular properties. VI. CONCLUSIONS Recent studies of the convergence of the perturbation expansion of the correlation energy by Cremer and He5 and Moller–Plesset perturbation theory, especially second- ” Olsen et al.10 suggest that Moller–Plesset (MPn) perturba- through fourth-order perturbation ͑MP2, MP3, MP4͒ theory, ” tion expansions are not always well behaved, even for mol- is one of the most widely used techniques for including the ecules whose wave functions are dominated by the Hartree– effects of electron correlation on molecular energies and Fock configuration. However, their studies, which were properties. Explicit in the use of perturbation theory in mo- based on full configuration interaction approaches, of neces- lecular calculations is the assumption that the MPn expan- sity used small basis sets. In a recent report9 and the present sions converge to the exact solution of the electronic Schro¨- dinger equation. In addition, it is usually assumed that higher work we used the well-known convergence characteristics of orders of perturbation theory, e.g., MP4 theory, are inher- the correlation consistent basis sets to determine the com- ently more accurate than lower orders of perturbation theory, plete basis set ͑CBS͒ limits for the spectroscopic constants of e.g., MP2 theory. To date, because of the limitations im- a representative set of diatomic molecules: BH, CH, HF, N2, posed by finite basis sets, neither of these assumptions has CO, and F2. For the exact perturbation expansions, we found been rigorously tested. In this paper we examined two as- the following: pects of this problem for MPn expansions of the molecular ͑a͒ MPn expansions of the spectroscopic constants are, at spectroscopic constants of a selection of first row diatomic best, slowly converging and will require the use of molecules: ͑i͒ the convergence of the individual perturbation higher than MP4 theory to obtain chemically accurate theory terms in the MPn expansion with basis set and ͑ii͒ the results. This is especially true for molecules identified convergence of the exact MPn expansions with increasing as Class B ͑oscillatory convergence behavior͒ by Cre- order of perturbation theory. Large correlation consistent ba- 5 mer and He, which includes HF, N2, CO, and F2 in the sis sets were used to obtain reliable estimates of the complete current set. basis set limits of the spectroscopic constants. ͑b͒ Use of higher orders of perturbation theory does not In our studies of the convergence of the perturbation guarantee improved accuracy in a calculation—the in- expansions of the spectroscopic constants with basis set, we trinsic errors for MP2 theory can be smaller than those found that the second-order contributions were, in general, for MP4 theory. For example, for HF, N , and CO, the strongly dependent on basis set. For BH, the dependence of 2 MP2 values for re , ␻e , and ␻exe are more accurate the higher order terms on basis set decreased with increasing than the MP4 values, substantially more so for CO. order of perturbation theory. However, for HF and CO, that Only for De is the MP4 method consistently more ac- was not the case; for these systems the dependence of the curate than the MP2 method. higher order contributions on basis set was often comparable to that for the lower order terms. The results were particu- The current study, combined with the previous 5,10 larly dramatic in HF where the fifth-order contributions, studies, raise serious questions about the use of Mo”ller– which were essentially zero for the cc-pVDZ set, increased Plesset perturbation theory to describe the effects of electron with basis set at a rate comparable to that for MP3, e.g., the correlation in atomic and molecular calculations. This is not

Downloaded 22 Feb 2001 to 192.101.100.146. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 T. H. Dunning, Jr. and K. A. Peterson 4771 a failure of the general approach used to describe electron optimization and vibrational analysis package written by J. F. Stanton and correlation because coupled cluster calculations, especially D. E. Bernholdt. 5 CCSD͑T͒ calculations, with large correlation consistent sets D. Cremer and Z. He, J. Phys. Chem. 100, 6173 ͑1996͒. 6 P. J. Knowles, K. Somasundram, and N. C. Handy, Chem. Phys. Lett. 113, can provide highly accurate molecular spectroscopic con- 8 ͑1985͒. stants ͑cf. Refs. 17, 23–25, 33, 34, 36–38͒. Rather, it is a 7 N. C. Handy, P. J. Knowles, and K. Somasundram, Theor. Chim. Acta 68, failure of Moller–Plesset perturbation theory itself. We do 87 ͑1985͒. ” 8 not yet know if other forms of perturbation theory suffer P. M. W. Gill and L. Radom, Chem. Phys. Lett. 132,16͑1986͒. 9 Intrinsic Errors in Correlated Electronic Structure Calculations on Mol- from this same problem. ecules, T. H. Dunning, Jr., K. A. Peterson, and D. E. Woon, Pacific- Chem’95, Honolulu, Hawaii, 17–22 December 1995 ͑unpublished͒. 10 ACKNOWLEDGMENTS J. Olsen, O. Christiansen, H. Kock, and P. Jo”rgensen, J. Chem. Phys. 105, 5082 ͑1996͒. 11 The authors wish to acknowledge the support of the O. Christiansen, J. Olsen, P. Jo”rgensen, H. Koch, and P.-A. Malmqvist, Chemical Sciences Division in the Office of Basic Energy Chem. Phys. Lett. 261, 369 ͑1996͒. 12 Sciences of the U.S. Department of Energy. The work was T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 ͑1989͒. 13 R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 96, carried out at Pacific Northwest National Laboratory, a mul- 6796 ͑1992͒. tiprogram national laboratory operated for the U.S. Depart- 14 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 ͑1993͒. ment of Energy by Battelle Memorial Institute under Con- 15 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 100, 2975 ͑1994͒. 16 tract No. DE-AC06-76RLO 1830. This research was also D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 ͑1995͒. 17 A. K. Wilson, T. van Mourik, and T. H. Dunning, Jr., J. Mol. Struct.: supported by Associated Western Universities, Inc., North- THEOCHEM 388, 339 ͑1996͒. west Division under Grant No. DE-FG06-89ER-75522 with 18 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 99, 1914 ͑1993͒. the U.S. Department of Energy. Additionally, we wish to 19 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, thank Dr. Angela Wilson, Dr. Tanja van Mourik, and Dr. 1930 ͑1993͒. 20 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, David F. Feller for their comments. 9790 ͑1993͒. 21 K. A. Peterson, D. E. Woon, and T. H. Dunning, Jr., J. Chem. Phys. 100, 1 GAUSSIAN 94, Revision D.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. 7410 ͑1994͒. 22 M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. D. E. Woon, J. Chem. Phys. 100, 2838 ͑1994͒. 23 Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 101, 8877 ͑1994͒. 24 Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A. K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 102, 2032 ͑1995͒. 25 Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 106, 4119 ͑1997͒. 26 Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, A. K. Wilson and T. H. Dunning, Jr., J. Chem. Phys. 106, 8718 ͑1997͒. 27 J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. D. Feller, J. Chem. Phys. 96, 6104 ͑1992͒. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh, PA, 1995. 28 J. M. L. Martin, Chem. Phys. Lett. 259, 669 ͑1996͒. 2 29 The CADPAC electronic structure program is maintained by R. D. Amos, K. A. Peterson and T. H. Dunning, Jr. ͑unpublished͒. University of Cambridge. 30 K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure 3 MOLPRO is a package of ab initio programs written by H.-J. Werner and P. IV. Constants of Diatomic Molecules ͑Van Nostrand, Princeton, 1979͒. J. Knowles with contributions from J. Almlo¨f, R. D. Amos, M. J. O. 31 J. L. Dunham, Phys. Rev. 41, 721 ͑1932͒. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. A. Peterson, R. M. Pitzer, 32 W. J. Lauderdale, J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, A. J. Stone, P. R. Taylor, and R. Lindh. Chem. Phys. Lett. 187,21͑1991͒. 4 33 ACES II is a computational chemistry package especially designed for CC K. A. Peterson and T. H. Dunning, Jr., J. Phys. Chem. 99, 3898 ͑1995͒. and MBPT energy and gradient calculations. Elements of this package are: 34 K. A. Peterson and T. H. Dunning, Jr., J. Mol. Struct.: THEOCHEM 400, the self-consistent field, integral transformation, correlation energy, and 93 ͑1997͒. gradient programs written by J. F. Stanton, J. Gauss, J. D. Watts, W. J. 35 A. K. Wilson and T. H. Dunning, Jr. ͑unpublished͒. 36 Lauderdale, and R. J. Bartlett; the VMOL integral and VPROPS property D. Feller and K. A. Peterson, J. Mol. Struct.: THEOCHEM 400,69 integral programs written by P. R. Taylor and J. Almlo¨f; a modified ver- ͑1997͒. 37 sion of the integral derivative program ABACUS written by T. Helgaker, H. J. M. L. Martin and P. R. Taylor, Chem. Phys. Lett. 225, 473 ͑1994͒. 38 J. Jensen, P. Jo”rgensen, J. Olsen, and P. R. Taylor and the geometry J. M. L. Martin and P. R. Taylor, Chem. Phys. Lett. 248, 336 ͑1996͒.

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