Approximating the basis set dependence of coupled cluster calculations: Evaluation of perturbation theory approximations for stable molecules

JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 18 8 NOVEMBER 2000 Approximating the basis set dependence of coupled cluster calculations: Evaluation of perturbation theory approximations for stable molecules Thom H. Dunning, Jr. a) and Kirk A. Peterson b) Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, c Richland, Washington 99352 Received 4 February 2000; accepted 17 August 2000 The coupled cluster CCSD T method provides a theoretically sound, accurate description of the electronic structure of a wide range of molecules. To obtain accurate results, however, very large basis sets must be used. Since the computational cost of CCSD T calculations formally increases with the seventh power of the number of basis functions (N 7 ), the CCSD T method can only be applied to a restricted range of molecules. In this work we show that the basis set dependence of the CCSD(T) method is well described by perturbation theory. Starting with CCSD T /aug-cc-pvtz calculations, use of the MP3 method to simulate the effect of increasing the basis set to aug-cc-pv5z leads to average absolute errors, relative to the full CCSD T /aug-cc-pv5z calculations, of less than 0.4 kcal/mol (D e ), 0.0002 Å (r e ), 2cm 1 ( e ), 0.1 kcal/mol (IP e ), and 0.2 kcal/mol (EA e ) for the test set of diatomic molecules considered here. Although the corresponding MP2 approximation does not provide this high level of accuracy, it also should be useful for many molecular studies. When properly implemented, the savings in computer time should be significant since the MP3 method formally scales as N 6, while the MP2 method scales as only N 5. 2000 American Institute of Physics. S0021-9606 00 31942-0 I. INTRODUCTION During the past decade it has been established that coupled cluster methods provide a theoretically sound as well as rapidly convergent description of the electronic structure of atoms and molecules. The coupled cluster method that includes all single and double excitations plus a perturbative estimate for triple excitations CCSD T 1 provides unrivaled accuracy for a wide range of molecules see the review by Lee and Scuseria, 2 as well as the more recent papers of Dunning and co-workers, 3 Martin and co-workers, 4 Helgaker and co-workers, 5 and Feller and co-workers 6. For molecules well described by a Hartree Fock wave function, the CCSD T method predicts bond dissociation energies, ionization potentials, and electron affinities to an accuracy of approximately 0.5 kcal/mol, bond lengths accurate to 0.0005 Å, and vibrational frequencies accurate to 5 cm 1. Further, Feller 7 has recently shown that, for N 2, CO, and HF, the differences between the bond dissociation energies calculated by full CI and the CCSD T method are 0.4 0.25, 0.4 0.3, and 0.05 0.15 kcal/mol. Thus for D e, the results obtained with the CCSD T method are converged to within the error bars noted above. To obtain high accuracy in CCSD T calculations, very a Current address: Office of Science, U.S. Department of Energy, 19901 Germantown Road, Germantown, MD 20874. b Also at: Department of Chemistry, Washington State University, Richland, WA 99352. c The Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the U.S. Department of Energy under Contract No. DE- AC06-76RLO 1830. large basis sets must be used and/or a series of calculations must be extrapolated to the complete basis set limit. This is unfortunate as CCSD T calculations formally scale as N 7, where N is the number of basis functions. Thus using a basis set with double the number of functions in the set or doubling the number of atoms in the molecule with the same basis set increases the cost of the calculation by two orders of magnitude. Although work is underway to reduce the dependence of CCSD T on N through a series of controlled approximations, 8 even in the best of circumstances it is unlikely that the dependence will be reduced below N 4 5 for many of the molecules of interest in chemistry. This steep dependence of the CCSD T method on the number of basis functions greatly restricts the range of applicability of this otherwise promising theoretical approach. Before continuing, one additional fact should be acknowledged. To achieve the accuracies stated above, it is necessary to include the effects of correlating the core electrons which requires even larger basis sets! as well as scalar and spin-orbit relativistic effects. For first row atoms, these corrections are small relative to the errors introduced by truncation of the basis set. For atoms further down in the periodic chart, these effects, especially relativistic effects, become much more important. Even in these cases, however, truncation of the basis set is still a major problem. It is the errors introduced by basis set truncation that are of interest here. In examining the rate of convergence of a number of molecular properties with basis set, e.g., D e, r e, and e,we noted that: 0021-9606/2000/113(18)/7799/10/$17.00 7799 2000 American Institute of Physics

7800 J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 T. H. Dunning, Jr. and K. A. Peterson TABLE I. List of molecules included in the current study. Neutrals AH AB Cations AH AB Anions AH AB Q L-Si Q L CCSD T ] Q Si CCSD T ] Q L MPn Q Si MPn, BH, CH, NH, OH, HF C 2,N 2,O 2,F 2, BF, CN, CO, NO BH,CH,NH,OH,HF O 2,CO,NO CH,NH,OH O 2,CN where Q refers to the property, L to the largest basis set used, S i to the series of smaller basis sets used, and MPn to nth order Møller Plesset perturbation theory L can also refer to the complete basis set limit; however, the analysis is then complicated by the differences in the extrapolation errors for the various methods. That is, the dependence of the property Q on the basis set was found to be approximately the same for perturbation theory methods as for the far more accurate CCSD T method. This suggests that it might be possible to obtain accurate CCSD T energies by performing CCSD T calculations in a small basis set and then correcting those energies by adding the difference between the energies obtained in MPn calculations with the same small basis set and the larger, more accurate basis set, i.e., E L CCSD T ] E Si CCSD T ] E L MPn E Si MPn. 2 This approximation is also a key assumption in the Gn methods (n 2,3) of Curtiss, Raghavarchari, Pople, and co-workers. 9 Since MP2 formally scales as N 5, MP3 as N 6, and MP4 as N 7, use of the approximation in Eq. 2 could result in significant computational savings if either the MP2 or MP3 methods result in acceptable errors. In this paper we examine the use of Eq. 2 to approximate CCSD T calculations. Specifically, we use the energies obtained from Eq. 2 to calculate the minimum energies (E e ), spectroscopic properties (D e,r e, e, e x e ), ionization potentials (IP e ), and electron affinities (EA e ) of a number of diatomic molecules. This set of molecules was selected to be representative of a wide range of molecule types: ionic and covalent, open-shell and closed-shell, and those well described or poorly described by Hartree Fock. The molecules included in the set are listed in Table I. For the present exploratory calculations, we take the large basis set L to be the aug-cc-pv5z basis set of Peterson and Dunning 10 and determine how well Eq. 2 reproduces the results of CCSD T /aug-cc-pv5z calculations when the small basis set ranges from aug-cc-pvdz to aug-cc-pvqz 11 ( S i ) and MPn ranges from MP2 to MP4. 1 In the next section we provide the computational details. In the following two sections we first assess the accuracy of the various combinations of perturbation theory method, basis set for the test set of molecules, and then determine the accuracy of the resulting recommended approach for all of the molecules in the test set. Finally, in the last section, we summarize the most important conclusions that can be drawn from the current work. II. COMPUTATIONAL DETAILS Calculations on the molecules listed in Table I used the augmented correlation consistent polarized valence (aug-cc-pvnz) basis sets of Kendall et al. 11 and Peterson and Dunning. 10 All CCSD T and closed-shell MPn calculations were carried out with the MOLPRO program package. 12 The MPn calculations on the open-shell atoms and molecules were carried out with ACES II. 13 The energies of the open-shell atoms and molecules were computed with the spin-restricted coupled cluster methods of Knowles et al. 14 and the restricted many-body perturbation theory of Lauerdale et al. 15 Spherical harmonics were used for the angular parts of the Gaussian basis functions. The calculations were performed on an SGI Origin 2000 at PNNL and on CRAY J90 s in DOE s National Energy Research Scientific Computing Center at Lawrence Berkeley National Laboratory. The energies of the diatomic molecules were computed at seven internuclear separations around the equilibrium internuclear distance. Potential energy curves were obtained by fitting polynomials of sixth order through the computed energies. A standard Dunham analysis 16 was employed to derive the minimum energy E e, bond dissociation energy D e, equilibrium distance r e, harmonic frequency e, and anharmonic correction e x e from the polynomial coefficients. The equilibrium ionization potentials, IP e, and electron affinities, EA e, are simply the differences in E e s for the ions and corresponding neutral molecules. CCSD T calculations were first carried out with the aug-cc-pv5z set. This defined the reference results. Then, the CCSD T /aug-cc-pv5z results were approximated using Eq. 2. The difference between these results and the reference results defined the approximation errors: E e, D e, r e, e, e x e, IP e, and EA e. We also carried out a corresponding series of calculations with the standard correlation consistent basis sets. 17,10 For molecules that are well described by the standard sets, the conclusions are essentially the same as those drawn here for the augmented sets. Therefore, these results are not reported. III. EVALUATION OF PERTURBATION THEORY APPROXIMATIONS: CCSD METHOD Although only the CCSD T method provides high accuracy for molecular calculations, it is important to also check the accuracy of the perturbation theory approximations to the CCSD method. The errors resulting from use of the various MPn approximations to describe the basis set dependence of the CCSD method are summarized in Table II for the total energies ( E e ) and spectroscopic constants

J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 Basis set dependence 7801 TABLE II. Errors in the approximate CCSD energies ( E e ) and spectroscopic constants ( D e, r e, e ) for the molecules included in the current study. Errors are relative to results obtained from CCSD calculations with the aug-cc-pv5z set. MP2 MP3 MP4 Molecule set avdz avtz avqz avdz avtz avqz avdz avtz avqz E e (me e ) A 4.593 2.704 1.113 1.738 0.487 0.096 1.902 0.485 0.124 AH 5.749 3.299 1.357 1.638 0.481 0.091 AB 11.713 6.468 2.751 2.853 1.087 0.201 10.774 2.532 0.685 A 5.125 3.142 1.615 1.984 0.724 0.205 2.990 0.891 0.259 AH 6.220 3.783 1.860 1.888 0.727 0.184 AB 15.253 7.120 3.505 3.453 1.773 0.476 14.017 3.368 0.886 D e (kcal/mol) AH 0.681 0.306 0.126 0.067 0.019 0.008 AB 1.747 0.720 0.350 0.484 0.103 0.046 4.166 0.925 0.273 AH 1.347 0.438 0.189 0.104 0.040 0.013 AB 3.139 1.983 0.995 0.849 0.244 0.160 6.968 1.746 0.480 r e (må) AH 1.41 0.25 0.08 0.33 0.11 0.02 AB 2.66 0.91 0.23 0.58 0.21 0.06 3.77 0.72 0.18 AH 1.92 0.35 0.10 0.48 0.17 0.04 AB 5.78 1.88 0.54 2.64 0.91 0.19 7.95 1.58 0.40 e (cm 1 ) AH 15.47 4.20 0.93 4.79 2.04 0.41 AB 16.69 5.23 1.32 3.81 1.52 0.35 26.15 5.35 1.32 AH 20.65 5.83 1.85 7.54 3.18 0.75 AB 23.79 7.18 1.86 12.26 3.99 1.15 46.72 8.91 2.57 ( D e, r e, e ). Both the average absolute errors and the maximum absolute errors are listed. In the tables we use the shorthand notation: avnz aug-cc-pvnz n d/d, t/t, q/q. Double excitations and products of double excitations are the dominant contributions to the CCSD wave function. The MP2 and MP3 methods involve only double excitations and, thus may be expected to provide reliable approximations to the CCSD method. The presence of triple excitations in the MP4 wave function can be expected to lead to a loss of accuracy. To check this expectation, the MP4 method was used to approximate the CCSD energies of the AB molecules as well as those of the first row atoms. The results, which are also reported in Table II, confirm this expectation. Errors in the MP4 approximation are usually larger than the errors resulting from use of the MP2 approximation and are always larger than from the MP3 approximation. This difference is most pronounced for the MP3 and MP4 maximum absolute errors. Below we only discuss the results obtained with the MP2 and MP3 approximations. Consider the errors in the total energies a very demanding criterion for assessing the accuracy of the perturbation theory approximations. First, we note that the error decreases significantly as the basis set is expanded from augcc-pvdz to aug-cc-pvtz to aug-cc-pvqz. This is to be expected at n 5, the error is exactly zero; see Eq. 2. Second, it is quite clear that the MP3 approximation is far more accurate than the MP2 approximation. Differences in the errors of factors of 5 or more are not uncommon. Finally, note that the errors in the atomic energies are approximately the same as those in the AH energies and about half those in the AB energies. Since the signs of the errors are the same, much of the error in the absolute energies will cancel when differences are taken; compare, e.g., the errors in E e versus those in D e. We see similar trends for the approximated spectroscopic constants as for the total energies. In general, we find the errors to be significantly larger for the AB molecules than for the AH molecules. For example, for the AH molecules the average absolute error in r e is 0.11 må for the MP3/augcc-pVTZ approximation, whereas it is twice that value 0.21 må for the AB molecules. We again find that the error decreases as n (avnz) increases. For D e, the error in the MP2 approximation for the AB molecules decreases by a

7802 J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 T. H. Dunning, Jr. and K. A. Peterson TABLE III. Errors in the approximate CCSD T total energies, E e, for the atoms and molecules included in the current study. Errors are relative to results obtained from CCSD T calculations with the aug-cc-pv5z set. Errors in total energies are given in millihartrees. MP2 MP3 MP4 Molecule set avdz avtz avqz avdz avtz avqz avdz avtz avqz A 2.813 2.225 0.976 0.674 0.059 0.050 0.122 0.058 0.020 AH 3.227 2.683 1.188 1.146 0.135 0.095 0.303 0.073 0.017 AB 4.163 4.642 2.235 4.762 0.764 0.337 3.266 0.702 0.169 A 3.880 2.508 1.366 1.335 0.084 0.067 0.189 0.083 0.033 AH 4.420 2.947 1.551 3.046 0.348 0.125 0.538 0.131 0.025 AB 8.113 4.927 2.807 6.764 1.277 0.534 6.876 1.562 0.413 factor of slightly more than 2 from the aug-cc-pvdz to the aug-cc-pvtz set and by another factor of 2 from the aug-ccpvtz to the aug-cc-pvqz set. A similar trend is observed for the MP3 approximation, although the overall decrease is a factor of 10, rather than a factor of 5. For both r e and e, the errors tend to decrease more rapidly with increasing n than for D e. The MP3 approximation is also found to be substantially more accurate than the MP2 approximation for the spectroscopic constants. For example, with the aug-cc-pvtz set, the average absolute error in D e for the MP3 approximation is 16 (AH) and 7(AB) times smaller than the corresponding errors for the MP2 approximation. The reductions are nearly as large for the maximum absolute errors; the factors are 11 and 8, respectively. The error ratios are smaller for r e and e, although still significant. The average absolute errors are decreased by factors ranging from 2.1 to 4.3, while the maximum absolute errors are reduced by factors ranging from 1.8 to 2.1. IV. EVALUATION OF PERTURBATION THEORY APPROXIMATIONS: CCSD T METHOD The errors resulting from use of the various MPn approximations to describe the basis set dependence of the CCSD T method are summarized in Tables III total energies: E e,iv spectroscopic constants: D e, r e, e, e x e, and V ionization potentials and electron affinities: IP e, EA e. The average absolute errors in D e, r e, and e are plotted in Figs. 1 and 2 for the AH and AB sets; the average absolute errors in the IP e s and EA e s are plotted in Figs. 3 and 4, respectively. Also included in the figures are the average absolute errors associated with the corresponding CCSD T /aug-cc-pvnz calculations heavy solid lines as well as the intrinsic errors relative to experiment discussed in the Introduction dashed lines. Consider first the errors in the total energies reported in Table III. As can be seen, the errors for the MP2 method are fairly large. Even for the aug-cc-pvqz basis set, the average absolute error for the AB molecule set exceeds 2 millihartrees (me h ) and the maximum error is nearly 3 me h. Again, it should be noted that the average absolute error for the AH molecule set is approximately the same as that for the A atom set 1.2 vs 1.0 me h and the average absolute error for the AB molecule set is approximately twice that of the A atom set 2.2 vs 1.0 me h. Thus the errors, although large, will again cancel when energy differences are taken. There is a dramatic improvement in both the average and maximum absolute errors listed in Table III when the MP3 method is used. For the AB molecule set, the average absolute error for the MP3 method with the aug-cc-pvtz set is three times smaller than for the MP2 method with the augcc-pvqz set. Use of the aug-cc-pvqz set reduces the error by another factor of 2. This trend continues with the MP4 method, although the decrease is in general less dramatic. One point that should be noted is that for both the MP3 and MP4 methods, the errors in the atomic calculations are markedly smaller than half the errors in the AB calculations. Thus there is less tendency for the errors to cancel when taking energy differences. One further point to note in Table III is that the maximum absolute errors are rarely more than twice the average absolute errors. The largest difference is for the calculations on the AH molecule set with the aug-cc-pvdz basis set, where the ratio is a factor of 2.7. This is encouraging. A method that yields maximum absolute errors not much larger than the average absolute errors is obviously superior to a method that yields maximum absolute errors that are much larger than the average absolute errors even if the average absolute errors are smaller. Now, let us compare the average absolute errors from CCSD T /aug-cc-pvnz calculations for D e, r e, and e with those obtained by using the various MPn approximations to describe the basis set dependence of the CCSD T method. That is, we will compare the results obtained using only the first term on the right hand side of Eq. 2 with the results obtained using the complete expression on the right hand side. For the AH molecules see Fig. 1, use of the MPn approximation reduces the error by approximately an order of magnitude in the worst case and by nearly two orders of magnitude in the best case. For D e, use of the MP2 approximation yields less accurate results than use of the MP3 approximation, which, in turn, yields less accurate results than the MP4 approximation. This seems intuitively reasonable the MP3 method includes the effects of interactions between double excitations as is the case in the CCSD part of the CCSD T method see the last section, while the MP4

J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 Basis set dependence 7803 TABLE IV. Errors in the approximate CCSD T spectroscopic constants ( D e, r e, e, e x e ) for the molecules included in the current study. Errors are relative to results obtained from CCSD T calculations with the aug-cc-pv5z set. MP2 MP3 MP4 Molecule set avdz avtz avqz avdz avtz avqz avdz avtz avqz D e (kcal/mol) AH 0.301 0.237 0.110 0.530 0.089 0.029 0.117 0.031 0.010 AB 0.862 0.163 0.129 2.854 0.486 0.152 1.873 0.425 0.119 AH 0.369 0.369 0.171 1.073 0.166 0.052 0.272 0.052 0.014 AB 1.481 0.317 0.279 3.675 0.902 0.280 4.097 1.019 0.288 r e (må) AH 0.80 0.14 0.04 0.43 0.02 0.02 0.32 0.08 0.02 AB 1.36 0.71 0.17 1.30 0.23 0.09 2.62 0.52 0.12 AH 1.29 0.19 0.05 0.83 0.03 0.03 0.48 0.12 0.03 AB 2.23 1.19 0.35 2.77 0.49 0.20 4.97 0.85 0.22 e (cm 1 ) AH 7.86 2.65 0.57 3.34 0.47 0.22 5.28 1.29 0.26 AB 8.34 3.40 0.73 8.65 1.96 0.73 18.81 3.67 0.81 AH 12.93 3.78 1.08 9.19 1.04 0.43 8.32 2.22 0.51 AB 13.66 4.40 1.11 15.47 5.46 1.82 36.71 7.24 2.11 e x e (cm 1 ) AH 0.61 0.17 0.11 0.46 0.13 0.03 0.38 0.11 0.03 AB 0.25 0.03 0.02 0.08 0.02 0.01 0.45 0.08 0.02 AH 0.97 0.35 0.16 0.96 0.24 0.09 0.85 0.25 0.06 AB 1.09 0.07 0.04 0.10 0.05 0.01 1.39 0.18 0.06 TABLE V. Errors in the approximate CCSD T ionization potentials ( IP e ) and electron affinities ( EA e ) for the molecules included in the current study. Errors are relative to results obtained from CCSD T calculations with the aug-cc-pv5z set. MP2 MP3 MP4 Molecule set avdz avtz avqz avdz avtz avqz avdz avtz avqz IP e in kcal/mol AH 0.440 0.270 0.130 0.609 0.113 0.040 0.089 0.021 0.007 AB 0.146 0.211 0.130 0.042 0.034 0.018 0.345 0.089 0.048 AH 1.611 0.809 0.250 1.271 0.185 0.051 0.235 0.029 0.013 AB 0.353 0.348 0.167 0.084 0.057 0.024 0.438 0.136 0.087 EA e in kcal/mol AH 0.492 0.326 0.150 1.014 0.267 0.112 0.401 0.096 0.033 AB 0.453 0.351 0.193 0.107 0.088 0.030 0.193 0.053 0.032 AH 0.553 0.452 0.208 1.584 0.360 0.145 0.704 0.163 0.064 AB 0.587 0.420 0.236 0.154 0.093 0.031 0.227 0.079 0.046

7804 J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 T. H. Dunning, Jr. and K. A. Peterson FIG. 1. Average absolute errors for the AH molecule set for MPn approximations to CCSD T /aug-cc-pv5z calculations: a D e ; b r e ; c e. Also plotted are the average absolute errors for CCSD T /aug-cc-pvnz calculations heavy solid lines as well as the intrinsic errors for the CCSD T method dashed lines ; see the text. method includes the effects of triple excitations, which are also included in the CCSD T method. For r e and e, on the other hand, the results obtained with the MP3 method are nearly as accurate as, and often more accurate than, those obtained with the MP4 method. For the AB molecule set, use of the MPn approximation also reduces the error by approximately an order of magnitude. For D e, the MP2 approximation actually leads to the most accurate results with the MP3 and MP4 approximations yielding very similar results the MP4 method provides slightly more accurate results than the MP3 method. For r e and e, the MP3 approximation yields the most accurate results with the MP2 and MP4 methods often providing similar results. It is surprising, and perhaps even counterintuitive, that for both the AH and AB molecule sets, the MP3 method yields results for r e and e as good as and often better than the MP4 method. However, in a study of the convergence of FIG. 2. Average absolute error for the AB molecule set for MPn approximations to CCSD T /aug-cc-pv5z calculations: a D e ; b r e ; c e. Also plotted are the average absolute errors for CCSD T /aug-cc-pvnz calculations heavy solid lines as well as the intrinsic errors for the CCSD T method dashed lines ; see text.

J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 Basis set dependence 7805 FIG. 3. Average absolute errors, IP e, for the a AH and b AB molecule sets for MPn approximations to CCSD T /aug-cc-pv5z calculations in kcal/mol. Also plotted are the average absolute errors for CCSD T /aug-cc-pvnz calculations heavy solid lines as well as the intrinsic errors for the CCSD T method dashed lines ; see text. the perturbation expansion for molecular spectroscopic constants, Dunning and Peterson 18 also noted a marked difference in the behavior of D e and r e, e. Equation 2 can also be recast as E L CCSD T ] E Si CCSD E L MPn E Si MPn E Si CCSD T E Si CCSD ; that is, we can regard the perturbation theory approximation to the basis set dependence of the energy as a correction to the CCSD energy, which is then corrected for the effect of triple excitations. If changes in the CCSD energy with basis set are substantially larger than the corresponding changes in the triples correction, one would expect the MP3 approximation to be more accurate than the MP4 approximation as shown in the last section. In Eq. 3, proper choice of S i is 3 required to ensure that both the MPn approximation and the triples correction are converged to the desired accuracy. For both the AH and AB molecule sets, there is a nearly exponential decrease in the average absolute error with increasing basis set size. The MP2 method tends to behave less monotonically than the MP3 and MP4 methods, although nonmonotonic behavior can also be observed for the MP3 method see, e.g., r e for the AH set. Many of the same conclusions can be drawn about the calculated ionization potentials and electron affinities see Figs. 3 and 4. Here, however, with the exception of the EA e s for the AH molecule set calculated with the aug-ccpvqz basis set, the MP2 approximation is markedly less accurate than the MP3 and MP4 approximations. In addition, the MP2 approximation often does not show significant im- FIG. 4. Average absolute errors, EA e, for the a AH and b AB molecule sets for MPn approximations to CCSD T /aug-cc-pv5z calculations in kcal/mol. Also plotted are the average absolute errors for CCSD T /aug-cc-pvnz calculations heavy solid lines as well as the intrinsic errors for the CCSD T method dashed lines ; see text.

7806 J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 T. H. Dunning, Jr. and K. A. Peterson provement with increasing basis set size increasing n. Since the errors in the CCSD T /aug-cc-pvnz calculations decrease steadily with increasing basis set size, the improvement resulting from use of the MP2 method to approximate the basis set dependence also decreases steadily with basis set. For the approximation in Eq. 2 to be most useful, the errors introduced by the approximation should be comparable to or even less than the intrinsic errors in the CCSD T method itself. The intrinsic errors D e, IP e, EA e : 0.5 kcal/ mol; r e : 0.0005 Å; e :5cm 1 are plotted as dashed lines in Figs. 1 4. For the AH molecule set, essentially all of the errors except those obtained with the aug-cc-pvdz set fall below the dashed lines. For the aug-cc-pvdz set, the approximation errors lie slightly above the intrinsic errors for both r e, e, and EA e. The same trend holds for the AB molecule set, although the errors resulting from use of the aug-cc-pvdz set are now well above the intrinsic errors for D e, r e, and e. The reason for the poor performance of the aug-cc-pvdz set is a result of the errors associated with the CCSD T /aug-cc-pvdz calculations. The average absolute errors for CCSD T calculations with the aug-cc-pvdz basis set are three to four times larger than those from calculations with the aug-cc-pvtz set. The use of perturbation theory to approximate the basis set dependence of the CCSD T method is simply not able to compensate for this increased error. V. RESULTS OBTAINED WITH MP3Õaug-cc-pVTZ APPROXIMATION From the results summarized in the preceding sections it would appear that a reasonable compromise between accuracy and computational cost in approximating the basis set dependence of the CCSD T method is to use the MP3 method with an aug-cc-pvtz basis set; that is, Ẽ av5z CCSD T ] E avtz CCSD T ] E av5z MP3 E avtz MP3. This approximation leads to average absolute errors that are less than the intrinsic errors associated with the CCSD T method itself. It replaces an N 7 CCSD T calculation with the aug-cc-pv5z basis set 127 functions with an N 7 CCSD T calculation with the aug-cc-pvtz set 46 functions plus two N 6 calculations, one with the aug-cc-pv5z set and the other with the aug-cc-pvtz set. CCSD T calculations with the aug-cc-pvtz set are as much as three orders of magnitude cheaper than such calculations with the aug-ccpv5z set and, so, more than compensate for the cost of the two additional N 6 calculations. The errors resulting from the use of Eq. 4, relative to those obtained from CCSD T /aug-cc-pv5z calculations, are listed in Tables VI ( D e, r e, e, e x e ) and VII ( IP e, EA e ) for all of the molecules considered here. Also included in Tables VI and VII are the errors for the base CCSD T /aug-cc-pvtz calculations. Comparing the columns labeled CCSD T and CCSD T MP3 allows 4 the impact of the MP3 approximation in Eq. 4 to be quickly quantified. The improvements are dramatic. For D e, the errors are reduced by a factor of 5 (C 2 ) to nearly a factor of 40 NH ; for IP e, the improvements span nearly the same scale, from a factor of 5 NH to 30 BH. For EA e, the improvements are not as large, but are still significant, varying from a factor of 3 CH to more than 15 CN. For both r e and e, substantial improvements are found, varying from a low of 4 e (CN) to a high of more than 300 r e (OH). All in all, there can be little doubt that the use of Eq. 3 dramatically decreases the errors in the uncorrected CCSD T /aug-ccpvtz calculations. Examining Table VI more carefully, we see that the largest errors resulting from use of the MPn approximation are associated with molecules that are poorly described by a Hartree Fock wave function. For D e, the largest error is for C 2 0.9 kcal/mol, a molecule that is known to have substantial multireference character in the wave function. C 2 is followed by O 2, which has an error of 0.65 kcal/mol, another difficult molecule to describe with a single reference wave function. For both r e and e, the largest errors are for CN 0.49 må, 5.45 cm 1, which, like C 2, is best described by a multireference wave function. In both cases CN is followed by O 2 0.45 må, 3.36 cm 1. The coupled cluster and perturbation theory methods used here are both based on Hartree Fock wave functions. However, past work has clearly established that the range of applicability of coupled cluster methods far exceeds that of perturbation theory methods. Thus it is not surprising that the MPn approximation to the basis set dependence of coupled cluster theory is less accurate for molecules that are poorly described by a Hartree Fock wave function. Although the above recommends use of the MP3 method to correct for basis set truncation in CCSD T calculations, it should be noted that the MP2 method, when combined with the aug-cc-pvtz set, also improves the accuracy of the base CCSD T /aug-cc-pvtz calculations. The average accuracies of the MP2 and MP3 approximations can be readily compared in Tables IV and V. From these tables, we see that the MP3 approximation is on the average significantly more accurate than the MP2 approximation. However, with the single exception of r e for the AB molecule set, the average absolute errors obtained using the MP2 approximation also fall below the intrinsic errors associated with the CCSD T method there are, however, larger variations in the errors themselves. The MP2 approximation, while clearly not as accurate as the MP3 approximation, may be adequate for many computational studies. The MP2 method scales as N 5 and approaches have already been developed that reduce this scaling to N 3, although with some loss of accuracy for smaller basis sets 19 see also Ref. 20. VI. CONCLUSIONS The CCSD T method a coupled cluster method that includes all single and double excitations plus a perturbative estimate of triple excitations provides an accurate description of the electronic structure of a broad range of molecules. To achieve high accuracy, however, large basis sets must be

J. Chem. Phys., Vol. 113, No. 18, 8 November 2000 Basis set dependence 7807 TABLE VI. Errors in the spectroscopic constants ( D e, r e, e, e x e ) obtained from CCSD T calculations with the aug-cc-pvtz set compared to the errors obtained from the same calculations with an MP3 correction. Errors are relative to results obtained from CCSD T calculations with the aug-cc-pv5z set. D e (kcal/mol) r e (må) e (cm 1 ) e x e (cm 1 ) Molecule CCSD T CCSD T MP3 CCSD T CCSD T MP3 CCSD T CCSD T MP3 CCSD T CCSD T MP3 BH 0.96 0.06 2.70 0.03 11.0 0.05 0.15 0.01 CH 1.31 0.06 2.24 0.02 14.0 0.21 1.16 0.12 NH 1.85 0.05 2.55 0.02 20.2 0.64 1.25 0.20 OH 1.93 0.11 3.12 0.01 24.0 1.04 2.31 0.17 HF 1.93 0.17 3.66 0.02 17.6 0.41 2.47 0.17 C 2 4.57 0.90 5.96 0.16 17.7 0.95 0.09 0.00 N 2 7.60 0.45 4.44 0.25 19.8 2.86 0.03 0.02 O 2 4.02 0.65 6.02 0.45 25.0 3.36 0.14 0.04 F 2 1.64 0.28 7.03 0.03 10.6 0.19 0.58 0.01 BF 3.32 0.26 7.23 0.21 11.6 0.74 0.04 0.00 CN 5.74 0.40 4.82 0.49 21.2 5.45 0.05 0.08 CO 5.51 0.47 5.08 0.07 19.3 0.79 0.11 0.00 NO 5.90 0.47 5.08 0.17 20.1 1.34 0.15 0.04 used. Since the computational cost of the CCSD T method increases as N 7, where N is the number of basis functions used in the calculations, CCSD T calculations becomes prohibitively expensive for large molecules. In this study we examined the use of perturbation theory to predict the effect of increasing the size of the basis set in CCSD T calculations, using the augmented correlation consistent sets of Dunning and co-workers. 10,11 We found that the spectroscopic properties (D e,r e, e, e x e ) for a set of diatomic molecules calculated with the aug-cc-pv5z basis set, as well as the associated ionization potentials (IP e ) and electron affinities (EA e ), were well reproduced by: Ẽ av5z CCSD T ] E avtz CCSD T ] E av5z MP3 E avtz MP3. The average absolute errors for the (AH AB) molecule set resulting from the use of this equation, relative to the full CCSD T /aug-cc-pv5z calculations, are: 0.33 kcal/mol (D e ), 0.15 må (r e ), 1.4 cm 1 ( e ), 0.06 cm 1 ( e x e ), 0.08 kcal/mol (IP e ), and 0.20 kcal/mol (EA e ). These errors are TABLE VII. Errors in the ionization potential ( IP e ) and electron affinity ( EA e ) obtained from CCSD T calculations with the aug-cc-pvtz set compared to the errors obtained from the same calculations with an MP3 correction. Errors are relative to results obtained from CCSD T calculations with the aug-cc-pv5z set. Molecule IP e (kcal/mol) EA e (kcal/mol) CCSD T CCSD T MP3 CCSD T CCSD T MP3 BH 0.60 0.02 CH 0.84 0.09 0.53 0.19 NH 0.85 0.16 1.92 0.25 OH 2.51 0.11 1.71 0.36 HF 2.35 0.18 O 2 0.78 0.00 0.99 0.09 CN 1.24 0.08 CO 1.38 0.06 NO 0.80 0.04 less than the intrinsic errors in the CCSD T method itself, namely, D e, IP e, EA e : 0.5 kcal/mol; r e : 0.0005 Å, and e : 5 cm 1. Since MP3 calculations scale as N 6, use of the above approximation should significantly increase the size of molecules that can be treated by the CCSD T method. The MP3 method also provides an excellent approximation of the basis set dependence of the CCSD method. The MP4 method, which includes triple excitations, does not provide as accurate a representation of the CCSD/aug-cc-pV52 energy as the MP3 method. This is at least partly responsible for the poor performance of the MP4 method in representing the basis set dependence of the CCSD T method. Although the MP2 approximation when combined with the aug-cc-pvtz set provides a less accurate description of the energy than does use of the MP3 approximation, it also significantly reduces the errors in the CCSD T /aug-ccpvtz calculations. The MP2 method scales as N 5, and variants with reduced scaling requirements currently exist. 19 The MP2, aug-cc-pvtz approximation may be adequate for many computational studies. Finally, it should be noted that, by carrying out a sequence of calculations with the augmented correlation consistent basis sets, the second term on the right hand side can be extrapolated to the complete basis set limit. In this case, the above equation leads to an approximation to E CBS CCSD T)], the CCSD T energy at the complete basis set limit. ACKNOWLEDGMENTS This work was supported by the Division of Chemical Sciences in the Office of Basis Energy Sciences of the U.S. Department of Energy at Pacific Northwest National Laboratory, a multiprogram national laboratory operated by Battelle Memorial Institute, under Contract No. DE-AC06-76RLO 1830. Computer resources were provided by the Division of Chemical Sciences and by the Office of Scientific Computing, at the National Energy Research Scientific Computing Center NERSC at Lawrence Berkeley National

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