3898
J. Phys. Chem. 1995,99, 3898-3901
Intrinsic Errors in Several ab Initio Methods: The Dissociation Energy of N2 Kirk A. Peterson* Department of Chemistry, Washington State University, and the Environmental Molecular Sciences Laboratory, Paci$c Northwest Laboratory, Richland, Washington 99352
Thom H. Dunning, Jr.+ Environmental Molecular Sciences Laboratory, Pacific Northwest Laboratory, Richland, Washington 99352 Received: November 23, 1994; In Final Form: January 30, 1995@
Using sequences of correlation consistent basis sets, complete basis set (CBS) limits for the dissociation energy D, of N2 have been estimated for a variety of commonly used electron correlation methods. After extrapolation to the CBS limit, the difference between theory and experiment corresponds to the errcr intrinsic to the chosen theoretical method. Correlated wave functions (valence electrons correlated only) for which intrinsic errors have been estimated include internally contracted multireference configuration interaction (CMRCI), singles and doubles coupled cluster theory with and without perturbative triple excitations [CCSD, CCSD(T)], and second-, third-, and fourth-order Moller-Plesset perturbation theory (MP2, MP3, MP4). For CMRCI and CCSD(T), D, converges smoothly from below the experimental value and yields the smallest intrinsic errors, -0.8 and - 1.6 kcallmol, respectively. In contrast, for MP2 and MP4, De exhibits fortuitously good agreement with experiment for small basis sets but leads to CBS limits that are 11.6 and 3.4 kcallmol larger than experiment, respectively. Correlation of the 1s core electrons is predicted to yield intrinsic errors of less than 1 kcaVmol for CMRCI and CCSD(T), while those for MP2 and MP4 increase still further.
Introduction The importance of accurately treating electron correlation effects to describe fundamental molecular properties and processes is widely known. However, the resulting accuracy of any correlated electronic structure calculation can be highly dependent on the one-particle basis set used to represent the molecular orbitals. In particular, the coupling between the correlation method and basis set can lead to erratic results that limit our understanding of the true errors associated with the chosen method. Only by approaching the complete basis set (CBS) limit can the intrinsic error of the theoretical method be assessed. Once these intrinsic errors or inherent accuracies are determined, the merits of different correlated methods can be unambiguously compared. In an ongoing series of benchmark calculation^,^-^ our research group has demonstrated how the CBS limit can be estimated using the correlation consistent basis sets of Dunning and co-workers.8-i The standard correlation consistent basis sets, denoted cc-pVxZ, are constructed by adding shells of correlating s, p, d, ... functions to the atomic Hartree-Fock orbitals, where each function within a shell contributes nearly an equal amount to the correlation energy. In this manner the correlation consistent basis sets are well defined with respect to increases in size and accuracy and might be expected to converge systematically to CBS limits for both Hartree-Fock and correlated wave functions. By exploiting the observed, nearly exponential convergence characteristics of systematic sequences of these one-particle basis sets, CBS limits have been estimated for a variety of molecular properties, which include total energies, dissociation energies, equilibrium structures, and the binding energies of weakly interacting systems. Systematic * To whom correspondence should be addressed. E-mail: ka-peterson@ pnl.gov. E-mail:
[email protected]. Abstract published in Advance ACS Abstracts, March 1. 1995. A
convergence has also been observed for vibrational frequencies and dipole moment matrix elements. Given the capability of accurately determining CBS limits for a given molecular property, it is of great interest to compare the intrinsic errors of different correlation methods that are commonly used in electronic structure calculations. In our previous benchmark studies of diatomic hydrides and homonuclear diatomics containing first and second row a t ~ m s , l - ~ > ~ the intrinsic errors of various configuration interaction (CI) methods were compared using the standard correlation consistent polarized valence basis sets. The N? molecule was included in one of these CI ~ t u d i e s . The ~ dissociation energy (De)of N? has been the subject of several recent high-quality ab initio s t ~ d i e s , ~ . since ' ~ - ' ~it presents a challenge to both the n-particle (correlation method) and one-particle basis sets. Intemally contracted multireference CI (CMRCI) calculations that employed a full valence complete active space (CAS) reference function gave the smallest intrinsic error for the De of N2 in the CI study of ref 3. Most of the remaining sources of error were attributed to the effects of neglecting the correlation of the 1s core electrons and higher excitations from the reference function. In the present Letter, the previous CMRCI results for the D, of NZ are compared to De's calculated by singles and doubles coupled cluster (CCSD),I6." CCSD with a perturbative estimate of connected triple excitations [CCSD(T)],'8.'9and second-, third-, and fourth-order Mgller-Plesset perturbation theory. All of these latter methods are based on a Hartree-Fock reference and were selected because they are commonly used in a wide variety of computational chemistry applications. For each theoretical method, the series of cc-pVxZ basis sets is extrapolated to the CBS limit and compared to experiment to yield the intrinsic error in D, for valence electron correlation.
Details of the Calculations The standard correlation consistent Gaussian basis sets* used in the present work correspond to primitive sets of (9s4pld),
0022-365419512099-3898$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Val. 99, No. 12, 1995 3899
Letters
TABLE 1: Equilibrium NZDissociation Energies De, Estimated CBS Limits: and Intrinsic E r r o d Calculated with 10 Correlated Electrons (kcaymol) method MP2
MP3 MP4 CCSD CCSD(T) CMRCI‘ CMRCI+Q
CC-pVDZ
CC-PVTZ
CC-PVQZ
cc-pv5z
est CBS limit
est intrinsic erroP
213.36 191.31 203.58 193.67 200.69 201.96 200.74
229.23 205.76 221.13 207.98 216.70 217.90 216.62
235.78 212.14 227.98 214.17 223.17 224.19 223.14
238.36 214.31 230.41 216.28 225.42 226.3 1 225.37
240.0 215.8 231.8 217.7 226.8 227.6 226.8
$11.6 -12.6 +3.4 - 10.7 -1.6
-0.8 -1.6
Obtained by separately fitting the calculated total energies to eq 1. See text. Defined as the difference between the estimated complete basis set limit and the experimental value (ref 33) of 228.4 kcal/mol. Reference 3.
--
240
- -
Expt’l(228.4 kcaUmol)
--....~l~~~.I..c - -
230
220
210
200
190
DZ
TZ
QZ
52
DZ
TZ
QZ
5Z
DZ
TZ
QZ
52
Figure 1. Calculated equilibrium dissociation energies De are shown as a function of the correlation consistent basis set used. The curves are fits to the exponential function of eq 1, and the dotted lines show the positions of the estimated complete basis set (CBS) limits for each correlation method. The difference between the CBS limit and the experimental value is the intrinsic error for that method. The uncontracted MRCUcc-pVQZ value for De is designated by the symbol F in the leftmost plot. (10s5p2dlf), (12s6p3d2flg), and (14s8p4d3f2glh) generally contracted to [3s2pld] (cc-pVDZ), [4s3p2dlfl (cc-pVTZ), [5s4p3d2flg] (cc-pVQZ), and [6s5p4d3f2glh] (cc-pVSZ), respectively. Complete details of the CMRCI calculations can be found in our previous work.3 Briefly, both the CI reference space and orbitals were obtained from full valence, complete active space self-consistent field (CASSCF) calculations20q2’(10 electrons in 8 orbitals). In order to make the MRCI more tractable, the doubly external configurations were internally contracted (CMRCI).22,23 To assess the effects of higher excitations, the multireference analog24 of the Davidson corr e ~ t i o n *(CMRCI+Q) ~ has also been investigated. Both the coupled cluster and Moller-Plesset calculations employed SCF orbitals. In each case, the computed De was calculated relative to the equilibrium bond distance for the respective basis set and theoretical method (re’s available on request). As in our previous work,3 the dissociated limits for the CMRCI and CMRCIfQ calculations were obtained by a supermolecule approach with r = 50 bohrs. For the calculation of the atomic N(4S) energies required for the coupled cluster and MollerPlesset dissociation energies, spin-restricted methods were employed. These corresponded to the partially spin-restricted coupled cluster method (RCCSD) of Knowles et and the restricted many-bodied perturbation theory (MBPT) of Lauderdale et ai.*’ D2h symmetry has been used throughout. All calculations were carried out with the MOLPRO suite of ab initio programs,28 except for the atomic MBPT results, which were determined using the ACES I1 package.29 Very often the basis set dependence of a given molecular property computed using correlation consistent basis sets has
been ~ b s e r v e d ’ - ~ , ~ - ’ to - ~ be ~ - ~well ~ described by a simple exponential function of the form
A(x) = A(m)
+ Be-”
(1)
where x is the cardinal number of the basis set (2, 3, 4, and 5 for DZ, TZ,QZ, and 5Z sets, respectively) and A(=) corresponds to the estimated CBS limit as x for the property A. The use of the cardinal number x for the functional parameter is a direct consequence of the “shell” structure of the basis sets. The exponential convergence of many properties is a reflection of the empirically observed exponential convergence of the total energy for the first- and second-row atoms.’.8 This procedure was used to estimate for each method complete basis set limits for the total energies. CBS limit values for De were then determined from the total energy limits.
-
00
Results and Discussion Calculated values of De for valence electron correlation are shown in Table 1 and displayed graphically in Figure 1. For each method, smooth convergence with basis set size is observed, and the complete basis set limits are estimated by fitting the total energies to eq 1. As observed in previous work, the CMRCI method yields a very accurate value for De as one approaches the basis set limit. Comparing the CBS limit to the experimental value33 of 228.4 kcal/mol yields an intrinsic error for CMRCI of just -0.8 kcal/mol. [This differs slightly from our previous result3of -0.6 kcal/mol, where the De’srather than the total energies were directly fit by eq 1.1 The uncontracted MRCI value for De with the cc-pVQZ basis set
Letters
3900 J. Phys. Chem., Vol. 99, No. 12, 1995 has been shown previously to be smaller than CMRCI by about 0.5 kcal/m01.~ This implies an intrinsic error of - 1.3 kcal/mol for a conventional MRCI calculation. Both CMRCI+Q and CCSD(T) yield intrinsic errors of just - 1.6 kcal/mol. Also shown in Table 1 and Figure 1 is the great importance of triple excitations on De in the CCSD calculations; the intrinsic error for CCSD is larger than that of CCSD(T) by 9.1 kcaumol. Our CCSD and CCSD(T) dissociation energies are in good agreement with the results of Bauschlicher and Partridge (BP),I5 where cc-pVxZ basis sets were also used. Their De values were about 0.4 kcal/mol smaller than ours, however, due to their use of the ROHF-CCSD(T) m e t h ~ d ~implemented "~~ in ACES I1 for the separated atom calculations. This method leads to slightly lower energies for the free atom compared to RCCSD(T), presumably due to some remaining spin contamination and the inclusion of determinants beyond the first-order interacting space. The work of BP also included CMRCI calculations with a cc-pV6Z basis set, which provides a check on our extrapolation procedure. Using our fits (DZ through 5Z) to eq 1, we predict De values for the CMRCI and CMRCIfQ methods with a cc-pV6Z basis set to be 227.2 and 226.3 kcal/mol, respectively. These differ from the values directly calculated by BP by less than 0.1 kcal/mol. While the dissociation energies calculated by multireference CI and coupled cluster both smoothly converge to CBS limits below the experimental value, the CBS limits determined from the Mdler-Plesset series exhibit an oscillatory behavior as shown in Figure 1. Compared to the experimental value, MP2 greatly overestimates De, MP3 underestimates it, and MP4 is again too large but smaller than MP2. While the MP3 results are generally in poor agreement with experiment, certain sizes of basis sets lead to MP2 and MP4 dissociation energies very close to the experimental value. In particular, at the MP2/ cc-pVTZ level of theory, De differs from experiment by just f 0 . 8 3 kcal/mol. Use of the cc-pVQZ basis set and MP4 results in an error of just -0.42 kcal/mol. However, in each case expansion of the basis set produces worse agreement with experiment. Situations such as these where less accurate wave functions yield better agreement with experiment severely limit our understanding of the correlation and basis set requirements of a particular property. At the estimated CBS limit, where the remaining errors are essentially due to just the correlation method, both MP2 and MP4 greatly overshoot the experimental value: MP2 is too large by +11.6 kcal/mol. while MP4 is in error by f 3 . 4 kcal/mol.
Conclusions By exploiting the systematic convergence characteristics of the correlation consistent basis sets, complete basis set limits for the dissociation energy of N2 have been estimated for several commonly used electron correlation methods. From these results, the intrinsic errors in De for valence electron correlation have been estimated to be (in kcal/mol) -0.8 (CMRCI), -1.6 (CMRCIfQ), -10.7 (CCSD), -1.6 [CCSD(T)], +11.6 (MP2), -12.6 (MP3), and +3.4 (MP4). As also noted in ref 15, the CCSD(T) method is observed to yield results comparable to the best multireference CI calculations for this system. Not surprisingly, due to the known difficulty in accurately describing the bonding in N2, the Moiler-Plesset methods yield results of much poorer quality. Both MP2 and MP4 greatly overestimate the experimental dissociation energy in the limit of a complete basis set (+11.6 and f 3 . 4 kcal/mol, respectively). However, since smaller basis sets actually yield better agreement with experiment for MP2 and MP4, the actual accuracy of these
methods would have been very difficult to assess without a means of systematically expanding the basis set toward the CBS limit. One of the main sources of intrinsic error in each of these methods is the neglect of core-core and core-valence correlation. Previous calculation^'^-'^^^^ have demonstrated that the contribution to De from correlating the 1s electrons is about 0.7- 1.3 kcal/mol. CMRCI and CCSD(T) c a l c ~ l a t i o n susing ~~ newly developed correlation consistent core-valence basis have recently confirmed these earlier results. Adding 1 kcaV mol to the CBS limits estimated from this work yields smaller net intrinsic errors for CCSD(T), CMRCI (and MRCI), and CMRCI+Q (all within 1 kcal/mol). For MP2 and MP4, however, the intrinsic errors increase still further over the valence-only results.
Acknowledgment. This work was supported by the Division of Chemical Sciences in the Office of Basic Energy Sciences of the U.S. Department of Energy at Pacific Northwest Laboratory, a multiprogram national laboratory operated by Battelle Memorial Institute, under Contract DE-AC06-76RLO 1830. K.A.P. acknowledges the support of the Associated Western Universities, Inc., Northwest Division under Grant DE-FG0689ER-75522 with the U.S. Department of Energy. Computational resources for this work were provided by the Division of Chemical Sciences and by the Office of Scientific Computing, Office of Energy Research, at the National Energy Research Supercomputing Center (Livermore, CA). The authors thank Drs. E. D. Glendening, D. E. Woon, and S. S. Xantheas for prepublication review of the manuscript and helpful comments. References and Notes (1) Woon, D. E.; Dunning, T. H.. Jr. J. Chem. Phys. 1993, 99, 1914. (2) Peterson, K. A,; Kendall, R. A.; Dunning, T. H., Jr. J. Chem. Phys. 1993, 99, 1930. (3) Peterson, K. A,; Kendall, R. A,: Dunning. T. H., Jr. J. Chem. Phys. 1993, 99, 9790. (4) Peterson, K. A,; Woon, D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1994, 100, 7410. ( 5 ) Woon, D. E. J. Chem. Phys. 1994, 100, 2838. (6) Woon, D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1994, 101, 8877. (7) Peterson, K. A,; Dunning. T. H.. Jr. J. Chem. Phys. 1995, 102, 2032. (8) Dunning, T. H.. Jr. J . Chem. Phys. 1989, 90. 1007. (9) Kendall. R. A,: Dunning, T. H., Jr.: Harrison, R. J. J. Chem. Phys. 1992, 96, 6796. (10) Woon. D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1993, 98, 1358. (11) Woon. D. E.; Dunning, T. H., Jr. J. Chem. Phjs. 1994, 100, 2975. (12) The correlation consistent and augmented correlation consistent basis sets for the first-row and second-row atoms (plus hydrogen and helium) may be downloaded via anonymous ftp from pnlg.pnl.gov in directory ccbasis. (13) Almlof. J.; DeLeeuw, B. J.; Taylor, P. R.; Bauschlicher, C. W.. Jr.; Siegbahn, P. Inr. J. Quantum Chem. Symp. 1989, 23, 345. (14) Werner, H.-J.; Knowles, P. J. J. Chem. Phys. 1991, 94, 1264. (15) Bauschlicher. C. W.. Jr.; Partridge. H. J . Chem. Phys. 1994, 100, 4329. (16) Purvis, G. D.: Bartlett, R. J. J. Chem. Phys. 1982, 76, 1910. (17) Hampel, C.: Peterson, K. A,; Werner, H.-J. Chem. Phys. Lett. 1992, 190, 1. (18) Ragavachari. K.: Trucks, G. W.: Pople. J. A,; Head-Gordon, M. Chem. Phys. Lett. 1989, 157, 479. (19) Deegan. M. J. 0.; Knowles, P. J. Chem. Phys. Lett. 1994, 227. 321. (20) Werner, H.-J.; Knowles, P. J. J. Chem. Phys. 1985, 82, 5053. (21) Knowles, P. J.: Werner, H.-J. Chem. Phys. Lett. 1985, 115, 259. (22) Werner, H.-J.; Knowles. P. J. J. Chem. Phys. 1988, 89, 5803. (23) Knowles, P. J.; Werner, H.-J. Chem. Phys. Lett. 1988. 145, 514. (24) Blomberg, M. R. A,; Siegbahn, P. E. M. J. Chem. Phys. 1983, 78. 5682. (25) Langhoff. S. R.; Davidson, E. R. Int. J . Quantum Chem. 1974, 8, 61. (26) Knowles. P. J.; Hampel. C.: Werner. H.-J. J. Chem. Phys. 1994. 99, 5219. (27) Lauderdale, W. J.; Stanton, J. F.; Gauss. J.; Watts, J. D.: Bartlett, R. J. Chem. Phys. Lett. 1991, 187, 21.
Letters (28) MOLPRO is a suite of ab initio programs written by H.-J. Werner and P. J. Knowles with contributions by J. Almldf, R. D. Amos, M. J. 0. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. A. Peterson, R.M. Pitzer, E.-A. Reinsch, A. J. Stone, and P. R. Taylor. (29) ACES I1 is a computational chemistry package especially designed for CC and MBPT energy and gradient calculations. Elements of this package are the SCF, integral transformation, correlation energy, and gradient programs written by J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett; the VMOL integral and VPROPS property integral programs written by P. R. Taylor and J. Almlof; a modified version of the integral derivative program ABACUS written by T. Helgaker, H. J. Jensen, P. Jorensen, J. Olsen, and P. R. Taylor; and the geometry optimization and vibrational analysis package written by J. F. Stanton and D. E. Bernholdt. (30) Feller, D. J. Chem. Phys. 1992,96, 6104.
J. Phys. Chem., Vol. 99,No. 12, 1995 3901 (31) Xantheas, S . S.; Dunning, T. H., Jr. J. Phys. Chem. 1993,97, 18. (32) Woon, D. E. Chem. Phys. Lett. 1993,204, 29. (33) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules: Van Nostrand: Princeton, 1979. (34) Rittby, M.: Bartlett, R. J. J. Phys. Chem. 1988,92, 3033. (35) Stanton, J. F.: Gauss, J.; Watts, J. D.; Bartlett, R. J. J. Chem. Phys. 1991,94, 4334. (36) Ahlrichs, R.; Scharf, P. S.; Jankowski,'K. Chem. Phys. 1985,98, 381. (37) Peterson, K. A.; Woon, D. E.; Dunning, T. H., Jr. Manuscript in preparation. (38) Woon, D. E.; Dunning, T. H., Jr. Manuscript in preparation. JP943 1346
