An examination of intrinsic errors in electronic structure methods using

An examination of intrinsic errors in electronic structure methods using the Environmental Molecular Sciences Laboratory computational results database and the Gaussian-2 set David Feller and Kirk A. Peterson Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, 906 Battelle Boulevard, MS K1-96, Richland, Washington 99352

~Received 28 August 1997; accepted 23 September 1997! The Gaussian-2 ~G2! collection of atoms and molecules has been studied with Hartree–Fock and correlated levels of theory, ranging from second-order perturbation theory to coupled cluster theory with noniterative inclusion of triple excitations. By exploiting the systematic convergence properties of the correlation consistent family of basis sets, complete basis set limits were estimated for a large number of the G2 energetic properties. Deviations with respect to experimentally derived energy differences corresponding to rigid molecules were obtained for 15 basis set/method combinations, as well as the estimated complete basis set limit. The latter values are necessary for establishing the intrinsic error for each method. In order to perform this analysis, the information generated in the present study was combined with the results of many previous benchmark studies in an electronic database, where it is available for use by other software tools. Such tools can assist users of electronic structure codes in making appropriate basis set and method choices that will increase the likelihood of achieving their accuracy goals without wasteful expenditures of computer resources. © 1998 American Institute of Physics. @S0021-9606~98!00601-1#

I. INTRODUCTION

The evolution of ab initio electronic structure theory as a practical tool for the understanding and prediction of molecular phenomena has been adversely impacted by the slow convergence of the mathematical expansions used for solving the many electron Schro¨dinger equation. Chief among these are the one-particle and the n-particle expansions, normally associated with the underlying Gaussian basis set and the degree of correlation recovery, respectively. Consequently, over the past 40 years a considerable effort has been invested in developing improved techniques for addressing the problems inherent to each expansion. In addition to the difficulties arising from slowly convergent expansions, users of electronic structure methods are also plagued by the general lack of formal error bars for computed properties that might provide an a priori guide to the accuracy of a calculation. In the absence of rigorous error limits, basis sets and levels of theory are typically benchmarked against the available experimental data for selected groups of molecules. It is then argued by inference that ‘‘similar’’ accuracy should be expected in chemically related systems. Due to the lack of high-quality experimental data on many systems of interest, this approach is often limited in its usefulness. Even in fortunate cases where reliable experimental information is available, it is frequently not clear what fraction of the apparent error in a computed property resulted from shortcomings in the Gaussian basis set, as opposed to the correlation treatment. Therefore, selecting appropriate combinations of basis sets and levels of theory is difficult for the increasing number of scientists who desire to use electronic structure theory in their research. A poor choice can result in inefficient use of valuable computer resources or in failure to 154

J. Chem. Phys. 108 (1), 1 January 1998

achieve the desired accuracy. Straightforward increases in the size of the one-particle basis set of Gaussian functions or the use of more sophisticated levels of theory can frequently lead to a seemingly paradoxical deterioration in the agreement with experiment. At present, the only hope for attacking these problems is through more extensive benchmarking efforts that approach completeness in both the one-particle and n-particle expansions for as large a set of molecules as possible. In the present work, we address the performance of the correlation consistent basis sets1 on the Gaussian-2 ~G2!2 collection of molecules using Hartree–Fock ~HF! theory and four widely used correlated methods. The G2 procedure consists of a sequence of smaller calculations that are combined in such a way as to approximate the results obtained from a single much larger calculation. The G2 method exploits the approximate separability of the one-particle and n-particle expansions. G2 is one of a number of procedures that are designed to achieve very high accuracy in computed energy differences. An earlier technique, known as Gaussian-1 ~G1! theory,3 was reported to achieve an accuracy of better than 62 kcal/mol for 32 experimentally well-characterized atomization energies. The set of atomization energies was increased to 56 in G2, with an additional 79 cases reported which had larger experimental uncertainties. To this set we have added the atomization energy of N2O, for which accurate experimental data exists. The G2 set also includes 38 adiabatic ionization potentials, 25 electron affinities, and seven proton affinities. An even larger collection of molecules has very recently been described by Curtiss et al.,4 but is beyond the scope of the present work. The correlation consistent basis sets were chosen for this study because the regularity of their convergence pattern

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D. Feller and K. A. Peterson: Errors in electronic structure methods

makes it easier to distinguish the errors found with a specific ~level of theory/basis set! combination from the intrinsic error associated with the same level of theory at the complete basis set ~CBS! limit. This ability is essential for unraveling the interdependent one-particle and n-particle expansion errors. The intrinsic error for a given level of theory will fluctuate from one chemical system to another, depending on the impact of the approximations in the chosen level of theory. What is desired is the identification of intrinsic errors in a sufficiently broad class of molecules so that researchers may begin to predict the likely error resulting from potential calculations under consideration. The choice of methods was dictated by a desire that our conclusions be as relevant as possible to as many researchers performing ab initio electronic structure calculations as possible. The five methods chosen are ~1! Hartree–Fock theory, ~2! Mo” ller–Plesset perturbation theory at second order ~MP2!, ~3! fourth-order @MP4~SDTQ!#, ~4! coupled cluster theory with single and double excitations ~CCSD!, and ~5! coupled cluster theory with perturbative inclusion of triple excitations @CCSD~T!#. Numerous studies have shown that while HF calculations yield reasonable geometries for molecules consisting of first- and second-row elements, it predicts thermochemical properties that are grossly in error. HF bond energies may differ from experiment by as much as 100–200 kcal/mol. Correlated methods, by accounting for instantaneous electron-electron repulsion, are ultimately capable of overcoming the inadequacies in HF theory, but at the expense of dramatically increased computational costs and a proportional decrease in the size of the systems that can be studied. For example, the computer time required for the correlated methods selected for the present study formally scale as N 5 – N 7 , where N is the number of basis functions. Thus as will be seen, the choice of basis sets and methods spans a wide range in terms of accuracy and in terms of computational requirements. Although any finite collection of compounds or reactions will have obvious limitations, the G2 set is an adequate starting point for small first- and second-row molecules. It has received widespread use by researchers attempting to extend the method or to develop totally new approaches. For example, Bauschlicher and Partridge5 recently proposed a modification of the G2 theory that involves density functional and CCSD~T! components. The parametrized configuration interaction scheme of Siegbahn et al.,6 which applies a multiplicative correction factor for the correlation contribution to the energy, was developed with the help of the G1 molecule set. Petersson and co-workers7,8 have also compared the quality of their complete basis set models against the G2 set. II. PROCEDURE

Originally, the correlation consistent basis sets supported H and B–Ne at the valence double, triple, and quadruple zeta levels.1 Subsequent extensions to this basis set family have included diffuse functions for improved descriptions of anions and molecular properties,9 core/valence functions for recovering core correlation effects,10,11 sets for second row

155

atoms12 and alkali and alkaline earth metal atoms,13 and valence quintuple and sextuple zeta sets for B–Ne.14,15 Because of the large number of molecules contained in the G2 set and the need to treat negatively charged and highly ionic systems, we limited our attention to the diffuse function augmented sets, aug-cc-pVxZ, x5D, T, and Q. Throughout the subsequent discussion of atomization energies, SD e , it should be kept in mind that for strongly bound molecules involving atoms with similar electronegativities the dissociation energy is relatively insensitive to the presence of diffuse functions. This is especially true for the larger sets. Therefore general comments concerning the augcc-pVTZ and aug-cc-pVQZ basis sets also apply to their smaller counterparts ~cc-pVTZ and cc-pVQZ basis sets!. In the majority of cases, the aug-cc-pVQZ basis set was the largest affordable set applied to the G2 molecules. However, in some cases, as a test of the accuracy of the complete basis set extrapolation, basis sets as large as augmented quintuple zeta, i.e., aug-cc-pV5Z, have been used. Because augmented correlation consistent basis sets are not available for the alkali and alkaline earth metals, we combined augmented sets on all other atoms with nonaugmented sets on Li, Be, Na, and Mg. For the sake of brevity, results obtained with these hybrid basis sets will be presented in combination with results from the pure aug-cc-pVxZ sets, under the same headings. Unless otherwise noted, all energies were computed at the optimized geometry for each specific basis set and level of theory. Most geometries were optimized with a gradient convergence criterion of 1.531025 E h /a 0 , corresponding to the ‘‘tight’’ criterion defined by GAUSSIAN-94.16 Due to the expense of the aug-cc-pVQZ basis set optimizations, they were performed with the default Gaussian gradient convergence criterion of 4.531024 E h /a 0 for molecules containing four or more atoms. Open shell calculations were performed with unrestricted Hartree–Fock ~UHF! wave functions as a starting point in order to permit direct comparisons with the original G2 results. Selected closed-shell diatomic calculations were performed with MOLPRO-96.17 HF and MP2 harmonic vibrational frequencies were obtained analytically with GAUSSIAN94, whereas numerical differencing techniques were required for both geometry optimizations and frequencies at all higher levels of theory. Except where noted, the frozen core approximation was used in this work. In order to accurately determine the effects of core/valence correlation on the properties of interest, it would have been necessary to carry out CBS extrapolations with extra tight functions added to the aug-cc-pVxZ series of basis sets. Because such aug-cc-pCVxZ basis sets would have been significantly larger than the already large sets we were currently using, the resulting calculations were judged to be prohibitively expensive. Alternatively, selected dissociation energies were corrected for the effects of core/ valence correlation by performing CCSD~T! calculations with the cc-pCVTZ ~or cc-pwCVTZ for second row elements! basis sets at the optimal CCSD~T!/aug-cc-pVTZ geometries. Comparisons of the core/valence corrections ob-

J. Chem. Phys., Vol. 108, No. 1, 1 January 1998


156


tained from this procedure with corresponding corrections obtained from larger basis sets and geometry reoptimization including core/valence effects18–21 suggest that the procedure is capable of recovering 80%–90% of the true effect. Dissociation energies were computed as the difference between the molecular energies at the respective optimized geometries and the sum of the atomic ground state energies, without correcting for basis set superposition error ~BSSE!. Although BSSE can often be substantial for the aug-ccpVDZ basis set, our aim was to report on the performance of the correlation consistent basis sets in the manner that people most frequently use them. Since methods designed to correct for BSSE, such as the counterpoise method of Boys and Bernardi,22 all require a significant additional investment in computer time, they are seldom used for strongly bound systems. No symmetry or equivalence restrictions were imposed in the atomic calculations. The results reported in this work were obtained from the already extensive body.10,12,23–30 of benchmark information on the chosen basis sets and levels of theory that are present in the chemistry literature, in addition to a large number of new calculations. Martin31,32 has also reported CCSD~T! atomization energies, geometries, and vibrational frequencies for a small subset of the present molecules that were obtained with the cc-pVxZ basis sets. All results are stored in the Environmental Molecular Sciences Laboratory ~EMSL! Computational Results Database, which has been described elsewhere.33 The database software makes it possible to compare results obtained from a series of calculations performed with various basis sets and methods against experiment or other high-level theoretical results. Ignoring literature citations, the EMSL Computational Results Database currently contains over 19 700 entries pertaining to energetics, molecular structure, and normal-mode frequencies.

III. COMPLETE BASIS SET ESTIMATES

Shortly after the introduction of the correlation consistent basis sets, it was observed that in most cases the total energies obtained from these sets converge with sufficient regularity that a simple three-parameter, exponential function provides a reasonable estimate of the CBS limit.23,26,34 The form of the function is given by E ~ x ! 5E CBS1be 2cx

~1!

where x is an index associated with each basis set, 25DZ, 35TZ, 45QZ, etc. In general, the three parameters E CBS b, and c are determined from a nonlinear least-squares fit. Energy differences, bond lengths, bond angles, and some one-electron properties19,35 were also found to converge with approximate exponential behavior, although exceptions are more common than with total energies. In the present work, we have chosen to extrapolate the individual total energies to the CBS limit, rather than extrapolate energy differences. Several recent studies have examined both approaches and found that they provide nearly identical results.30,36

Other functional forms have also been proposed. Woon and Dunning25 used a combined Gaussian/exponential function of the form: E ~ x ! 5E CBS1be 2 ~ x21 ! 1ce 2 ~ x21 ! ** 2 More recently, Martin from the expression:

31,32

~2!

reported CBS estimates obtained

DE5a s n s 1b p n p 1 ~ n s 1n p 1n lp! c pair ,

~3!

where n s , n p , and n lp are the number of s and p bonds and lone pairs and the coefficients a s , b p , and c pair are specific to a given combination of basis set and level of theory. The same author examined expansions in inverse powers of l max the maximum angular momentum present in the basis set.37 Superior agreement with experiment, compared to the exponential CBS estimates, were found for 13 total atomization energies. The same approach was used by Martin37 to compute the atomization energy of HCN and by Martin and Lee38 to study NH3. Their work relied on expressions of the form: E ~ l max! 5E CBS1

B C . 41 ~ l max10.5! ~ l max10.5! 6

~4!

or a simpler two-parameter expression with C50. Wilson and Dunning39 also found that expressions of the general form: 2! E ~ 2 ! ~ l max! 5E ~CBS 1

1

B C 1 ~ l max1d ! m ~ l max1d ! m11

D ~ l max1d ! m12

~5!

provided improved estimates of the CBS limit for the MP2 correlation energy. The best agreement between the correlation corrections predicted by Eq. ~5! and the ‘‘exact’’ MP2R12 values of Klopper40 were obtained with values of m 54, d51, and D50. All of these approaches are based on the asymptotic limit of the two-electron cusp.41 The performance of the exponential CBS extrapolation, or any extrapolation constructed from a sequence of finite basis set values, can be judged from several perspectives. Of primary importance is the accuracy and efficiency of the extrapolation, i.e., how well does it reproduce the best available estimate of the CBS limit for the property in question, using the smallest possible investment in computer resources. A related but subtlely different criterion is the degree to which the functional form is able to fit all of the available data when the number of data points exceeds the number of adjustable parameters in the fit. For total energies, both criteria are illustrated in Table I, where CCSD~T! values predicted by DZ→QZ and DZ→6Z exponential extrapolations are compared with the actual computed values for HF, N2, and CO. All of the fits are seen to reproduce the computed energies quite well, up through sextuple zeta. The largest error in the DZ→QZ fit, measured with respect to the computed CCSD~T! energies, was only 1.4 mE h , compared to an overall decrease in the total energy over these five basis sets which is three orders of magnitude larger. The errors in the DZ→5Z fit increase slightly with




157

TABLE I. Comparison of predicted and calculated frozen core CCSD~T! energies for HF, N2, and CO.a HF 1 S 1 Basis set

E CCSD~T!

DZ→QZ Exp. Fit

Error

DZ→6Z Exp. Fit

Error

TZ→QZ 1 1/(l max1 2)4

Error

cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z cc-pV6Z Est. CBS

2100.2282 2100.3384 2100.3732 2100.3852 2100.3891 2100.3908b

2100.2282 2100.3384 2100.3732 2100.3842 2100.3877 2100.3893

0.0000 0.0000 0.0000 10.0010 10.0014 10.0015

2100.2282 2100.3381 2100.3737 2100.3852 2100.3889 2100.3907

0.0000 10.00003 20.0005 0.0000 10.0008 10.0001

2100.1824 2100.3384 2100.3732 2100.3843 2100.3887 2100.3933

10.0458 0.0000 0.0000 10.0009 10.0004 20.0025

N2 1 S 1 g Basis set cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z cc-pV6Z Est. CBS

2109.2765 2109.3739 2109.4044 2109.4142 2109.4178 2109.4199b

2109.2765 2109.3739 2109.4044 2109.4139 2109.4169 2109.4182

0.0000 0.0000 0.0000 10.0003 10.0009 10.0017

2109.2765 2109.3738 2109.4046 2109.4144 2109.4175 2109.4189

0.0000 10.0001 20.0002 20.0002 10.0003 10.0010

2109.2372 2109.3739 2109.4044 2109.4141 2109.4180 2109.4220

10.0393 0.0000 0.0000 10.0001 20.0002 20.0021

CO 1 S 2 Basis set cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z cc-pV6Z Est. CBS

2113.0550 2113.1556 2113.1879 2113.1982 2113.2018 2113.2038b

2113.0550 2113.1556 2113.1879 2113.1983 2113.2016 2113.2032

0.0000 0.0000 0.0000 20.0001 10.0002 10.0006

2113.0550 2113.1556 2113.1879 213.1982 2113.2018 2113.2038

0.0000 0.0000 0.0000 20.0002 10.0001 10.0005

2113.0180 2113.1556 2113.1879 2113.1982 2113.2018 2113.2038

10.0442 0.0000 0.0000 0.0000 20.0005 20.0027

a

Energies are in hartrees. CCSD~T! energies are taken from Wilson et al., Ref. 39. The exponential function used in the fit was of the form: E(x)5E CBS 1 1 1be 2cx , x52 ~DZ!, 3~TZ!, etc. The 1/(l max1 2)4 fit used a function of the form: E(l max)5ECBS1b/(l max1 2)4. b Estimate obtained from an exponential extrapolation of the QZ, 5Z, and 6Z energies.

increasing basis set size and are of varying sign. CBS energies were also estimated by extrapolating the QZ, 5Z, and 6Z energies. If we assume that these values represent our best current estimates of the true basis set limit, the errors in the DZ→QZ CBS energies range from 10.6(CO) to 11.7(N2)mE h and from 10.1(HF) to 11.0(N2)mE h with the DZ→6Z fits. The positive sign of the errors reflects the fact that the true growth in the correlation energy as a function of x falls off less rapidly than e 2x , thus causing the exponential fits to underestimate the true limit. Based on the limited MP2-R12 results of Klopper40 it appears that the final 2% to 3% of the MP2 correlation energy is only recovered with very extensive basis sets, well in excess of sextuple zeta. This basis set regime is not well described by the simple exponential CBS extrapolation, but it is hoped that such effects are relatively small in size. Calculations with quintuple and sextuple zeta basis sets are currently so expensive that they are only feasible for small systems. In this work we have focused on results obtained with the augmented double through augmented quadruple zeta sets which, while large by the standards of just a few years ago, are still tractable for the G2 set of molecules. We will use the notation ‘‘CBS~aDTQ/e 2x )’’ to denote extrapolated results obtained from an exponential fit of aug-cc-pVDZ through aug-cc-pVQZ values. Similarly, ‘‘CBS~aTQ5/e 2x )’’ denotes results obtained from the augmented triple through quintuple zeta basis sets. Complete basis set estimates were also obtained by fit-

ting the TZ and QZ energies using only the first two terms of Eq. ~4!, i.e., setting C50. As suggested by Martin,37 the DZ energies were not included in the fit because this basis set is too small to yield energies that are dominated by 1/( l max1 21)4 convergence. In Table I the l max fit is shown to predict CBS energies that are appreciably lower than any of the exponential values. In the absence of independent estimates of the CBS limits, e.g., CCSD~T!-R12 energies, it is difficult to decide whether the CBS~aDTQ5-Exp! or l max estimates are superior. Ignoring the DZ energies, both fits display comparable deviations with respect to the computed energies. CBS~aDTQ/e 2x ) energies for 135 of the 221 chemical systems considered in this study are presented in Table II. Resource constraints prevented treating the entire collection of molecules at such a high level of theory. In those instances where extrapolation to the complete basis set limit was not available, the highest level basis set used to treat each system is indicated in Table II. Comparisons between CBS(aDTQ/e 2x ) energies and directly computed energies or extrapolated values obtained from larger basis set calculations, such as those provided in Table I, are helpful in establishing the degree of absolute accuracy one can expect from the exponential fit. Towards this end, we have performed aug-cc-pV5Z calculations on 35 of the systems in Table II and used these additional energies to derive the CBS(aTQ5/e 2x ) energies listed in Table III. The mean absolute deviation, e MAD , based on all 35 available cases is 0.4mE h for Hartree–Fock theory and 1.1mE h



158


TABLE II. Estimated complete basis set, frozen core energies obtained from an exponential fit of aug-cc-pVDZ through aug-cc-pVQZ energies.a Molecule 2

H ( S) H2 ( 1 S 1 g ) Li ( 2 S) Be ( 1 S) B ( 2 P) C ( 3 P) N ( 4 S) O ( 3 P) F ( 2 P) Na ( 2 S) Mg ( 1 S) Al ( 2 P) Si ( 3 P) P ( 4 S) S ( 3 P) Cl ( 2 P) LiH ( 1 S 1 ) BeH ( 2 S 1 ) BeH2 ( 1 S 1 g ) BH ( 1 S 1 ) BH2 ( 2 A 1 ) BH3 ( 1 A 18 ) CH ( 2 P) CH2 ( 3 B 1 ) CH2 ( 1 A 1 ) CH3 ( 2 A 92 ) CH4 ( 1 A 18 ) NH ( 3 S 2 ) NH2 ( 2 B 1 ) NH3 ( 1 A 1 ) OH ( 2 P) H2O ( 1 A 1 ) HF ( 1 S 1 ) NaH ( 1 S 1 ) MgH ( 2 S 1 ) MgH2 ( 1 A 18 ) AlH ( 1 S 1 ) AlH2 ( 2 A 1 ) AlH3 ( 1 A 18 ) SiH ( 2 P) SiH2 ( 1 A 1 ) SiH2 ( 3 B 1 ) SiH3 ( 2 A 29 ) SiH4 ( 1 A 18 ) PH ( 3 S 2 ) PH2 ( 2 B 1 ) PH3 ( 1 A 1 ) SH ( 2 P) H2S ( 1 A 1 ) HCl ( 1 S 1 ) Li2 ( 1 S 1 g ) LiN ( 3 S 2 ) LiO ( 2 P) LiOH ( 1 S 1 ) LiF ( 1 S 1 ) LiCl ( 1 S 1 ) BeO ( 1 S 1 ) BeOH ( 1 A 8 ) BeF ( 2 S 1 ) BeS ( 1 S 1 ) BeCl ( 2 S 1 ) BO ( 2 S 1 ) HBO ( 1 S)

E HF

E MP2

E MP4

E CCSD

E CCSD~T!

20.5000 21.1336 27.4327 214.5730 224.5338 237.6943 254.4051 274.8194 299.4169 2161.8588 2199.6144 2241.8806 2288.8593 2340.7192 2397.5137 2459.4902 27.9873 215.1537 aVDZ 225.1320 aVDZ aVDZ 238.2848 238.9411 238.8963 239.5811 240.2170 254.9870 255.5930 256.2257 275.4290 276.0688 2100.0719 2162.3923 aVDZ aVDZ 2242.4644 aVDZ aVDZ 2289.4430 2290.0355 2290.0294 2290.6472 aVDZ 2341.3028 2341.8922 aVDZ 2398.1108 2398.7200 2460.1128 214.8719 261.8409 282.3066 282.9617 2106.9949 2467.0564 289.4554 aVDZ 2114.1735 2412.1483 aVDZ 299.5683 aVDZ

NA 21.1672 NA 214.6034 224.5828 237.7693 254.5131 274.9854 299.6512 NA 2199.6391 2241.9201 2288.9211 2340.8107 2397.6468 2459.6747 28.0172 215.1899 aVDZ 225.2129 aVDZ aVDZ 238.3973 239.0707 239.0493 239.7544 240.4324 255.1403 255.7984 256.4855 275.6568 276.3636 2100.3846 2162.4210 aVDZ aVDZ 2242.5326 aVDZ aVDZ 2289.5323 2290.1564 2290.1286 2290.7760 aVDZ 2341.4208 2342.0457 aVDZ 2398.2828 2398.9348 2460.3465 214.8945 262.0034 282.5447 283.2679 2107.3165 2467.2929 289.7887 aVDZ 2114.4962 2412.3902 aVDZ 299.9027 aVDZ



NA NA NA NA 224.6022 237.7890 254.5290 275.0025 299.6649 NA NA 2241.9345 2288.9392 2340.8302 2397.6721 2459.7023 NA 215.1991 aVDZ 225.2369 aVDZ aVDZ 238.4221 239.0908 239.0759 239.7765 240.4551 255.1604 255.8181 256.5015 275.6729 276.3732 2100.3907 NA aVDZ aVDZ 2242.5516 aVDZ aVDZ 2289.5563 2290.1837 2290.1515 2290.8017 aVDZ 2341.4465 2342.0748 aVDZ 2398.3110 2398.9636 2460.3729 NA 262.0224 282.5584 283.2727 2107.3192 2467.3168 289.7900 aVDZ 2114.5018 2412.4139 aVDZ 299.9139 aVDZ




159

TABLE II. ~Continued.! Molecule 1

BS ( S ) HBS ( 1 S 1 ) BCl ( 1 S 1 ) C2 ( 1 S 1 g ) C2H ( 2 S 1 ) C2H2 ( 1 1 g ) C2H3 ( 2 A 8 ) C2H4 ( 1 A 8g ) C2H5 ( 2 A 28 ) C2H6 ( 1 A 82 ) CN ( 2 S 1 ) HCN ( 1 S 1 ) H2CNH H3CNH2 CO ( 1 S 1 ) HCO ( 2 A 8 ) H2CO ( 1 A 1 ) H3COH CF ( 2 P) HCF ( 1 A 8 ) H3CF ( 1 A 1 ) CSi ( 3 P) CP ( 2 S 1 ) HCP ( 1 S 1 ) H2CPH ( 2 A 8 ) H3CPH2 CS ( 1 S 1 ) H2CS ( 1 A 1 ) H3CSH ( 1 A 8 ) CCl ( 2 P) CH3Cl ( 1 A 1 ) N2 ( 1 S 1 g ) N2H4 ( 1 A) NO ( 2 P) HNO ( 1 A 8 ) NF ( 3 S 2 ) NSi ( 2 S 1 ) NP ( 1 S 1 ) NS ( 2 P) O2 ( 3 S 2 g ) HO2 ( 1 A 9 ) H2O2 ( 1 A) HOF ( 1 A 8 ) NaO ( 2 P) HONa ( 1 S) MgO ( 1 S 1 ) MgOH ( 2 S) SiO ( 1 S 1 ) PO ( 2 P) HPO ( 1 A 8 ) SO ( 3 S 2 ) ClO ( 2 P) HOCl ( 1 A 8 ) F2 ( 1 S 1 g ) NaF ( 1 S 1 ) MgF ( 2 S 1 ) AlF ( 1 S 1 ) SiF ( 2 P) FSiH3 ( 1 A 1 ) PF ( 3 S 2 ) SF ( 2 P) ClF ( 1 S 1 ) Na2 ( 1 S 1 g ) NaCl ( 1 S 1 ) 2

E HF

E MP2

E MP4

E CCSD

E CCSD~T!

aVDZ aVDZ 2484.1628 275.4076 aVDZ 276.8570 aVDZ 278.0713 AVDZ 279.2678 292.2446 292.9187 aVDZ aVDZ 2112.7940 2113.3066 2113.9256 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2435.4634 aVDZ aVDZ aVDZ 2499.1569 2108.9976 2111.2386 2129.3136 2129.8552 aVDZ aVDZ 2395.1918 aVDZ 2149.6975 aVDZ aVDZ aVDZ aVDZ aVDZ 2274.3894 aVDZ 2363.8556 2415.6387 aVDZ 2472.4222 2534.3187 2534.9365 2198.7801 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2558.9216 2323.7168 2621.4603

aVDZ aVDZ 2484.4603 275.7892 aVDZ 277.1958 aVDZ 278.4379 aVDZ 279.6697 292.5568 293.2970 aVDZ aVDZ 2113.1893 2113.7155 2114.3646 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2435.7015 aVDZ aVDZ aVDZ 2499.5772 2109.4088 2111.7254 2129.7552 2130.3384 aVDZ aVDZ 2395.5747 aVDZ 2150.1807 aVDZ aVDZ aVDZ aVDZ aVDZ 2274.7520 aVDZ 2364.2319 2416.0366 aVDZ 2472.8456 2534.7615 2535.4353 2199.3714 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2559.4351 2323.7367 2621.6965

aVDZ aVDZ 2484.4976 275.8253 aVDZ 277.2199 aVDZ aVTZ aVDZ aVTZ 292.5833 293.3174 aVDZ aVDZ 2113.2098 2113.7383 2114.3902 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2435.7381 aVDZ aVDZ aVDZ 2499.6188 2109.4259 2111.7540 2129.7793 2130.3597 aVDZ aVDZ 2395.5996 aVDZ 2150.2012 aVDZ aVDZ aVDZ aVDZ aVDZ 2274.7788 aVDZ 2364.2544 2416.0633 aVDZ 2472.8755 2534.8019 2535.4647 2199.3934 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2559.4693 2323.7423 2621.7202

aVDZ aVDZ 2484.4834 275.7734 aVDZ 277.1997 aVDZ aVTZ aVDZ aVTZ 292.5809 293.2928 aVDZ aVDZ 2113.1848 2113.7146 2114.3662 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2435.7085 aVDZ aVDZ aVDZ 2499.5998 2109.3985 aVTZ 2129.7521 2130.3350 aVDZ aVDZ 2395.5633 aVDZ 2150.1753 aVDZ aVDZ aVDZ aVDZ aVDZ 2274.7230 aVDZ 2364.2252 216.0358 aVDZ 2472.8499 2534.7840 2535.4456 2199.3678 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2559.4449 2323.7441 2621.7092

aVDZ aVDZ 2484.4998 275.8091 aVDZ 277.2178 aVDZ 278.4709 aVDZ aVTZ 292.6012 293.3126 aVDZ aVDZ 2113.2036 2113.7338 2114.3862 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2435.7324 aVDZ aVDZ aVDZ 2499.6185 2109.4190 2111.7525 2129.7730 2130.3575 aVDZ aVDZ 2395.5889 aVDZ 2150.1959 aVDZ aVDZ aVDZ aVDZ aVDZ 2274.7482 aVDZ 2364.2453 2416.0577 aVDZ 2472.8726 2534.8056 2535.4680 2199.3896 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2559.4661 NA 2621.7204



160


TABLE II. ~Continued.! Molecule 1

MgS ( S ) MgCl ( 2 S 1 ) AlCl ( 1 S 1 ) Si2 ( 3 S 2 g ) Si2H6 SiS ( 1 S 1 ) SiCl ( 2 P) SiH3Cl ( 1 A 1 ) P2 ( 1 S 1 g ) P2H4 ( 1 A) PS ( 2 P) PCl ( 3 S 2 ) S2 ( 3 S 2 g ) S2H2 ( 1 A) ClS ( 2 P) Cl2 ( 1 S 1 g ) CO2 ( 1 S 1 g ) OCS ( 1 S) CS2 ( 1 S 1 ) O3 ( 1 A 1 ) SiO2 ( 1 S 1 g ) SO2 ( 1 A 1 ) ClO2 ( 1 B 1 ) N2O ( 1 S) Li1 ( 1 S) Be1 ( 2 S) B1 ( 1 S) C1 ( 2 P) N1 ( 3 P) O1 ( 3 P) F1 ( 2 P) Na1 ( 1 S) Mg1 ( 2 S) Al1 ( 1 S) Si1 ( 2 P) P1 ( 3 P) S1 ( 4 S) Cl1 ( 3 P) 2 CH1 4 ( B 1) 1 2 NH3 ( A 29 ) 1 NH1 4 ( A 1) 1 3 2 OH ( S ) H2O1 ( 2 B 1 ) H3O1 ( 1 A 1 ) HF1 ( 2 P) 2 SiH1 4 ( A 8) 1 1 SiH5 ( A 8 ) PH1 ( 2 P) 1 PH1 2 ( A 1) 2 ( A 1) PH1 3 1 PH1 4 ( A 1) SH1 ( 3 S 2 ) 2 SH1 2 ( B 1) 1 2 SH2 ( A 1 ) 1 SH1 3 ( A 1) 1 2 HCl ( P) H2Cl1 ( 1 A 1 ) 2 C2H1 2 ( P) C2H1 3 2 C2H1 4 ( B 3u ) 1 2 1 CO ( S ) CS1 ( 2 S 1 ) 2 1 N1 2 ( Sg ) 2 N1 2 ( P u) 1

E HF

E MP2

E MP4

E CCSD

aVDZ aVDZ aVDZ 2577.7856 aVDZ 2686.5163 aVDZ aVDZ 2681.5019 aVDZ aVDZ aVDZ 2795.1093 aVDZ aVDZ 2919.0077 2187.7302 aVDZ aVDZ 2224.3838 aVDZ 2547.3267 aVDZ 2183.7748 27.2364 214.2774 224.2378 237.2972 253.8945 274.3778 298.8418 2161.6767 2199.3717 2241.6746 2288.5786 2340.3558 2397.1743 2459.0579 aVDZ aVDZ 256.5699 275.0117 275.6667 276.3487 299.5482 aVDZ aVDZ 2340.9474 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2399.0018 aVDZ 2460.3322 aVDZ aVDZ aVDZ 2112.3167 aVDZ aVTZ aVDZ

aVDZ aVDZ aVDZ 2577.9610 aVDZ 2686.8112 aVDZ aVDZ 2681.8125 aVDZ aVDZ aVDZ 2795.4649 aVDZ aVDZ 2919.4496 2188.4011 aVDZ aVDZ 2225.2312 aVDZ 2548.0650 aVDZ 2184.4826 NA NA 224.2787 237.3544 253.9755 274.4895 299.0095 NA NA 2241.7044 2288.6240 2340.4243 2397.2735 2459.1991 aVDZ aVDZ 256.8209 275.1754 275.8896 276.6344 299.7827 aVDZ aVDZ 2341.0468 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2399.2101 aVDZ 2460.5630 aVDZ aVDZ aVDZ 2112.6632 aVDZ aVTZ aVDZ

aVDZ aVDZ aVDZ 2577.9950 aVDZ 2686.8459 aVDZ aVDZ 2681.8458 aVDZ aVDZ aVDZ 2795.5076 aVDZ aVDZ 2919.4977 2188.4229 aVDZ aVDZ 2225.2569 aVDZ 2548.0987 aVDZ 2184.5082 NA NA 224.2962 237.3737 253.9930 274.5033 295.0244 NA NA 2241.7131 2288.6377 2340.4421 2397.2929 2459.2244 aVDZ aVDZ 256.8391 275.1928 275.9063 276.6469 299.7972 aVDZ aVDZ 2341.0697 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2460.5911 aVDZ aVDZ aVDZ 2112.6836 aVDZ aVTZ aVDZ

aVDZ aVDZ aVDZ 2577.9854 aVDZ 2686.8243 aVDZ aVDZ 2681.8190 aVDZ aVDZ aVDZ 2795.4833 aVDZ aVDZ 2919.4743 2188.3805 aVDZ aVDZ 2225.1863 aVDZ 2548.0445 aVDZ 2184.4487 NA NA 224.3013 237.3758 253.9940 274.5020 299.0224 NA NA 2241.7151 2288.6389 2340.4414 2397.2892 2459.2202 aVDZ aVDZ 256.8315 275.1904 275.9013 276.6382 299.7926 aVDZ aVDZ 2341.0693 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVTZ aVDZ aVDZ aVDZ 2112.6704 aVDZ aVTZ aVDZ

E CCSD~T! aVDZ aVDZ aVDZ 2577.9981 aVDZ 2686.8449 aVDZ aVDZ 2681.8434 aVDZ aVDZ aVDZ 2795.5069 aVDZ aVDZ 2919.4974 2188.4123 aVDZ aVDZ 2225.2367 aVDZ 2548.0833 aVDZ 2184.4877 NA NA NA 237.3771 253.9959 274.5041 299.0251 NA NA NA 2288.6402 2340.4441 2397.2937 2459.2258 aVDZ aVDZ 256.8390 275.1942 275.9066 276.6463 299.7977 aVDZ aVDZ 2341.0730 aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ 2460.5919 aVDZ aVDZ aVDZ 2112.6883 aVDZ aVTZ aVDZ




161

TABLE II. ~Continued.! Molecule O1 2

( P g) FCl1 ( 2 P) 2 P1 2 ( P u) 2 S1 2 ( P g) 2 ( P g) Cl1 2 C2 ( 4 S) C2 ( 2 P) F21 ( 1 S) Si2 ( 4 S) P2 ( 3 P) S2 ( 2 P) Cl2 ( 1 S) CH2 ( 3 S 2 ) 2 CH2 2 ( B 1) 1 CH2 ( A 1) 3 2 2 NH ( P) 1 NH2 2 ( A 1) OH2 ( 1 S 1 ) SiH2 ( 3 S 2 ) 2 SiH2 2 ( B 1) 2 1 SiH3 ( A 1 ) PH2 ( 2 P) 1 PH2 2 ( A 1) SH2 ( 1 S 1 ) CN2 ( 1 S 1 ) NO2 ( 3 S 2 ) 2 O2 2 ( P g) 2 3 2 PO ( S ) 2 S2 2 ( P g) 2 1 Cl2 2 ( Su )

a

2

E HF aVDZ aVDZ aVDZ aVDZ aVDZ 237.7111 274.7984 299.4602 2288.8903 2340.7077 2397.5462 2459.5775 238.3004 238.9041 239.5266 254.9336 255.5535 275.4195 2289.4706 2290.0571 2290.6526 2341.2963 2341.8949 2398.1501 292.3509 2129.2976 2149.6622 2415.6655 2759.1387 2919.0890

E MP2 aVDZ aVDZ aVDZ aVDZ aVDZ 237.8168 275.0424 299.7903 2288.9731 2340.8328 2397.7234 2459.8152 238.4441 239.0912 239.7587 255.1518 255.8348 275.7371 2289.5779 2290.1948 2290.8208 2341.4562 2342.0926 2398.3725 292.7315 2129.7513 2150.1939 2416.0754 2759.5263 2919.5398

E MP4 aVDZ aVDZ aVDZ aVDZ aVDZ 237.8332 275.0569 299.7981 2288.9900 2340.8551 2397.7482 2459.8388 238.4635 239.1116 239.7783 255.1697 255.8499 275.7469 2289.5997 2290.2201 2290.8491 2341.4815 2342.1173 2398.3975 292.7510 2129.7753 2150.2161 2416.1028 2759.5694 2919.5888

E CCSD aVDZ aVDZ aVDZ aVDZ aVDZ 237.8293 275.0458 299.7799 2288.9854 2340.8489 2397.7394 2459.8270 238.4585 239.1039 239.7669 255.1619 255.8319 275.7266 2289.6053 2290.2160 2290.8416 2341.4741 2342.1100 2398.3855 292.7238 2129.7515 2150.1862 2416.0764 2759.5434 2919.5652

E CCSD~T! aVDZ aVDZ aVDZ aVDZ aVDZ 237.8346 275.0556 299.7913 2288.9908 2340.8565 2397.7490 2459.8383 238.4654 239.1127 239.7775 255.1721 255.8451 275.7402 2289.6023 2290.2227 2290.8485 2341.4831 2342.1182 2398.3974 292.7453 2129.7733 2151.2101 2416.0982 2759.5687 2919.5887

Energies ~in hartrees! were obtained from an exponential extrapolation of the form E(x)5E ` 1be 2cx . For group IA and IIA elements, where no augmented basis sets were available, the energies were obtained from cc-pVDZ through cc-pVQZ sets. All open-shell calculations used unrestricted Hartree–Fock wave functions. Atomic calculations did not impose symmetry equivalence restrictions. NA5not applicable. aVDZ5the highest level calculations used the aug-cc-pVDZ basis. aVTZ5the highest level calculations used the aug-cc-pVTZ basis.

averaged over all correlated methods. Among the correlated methods, the error in the extrapolated MP2 energies was twice as large as the error for the other correlated methods. CCSD~T! showed the smallest e MAD at 0.9mE h . The maximum observed errors were 1.2mE h ~CO, Hartree–Fock! and 7.9mE h ~S atom, MP2!. At the Hartree–Fock level, the CBS(aDTQ/e 2x ) procedure slightly overestimates the true CBS limit in most cases. For correlated methods this behavior is partially offset by the tendency of the fit to underestimate the correlation contribution to the total energy. However, since the latter effect is usually larger in magnitude, the CBS(aDTQ/e 2x ) energy estimates for correlated methods almost always underestimate results obtained from still larger basis sets. Since total energies are typically of less interest to chemists than energy differences, we also examined the effectiveness of the CBS(aDTQ/e 2x ) extrapolation in reproducing the latter. For the 35 systems described previously, e MAD with respect to experimental atomization energies dropped from 2.21 kcal/mol at the CCSD~T!/aug-cc-pVQZ level to 1.23 kcal/mol at the CCSD~T!aug-ccp-pV5Z level. The corresponding CBS(aDTQ/e 2x ) value of e MAD was 0.71 kcal/ mol, in good agreement with the CBS(aTQ5/e 2x ) value of

0.88 kcal/mol, suggesting that the extrapolation based on double through quadruple zeta basis sets recovers the majority of the remaining basis set truncation error.

IV. ATOMIZATION ENERGIES

In the G1 and G2 procedures, atomization energies are compared directly with experimental values that include the zero-point vibrational contributions, i.e., SD 0 , that were extrapolated to 0 K. The original G1 and G2 methods relied on scaled RHF/6-31G* frequencies in order to compute the zero-point energies ~ZPEs!. In a subsequent paper, Curtiss et al.42 examined the use of scaled MP2 frequencies, but found ‘‘no overall improvement.’’ They attributed this to the fact that both RHF and MP2 frequencies were scaled to reproduce experimental fundamental frequencies. Bauschlicher and Partridge5 suggested replacing the MP2 geometry optimization and self-consistent field ~SCF! frequencies with nonlocal density functional calculations. The density functional frequencies were scaled by 0.989, but instead of trying to reproduce fundamental frequencies, the scale factor was chosen to improve the agreement between the theoretical



162


TABLE III. Estimated complete basis set, frozen core energies obtained from an exponential fit of aug-cc-pVTZ through aug-cc-pV5Z energies.a Molecule 2

H ( S) H2 ( 1 S 1 g ) Li ( 2 S) Be ( 1 S) B ( 2 P) C ( 3 P) N ( 4 S) O ( 3 P) F ( 2 P) Na ( 2 S) Mg ( 1 S) Al ( 2 P) Si ( 3 P) P ( 4 S) S ( 3 P) Cl ( 2 P) LiH ( 1 S 1 ) H2O ( 1 A 1 ) HF ( 1 S 1 ) b HCl ( 1 S 1 ) Li2 ( 1 S 1 g ) CO ( 1 S 1 ) c N2 ~1S1 g! 2 2 O2 ( S g ) F2 ( 1 S 1 g ) ClF ( 1 S 1 ) SiO ( 1 S 1 ) SiS ( 1 S 1 ) C2 ( 4 S) O2 ( 2 P) F2 ( 1 S) Si2 ( 4 S) P2 ( 3 P) S2 ( 2 P) Cl2 ( 1 S) SH2 ( 1 S 1 )

E HF

E MP2

E MP4

E CCSD

E CCSD~T!

20.5000 21.1337 27.4327 214.5730 224.5332 237.6941 254.4047 274.8191 299.4169 2161.8587 2199.6151 2241.8807 2288.8588 2340.7195 2397.5135 2459.4900 27.9874 276.0685 2100.0718 2460.1130 214.8719 2112.7928 2108.9970 2149.6963 2198.7811 2558.9225 2363.8583 2686.5171 237.7106 274.7981 299.4599 2288.8904 2340.7078 2397.5459 2459.5770 2398.1500

NA 21.1675 NA 214.6032 224.5821 37.7693 254.5140 274.9874 299.6545 NA 2199.6398 2241.9205 2288.9215 2340.8136 2397.6495 2459.6779 28.0175 276.3650 2100.3870 2460.3509 214.8945 2113.1902 2109.4100 2150.1836 2199.3791 2559.4429 2364.2376 2686.8162 237.8172 275.0444 299.7927 2288.9743 2340.8352 2397.7258 2459.8169 2398.3758

NA 21.1738 NA 214.6148 224.5979 237.7862 254.5284 275.0025 299.6661 NA 2199.6469 2241.9312 2288.9374 2340.8306 2397.6674 2459.7024 28.0238 276.3750 2100.3932 2460.3745 214.9014 2113.2098 2109.4264 2150.2028 2199.3978 2559.4729 2364.2575 2686.8481 237.8334 275.0576 299.7984 2288.9912 2340.8562 2397.7486 2459.8377 2398.3986

NA 21.1744 NA 214.6191 224.6003 237.7865 254.5264 274.9994 299.6610 NA 2199.6489 2241.9323 2288.9364 2340.8265 2397.6674 2459.6956 28.0246 276.3644 2100.3825 2460.3646 214.9038 2113.1840 2109.3982 2150.1757 2199.3710 2559.4480 2364.2285 2686.8266 237.8294 275.0460 299.7796 2288.9863 2340.8500 2397.7399 2459.8256 2398.3865

NA NA NA NA 224.6018 237.7890 254.5295 275.0036 299.6664 NA NA 2241.9336 2288.9394 2340.8314 2397.6721 459.7037 NA 276.3734 2100.3912 2460.3750 NA 2113.2034 2109.4193 2150.1961 2199.3936 2559.4699 2364.2489 2686.8476 237.8348 275.0562 299.7916 2288.9918 2340.8577 2397.7498 2459.8375 2398.3988

Energies ~in hartrees! were obtained from an exponential extrapolation of the form E(x)5E ` 1be 2cx . For group IA and IIA elements, where no augmented basis sets were available, the energies were obtained from cc-pVTZ through cc-pV5Z sets. All open-shell calculations used unrestricted Hartree–Fock wave functions. Atomic calculations did not impose symmetry equivalence restrictions. b Calculations with the cc-pV6Z basis set, Ref. 39, yielded ~nonextrapolated! energies of: 2100.0712 ~HF!, 2100.3837 ~MP2!, 2100.3910 ~MP4!, 2100.3805 ~CCSD!, 2100.3891 @CCSD~T!#. c Calculations with the cc-pV6Z basis set, Ref. 39, yielded ~nonextrapolated! energies of 2108.9965 ~HF!, 2109.4072 ~MP2!, 2109.3924 ~MP4!, 2109.3968 ~CCSD!, 2109.4178 @CCSD~T!#. a

ZPEs and experimental values for 41 molecules. As pointed out by Grev et al.,43 the experimental zero-point energy is not given by 1/2S n 1 . In order to differentiate ZPE errors from errors caused by basis set truncation or limitations in the degree of correlation recovery, we have chosen to compare our results with SD e , i.e., atomization energies from which zero-point energies have been removed. Whenever possible, experimental ZPEs were used to derive SD e from SD 0 . For diatomics, the ZPE was computed as 1/2v e 21/4v e x e , where v e and v e x e were taken from Huber and Herberg.44 A limited number of experimental ZPEs were available for polyatomic species, as well.45–50 For molecules lacking experimental ZPEs, we used 1/2S v i , where the harmonic frequencies were taken from CCSD~T! calculations performed with the largest

basis set indicated in the Appendix. The Appendix lists all of the zero-point energies determined in this study, along with the corresponding experimental values, where available. Agreement between the G2 ZPEs and the best values is generally good, although differences of more than 0.5 kcal/mol are common, especially for the larger systems where differences can approach 2 kcal/mol. Clearly, in order for G2 to provide a mean absolute deviation for atomization energies on the order of 1 kcal/mol, the empirically determined ‘‘higher-order correction’’ must be accounting for some of the errors in the RHF/6-31G* zero-point energies. An additional correction was applied to the experimental atomization energies in order to take into account the lack of atomic spin–orbit effects in the results reported here. Our atomic energies correspond to an average over the possible




163

FIG. 1. Atomization energy mean absolute deviations ~kcal/mol! and standard deviations with respect to experimentally derived values for 66 cases comprised of the 55 entries in Table III of the paper by Curtiss et al. ~Ref. 2!, plus ~1! O3 from Table IX with a listed uncertainty of 60.5 kcal/mol; ~2! SiH from Table VIII with a listed uncertainty of 60.6 kcal/mol; ~3! C2 from Table IX with a listed uncertainty of 60.9 kcal; ~4! CH3NH2 from Table IX with a listed uncertainty of 60.1 kcal/mol; ~5! SH with an uncertainty of 60.7 kcal/mol; ~6! AlH with an uncertainty of 60.2 kcal/mol; ~7! PO from Table VIII with a listed uncertainty of 60.9 kcal/mol; ~8! SiS to which Huber and Herzberg assign an uncertainty of 60.5 kcal/mol; ~9! HNO; ~10! H2, and ~11! N2O. All results were obtained from frozen core calculations performed at the respective optimized geometries using the aug-cc-pVDZ, aug-cc-pVTZ, and aug-ccpVQZ basis sets. The complete basis set ~CBS! results were obtained from an exponential extrapolation of the total energies. A negative value for a maximum error is associated with an underestimation of the binding energy.

spin multiplets. In the case of a diatomic like HF, the ‘‘adjusted’’ experimental dissociation energy is given by 2 3 2 2 1 5D expt D adjusted e e 11/3@ E( P 2 )2E( P 2 ) # . For the P states of molecules like CH and OH there is an additional molecular spin–orbit correction due to the splitting of the 2 P 21 and 2 P 32 states. Corrections were taken from the atomic and molecular values given by Dunning and co-workers,25,26 which were based on the experimental values of Herzberg51 and Moore.52 In Fig. 1 the e MAD’s and standard deviations ~s’s! are displayed for a collection of 66 atomization energies as a function of basis set size and level of theory. The 66 cases are comprised of the 55 G2 atomization energies for which there was accurate experimental data ~see Table III of Curtiss et al.2! and 11 additional cases for which the experimental data was equally good or only slightly poorer in quality. The reader should note that the vertical scales differ among the five plots. Due to their high cost, aug-cc-pVQZ and CBS results were obtained for only 59 of the 66 cases. While this limitation will affect the precise numeric values of e MAD and s, the G2 and aug-cc-pVTZ data suggests that the reduced number of atomization energies should yield statistics within 63% of the complete set of 66.

We encountered UHF instabilities when optimizing the P state of PO. As a result, optimal restricted open-shell geometries were used when evaluating the UHF energies. Several qualitative features are evident. First, the Hartree–Fock results are quite poor, with maximum errors as large as 2199 kcal/mol. All deviations are negative, where a negative sign indicates underestimation of the binding energy. Increasing the basis set beyond the double zeta level provides some improvement, but even at the CBS limit e MAD is still on the order of 60 kcal/mol. The second obvious feature of Fig. 1 is that a small amount of correlation recovery ~via second-order perturbation theory! has a profound affect on improving the level of agreement with experiment. Over the range of the three basis sets we have considered, e MAD has been reduced relative to the HF values by somewhere between a factor of 5 and a factor of 10. However, at the double zeta level the maximum error ~285 kcal/mol for the dissociation of C2H6! is still very large. As the basis set increases in size, the maximum negative error monotonically decreases while the maximum positive error monotonically increases. These observations suggest that MP2 may be a relatively economical method that is 2



164


capable of providing moderately accurate atomization energies for a specific class of molecules, but it is essential that the class be thoroughly benchmarked because of the widely varying performance. MP4, which scales in computational costs as N7, displays the same qualitative trends as MP2, but e MAD for the triple and quadruple zeta basis sets is ;70% smaller than the MP2 values and the maximum errors also tend to be smaller. CCSD was found to be noticeably less accurate than MP2, even though it is computationally more expensive. Unlike the perturbation theory expansions, CCSD maximum errors proved to be uniformly negative and systematically decreased as the basis set was improved. However, the intrinsic error for this method is larger than the intrinsic error for MP4 and the large size of the maximum errors make it less desirable on average. CCSD~T! is the highest level correlated treatment used in the present study. As expected, the corresponding intrinsic error, as measured by the CBS mean absolute deviation, is the smallest of the five methods tested. Using the CBS(aDTQ/e 2x ) estimate of the CBS limit reduces e MAD from 2.7 kcal/mol at the aug-cc-pVQZ level to 1.3 kcal/mol. The latter value comes very close to meeting the commonly used definition of ‘‘chemical accuracy’’ (61 kcal/mol). The extrapolation based on 1/(l max1 21)4 yields a smaller mean absolute deviation of 1.0 kcal/mol. Although agreement with experiment is better for the l max fit, it is not possible to conclude that this approach provides a better estimate of the CBS limit because ~1! core/valence effects have yet to be included, ~2! the intrinsic error of the CCSD~T! method is nonzero, and ~3! the energy differences under discussion are approaching the limit of the experimental accuracy for this set of molecules. Further comparisons with CCSD~T!-R12 atomization energies for a large body of molecules will be necessary in order to quantitatively assess the accuracy of the various CBS estimates. For the sake of comparison, the G2 value of e MAD ~1.4 kcal/mol! is shown in Fig. 1 as a horizontal line in the CCSD~T! plot. The G2 maximum errors are not shown in Fig. 1 in order to prevent the figure from becoming overly cluttered. They were on the order of 65 kcal/mol, somewhat larger than the CCSD~T! values. The CCSD~T! mean absolute deviation, standard deviation, and maximum errors for this method all exhibit wellbehaved, monotonic convergence as a function of the basis set size. It is interesting to note that at the double zeta level a fortuitous cancellation of error results in the MP2 mean absolute deviation being 3 kcal/mol less than the CCSD~T! value, with nearly identical maximum errors. In other studies, the internally contracted, complete active space configuration interaction ~iCAS-CI! method of Werner and Knowles53 has produced comparable or slightly smaller errors with respect to experiment. However, the N! increase in the number of reference space configurations makes this method extremely difficult to apply to molecules such as C2H6 without resorting to a selection technique to help reduce the size of the reference space.

In order to further decrease e MAD , it is necessary to include core/valence effects. Although core/valence effects are typically small relative to the magnitude of the total atomization energy, with CCSD~T!/CBS mean absolute deviations on the order of 1 kcal/mol, this correction becomes important. For example, the core/valence correction to the binding energy of N229,54 and CO ~Ref. 20! are on the order of 1 kcal/mol. Martin37,55 has reported a core/valence correction on the order of 1.9–2.4 kcal/mol for the atomization of C2H2. In order to gauge the impact of core/valence effects on atomization energies we have applied the approximate procedure discussed earlier to 31 of the systems for which CBS estimates were available. We have chosen to only apply core/ valence corrections at the CCSD~T! level of theory because these corrections usually increase binding energies and frozen core perturbation theory frequently overestimates the experimental binding energy. Thus the addition of core/valence corrections would actually worsen the level of agreement with experiment for second- and fourth-order perturbation theory results. We find that when core/valence effects are added to the CBS~aDTQ/e 2x ) SD e ’s, the mean absolute deviation drops from 1.32 to 0.89 kcal/mol and the deviation for the l max CBS values falls to 0.70 kcal/mol, with a significant number of the latter now slightly overestimating the experimental numbers. If it were feasible to determine accurate core/ valence contributions near the CBS limit for all of the systems in this study, it seems likely that the two approaches for estimating the CBS atomization energies would fall within 0.3 kcal/mol of each other. Although the magnitude of core/valence corrections for electron affinities, proton affinities, and ionization potentials is usually smaller than their effect on atomization energies, the inclusion of this correction has been shown to also improve CCSD~T! predictions of these properties.30,36,38 G2 theory lacks an explicit treatment of core/valence effects. However, the magnitude of the core/valence effect is such that the so-called ‘‘higher level correction’’ must partially compensate for this omission. Agreement between G2 atomization energies and estimated complete basis set CCSD~T! values are reasonably good, especially for the 55 cases used in establishing the empirical higher-order correction. If the sample set is expanded to include other G2 molecules for which the experimental uncertainty is larger, the differences in the mean absolute deviations increase slightly. Xantheas et al.56 have very recently reported restricted open-shell CCSD~T! @RCCSD~T!# and multireference Cl calculations on the dissociation energy of NF (X 3 S 2 ), one of the G2 molecules for which the experimental uncertainty is large.44,57–60 Their CBS limits, based on calculations up through cc-pV6Z or aug-cc-pV5Z are 76.3 and 76.5 kcal/ mol, respectively, with the RCCSD~T! method. The Cl values, corrected for the effects of unlinked clusters,6 were 75.7 and 75.8 kcal/mol with the same two basis set sequences. Core/valence correlation had minimal effect on the computed binding energy. Our present unrestricted findings are in close agreement with the restricted open-shell results, and can be




165

FIG. 2. Adiabatic electron affinity mean absolute deviations ~kcal/mol! and standard deviations with respect to experimentally derived values for the 25 cases in the G2 set. All results were obtained from frozen core calculations performed at the respective optimized geometries. The complete basis set ~CBS! results were obtained from an exponential extrapolation of the total energies, as described in the text.

compared to the G2 D e value of 77.7 kcal/mol. Given the high level of agreement between a large body of CCSD~T! and reliable experimental dissociation energies, it seems likely that the true D e binding energy is near 76 kcal/mol with an uncertainty of 61 kcal/mol. Another G2 diatomic with somewhat larger experimental uncertainty is LiCl (X 1 S 1 ). The JANAF tables59 list a value of D 0 5113.063 kcal/mol (D e 5113.9), whereas the older Huber and Herzberg44 value is D 0 5111.6 kcal/mol (D e 5112.5). Our frozen core CCSD~T!/CBS value is D e 5114.1 kcal/mol, slightly larger than the G2 value of 113.0 kcal/mol. Core/valence calculations with the augmented triple zeta weighted core/valence sets11 yielded corrections of 0.9 kcal/mol to the binding energy, for a best value of D e 5115.0 kcal/mol. V. ELECTRON AFFINITIES

Electron affinities ~EAs! have a long-established reputation for being very difficult to compute with high accuracy. Previous studies9,62–66 have served to establish the importance of using large, diffuse function augmented basis sets and extensive correlation recovery when calculating this property. Compared to other widely reported energies, e.g., D e ’s or ionization potentials ~IPs!, the notoriety attached to electron affinities is likely due to the relatively small size of the latter, which are seldom larger than 50 kcal/mol, while

many D e ’s exceed 100 kcal/mol. Consequently, errors of the same magnitude in electron affinities and dissociation energies are viewed quite differently. As we will see in the following discussion, errors in electron affinities are actually on the order of a third the size of the corresponding errors in atomization energies. Plots of the basis set/level of theory convergence, corresponding to the SD e plots in Fig. 1, are shown in Fig. 2 for the 25 adiabatic electron affinities contained in the G2 set. The CCSD~T! data for the methyl radical (CH3•) was taken from the work of Dixon et al.30 HF mean absolute deviations show a slight increase as one approaches the complete basis set limit, whereas the e MAD’s decreased for atomization energies. At 40–50 kcal/ mol, the maximum errors are significantly larger than the average value of the property being computed ~33.9 kcal/ mol!, as was also the case for SD e . Trends in the EA maximum positive and negative errors at the MP2 level show a strong similarity to the corresponding plot in Fig. 1. Once again, MP2 displays a pronounced tendency to overestimate the property in question as the basis set approaches completeness. The MP2 mean absolute error reaches a minimum for the aug-cc-pVTZ basis set and then increases slightly for larger sets. Cancellation of errors at the MP2/aug-cc-pVDZ level of theory results in a value of e MAD that is as good, or better than, errors obtained from more sophisticated correlation treatments. While the maximum positive and negative



166


FIG. 3. Adiabatic ionization potential mean absolute deviations ~kcal/mol! and standard deviations with respect to experimentally derived values for the 38 cases in the G2 set. All results were obtained from frozen core calculations performed at the respective optimized geometries.

errors for MP4 atomization energies were similar in size at the CBS limit, that is not the case for electron affinities, where large overestimations were observed. Unlike the perturbation theory approaches, both of the coupled cluster methods exhibited strong, monotonic convergence to the CBS limit. In general, CCSD and CCSD~T! tended to underestimate the experimental electron affinities, with the exception of Cl2. The intrinsic errors, as measured by e MAD , are 28.2 ~HF!, 3.5 ~MP2!, 2.5 ~CCSD!, and 0.8 @CCSD~T!# kcal/mol. The CBS extrapolation based on 1/(l max1 21)4 behavior yields a CCSD~T! mean absolute deviation of 0.7 kcal/mol. Because the G2 ‘‘higher level correction’’ was not parametrized with regard to electron affinities, it is perhaps not surprising that the G2 mean absolute deviation for the 25 EAs ~1.4 kcal/mol! is roughly twice the size of the CCSD~T! complete basis set limit, whereas for atomization energies the two were within 17% of each other. The associated G2 maximum errors are 22.9 kcal/mol (CH2) and 3.3 kcal/mol (SH2). Finally, comparing e MAD convergence patterns for EA and SD e across the four correlated methods, one finds that, method-by-method, the mean absolute deviations for electron affinities are less than half the size of the errors for atomization energies. Similar findings, suggesting that SD e is the most challenging of the four energetic quantities examined in this work, will be presented for ionization potentials and proton affinities. Because we have chosen to focus

solely on the aug-cc-pVxZ family of basis set sets, it may not be apparent that while diffuse functions have little impact on most dissociation energies, their presence can easily cause a 40–50 kcal/mol shift in EA. In this respect, electron affinities live up to their reputation for difficulty. It should also be noted that atomization energies refer to complete separation of the molecule into its constituent atoms, a process that substantially reduces the number of correlating electron pairs in the atomic asymptotes, whereas the EA involves the addition of just a single electron to the system. VI. IONIZATION POTENTIALS

The G2 set contains 38 adiabatic ionization potentials (IPe ’s). Despite perceptions that ionization potentials are much easier to compute than electron affinities, the IPe convergence patterns shown in Fig. 3 are very similar to the EAe results in Fig. 2. The largest differences occur at the HF level, where the ionization potential e MAD’s are 20%–50% smaller than the electron affinity values. Perturbation theory does a better job of describing the X→X 1 energy difference than the corresponding X→X 2 electron affinity energy difference. Although the mean absolute deviations are of the same order of magnitude, the maximum IPe errors are significantly smaller. As was the case for atomization energies and electron affinities, second-order perturbation theory appears to be the most cost effective of the four correlated methods examined




167

FIG. 4. Adiabatic proton affinity mean absolute deviations ~kcal/mol! and standard deviations with respect to experimentally derived values for the seven cases in the G2 set. All results were obtained from frozen core calculations performed at the respective optimized geometries.

in this study for use with the aug-cc-pVDZ basis set. For the previously discussed properties, CCSD exhibited larger e MAD values than both second-order and fourth-order perturbation theory. In this case, the errors were comparable. The G2 mean absolute deviation for the 38 IPe ’s was 1.6 kcal/ mol. VII. PROTON AFFINITIES

The G2 set contains only seven proton affinities (PA0). The proton affinity is defined as the negative of the enthalpy change for the reaction X1H1→XH1. However, as with the three properties already discussed, we will limit ourselves to a discussion of the properties with the vibrational contribution removed, PAe . It is evident from Fig. 4, that proton affinities are considerably easier to compute than atomization energies, electron affinities, or ionization potentials. On average, even Hartree–Fock theory is capable of yielding PAe ’s within 64 kcal/mol. All of the correlated methods produce values that are within several kcal/mol of each other. This corresponds to a percentage variation of 61%. The G2 and CCSD~T! mean absolute deviations were both 1 kcal/ mol. The MP2, MP4, CCSD, and CCSD~T! proton affinity data for NH3 and H2O were taken from the work of Peterson et al.,36 who also examined the effects of core/valence correlation for this property and found them to be small~;0.1

kcal/mol!. The same authors suggested that the currently accepted experimental value of PA298 0 for water is too high by 1.5 kcal/mol. VIII. GEOMETRIES

Statistics were also collected on the performance of the five levels of theory in predicting molecular structures. There are numerous reports in the literature focusing on particular classes of molecules. A good summary of the typical accuracy to be expected from low-level methods ~HF, MP2! used in conjunction with small-to-medium sized basis sets is presented by Hehre et al.67 They report mean absolute deviations of 0.014 Å for AH bonds and 0.029 Å for AB bonds at the RHF/3-21G level of theory, although the exact number of comparisons are not given. At the MP2/6-31G* level these values fall slightly to 0.014 Å~AH! and 0.023 Å~AB!. Comparable results for the 184 molecules in the G2 set for which experimental structures were available are shown in Fig. 5. In general, HF errors in bond lengths and bond angles actually worsen slightly with increasing basis set size. Although correlated methods do not display the same undesirable trend, the maximum errors can still be quite large. Some of this error may be attributed to the uncertainties associated with defining a bond length based on the experimental techniques of x-ray electron, or neutron diffraction and microwave spectroscopy. The value predicted by our calculations corresponds to r e , the bottom of the potential well.



168


FIG. 5. Bond length and bond angle mean absolute deviations ~Å and degrees! and standard deviations with respect to experiment for the 184 molecules in the G2 set. All results were obtained from frozen core calculations.

Unfortunately, r e is a difficult quantity to obtain experimentally for polyatomic molecules. For a discussion of the implications for comparing experimental and theoretical structural parameters, the reader is referred to the work of Burkert and Allinger.68 These authors conclude that the level of inherent uncertainty amounts to at least several thousandths of an angstrom for most organic compounds. The CCSD~T! mean absolute deviations obtained with the aug-cc-pVQZ basis set were 0.009 Å~AH!, 0.007 Å~AB!, and 0.4°~HAH!. There were insufficient ABC bond angles for obtaining meaningful statistics. Although we have not chosen to extrapolate the individual bond lengths and bond angles to the CBS limit, in many cases the exponential functional form provides a reasonable fit to the data and can be used for obtaining an estimate of the basis set truncation error. Many diatomic examples are already available in the literature on the correlation consistent basis sets. Although CBS estimates of bond lengths based on an exponential fit are often useful in gauging the approximate degree of convergence in the computed value, they tend not to be as accurate as the energy estimates. For example, the CCSD~T!/aug-cc-pVQZ bond length of CO is 1.1318 Å.20 Extrapolating to the CBS limit gives r e 51.1293 Å, only 0.0010 Å longer than the experimental value of 1.1283 Å. However, much larger basis set calculations ~using the cc-pV6Z set! place the true CBS limit

nearer to 1.1305 Å. The CBS~aDTQ/e 2x ) extrapolation was, therefore, successful in reducing the aug-cc-pVQZ error ~0.0013 Å! by almost half. However, other corrections are often as large or larger than the basis set truncation error in r e at the quadruple zeta level. Peterson and Dunning20 examined the core/valence correlation contribution to the bond length of CO and found it to be 20.0026 Å, or nearly twice the size of the frozen core quadruple zeta basis set truncation error.

IX. VIBRATIONAL FREQUENCIES

A comparison of theoretical harmonic frequencies, v e , and known experimental frequencies, v e ~when available! or v fund , was also performed. In the original G2 procedure a scale factor of 0.8929 was applied to the HF/6-31G* frequencies. We have chosen to use a slightly cruder scale factor of 0.9 for our HF frequencies. The findings for three types of vibrations ~bends, stretches, and general deformations! are shown in Fig. 6. At the HF level of theory mean absolute deviations are in the 60– 70 cm21 range for bends, near 120 cm21 for stretches, and in the 60– 80 cm21 range for deformations. The aug-cc-pVQZ data points in Fig. 6 for bends are probably not statistically significant because they are based on only seven comparisons.




169

FIG. 6. Normal-mode frequency mean absolute deviations (cm21) and standard deviations with respect to experiment for the molecules in the G2 set.

Unscaled MP2 shows poorer agreement with experiment than scaled HF theory. For example, for stretches the value of e MAD increases by 55 cm21 ~or 47%! at the double zeta level. MP4~SDTQ! and CCSD both correct the MP2 tendency to overestimate v e , and CCSD~T! provides additional small improvements. With the quadruple zeta basis set the mean absolute deviations for stretching modes were 119 cm21 ~HF!, 123 cm21 ~MP2!, 111 cm21 ~MP4!, 67 cm21 ~CCSD!, and 43 cm21 ~CCSD~T!!. A large number of studies15,19,20,25 on diatomics have shown that, in the complete basis set limit, frozen core CCSD~T! theory is capable of reproducing v e ’s to approximately 610 cm21. Indeed, if the number of comparisons is limited to well-characterized diatomics and the few polyatomics for which experimentally derived harmonic frequencies are available, « MAD drops to 11.8 cm21. Similar studies23,26,69 using the iCAS-Cl method report comparable levels of agreement. As with the structural parameters, no attempt was made to extrapolate the DZ, TZ, and QZ frequencies to the CBS limit. In previous studies, the basis set truncation error in v e at the aug-cc-pVQZ level has generally been small. For example, the CCSD~T!/aug-cc-pVQZ value of v e for CO is 2160.1 cm21. The CBS limit, based on an extrapolation using up through sextuple zeta calculations, is 2166 cm21. For this same molecule, Peterson and Dunning20 have reported a core/valence correction of 110 cm21, which causes the

computed value to slightly overshoot experiment. X. SUMMARY AND CONCLUSIONS

Hartree–Fock and correlated calculations were performed on the 220 G2 chemical systems and N2O with the diffuse function augmented correlation consistent basis sets. Extrapolations to the complete basis set limit were then used to estimate the intrinsic errors associated with a variety of energetic properties, including SD e , EAe , IPe , and PAe . Structural and vibrational properties were also examined. The largest correlated calculations in the present study involved more than 430 basis functions. On the basis of chemical systems for which accurate experimental information is available, atomization energies (SD e ) were the most difficult property for theory to reproduce. Within the frozen core approximation, the highest level of theory ~CCSD~T!! yielded mean absolute deviations of 19.2 ~aVDZ!, 6.2 ~aVTZ!, 2.8 ~aVQZ! and 1.6 ~aV5Z! kcal/ mol. At the complete basis set limit the error falls in the range of 1.1–1.3 kcal/mol, depending upon the formula used to generate the estimate. Inclusion of core/valence correlation with the cc-pCVTZ and cc-pwCVTZ basis sets reduced the error by 0.4 kcal/mol, to 0.7–0.9 kcal/mol. In the complete basis set limit, the core/valence correction is likely to be somewhat larger



170


~;0.5 kcal/mol!. For the same collection of molecules, the G2 method had a mean absolute deviation of 1.4 kcal/mol. Presumably, as a larger number of molecules outside of the G2 calibration set were sampled, the value of « MAD obtained from G2 would increase. Electron affinities, proton affinities, and ionization potentials displayed somewhat smaller mean absolute deviations of 0.7 to 0.8 kcal/mol at the CCSD~T!/CBS level, compared to a G2 « MAD value of 1.4 kcal/mol for EAe . This improved performance obviously comes at the expense of much longer computer runs. Bond lengths between nonhydrogen atoms were reproduced to an accuracy of 60.01 Å at the CCSD~T!/aug-cc-pVQZ level and the error in stretching frequencies amounted to 3 cm21. As a result of fortuitous cancellation of error, the MP2/ aug-cc-pVDZ level of theory provided energetics in better agreement with experiment than more sophisticated ~and expensive! correlation treatments. Among the correlated methods, only CCSD~T! showed uniformly decreasing maximum positive and negative errors and standard deviations as the basis set expansion approached completeness. CCSD, without the inclusion of triple excitations, was typically worse than MP2 for atomization energies and electron affinities. With the exception of the CCSD~T! method, « MAD often displayed irregular convergence to the CBS limit, thus making it difficult to generalize about accuracy. This is a result of the interplay between the intrinsic error of the correlation method and the convergence error associated with the basis set expansion. The smallest errors were not always obtained with the largest basis sets. In order to more easily exploit all of the data on the 15 combinations of basis sets and methods,

the information compiled in this study has been stored in the EMSL Computational Results Database, which has been described elsewhere.33 The database and it’s accompanying software make it possible to extract from this large body of information, specific basis sets and methods that efficiently handle particular classes of compounds for any of the properties contained in the database. By combining information about basis set/method performance with information on hardware and software performance70 the software can construct recommendations as to appropriate basis sets and methods that will increase the likelihood that a calculation will achieve the desired accuracy. ACKNOWLEDGMENTS

This research was supported by the U. S. Department of Energy under Contract No. DE-AC06-76RLO 1830. We thank Dr. Thom Dunning, Jr. and co-workers for early access to their results and we acknowledge the support of the Division of Chemical Sciences, Office of Basic Energy Sciences and the Environmental Technology Partnerships program. Sharee Johnson and Ben Feller are thanked for their assistance in compiling portions of the Computational Results Database. We also thank Dr. Michel Dupuis for a critical reading of this work prior to publication. Portions of this work were completed with the computer resources at the National Energy Research Supercomputer Center with a grant provided by the Scientific Computing Staff, Office of Energy Research, U. S. Department of Energy. The Pacific Northwest National Laboratory is a multiprogram national laboratory operated by Battelle Memorial Institute.

APPENDIX ZERO-POINT VIBRATIONAL ENERGIES ~kcal/mol! FOR THE G2 SET.a Molecule

Expt.b

G2c

Basis

HFd

MP2

MP4

CCSD

H2 ( 1 S 1 g )

6.21

5.93

LiH ( 1 S 1 )

1.99

1.81

BeH ( 2 S 1 )

2.92

2.74

aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ

5.87 5.90 5.90 1.80 1.84 1.84 2.72 2.72 2.73 7.57 7.52 3.19 3.20 3.20 8.46 15.32 3.91 3.90 3.91 10.21 10.21 10.22 9.97 9.97 9.97 17.35 17.35

6.38 6.46 6.46 1.97 2.02 2.03 2.98 3.00 3.04 8.29 8.24 3.44 3.47 3.48 9.21 16.65 4.19 4.23 4.23 11.07 11.10 11.12 10.63 10.69 10.71 18.52 18.97

6.26 6.34 6.34 1.93 2.00 2.00 2.92 2.95 2.97 8.20 8.15 3.34 3.38 3.39 9.06 16.43 4.06 4.10 4.11 10.89 10.92 10.95 10.40 10.46 10.50 18.68 18.75

6.21 6.29 6.29 1.92 1.98 1.99 2.89 2.92 2.94 8.17 8.12 3.33 3.37 3.39 9.04 16.42 4.05 4.09 4.11 10.86 10.90 10.92 10.41 10.50 10.52 18.65 18.74

BeH2 ( 1 S 1 g ) 1

BH ( S ) 1

7.60 3.35

3.21

BH2 ( 2 A 1 ) BH3 ( 1 A 81 ) CH ( 2 P)

4.04

8.59 15.52 3.90

CH2 ( 3 B 1 )

10.55

10.32

CH2 ( 1 A 1 )

10.33

10.07

CH3 ( 2 A 29 )

17.35

CCSD~T! NA NA NA NA NA NA 2.89 2.91 2.93 8.16 8.21 3.32 3.36 3.37 9.01 16.37 4.03 4.07 4.08 10.81 10.84 10.86 10.34 10.42 10.46 18.57 18.64




171

~Continued.! Molecule

Expt.b

G2c

Basis

HFd

MP2

MP4

CCSD

CCSD~t!

CH4 ( A 81 )

27.71

26.77

27.84

4.51

28.24 28.94 4.79 4.85 4.87 12.10 12.19 12.23 21.58 21.70 5.39 5.43 5.45 13.39 13.43 13.49 5.83 5.89 5.91 1.64 1.66 1.67 2.18 2.22 5.96 2.38 2.44 2.46 6.49 6.58 11.73 11.90 2.95 2.96 3.00 7.47 7.54 7.76 13.54 19.23 3.44 3.47 3.48 8.62 15.30 3.92 3.95 3.95 9.62 9.69 9.70 4.32 4.35 4.35 0.48 0.49 0.49 0.95 0.98 0.99 1.15 1.19 7.88 8.00

28.00

4.64

26.37 26.48 4.50 4.51 4.52 11.47 11.48 11.48 20.53 20.57 5.19 5.19 5.20 13.01 12.98 13.01 5.75 5.75 5.75 1.50 1.51 1.51 1.96 1.99 5.45 2.18 2.17 2.23 5.96 6.01 10.78 6.01 2.70 2.73 2.73 6.89 6.94 7.16 12.53 18.34 3.20 3.21 3.21 8.00 14.31 3.65 3.65 3.65 9.09 9.05 9.06 4.03 4.03 4.04 0.43 0.43 0.43 0.87 0.89 0.89 1.10 1.13 7.58 7.74

27.92

NH ( 3 S 2 )

aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ

4.66 4.73 4.79 11.90 12.20 12.03 21.39 21.51 5.30 5.34 5.36 13.31 13.35 13.49 5.80 5.85 5.88 1.61 1.63 1.64 2.13

4.63 4.65 4.72 11.91 12.05 12.09 21.52

4.59 4.67 4.68 11.82 11.94 11.95 21.37

5.31 5.37 5.40 13.44 13.54 13.61 5.88 5.96 5.99 1.59 1.61 1.62 2.08

5.27 5.32 5.34 13.34 13.41 13.52 5.83 5.90 5.92 1.59 1.61 1.62 2.08

5.87 2.34 2.40 2.42 6.36

5.84 2.33 2.39 2.42 6.32

11.54

11.51

5.83 2.32 2.38 2.41 6.29 6.39 11.47

2.88 2.92 2.94 7.32 7.39 7.59 13.30 19.52 3.36 3.44 3.46 8.44 15.04 3.84 3.86 3.87 9.46 9.53 9.54 4.27 4.29 4.28 0.49 0.49 0.49 0.93 0.96 0.97 1.14 1.18 7.80 7.92

2.87 2.92 2.94 7.30 7.38 7.56 13.28 19.51 3.35 3.39 3.41 8.43 15.06 3.84 3.87 3.88 9.50 9.58 9.60 4.27 4.31 4.31 0.49 0.49 0.50 0.90 0.94 0.96 1.16 1.19 7.95 8.09

2.86 2.90 2.92 7.26 7.33 7.52 13.21 19.42 3.33 3.36 3.38 8.37 14.95 3.82 3.84 3.85 9.43 9.50 9.52 4.25 4.28 4.27 0.49 0.49 0.50 0.89 0.93 0.95 1.15 1.20 7.87 8.00

1

NH2 ( 2 B 1 )

11.52

NH3 ( 1 A 1 )

20.73

OH ( 2 P)

5.28

5.10

H2O ( 1 A 1 )

13.25

12.87

HF ( 1 S 1 )

5.85

5.56

NaH ( 1 S 1 )

1.66

1.53

MgH ( 2 S 1 )

2.10

1.96

MgH2 ( 1 A 1 ) AlH ( 1 S 1 )

2.38

5.45 2.26

AlH2 ( 2 A 1 )

6.06

AlH3 ( 1 A 18 )

10.90

SiH ( 2 P)

2.89

2.78

SiH2 ( 1 A 1 )

7.09

SiH2 ( 3 B 1 ) SiH3 ( 2 A 29 ) SiH4 ( 1 A 81 ) PH ( 3 S 2 )

3.36

7.33 12.81 18.76 3.27

PH2 ( 2 B 1 ) PH3 ( 1 A 1 ) SH ( 2 P)

3.81

8.19 14.66 3.70

H2S ( 1 A 1 )

9.40

9.21

HCl ( 1 S 1 )

4.24

4.07

Li2 ( 1 S 1 g )

0.50

0.43

LiN ( 3 S 2 )

LiO ( 2 P) LiOH ( 1 S 1 )

0.85

1.21

1.18 7.44



172


~Continued.! Expt.b

G2c

Basis

HFd

MP2

MP4

CCSD

CCSD~t!

LiF ( S )

1.30

1.32

LiCl ( 1 S 1 )

0.92

0.80

BeO ( 1 S 1 )

2.13

2.22

BeOH ( 1 A 8 ) BeF ( 2 S 1 )

1.78

7.53 1.68

BeS ( 1 S 1 )

1.42

1.37

BeCl ( 2 S 1 ) BO ( 2 S 1 )

1.21 2.69

1.09 2.67

aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVDZ aVDZ aVTZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ

1.20 1.20 1.21 0.80 0.82 0.83 2.15 2.46 2.47 7.71 1.61 1.68 1.36 1.37 1.37 1.07 2.63 2.67 2.97 8.59 1.61 7.40 1.08 1.09 1.10 2.45 2.45 2.45 8.31 16.21 16.46 21.60 30.51 30.50 35.20 44.26 2.55 2.60 2.61 9.94 10.00 24.12 38.32 3.09 3.11 3.12 7.95 7.98 16.18 16.19 30.83 1.72 7.49 23.46 1.03 1.29 8.44 20.16 32.32 1.81 1.83 1.83 14.92 27.40

1.23 1.25 1.26 0.86 0.90 0.90 1.88 1.99 2.01 7.97 1.70 1.76 1.40 1.41 1.43 1.18 2.65 2.71 2.73 8.83 1.70 7.78 1.18 1.22 1.22 2.66 2.68 2.70 10.72 15.90 16.61 24.02 32.13 33.56 37.61 47.11 4.06 4.14 4.17 9.79 9.98 25.10 40.43 2.96 3.02 3.04 8.21 8.28 16.77 16.93 32.37 1.76 7.82 24.81 1.22 1.71 8.36 21.19 34.39 2.03 1.85 1.86 15.64 29.08

1.22 1.24 1.26/Z 0.86 0.90 0.90 1.15 1.48 1.56 7.91 1.68 1.73 1.32 1.35 1.37 1.17 2.56 2.63 2.65 8.65 1.64 7.60 1.15 1.19 1.19 2.46 2.48 2.50 10.32 15.61 16.46 23.72 31.74 31.77

1.26 1.28 1.29 0.86 0.90 0.91 2.10 2.21 2.24 8.02 1.72 1.75 1.41 1.44 1.46 1.17 2.67 2.75 2.82 8.96 1.69 7.75 1.16 1.20 1.20 2.65 2.68 2.70 8.01 16.05 16.87 23.19 31.98

1.24 1.27 1.28 0.86 0.90 0.90 1.95 2.07 2.10 7.95 1.70 1.75 1.37 1.39 1.42 1.17 2.59 2.67 2.69 8.82 1.65 7.63 1.15 1.18 1.19 2.59 2.63 2.65 7.79 15.90 16.75 22.96 31.67

46.57 3.79 3.88 3.91 9.69 9.88 24.83 40.17 2.79 2.86 2.88 7.97 8.05 16.48

46.79 3.02 3.08 3.11 10.03

46.46 2.97 3.02 3.04 9.83

25.12 40.16 3.10 3.17 3.19 8.18 8.30 16.81

24.83 39.94 3.01 3.07 3.09 8.04 8.14 16.59

32.02 1.68 7.57 24.46

32.28 1.76 7.69 24.67

32.01 7.59 24.48

8.23

8.52

8.33

33.94 1.83 1.67 1.69

1.63 1.89 1.91

1.79 1.81 1.83

Molecule 1

1

HBO ( 1 S) BS ( 2 S 1 ) HBS ( 1 S) BCl ( 1 S 1 )

1.20

8.65 1.61 7.39 1.08

2.64

2.48

C2H ( 2 S 1 ) C2H2 ( 1 S 1 g )

16.46

8.41 16.50

C 2H 3 ( 2 A 8 ) C2H4 ( 1 A 8g )

31.47

21.69 30.69

C2H5 ( 2 A 28 ) C2H6 ( 1 A 1g ) CN ( 2 S 1 )

46.4 2.95

35.50 44.68 2.53

HCN( 1 S 1 )

9.95

10.08

H2CNH( 1 A 8 ) H3CNH2 CO( 1 S 1 ) p

3.09

24.25 38.61 3.11

HCO( 2 A 8 )

8.16

8.06

H2CO( 1 A 1 )

16.53

16.36

C2 ( 1 S 1 g )

H3COH CF( 2 P) HCF( 1 A 8 ) CH3F( 1 A 1 ) CSi( 3 P) CP( 2 S 1 ) HCP( 1 S 1 ) H2CPH( 1 A 8 ) H3CPH2( 1 A 8 ) CS( 1 S 1 )

H2CS( 1 A 1 ) CH3SH( 1 A 8 )

1.68

1.77

1.83

31.01 1.80 7.61 23.78 1.05 1.29 8.57 20.39 32.86 1.82

15.17 27.80




173

~Continued.! Molecule

Expt.b

G2c

Basis

HFd

MP2

MP4

CCSD

CCSD~t!

CCl( P) CH3Cl( 1 A 1 ) N2( 1 S 1 g )

1.23 23.45 3.36

1.11 22.80 3.52

2.71

32.53 2.84

HNO( 1 A 8 )

8.56

9.00

1.63

1.62

NSi( 2 S 1 ) NP( 1 S 1 )

1.64 1.91

1.17 2.03

1.21 23.56 3.11 3.12 3.14 33.27 5.18 4.90 4.80 8.42 8.50 1.53 1.60 1.84 1.53 1.61 1.64

NS( 2 P) O2( 3 S 2 g )

1.74 2.25

1.14 2.55

HO2 H2O2( 1 A) HOF( 1 A 8 ) NaO( 2 P) HONa( 1 S) MgO( 1 S 1 )

1.12

8.79 16.42 8.73 0.75 6.74 0.98

MgOH( 2 S) SiO( 1 S 1 )

1.77

6.48 1.79

1.25 23.84 3.08 3.13 3.15 33.62 5.24 4.77 4.61 8.63 8.70 1.60 1.66 2.00 1.60 1.69 1.71 1.83 2.04 2.08 2.07 8.99 16.49 8.67 0.71 6.99 1.44 1.44 1.45 6.80 1.58 1.68 1.70

1.22 23.70 3.42 3.46 3.48 33.61 2.77 2.84 2.87 8.81

NF( 3 S 2 )

1.11 22.47 3.52 3.51 3.51 32.37 2.84 2.85 2.85 8.96 8.94 1.59 1.62 1.05 2.00 2.04 2.05 1.14 2.55 2.52 2.53 8.91 16.65 8.82 0.67 6.96 1.02 1.04 1.05 6.79 1.71 1.80 1.81 1.75 6.07 1.65 1.72 1.92 1.06 1.09 1.09 8.18 8.21 1.56 1.64 1.63 0.71 0.96 1.03 1.09 16.13 1.12 1.15 1.17 1.18 0.20 0.20 0.20 0.46 0.46 0.46 0.69 0.58 0.58

1.20 23.51 3.32 3.34 3.37

N2H4( 1 A) NO( 2 P)

aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ

2

PO( 2 P) HPO( 1 A 8 ) SO( 3 S 2 )

1.79 6.28 1.68

ClO( 2 P)

1.06

HOCl( 1 A 8 )

8.11

F2( 1 S 1 g )

1.30

1.59

NaF( 1 S 1 ) MgF( 2 S 1 ) AlF( 1 S 1 ) SiF( 2 P) FSiH3( 1 A 1 ) PF( 3 S 2 ) SF( 2 P) ClF( 1 S 1 )

0.76 1.01 1.14 1.22

0.76 0.99 1.02 1.17 16.75 1.17 1.18 1.17

1.21 1.12

Na2( 1 S 1 g )

0.20

NaCl( 1 S 1 )

0.46

MgS( 1 S 1 ) MgCl( 2 S 1 ) AlCl( 1 S 1 )

0.75 0.66 0.69

0.69 0.69 0.61

6.09 1.53 1.62 1.64 1.21 1.21 1.22 8.20 8.29 1.33 1.43 1.43 0.75 1.00 1.06 1.13 17.33 1.14 1.17 1.12 1.14 0.22 0.22 0.22 0.49 0.50 0.50 0.77 0.63 0.64

1.59

2.88 2.86 2.78 8.58 8.78 1.54

1.57 1.90 1.98 2.00

1.57 1.81 1.88 1.91

2.07 2.06 2.09 8.86 16.31 8.43 0.70 6.86 1.59 1.57 1.58 6.76 1.35 1.51 1.55

2.35 2.37 2.40 8.92 16.70 8.72 0.71 7.09 1.01 1.04 1.05 6.87 1.70 1.82 1.84 see footnotee

2.24 2.25 2.26 8.76 16.37 8.49 0.70 6.98 1.06 1.10 1.13 6.81 1.63 1.74 1.76

1.50 1.57 1.59 1.19 1.22 1.20 8.24 8.32 1.20 1.30 1.30 0.74

1.59 1.69 1.71 1.15 1.22 1.24 8.29 8.41 1.32 1.44 1.45 0.76

1.53 1.62 1.64 1.13 1.20 1.22 8.15 8.25 1.18 1.31 1.32 0.76

1.07 1.09 0.22 0.22 0.22 0.49 0.50 0.50

1.12 1.15 0.21 0.22 0.22 0.49 0.50 0.50

1.08 1.10 0.21 0.22 0.22 0.49 0.50 0.50



174


~Continued.! Molecule Si2(

3

S2 g )

Si2H6( 1 A 1g ) SiS( 1 S 1 )

Expt.b

G2c

0.73

0.72

1.07

29.36 1.04

SiCl( 2 P) SiH3Cl( 1 A 1 ) P2 ( 1 S 1 g )

1.11

0.68 15.88 1.16

P2H4 ( 1 A) PCl ( 3 S 2 ) S2 ( 3 S 2 g )

0.82 1.04

21.70 0.72 1.04

S2H2 ( 1 A) SCl ( 2 P) Cl2 ( 1 S 1 g )

0.80

11.33 0.76 0.77

CO2 ( 1 S 1 g )

7.24

7.14

0.76

OCS ( 1 S) CS2 ( 1 S 1 g ) O3 ( 1 A 1 ) SiO2 ( 1 S 1 g ) SO2 ( 1 A 1 ) ClO2 ( 2 B 81 ) N2O ( 1 S) 2 CH1 4 ( B 1) 1 2 NH3 ( A 29 ) 1 NH1 4 ( A 1) 1 2 OH ( P)

5.53 4.08 4.90

4.53

4.39

4.30 aVDZ 3.67 6.90 22.43 19.61 29.85 4.28

H2O1 ( 2 B 1 )

11.26

H3O1 ( 1 A 1 ) HF1 ( 2 P)

20.58 4.03

2 SiH1 4 ( A 8) 1 ( A 8) SiH1 5 1 2 PH ( P)

16.91 23.01 3.39

PH1 2 PH1 3 PH1 4 SH1

3.29

( 1A 1) ( 2A 1) ( 1A 1) ( 3S 2)

8.46 14.71 21.74 3.70

2 SH1 2 ( B 1) 1 2 SH2 ( A 1 ) 1 SH1 3 ( A 1)

HCl1 ( 2 P) H2Cl1 ( 1 A 1 ) 2 C 2H 1 2 ( P)

9.09 8.48 16.23 3.78

3.75 9.44

15.86

Basis

HFd

MP2

MP4

CCSD

CCSD~t!

aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVDZ 4.31 aVDZ aVDZ aVTZ aVDZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVDZ aVDZ aVTZ aVDZ aVDZ aVDZ aVTZ aVDZ aVDZ aVTZ aVQZ aVDZ

0.73 0.74 0.74 28.88 1.03 1.05 0.65 15.47 1.16 1.17 1.17 21.14 0.69 1.03 1.05 1.05 11.07 0.74 0.76 0.79 0.79 7.23 7.22 7.23 5.57 4.09 4.93 4.89 4.17 3.85 3.52 6.96 6.96 22.56 19.51 29.61 4.31 4.31 4.31 11.38 11.36 20.73 4.17 4.26 4.26 16.81 22.55 3.25 3.25 3.25 8.12 14.22 21.10 3.48 3.49 8.68 8.18 15.50 15.61 3.60 9.05 9.12 9.13 15.50

0.72 0.74 0.75 31.10 1.04 1.06 0.72 16.64 1.01 1.04 1.05 22.44 0.75 0.98 1.02 1.03 11.66 0.79 0.77 0.82 0.83 7.14 7.21 7.25 5.67 4.31 5.96 5.93 4.05 3.53 3.08 6.59 6.69 24.33 20.82 31.17 4.53 4.57 4.59 11.86 11.93 21.65 4.30 4.47 4.46 18.44 24.61 3.49 3.50 3.51 8.69 15.17 22.46 3.72 3.74 9.26 8.83 16.42 16.54 3.87 9.62 9.70 9.72 15.69

0.70 0.72 0.73 30.63 0.97 1.01

0.73 0.75 0.76 30.66 1.06 1.09

0.70 0.73 0.74 30.47 1.02 1.05

1.01 1.03 1.05

1.12 1.15 1.17

1.07 1.10 1.12

0.96 1.00 1.01

1.02 1.05 1.07

0.98 1.01 1.03

0.73 0.78 0.80 6.86 6.95 6.96

0.75 0.80 0.82 7.29 7.40 7.42

0.72 0.77 0.79 7.11 7.20 7.24

4.72 4.81 3.77 4.14 3.18 6.34 6.43 24.05 20.67 31.04 4.45 4.51 4.52 11.79 11.87 21.69 4.28 4.46 4.45

4.60 4.70 4.16 3.91

4.01 4.17 4.00

6.89

6.65

24.03

23.90

31.18 4.42 4.49 4.51 11.81 11.92 21.82 4.30 4.50 4.49

31.00 4.39 4.45 4.47 11.73 11.83 21.70 4.26 4.45 4.49

25.05 3.41 3.42 3.43 8.52

3.40 3.42 3.43 8.51

3.38 3.40 3.41 8.46

22.15 3.63 3.67 9.11

22.03 3.61 3.63 9.05 8.67 16.17 16.29 3.81 9.54 9.62

22.13 3.65 3.66 9.12 8.73 16.25 16.34 3.84 9.58 9.64 9.63

16.28 16.36 3.83 9.60 9.70




175

~Contiunued.! Expt.b

G2c

Basis

HFd

MP2

MP4

CCSD

CCSD~t!

C2H1 3 f 2 C2H1 4 ( B 3u ) 1 2 1

1.98 3.14

1.59 3.27

2 N1 2 ( P u) 2 ( P g) O1 2 ClF1 ( 2 P) 2 P1 2 ( P u) 1 2 S2 ( P g ) 2 Cl1 2 ( P g) CH2 ( 3 S 2 )

2.70 2.71 1.24 0.89 1.13 0.92 4.32

3.03 3.19 1.27 1.03 1.22 0.95 3.16

21.28 31.03 4.07 4.10 4.15 2.05 2.96 2.99 2.48 2.08 1.41 0.99 1.01 0.85 3.87 3.92 3.94 10.00 10.12 10.15 18.04 4.63 4.67 4.63 11.70 11.81 5.38 5.42 5.45 2.68 2.72 2.74 6.88 6.95 12.34 3.30 3.32 3.34 8.25 3.86 3.88 3.89 2.78 2.83 2.84 2.02 2.03 2.04 1.49 1.57 1.59 1.34 1.45 1.47 0.80 0.84 0.85 0.37 0.38 0.39

21.29

CS1 ( 2 S 1 ) 2 1 N1 2 ( Sg )

21.07 29.51 3.06 3.07 3.11 1.61 3.27 3.26 3.02 3.23 1.72 1.09 1.22 0.79 3.49 3.50 3.50 9.14 9.16 9.16 16.73 4.36 4.38 4.40 11.17 11.20 5.22 5.22 5.23 2.46 2.48 2.48 6.27 6.32 11.37 3.02 3.03 3.04 7.62 3.57 3.57 3.57 2.97 2.98 2.98 2.02 2.01 2.01 1.81 1.82 1.82 1.42 1.50 1.51 0.81 0.83 0.83 0.32 0.32 0.32

21.63

3.15

aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVDZ aVDZ aVTZ aVQZ aVDZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ aVDZ aVTZ aVQZ

21.28

CO ( S )

21.38 29.77 3.05

3.87 3.89 3.93 2.05 2.79 2.80 2.61 2.27 1.37 0.97 1.03 0.83 3.73 3.80 3.81 9.72 9.74

3.21 3.24 3.31 1.99 3.20 3.24 2.80 2.88 1.27 1.05 1.15 0.88 3.70 3.78 3.79 9.72

3.19 3.20 3.28 1.96 3.10 3.13 2.69 2.73 1.25 1.00 1.09 0.84 3.67 3.74 3.75 9.63

17.70 4.50 4.54 4.52 11.46 11.58 5.25 5.28 5.31 2.60 2.71 2.68 6.72 6.79 12.08 3.21 3.23 3.25 8.07 3.76 3.79 3.80 2.73 2.77 2.79 1.91 1.93 1.94 1.42 1.47 1.49 1.27 1.38 1.41 0.77 0.81 0.83 0.36 0.37 0.38

17.81 4.51 4.59 4.62 11.61 11.76 5.34 5.40 5.43 2.60 2.65 2.68 6.70 6.78 12.08 3.21 3.24 3.26 8.08 3.77 3.80 3.81 2.97 3.02 3.04 2.01 2.03 2.05 1.63 1.70 1.72 1.41 1.51 1.54 0.81 0.84 0.86 0.35 0.37 0.37

9.80 17.66 4.46 4.53 4.55 11.47 11.61 5.27 5.32 5.36 2.58 2.63 2.66 6.65 6.73 11.99 3.18 3.21 3.23 8.02 3.74 3.77 3.78 2.88 2.93 2.95 1.95 1.97 2.01 1.52 1.59 1.61 1.35 1.46 1.48 0.78 0.81 0.83 0.35 0.37 0.27

Molecule

2 CH2 2 ( B 1)

8.79

1 CH2 3 ( A 1) NH2 ( 2 P)

16.64 3.98

1 NH2 2 ( A 1)

10.69

OH2 ( 1 S 1 )

5.29

4.77

SiH2 ( 3 S 2 )

3.11

2.47

2 SiH2 2 ( B 1)

6.38

1 SiH2 3 ( A 1) PH2 ( 2 P)

11.63 2.98

1 PH2 2 ( A 1) SH2 ( 1 S 1 )

7.62 3.70

CN2 ( 1 S 1 )

3.01

NO2 ( 3 S 2 )

1.94

2.09

2 O2 2 ( 1P g )

1.55

1.87

PO2 ( 3 S 2 ) 2 S2 2 ( P g)

2 1 Cl2 2 ( Su )

1.49

0.86

0.83

0.32

Computed as 0.5S v ei . 1 1 Experimental zero-point energies for diatomics were defined as 2 v e 2 4 v e x e , where v e and v e x e are taken from Huber and Herberg, Ref. 44. For polyatomics, the values are taken from Refs. 45–50. c HF/6-31G* scaled by 0.8929. d Scaled by 0.9. e UHF instabilities made it impossible to obtain meaningful zero-point frequencies for PO. f The 2 B 3u planar configuration is the lowest energy form at the UHF level. This geometry produces a single imaginary frequency of 2101 cm21 at the MP2 and higher levels of theory, where a D 2 symmetry, twisted form ( 2 B 1 ) is lowest. a

b



176


T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 ~1989!. L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J. Chem. Phys. 94, 7221 ~1991!. 3 L. A. Curtiss, C. Jones, G. W. Trucks, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 93, 2537 ~1990!. 4 L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 103, 1063 ~1997!. 5 C. W. J. Bauschlicher and H. Partridge, J. Chem. Phys. 103, 1788 ~1995!. 6 P. E. Siegbahn, M. Svensson, and P. J. E. Boussard, J. Chem. Phys. 102, 5377 ~1995!. 7 J. A. Montgomery, Jr., J. W. Ochterski, and G. A. Petersson, J. Chem. Phys. 101, 5900 ~1994!. 8 J. W. Ochterski, J. A. Montgomery, Jr., and G. A. Petersson, J. Chem. Phys. 104, 2598 ~1996!. 9 R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 96, 6796 ~1992!. 10 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 ~1995!. 11 K. A. Peterson and T. H. Dunning, Jr. ~to be published!. 12 D. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 ~1993!. 13 D. E. Woon and T. H. Dunning, Jr. ~to be published!. 14 D. E. Woon, K. A. Peterson, and T. H. Dunning, Jr. ~to be published!. 15 A. K. Wilson, T. v. Mourik, and T. H. Dunning, Jr., J. Mol. Struct.: THEOCHEM 388, 339 ~1996!. 16 GAUSSIAN 94, Revision D.3, M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. A. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andreas, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. J. P. Stewart, M. HeadGordon, C. Gonzalez, and J. A. Pople, Gaussian, Inc. Pittsburgh, PA ~1996!. 17 MOLPRO, H. J. Werner, P. J. Knowles, J. Almlof, R. D. Amos, M. J. O. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. A. Peterson, R. M. Pitzer, A. J. Stone, P. R. Taylor, and R. Lindh, Universita¨t Bielefeld, Bielefeld, Germany, University of Sussex, Falmer, Brighton, England ~1996!. 18 J. M. L. Martin, Chem. Phys. Lett. 259, 669 ~1996!. 19 D. Feller and K. A. Peterson, J. Mol. Struct.: THEOCHEM 400, 69 ~1997!. 20 K. A. Peterson and T. H. Dunning, Jr., J. Mol. Struct.: THEOCHEM 400, 93 ~1997!. 21 T. H. Dunning, Jr., K. A. Peterson, and D. E. Woon ~to be published!. 22 S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 ~1970!. 23 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 99, 1914 ~1993!. 24 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 100, 2975 ~1994!. 25 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 101, 8877 ~1994!. 26 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, 1930 ~1993!. 27 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, 9790 ~1993!. 28 K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 102, 2032 ~1995!. 29 C. W. J. Bauschlicher and H. Partridge, J. Chem. Phys. 100, 4329 ~1994!. 30 D. A. Dixon, D. Feller, and K. A. Peterson, J. Phys. Chem. ~to be published!. 31 J. M. L. Martin, J. Chem. Phys. 97, 5012 ~1992!. 32 J. M. L. Martin, J. Chem. Phys. 100, 8186 ~1994!.

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1

33

2

34



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RECOVERING INTRINSIC IMAGES USING AN ... - CiteSeerX

An examination of disparities in cancer incidence in Texas using ...

RECOVERING INTRINSIC IMAGES USING AN ... - CiteSeerX

Unicollaboration Research Methods Seminar Using Electronic ... - UV

Examination of Parameters Evaluation Methods in Computational

An Investigation of Electronic Structure and Aromaticity in Medium ...