摘要:

Improved coefficients for the scaling all correlation and multi-coefficient correlation methods Christine M. Tratz, Patton L. Fast and Donald G. Truhlar Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431. E-mail: truhlar@umn.edu Received 12th October 1999, Accepted 8th November 1999, Published 19th November 1999 We have re-optimized the coefficients for ten scaling all correlation (SAC) methods, five empirical infinite basis (EIB) methods, and 18 multi-coefficient (MC) correlation methods, including the special cases of multi-coefficient SAC and multi-coefficient Gaussian-2 and Gaussian-3. The new parameterization is based on a training set of 82 atomization energies except for multicoefficient Gaussian-2, which is restricted to H and the first period (nuclear charge £ 9) and is based on a training set of 52 atomization energies.Each method may be employed with or without including core-correlation effects, which are based on a new set of parameters optimized on a 123-molecule training set. The mean unsigned error in the atomization energies of the 82- molecule set is reduced on average by 20% when the new parameters used here are adopted. In recent papers we presented optimized coefficients for 46 methods of the SAC, EIB, and MC type.16,18,24,25 In 44 cases the empirical parameters were based on 49 atomization energies of molecules containing H, C, N, O, F, Si, P, S, and Cl. In the other two cases (two versions of MCG2), the coefficients were based on the 31-molecule first-period training set24 containing only the first-period atoms Be, Li, C, N, O, and F and H.In 44 of the parameterizations, core-correlation energy was based on a simple scheme38 involving the number of bonds to atoms of each atomic number. The parameters of that theory were estimated based on a separate 72-molecule training set. At this point it is convenient to systematize the notation. In some previous calculations we included spin–orbit and/or core-correlation effects, and in some we did not. The SAC calculations of ref. 16 included the spin–orbit term, and the parameters of that study will hence forth be labeled version 1s, i.e., v1s, to denote this fact. In contrast the parameters in ref. 18 were determined in the presence of both spin–orbit and core correlation terms, and will be denoted v1sc.The parameters for other methods in ref. 18 were also determined in the presence of spin–orbit and core correlation terms (ESO and ECC) and will be labeled v1sc for consistency (although those methods have been parameterized only one way so far). In ref. 24 we presented two multi-coefficient parameterizations of Gaussian-2, one called MCG2 that included spin–orbit and core correlation, and one called minimal MCG2 (MMCG2) that did not include either of these effects; these parameterizations will be henceforth labeled MCG2v1sc instead of MCG2 and MCG2v1m instead of MMCG2. Similarly the MCG3 and MMCG3 results of ref. 25 will henceforth be labeled MCG3v1sc and MCG3v1m.One can also denote G2 and G3 as versions 2 and 3 of the Gaussian-x method. Technically G2 is an m (minimal) method, and G3 is an sc (spin–orbit-core-correlation) method. The purpose of the present communication is to update the parameterizations of 37 of the methods of refs. 16, 18, 24, and 25 to a new 82-molecule training set (except for MCG2, which is restricted to elements H–F and is therefore parameterized on a 52-molecule first-period training set 1. Introduction In the early days of quantum chemistry, there was a rigid distinction between ab initio electronic structure methods and semiempirical methods. The former were exemplified by Hartree–Fock theory,1 Hartree–Fock–Slater theory,2 Møller–Plesset perturbation theory,3 and coupled-cluster theory,4 whereas the latter were exemplified by the semiempirical versions of neglect-of-differential-overlap molecular orbital theory.5–7 Later work introduced hybrid methods combining ab initio and semiempirical elements into a single calculation, as in scaling external correlation8,9 (SEC), scaling all correlation10–18 (SAC), Møller–Plesset perturbation theory with bond additivity corrections19,20 (BAC-MP4), the higher level correction (HLC) of Gaussian-x theory21,22 (x = 1, 2, 3), and empirical coefficients in hybrid density functional theories.23 In recent work18,24,25 we have presented a systematic approach to creating hybrid ab initio/empirical methods by combining basis set extrapolations26–37 with the scaling of individual components of the correlation energy.8–18 The coefficients used for the basis set extrapolations and the scaling of correlation energy are determined empirically from experimental data.Most of the calculations have been based on Dunning’s correlation-consistent basis sets.27,38–40 The resulting procedure is called the multi-coefficient correlation method (MCCM), and up to 10 empirical coefficients have been employed. There are three kinds of MCCM methods, namely Colorado (CO), Utah (UT), and New Mexico (NM).18 The SAC method is a one-coefficient special case of MCCM, and some other special cases also received their own label, namely the empirical infinite basis (EIB) method in which one extrapolates the basis set but does not scale the correlation energy, the multi-coefficient SAC (MCSAC) method in which one scales individual components of the correlation energy based on calculations with a single basis set, and the multi-coefficient Gaussian-2 (MCG2) and multi-coefficient Gaussian-3 (MCG3) methods that are based on the basis sets of Pople and coworkers rather than the basis sets of Dunning and coworkers.27,38–40 Curtiss et al.have also developed a method of this type, which they called Gaussian-3 based on scaling41 (G3S). PhysChemComm, 1999, 14that contains no atom heavier than F) and to update the core-correlation parameterization of ref. 42 to a 123- molecule training set. We will give two sets of new coefficients for each method, one determined in the presence of spin–orbit but not core correlation terms and labeled v2s and one in the presence of both spin–orbit and core correlation terms and labeled v2sc.(None of the current methods include scalar relativistic effects,37,43–45 but if later parameterizations include such terms they could be labeled with "sr" or "src".) The new v2s and v2sc parameters make all the version 1 parameterizations,16,18,24,25 as well as some of the earlier SAC parameterizations,15 essentially obsolete, although specialists might occasionally want to take advantage of the fact that parameterizations based on smaller training sets can actually be more reliable if one is concerned only with molecular types that were well represented in the original training set.For most applications though, it is better to use the parameterization based on the broadest, most representative set of data, and that is the motivation for the consistent parameterization of all methods presented here. 2. Basis sets, correlation methods, and data sets For convenience, all basis sets18,25,27,38–40,46,47 used in the present paper will use the abbreviations given in Table 1. In addition we will use standard abbreviations for electronic structure methods. All calculations in this paper are based on the following methods: HF MP2 MP4D MP4SDQ MP4 CCD CCSD CCSD(T) QCISD QCISD(T) quadratic configuration interaction with single and double excitations plus two In principle, any reasonable geometries can be used with the present methods, but for the results presented here, we use MP2/pDZ geometries for SAC, EIB, and MC methods (excluding MCG2 and MCG3), and we use MP2(full)/Dd geometries for G2, MCG2, G3, and MCG3.Hartree–Fock46 Møller–Plesset (MP) perturbation theory, second order46 MP perturbation theory, fourth order, with double excitations46 MP perturbation theory, fourth order, with single, double, and quadruple excitations46 full fourth-order MP perturbation theory, i.e., MP4SDQ plus triple excitations46 coupled-cluster theory with double excitations4,48,49 coupled-cluster theory with single and double excitations50 CCSD plus two quasiperturbative terms involving triple excitations51 quadratic configuration interaction with single and double excitations52 quasiperturbative terms involving triple excitations52 Table 1 Abbreviations and references for the basis sets Abbreviation Full notation cc-pVDZa pDZ 6-31G(d)b Dd 6-31+G(d)b D+d 6-31G(2df,p)b D2dfp aug"-cc-pVDZc pDZ+ cc-pVTZa pTZ 6-311G(d,p)b Tdp 6-311+G(d,p)b T+dp 6-311G(2df,p)b T2dfp 6-311+G(3df,2p)b T+3df2p modified G3larged MG3 a Ref.27 and 38. b Ref. 46. c Ref. 18, 39 and 40. d Ref. 25; the MG3 basis set is the same basis set as the one referred to as G3MP2large in ref. 47. In the previous papers we used 49 zero-point exclusive atomization energies of molecules containing H, C, N, O, F, Si, P, S, and Cl, with the exception of the MCG2 versions in which cases we used a 31-molecule training set24 containing only H and the first-period atoms Be, Li, C, N, O, and F to parameterize the methods.Throughout the rest of this paper we will refer to these original training sets as version 1, i.e., v1. For the present paper we have extended the training sets to 82-molecules for the first-andsecond-period training set and 52-molecules for the firstperiod training set. The extension was accomplished by judicious choice of molecular types that were not well represented in the v1 training set. The first-and-secondperiod training set contains molecules composed of H, C, N, O, F, Si, P, S, and Cl. The first-period training set contains the 48-molecule subset of molecules containing only H, C, N, O, and F plus 4 additional compounds24 containing H, Li, Be, and F.The BeH, Li2, LiH, and LiF molecules are not contained in the 82-molecule training set because the pDZ and pTZ basis sets for Be and Li are currently unavailable. The new training sets will be referred to as version 2, denoted by v2. The 33 new experimental zero-point exclusive atomization energies were calculated from f H D 0,298 data.53,54 The procedure for changing the f H D 0,298 data to De is given in ref. 16. The full 82-molecule training set is given in Table 2 . The coefficients for the different methods were determined by a linear least squares fit to the 82-molecule training set, except for MCG2v2s and MCG2v2sc which were fitted to the 52-molecule first-period training set.The original42 values for ECC were obtained by using literature values of the core correlation contribution to the atomization energy for a 72 molecule training set containing 13 elements. We have extended the training set by adding an additional 51 molecules. These 51 molecules consist of all the molecules in the new 82-molecule training set that were not included in the original 72-molecule core correlation training set. The core-correlation binding energies for the 51 new molecules as well as the 72 original molecules have been calculated with the following formula DCC(MP2/G3large) = De(MP2(full)/G3large)–De(MP2/MG3) (1)Table 2 Experimental atomization energies (kcal mol–1) Molecule Molecule De CH 84.00 N2 CH2 (3B1) 190.07 H2 NNH2 NO CH2 (1A1) CH3 NH CH4 O2 HOOH F2 CO2 Si2 SP22 Cl2 NH2 NH3 OH OH2 FH SiH2 (1A1) SiH2 (3B1) SiH3 SiH4 PH2 PH3 SH2 SiO SC SO ClO ClF Si2H6 CH3Cl CH3SH ClH HCCH H2CCH2 H3CCH3 CN HOCl SO2 AlCl3 AlF3 BCl3 BF3 C2Cl4 181.51 307.65 420.11 83.67 181.90 297.90 106.60 232.55 141.05 151.79 131.05 227.37 322.40 153.20 242.55 182.74 106.50 405.39 563.47 712.80 180.58 313.20 259.31 278.39 373.73 HCN CO HCO H2CO C2F4 H3COH 512.90 C3H4 a Bicyclobutane. b Cyclobutene.where De is the zero-point exclusive atomization energy, and the geometries for all the calculations are MP2(full)/6- 31G(d) optimized geometries.When a literature value as well as an MP2/G3large value is available, we averaged the two values and used that value in the data set. All electronic structure information, optimized geometries, electronic energies, and G2 energies for the 86 molecules and 11 atoms were obtained using the GAUSSIAN9855 electronic structure package. CC SO 3. Theory and results Since the details of all the methods used in this paper have been published elsewhere10–12,15,16,18,24,25,35,42 we will only give a short summary of each of the methods here. The reader is referred to the original papers for specific details. The core-correlation contribution to the electronic energy has been re-optimized over the new 123-molecule core correlation training set.The mean signed errors (MSEs), and mean unsigned errors (MUEs) in DCC (which is the contribution of ECC to the dissociation energy) for molecules containing each atom type is given in Table 3. The mean signed value (MSV) and mean unsigned value (MUV) for all molecules containing a certain atom type are also given in Table 3. The energy for SAC methods is calculated from E = E( ) HF c E correlated E HF + - [ ] ( ) ( )+ E + lE (2) and the energy for all of the other methods can be calculated from Molecule De De C4H4O 993.74 C4H4S 962.73 1071.57 987.20 1001.61 1237.69 C4H5N C4H6a C4H6b C5H5N C2H 267.83 CCl4 CF3CN CF4 CH2OH CH3CN CH3NH2 312.74 639.85 476.32 409.76 615.84 582.56 601.27 343.18 457.50 125.33 109.61 CH3NO2 CHCl3 CHF3 ClF3 H2 H2C2H 445.79 HCOOCH3 HCOOH NF3 PF3 SH 785.26 500.98 204.53 363.87 86.98 384.94 574.35 SiCl4 SiF4 228.46 438.60 155.22 119.99 268.57 38.20 389.14 71.99 117.09 101.67 57.97 192.08 171.31 125.00 64.49 61.36 530.81 394.64 473.84 164.36 257.86 306.26 426.50 322.90 470.04 466.28 583.96 682.74 E (3) l + E = i i CC SO 0E c i�1 ( ) HF +åc DE + E where ESO and ECC are the spin–orbit and core-correlation contributions to the electronic energy, l = 1 for version 2sc parameters, l = 0 for version 2s parameters, E(HF) is the Hartree–Fock energy, and Ei is an energy difference which can be determined from the different coefficient trees given in Fig.1–3. A vertical line in Fig. 1–3 represents the energy increment when increasing the one-electron basis set with a fixed level of electron correlation. A horizontal line in Fig. 1–3 is the change in energy increment due to increasing the level of the treatment of the correlation energy with one basis set as compared to the increment obtained with a smaller basis set. An exception to this is the first horizontal line which represents the increase in basis set at the Hartree–Fock level. Comparing the current coefficient trees to the original18 ones, one can see that the MP4D calculations have been removed. We no longer have a separate term for MP4D because currently available programs do not allow us to obtain the MP4D gradient, and they do not allow us to obtain the MP4D energy as a single-level calculation.Therefore, we have left it out of all the new parameterizations. In addition we have removed the MP4/D2dfp term from the MCG3 coefficient tree. We have found that the MP4D term does not significantly improve our overall fit; therefore, we have omitted this term in version 2. In the present paper, ESO is taken from the literature as in previous work.16,18,24,25 However, in principle it could alsoTable 3 Parameters, mean signed error (MSE) and mean unsigned error (MUE) in kcal mol–1 and the number of data points for each atom Atom type HLi Be BCNOFAl Si PSCl Entire data set 123 0.04 0.26 1.65 1.71 a Number of molecules containing this atom type.b Mean errors for molecules containing this atom type. c Mean signed and unsigned value of the core-correlation contribution DCC to the atomization energy for all molecules containing this atom type. Fig. 1 Coefficient tree for EIB and MC methods (version 2). be calculated. The mean unsigned contribution of ESO to the atomization energy for the 82 molecules considered here is 0.71 kcal mol–1 (which comes from a mean unsigned value of 0.80 kcal mol–1 for 73 molecules for which it is nonzero and 0.00 kcal mol–1 for the other 9 molecules.) In addition to ESO which is a vector relativistic contribution, one should also consider the scalar nonrelativistic effects.37,43–45 These are not included explicitly here, and therefore their effects a implicitly included in the terms with semiempirical coefficients. In a separate study we have calculated the scalar relativistic contribution to the atomization energy for a 60-molecule subset of the present 82-molecule data set, and the mean unsigned value was only 0.35 kcal mol–1.56 The coefficients for version 2sc are given in Table 4.Removing ECC from eqn. (2) and eqn. (3) creates v2s, the optimized coefficients for which are given in Table 4. The mean signed error (MSE), the mean unsigned error (MUE), and the root-mean-square error (RMSE) for all of the methods including explicit core correlation terms are given in Table 5. The mean errors for methods treating core correlation implicitly are given in Table 6.An estimation of the computational cost for all of the methods is given in Tables 5, 7, and 8. The cost is defined as the mean CPU time on an SGI Origin 2000 computer with R12000 processors for the seven most expensive molecules, C No. dataa Z DZ 75677 51 19 26 207 136 10 1 0.0000 3 0.4378 4 0.3345 5 0.3264 6 0.3184 7 0.2137 8 0.2004 9 0.2513 13 –0.0751 14 0.0654 15 0.1893 16 0.3495 17 0.8320 16 Fig. 2 Coefficient tree for MCG2. C4H4O, C4H4S, C4H5N, C5H5N, CF3CN, and SiCl4, relative to the mean CPU time (147 s) on the same computer for MP2/pDZ calculations for the same seven molecules. The MCG2 method is not included in Tables 5 and 6 because we have parameterized this method only for molecules containing atoms no heavier than F.Table 7 gives the mean errors for the two old and two new versions of MCG2, for the G2 and G3 methods, and for six methods whose overall efficiency (i.e., performance/cost ratio) is very good for the full test set. The mean errors in Table 7 are based on the 48-molecule first-period-atom subset of the full 82-molecule first-and-second-period training set. Recall that BeH, Li2, LiH, and LiF were included in the MCG2 training set, but they are not included in the calculation of the mean errors in Table 7 because we want a consistent comparison of methods over a common data set. For the methods that contained Li and Be in the original training set, the mean errors over the 52-molecule firstperiod data are given in parentheses.The cost given in Table 7 is the mean CPU time on an SGI Origin 2000 computer with R12000 processors for the four most expensive molecules, C4H4O, C4H5N, C5H5N, and CF3CN, that have no atom with Z > 9, relative to the mean CPU time (126 s) for an MP2/pDZ calculation on the same computer for the same molecules. 2Cl4, MUVc MSVc MUEb MSEb 1.79 0.87 0.87 1.30 2.81 2.04 1.48 1.47 1.07 0.93 0.80 1.75 2.37 1.69 0.87 0.87 1.30 2.81 2.04 1.48 1.47 0.54 0.67 0.76 1.75 2.37 0.22 0.32 0.12 0.22 0.19 0.18 0.22 0.32 0.48 0.53 0.24 0.26 0.49 0.08 –0.07 –0.02 –0.01 0.04 0.01 0.00 –0.02 –0.07 –0.02 0.04 –0.03 0.06 Fig.3 Coefficient tree for MCG3 (version 2).Table 4 Coefficients optimized in this work Methods Version c or c0 SAC-MP2/pDZ SAC-MP2/pTZ SAC-MP4SDQ/pDZ SAC-MP4SDQ/pTZ SAC-MP4/pDZ SAC-MP4/pTZ SAC-QCISD/pDZ SAC-CCD/pDZ SAC-CCSD/pDZ SAC-CCSD/pTZ SAC-CCSD(T)/pDZ SAC-CCSD(T)/pTZ EIB-MP2 EIB-MP4SDQ EIB-MP4 EIB-CCSD EIB-CCSD(T) MCSAC-MP4/pTZ MCSAC-CCSD/pDZ MCSAC-CCSD/pTZ 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc 2s 2sc MCSAC-MP4SDQ/pDZ 2s 2sc MCSAC-MP4SDQ/pTZ 2s 2sc 2s 2sc 2s 2sc 2s 2sc MCSAC-CCSD(T)/pTZ 2s MCCM-CO-MP2 2sc 2s 2sc MCCM-CO-MP4 MCCM-CO-CCSD MCCM-CO-MP4SDQ 2s 2sc 2s 2sc 2s 2sc MCCM-UT-MP4 MCCM-UT-CCSD MCCM-CO-CCSD(T) 2s 2sc MCCM-UT-MP4SDQ 2s 2sc 2s 2sc 2s 2sc MCG2 MCG3 MCCM-UT-CCSD(T) 2s 2sc MCCM-NM-CCSD(T) 2s 2sc 2s 2sc 2s 2sc c8 c7 c6 c5 c4 c3 c2 c1 1.2373 1.2181 1.0138 0.9970 1.4431 1.4209 1.1933 1.1737 1.3362 1.3156 1.0788 1.0610 1.4622 1.4398 1.4996 1.4766 1.4727 1.4501 1.2178 1.1979 1.3774 1.3562 1.1232 1.1047 1.0000 0.8145 1.0000 1.2583 1.0000 0.8352 1.0000 1.1165 1.0000 1.8558 1.0000 2.0148 1.0000 1.8424 1.0000 1.8636 1.0000 1.3542 1.0000 1.3942 1.0000 1.3725 1.0000 1.2583 1.0000 1.9798 1.0000 2.1097 1.0000 1.9632 1.0000 1.9577 1.0000 1.5845 1.0000 1.6002 1.0000 1.5899 1.0000 1.4580 1.0000 1.3727 0.9387 1.0000 1.3573 0.9644 1.0000 1.1299 0.7626 1.0000 1.1169 0.7875 1.0000 1.0880 1.0538 0.9363 1.0000 1.0775 1.0621 0.8829 1.0000 1.3852 0.9121 1.0000 1.3699 0.9358 1.0000 1.1388 0.7353 1.0000 1.1257 0.7569 1.0000 1.0769 0.9761 1.4314 1.0000 1.0712 0.9688 1.2601 0.9887 1.0828 0.7768 2.7893 0.9888 1.1177 0.7671 2.7028 0.9633 1.3834 0.9872 2.6535 0.5145 2.6409 0.9615 1.4505 1.0035 2.5157 0.5870 2.8439 0.9964 1.5157 0.8123 2.1066 1.2808 3.6512 1.8043 5.0707 0.9886 1.5656 0.8568 2.0846 1.2148 3.5897 1.5567 3.9450 0.9635 1.4008 1.0217 2.5017 0.5515 2.4227 0.9629 1.4633 1.0392 2.3290 0.6254 2.5222 0.9981 1.5432 0.9097 1.7613 0.9684 1.3340 2.1823 1.2716 0.9914 1.5839 0.9477 1.7238 0.9633 1.5365 1.8210 0.7451 0.9903 1.2975 1.0099 1.8153 0.7872 0.9905 1.3579 1.0280 1.6130 0.8807 1.0009 2.5466 0.8862 2.9916 1.0962 1.7254 1.0002 2.6548 0.9165 2.8444 1.1591 1.5546 0.9935 1.2997 1.0332 1.6755 0.7675 0.9941 1.3580 1.0512 1.4689 0.8502 1.0112 1.5307 0.9101 1.4394 1.0473 2.4238 1.0099 1.5644 0.9413 1.2580 1.1002 2.1652 1.0126 1.5334 0.9156 1.3905 1.0280 2.3481 0.2140 1.0114 1.5672 0.9471 1.2062 1.0798 2.0852 0.2262 0.9932 0.6787 1.1695 0.9901 0.9980 4.4804 0.5096 4.2274 1.2598 0.9900 0.7229 1.1715 0.9504 1.0366 4.1810 0.4366 3.8767 1.2064 1.0096 1.2246 1.0637 1.1363 1.1506 0.5224 1.3219 1.3980 1.0080 1.2750 1.0661 1.0998 1.1330 0.8006 1.1689 1.3192Table 5 Mean errors in kcal mol–1 for methods including explicit core correlation terms (82-molecule test set) Method SAC-MP2/pDZ SAC-MP2/pTZ SAC-MP4SDQ/pDZ SAC-MP4SDQ/pTZ SAC-MP4/pDZ SAC-MP4/pTZ SAC-QCISD/pDZ SAC-CCD/pDZ SAC-CCSD/pDZ SAC-CCSD/pTZ SAC-CCSD(T)/pDZ SAC-CCSD(T)/pTZ EIB-MP2 EIB-MP4SDQ EIB-MP4 EIB-CCSD EIB-CCSD(T) MCSAC-MP4SDQ/pDZ MCSAC-MP4SDQ/pTZ MCSAC-MP4/pTZ MCSAC-CCSD/pDZ MCSAC-CCSD/pTZ MCSAC-CCSD(T)/pTZ MCCM-CO-MP2 MCCM-CO-MP4SDQ MCCM-CO-MP4 MCCM-CO-CCSD MCCM-CO-CCSD(T) MCCM-UT-MP4SDQ MCCM-UT-MP4 MCCM-UT-CCSD MCCM-UT-CCSD(T) MCCM-NM-CCSD(T) MCG3 Gaussian-x Version Coeffs.2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc 2sc 1sc3 111111111111111111111111222222222223233422223344688 1066885667556677894 MUE 1st 2nd Olda Newb period period All 10.39 11.46 10.35 11.29 10.82 9.27 15.79 13.09 9.46 11.89 6.58 6.91 6.78 6.24 6.71 5.82 12.78 9.10 7.10 8.62 4.91 10.66 5.72 9.34 7.22 4.89 10.56 5.57 9.43 7.17 3.60 5.52 3.33 5.68 4.37 3.62 5.55 3.40 5.64 4.40 5.49 10.33 6.31 9.17 7.44 5.44 10.82 6.81 8.84 7.60 2.58 4.35 2.98 3.63 3.29 2.53 5.41 4.02 3.04 3.69 4.44 10.46 5.09 9.30 6.87 4.20 10.26 4.31 9.87 6.64 5.57 11.44 6.21 10.47 7.94 5.40 11.10 5.59 10.77 7.69 4.70 10.66 5.25 9.69 7.10 4.44 10.29 4.37 10.21 6.79 4.00 5.58 3.19 6.55 4.64 3.87 5.45 2.84 6.72 4.51 4.23 9.77 4.99 8.60 6.46 4.25 9.70 4.87 8.72 6.44 1.75 4.10 1.82 3.76 2.70 1.88 4.48 2.33 3.59 2.93 6.33 7.42 6.64 6.52 6.76 4.97 14.49 9.41 7.01 8.80 4.38 4.44 4.09 4.68 4.40 3.50 7.06 4.41 5.45 4.93 1.90 2.84 2.46 1.82 2.28 1.93 4.22 3.39 1.81 2.85 4.97 4.89 4.54 5.38 4.94 3.74 8.22 5.07 6.03 5.55 1.62 1.92 1.55 1.76 1.74 1.45 3.13 1.81 2.29 2.13 5.86 9.69 6.11 9.23 7.40 4.92 10.45 5.50 9.38 7.14 2.94 4.33 2.97 4.03 3.50 2.94 6.10 4.90 2.95 4.21 2.43 4.31 2.83 3.56 3.19 2.07 4.83 2.63 3.90 3.18 5.42 9.36 5.53 9.06 7.01 4.56 10.02 5.07 9.08 6.75 2.57 4.08 2.47 3.95 3.18 2.67 5.46 4.17 3.02 3.79 1.87 3.80 1.99 3.41 2.65 1.50 4.24 2.07 3.26 2.60 4.90 6.14 4.57 6.65 5.40 3.40 12.91 6.32 8.31 7.23 2.18 2.60 2.16 2.60 2.35 1.90 5.56 3.40 3.25 3.38 1.33 1.85 1.50 1.45 1.54 1.29 3.28 1.88 1.98 2.09 1.84 2.26 1.94 2.14 2.01 1.49 3.90 2.42 2.54 2.46 0.92 1.40 1.09 1.10 1.11 0.78 2.05 1.29 1.21 1.29 2.49 2.98 2.94 2.27 2.68 2.34 4.83 3.66 2.68 3.34 2.00 2.12 2.15 1.80 2.05 1.95 4.13 2.61 2.93 2.83 2.28 2.82 2.66 2.26 2.50 1.89 5.09 3.41 2.74 3.18 1.24 1.68 1.45 1.29 1.41 1.08 2.57 1.60 1.66 1.68 1.29 1.64 1.46 1.32 1.43 1.03 2.93 1.52 2.08 1.80 1.01 1.21 1.09 1.13 1.09 0.92 2.37 1.24 1.78 1.50 0.97 1.70 1.21 1.35 1.26 RMSE MSE All All 12.87 14.68 8.58 11.10 11.05 11.05 6.00 6.00 11.01 11.05 4.66 5.03 10.74 10.84 11.92 11.95 11.00 11.08 6.13 6.18 10.62 10.62 4.19 4.24 8.50 11.77 5.40 7.00 3.45 4.06 5.87 7.79 2.29 3.05 10.46 10.75 4.90 5.69 4.62 5.09 10.07 10.41 4.34 4.95 3.87 4.27 6.67 10.13 3.21 4.82 2.33 3.29 2.64 3.46 1.54 1.95 3.98 4.82 3.21 4.07 3.40 4.34 1.87 2.33 1.85 2.69 1.45 2.22 1.69 –3.83 2.10 –2.32 3.62 –1.75 –1.94 –0.80 –0.71 –2.19 –1.38 –1.08 0.53 –1.23 –2.47 –1.79 –2.44 –1.32 –2.42 –0.50 –1.13 –1.55 –1.74 –0.59 –0.04 –1.69 5.29 –0.49 –0.28 –0.51 1.33 –0.34 –0.78 –0.13 0.54 –2.27 –2.12 –1.15 1.28 –1.03 –0.76 –2.01 –1.61 –0.97 1.00 –0.77 0.00 –1.51 4.73 –0.58 1.63 –0.30 1.13 –0.44 0.76 –0.11 0.57 –0.38 0.92 –0.27 –0.34 –0.22 0.31 0.03 0.13 0.03 –0.23 0.05 0.35 –0.44 Costc 1 192 41 11 267758 277 17 430 20 44 278 285 4472 41 2678 277 430 20 44 278 285 447 21 30 27 36 37 25 86a This column refers to the 49 original molecules.b This column refers to the 33 new molecules. c Cost is the mean CPU time on an SGI Origin 2000 computer with R12000 processors for the seven most expensive molecules, C2Cl4, C4H4O, C4H4S, C4H5N, C5H5N, CF3CN, SiCl4, relative to the mean CPU time (147 s) for an MP2/pDZ calculation on the same computer for the same seven molecules. This cost function is the same for versions 1sc and 2sc.Table 6 Mean errors in kcal mol–1 for methods treating core correlation implicitly (82-molecule test set) Method SAC-MP2/pDZ SAC-MP2/pTZ SAC-MP4SDQ/pDZ SAC-MP4SDQ/pTZ SAC-MP4/pDZ SAC-MP4/pTZ SAC-QCISD/pDZ SAC-CCD/pDZ SAC-CCSD/pDZ SAC-CCSD/pTZ SAC-CCSD(T)/pDZ SAC-CCSD(T)/pTZ EIB-MP2 EIB-MP4SDQ EIB-MP4 EIB-CCSD EIB-CCSD(T) MCSAC-MP4SDQ/pDZ MCSAC-MP4SDQ/pTZ MCSAC-MP4/pTZ MCSAC-CCSD/pDZ MCSAC-CCSD/pTZ MCSAC-CCSD(T)/pTZ MCCM-CO-MP2 MCCM-CO-MP4SDQ MCCM-CO-MP4 MCCM-CO-CCSD MCCM-CO-CCSD(T) MCCM-UT-MP4SDQ MCCM-UT-MP4 MCCM-UT-CCSD MCCM-UT-CCSD(T) MCCM-NM-CCSD(T) MCG3 Gaussian-x 2 1.16 3.11 1.95 1.97 1.95 a This column refers to the 49 original molecules. b This column refers to the 33 new molecules.4. Discussion First of all we compare Table 5 to Table 6. We see that the inclusion of core correlation by our simple scheme makes a small but significant improvement in most cases. However the methods that neglect core correlation have the advantage of simplicity. After considerable thought we decided to present both sets of coefficients and results since Version Coeffs. 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 1s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 2s 1m 111111111111111111111111222222232234686856567892 MUE 2nd period All 1st period Olda Newb 11.05 10.80 9.38 11.58 6.05 6.68 6.65 8.16 9.51 7.43 10.05 7.24 6.10 4.77 6.39 4.58 9.11 7.49 9.15 7.48 3.64 3.35 3.26 3.44 9.52 7.22 10.61 6.95 10.81 8.32 11.47 7.76 9.93 7.44 10.90 7.03 7.02 5.07 7.48 4.90 8.77 6.61 9.48 6.55 4.10 3.01 4.24 2.91 6.24 6.48 5.50 4.99 1.85 2.26 6.20 5.54 2.40 2.22 9.25 7.54 4.13 3.65 3.65 3.33 9.16 7.20 4.06 3.36 3.40 2.74 6.21 5.14 2.94 2.66 1.65 1.64 2.51 2.36 1.20 1.19 2.42 2.85 1.72 2.05 2.51 2.73 1.36 1.43 1.37 1.44 1.32 1.19 1.52 1.46 10.51 12.70 6.88 8.70 5.90 5.22 3.68 3.16 6.38 6.34 3.01 3.44 5.51 4.30 6.59 5.18 5.61 4.25 3.55 2.96 5.06 4.49 2.08 1.80 6.39 4.49 2.38 4.97 1.89 6.28 3.11 2.96 5.72 2.65 2.12 4.45 2.47 1.53 2.30 1.17 3.14 2.23 2.90 1.45 1.46 1.05 1.32 10.33 11.49 9.39 14.84 6.51 6.92 5.89 11.53 4.92 11.15 4.82 10.84 3.89 6.08 3.73 5.86 5.45 10.52 5.46 10.49 2.48 4.62 2.44 4.93 4.62 11.10 4.19 11.05 5.86 11.97 5.39 11.27 4.91 11.18 4.43 10.87 4.35 6.13 4.06 6.15 4.20 10.18 4.27 9.93 1.96 4.58 1.88 4.43 6.02 7.17 4.93 5.08 1.72 3.06 5.55 5.53 2.07 2.44 5.93 9.94 2.96 4.68 2.44 4.64 5.51 9.69 2.61 4.46 1.85 4.05 4.74 5.72 2.45 2.97 1.37 2.03 2.14 2.69 0.97 1.52 2.65 3.14 2.04 2.07 2.46 3.13 1.25 1.69 1.28 1.68 1.06 1.37 0.99 2.15 they can both be useful.For the rest of the discussion, though, we will focus on versions 2sc, i.e., the new versions that include core-correlation energy explicitly. The mean unsigned error is plotted versus the cost for version 2sc of the present methods and version 3 of the Gaussian-x method is shown in Fig. 4. The circled points denote the methods with the best performance/price ratio. RMSE MSE All All 12.81 14.18 8.49 10.43 11.24 11.31 6.36 6.43 11.08 11.08 4.76 4.84 11.01 11.29 12.15 12.28 11.24 11.48 6.55 6.79 10.73 10.80 4.41 4.42 8.12 6.08 3.48 6.62 2.82 10.53 5.06 4.75 10.16 4.52 3.92 6.35 3.57 2.48 3.06 1.66 4.19 3.27 3.68 1.87 1.85 1.54 1.89 2.62 –3.79 1.31 –2.27 2.84 –1.70 –2.78 –0.74 –1.54 –2.14 –2.21 –1.02 –0.30 –1.18 –3.31 –1.75 –3.28 –1.27 –3.26 –0.44 –1.98 –1.49 –2.58 –0.53 –0.86 –1.64 –0.48 –0.47 –0.34 –0.11 –2.27 –1.14 –1.01 –2.02 –0.96 –0.73 –1.46 –0.64 –0.30 –0.51 –0.11 –0.46 –0.33 –0.30 –0.02 –0.02 0.05 –0.11 –0.29Fig.4 Mean unsigned error for the 82-molecule test set as a function of cost for G3 and all SAC, EIB, MCSAC, MCCM and MCG3 v2sc methods. The cost is represented by the mean CPU time on an SGI Origin 2000 computer with R12000 processors for the seven most expensive molecules, C2Cl4, C4H4O, C4H4S, C4H5N, C5H5N, CF3CN, SiCl4.Squares: N5 methods; circles: N6 methods; diamonds: N7 methods. Circled symbols are the ones deemed to be particularly efficient. The computer time for methods including triple excitations, i.e., MP4, QCISD(T), and CCSD(T), scales57 as N7 where N is the number of electrons, whereas methods that do not have higher levels than MP2 scale57 as N5. Intermediate methods, for which the highest level is MP4SDQ, QCISD, CCD, or CCSD, scale57 as N6. It is useful to find efficient methods for each level of scaling since methods with an N7 component tend to be most accurate when affordable, but only N5 methods are affordable for very large molecules. We will pick out the most efficient methods on the basis of Fig.4; using a very similar figure (not shown) for version 2s leads to the same conclusions. The most efficient methods are defined as the ones with the best performance/price ratio. As far as N5 methods (solid squares), the most efficient methods appear to be SACMP2/pDZ and MCCM-CO-MP2. The most efficient of the N6 methods (solid circles) are SAC-MP4SDQ/pDZ, MCCM-UT-MP4SDQ, and MCCM-UT-CCSD. Even though the MCCM-UT-MP4SDQ and MCCM-UT-CCSD methods have very similar performance/price ratios, we include both the MP4SDQ and CCSD methods in the list of select methods because the MP4SDQ appears slightly more efficient whereas CCSD may be more advantageous for calculations on systems with some multi-reference character.For example, experience indicates that the coupled-cluster methods with odd excitations alleviate some of the problems of spin contamination. Therefore, if one wanted to calculate the energy or optimize the geometry of a transition state, the MCCM-UT-CCSD method would be a better choice than the MCCM-UTMP4SDQ method. For the sane reason, one might want to consider using the SAC-CCSD/pDZ method. On the basis of Fig. 4, it would appear that out of all of the N7 methods (solid diamonds) where parameterization includes the second period, only the MCG3 method is competitive on a performance/price basis. The other N7 methods are all very expensive and do not improve in accuracy as compared to MCG3.In Tables 5 and 6 there are 46 cases where we can compare a version 2 parameterization to version 1 of the same method. It is particularly interesting to compare the performance for the "new" molecules, i.e., those in version 2 of the training set that were not in version 1. In 25 of the 46 cases, the mean unsigned error for these 33 molecules decreases by 10% or more, and the average improvement over these 25 cases is 36%, whereas the average improvement in the MUE over all 46 cases is 20%. As a result of these improvements we believe that the new parameters are more robust and should be preferred for general applications. Table 8 shows the mean errors in single-level calculations by various single-level methods that are components of the multilevel methods used here.A comparison of these results to those in Tables 5 and 6 is given in Fig. 5. This comparison shows that the multilevel methods reduce the errors considerably compared to single-level methods, so that even the less efficient methods in Fig. 4 are still more efficient than traditional calculations based on a single level without extrapolation. Fig. 5 Mean unsigned error for the 82-molecule test set as a function of cost for G3, all SAC, EIB, MCSAC, MCCM and MCG3 v2sc methods and for single-level methods. The cost function is the same as the one used in Table 5, Table 8 and Fig. 4. Solid circles: multilevel methods; open circles: single-level methods. The performance over the 48-molecule non-metallic firstperiod data set of the selected methods in Table 7 is shown in Fig.6 (for MCG2 we show only version 2s). To guide the eye, Fig. 6 also shows a straight line through the MCCM-UT-MP4SDQ and MCG2 points. The most efficient methods lie to the left of this line. By this criterion the three most efficient methods are SAC-MP2/pDZ, SACMP4SDQ/pDZ, and MCG3 followed by MCG2 and MCCM-UT-MP4SDQ. 5. Summary We have updated our recent parameterizations of the corecorrelation energy, scaling-all-correlation energy, and multi-coefficient correlation methods to larger training sets. The best single methods for calculating atomization energies are the multi-coefficient Gaussian-3 and multicoefficient Gaussian-2 methods, both of which scale in cost as N7 where N is the number of electrons.Among N6 scaling methods, we find the MCCM-UT-MP4SDQ and MCCM-UT-CCSD methods, in which one carries out a polarized double zeta MP4SDQ or CCSD calculation and a polarized triple zeta MP2 calculation, to be particularly efficient.Table 7 Mean errors in kcal mol–1 for selected methods applied to the subset of molecules composed of non-metallic atoms with Z. 9 (48-molecule set) Version Method SAC-MP2/pDZ MCCM-CO-MP2 MCCM-UT-CCSD MCG3 Gaussian-x MCG2 2s 2s SAC-MP4SDQ/pDZ 2sc MCCM-UT-MP4SDQ 2sc 2sc 2sc 231m 1sc 2sc 2s 0.77 (0.79) –0.03 (0.00) 1.02 (1.03) 123 a Cost is the mean CPU time on an SGI Origin 2000 computer with R12000 processors for the four most expensive molecules in this data set, i.e., C4H4O, C4H5N, C5H5N, CF3CN, relative to the mean CPU time (126 s) for an MP2/pDZ calculation on the same computer for the same four molecules.b Values in parentheses are for the full 52-molecule first-period data set. Table 8 Mean errors in kcal mol–1 for standard, single-level methods (0 coefficients, 82-molecule test set) Method MP2/pDZ HF/pDZ+ MP4SDQ/pDZ CCD/pDZ QCISD(T)/Dd QCISD/pDZ MP4SDQ/D2dfp MP4/pDZ MP2/MG3 MP2/T+3df2p CCSD/pDZ QCISD(T)/Tdp MP2/pTZ CCSD(T)/pDZ MP4SDQ/pTZ MP4/T2dfp MP4/pTZ CCSD/pTZ CCSD(T)/pTZ 12.93 a Cost calculated the same way as for Table 5. Fig. 6 Mean unsigned error for the 52-molecule first-period test set as a function of cost for G2, G3, MCG2 and the six other methods in Table 7.The cost function is the same as in Table 8. Squares: N5 methods; circles: N6 methods; diamonds: N7 methods except G2 and G3; up triangle: G3; down triangle: G2. The straight line is shown just to guide the eye. RMSE MSE MUE Coeffs. 10.51 4.45 5.72 2.94 2.66 1.09 1.95 (1.87)b 12.14 5.60 7.20 4.46 3.60 1.44 2.62 (2.54) 1.62 1.84 (1.77) 1.90 (1.85) 1.06 (1.08) –0.55 –1.51 3.15 0.17 0.61 0.20 –0.11 (–0.08) –0.51 0.15 (0.26) 0.54 (0.58) –0.06 (0.01) 1.21 1.23 (1.17) 1.14 (1.12) 0.80 (0.84) 141558249999 Costa MUE MSE RMSE 27.08 128.21 40.05 43.40 35.55 40.85 10.34 33.33 8.48 10.51 41.53 27.79 6.97 35.69 19.36 12.06 9.00 21.13 30.70 149.20 46.45 50.42 40.66 47.71 12.26 38.43 10.51 15.44 48.46 32.47 8.79 41.55 22.81 16.55 10.60 25.12 15.23 –27.08 –128.21 –40.05 –43.40 –35.55 –40.85 –9.88 –33.33 –1.85 –2.18 –41.53 –27.79 –3.17 –35.69 –19.36 –11.93 –9.00 –21.13 –12.93 1125579 11 12 13 14 15 19 31 41 65 267 277 430 If one can tolerate a larger error or cannot afford the triple zeta calculation, scaling all the correlation energy in a double zeta MP4SDQ calculation provides an attractive alternative.Finally, among N5 scaling methods we single out SAC-MP2 with the double zeta basis set and MCCMCO-MP2.References 1 C. C. J. Roothaan, Rev. Mod. Phys., 1951, 23, 69. 2 J. C. Slater, Phys. Rev., 1951, 81, 385. 3 J. A. Pople, J. S. Binkley and R. Seeger, Int. J. Quantum Chem. Symp., 1976, 10, 1. 4 J. Cizek, Adv. Chem. Phys., 1969, 14, 33. 5 M. J. S. Dewar, Faraday Discuss. Chem. Soc., 1977, 62, 197. 6 J. J. P. Stewart, Rev. Comput. Chem., 1990, 1, 45. 7 M. C. Zerner, Rev. Comput. Chem., 1991, 2, 313. 8 F. B. Brown and D. G. Truhlar, Chem. Phys. Lett., 1985, 117, 307. 9 R. Steckler, D. W. Schwenke, F. B. Brown and D. G. Truhlar, Chem. Phys. Lett., 1985, 121, 475. Costa 1 323 34 46 32 153 107 123 123 12310 M. S. Gordon and D. G. Truhlar, J. Am. Chem. Soc., 1986, 108, 5412. 11 M. S. Gordon and D. G.Truhlar, Int. J. Quantum Chem., 1987, 31, 81. 12 M. S. Gordon, K. A. Nguyen and D. G. Truhlar, J. Phys. Chem., 1989, 93, 7356. 13 P. E. M. Siegbahn, M. R. A. Blomberg and M. Svensson, Chem. Phys. Lett., 1994, 223, 5. 14 P. E. M. Siegbahn and R. H. Crabtree, J. Am. Chem. Soc., 1996, 118, 4442. 15 I. Rossi and D. G. Truhlar, Chem. Phys. Lett., 1995, 234, 64. 16 P. L. Fast, J. C. Corchado, M. L. Sánchez and D. G. Truhlar, J. Phys. Chem. A, 1999, 103, 3139. 17 J. C. Corchado and D. G. Truhlar, ACS Symp. Ser., 1998, 712, 106. 18 P. L. Fast, J. C. Corchado, M. L. Sánchez and D. G. Truhlar, J. Phys. Chem. A, 1999, 103, 5129. 19 P. Ho, M. E. Coltrin, J. S. Binkley and C. F. Melius, J. Phys. Chem., 1986, 90, 3399. 20 P. Ho and C. F. Melius, J.Phys. Chem., 1990, 94, 5120. 21 K. Raghavachari and L. A. Curtiss, in Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, ed. S. R. Langhoff; Kluwer, Dodrecht, 1995, p. 139. 22 L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov and J. A. Pople, J. Chem. Phys., 1988, 109, 7764. 23 A. D. Becke, J. Chem. Phys., 1993, 98, 5648. 24 P. L. Fast, M. L. Sánchez, J. C. Corchado and D. G. Truhlar, J. Chem. Phys., 1999, 110, 11679. 25 P. L. Fast, M. L. Sánchez and D. G. Truhlar, Chem. Phys. Lett., 1999, 306, 407. 26 M. R. Nyden and G. A. Petersson, J. Chem. Phys., 1981, 75, 1843. 27 T. H. Dunning, Jr., J. Chem. Phys., 1989, 90, 1007. 28 D. Feller, J. Chem. Phys., 1992, 96, 6104. 29 J. W. Ochterski, G.A. Petersson and J. A. Montgomery, Jr., J. Chem. Phys., 1996, 104, 2598. 30 J. M. L. Martin and P. R. Taylor, Chem. Phys. Lett., 1996, 248, 336. 31 J. M. L. Martin, Chem. Phys. Lett., 1996, 259, 669. 32 T. Helgaker, W. Klopper, H. Koch and J. Noga, J. Chem. Phys., 1997, 106, 9639. 33 A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, H. Koch, J. Olsen and A. K. Wilson, Chem. Phys. Lett., 1998, 286, 243. 34 D. G. Truhlar, Chem. Phys. Lett., 1998, 294, 45. 35 P. L. Fast, M. L. Sánchez and D. G. Truhlar, J. Chem. Phys., 1999, 111, 2921. 36 J. A. Montgomery, M. J. Frisch, J. W. Ochterski and G. A. Petersson, J. Chem. Phys., 1999, 110, 2822. 37 C. W. Bauschlicher, Jr., Theor. Chem. Acc., 1999, 101, 421. 38 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys., 1993, 98, 1358. 39 R. A. Kendall, T. H. Dunning, Jr., J. Chem. Phys., 1992, 96, 6796. 40 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys., 1994, 100, 2975. 41 L. A. Curtiss, K. Raghavachari, P. C. Redfern and J. A. Pople, preprint. 42 P. L. Fast and D. G. Truhlar, J. Phys. Chem. A, 1999, 103, 3802. 43 J. M. L. Martin and G. de Oliveira, J. Chem. Phys., 1999, 111, 1843. 44 R. L. Martin, J. Phys. Chem., 1983, 87, 750. 45 G. S. Kedziora, J. A. Pople, V. A. Rassolov, M. A. Ratner, P. C. Redfern and L. A. Curtiss, J. Chem. Phys., 1999, 110, 7123. 46 W. J. Hehre, L. Radom, P.v.R. Schleyer and J. A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986. 47 L. A. Curtiss, P. C. Redfern, K. Raghavachari, V. Rassolov and J. A. Pople, J. Chem. Phys., 1999, 110, 4703. 48 J. A. Pople, R. Krishnan, H. B. Schlegel and J. S. Binkley, Int. J. Quantum Chem. Symp., 1978, 14, 545. 49 R. J. Bartlett, Annu. Rev. Phys. Chem., 1981, 32, 359. 50 G. D. Purvis and R. J. Bartlett, J. Chem. Phys., 1982, 76, 1910. 51 K. Raghavachari, G. W. Trucks, J. A. Pople and M. Head-Gordon, Chem. Phys. Lett., 1989, 157, 479. 52 J. A. Pople, M. Head-Gordon and K. Raghavachari, J. Chem. Phys., 1987, 87, 5968. 53 G. A. Petersson, D. K. Malick, W. G. Wilson, J. W. Ochterski, J. A. Montgomery, Jr. and M. J. Frisch, J. Chem. Phys., 1998, 109, 10570. 54 J. W. Ochterski, G. A. Petersson and K. B. Wiberg, J. Am. Chem. Soc., 1995, 117, 11299. 55 J. Frisch et al., GAUSSIAN98 (Revision A.7), Gaussian, Inc., Pittsburgh PA, 1998. 56 J. M. L. Martin, A. Sundermann, P. L. Fast and D. G. Truhlar, to be published. 57 K. Raghavachari and J. B. Anderson, J. Phys. Chem., 1996, 100, 12960. Paper 9/08207G PhysChemComm © The Royal Society of Chemistry 1999

ISSN:1460-2733    

DOI:10.1039/a908207g

出版商:RSC    

年代:1999

数据来源: RSC