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JOURNAL OF CHEMICAL PHYSICS

VOLUME 113, NUMBER 4

22 JULY 2000

Thermochemical analysis of core correlation and scalar relativistic effects on molecular atomization energies Jan M. L. Martina) and Andreas Sundermann Department of Organic Chemistry, Kimmelman Building, Room 262, Weizmann Institute of Science, 76100 Rehគ ovot, Israel

Patton L. Fast and Donald G. Truhlar Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431

共Received 13 January 2000; accepted 27 April 2000兲 Core correlation and scalar relativistic contributions to the atomization energy of 120 first- and second-row molecules have been determined using coupled cluster and averaged coupled-pair functional methods and the MTsmall core correlation basis set. These results are used to parametrize an improved version of a previously proposed bond order scheme for estimating contributions to atomization energies. The resulting model, which requires negligible computational effort, reproduces the computed core correlation contributions with 88%–94% average accuracy 共depending on the type of molecule兲, and the scalar relativistic contribution with 82%–89% accuracy. This permits high-accuracy thermochemical calculations at greatly reduced computational cost. © 2000 American Institute of Physics. 关S0021-9606共00兲30328-2兴

I. INTRODUCTION

The core correlation contribution was evaluated in that paper using the CCSD共T兲 共coupled cluster with all single and double substitutions and quasiperturbative correction for connected triple excitations兲 electron correlation method6 and the MTsmall core correlation basis set.5 Methods that neglect connected triple excitations such as CCSD or the inexpensive MP2 method were shown there to underestimate the core correlation contribution by 50% or more, while the MTsmall basis set was found to be the smallest one that consistently reproduced near-basis-set-limit values for core correlation contributions to molecular atomization energies. Hence no appreciable reduction in CPU time for this contribution can be achieved without compromising accuracy. And because of the asymptotic n 3 N 4 共with n the number of electrons correlated and N the number of molecular orbitals included兲 CPU time dependence of the CCSD共T兲 method, the total CPU time for calculations on second-row systems will be dominated by the core correlation step to an even greater extent than for first-row systems like benzene. In short, a reliable approximation to the core correlation 共and, if possible, scalar relativistic兲 contributions to the binding energy would greatly increase the efficiency of accurate computational thermochemistry schemes. A simple bond-order approximation to the core correlation effects on binding energies has recently been proposed;7 it involves parametrizing the core correlation contribution to the binding energy in terms of the number of heavy-atom bonds in a molecule. The original parametrization7 was considered valid for B, C, N, O, F, and Si, and it was later updated to include Li, Be, Al, P, S, and Cl.8 It was originally7 carried out against a collection of calculated data at a variety of levels of theory for 72 molecules, later augmented8 with relatively low-level calculations for 51 additional molecules.

Most ab initio electronic structure calculations that are performed today are nonrelativistic and include only the correlation of the valence electrons. The latter approximation is often referred to as the frozen core 共FC兲 approximation. In order to obtain quantitatively correct electronic energies one needs to include the relativistic effects and the contribution of the inner-shell 共‘‘core’’兲 electrons to the correlation energy. To include the core correlation contribution one needs to perform an electronic structure calculation without the use of the frozen core approximation and add extra basis functions in the core region 共e.g., Refs. 1 and 2兲; standard basis sets3,4 do not include these. Explicitly including the correlation of core electrons significantly increases the computational cost, due both to the larger number of many-electron configurations that need to be considered and also to the need for larger one-electron basis sets. An example will illustrate this point. In the recent W1 and W2 theory paper,5 a W1 calculation of the total atomization energy of benzene was presented which reproduced the experimentally derived total atomization energy 共1368.2 kcal/mol兲 to within better than 0.1 kcal/ mol. While the core correlation and scalar relativistic contributions 共7.1 and ⫺1.0 kcal/mol, respectively兲 constitute only a small proportion of the total, they clearly cannot be neglected for even medium-accuracy thermochemical work. Yet while the entire valence correlation treatment required about 19 h of CPU time on an SGI Octane workstation, the inner-shell correlation calculation required 68 h on the same platform, followed by an additional 5 h for the scalar relativistic contribution. a兲

Electronic mail: [email protected]

0021-9606/2000/113(4)/1348/11/$17.00

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J. Chem. Phys., Vol. 113, No. 4, 22 July 2000

Core and relativistic effects

In the present article, we have calculated core correlation contributions for 120 molecules at one consistent level of theory, namely CCSD共T兲/MTsmall. The parameters in the bond-order model have been reoptimized in several ways with this new additional data, and a revision of the model itself is proposed. In addition we consider a similar treatment of scalar relativistic effects. Relativistic effects may be divided into vector terms 共depending on spin and orbital angular momentum兲 and scalar terms 共depending primarily on electron speed兲. 共Although the division is unique only to first order, that should be sufficient for the first- and second-row elements considered here.兲 In most first- and second-row cases, the spin-orbit coupling contributions are trivially calculated from atomic fine structures, and we will not consider them further here. The scalar relativistic terms considered in this article are those associated with the first-order Darwin and mass-velocity terms in the Hamiltonian.9,10 Comparison with more rigorous relativistic calculations has shown11,12 that while this approximation incurs small errors for third-row main group compounds,13 it is quite sufficient for first- and second-row compounds which are the only ones considered in the present article. II. DATA SETS AND COMPUTATIONAL METHODOLOGY

We will use the following definition of the core correlation contribution to the binding energy 共total atomization energy兲, D (c) : minus the difference between the core correlation energy of a molecule and the core correlation energy of all its constituent atoms,

冋

D (c) ⫽⫺ E 共 L共Full兲/Bcore ;Re兲 ⫺

兺A E 共 L共Full兲/Bcore ;atom A 兲

冉

⫺ E 共 L共FC兲/Bcore ;Re兲 ⫺

兺A E 共 L共FC兲/Bcore ;atom A 兲

冊册

,

共1兲

where L is the many-electron level of theory, ‘‘Full’’ denotes that both the core and valence electrons are correlated, ‘‘FC’’ denotes that the frozen core approximation is used 共i.e., all inner-shell orbitals are constrained to be doubly occupied兲, Re denotes the computed equilibrium geometry at some appropriate level of theory, and Bcore is a one-electron basis set with added inner-shell correlation functions 共i.e., ‘‘polarization’’ functions with spatial extent similar to the occupied core orbitals兲 as well as added radial flexibility in the innershell orbitals. In the same manner we will use the following definition of the scalar relativistic contribution to the binding energy, D (r) : minus the difference between the scalar relativistic energy of a molecule and that of all its constituent atoms, D (r) ⫽⫺ 关 E (r) 共 Re兲 ⫺E (r) 共 atoms兲兴 .

共2兲

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In the present work, E (r) is the expectation value of the firstorder Darwin and mass-velocity operators evaluated using an averaged coupled-pair functional14,15 共ACPF兲 wave function with basis MTsmall. All electrons are correlated in this calculation. In the rest of this paper D (c) and D (r) will simply be called the core correlation binding energy and the relativistic binding energy, respectively. Two databases of core correlation binding energies have been assembled previously; the first7 included 72 molecules and was based on values compiled from the literature.16–27 The second8 includes these 72 molecules plus an additional 51 molecules; core correlation binding energies for all these 123 molecules were calculated by applying Eq. 共1兲 at the MP2 关second-order Mo” ller-Plesset perturbation theory3兴 level with the G3large28 basis set. All of the geometries were optimized at the MP2共Full兲/6-31G(d) level of theory, which is henceforth denoted ‘‘//M.’’ This level of reference geometry, MP2共Full兲/6-31G(d), is used in the popular G229 and G328 computational thermochemistry models, as well as in the parametrization of the MCG2 and MCG3 methods.30 However, it exhibits known deficiencies for molecules with even very mild nondynamical correlation effects 共for which MP2 is inadequate兲 and for polarly bound second-row systems 共where ‘‘inner polarization effects’’ 31 require the addition of high-exponent d functions兲. In the present article we have created a new database of core correlation values for 125 molecules. For all molecules except five, CCSD共T兲/MTsmall calculations of the core correlation binding energy, as well as ACPF/MTsmall calculations of the Darwin and mass-velocity corrections, were carried out at reference geometries obtained using the B3LYP and 共Becke three-parameter hybrid exchange,32 33 Lee–Yang–Parr correlation兲 density functional method and the cc-pVTZ⫹1 basis set.31 This latter basis set consists of the standard cc-pVTZ basis set of Dunning and co-workers,4 augmented with a single high-exponent d function on every second-row atom to accommodate inner polarization effects 共not to be confused with inner-shell correlation兲. Such geometries, used in W1 theory5 and denoted ‘‘//B’’ throughout the article, are generally within about 0.003 Å of experiment.34 关Note that for open-shell species, the restricted open-shell RCCSD共T兲 method as defined in Ref. 35 has been used throughout. Throughout this article we use the abbreviation CC to denote RCCSD共T兲.兴 In short, D (c) 共 CC//B兲 ⫽D e 关 CC共Full兲/MTsmall//B兴 ⫺D e 关 CC共FC兲/MTsmall//B兴 ,

共3兲

where D e denotes the zero-point-exclusive 共‘‘bottom of the well’’兲 atomization energy, and D (r) ⫽⫺ 关 E (r) 共 molecule,ACPF共Full兲/MTsmall//B兲 ⫺E (r) ⫺ 共 atoms,ACPF共Full兲/MTsmall兲兴 .

共4兲

For the five remaining molecules, namely PO, HPO, PO2, HPO2, and PO3 , D (c) and D (r) values were taken from a very recent study by Bauschlicher.36 These results were obtained at levels of theory that meet or exceed the standards for the other 120 molecules, and they should be quite close to CCSD共T兲/MTsmall//B results.

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For comparison purposes, we have also calculated D (c) at the MP2/G3large level at MP2/6-31G* reference geometries for the 125-molecule data set. This difference can be written as D (c) 共 MP2///M兲 ⫽D e 共 MP2共Full兲/G3large//M兲 ⫺D e 共 MP2共FC兲/G3large//M兲 ,

共5兲

where MP2//M is used in Eq. 共5兲 and throughout the article to denote that D (c) is calculated at the MP2/G3large//M level of theory, and D e denotes the zero-point-exclusive 共‘‘bottom of the well’’兲 atomization energy. In addition, in order to gauge dependence of our results on the level of theory used for the reference geometry, we have calculated D (c) for a 98-molecule subset of the 125-molecule data set at the CCSD共T兲/MTsmall level using MP2/6-31G* reference geometries:

共6兲

where CC//M is used in Eq. 共6兲 and throughout the article to denote that D (c) is calculated at the CC/MTsmall//M level of theory. Literature values and CC//B, CC//M, and MP2//M values for the core-correlation binding energies of all molecules in the 125-molecule data set are given in Table I. The relativistic binding energy, D (r) , has been calculated for 120- and 98-molecule subsets of the 125-molecule data set using Eq. 共2兲 at the ACPF/MTsmall//B3LYP/cc-pVTZ⫹1 共abbreviated ACPF//B兲 and ACPF/MTsmall//MP2/6-31G* 共abbreviated ACPF//M兲 levels of theory, respectively. The remaining five relativistic binding energies, namely PO, PO2, PO3, POH, and PO2H were taken from the literature.36 The ACPF//B, ACPF//M, and literature values of D (r) are given in Table I. From the scalar relativistic contributions in Table I it is easy to see that the scalar relativistic contributions exhibit a very weak dependence on the reference geometry. The MP2//M calculations were carried out with 37 GAUSSIAN98 in Minnesota. The CC//B, CC//M, and scalar relativistic calculations were carried out with MOLPRO38 at the Weizmann Institute.

A. Original method: Based on number of bonds

In the original bond-order scheme,7,8 the core correlation binding energy is approximated by a sum over all the atoms, ␣, in a given molecule

兺␣ D Z(c) n ␣ , ␣

兺␣ D Z(r) n ␣ ,

共8兲

␣

where D Z(r) denotes the average scalar relativistic contribu␣ tion to the binding energy per bond. Although D (r) is not strictly zero for hydrogen, lithium, and beryllium atoms, it is so small that we set it equal to zero in the fit to ensure a more physical fit. This fit of the scalar relativistic effects will be referred to as version-1.

Note that n ␣⫽

兺

␤⫽␣

共9兲

B ␣␤

where B ␣␤ is the formal bond order 共as derived from the dominant Lewis structure兲 between atoms ␣ and ␤. Thus, Eqs. 共6兲 and 共7兲 may be rewritten D (x) ⫽

兺␣ ␤兺⫽ ␣ D Z(x) B ␣␤ , ␣

共10兲

where x⫽c or r. A natural generalization is D (x) ⫽

兺␣ ␤兺⫽ ␣ D Z(x)共 B ␣␤ 兲 B ␣␤ , ␣

共11兲

that is, we allow the coefficients D Z(x) (B ␣␤ ) to depend on the ␣ formal bond order B ␣␤ . We will refer to this added flexibility as ‘‘splitting the coefficients.’’ In fact, Eq. 共11兲 is recovered from a bond-equivalent model of the type D (x) ⫽

(x) 共 B ␣␤ 兲 B ␣␤ 兺␣ ␤兺⫽ ␣ D ␣␤

共12兲

共in which the parameters are dependent both on the formal bond type and on the formal bond order兲 upon introducing— for the purpose of reducing the number of adjustable parameters—the Mulliken-type approximation

III. PARAMETRIZATION

D (c) ⫽

D (r) ⫽

B. New method: Based on bond order

D 共 c 兲 共 CC//M兲 ⫽D e 共 CC共Full兲/MTsmall//M兲 ⫺D e 共 CC共FC兲/MTsmall//M兲 ,

bond order scheme because, in counting bonds, a double bond is counted twice and a triple bond is counted three times. In the same manner the relativistic binding energy is parametrized by

共7兲

where D Z(c) is the average core-correlation contribution to the ␣ binding energy per bond by a given atom ␣ with atomic number Z ␣ , and n ␣ is the number of bonds that atom ␣ makes with other atoms in the molecule. Since the innershell correlation energy of the hydrogen atom is trivially zero (c) 共it has no core兲, we set D H equal to zero. We call this a

(x) ⬇ 21 共 D Z(x) ⫹D Z(x) 兲 . D ␣␤ ␣ ␤

共13兲

After extensive testing we found that there is no benefit in splitting the coefficients for core correlation energy for Be, B, C, and P; in addition, our data base contains only single bonds for Li, F, Al, and Cl. Therefore we only retained a (c) (2) single D Z(x) for these atoms. For nitrogen we set D N ␣ (c) (c) (c) ⫽D N (3), denoted D N (2,3), but we allowed D N (1) to be different. For Si we also distinguished separate values for (c) (c) D Si (1) and D Si (2,3). Note that we will still count double bonds as two and triple bonds as three; however, the D Z(c) ␣ values will be different for singly bonded N and Si and for N and Si forming multiple bonds. The final fits for the core

J. Chem. Phys., Vol. 113, No. 4, 22 July 2000

Core and relativistic effects

TABLE I. Literature values and MP2//M, CC//M, and CC//B calculations of the core-correlation contribution to the binding energy and ACPF//M and ACPF//B calculations of the scalar relativistic contribution to the atomization energy 共kcal/mol兲. D (c)

Molecule AlCl AlF AlF3 AlH AlH2 AlH3 BeCl BeF BeH BeH2 BeO BeOH BeS BF BF2 BF3 BH BH2 BH3 BO C2Cl2 C2ClF C2F2 C2F4 C2H C2H2 C2H3 C2H4 C2H5 C2H6 cyclo-C3H2( 1 A 1 ) cyclo-C3H2( 3 B 1 ) cyclo-C3H3 cyclo-C3H⫹ 3 cyclo-C3H4 cyclo-C3H6 C4H6 共bicyclobutane兲 C4H6 共trans-butadiene兲 C6H6 CCH2 CF4 CH CH2( 1 A 1 ) CH2( 3 B 1 ) cyclo-CH2CH2C( 1 A 1 ) cyclo-CH2CH2C( 3 B 1 ) CH2OH CH3 CH3Cl CH3CN CH3F CH3NH2 CH3OH CH3SH CH4 Cl2 ClCN ClF ClO CN CO

Literature valuea 0.30 ⫺0.18

0.50 0.89

0.70 1.92 0.16 0.79 1.05

2.05 2.44 2.21 2.36 2.42 2.61 3.24 3.18 3.60 3.80 4.07

1.86 0.14 0.45 0.79 3.38 3.39 1.04

2.05 1.25 0.48 1.76 0.08 1.18 0.96

D (r)

MP2/ G3large// MP2共Full兲/ 6-31G*

CCSD共T兲/ MTsmall// MP2共Full兲/ 6-31G*

CCSD共T兲/ MTsmall// B3LYP/ cc-pVTZ⫹1

ACPF/ MTsmall// MP2共Full兲/ 6-31G*

ACPF/ MTsmall// B3LYP/ cc-pVTZ⫹1

1.35 0.71 ⫺0.08 ⫺0.02 ⫺1.15 ⫺0.87 0.96 0.58 0.02 0.33 0.86 1.00 1.21 0.66 1.08 1.79 0.15 0.39 0.65 0.99 3.85 3.40 2.92 3.24 1.90 2.07 2.04 2.13 2.12 2.29 2.94 2.82 3.18 3.20 3.40 3.59 4.87 4.47 7.02 1.68 1.96 0.19 0.46 0.62 2.94 2.82 1.44 0.93 1.72 2.94 1.26 1.91 1.56 1.93 1.16 1.05 2.42 0.48 0.70 1.40 1.07

0.23 0.23 ⫺0.93 ⫺0.19 ⫺0.92 ⫺1.02 0.88 1.10 0.50 0.88 1.49 1.46 1.04 0.61

0.19 0.24 ⫺0.78 ⫺0.19 ⫺0.91 ⫺0.97 0.84 1.14 0.51 0.89 1.69 1.49 1.08 0.67 1.30 1.77 0.19 0.82 1.09 1.46 2.83 2.82 2.82 2.57 2.08 2.40 2.15 2.30 2.24 2.34 2.95 3.07 3.37 3.26 3.55 3.61 4.74 4.63 7.09 1.73 1.04 0.14 0.38 0.79 2.92 3.07 1.35 1.04 1.20 2.87 1.12 1.72 1.38 1.36 1.21 0.16 1.77 0.08 0.20 1.11 0.91

⫺0.251 ⫺0.280 ⫺1.326 ⫺0.085 ⫺0.381 ⫺0.452 ⫺0.271 ⫺0.226 ⫺0.022 ⫺0.022 ⫺0.108 ⫺0.330 ⫺0.254 ⫺0.156

⫺0.253 ⫺0.281 ⫺1.314 ⫺0.085 ⫺0.380 ⫺0.445 ⫺0.274 ⫺0.225 ⫺0.022 ⫺0.022 ⫺0.124 ⫺0.324 ⫺0.253 ⫺0.154 ⫺0.438 ⫺0.682 ⫺0.019 ⫺0.067 ⫺0.075 ⫺0.220 ⫺0.837 ⫺0.767 ⫺0.700 ⫺1.025 ⫺0.269 ⫺0.268 ⫺0.301 ⫺0.326 ⫺0.369 ⫺0.389 ⫺0.390 ⫺0.495 ⫺0.496 ⫺0.443 ⫺0.525 ⫺0.581 ⫺0.746 ⫺0.651 ⫺0.989 ⫺0.193 ⫺0.818 ⫺0.037 ⫺0.084 ⫺0.150 ⫺0.445 ⫺0.523 ⫺0.444 ⫺0.171 ⫺0.422 ⫺0.411 ⫺0.370 ⫺0.458 ⫺0.454 ⫺0.599 ⫺0.192 ⫺0.148 ⫺0.434 ⫺0.123 ⫺0.192 ⫺0.170 ⫺0.142

1.70 0.19 0.81 1.08

2.20 2.28 2.20 2.25 2.22 2.33 2.85 2.99

3.47 3.60 4.71

1.66 1.01 0.14 0.38 0.79 2.88 3.06 1.30 1.03 1.22 2.67 1.70 1.36 1.37 1.20 0.16 0.06 0.16 1.28 0.77

⫺0.683 ⫺0.019 ⫺0.067 ⫺0.076

⫺0.258 ⫺0.281 ⫺0.296 ⫺0.330 ⫺0.369 ⫺0.389 ⫺0.393 ⫺0.499 ⫺0.531 ⫺0.582 ⫺0.746 ⫺0.198 ⫺0.817 ⫺0.037 ⫺0.084 ⫺0.150 ⫺0.445 ⫺0.528 ⫺0.438 ⫺0.171 ⫺0.427 ⫺0.423 ⫺0.455 ⫺0.451 ⫺0.601 ⫺0.192 ⫺0.147 ⫺0.110 ⫺0.174 ⫺0.160 ⫺0.150

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J. Chem. Phys., Vol. 113, No. 4, 22 July 2000 TABLE I. 共Continued.兲 D (c)

Molecule CO2 CS CSiH4 CSiH6 F2 FCCH FCN FH FLi H2CCCH2 H2CO H2O H2O2 H2SiO HBO HCCCl HCl HCN HCO HCOOH HCP HLi HNO HOCl HPO LiCl LiN LiOH N2 N2H4 N2O NH NH2 NH3 NO O2 OH P2 PH PH2 PH3 PN PO PO2 PO2H PO3 S2 SH SH2 Si2 Si2H4 Si2H6 SiF SiF⫹ SiF4 SiH SiH2( 1 A 1 ) SiH2( 3 B 1 ) SiH3 SiH4 SiO SiS SO SO2 a

Literature valuea 1.78 0.73 ⫺0.07 2.43 0.18 1.13 1.32 0.38 1.27 0.09

0.15 1.67 1.46

0.32 0.48 0.31 0.61

0.85 1.68 1.26 0.11 0.31 0.66 0.42 0.30 0.14 0.60

0.30 0.71 0.66 0.91 1.14 1.07

0.34

0.57 0.67 ⫺0.04 ⫺0.11 ⫺0.56 ⫺0.22 ⫺0.31 0.72 0.77 0.40

MP2/ G3large// MP2共Full兲/ 6-31G* 2.16 1.54 1.00 0.93 0.16 2.48 1.92 0.26 0.88 3.36 1.39 0.48 0.70 ⫺0.14 1.50 2.94 0.54 1.60 1.19 2.00 1.79 0.25 0.83 0.81 0.59 1.01 0.71 1.31 1.00 1.51 1.80 0.18 0.44 0.79 0.95 0.44 0.21 1.29 ⫺0.18 ⫺0.02 0.21 1.11 0.81 0.74 1.38 1.25 1.24 0.26 0.65 1.00 0.00 ⫺0.30 0.39 0.57 0.85 ⫺0.16 ⫺0.15 ⫺1.14 ⫺0.85 ⫺0.63 0.96 1.59 0.63 1.12

D (r)

CCSD共T兲/ MTsmall// MP2共Full兲/ 6-31G*

CCSD共T兲/ MTsmall// B3LYP/ cc-pVTZ⫹1

ACPF/ MTsmall// MP2共Full兲/ 6-31G*

ACPF/ MTsmall// B3LYP/ cc-pVTZ⫹1

⫺0.458 ⫺0.150

0.25 0.32 0.25

1.69 0.72 0.95 0.93 ⫺0.07 2.60 1.70 0.17 1.02 3.58 1.27 0.36 0.36 0.03 1.82 2.61 0.15 1.62 1.13 1.59 1.40 0.32 0.44 0.28

⫺0.002 ⫺0.235 ⫺0.275

0.49 0.98 1.35 0.55 0.95 0.88 0.11 0.31 0.61 0.45 0.13 0.14 0.46 0.08 0.15 0.21 0.47

0.61 0.62 1.43 0.78 0.97 1.18 0.11 0.31 0.62 0.44 0.25 0.14 0.63 0.08 0.15 0.21 0.74

⫺0.316 ⫺0.087 ⫺0.289 ⫺0.127 ⫺0.458 ⫺0.416 ⫺0.067 ⫺0.150 ⫺0.252 ⫺0.163 ⫺0.128 ⫺0.120 ⫺0.221 ⫺0.136 ⫺0.293 ⫺0.470 ⫺0.174

0.39 0.13 0.24 0.17

0.43 0.12 0.24 0.15 ⫺0.50 ⫺0.58 0.24 0.29 0.08 ⫺0.05 ⫺0.11 ⫺0.61 ⫺0.45 ⫺0.37 0.66 0.47 0.42 0.83

⫺0.267 ⫺0.191 ⫺0.409 ⫺0.158

⫺0.454 ⫺0.146 ⫺0.787 ⫺0.906 0.030 ⫺0.478 ⫺0.382 ⫺0.196 ⫺0.224 ⫺0.464 ⫺0.323 ⫺0.264 ⫺0.345 ⫺0.786 ⫺0.224 ⫺0.552 ⫺0.257 ⫺0.206 ⫺0.264 ⫺0.575 ⫺0.323 ⫺0.002 ⫺0.242 ⫺0.285 ⫺0.421b ⫺0.312 ⫺0.086 ⫺0.287 ⫺0.110 ⫺0.463 ⫺0.411 ⫺0.067 ⫺0.149 ⫺0.252 ⫺0.163 ⫺0.149 ⫺0.120 ⫺0.217 ⫺0.136 ⫺0.291 ⫺0.462 ⫺0.158 ⫺0.275b ⫺0.856b ⫺0.731b ⫺1.152b ⫺0.273 ⫺0.191 ⫺0.406 ⫺0.157 ⫺1.171 ⫺1.384 ⫺0.274 ⫺0.011 ⫺1.880 ⫺0.102 ⫺0.218 ⫺0.573 ⫺0.623 ⫺0.703 ⫺0.244 ⫺0.264 ⫺0.311 ⫺0.707

1.52 0.67 ⫺0.09 2.45 0.16 1.07 1.20 0.35 0.32 ⫺0.11 0.15 1.44 1.05 1.51

⫺0.59 0.22 0.24 ⫺0.04 ⫺0.11 ⫺0.60 ⫺0.46 ⫺0.39 0.52 0.31 0.55

All literature values are given in Refs. 5, 16–27, and 36. Literature values 共Ref. 36兲.

b

0.040 ⫺0.488 ⫺0.195 ⫺0.223 ⫺0.325 ⫺0.263 ⫺0.334 ⫺0.798 ⫺0.257 ⫺0.219 ⫺0.267 ⫺0.575

⫺1.385 ⫺0.272 ⫺0.009 ⫺0.102 ⫺0.220 ⫺0.572 ⫺0.625 ⫺0.706 ⫺0.248 ⫺0.290 ⫺0.643

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Core and relativistic effects

TABLE II. Comparison of basis sets for selected core-correlation binding energies,a in kcal/mol. Molecule

CCSD共T兲/G3large

CCSD共T兲/MTsmall

CCSD共T兲/ACVQZb

Cl2 ClCN ClF CS HOCl OCS SO

0.69 2.46 0.42 1.23 0.69 2.60 0.91

0.18 1.71 0.09 0.66 0.29 1.58 0.42

0.19 1.76 0.08 0.75 0.31 1.68 0.46

a

All values taken from Ref. 5. ACVQZ or equivalent.

b

correlation binding energy based on Eqs. 共7兲 and 共11兲 will be denoted version 3 共v3兲 and version 4 共v4兲, respectively. For the scalar relativistic contributions we retained only a single D Z(r) for H, Li, Be, B, F, Al, and Cl, while introduc␣ (r) (1), ing the new parameters D C(r) (1), D C(r) (2,3), D N (r) (r) (r) (r) (r) D N (2,3), D O (1), D O (2,3), D Si (1), D Si (2,3), D P(r) (1), D P(r) (2,3), D S(r) (1), and D S(r) (2,3). As noted above we will still count double bonds as two and triple bonds as three; however, the D Z(r) values will be different for C, N, O, Si, P, ␣ and S atoms exhibiting multiple-bond character than for singlebond character. For conjugated systems we use the major resonance structure for counting single, double, and triple bonds. For example, in C6H6 共benzene兲 we count 12 for the three double bonds and 12 for the 6 C–H and 3 C–C single bonds. This fit to the scalar relativistic effects will be referred to as version-2.

IV. CALCULATIONS AND RESULTS

Evaluating the inner-shell correlation effects with the G3large basis set greatly overestimates the inner-shell correlation energy. This overestimation is exhibited in Table II with values taken from Ref. 5, which were computed at the CCSD共T兲 level using an ACVQZ basis set2 for first-row atoms and the MTavqz basis set1,5 共of equivalent quality兲 for second-row atoms. These basis sets, which include up to g functions and have 109 basis functions per first-row atom and 144 basis functions per second-row atoms, are much more extended than either the MTsmall 共62 and 79 contracted basis functions for first and second row atoms兲 or the G3large 共42 and 66 contracted basis functions for first and second row atoms兲 basis sets—both of which only include up to f functions—and are known to be very close to the basis set limit for the D (c) contributions 共see e.g. Refs. 5 and 21兲. The deviation of the CCSD共T兲/G3large or CCSD共T兲/ MTsmall results from the CCSD共T兲/兵ACVQZ,MTavqz其 D (c) values thus seems to be a reasonable error estimate. We find that the mean unsigned error 共MUE兲 for the G3large basis set is 0.54 kcal/mol, while that for the MTsmall basis set is an order of magnitude lower, 0.05 kcal/mol. From this comparison we conclude that the G3large basis is simply too small to capture the inner-shell correlation effects quantitatively. Therefore, any parametrizations we will do with G3large data are for comparison purposes only. We do not recom-

1353

mend using any of the G3large parametrizations, denoted MP2//M in tables, given in this article, except perhaps as a shortcut in G3 calculations. The number, n ␣ , of bonds for each heavy atom 共atoms heavier than H兲 in a molecule was counted for each of the 125 molecules. We used linear regression techniques and Eq. 共7兲 to obtain D Z(c) values. The D Z(c) values based on our most ␣ ␣ accurate calculations—as well as standard deviations from the least-squares fitting—are given in Table III, where they are labeled v3 共version-3兲. We consider these to be more accurate than the version-1 共Ref. 7兲 and version-2 共Ref. 8兲 coefficients published previously. Recall that the v3 values are obtained by fitting the 125 most accurate values of D (c) 关 120D (c) 共CC//B兲 values and the 5 D (c) literature values,36 namely PO, PO2, PO3, POH, PO2H, and PO3H兴. To test the geometry sensitivity 共GS兲 of the core correlation binding energy we calculated D Z(c) values for a 125 molecule data set ␣ composed of 98 CC//M values, 22 CC//B values, and 5 literature values 共see Table I兲. For comparing the level of core correlation treatment used here to G3 theory, the MP2//M column of Table IV gives the D Z(c) values obtained from ␣ fitting a 125-molecule data set composed of 125 MP2//M values. The number of molecules containing atoms of each atomic number are also given in Table III, as are the mean unsigned errors 共MUE兲 and mean unsigned values 共MUV兲 over these respective subsets. The D Z(c) values for the version-4 core correlation are ␣ given in Table IV along with the MUEs, MUVs, and the number of molecules in the database containing each unique atom type being retained. We have fit the scalar-relativistic contribution to the binding energy in the same manner as above. For this fit we use the relativistic binding energies for 120 molecules calculated by ACPF//B and 5 literature values,36 namely for PO, PO2, PO3, POH, and PO2H. The D Z(r) values thus obtained ␣ are given in Table V where they are called v1 共version-1 of relativistic parameters兲. As a test of geometry sensitivity, Table V also lists coefficients for a fit to 98 CC//M values, 22 CC//B values, and 5 literature values36 共see Table I兲. The MUEs and MUVs for each atom type are also given in Table V. The D Z(r) values for the version-2 scalar-relativistic ef␣ fects are given in Table VI, again along with the MUEs, MUVs, and the number of molecules in the database for each species.

V. DISCUSSION A. Original fitting analysis

1. Core correlation

The results in Tables I and in Table IV are very informative. In particular they show that the MP2/G3large//MP2/ 6-31G* method, which is an intrinsic part of the 28 GAUSSIAN-3 共G3兲 method, shows systematic deviations from the more accurate CCSD共T兲/MTsmall//B3LYP/ cc-pVTZ⫹1 results, especially for Be, N, O, P, S, and Cl.

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Martin et al.

J. Chem. Phys., Vol. 113, No. 4, 22 July 2000 TABLE III. Mean unsigned errors and mean unsigned values and parameters, D Z(c)␣ kcal/mol, optimized in this work. D Z(c)␣

v3 Atom type H Li Be B C N O F Al Si P S Cl Full data set H,B-F,P-Cl H,B-F only H,C,N,O,F Hydrocarbons

D Z(c)␣

MUEa

MUVa

GSb

MP2/ G3large

No. of molecules

关0兴c 0.668共94兲d 0.616共53兲 0.474共34兲 0.297共4兲 0.140共15兲 0.147共16兲 0.081共24兲 ⫺0.311(43) ⫺0.056(13) 0.113共20兲 0.086共27兲 0.150共49兲

0.12 0.30 0.29 0.18 0.11 0.10 0.18 0.20 0.22 0.18 0.09 0.17 0.14 0.15 0.12 0.13 0.12 0.07

1.40 0.80 1.09 1.14 2.15 1.06 0.91 1.13 0.54 0.40 0.69 0.63 1.06 1.23 1.45 1.72 1.81 2.69

关0兴c 0.708共99兲d 0.595共56兲 0.442共34兲 0.295共4兲 0.120共16兲 0.118共16兲 0.078共25兲 ⫺0.335(45) ⫺0.058(14) 0.112共21兲 0.071共29兲 0.155共51兲

关0兴c 0.540共133兲 0.260共75兲 0.270共45兲 0.289共5兲 0.213共21兲 0.176共22兲 0.247共33兲 ⫺0.249(61) ⫺0.024(18) 0.156共29兲 0.313共39兲 0.651共69兲

78 5 7 8 51 16 30 20 6 16 11 9 13 125 91 63 55 24

a

mean unsigned error and mean unsigned value for the subset of molecules containing the specified atoms. test for sensitivity to the reference geometry 共see text兲. c Constrained to zero. d Standard deviations on last digits given in parentheses. b

The CCSD共T兲/MTsmall//MP2/6-31G* results provide an interesting intermediate case since they use the same geometries as the MP2/G3large results but the same electronic structure level as our best results. Table I shows that the CCSD共T兲/ results often differ significantly between the MP2/ TABLE IV. The version-4 D Z(c)␣ parameters and the associated mean unsigned errors, mean unsigned values 共kcal/mol兲, and number of molecules containing the particular atom type. Atom type H Li Be B C N共1兲 N共2,3兲 O共1兲 O共2,3兲 F Al Si共1兲 Si共2,3兲 P S共1兲 S共2,3兲 Cl Full data set H,B–F,P–Cl H,B–F H,C,N,O,F a

v4

MUEa

MUVa

No. of molecules

关0兴b 0.651共90兲c 0.623共51兲 0.477共32兲 0.297共4兲 0.162共26兲 0.133共18兲 0.179共28兲 0.124共20兲 0.088共23兲 ⫺0.314(41) ⫺0.092(17) 0.027共28兲 0.115共20兲 0.121共63兲 0.064共30兲 0.146共47兲

0.12 0.28 0.28 0.19 0.11 0.10 0.08 0.19 0.14 0.21 0.23 0.16 0.15 0.09 0.06 0.20 0.14 0.14 0.12 0.12 0.11

1.40 0.80 1.09 1.14 2.15 0.68 1.26 0.90 0.96 1.13 0.54 0.40 0.46 0.69 0.64 0.66 1.06 1.23 1.45 1.72 1.81

78 5 7 8 51 7 10 14 21 20 6 13 6 11 4 6 13 125 91 63 55

mean unsigned error and mean unsigned value for the subset of molecules containing the specified atoms. b Constrained to zero. c Standard deviations on last digits given in parentheses.

6-31G* and B3LYP/cc-pVTZ⫹1 reference geometries, yet Table IV shows that on the average these deviations tend to wash out, and the average values are very similar. Thus the deviations of the CCSD共T兲/MTsmall//B3LYP/cc-pVTZ⫹1 results from the MP2/G3large//MP2/6-31G* results are primarily due to the higher level of electron correlation and better basis set, not to the reference geometries.

TABLE V. Mean unsigned errors and mean unsigned values and parameters, D Z(r)␣ kcal/mol, optimized in this work. v1 Atom type H Li Be B C N O F Al Si P S Cl Full data set No Al or Si First row only a

D Z(r)␣ 关0兴 关0兴c 关0兴c 关0兴c ⫺0.043(3) d ⫺0.037(11) ⫺0.093(12) ⫺0.198(17) ⫺0.180(32) ⫺0.151(10) ⫺0.094(15) ⫺0.075(20) ⫺0.166(36) c

MUEa 0.07 0.06 0.07 0.05 0.07 0.09 0.11 0.13 0.10 0.20 0.18 0.18 0.10 0.11 0.09 0.07

MUVa

GSb

No.

0.38 0.18 0.18 0.23 0.45 0.26 0.38 0.52 0.46 0.63 0.46 0.35 0.37 0.39 0.35 0.32

关0兴 关0兴c 关0兴c 关0兴c ⫺0.043共3兲d ⫺0.038(11) ⫺0.090(11) ⫺0.198(17) ⫺0.183(32) ⫺0.151(10) ⫺0.096(15) ⫺0.071(20) ⫺0.164(36)

78 5 7 8 51 16 30 20 6 16 11 9 13 125 103 72

c

Mean unsigned error and mean unsigned value for the subset of molecules containing the specified atoms. Test for sensitivity to the reference geometry 共see text兲. c Constrained to zero. d Standard deviations on last digits given in parentheses. b

J. Chem. Phys., Vol. 113, No. 4, 22 July 2000

Core and relativistic effects

TABLE VI. The version-2 D Z(r)␣ parameters and the associated mean unsigned errors, mean unsigned values 共kcal/mol兲, and number of molecules containing the particular atom type. Atom type H Li Be B C共1兲 C共2,3兲 N共1兲 N共2,3兲 O共1兲 O共2,3兲 F Al Si共1兲 Si共2,3兲 P共1兲 P共2,3兲 S共1兲 S共2,3兲 Cl Full data set No Al or Si First row only Hydrocarbons

D Z(r)␣

MUE

MUV

No.

关0兴 关0兴 关0兴 ⫺0.030(18) ⫺0.048(3) ⫺0.036(5) ⫺0.079(14) ⫺0.024(10) ⫺0.130(16) ⫺0.074(11) ⫺0.185(13) ⫺0.186(23) ⫺0.196(9) ⫺0.060(15) ⫺0.200共26兲 ⫺0.056(14) ⫺0.213(35) ⫺0.059(17) ⫺0.169(26)

0.04 0.04 0.06 0.02 0.03 0.04 0.02 0.04 0.08 0.08 0.11 0.10 0.16 0.12 0.12 0.10 0.04 0.09 0.12 0.07 0.05 0.04 0.02

0.38 0.18 0.18 0.23 0.47 0.46 0.25 0.27 0.48 0.42 0.52 0.46 0.72 0.57 0.58 0.52 0.48 0.33 0.37 0.39 0.34 0.32 0.40

78 5 7 8 47 32 7 10 14 21 20 6 13 6 7 8 4 6 13 125 103 72 24

The question of the relative accuracy of RCCSD共T兲 and MP2 was also addressed 共for a smaller sample of molecules兲 in a previous paper.5 That paper showed that CCSD and MP2 calculations systematically but similarly underestimate the CCSD共T兲 core correlation contributions, suggesting that the leading error in the MP2 treatment is the neglect of connected triple excitations. Furthermore, that paper showed that the G3large basis set, which is smaller than the MTsmall basis set, can seriously overestimate the core correlation contribution. Combining MP2 with G3large, these two sources of error partly cancel, making MP2/G3large calculations more accurate than they would be in the absence of cancellation, yet still unreliable. Table III reveals that the simple model recovers about 88% of the core correlation binding energy over the entire sample of 125 molecules. For the subset of 91 compounds that contain only H, B-F, P, S, and Cl 共i.e., the ‘‘biological/ organic’’ elements兲, this proportion rises to almost 92%. Restriction to HCNOF compounds leads to a slight further improvement to 93.5%. It was previously suggested5 that core correlation binding energies in hydrocarbons appear to obey additivity approximations quite well; and indeed, for the 24 hydrocarbons in the sample, the present model accounts for no less than 97.2% of the variation in the core correlation energies. The D Z(c) values for the first row reveal the expected ␣ decreasing trend from left to right in the periodic table 共corresponding to the increasing 1s – 2s gap兲, with the exception (c) (c) which is slightly larger than D N . For the second of D O (c) row, the D Z change sign from left to right in the periodic ␣ table. Two tentative explanations are offered for this at first sight peculiar phenomenon: 共a兲 because of the large covalent

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radii of Al and Si and their low electronegativity, typical bonds to these elements are so long that the valence electrons are, on average, further separated from the core in a typical molecule than in the free atom, leading to ‘‘antibonding’’ core correlation contributions 共see also Sec. VII, where the variation of the core correlation contribution with the geometry is considered兲; 共b兲 while in the first row the (2s) and (2p) valence orbitals have similar spatial extents, in the second row the (3p) is considerably more diffuse than the (3s), to an extent that increases from right to left in the periodic table. Since for Al and Si compounds, hybridization on binding requires a net transfer of electrons from (3s) to (3p), the ‘‘atom-in-the-molecule’’ has a more diffuse valence shell than the original atom, leading to a reduction in the corevalence correlation energy. Consideration of the MUE/MUV ratios for the sets of compounds containing a particular element, as well as the standard deviations on the D Z(c) obtained from the fitting, ␣ suggests that the model is least effective near the left end of each row, which translates in the improved performance noted in the previous paragraph when Li, Be, Al, and Si compounds are omitted from consideration. The fact that the distinction between D N(1) and D N(2,3) is statistically significant, and that between D P(1) and D P(2,3) is not, probably mainly reflects the comparative dearth of species with ‘‘proper’’ multiple bonds to P in our sample. An interesting way to illustrate the success of the present approximation is to compare the quality of the predictions from Eq. 共7兲, which has negligible cost, to those from full MP2/G3large calculations, which have reasonably large cost. Comparing the MP2//M calculations to the best estimates results in a MUE of 0.54 kcal/mol, whereas our v3 fit results in an MUE of 0.15 kcal/mol.

2. Relativistic effects

Kedziora et al.39 previously obtained relativistic corrections for a set of 148 molecules—including many of those in our sample—using a second-order perturbation treatment of the Dirac–Coulomb Hartree–Fock method. Our own Hartree–Fock-level MVD 共mass-velocity and Darwin兲 corrections are in very good agreement with these values; however, we find that introduction of electron correlation reduces these contributions by about 20%–25% on average. This is consistent with the findings of Bauschlicher11 using CCSD共T兲 and the Douglas–Kroll approximation40 for CF, CF4, and SiF4 and those of de Oliveira et al.41 for atomic electron affinities. Table V shows that the average relativistic binding energy is negative for every atomic number, so the relativistic effect is more systematic than the core correlation one. 共The H, Li, Be, and B parameters are so small that no statistically significant fit could be obtained for them, hence they were constrained to zero in this fit.兲 Comparison of the MUV values in Table VI to those in Table IV shows that on average the magnitude of the relativistic binding energy is 3.6 times

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J. Chem. Phys., Vol. 113, No. 4, 22 July 2000

smaller than the magnitude of the core correlation binding energy. This ratio, however, largely reflects the preponderance of hydrogen and first-row atoms in our sample. For typical second-row compounds, the scalar relativistic contribution tends to be comparable to or even more important than the inner-shell correlation contribution. For example, if we assume a typical bond order to Si of 4, then the typical relativistic binding energy for Si is ⫺0.6 kcal/mol. As expected, the effect is smaller for the first period, e.g., ⫺0.12 kcal/mol for trivalent N. Finally, we note from Table V that the geometry dependence of the scalar relativistic corrections is very weak, and definitely much weaker than for the core correlation binding energies. B. New fits based on bond-order-dependent parameters

These new fits are motivated in particular by recognizing that bond orders are correlated with hybridization state. For example, carbon atoms making double bonds usually have sp 2 hybridization. The hybridization in turn should be directly correlated with the strength of core–valence interactions and even more so with relativistic effects since s orbitals penetrate closer to the nucleus than p orbitals. Indeed, in preliminary work not presented here, we obtained very good fits by letting the parameters depend on hybridization state. Sometimes, however, ambiguities arise in assigning hybridization states—especially for radicals—whereas assigning formal bond orders was already required even with the original method. Consequently no new ambiguities are added by letting the parameters depend on formal bond orders. For core correlation, Eq. 共11兲 leads to only a rather small change in the overall mean unsigned error 共MUE兲, from 0.15 kcal/mol 共version 3兲 to 0.14 kcal/mol 共version 4兲. 共For HCNOF compounds, the corresponding change is from 0.12 to 0.11 kcal/mol.兲 On the other hand, as anticipated in the previous paragraph, Eq. 共11兲 provides significant improvement over Eq. 共7兲 for relativistic effects, reducing the MUE from 0.11 kcal/ mol 共version 1兲 to 0.07 kcal/mol 共version 2兲. For the subset of 72 first-row compounds, the MUE drops from 0.07 kcal/ mol to only 0.04 kcal/mol! In addition, the variation for boron no longer ‘‘drowns in the noise’’ and a statistically significant D B(r) parameter could be fitted. The model accounts for 82% of the variation in the relativistic binding energies over the entire sample, which goes up to 85% for compounds of the ‘‘bio-organic’’ elements H, B–F, P–Cl, and 89% if only first-row compounds are considered. For hydrocarbons, no less than 93.5% is recovered. Inspection of the coefficients in Table VI reveals some interesting trends. For instance, the Atom共1兲 parameters are systematically larger in absolute value than the corresponding Atom共2,3兲 parameters, a trend which is particularly pronounced for the second row. This reflects the proportionally

larger s character of single compared to multiple bonds—it is recalled that the s orbitals will bear the brunt of the relativistic effect. Furthermore, the parameters for B–F are themselves fitted very well (R⫽0.996) by D Z(r) ⫽0.0002Z 3.1543; ␣ while the numerical values of these fit parameters should be taken with a grain of salt, they do suggest the type of power dependence on Z one theoretically expects for the scalar relativistic contribution. Interestingly, no such dependence appears to exist for the second-row parameters, which only display weak variation in terms of Z. A fit based on atomic hybridization states yielded no appreciable improvement over the present bond-order-based one. The former requires assigning hybridization states as well as bond orders, which introduces an additional source of ambiguity in less clear-cut cases. Since, on the other hand, the bond-order-dependent v2 fit manifestly performs better than the v1 fit, and since there is an obvious theoretical justification for this in terms of the dependence of relativistic effects on the amount of s character in a bond, we conclude that, at this stage, v2 represents the optimum compromise between simplicity and performance for relativistic contributions. This model is seen in Table VI to capture 80% of the variation in the relativistic contributions, Thus, while for core correlation effects arguments could be found in favor of either v3 共smaller number of fitting parameters兲 or v4 共slightly better performance without additional effort兲, we recommend only v2 for the relativistic treatment.

VI. FUTURE DIRECTIONS

In general, for predicting potential energy surfaces, one wants the various contributions to the energy to be smooth, analytic functions of geometry. Let R denote a set of internal coordinates for a molecule, and let Re denote the equilibrium geometry. One might attempt to fit the R dependence found here by simple functions that pass through the values predicted here both for the equilibrium geometry of the molecule and also for the relevant subsystems. For example, for H2O the fit would pass through the H2O value for water’s equilibrium structure, would pass through the OH value for the equilibrium structure of hydroxyl infinitely separated from H, and would be zero for O⫹2H. However, it is not clear what the optimum form of such a function should be. In particular we have found that although the formal bond orders 共which are real bond orders rounded to integers兲 provide a good fit to the total core-correlation binding energies at the equilibrium geometries, the unrounded bond orders42 共which are easily computed from Hartree–Fock wave functions at various geometries兲 do not provide a good fit to the details of the geometry dependence. The geometry dependence of the core correlation binding energy is actually quite intriguing. Figures 1–3 show this geometry dependence of the core-correlation energy as defined by

J. Chem. Phys., Vol. 113, No. 4, 22 July 2000

Core and relativistic effects

FIG. 1. The RCCSD共T兲/MTsmall core-correlation contribution, V (c) 共kcal/ mol兲, to the N2 potential curve as a function of 1/R NN (Å ⫺1 ) is plotted on the primary axis 共⽧兲, while the RHF/6-31G(d) 1s – 2s gap, ⌬⌬␧ 共kcal/ mol兲, for N2 as a function of 1/R NN is plotted on the secondary axis 共䊏兲.

冋

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FIG. 3. The RCCSD共T兲/MTsmall core-correlation contribution, V (c) 共kcal/ mol兲, to the HF potential curve as a function of 1/R HF (Å ⫺1 ) is plotted on the primary axis 共⽧兲, while the RHF/6-31G(d) 共䊏兲 and UHF/6-31G(d) 共䊐兲 1s – 2s gap, ⌬⌬␧ 共kcal/mol兲, for HF as a function of 1/R HF is plotted on the secondary axis. The UHF instability occurs at r⫽1.3 Å.

V (c) ⫽⫺ E 共 RCCSD共T兲/MTsmall;R兲 ⫺

⌬⌬␧ 共 R兲 ⫽⌬␧ 共 dissociated兲 ⫺⌬␧ 共 R兲 ,

兺A E 共 RCCSD共T兲/MTsmall;atom A 兲

冉

where ⌬␧⫽␧ 2s ␣ ⫺␧ 1s ␤

⫺ E 共 RCCSD共T兲共FC兲/MTsmall;R兲 ⫺

兺A E 共 RCCSD共T兲共FC兲/MTsmall;atom A 兲

共15兲

冊册

. 共14兲

We see three interesting trends: (i) the sign depends on geometry; (ii) the dependence on geometry is steep 共tending towards ⬀R⫺1 for shorter R兲; and (iii) the decay to zero as a bond is broken is nonmonotonic. Is there a simple way to understand these trends, e.g., can we relate it to shifts in core orbital energies? Figures 1–3 show one function of orbital energies that does correlate with V (c) (R), namely the 1s – 2s gap function defined by

共16兲

and ␧ ␮ is the orbital energy of orbital ␮, and, at the dissociation limit where the atoms have an odd number of electrons and the orbitals for ␣ and ␤ spins have different energies, the lower energy one is labeled ␣. Interestingly, this function goes through zero near the geometry where V (c) goes through zero. Thus it may ultimately be possible to understand geometric trends in core correlation energies in terms of orbitals and orbital energies, which are much less expensive to compute than correlation energies. This appears to be a promising area for future research. For second row atoms, one would generalize this to the HSVO–LVO 共highest subvalence orbital-lowest valence orbital兲 gap. While neither N2 nor F2 display UHF instability over the range of internuclear distances shown in the figures, the orbitals of HF become unstable with respect to breaking of spin symmetry at about 1.3 Å, and consequently ⌬⌬␧ bifurcates at this point into RHF and UHF solutions. From Fig. 3, it appears that the core correlation energies—which, we recall, were obtained from an RHF reference wave functions— vary more similarly to ⌬⌬␧ for the RHF solution than to its UHF counterpart. VII. CONCLUSION

FIG. 2. The RCCSD共T兲/MTsmall core-correlation contribution, V (c) 共kcal/ mol兲, to the F2 potential curve as a function of 1/R FF (Å ⫺1 ) is plotted on the primary axis 共⽧兲, while the RHF/6-31G(d) 1s – 2s gap, ⌬⌬␧ in kcal/mol, for F2 as a function of 1/R FF is plotted on the secondary axis 共⽧兲.

Among the oldest methods for estimating binding energies of molecules are bond equivalent methods, of which Benson group equivalents43 are a more sophisticated variant. Inner-shell correlation contributions to the molecular binding energy amount to only about 1% of the total; in addition, they are 共because of the ‘‘tight’’ nature of core orbitals兲 presumably more local in character than the valence binding energy. Hence it seems reasonable that they should also be amenable to such a treatment, and we have found that this is indeed the case. In the present article we have presented a

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new, more accurate set of parameters for estimating the core correlation binding energy for molecules containing H, Li, Be, B, C, N, O, F, Al, Si, P, S, and Cl. In addition, we have extended the treatment to include scalar relativistic effects. The mean unsigned error in the estimate of the sum of these contributions is a factor of 6–12 smaller than the values being fit. Thus one can eliminate most of the error caused by these effects in thermochemical calculations at negligible cost. ACKNOWLEDGMENTS

The authors are grateful to Peter R. Taylor for comments on an early draft. J. M. is the incumbent of the Helen and Milton A. Kimmelman Career Development Chair. Research at the Weizmann Institute was supported by the Minervagesellschaft fu¨r die Forschung, Munich, Germany. Research at the University of Minnesota was supported in part by the U. S. Department of Energy, Office of Basic Energy Sciences. J. M. L. Martin and P. R. Taylor, Chem. Phys. Lett. 225, 473 共1995兲; J. M. L. Martin, ibid. 242, 343 共1995兲. 2 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 共1995兲. 3 W. J. Hehre, L. Radom, P. von Rague´ Schleyer, and J. A. Pople, ab initio Molecular Orbital Theory 共Wiley, New York, 1986兲. 4 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 共1989兲 and subsequent papers. 5 J. M. L. Martin and G. de Oliveira, J. Chem. Phys. 111, 1843 共1999兲. 6 K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 473 共1989兲. 7 P. L. Fast and D. G. Truhlar, J. Phys. Chem. A 103, 3802 共1999兲. 8 C. M. Tratz, P. L. Fast, and D. G. Truhlar, Phys. Chem. Comm. http:// www.rsc.org/is/journals/current/PhysChemComm/pccpub.htm 共1999兲. 9 R. D. Cowan and M. Griffin, J. Opt. Soc. Am. 66, 1010 共1976兲. 10 R. L. Martin, J. Phys. Chem. 87, 750 共1983兲. 11 C. W. Bauschlicher, Jr., J. M. L. Martin, and P. R. Taylor, J. Phys. Chem. A 103, 7715 共1999兲. 12 C. W. Bauschlicher, Jr., J. Phys. Chem. A 104, 2281 共2000兲. 13 C. W. Bauschlicher, Jr., Theor. Chem. Acc. 101, 421 共1999兲. 14 R. J. Gdanitz and R. Ahlrichs, Chem. Phys. Lett. 143, 413 共1988兲. 15 P. G. Szalay, H. Lischka, and A. Karpfen, J. Phys. Chem. 93, 6629 共1989兲. 16 R. S. Grev and H. F. Schaefer III, J. Chem. Phys. 97, 8389 共1992兲. 17 A. D. Pradhan, H. Partridge, and C. W. Bauschlicher, Jr., J. Chem. Phys. 101, 5900 共1994兲. 18 J. A. Montgomery, Jr., J. W. Ochterski, and G. A. Petersson, J. Chem. Phys. 101, 5900 共1994兲. 1

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