Computational details

Most electronic structure calculations reported in this work were carried out using MOLPRO 97.3[25] and MOLPRO 98.1[26] running on a Silicon Graphics (SGI) Octane workstation and on the SGI Origin 2000 of the Faculty of Chemistry. The full CCSDT (coupled cluster with all connected single, double, and triple substitutions[27]) calculations were carried out using ACES II[28] running on a DEC Alpha 500/500 workstation.

SCF and valence correlation calculations were carried out using correlation consistent[29,30] polarized n-tuple zeta (cc-pVnZ, or VnZ for short) (n=D, T, Q, 5, 6) and augmented correlation consistent[31] polarized n-tuple zeta (aug-cc-pVnZ, or AVnZ for short) (n=D, T, Q, 5, 6) basis sets of Dunning and coworkers. The maximum angular momentum parameter l, which occurs in the extrapolation formulas for the correlation energy, is identified throughout with the nin VnZ and AVnZ. Except for the calculation of the electron affinity of hydrogen, regular VnZ basis sets were used throughout on hydrogen atoms.

Most valence correlation calculations were carried out using the CCSD (coupled cluster with all single and double substitutions[32]) and CCSD(T) (i.e. CCSD followed by a quasiperturbative estimate of the effect of connected triple excitations[33,34]) electron correlation methods. The CCSD(T) method is known[35] to be very close to an exact solution within the given one-particle basis set if the wave function is dominated by dynamical correlation.

Where possible, imperfections in the treatment of connected triple excitations were estimated by comparing with full CCSDT calculations. The effect of connected quadruple and higher excitations were estimated by small basis set FCI (full configuration interaction) calculations -- which represent exact solutions with a finite basis set.

Inner-shell correlation contributions were evaluated by taking the difference between valence-only and all-electron CCSD(T) calculations in special core-correlation basis sets. For first-row compounds, both Dunning's ACVQZ (augmented correlation consistent core-valence quadruple zeta[36]) basis set and the Martin-Taylor (MT) core correlation basis sets[37,15] were considered; for second-row compounds only the MT basis sets. The latter are generated by completely decontracting a CVnZ or ACVnZ basis set, and adding one tight p function, three high-exponent d functions, two high-exponent f functions, and (in the case of the MTv5z basis set) one high-exponent g function to the basis set. The additional exponents were derived from the highest ones already present for the respective angular momenta, successively multiplied by 3.0. The smallest such basis set, MTvtz (based on VTZ) is also simply denoted MT.

Scalar relativistic corrections were calculated at the ACPF (averaged coupled pair functional[38]) level as expectation values of the first-order Darwin and mass-velocity terms[39,40]. An idea of the reliability of this approach is given by comparing a very recent relativistic (Douglas-Kroll[41]) coupled cluster calculation[42] of the relativistic contribution to TAE[SiH₄], -0.67 kcal/mol, with the identical value of -0.67 kcal/mol obtained by means of the present approach. For GaCl, GaCl₂, and GaCl₃ -- where relativistic effects are an order of magnitude stronger than even in the second-row systems considered here -- Bauschlicher[43] found that differences between Douglas-Kroll calculations and the presently followed approach amounted to 0.12 kcal/mol or less on the binding energy.

Spin-orbit coupling constants were evaluated at the CASSCF-CI level using the spdf part of the MTav5z basis set. (For a recent review of the methodology involved, see Ref.[44].)

Density functional calculations for the purposes of obtaining certain reference geometries and zero-point energies were carried out using the Gaussian 98[45] package. Both the B3LYP (Becke 3-parameter[46]-Lee-Yang-Parr[47]) and B3PW91 (Becke 3-parameter[46]-Perdew-Wang-1991[48]) exchange-correlation functionals were considered.

Most geometry optimizations were carried out at either the CCSD(T)/VQZ+1 or the B3LYP/VTZ+1 (in some cases B3PW91/VTZ+1) levels of theory, where the notation VnZ+1 indicates the addition to all second-row atoms of a single high-exponent d-type `inner polarization function'[49,23] with an exponent equal to the highest d exponent in the Dunning V5Z basis set. In the past this was found[49,23,50,51] to recover the largest part of the effects of inner polarization on geometries and vibrational frequencies. (We note that for molecules consisting of first-row atoms only, the VnZ+1 basis sets are equivalent to regular VnZ basis sets.)

Past studies[52,53,54] of the convergence behavior of the SCF energy have shown it to be very well described by a geometric extrapolation of the type first proposed by Feller[55], A+B/C^l. Clearly, for this purpose a succession of three SCF/AVnZ basis sets is required.

For the valence correlation CCSD and (T) energies, two extrapolation formulas were considered. The first, $A+B/(l+1/2)^\alpha$, was proposed by Martin[13] -- the philosophy being that using the extrapolation exponent as an adjustable parameter would enable inclusion of higher-order terms in the asymptotic expansion

A/(L+1)³ + B/(L+1)⁴ + C/(L+1)⁵ + ... (1)

while the denominator shift of 1/2 was a compromise -- for identification of the l in cc-pVlZ with L -- between hydrogen and nonhydrogen atoms. The second formula, simply A+B/l³, was proposed by Helgaker and coworkers[56] -- where l was identified with L-1 throughout. Halkier et al.[56] already noted that in terms of the extrapolated energy using A+B/(l+C)^D, the parameters C and D were very strongly coupled, and that it only made sense to vary one of them.

The combination of treatments for SCF, CCSD valence correlation, (T), imperfections in the T treatment, and connected quadruple and higher excitations is compactly denoted here by W[p5;p4;p3;p2;p1], in which p1 denotes the basis sets involved in the SCF extrapolation, p2 the basis sets involved in the CCSD extrapolation, p3 those in the (T) extrapolation (which may or may not be different from p2), p4 (if present) the basis sets used in correcting for imperfections in the treatment of connected triple excitations, and p5 (if present) those involved in evaluating the effect of connected quadruple and higher excitations. If any of the p's consists of a single index, a simple additivity approximation is implied; two indices denote a two-parameter extrapolation of the type A+B/l³, while three indices indicate a three-parameter extrapolation of the type $A+B/(l+1/2)^\alpha$ in the case of correlation contributions, and A+B/C^l in the case of SCF contributions. For example, the level of theory used in the previous work of Martin and Taylor would be W[TQ5;TQ5;TQ5] in the present notation, while W[D;Q;TQ5;TQ5;TQ5] indicates W[TQ5;TQ5;TQ5]+CCSDT/AVQZ-CCSD(T)/AVQZ+FCI/AVDZ-CCSDT/AVDZ.