The use of the smaller MT basis set for the scalar relativistic contributions is found to have an effect of about 0.01 kcal/mol or less, with 0.02 kcal/mol being the largest individual cases. This approximation can therefore safely be made.
Using the same MT basis set for the core correlation contribution on average affects energetics by 0.03 kcal/mol, the largest individual effects being 0.07 kcal/mol for H2S, and 0.08 kcal/mol for OCS.
Even so, in fact, the core correlation calculations are quite CPU-time consuming, particularly for second-row compounds, due to the large number of electrons being correlated. Any further reduction would obviously be welcome -- it should be noted that the MT basis set was developed not with efficiency, but with saturation (in the core-valence correlation energy) in mind. Further experimentation revealed that the tightest p, d, and f functions could safely be eliminated, but that further basis set reductions adversely affect the quality of the core correlation contribution computed. The reduced basis set shall be denoted as MTsmall, and in fact consists of a completely decontracted cc-pVTZ basis set with two tight d and one tight f functions added. Since this basis set only has about half the basis functions of the ACVQZ basis set per heavy atom, it represents a very significant CPU time savings (about 16 times) in a CCSD(T) calculation. The only molecule for which we see a substantial difference with the MT basis set is SO2, for which Bauschlicher and Ricca[21] previously noted that the inner-shell correlation contribution is unusually sensitive to the basis set.
For the evaluation of the Darwin and mass-velocity corrections, differences with the larger MT basis set are less than 0.01 kcal/mol across the board.
A further reduction in CPU time for the core correlation contribution would have been achieved if MP2 or even CCSD calculations could be substituted for their CCSD(T) counterparts. However, as seen from Table VII, CCSD underestimates the CCSD(T) core correlation contributions for several molecules by as much as 50%. The behavior of MP2 is quite similar and the MP2-CCSD differences are substantially smaller than the (T) contribution, suggesting that it is the treatment of connected higher excitations that is the issue. Predictably, the largest (T) effects on the core correlation contribution occur in molecules where connected triple excitations are important for the valence binding energy as well, e.g. SO2, F2, Cl2, N2. Conversely, in CH3 or CH4, which have quite small (T) contributions to the binding energy, CCSD does perform excellently for the core correlation contribution. In PH3 and H2S, on the other hand, substantial errors in the core correlation are seen even as the (T) contribution to the valence correlation binding energy is quite small -- it should be noted, however, that both the absolute inner-shell correlation energy and the (T) contribution to it are much more important in these second-row systems than in their first-row counterparts.
One may rightly wonder whether the inner-shell contributions are in fact converged at the CCSD(T) level. Unfortunately, if a more elaborate treatment is already impractical for the valence correlation, this would a fortiori be true for the inner-shell correlation. We did, however, carry out a CCSDT/MTsmall calculation on the N2 molecule, which we chose as a representative case of failure of the CCSD approach for core correlation. The resulting CCSDT level core contribution, 0.87 kcal/mol, is only 0.05 kcal/mol larger than the CCSD(T) value of 0.82 kcal/mol, to be compared with 0.42 kcal/mol at the MP2 and 0.52 kcal/mol at the CCSD level. It cannot be ruled out a priori that connected quadruple and higher excitations might contribute to the inner-shell correlation energy. However, since apparently their importance for the valence correlation binding energy is very small (otherwise a treatment that completely ignores them would not yield the type of agreement with experiment found in this work), it seems unlikely that they would greatly contribute to the inner-shell correlation energy.
The ``G3large'' basis set used to evaluate, among other things, inner-shell correlation effects in G3 theory[3] is still smaller than the MTsmall basis set, and its performance therefore is certainly of interest. Alas, in Table VII it is seen that in many cases it seriously overestimates the inner-shell correlation energy, almost certainly because of basis set superposition error which[21] is apparently more of an issue for inner-shell correlation energies than for their valence counterparts. In G3 theory, the inner-shell correlation is evaluated at the MP2 level: hence the two errors cancel to a substantial extent.