Use of empirical extrapolation exponents

Truhlar[73] considered the use of L-extrapolation formulas with empirical exponents, carried out from cc-pVDZ and cc-pVTZ calculations, as an inexpensive alternative to very large basis set calculations.

We will investigate here a variant of this suggestion adapted to the present framework. The valence correlation component to TAE will indeed be extrapolated using the formula $A+B/l^{\beta}$, in which $\beta$ is now an empirical parameter -- we will denote this W[Q5;Q5;TQ5]$\beta$ and the like.

We then add in all the further corrections (core correlation, scalar relativistics, spin-orbit) that occur in W2 theory, and try to determine $\beta$ by minimizing MAE with respect to the experimental TAE values for our `training set'. Not surprisingly, for W[Q5;Q5;TQ5]$\beta$ this yields an optimum exponent ($\beta$=2.98) which differs insignificantly from the `ideal' value of 3.0. Alternatively, $\beta$ could be optimized for the best possible overlap with the W[56;56;Q56] results: in fact, the same conclusion is obtained, namely that making $\beta$ an empirical parameter does not improve the quality of the results.

For W[TQ;TQ;DTQ] however, the situation is rather different. The optimum exponent $\beta$ is found to be 3.18 if optimized against the experimental TAE values, and 3.16 (insignificantly different) if optimized against the W[Q5;Q5;TQ5] results. In both cases, MAE drops to 0.30 kcal/mol, and on average no more overestimation occurs.

W[TQ;TQ;DTQ]3.18 represents a significant savings over W2 theory. Its time-determining step in molecules with many electrons will be the evaluation of the parenthetical triples in the AVQZ+2d1f basis set -- their elimination would be most desirable.

A natural suggestion would then be W[DT;TQ;DTQ]$\beta$. Optimization of $\beta$ against the experimental TAE values yields $\beta$=3.26; the not greatly different $\beta$=3.22 is obtained by minimization of the deviation from W[Q5;Q5;TQ5] results for the training set. Since the latter does not explicitly depend on experimental results and therefore minor changes in the computational protocol do not require recalculation for the entire `training set', we will opt for the latter alternative.

In either case, we obtain MAE=0.30 kcal/mol -- for a calculation that requires not more than an AVTZ+2d1f basis set for the largest CCSD(T) calculation, and an AVQZ+2d1f basis set for the largest CCSD calculation. Again, the latter is amenable to a direct algorithm.