Statistical Analysis of Semiclassical Dispersion Corrections

Semiclassical dispersion corrections developed by Grimme and coworkers have become indispensable in applications of Kohn-Sham density functional theory. We present an in-depth assessment of the fit parameters present in semiclassical (D3-type) dispersion corrections by means of a statistically rigorous analysis. We find that the choice of the cost function generally has a small effect on the empirical parameters of D3-type dispersion corrections with respect to the reference set under consideration. Only in a few cases, the choice of cost function has a surprisingly large effect on the total dispersion energies. In particular, the weighting scheme in the cost function can significantly affect the reliability of predictions. In order to obtain unbiased (data-independent) uncertainty estimates for both the empirical fit parameters and the corresponding predictions, we carried out a nonparametric bootstrap analysis. The mean prediction uncertainty obtained by bootstrapping is not much larger than previously reported error measures. On the basis of a jackknife analysis, we find that the original reference set is slightly skewed, but our results also suggest that this feature hardly affects the prediction of dispersion energies. However, a rigorous analysis of error accumulation arising from different parameterizations reveals that error cancellation does not necessarily occur, leading to a monotonically increasing deviation in the dispersion energy with increasing molecule size. We discuss this issue in detail at the prominent example of the C60 buckycatcher. We find deviations between individual parameterizations of several tens of kcal/mol in some cases. Hence, in combination with any calculation of dispersion energies, we recommend to always determine the associated uncertainties for which we will provide a software tool.