Scaling factors for ab initio vibrational frequencies: comparison of uncertainty models for quantified prediction

Bayesian Model Calibration is used to revisit the problem of scaling factor calibration for semi-empirical correction of ab initio calculations. A particular attention is devoted to uncertainty evaluation for scaling factors, and to their effect on prediction of observables involving scaled properties. We argue that linear models used for calibration of scaling factors are generally not statistically valid, in the sense that they are not able to fit calibration data within their uncertainty limits. Uncertainty evaluation and uncertainty propagation by statistical methods from such invalid models are doomed to failure. To relieve this problem, a stochastic function is included in the model to account for model inadequacy, according to the Bayesian Model Calibration approach. In this framework, we demonstrate that standard calibration summary statistics, as optimal scaling factor and root mean square, can be safely used for uncertainty propagation only when large calibration sets of precise data are used. For small datasets containing a few dozens of data, a more accurate formula is provided which involves scaling factor calibration uncertainty. For measurement uncertainties larger than model inadequacy, the problem can be reduced to a weighted least squares analysis. For intermediate cases, no analytical estimators were found, and numerical Bayesian estimation of parameters has to be used.