Semi-empirical correction of ab initio harmonic properties by scaling factors: a validated uncertainty model for calibration and prediction

Bayesian Model Calibration is used to revisit the problem of scaling factor calibration for semi-empirical correction of ab initio harmonic properties (e.g. vibrational frequencies and zero-point energies). A particular attention is devoted to the evaluation of scaling factor uncertainty, and to its effect on the accuracy of scaled properties. We argue that in most cases of interest the standard calibration model is not statistically valid, in the sense that it is not able to fit experimental calibration data within their uncertainty limits. This impairs any attempt to use the results of the standard model for uncertainty analysis and/or uncertainty propagation. We propose to include a stochastic term in the calibration model to account for model inadequacy. This new model is validated in the Bayesian Model Calibration framework. We provide explicit formulae for prediction uncertainty in typical limit cases: large and small calibration sets of data with negligible measurement uncertainty, and datasets with large measurement uncertainties.