Free-energy perturbation in the exchange-correlation space accelerated by machine learning: Application to silica polymorphs

We propose a free-energy-perturbation approach accelerated by machine-learning potentials to efficiently compute transition temperatures and entropies for all rungs of Jacob's ladder. We apply the approach to the dynamically stabilized phases of SiO$_2$, which are characterized by challengingly small transition entropies. All investigated functionals from rungs 1-4 fail to predict an accurate transition temperature by 25-200%. Only by ascending to the fifth rung, within the random phase approximation, an accurate prediction is possible, giving a relative error of 5%. We provide a clear-cut procedure and relevant data to the community for, e.g., developing and evaluating new functionals.