Free-energy calculations in solids from first-principles molecular dynamics: Vacancy formation in sodium

Thermodynamic integration methods are applied to obtain the classical free energy of solids via first-principles (or ab initio$)--- molecular-dynamics simulations based upon the Car-Parrinello strategy. The methods introduced are particularly well suited to the calculation of the formation free energy of defects and of the absolute (classical) free energy of crystals. We find that specific techniques have to be used to ensure canonical sampling, particularly for the case of the perfect-crystal calculations. The method is illustrated with calculations of the free energy of formation of the sodium vacancy and of the free energy of the sodium crystal.