Harmonically Assisted Methods for Computing the Free Energy of Classical Crystals by Molecular Simulation: A Comparative Study

Four methods for calculation of the classical free energy of crystalline systems are compared with respect to their efficiency and accuracy. Two of the methods involve thermodynamic integration along an unphysical path (λ integration, λI), and two involve integration in temperature from the low-temperature harmonic limit (T integration, TI). Specifically, the methods considered are (1) Frenkel-Ladd integration from a noninteracting Einstein crystal reference (ECR-λI); (2) conventional integration in temperature (Conv-TI); (3) integration from an interacting quasi-harmonic reference (QHR-λI); and (4) temperature integration using harmonically mapped averaging to evaluate the integrand (HMA-TI). The latter two methods are “harmonically assisted”, meaning that they exploit the harmonic nature of the crystal to greatly reduce fluctuations in the relevant averages. This feature allows them to deliver a result of much higher precision for a given computational effort, compared to ECR-λI and Conv-TI, and with no less accuracy. Regarding the harmonically assisted methods, HMA-TI has several advantages over QHR-λI with respect to the simplicity of the integration path (which promotes a more accurate result), ease of implementation, and usefulness of the data recorded along the integration path.