Very fast averaging of thermal properties of crystals by molecular simulation

Abstract

Knowledge of approximate harmonic behavior of crystals is introduced into a new "mapped averaging" framework to yield alternative expressions for the thermodynamic properties of crystalline systems. The expressions separate the known harmonic behavior from residual averages, which thus encapsulate anharmonic contributions to the properties. With harmonic contributions removed, direct measurement of these anharmonic contributions by molecular simulation can be accomplished without contamination by noise produced by the already-known harmonic behavior. We show with application to the Lennard-Jones model that first-derivative properties (pressure, energy) can be obtained to a given precision via this harmonically mapped averaging at least 10 times faster than by using conventional averaging, and second-derivative properties (e.g., heat capacity) are obtained at least 100 times faster; in more favorable cases, the speedup exceeds a millionfold. Free-energy calculations are accelerated by 50 to 1000 times. Data obtained using these formulations are rigorous and not subject to any added approximation, and in fact are less sensitive to inaccuracies relating to finite-size effects, potential truncation, equilibration, and similar considerations. Moreover, the approach does not require any alteration in how sampling is performed during the simulation, so it may be used with standard Monte Carlo or molecular dynamics methods. However, the mapped averages do require evaluation of first and second derivatives of the intermolecular potential, for evaluation of first and second thermodynamic-derivative properties, respectively. Apart from its usefulness to simulation, the formalism developed here may constitute a basis for new theoretical treatments of crystals.

Accurate and precise ab initio anharmonic free-energy calculations for metallic crystals: Application to hcp Fe at high temperature and pressure

“…As shown by our earlier work with classical models 46,48,53 , the anharmonic FE has much smaller finitesize effects compared to the quasiharmonic part. The current work shows that this conclusion is still valid for DFT-type models (at least for iron).…”

“…This feature is particularly useful for phase stability analysis where the FE along a wide range of states is required. In addition, we have observed (here and in previous work 46,48 ) that the integrand of this method has small curvature along the path; hence, averages at just a few points along the path can be sufficient to yield an accurate result.…”

“…LT (with both Conv and HMA) shows a much smaller (statistically flat) dependence on ∆t, compared to AT. On the other hand, the uncertainty does not show strong dependence on the thermostat type (for a given averaging method), but it is strongly dependent on the form of the average, with HMA about seven times more precise than Conv (the improvement increases with decreasing temperature 46 ). In addition to its low uncertainty, the HMA formulation suppresses the effect of ∆t on accuracy (especially with AT), allowing for use of larger ∆t for a given thermostat.…”

“…Using the recently developed HMA-TI approach [46][47][48] to determine the baseline temperature dependence of the anharmonic FE, providing results with order(s) of magnitude CPU speed-up (≈ 15-150×, compared to conventional TI).…”

“…This approach can be useful in other contexts, such as the stochastic self-consistent harmonic approximation method (SSCHA) 33,34 . However it cannot be applied here, because U As has been discussed elsewhere [46][47][48] , a new harmonicmapping framework provides a formulation that is still consistent with Eq. (5) while exhibiting much better statistical features:…”