Reformulation of expressions for thermoelastic properties of crystals using harmonic mapping

Abstract

We introduce alternative statistical mechanics expressions for thermoelastic properties of monatomic crystals, in the classical canonical ensemble, using the mapped-averaging framework. Two reference models are examined as a basis for the coordinate mapping: ideal gas (IG) and quasiharmonic (QH) crystal. The former prescribes affine mapping and yields the conventional expressions for the properties, while the latter has, in addition, a nonaffine component and yields alternative formulas, denoted as ``harmonically mapped averaging'' (HMA). While the (strain-dependent) affine mapping is universal and does not require system-specific information, the HMA method uses a system- (and strain-) dependent matrix to specify the mapping; this mapping matrix is derived from the force-constant matrix of the QH reference and its variation with strains. In practice, implementing HMA (and affine) schemes depends on the availability of analytical (Lagrangian) derivatives of potential energy with respect to strains (e.g., Born term) and coordinates (e.g., Hessian matrix); however, when these derivatives are unavailable, numerical derivatives can be computed via finite difference by moving atoms according the mapping matrices, while straining the crystal. In no circumstances does mapped averaging affect sampling of configurations; it involves only the calculation of averages. Results show that HMA provides CPU speedup ranging from a factor of 4, to over two orders of magnitude, compared to the conventional formulas (especially at low temperature), with the best performance obtained with isotropic-deformation properties. In addition to its application to molecular simulation, HMA can open avenues for developing new theories for thermoelastic properties.