Thermal Gradient Approach for the Quasi-harmonic Approximation and Its Application to Improved Treatment of Anisotropic Expansion

Abstract

We present a novel approach to efficiently implement thermal expansion in the quasiharmonic approximation (QHA) for both isotropic and more importantly, anisotropic expansion. In this approach, we rapidly determine a crystal's equilibrium volume and shape at a given temperature by integrating along the gradient of expansion from zero Kelvin up to the desired temperature. We compare our approach to previous isotropic methods that rely on a brute-force grid search to determine the free energy minimum, which is infeasible to carry out for anisotropic expansion, as well as quasi-anisotropic approaches that take into account the contributions to anisotropic expansion from the lattice energy. We compare these methods for experimentally known polymorphs of piracetam and resorcinol and show that both isotropic methods agree to within error up to 300 K. Using the Grüneisen parameter causes up to 0.04 kcal/mol deviation in the Gibbs free energy, but for polymorph free energy differences there is a cancellation in error with all isotropic methods within 0.025 kcal/mol at 300 K.Anisotropic expansion allows the crystals to relax into lattice geometries 0.01-0.23 kcal/mol lower in energy at 300 K relative to isotropic expansion. For polymorph free energy differ-ences all QHA methods produced results within 0.02 kcal/mol of each other for resorcinol and 0.12 kcal/mol for piracetam, the two molecules tested here, demonstrating a cancellation of error for isotropic methods.We also find that when expanding in more than a single volume variable, there is a nonnegligible rate of failure of the basic approximations of QHA. Specifically, while expanding into new harmonic modes as the box vectors are increased, the system often falls into alternate, structurally distinct harmonic modes unrelated by continuous deformation from the original harmonic mode. arXiv:1712.00936v3 [cond-mat.mtrl-sci]

Adding Anisotropy to the Standard Quasi-Harmonic Approximation Still Fails in Several Ways to Capture Organic Crystal Thermodynamics

“…Further quantification of the numerical and structural instability is discussed in the work where we initially presented the gradient method. 25…”

“…Our gradient approach identifies high temperature free energy minimum that are 0.01-0.23 kcal/mol lower than the minimum identified with an isotropic model, altering the polymorph free energy differences of pirac-etam and resorcinol by 0.02-0.12 kcal/mol for unit cells of 4-8 molecules. 25 Our fully anisotropic approach and even a simplified 1D-anisotropic approach outperformed the accuracy of current quasi-anisotropic methods that only consider the anisotropy due to the potential energy as a function of crystal lattice vectors. [26][27][28][29][30] In this paper we re-evaluate the 10 of the 12 polymorph pairs previously studied with MD and isotropic QHA to determine whether an anisotropic thermal expansion model can improve the performance of QHA relative to MD.…”

“…24,[33][34][35][36] This assumption can introduce errors, but we have shown for polymorph free energy differences that those errors are generally less than 0.02 kcal/mol. 25 This is smaller than the experimental error in most cases, so we will exclusively use the Grüneisen parameter approach for QHA calculations in this paper when calculating polymorph free energy differences.…”

“…For specifications on how we chose certain parameters we refer the reader back to our previous publications that present the methods for MD 24,31 and QHA. 25…”

“…Results for the graphed intermediate points between Runge-Kutta steps were calculated with a third order spline that was fit to the lattice parameters with temperature. 25,42…”