Enhancing Efficiency and Scope of First-Principles Quasi-Harmonic Approximation Methods Through the Calculation of Third-Order Elastic Constants

In this paper we present a computational strategy to calculate from first principles the coefficient of thermal<br/>expansion and elastic constants of a material over meaningful intervals of temperature and pressure. This computational strategy relies on a novel implementation of the quasi-harmonic approximation (QHA) to calculate isothermal-isochoric linear and nonlinear elastic constants of a material in a reference state, followed by the use of elementary equations of nonlinear continuum mechanics to extrapolate values of lattice parameters and elastic constants of the material in an arbitrary deformed state. Our implementation of QHA relies on finite deformations, the use of non-primitive supercells to describe a material, a recently proposed technique to calculate generalized mode Grüneisen parameters, and the numerical differentiation of the second Piola-Kirchhoff (thermal) stress tensor to calculate both second-order elastic constants (SOECs) and third-order elastic constants (TOECs) (at finite temperature). We remark that our implementation of QHA involves a manageable computational workflow that, measured in terms of number of deformed configurations, requires only 4, 6, and 10 configurations (including reference state) to calculate the isothermal SOECs of a material with the cubic, hexagonal, and orthorhombic symmetry, respectively. In this paper, we first focus on conceptual and technical details of our methods, then we present the results of selected applications, and we conclude with a discussion of the advantages and limits of our methods. Overall, we envision that the methods presented in<br/>this paper have the potential to be used for the following purposes. One, the calculation from first principles<br/>of nonlinear elastic constants (TOECs and potentially fourth-order elastic constants) of materials at finite<br/>temperature. TOECs are important coefficients characterizing the non-linear mechanical response of a material subjected to a deformation, and thereby related to properties such as sound attenuation and yield strength. Our QHA method can be used to calculate these nonlinear elastic coefficients at finite temperature for a variety of materials, for which experimental data are still missing or difficult to obtain. Two, the calculation from first principles of isothermal SOECs of materials with the orthorhombic, monoclinic, or triclinic symmetry, i.e. classes of materials that are within the reach of our QHA approach, which is suitable to be applied to study thermoelastic properties of low-symmetry materials. Three, the "rapid" calculation from first principles of<br/>the thermal expansion coefficient and SOECs of a material over finite intervals of temperature and pressure (by<br/>using our strategy combining QHA calculations and nonlinear continuum mechanics). We remark that our computational strategy is applicable to materials of arbitrary symmetry and complexity, and therefore it could be used to build databases of materials properties at finite temperature, or to investigate the thermoelastic and mechanical properties of, for example, minerals of geological relevance or metal alloys for structural applications.