Symmetry-general least-squares extraction of elastic coefficients from ab initio total energy calculations

Abstract

A symmetry-general scheme for the simultaneous least-squares extraction of the elastic coefficients and of the residual strain components from ab initio total energy calculations on crystal structure models of materials is proposed. It is quite efficient and avoids error propagation. An appropriate, but usually singular, set of normal equations is first formulated in a triclinic framework, with 21 stiffness coefficients and 6 residual strain components. Rank reduction of this $27\ifmmode\times\else\texttimes\fi{}27$ least-squares system of normal equations is then performed through systematic implementation of the constraints corresponding to the known symmetry of the material. A regular $p\ifmmode\times\else\texttimes\fi{}p$ matrix is obtained through this process, where p is the total number of independent coefficients and components. This computationally robust approach to the extraction of elastic coefficients and their standard deviations can be used to analyze any number of adequately selected and weighted values of the total energy that is larger than the number of independent parameters. It also provides values for the minimum energy and for the corresponding cell data, again with standard errors. The present work enables the automated calculation of elastic coefficients from crystals with any symmetry through a single logical flow. Examples are given for a few cubic, hexagonal, rhombohedral, tetragonal, and orthorhombic materials with known experimental stiffness values. It would be difficult to exaggerate the convenience of the automated implementation of this symmetry-general approach based on total energy calculations.