Symmetry-General Least-Squares Extraction of Elastic Coefficients From Ab Initio Total Energy Calculations
Yvon Le Page, Paul Saxe
Physical Review B Condensed Matter 63, 174103 (2001)
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 27×27 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×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.