A high-efficient strain-stress method for calculating higher-order elastic constants from first-principles

Though the method for calculating higher-order elastic constants (HOECs) have a long history, there still exist some unsolved issues (e.g. low efficiency), making the development of HOECs much slower than second-order elastic constants. In this paper, we present a general and efficient strain-stress method (SSM) for calculating HOECs from first-principles. In this method, the required number of strain modes is sharply cut down, which ensures a higher efficiency. The time spent on traditional methods is about 3-5 times that of SSM for calculating TOECs of diamond. By taking the HOECs into consideration, the convergence against maximum strain in SSM gets improved significantly, and the results of diamond, gold and magnesium obtained in SSM agree well with previous calculations by other methods. To accelerate the development of the calculation tools for HOECs, we present an algorithm, as well as an open source code, to deduce the strain modes and corresponding coefficients. Specially, we give an explicit expression of strain modes and corresponding coefficients for calculating TOECs in arbitrary symmetry and fifth-order elastic constants in CI (Laue group) symmetry. In addition, we make some extension, e.g. high-accurate numerical differentiation formula, of some existing methods.