Elastic tensors from first principles: the Born perturbation expansion and long-range multipolar electrostatics

Mechanical and elastic properties of materials are among the most fundamental quantities for many engineering and industrial applications. Here, we present an efficient and accurate approach for calculating the elastic tensor of crystalline solids based on interatomic force constants and Born perturbation theory. Importantly, in the long-wavelength limit, the induced long-range electrostatic interactions couple with the propagation of elastic waves. In order to obtain the elastic tensor under correct electrical boundary conditions, a separate treatment on the long-range multipolar effects up to at least octupole term is required. We implement such formulation in the first-principles Quantum ESPRESSO distribution, and perform an extensive validation against conventional finite-difference calculations and experimental measurements for Si, NaCl, GaAs and BaTiO₃. This approach greatly simplifies the calculation of elastic tensors of low-dimensional materials, and will help the characterization of novel functional materials.