Elastic constants and bending rigidities from long-wavelength perturbation expansion

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 and bending rigidity tensors of crystalline solids based on interatomic force constants and long-wavelength perturbation theory. In the long-wavelength limit, lattice vibrations will induce macroscopic electric fields which further couple with the propagation of elastic waves, and a separate treatment on the long-range electrostatic interactions is thereby needed to obtain elastic properties under the correct electrical boundary conditions. To achieve this, a cluster expansion model of the charge density response and dielectric screening function in the long-wavelength limit has been developed to efficiently extract the high-order multipole and dielectric tensors. We implement the proposed method in a first-principles framework and perform extensive validations on silicon, NaCl, GaAs and rhombohedral BaTiO$_3$ as well as monolayer graphene, hexagonal BN, MoS$_2$ and InSe, in good agreement with other theoretical approaches and experimental measurements. Surprisingly, we find that the multipolar interactions up to at least octupoles are necessary to converge the short-circuit elastic tensor of bulk materials, while the higher orders beyond the octupole interactions are required to converge the bending rigidity tensor of 2D crystals. Our approach greatly simplifies the calculations of bending rigidities and will enable the characterization of mechanical properties of novel functional materials. The dataset uploaded here contains essential data for reproducing the main results of this work. These data include the modified q2r and matdyn code of Quantum ESPRESSO distribution, pseudopotentials used in this work, optimized crystal structures, dielectric properties, dynamical matrices, phonon dispersions and elastic properties.