Elastic constants and bending rigidities from long-wavelength perturbation expansions

Mechanical and elastic properties of materials are among the most fundamental quantities for many engineering and industrial applications. Here, we present a formulation that is efficient and accurate for calculating the elastic and bending rigidity tensors of crystalline solids, leveraging interatomic force constants and long-wavelength perturbation theory. Crucially, in the long-wavelength limit, lattice vibrations 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 required to obtain elastic properties under the appropriate electrical boundary conditions. A cluster expansion of the charge-density response and dielectric screening function in the long-wavelength limit has been developed to efficiently extract multipole and dielectric tensors of arbitrarily high order. We implement the proposed method in a first-principles framework and perform extensive validations on silicon, Na⁢Cl , Ga⁢As and rhombohedral Ba⁢Ti⁢O3 as well as monolayer graphene, hexagonal BN , Mo⁢S2 , and In⁢Se , obtaining good to excellent agreement with other theoretical approaches and experimental measurements. Notably, we establish that multipolar interactions up to at least octupoles are necessary to obtain the accurate short-circuit elastic tensor of bulk materials, while higher orders beyond octupole interactions are required to converge the bending rigidity tensor of two-dimensional crystals. The present approach greatly simplifies the calculations of bending rigidities and will enable the automated characterization of the mechanical properties of novel functional materials.