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Elastic Constants and Bending Rigidities from Long-Wavelength Perturbation Expansions

Changpeng Lin1,*, Samuel Poncé2,3, Francesco Macheda4,5, Francesco Mauri4,5, and Nicola Marzari1,6

  • *Contact author: changpeng.lin@epfl.ch
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PRX Energy 5, 013012 – Published 19 March, 2026

DOI: https://doi.org/10.1103/hc53-g1p3

Abstract

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, NaCl, GaAs and rhombohedral BaTiO3 as well as monolayer graphene, hexagonal BN, MoS2, and InSe, 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.

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Popular Summary

Mechanical properties underpin many energy technologies, including solid-state batteries, piezoelectrics, superconductors, and flexible electronics. With the growing use of two-dimensional materials in these applications, bending rigidity has emerged as a crucial yet not fully understood property. Conventionally, bending rigidities are calculated from elastic constants based on continuum models and isotropic approximation. However, the validity of such frameworks become questionable due to the breakdown of continuum mechanics at the nanoscale.

Here, the authors show how Huang's atomic theory of elasticity, which links macroscopic elastic constants to the long-wavelength behavior of lattice vibrations, can be implemented into a computationally actionable framework and integrated into modern first-principles workflows. They further extend Huang's theory to compute the complete fourth-rank bending rigidity tensor of two-dimensional materials, beyond continuum and isotropic approximations. The derived microscopic expression is analytical in terms of interatomic force constants, and bending rigidities are obtained at no additional computational cost. In addition, to obtain mechanical properties under the correct electrical boundary conditions, the authors propose a novel cluster expansion approach that enables the accurate, efficient, and automated extraction of multipolar and dielectric properties up to an arbitrarily high order. These developments advance the realistic modeling of next-generation flexible electronics and electromechanical devices.

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PRX Energy 5, 013012– Published 19 March, 2026
  • Received 25 December 2024
  • Revised 19 December 2025
  • Accepted 18 February 2026
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