We demonstrate that the frozen-ion contribution to the flexoelectric coefficient is given solely in terms of the sum of third moments of the charge-density distortions induced by atomic displacements, even for ferroelectric or piezoelectric materials. We introduce several practical supercell-based methods for calculating these coefficients from first principles, and demonstrate them by computing the coefficients for C, Si, MgO, NaCl, SrTiO₃, BaTiO₃, and PbTiO₃. Three important subtleties associated with pseudopotentials, the treatment of surfaces, and the calculation of transverse components are also discussed.
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Hong et al. (2011) studied this question.