First-principles theory of frozen-ion flexoelectricity
  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, SrTiO$_3$, BaTiO$_3$, and   PbTiO$_3$. Three important subtleties associated with pseudopotentials, the treatment of surfaces, and the calculation of transverse components are also discussed. 
Jiawang Hong and David Vanderbilt
cond-mat.mtrl-sci