Mixed-space approach for calculation of vibration-induced dipole-dipole interactions

By explicitly taking into account the effects of vibration-induced dipole-dipole interactions between periodic supercells, we derive an efficient formulation to calculate the phonon frequencies of an ionic crystal. We demonstrate that the vibration-induced dipole-dipole interactions lead to a constant contribution to the interatomic force constant in real space. It recovers the result of Cochran and Cowley [Cochran and Cowley, J. Phys. Chem. Solids 23, 447 (1962)] at the long wavelength limit. Using MgO as the prototype, we demonstrate that a 16-atom $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ supercell of the primitive unit cell is sufficient to obtain the phonon dispersions when the dipole-dipole interactions are considered. We find that not properly taking into account the dipole-dipole interaction leads to oscillations along the (0,0,q) direction for the longitudinal optical phonon dispersion in a 128-atom elongated $1\ifmmode\times\else\texttimes\fi{}1\ifmmode\times\else\texttimes\fi{}16$ supercell with a cubic structure.