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 $2\times 2\times 2$ 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 $1\times 1\times 16$ supercell with a cubic structure.

Figure 1

Phonon dispersions of MgO calculated with the supercell method and with linear response (Courtesy of Comput. Phys. Commun. and Alfè[19]). Pink dotted and green dashed lines correspond to calculations with a 4 × 4 × 4 supercell (128 atoms) and a 4 × 4 × 4 equispaced grid of q points (linear response). Blue dashed and red solid lines correspond to calculations with an 8 × 8 × 8 supercell (1024 atoms) and an 8 × 8 × 8 grid of q points with the two methods, respectively. Experimental data are displayed with blue stars. For interpretation of the references and other legends in the figure, the reader is referred to Fig. 3 in the paper by Alfè.[19]

Figure 2

Phonon dispersions of MgO calculated with a 16-atom 2 × 2 × 2 supercell of the primitive unit cell. The solid (dashed) lines represent the present calculation with (without) considering the term represented by Eq. (16), which is the contribution to the IFC by the dipole-dipole interaction between the supercells. The open circles (red) are inelastic neutron scattering data from Sangster et al.[28]

Figure 3

(0,0,$q$) Phonon dispersions of MgO calculated with a 128-atom 1 × 1 × 16 elongated supercell of the 8-atom cubic unit cell. The solid (dashed) lines represent the present calculation with (without) considering the term represented by Eq. (16), which is the contribution to the IFC of the dipole-dipole interaction between the supercells. The open circles (red) are inelastic neutron scattering data from Sangster et al.[28]