Crystal-structure calculations with distorted ions

We present the polarization-included electron-gas (PEG) model for crystal structures, which is similar to the modified-electron-gas (MEG) model for crystal structures, but in which the anions can distort from spherical symmetry. This nonspherical distortion is important when the anions occupy low-symmetry positions. For ${\mathrm{SiO}}_2$ quartz, ${\mathrm{SiO}}_2$ cristobalite, ${\mathrm{BeF}}_2$ quartz, and the zeolite sodalite, which have open crystal structures, the structures and energies calculated with the PEG model are in much better agreement with experiment than those calculated with the MEG model. The improved structural results are due mainly to smaller and more accurate cation-anion-cation bond angles. For ${\mathrm{SiO}}_2$ stishovite, ${\mathrm{TiO}}_2$ rutile, and ${\mathrm{Mg}}_2$ ${\mathrm{SiO}}_4$ spinel, which have more closely packed crystal structures, the structures are modeled well with both the PEG and MEG models, but the energies are more accurately calculated with the PEG model. The improved results for the energies are due to the stronger bonds formed when charge density moves into the bonding regions. Electron-distribution plots are in good agreement with those from accurate band-structure calculations for the cristobalite and stishovite phases of silica. The electron-distribution plots show that the nonspherical distortions increase from ${\mathrm{BeF}}_2$ to ${\mathrm{TiO}}_2$ to ${\mathrm{SiO}}_2$, demonstrating that the extent of covalent bonding increases from ${\mathrm{BeF}}_2$ to ${\mathrm{TiO}}_2$ to ${\mathrm{SiO}}_2$, in agreement with electronegativity differences. We find that covalent effects are not as important in ${\mathrm{MgSiO}}_3$ perovskite as they are in the silica polymorphs quartz, cristobalite and stishovite, and ${\mathrm{Mg}}_2$ ${\mathrm{SiO}}_4$ spinel.