Polymorphism in silica studied in the local density and generalized-gradient approximations

The crystal structures of a large number of silica polytypes (alpha- and beta-quartz, alpha- and beta-cristobalite, beta-tridymite, keatite, coesite and stishovite) have been studied using density functional theory, both in the local density approximation and including generalized-gradient corrections to the exchange-correlation functional. All crystal structures have been optimized by minimizing the total energy with respect to all lattice parameters and to the atomic coordinates within the unit cell (up to 40 structural parameters in the case of coesite). The alpha --> beta transitions in quartz and cristobalite have been studied in detail, including different variants proposed for the structure of beta-cristobalite. The tetragonal (I (4) over bar 2d) and simple cubic (P2(1)3) structures are found to be energetically almost degenerate near the equilibrium volume. On volume expansion both structures converge towards the idealized highly symmetric Fd3m structure. A similar continuous transition from a more compact orthorhombic (C222(1)) to a highly symmetric hexagonal (P6(3)/mmc) variant is also proposed for beta-tridymite. For coesite two monoclinic variants (with C2/c and P2(1)/c space-group symmetries, respectively) have been examined and found to be energetically degenerate to within 1 meV per SiO2 unit. It is shown that within the local density approximation (LDA) the equilibrium atomic volume of all polytypes is predicted with an accuracy better than one per cent. The LDA also leads to excellent structural predictions and to accurate values of the bulk modulus. Corrections in the framework of the generalized-gradient approximation (GGA) lead to substantially larger equilibrium volumes, although at fixed volume the LDA and GGA lead to identical crystal structures. The increased volume also leads to less accurate structural parameters. However, we find that gradient corrections are essential for achieving accurate structural energy differences between the tetrahedrally coordinated phases found at larger atomic volumes tall polytypes except stishovite) and the octahedrally coordinated high-pressure polymorphs (stishovite and post-stishovite phases).