Exploiting superspace to enable DFT calculations of modulated structures with disordered sites using the example of mullite

Modulated structures with a harmonic occupational modulation are not suitable for force field methods (including molecular dynamics) or density functional theory (DFT) calculations because fully occupied sites are required. Instead of assuming a random distribution with some constraints, which is a common approach, the harmonic occupational modulation function from a refinement in superspace can be replaced by a block wave function resulting in a model, that represents the hypothetical completely ordered structure with fully occupied sites. If the modulation wave vector corresponds to a commensurate modulation, the corresponding model in physical space can be used for DFT calculations, otherwise a commensurate approximation is needed. As an example, the present approach is applied to the modulated structure of mullite to determine the ideal Al/Si ordering for different compositions and supercell sizes. For the analyzed composition, the calculations yield consistently the same Al/Si ordering pattern. The results are also in qualitative agreement with the refinement of the not fully ordered structure, i.e. the volumes of tetrahedra with Al/Si disorder follow the trend of the Al/Si ordering pattern from the DFT calculations. The example shows that the approach allows to determine structural parameters, that are very difficult or even impossible to access experimentally.