The use of symmetry in reciprocal space integrations. Asymmetric units and weighting factors for numerical integration procedures in any crystal symmetry

A systematic collection of spatial domains for reciprocal space integrations is derived for all possible crystal symmetries. This set can be used as a simpler alternative to the conventional Brillouin zones. The analysis is restricted to integrations where the function in the integrand satisfies inversion symmetry in k space. In this case only 24 different spatial domains have to be defined in order to allow for k space integrations in the 230 different crystal symmetries. A graphic representation of the asymmetric unit for each of the 24 integration domains is given. Special positions and the associated weighting factors required for numerical integrations in theoretical solid-state approaches are tabulated.