Bandgaps of insulators from moment-functional based spectral density-functional theory

Within the method of spectral moments it is possible to construct the spectral function of a many-electron system from the first $2P$ spectral moments ($P=1,2,3,\dots$). The case $P=1$ corresponds to standard Kohn-Sham density functional theory (KS-DFT). Taking $P>1$ allows us to consider additional important properties of the uniform electron gas (UEG) in the construction of suitable moment potentials for moment-functional based spectral density-functional theory (MFbSDFT). For example, the quasiparticle renormalization factor $Z$, which is not explicitly considered in KS-DFT, can be included easily. In the 4-pole approximation of the spectral function of the UEG (corresponding to $P=4$) we can reproduce the momentum distribution, the second spectral moment, and the charge response acceptably well, while a treatment of the UEG by KS-DFT reproduces from these properties only the charge response. For weakly and moderately correlated systems we may reproduce the most important aspects of the 4-pole approximation by an optimized two-pole model, which leaves away the low-energy satellite band. From the optimized two-pole model we extract \textit{parameter-free universal} moment potentials for MFbSDFT, which improve the description of the bandgaps in Si, SiC, BN, MgO, CaO, and ZnO significantly.