Bloch's theorem in orbital-density-dependent functionals: Band structures from Koopmans spectral functionals

Koopmans-compliant functionals provide an orbital-density-dependent framework for an accurate evaluation of spectral properties; they are obtained by imposing a generalized piecewise-linearity condition on the total energy of the system with respect to the occupation of any orbital. In crystalline materials, due to the orbital-density-dependent nature of the functionals, minimization of the total energy to a ground state provides a set of minimizing variational orbitals that are localized and thus break the periodicity of the underlying lattice. Despite this, we show that Bloch symmetry can be preserved and it is possible to describe the electronic states with a band-structure picture, thanks to the Wannier-like character of the variational orbitals. We also present a method to unfold and interpolate the electronic bands from supercell ($\mathrm{\Gamma }$-point) calculations, which enables us to calculate full band structures with Koopmans-compliant functionals. The results obtained for a set of benchmark semiconductors and insulators show very good agreement with state-of-the-art many-body perturbation theory and experiments, underscoring the reliability of these spectral functionals in predicting band structures.

Figure 1

Schematic representation of a two-dimensional two-band model showing the connection between the primitive and supercell Wannier representations. In the primitive picture with a $2\times 2$ sampling of the BZ, the WFs are identified with the pair of labels $\{\mathit{R},n\}$ (red labels): the cell index $\mathit{R}$ taking four values and the band index $n$ taking two values. In the $2\times 2$ supercell with $\mathrm{\Gamma }$ sampling of the BZ, the eight WFs are labeled by only one quantum number (black labels), i.e., the supercell band index $\alpha$ running over the eight states.

Figure 2

PBE (red circles) and Koopmans (green lines) band structure of Si, C, and BN. The zero of the energy is set at the PBE Fermi level. The blue lines represent the PBE interpolated band structures as calculated by the Wannier90 code. In the table at the bottom, we report the initial projections for the Wannierization procedure.

Figure 3

As in Fig. 2, here we report the PBE (blue lines) and KC (green lines) band structures of Ge and GaAs. The table at the bottom contains the initial projections for the Wannierization procedure.

Figure 4

As in Fig. 2, here we report the PBE (blue lines) and KC (green lines) band structures of MgO and LiF. The table at the bottom contains the initial projections for the Wannierization procedure.