Self-consistent hybrid functional for condensed systems

A self-consistent scheme for determining the optimal fraction of exact exchange for full-range hybrid functionals is presented and applied to the calculation of band gaps and dielectric constants of solids. The exchange-correlation functional is defined in a similar manner to the PBE0 functional, but the mixing parameter is set equal to the inverse macroscopic dielectric function and it is determined self-consistently by computing the optimal dielectric screening. We found excellent agreement with experiments for the properties of a broad class of systems, with band gaps ranging between 0.7 and 21.7 eV and dielectric constants within 1.23 and 15.9. We propose that the eigenvalues and eigenfunctions obtained with the present self-consistent hybrid scheme may be excellent inputs for $G_0{W}_0$ calculations.

Figure 1

Diagram of the self-consistent hybrid scheme. The potential used in the solution of the generalized Kohn-Sham equation is defined in Eq. (7). ${\epsilon }_{\infty }$ is the static dielectric constant.

Figure 2

Convergence of the value of the static dielectric constant ${\varepsilon }_{\infty }$ in the sc-hybrid scheme is shown for four prototypical semiconductors, Si, C, SiC, and Ge. The blue dashed line and triangles indicate the iterative procedure that starts with no inclusion of exact exchange: $\alpha =0$ (sc-hybrid@PBE); the red solid line and circles correspond to the iterative procedure started with a quarter of exact exchange $\alpha =0.25$ (sc-hybrid@PBE0). The blue and red arrows shown in the first panel indicate the PBE and PBE0 values of ${\epsilon }_{\infty }$. The solid black lines represent the value of the experimental macroscopic dielectric constant.

Figure 3

Total energy of MgO as a function of the lattice constant (Å). The black dotted vertical line indicates the experimental value extrapolated to 0 K, where the zero-point anharmonic contribution has been removed. The colored arrows point to the minima of each surface. The PBE and PBE0 total energy curves were shifted by –0.027 and –0.006 hartrees, respectively, in order to fit on the same plot.