Full determination of the mixing parameter in hybrid functionals

We propose a new method to fully determine the mixing parameter in the hybrid exchange-correlation functional. The mixing parameter is calculated based on the condition that the electron density of a system with a fractional electron number N-a (0<a<1) is a linear combination of the electron densities of systems with N and N-1 electrons. We derive the nonlinear equation for solving the mixing parameter and propose a simple approximation to much simplify this equation. The obtained mixing parameter is a three-dimensional function in the real space and is shown to become one in the one-electron regions. The method can also be generalized to spin-polarized systems to produce different mixing parameters for different spins. This new method is tested on both neutral molecules and anions, and the predictions of the ionization energies are much improved. This work shows a promising route to fully determine the mixing parameter in the hybrid functionals.