The factorization ansatz for non-local approximations to the exchange-correlation hole

Among the various types of approximations to the exchange-correlation energy (E_XC), the completely non-local approach is one of the lesser explored approximation schemes. It has not yet reached the predictive power of the widely used generalized gradient approximations, meta-generalized gradient approximations, hybrids, etc. In non-local functionals pursued here, the electron density at every point in space is employed to express the exchange-correlation energy per particle ϵ_XC(r) at a given position r. Here, we use the non-local, spherical-averaged density ρ(r,u)=∫dΩ_u4πρ(r+u) as a starting point to construct approximate exchange-correlation holes through the factorization ansatz ρ_XC(r, u) = f(r, u)ρ(r, u). We present upper and lower bounds to the exchange energy per particle ϵ_X(r) in terms of ρ(r, u). The factor f(r, u) is then designed to satisfy various conditions that represent important exchange and correlation effects. We assess the resulting approximations and find that the complex, oscillatory structure of ρ(r, u) makes the construction of a corresponding f(r, u) very challenging. This conclusion, identifying the main issue of the non-local approximation, is supported by a detailed analysis of the resulting exchange-correlation holes.