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 epsilon(XC)(r) at a given position r. Here, we use the non-local, spherical-averaged density rho(r,u) = integral d Omega(u)/4 pi rho(r + u) as a starting point to construct approximate exchange-correlation holes through the factorization ansatz rho(XC)(r, u) = f(r, u)rho(r, u). We present upper and lower bounds to the exchange energy per particle epsilon(X)(r) in terms of rho(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 rho(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. Published under an exclusive license by AIP Publishing.

Automated summary

Completely non-local approach, as one type of exchange-correlation energy approximation, has not been explored as much as other schemes. This approach uses electron density at every point to express the exchange-correlation energy, and constructs approximate exchange-correlation holes to handle the non-locality. However, the complex structure of the density makes it challenging to construct the corresponding function.