content_image/362148.jpg content_image/362149.jpg content_image/362150.jpg
This approximation requires that |〈 t | W$_{t}$ (0) | t 〉 / Δ$_{t}$ | ≪ 1. Section 11 provides numerical results for atoms where we show that this expansion of the square root is actually quite accurate. Since screening is rather weak in the localized atomic limit, we expect this approximation to work even better in extended solids which provide much more effective screening of the Coulomb interaction. 

The third approximation is to replace the matrix element 〈 t | W$_{t}$ (0) | t 〉 by 〈 t | W (0) | t 〉 . This also creates an error of O ( 〈 W 〉 $^{2}$/ Δ) and should be applicable in the same cases as the previous approximation. The result is 

<img src='content_image/362148.jpg'>

Completeness ∑ c | ψ$_{c}$ 〉〈 ψ$_{c}$ | = I − ∑ v | ψ$_{v}$ 〉〈 ψ$_{v}$ | permits us to rewrite Φ stat c using only valence states. Putting E$_{X}$ back and dropping terms of order O ( 〈 W 〉 $^{2}$/ Δ), 

<img src='content_image/362149.jpg'>

where W pol = W − V is the induced (polarization) part of W . The first term is the Coulomb hole energy (COH) and the second term the screened exchange energy (SEX). Differentiating versus the non-interacting density matrix ρ$_{0}$ yields the static self-energy Σ stat xc = δ Φ stat xc /δρ$_{0}$ , 

<img src='content_image/362150.jpg'>

This is the COHSEX self-energy if we drop the derivative term δW/δρ$_{0}$ . Ignoring the derivatives underlies the successful BSE approach for optical excitations [46] and excited states forces [55]: the successes of the BSE approach suggests that these terms are not significant in practice. In brief, the COHSEX self-energy of Eq. (52) is approximately associated with and derived from the exchange-correlation energy of Eq. (51). The key approximations underlying COHSEX are the static screening limit and the assumption that the matrix element 〈 t | W (0) | t 〉 for a transition t is much smaller than its energy Δ$_{t}$. Given W , Eqs. (51) and (52) depend only on the valence states through ρ$_{0}$ . Having the energy functional available in Eq. (51), we also know what missing terms we must add to the COHSEX self-energy of Eq. (52) in order to make it a variational scheme. 

To improve COHSEX, we step backwards through the approximations. The first stop is to use Eq. (46): this is still a static approximation but doesn’t assume that 〈 W 〉 / Δ is small. To go beyond Eq. (46), we need dynamic screening. In order to avoid the O ( N $^{6}$) scaling of the direct implementation of Eq. (44), we can contemplate the following scheme: we relate W$_{t}$ (Δ$_{t}$) by to W (Δ$_{t}$) via Eq. (47) and then approximate W by using sum rules as per plasmon-pole models to approximate the frequency dependence of W ( ω ) [6, 56– 58]. Alternatively, we can use model dielectric screening functions [59–61] to construct approximate W ( ω ) and then use Eq. (47) to find W$_{t}$ . Regardless of the specifics, having a total energy expression allows us to find the associated self-energy through differentiation. 

We conclude this section with some observations on what is and is not included when using the approximate expressions derived in this and the previous section. We expect these approximations to certainly include static and localized Coulombic effects automatically. This includes screening effects of the medium as well as localized Coulombic physics: even 

