content_image/362243.jpg content_image/362244.jpg content_image/362245.jpg content_image/362246.jpg content_image/362247.jpg
where v, v ′ label valence states and c, c ′ label conduction states. As a check, the diagonal cases v = v ′ or c = c ′ yield the same result as the ϵ$_{n}$ derivative as necessary by first order perturbation theory. Setting all the derivatives to zero is equivalent to solving the LSS Eq. (17) with no constraints on U$_{0}$ . This turns out to yield unphysical results as explained in Section 8. 

## 7. Φ GW xc REWRITTEN EXACTLY WITH PLASMON AND INTERBAND ENER- GIES 

In this section, we rewrite Φ GW xc exactly in terms of a sum over plasmon and interband transition energies. This exact expression has been reported by other workers using different methods [21]. (We found this result contemporaneously and independently.) Our derivation is based on contour methods which we present for completeness before moving on to the implications of this exact expression. 

We start with the result of Eq. (23). The time-ordered RPA ε ( ω ) is given by 

<img src='content_image/362243.jpg'>

which has poles at the interband energies ω = Δ$_{t}$. The poles of the inverse dielectric function ε ( ω ) − 1 occur at the plasma frequencies which are the natural modes of collective charge oscillation. The matrix ε ( ω ) − 1 is given by 

<img src='content_image/362244.jpg'>

(The ω$_{p}$ have infinitesimal negative imaginary parts.) The plasma frequencies ω$_{p}$ and mode functions | p 〉 obey the RPA/Casida Hermitian eigenvalue equations [40] 

The column vectors C$_{t,p}$ are orthonormal and the | p 〉 are given by 

<img src='content_image/362245.jpg'>

<img src='content_image/362246.jpg'>

We see that the ω$_{p}$ are the eigenvalues of the square root of the Ω 2 matrix defined as 

<img src='content_image/362247.jpg'>

Before going forward, we make a clarification concerning our use of the terms “plasmon” or “plasma” modes or frequencies. In this work, these terms refer to solutions of the Casida/RPA Eq. (36) so that the ω$_{p}$ are real frequencies. Of course, a plasmon excita- tion in an actual material has finite life time due to damping processes. Thus, we are 

