content_image/361941.jpg content_image/361942.jpg content_image/361943.jpg content_image/361944.jpg content_image/361945.jpg content_image/361946.jpg
or in matrix form 

<img src='content_image/361941.jpg'>

The direction of approach of the limit, t → 0 + versus t → − 0 $^{+}$, is immaterial as both yield the same answer for real valued Green’s functions G$_{0}$ ( ω ) due to time reversal symmetry. We then have our desired result: Φ GW xc written in terms of the time-ordered ε ( ω ), 

<img src='content_image/361942.jpg'>

This expression is our main workhorse. In Sections 7 and 9, we evaluate this integral exactly and approximately to generate alternative forms for Φ GW xc . 

## 6. DERIVATIVES OF F AND Φ GW xc 

In this section, we provide expressions for the derivatives of the exchange-correlation functional Φ GW xc [ G$_{0}$ ] and the total energy functional F [ G$_{0}$, G$_{0}$ ] versus the wave functions ψ$_{n}$ and eigenenergies ϵ$_{n}$ or equivalently the potential U$_{0}$ that determine G$_{0}$ . The ϵ$_{n}$ derivatives are used in Section 8 to prove the unboundedness of the energy functional. Separately, these derivative expressions can prove useful to investigators contemplating variational approaches that extremize F [ G$_{0}$, G$_{0}$ ] which require analytical derivatives. 

We begin with variations of the eigenenergies ϵ$_{n}$ . The derivative of G$_{0}$ is 

<img src='content_image/361943.jpg'>

As per Section 4, the only non-zero contribution to the variation of F [ G$_{0}$, G$_{0}$ ] when changing ϵ$_{n}$ comes from the exchange-correlation functional Φ GW xc so 

<img src='content_image/361944.jpg'>

We turn this into a contour integral over the upper complex ω half plane. Using Eq. (12), we get contributions from the poles of Σ$_{xc}$( ω ) that are above the real axis and a possible contribution from the pole of ∂G$_{0}$/∂ϵ$_{n}$ if n is occupied (if ϵ$_{n}$ < µ ). 

To organize the process, we write Σ$_{xc}$( ω ) in the general form of a sum over poles 

Here, Σ$_{x}$ is the Fock (bare) exchange operator 

<img src='content_image/361945.jpg'>

<img src='content_image/361946.jpg'>

