content_image/359559.jpg content_image/359566.jpg content_image/359567.jpg content_image/359568.jpg content_image/359569.jpg content_image/359570.jpg content_image/359571.jpg content_image/359572.jpg content_image/359573.jpg content_image/359574.jpg
We arrive at the desired derivative 

<img src='content_image/359559.jpg'>

It is clear that for unoccupied states the derivative is always positive and for filled states it is always negative. Repeating the derivation with the exact W produces the same conclusion which is thus inherent to the GW form of the self-energy when G is non-interacting. We discuss the implications of this result in Section 8. 

For completeness, we provide the derivatives of F versus the wave functions ψ$_{n}$ ( x ) and versus the potential U$_{0}$ . For the wave function derivatives we have 

The functional derivative of G$_{0}$ is 

<img src='content_image/359566.jpg'>

<img src='content_image/359567.jpg'>

We insert this into the integral and perform the contour integral. For the general case, 

<img src='content_image/359568.jpg'>

<img src='content_image/359569.jpg'>

and expressions for the GW -RPA approximation are found by performing the substitutions of Eqs. (26,27). For the derivative versus U$_{0}$ , it is convenient to express the potential in the one-particle eigenbasis so U$_{0}$$_{j,k}$ = 〈 j | U$_{0}$ | k 〉 . The variation δU$_{0}$ is then 

<img src='content_image/359570.jpg'>

Inserting this into Eq. (17) and performing the contour integral and some algebraic manip- ulations yields 

<img src='content_image/359571.jpg'>

<img src='content_image/359572.jpg'>

<img src='content_image/359573.jpg'>

<img src='content_image/359574.jpg'>

