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Hedin's equations are an iterative scheme to calculate the Green's function $G$, the self-energy $\Sigma$, the vertex $\Gamma$, the polarizability $\chi$, and the screened interaction $W$.

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However, is there a nice identity that gives me the ground state energy only in terms of $G$, $\Sigma$, $\Gamma$, $\chi$, and/or $W$?

I know that there are approximations to Hedin's equations, like the GW approximation or RPA. There we can write the ground state energy as: $$ E_0^{\text{RPA}} = \frac{1}{2\pi}\int \text d \omega \; \text{Tr} \ln\big(1-\chi(\text i \omega) V\big), $$ where $V$ is the Coulomb operator.

What if I don't make any approximations but solve Hedin's equations such that I have all five quantities $G$, $\Sigma$, $\Gamma$, $\chi$, and $W$ (I know that this is not possible in real life), how do I find the ground state energy?

Just from the diagrammatics I guess that it must be something like $$ E_0 = G\Sigma $$ or so... . Can someone help me out?

PS: I know that there is the Galitskii-Migdal formula such that I can write the ground state energy in terms of the Green's function. But there are operators in the integrand thus the equation is not practical for an implementation into a computer code.

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Yes you are right that total free energy can be symbolically written in the form $\Sigma G$. Refer to chapter 3 of Mahan's Many Particle Physics for details.

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    $\begingroup$ Rather than referring to a textbook, could you summarize the key parts from Ch. 3 of the text that you think is necessary to address thyme's question? $\endgroup$
    – Kyle Kanos
    Apr 5, 2017 at 9:59

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