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 $W$.

enter image description here

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.


1 Answer 1


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.

  • 3
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.