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I have a Hamiltonian in the momentum space which has a strong non-local electron-electron interaction.

I know that I have to find its corresponding exchange self-energy and solve the Dyson equation self-consistently.

$$G = G_0 * \Sigma_I * G$$

which $\Sigma$ is a function of $G$. $\quad$ ($\Sigma(G)$)

So how does this self-energy look like in the momentum space. If there is not a specific self-energy, how should I start to make it step by step.

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