# Poles of the self-energy in atoms and molecules

We can use the method of the Green functions to calculate ioniation potentials and electron affinities of atoms and molecules. These quantities can be determined if we know the self-energy ##\Sigma(E)##, that is a function of the energy ##E##. A matrix element of the self-energy in the basis of spin-orbitals has a spectral form $$\Sigma_{pq}(E) = \Sigma_{pq}(\infty) + \sum_{\mu}\frac{f_{pq\mu}}{E - M_{\mu} + i\eta} + \sum_{\nu}\frac{g_{pq\nu}}{E - N_{\nu} - i\eta},$$where ##\Sigma_{pq}(\infty)## is a E-independent term and ##\eta## is an infinitesimal positive. We can see that the self-energy has poles in the sets ##\{M_{\mu}\}## and ##\{N_{\nu}\}##, but this poles are separeted by a interval ##E = E_{0}##. It means that there aren't poles of the first sum between two poles of the second sum, and vice-versa. If we call the poles of the first sum by ##\Sigma_{+1},\ \Sigma_{+2},\ ...## in the increasing order and the poles of the second sum in decreasing order by ##\Sigma_{-1},\ \Sigma_{-2},\ ...##, and plot a graph of the self-energy, we espect that it has a form like:

The article I am reading say that, to a closed-shell system, the interval between the poles ##\Sigma_{-1}## and ##\Sigma_{+1}## conteim the energies of the outer valence electrons. He say that is easy to see looking to the perturbation expansion of the self-energy, that is similar to the expansion of the Green function using Feynman diagrams, but I can't see why. Does anyone have any idea?

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