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 selfenergy ##\Sigma(E)##, that is a function of the energy ##E##. A matrix element of the selfenergy in the basis of spinorbitals 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 Eindependent term and ##\eta## is an infinitesimal positive. We can see that the selfenergy 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 viceversa. 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 selfenergy, we espect that it has a form like:
The article I am reading say that, to a closedshell 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 selfenergy, that is similar to the expansion of the Green function using Feynman diagrams, but I can't see why. Does anyone have any idea?
The article I am reading say that, to a closedshell 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 selfenergy, 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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