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A method to evaluate the particle-phonon coupling (PC) corrections to thesingle-particle energies in semi-magic nuclei, based on the direct solution ofthe Dyson equation with PC corrected mass operator, is presented. It is usedfor finding the odd-even mass difference between even Pb and Sn isotopes andtheir odd-proton neighbors. The Fayans energy density functional (EDF) DF3-a isused which gives rather highly accurate predictions for these mass differencesalready at the mean-field level. In the case of the lead chain, account for thePC corrections induced by the low-laying phonons $2^+_1$ and $3^-_1$ makesagreement of the theory with the experimental data significantly better. Forthe tin chain, the situation is not so definite. In this case, the PCcorrections make agreement better in the case of the addition mode but theyspoil the agreement for the removal mode. We discuss the reason of such adiscrepancy.

Including particle-vibration coupling in the Fayans functional. Odd-even mass differences of semi-magic nuclei