  A method is developed to consider the particle-phonon coupling (PC) effects
in the calculation of the odd-even double mass differences (DMD) in semi-magic
nuclei starting from the free $NN$-potential. The PC correction $\delta
\Sigma^{\rm PC}$ to the mass operator $\Sigma$ is found in
$g_L^2$-approximation, $g_L$ being the vertex of creating the $L$-phonon. The
tadpole term of the operator $\delta \Sigma^{\rm PC}$ is taken into account.
The method is based on a direct, without any use of the perturbation theory,
solution of the Dyson equation with the mass operator
$\Sigma(\epsilon){=}\Sigma_0{+}\delta \Sigma^{\rm PC}(\epsilon)$ for finding
the single-particle energies and $Z$-factors. In its turn, they are used as an
input for finding different PC corrections to the DMD values. Results for a
chain of even semi-magic nuclei $^{200-206}$Pb show that the inclusion of the
PC corrections makes agreement with the experimental data significantly better.
