Phonon-mediated electron attraction ($V_{\mathrm{eff}}$)

For electron energy transfer $\Delta$ and phonon energy ${\Omega }_q=\hslash {\omega }_q$, the symmetrized pair amplitude is $V_{\mathrm{eff}}=2|M_q{|}^2{\Omega }_q/({\Delta }^2-{\Omega }_q^2)$. It is attractive for $|\Delta |<{\Omega }_q$. This interaction is retarded, and screened Coulomb repulsion must also be considered.