From electron to small polaron: An exact cluster solution

We perform the exact diagonalization of two, four, and six vibrating molecules strongly coupled with one electron to show that a small polaron describes both adiabatic and nonadiabatic regimes. Electron spectra are derived and compared with the adiabatic Holstein approximation and with the Lang-Firsov canonical transformation in a wide range of the adiabatic parameter and the coupling strength. We show that the self-consistent Migdal approximation is violated at an intermediate value of the coupling constant {λ}{}1 and the adiabatic or nonadiabatic small polaron is the ground state for the intermediate and strong coupling {λ}>1. The phonon frequencies are calculated for a wide range of the coupling, including the intermediate region.