Electronic and phononic transitions in the two-site Holstein model

They consider the Hamiltonian for a dimer including all the electronic one- and two-body terms for a single orbital, the free phonon term for a single frequency Ω and an electron-phonon coupling g of the Holstein type. The bare electronic interaction parameters were evaluated in terms of Wannier functions, built from atomic Gaussian orbitals. An effective polaronic Hamiltonian was obtained by a displaced-oscillator transformation, followed by evaluation of the phononic terms over a squeezed-phonon variational wavefunction. Its eigenvalues and eigenvectors have been obtained in explicit from for N = 1, 2, 3, 4 electrons. For each N and over a range of dimer length values, the ground state for given g and Ω was identified by optimizing the orbital shape and the squeezing effect strength. They find, for any N if g ≠ 0, an abrupt transition from a delocalized to a localized polar phase, due to a discontinuous collapse of the squeezing effect above a critical dimer length. Also, in the case N = 2, and for a different dimer length, they find a second transition from a singlet to a triplet ground state that depends also on g. All the effective electronic interactions are appreciably renormalized across the transitions, as the equilibrium shape of the wavefunctions strongly depends on both the squeezing effect and the type of ground state