Electronic and phononic states of the Holstein-Hubbard dimer of variable length

Abstract

We consider a model Hamiltonian for a dimer of length $a$ including all the electronic one- and two-body terms consistent with a single orbital per site, a free Einstein phonon term for a frequency $\ensuremath{\Omega},$ and an electron-phonon coupling ${g}_{0}$ of the Holstein type. The bare electronic interaction parameters were evaluated in terms of Wannier functions built from Gaussian atomic orbitals. An effective polaronic Hamiltonian was obtained by an unrestricted displaced-oscillator transformation, followed by evaluation of the phononic terms over a squeezed-phonon variational wave function. For the cases of quarter-filled and half-filled orbitals, and over a range of dimer length values, the ground state for given ${g}_{0}$ and $\ensuremath{\Omega}$ was identified by simultaneously and independently optimizing the orbital shape, the phonon displacement, and the squeezing effect strength. As $a$ varies, we generally find discontinuous changes of both electronic and phononic states, accompanied by an appreciable renormalization of the effective electronic interactions across the transitions, due to the equilibrium shape of the wave functions strongly depending on the phononic regime and on the type of ground state.