The Holstein-Hubbard dimer of variable length as the building block of the CuO plane: Electronic and phononic transitions

A polaronic Hamiltonian is obtained for a dimer of length a, including an Einstein phonon of frequency Ω, a Holstein coupling g0, and all the electronic one- and two-body terms consistent with a single orbital per site (the latter evaluated in terms of Wannier functions built from Gaussian atomic orbitals). Its ground state for given a, g0, and Ω is identified by independent and simultaneous optimization of the electronic and phononic parameters. As a varies, we find discontinuous changes of both electronic and phononic states, with interesting physical implications.