Non-adiabatic electron-phonon interaction: Investigation using continued fractions

The eigenvalue problem of the extended dimer Hubbard Hamiltonian with linear local electron-phonon interaction is investigated. The Hamiltonian is simplified via a transformation of the phonon operators. In this way the eigen value problem is reduced to a one-phonon-mode case. This leads to a three-dimensional matrix equation, which is solved using a matrix continued fraction procedure. As a generalization we consider a model in which the dimer couples irreducibly to two phonon modes. Again we deduce a matrix equation, but in this case it is - in principle - of infinite dimension. To solve this equation with the help of the matrix continued fraction procedure we have to use two “cut-off” parameters. The influence of the approximation parameters is explained. A special case of the last model proves to be the E-e-Jahn-Teller Hamiltonian which leads to an ordinary continued fractions problem.