Infinite-order non-Born-Oppenheimer perturbation theory for systems with intersecting potentials

We analyze models for two intersecting electronic states coupled by one nuclear coordinate. The corresponding model-Hamiltonians depend on a perturbation parameter α. Its variation allows us to switch on the nonadiabatic coupling continuously. We derive analytical expressions for the eigenstates and energies as function of α in the diabatic representation. We rederive the expression for the energy eigenvalues by complete resummation of the Brillouin-Wigner perturbation series in the Born-Oppenheimer basis. We find that the divergences of the nonadiabatic coupling cancel strictly order by order. The admixture of high energy components is shown explicitly.