A Born–Oppenheimer Expansion in a Neighborhood of a Renner–Teller Intersection

Abstract

We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner–Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of \({\epsilon}\) , where \({\epsilon^4}\) is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter \({\tilde{b}}\) , which is equivalent to the parameter originally used by Renner. For \({0< \tilde{b}< 1}\) , we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near \({\tilde{b}=0.925}\) . We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for \({0< \tilde{b}< 0.925}\) and non-degenerate for \({0.925< \tilde{b}< }\) .