An integrable model of a planar tri-atomic molecule

Abstract

A model of a planar tri-atomic molecule is presented, which is integrable in the Born–Oppenheimer adiabatic approximation. The molecular Hamiltonian is the sum of a nuclear vibrational energy operator and an electronic Hamiltonian, where vibrations of nuclei are defined to be motions with vanishing total angular momentum in the center-of-mass system, and where the electronic Hamiltonian is assumed to be a traceless 2 × 2 Hermitian matrix defined on Ṙ3, the shape space of the planar three-body system. Once an eigenvalue of the electronic Hamiltonian is chosen, vibrational-electronic interaction is introduced through covariant differential operators acting on sections of the eigen-line bundle associated with the chosen eigenvalue. The Hamiltonian for nuclear motion coupled with electronic state is then described in terms of these covariant differential operators together with the chosen eigenvalue as a potential for nuclear motion. The eigenvalues of the nuclear Hamiltonian are evaluated for bound states. In the case that vibrational-electronic interaction is restricted to small vibrational-electronic one around a symmetric configuration of the nuclei, a remark is made on a relation to a well-known Hamiltonian describing the dynamic Jahn–Teller effect for a planar tri-atomic molecule X3.