0 Born - Oppenheimer Approximation near Level Crossing

We consider the Born-Oppenheimer problem near conical intersection in two dimensions. For energies close to the crossing energy we describe the wave function near an isotropic crossing and show that it is related to generalized hypergeometric functions 0 F 3. This function is to a conical intersection what the Airy function is to a classical turning point. As an application we calculate the anomalous Zeeman shift of vibrational levels near a crossing. 1 Introduction. The Born-Oppenheimer (BO) problem [1] is concerned with the analysis of Schrödinger type operators where the small electron to nucleon mass ratio, plays the role of the semiclassical parameter. [2–9]. The theory identifies distinct energy scales: The electronic scale which, in atomic units, is of order one and the scale of nuclear vibrations which is of order (1/M) 1/2 in these units. M is the nucleus to electron mass ratio. The identification of the electrons as the fast degrees of freedom is central to the theory. The clean splitting between fast and slow degrees of freedom fails near eigenvalue crossing of the electronic Hamiltonian where there is strong mixing between electronic and vibrational modes. This lies at the boundary of the conventional BO theory. Since the coupling between different electronic energy surfaces becomes infinite near a crossing, the nuclear wave function does not reduce to a solution of a scalar (second order) Schrödinger equation. We describe the (double surface) nuclear wave function near an isotropic conical crossing, for energies close to the energy of the crossing. The strong mixing of the electronic and nuclear degrees of freedom near crossing leads an anomalous Zeeman effect. To describe the anomaly recall that the Zeeman splitting in molecules is reduced compared to the Zeeman splitting of atoms. It is convenient to parameterize the reduction by a parameter γ so that the Zeeman splitting is of the form (and order) M −γ B with B the external magnetic field. The low lying vibrational levels have large reduction, γ = 1. This is what one expects from nuclei whose magnetic moments are by factor M smaller than the Bohr magneton. Levels near the crossing energy can have a small reduction expressed by the fact that γ < 1. For the isotropic situation we calculate γ = 1/6, so that the Zeeman shift is anomalously large, by a factor of about 2000, than the normal Zeeman splitting of molecular levels. More precisely, …