Abstract: In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. We develop multiscale surface hopping methods for the SchrÃ¶dinger equation with conical crossings. The first approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau-Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. We also develop a multiscale coupling method that combines the Gaussian beam method away from the hopping zone and a direct Schrodinger solver in the hopping zone, in order to capture accurately phase information. |