New Surface Hopping Methods in Quantum Dynamics

In molecular dynamics, the Born-Oppenheimer approximation yields a system of Schrodinger equations to characterize the evolution of the wave functions that can describe quantum transitions-referred as surface hopping-between different electronic potential energy surfaces. In this work, we develop three numerical methods for solving the so-called surface hopping problem.;The first one is an Eulerian surface hopping method for the system of Schrodinger equations with conical crossing potentials. It is based on the semi-classical 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 is formulated as a hopping condition for flux, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. The advantage of an Eulerian method is that it relies on a fixed number of partial differential equations with a uniform in time computational accuracy.;The second one is a new method to compute the transition rate between the energy surfaces, which can be used for solving the system of Schrodinger equations with thin barriers. This method generalizes the first method, which relies on the Landau-Zener formula to obtain the transition coefficients.;The third one is a hybrid method coupling a Schrodinger solver and a Gaussian beam method for the numerical simulation of quantum tunneling through surface hopping across electronic potential energy surfaces. The idea is to use a Schrodinger solver near potential barriers or zones where potential energy surfaces cross, and a Gaussian beam method-which is much more efficient than a direct Schrodinger solver-elsewhere. Numerical examples show that this method indeed captures quantum tunneling and surface hopping accurately, with a computational cost much lower than a direct quantum solver in the entire domain.;In Chapter 5, the derivation of the Landau-Zener formula for a two-colliding-atom system is discussed in details. The use of the Wigner transform provides a clearer explanation of the limiting process. An interesting third order ODE system is solved numerically and analytically. A numerical way to generalize the Landau-Zener formula to include the interference phenomena is also explained with numerical examples.