Computational Error Estimates for Born–Oppenheimer Molecular Dynamics with Nearly Crossing Potential Surfaces

Abstract

This thesis consists of two papers that concern error estimates for the Born-Oppenheimer molecular dynamics, and adaptive algorithms for the Car-Parrinello and Ehrenfest molecular dynamics. In Paper I, we study error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces. The paper first proves an error estimate showing that the difference of the values of observables for the time- independent Schrodinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states and the electron/nuclei mass ratio. Then we present a numerical method to determine the probability to be in excited states, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem. In Paper II, we present an approach, motivated by the Landau-Zener probability estimation, to systematically choose the artificial electron mass parameter appearing in the Car-Parrinello and Ehrenfest molecular dynamics methods to achieve both good accuracy in approximating the Born-Oppenheimer molecular dynamics solution, and high computational efficiency. This makes the Car- Parrinello and Ehrenfest molecular dynamics methods dependent only on the problem data.