Error Estimation and Adaptive Methods for Molecular Dynamics

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 Schrödinger 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.