# Numerical Analysis: Numerical methods for molecular dynamics

The goal of this project is to develop numerical methods for deterministic and stochastic molecular dynamics simulations that include accurate error estimates. Molecular dynamics is a computational method to study molecular systems in material science and chemistry, e.g. to construct and understand new materials and determine biochemical reactions in drug design. An important question in classical molecular dynamics simulations is how to verify the accuracy computationally, without solving a Schrödinger equation. In our report [How accurate is Born-Oppenheimer molecular dynamics for near crossings of potential surfaces?, arXiv: 1305.3330] we use a combination of analysis and computations to determine this accuracy in three steps:

- We study the time-independent Schrödinger equation as the reference model, including excited electron states with near crossing potential surfaces and estimate the accuracy of observables as a function of the probability to be in excited states, using Egorov’s theorem in Fourier analysis and assuming that space-time averages of the molecular dynamics observable converge in distributional sense with a rate related to the maximal Lyapunov exponent,
- We use stability analysis of a perturbed eigenvalue problem to estimate the probability to be in excited electron states, based on perturbations related to (Landau-Zener like) dynamic transition probabilities,
- We compute Ehrenfest molecular dynamics to estimate the dynamic transition probability.

We also have related computational results for Ehrenfest dynamics with adaptive mass algorithms, based on the algorithm in step (2-3).