## How accurate is Born-Oppenheimer molecular dynamics for crossings of potential surfaces ?    [PDF]

Håkon Hoel, Ashraful Kadir, Petr Plecháč, Mattias Sandberg, Anders Szepessy
The difference of the value of observables for the time-independent Schr\"odinger equation, with matrix valued potentials, and the values of observables for {\it ab initio} Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states. We present a method to determine the probability to be in excited states from Landau-Zener like dynamic transition probabilities, based on Ehrenfest molecular dynamics and stability analysis of a perturbed eigenvalue problem. A perturbation $p_E$, in the dynamic transition probability for a time-dependent Schr\"odinger WKB-transport equation, yields through resonances a larger probability of the order $\mathcal O(p_E^{1/2})$ to be in an excited state for the time-independent Schr\"odinger equation, in the presence of crossing or nearly crossing electron potential surfaces. The stability analysis uses Egorov's theorem and shows that the approximation error for observables is $\BIGO(M^{-\gamma/2} + p_E^{1/2})$ for large nuclei-electron mass ratio $M$, provided the molecular dynamics has an ergodic limit which can be approximated with time averages over the period $T$ and convergence rate $\mathcal O(T^{-\gamma})$, for some $\gamma>0$. % $\mathcal O(T^{-\gamma})$ for some $\gamma>0$. Numerical simulations verify that the transition probability $p_E$ can be determined from Ehrenfest molecular dynamics simulations.
View original: http://arxiv.org/abs/1305.3330