  The difference of the values of observables for the time-independent
Schroedinger 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. The paper first proves an error estimate (depending
on the electron/nuclei mass ratio and the probability to be in excited states)
for this difference of microcanonical observables, assuming that molecular
dynamics space-time averages converge, with a rate related to the maximal
Lyapunov exponent. The error estimate is uniform in the number of particles and
the analysis does not assume a uniform lower bound on the spectral gap of the
electron operator and consequently the probability to be in excited states can
be large. A numerical method to determine the probability to be in excited
states is then presented, based on Ehrenfest molecular dynamics and stability
analysis of a perturbed eigenvalue problem.
