  It is known that ab initio molecular dynamics based on the electron ground
state eigenvalue can be used to approximate quantum observables in the
canonical ensemble when the temperature is low compared to the first electron
eigenvalue gap. This work proves that a certain weighted average of the
different ab initio dynamics, corresponding to each electron eigenvalue,
approximates quantum observables for any temperature. The proof uses the
semiclassical Weyl law to show that canonical quantum observables of
nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be
approximated by ab initio molecular dynamics with the error proportional to the
electron-nuclei mass ratio. The result covers observables that depend on
time-correlations. A combination of the Hilbert-Schmidt inner product for
quantum operators and Weyl's law shows that the error estimate holds for
observables and Hamiltonian symbols that have three and five bounded
derivatives, respectively, provided the electron eigenvalues are distinct for
any nuclei position and the observables are in the diagonal form with respect
to the electron eigenstates.
