The standard results show that the ab initio molecular dynamics on the electron ground state approximates quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. In this talk we discuss connection between ab initio molecular dynamics and quantum observables using the tools of semi-classical analysis and Weyl quantization. We present error analysis of such approximations for quantum observables in nuclei-electrons systems. We derive a certain weighted average of different dynamics that approximates quantum observables at any temperature provided the electron eigenvalue surfaces do not cross. (joint work with A. Kammonen, M. Sandberg, A. Szepessy)