Stochastic and Deterministic Molecular Dynamics Derived from the Time-independent Schrödinger Equation

Ehrenfest, Born-Oppenheimer, Langevin and Smoluchowski dynamics are shown to be accurate approximations of time-independent Schrödinger observables for a molecular system avoiding caustics, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding Hamilton-Jacobi equation, bypasses the usual separation of nuclei and electron wave functions, includes crossing electron eigenvalues, and gives a different perspective on the BornOppenheimer approximation, Schrödinger Hamiltonian systems, stochastic electron equilibrium states and numerical simulation in molecular dynamics modeling.