On the breakdown of the Ehrenfest method for molecular dynamics on surfaces

Due to a continuum of electronic states present in periodic systems, the description of molecular dynamics on surfaces poses a serious computational challenge. One of the most used families of approaches in these settings are friction theories, which up to a random fluctuating force term are based on the Ehrenfest approach. Yet, a mean-field treatment of electronic degrees of freedom in the Ehrenfest method makes this approach inaccurate in some cases. Our aim is to clarify when Ehrenfest breaks down for molecular dynamics on surfaces. Answering this question provides limits of applicability for more approximate friction theories derived from Ehrenfest. We assess the Ehrenfest method on one-dimensional, numerically exactly solvable models with a large but finite number of electronic states. Using the Landau-Zener formula and the Massey parameter, an expression that determines when Ehrenfest breaks down is deduced.