At the interface between quantum chemistry and mathematics : the surface hopping algorithms

Starting with the formalism of quantum physics, we shall give an overview of some mathematical questions which raise in the context of the description of molecular dynamics. The mathematical research about these issues, in particular the concept of adiabaticity and the Landau-Zener formula, originates in the 30's and has known developments in the last decades. Simultaneously and mainly since the 70's, chemists have developed a range of powerful technics which allow them to solve molecular dynamics with those surface hopping algorithms. We shall explain how a rigorous mathematical analysis can be performed to construct a surface hopping algorithm which enjoy a proof of convergence and an error estimate. During the talk, we shall focus on the ideas and their construction, and shall not enter into technical issues and proofs.