Compactness of molecular reaction paths in quantum mechanics
Spectral Methods in Mathematical Physics
04 April 15:30 - 16:30
Ioannis Anapolitanos - Karlsruhe Institute of Technology
We study isomerizations in quantum mechanics. We consider a neutral molecule composed of N quantum electrons and M classical nuclei and assume that the first eigenvalue of the corresponding N-particle Schrödinger operator possesses two local minima with respect to the locations of the nuclei. An isomerization is a mountain pass problem between these two local configurations, where one minimizes over all possible paths the highest value of the energy along the paths. I will discuss the problem in the particular case of a molecule composed of two rigid sub-molecules that can move freely in space: in this case under appropriate assumptions on the multipoles of the two molecules, we are able to prove that the distance between them stays bounded during the whole chemical reaction. We obtain a critical point at the mountain pass level, which is called a transition state in chemistry.
Our method required to study the critical points and the Morse indices of the classical multipole interactions, as well as to improve existing results about the van der Waals force, which is a purely quantum effect. In the first half of the talk I will explain background related to multipole moments and van der Waals forces.
The talk is based on a joint work with Mathieu Lewin.
University of Zurich, UZH