Compactness of Molecular Reaction Paths in Quantum Mechanics
Abstract
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 Nparticle 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 path. Here we state a conjecture about the compactness of minmaxing sequences of such paths, which we then partly solve in the particular case of a molecule composed of two rigid submolecules that can move freely in space. More precisely, 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 in chemistry is called a transition state. Our method requires us to study the critical points and the Morse indices of the classical multipole interactions, as well as to generalize existing results about the van der Waals force. This paper generalizes previous works by the second author in several directions.
 Publication:

Archive for Rational Mechanics and Analysis
 Pub Date:
 December 2019
 DOI:
 10.1007/s00205019014755
 arXiv:
 arXiv:1809.06110
 Bibcode:
 2019ArRMA.236..505A
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Spectral Theory
 EPrint:
 Final version to appear in Arch. Rat. Mech. Anal