Molecules as metric measure spaces with Kato-bounded Ricci curvature

TL;DR

This paper models molecules as metric measure spaces with Kato-bounded Ricci curvature, demonstrating their stochastic completeness and heat smoothing properties, linking geometric analysis with quantum molecular models.

Contribution

It introduces a novel geometric framework for molecules using metric measure spaces with Kato-bounded Ricci curvature, connecting probabilistic and analytical properties.

Findings

The metric measure space has a Bakry-Emery-Ricci tensor bounded by a Kato class function.

The space is stochastically complete, ensuring nonexplosive Brownian motion.

The heat semigroup exhibits an $L^{ abla}$-to-Lipschitz smoothing property.

Abstract

Set $\Psi:=-\log(\tilde{\Psi})$, with $\tilde{\Psi}>0$ the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $\Sigma\subset \IR^{3n}$ be the set of singularities of the underlying Coulomb potential. We show that the metric measure space $\IMM$ given by $\IR^{3n}$ with its Euclidean distance and the measure $$ \mu(dx)=e^{-2\Psi(x)}dx $$ has a Bakry-Emery-Ricci tensor which is absolutely bounded by the the function $x\mapsto |x-\Sigma|^{-1}$, which we show to be an element of the Kato class induced by $\IMM$. In addition, it is shown $\IMM$ is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive, and that the heat semigroup of $\IMM$ has the $L^{\infty}$-to-Lipschitz smoothing property. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for $e^{\Psi}$ by Fournais/S\o rensen, as well as Aizenman/Simon's Harnack inequality for Schr\"odinger operators. Moreover, our results suggest to study general metric measure spaces having a Ricci curvature which is synthetically bounded from below/above by a function in the underlying Kato class.