Parameterizing Intersecting Surfaces via Invariants

TL;DR

This paper presents numerical methods for reconstructing intersecting hypersurfaces from data, motivated by applications in molecular chemistry, demonstrating convergence, stability, and practical effectiveness in modeling electronic states.

Contribution

It introduces and analyzes companion matrix methods for hypersurface reconstruction from unordered data, with applications to molecular energy surfaces.

Findings

Methods accurately reconstruct energy surfaces in chemistry applications.

The approach demonstrates predicted convergence and stability behaviors.

Successful numerical experiments on Graphene and SO2 molecules.

Abstract

We introduce and analyze numerical companion matrix methods for the reconstruction of hypersurfaces with crossings from smooth interpolants given unordered or, without loss of generality, value-sorted data. The problem is motivated by the desire to machine learn potential energy surfaces arising in molecular excited state computational chemistry applications. We present simplified models which reproduce the analytically predicted convergence and stability behaviors as well as two application-oriented numerical experiments: the electronic excited states of Graphene featuring Dirac conical cusps and energy surfaces corresponding to a sulfur dioxide ($SO_2$) molecule in different configurations.