Algebraic Varieties in Quantum Chemistry

TL;DR

This paper applies algebraic geometry to coupled cluster theory in quantum chemistry, modeling electronic eigenstates with polynomial varieties and analyzing their properties to improve solution methods.

Contribution

It introduces algebraic geometric frameworks for coupled cluster equations, including truncation varieties and their degrees, advancing the mathematical understanding of quantum many-body problems.

Findings

Truncation varieties generalize Grassmannians in CC theory.

Derived Hamiltonians for polynomial systems.

Analyzed CC degrees and solution strategies.

Abstract

We develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schr\"odinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pl\"ucker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations.