Stiefel and Grassmann manifolds in Quantum Chemistry

TL;DR

This paper explores the geometric structure of Stiefel and Grassmann manifolds in quantum chemistry, revealing their properties as analytic homogeneous spaces and Finsler manifolds, which support advanced results on Hartree-Fock equations.

Contribution

It establishes new geometric insights into these manifolds, showing they are analytic homogeneous spaces and complete Finsler manifolds, aiding in solving Hartree-Fock equations.

Findings

Proved manifolds are analytic homogeneous spaces.

Showed manifolds are submanifolds of bounded operators.

Established manifolds are complete Finsler manifolds.

Abstract

We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.