Methods of Modern Differential Geometry in Quantum Chemistry: TD Theories on Grassmann and Hartree-Fock Manifolds

TL;DR

This paper applies modern differential geometry methods to analyze time-dependent Hartree-Fock theory within the framework of Grassmann and Hartree-Fock manifolds, revealing geometric and perturbative insights into transition energy calculations.

Contribution

It introduces a geometric perspective on TD-HF theory using Grassmann and Hartree-Fock manifolds, providing new insights into the dependence of transition energies on matrix elements.

Findings

Time-dependent HF theory's success is linked to incorrect energy dependence on constraining matrix elements.

Comparison of linearized Schrödinger equations shows geometric constraints affect transition energies.

Perturbation analysis clarifies the role of the constraining matrix in energy calculations.

Abstract

Hamiltonian and Schrodinger evolution equations on finite-dimensional projective space are analyzed in detail. Hartree-Fock (HF) manifold is introduced as a submanifold of many electron projective space of states. Evolution equations, exact and linearized, on this manifold are studied. Comparison of matrices of linearized Schrodinger equations on many electron projective space and on the corresponding HF manifold reveals the appearance in the HF case a constraining matrix that involves matrix elements of many-electron Hamiltonian between HF state and double excited determinants. Character of dependence of transition energies on the matrix elements of constraining matrix is established by means of perturbation analysis. It is demonstrated that success of time-dependent HF theory in calculation of transition energies is mainly due to the wrong behavior of these energies as functions of matrix elements of constraining matrix as compared with the exact transition energies