Quantum Crystallography: Projectors and kernel subspaces preserving N-representability

TL;DR

This paper introduces a mathematical approach using projector triple products and a new form of Clinton equations to decompose density matrices in quantum crystallography while maintaining N-representability, aiding analysis of large molecules.

Contribution

It presents a novel method for decomposing projector matrices into kernel components that preserve N-representability, extending the application of Clinton equations to quantum crystallography.

Findings

Derived a new mathematical framework for projector decomposition.

Extended Clinton equations for large molecule quantum crystallography.

Ensured N-representability in density matrix analysis.

Abstract

Consider a projector matrix P, representing the first order reduced density matrix in a basis of orthonormal atom-centric basis functions. A mathematical question arises, and that is, how to break P into its natural component kernel projector matrices, while preserving N-representability of P. The answer relies upon 2- projector triple products, P'jPP'j. The triple product solutions, applicable within the quantum crystallography of large molecules, are determined by a new form of the Clinton equations, which - in their original form - have long been used to ensure N-representability of density matrices consistent with X-ray diffraction scattering factors.