A density-matrix derivation of the Hartree--Fock equations in a nonorthogonal atomic-orbital basis

TL;DR

This paper offers a pedagogical derivation of the Hartree--Fock equations using a density-matrix formalism in a nonorthogonal atomic-orbital basis, connecting traditional theory with modern response methods.

Contribution

It introduces an alternative derivation of Hartree--Fock equations via exponential parametrization of the density matrix in a nonorthogonal basis.

Findings

Standard AO Hartree--Fock stationarity condition derived from density-matrix formalism.

Provides a compact link between elementary Hartree--Fock theory and modern response/density-matrix methods.

Facilitates understanding of linear-scaling formulations in quantum chemistry.

Abstract

We present a pedagogical derivation of the Hartree--Fock equations using the second-quantization atomic-orbital density-matrix formalism developed by Kj{\ae}rgaard, J{\o}rgensen, Olsen, Coriani, and Helgaker for AO-based response theory. The purpose is to introduce an alternative derivation of the Hartree--Fock equation, showing that the standard AO Hartree--Fock stationarity condition follows naturally from the exponential parametrization of the one-particle density matrix in a nonorthogonal AO basis. This route provides a compact bridge between elementary Hartree--Fock theory and the density-matrix machinery used in modern response theory and linear-scaling formulations.