Generalization of Cramer's rule and its application to the projection of Hartree-Fock wave function

TL;DR

This paper generalizes Cramer's rule and applies it to nuclear spectra calculation, Hartree-Fock wave functions, and derivations of L"owdin formula and Thouless theorem, offering new elementary methods for projection operators.

Contribution

It introduces a generalized Cramer's rule and applies it to nuclear physics and wave function projections, providing simplified derivations and new computational tools.

Findings

Generalized Cramer's rule for linear algebra

Application to nuclear spectra via Hill-Wheeler projection

Elementary derivation of projection operators

Abstract

We generalize the Cramer's rule of linear algebra. We apply it to calculate the spectra of nucleus by applying Hill-Wheeler projection operator to Hartree-Fock wave function, and to derive L\"owdin formula and Thouless theorem. We derive by an elementary method the infinitesimal or L\"owdin projection operators and its integral representation to be useful for the projection of Slater determinant.