On the interpretative aspects of second‐order reduced density matrices

A division of the 2‐matrix Γ, for a system of N identical fermions, into 2 parts\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{*{20}c} \Gamma \hfill & = \hfill & {\Gamma ^0 \, + \,\Gamma '} \hfill \\ {} \hfill & = \hfill & {\frac{1}{2}\left[ {\gamma ^0 (1|1')\gamma ^0 (2|2') ‐ \gamma ^0 (1|2')\gamma ^0 (2|1')} \right] + \Gamma '} \hfill \\ \end{array} $$\end{document}is proposed in such a way that Γ has the same 1‐matrix γ as does Γ and Γ has a vanishing 1‐matrix (and therefore vanishing trace). This is accomplished by evaluating the natural spin orbital (NSO) occupation numbers Γ of Γ from\documentclass{article}\pagestyle{empty}\begin{document}$$ n_i^0 = {{(N ‐ 1)n_i } \mathord{\left/ {\vphantom {{(N ‐ 1)n_i } {\left( {\sum\limits_{i \ne i} {n_i^0 } } \right)}}} \right. \kern‐\nulldelimiterspace} {\left( {\sum\limits_{i \ne i} {n_i^0 } } \right)}} $$\end{document}(where the n i are NSO occupation numbers of γ); and letting γ 0 have the same NSO's as does γ. The physical interpretation of Γ should be easier with this decomposition, since Γ is completely determined by γ and has a Hartree‐Fock‐like form, while Γ corrects the pair density of Γ without disturbing its 1‐density.