A natural range separation of the Coulomb hole into two components, one of them being predominant at long interelectronic separations and the other at short distances , is exhaustively analyzed throughout various examples that put forward the most relevant features of this approach and how they can be used to develop efficient ways to capture electron correlation. We show that , which only depends on the first-order reduced density matrix, can be used to identify molecules with a predominant nondynamic correlation regime and differentiate between two types of nondynamic correlation, types A and B. Through the asymptotic properties of the hole components, we explain how can retrieve the long-range part of electron correlation. We perform an exhaustive analysis of the hydrogen molecule in a minimal basis set, dissecting the hole contributions into spin components. We also analyze the simplest molecule presenting a dispersion interaction and how helps identify it. The study of several atoms in different spin states reveals that the Coulomb hole components distinguish correlation regimes that are not apparent from the entire hole. The results of this work hold out the promise to aid in developing new electronic structure methods that efficiently capture electron correlation.
A remarkable exception to this rule occurs in molecules that HF dissociates into fragments with the wrong number of electrons. See the last two examples given in this paper.