The knowledge of the properties and the applicability of domain‐restricted matrices has provided important advances in the description of molecular electronic distributions. In this work, we perform a critical comparison between two different methods: the domain‐averaged Fermi hole (DAFH) approach and the domain‐restricted first‐order reduced density matrix (DRRDM) one, focusing our study on both physical and mathematical points of view. Our results permit to show that both methods have a markedly different behavior at correlated wave function level: the DAFH approach provides information related with electron cloud localization and populations while the DRRDM is a true density matrix. To exemplify the theoretical discussion, we present a numerical test example and a simple analytical model that show such features. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011