Decomposition of the Electronic Energy in Terms of Density, Density Coherence, and the Connected Part of the Two-Body Reduced Density Matrix

We analyzed static and dynamic electron correlation by decomposing the total electronic energy of calculations by restricted Hartree-Fock theory, complete active-space self-consistent field (CASSCF) theory, and multireference configuration interaction (MRCI). We used three different schemes to break down the relative energy contributions to the potential energy curves for the dissociation of H₂, F₂, and N₂. The first decomposition scheme involves the classical and nonclassical components of the energy. The second and third recognize the part of the energy that is not expressible in terms of the one-body reduced density matrix; this is called the connected energy. The unconnected component is further decomposed into a part calculable from the density and the part calculable from the density coherence. The first decomposition scheme shows that the sum of the one-electron energy and the classical two-electron energy contains a negligible portion of the static correlation. This quantity has a relatively small variance between the three levels, especially for CASSCF and MRCI. This provides an explanation of why multiconfiguration pair-density functional theory and multiconfiguration density-coherence functional theory are able to improve the CASSCF energy. The latter two decompositions show that the connected energy contains a significant portion of static correlation. The energy representable by either the density or the density coherence is significantly different at the three levels. Mixing the density and density coherence between different methods may lead to a systematic error in the bond dissociation energy and the equilibrium bond distance, indicating that the density energy component and the density coherence energy component both include a significant amount of both static and dynamic correlation. These wave function decompositions can be useful for developing new functionals for density functional theory, density-coherence functional theory, density matrix functional theory, and pair-density functional theory and for guiding expectations for these theories.