Correlation holes for the helium dimer

Measuring correlated electron motion in atoms with the momentum-balance density

Three new measures of relative electron motion are introduced: equimomentum, antimomentum, and momentum-balance. The equimomentum is the probability that two electrons have the exact same momentum, whereas the antimomentum is the probability that their momenta are the exact opposite. Momentum-balance (MB) is the difference between the equimomentum and antimomentum and, therefore, indicates if equal or opposite momentum is more probable in a system of electrons. The equimomentum, antimomentum, and MB densities are also introduced, which are the local contribution to each quantity. The MB and MB density of the extrapolated-full configuration interaction wave functions of atoms of the first three rows of the periodic table are analyzed, with a particular focus on contrasting the correlated motion of electrons with opposite-spin and parallel-spin. Coulomb correlation between opposite-spin electrons leads to a higher probability of equimomentum, whereas Fermi correlation between parallel-spin electrons leads to a higher probability of antimomentum. The local contribution to MB, given an electron is present, is a minimum at the nucleus and generally increases as the distance from the nucleus increases. There are also interesting similarities between the effects of Fermi correlation and Coulomb correlation (of opposite-spin electrons) on MB.

Singling Out Dynamic and Nondynamic Correlation

The correlation part of the pair density is separated into two components, one of them being predominant at short electronic ranges and the other at long ranges. The analysis of the intracular part of these components permits to classify molecular systems according to the prevailing correlation: dynamic or nondynamic. The study of the long-range asymptotics reveals the key component of the pair density that is responsible for the description of London dispersion forces and a universal decay with the interelectronic distance. The natural range-separation, the identification of the dispersion forces, and the kind of predominant correlation type that arise from this analysis are expected to be important assets in the development of new electronic structure methods in wave function, density, and reduced density-matrix functional theories.

Electron-Pair Distribution in Chemical Bond Formation

The chemical formation process has been studied from relaxation holes, Δh(u), resulting from the difference between the radial intracule density and the nonrelaxed counterpart, which is obtained from atomic radial intracule densities and the pair density constructed from the overlap of the atomic densities. Δh(u) plots show that the internal reorganization of electron pairs prior to bond formation and the covalent bond formation from electrons in separate atoms are completely recognizable processes from the shape of the relaxation hole, Δh(u). The magnitude of Δh(u), the shape of Δh(u) ∀ u < R_eq, and the distance between the minimum and the maximum in Δh(u) provide further information about the nature of the chemical bond formed. A computational affordable approach to calculate the radial intracule density from approximate pair densities has been also suggested, paving the way to study electron-pair distributions in larger systems.

Electron-pair density relaxation holes

The electron-pair density relaxation hole has been defined as the electron-pair density of the real molecule minus the electron-pair density of a reference system consisting of overlapping, spherically averaged, undeformed atoms, positioned at the molecular nuclear coordinates. We have shown how it can be calculated from one- and two-electron reduced density matrices expanded in a Gaussian type basis set. Analysis of the calculated radial electron-pair density holes, from full configuration interaction one- and two-electron reduce density matrices, for the ground states of the hydrogen molecule, the helium dimer and the lithium and beryllium hydrides reveal that the different types of bonding interactions yield distinctively visually recognizable different topological patterns of the electron-pair density relaxation hole.