The Laplacian of the intracule and extracule densities and their relationship to the shell structure of atoms

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.

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.

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.

Comparative electronic analysis between hydrogen transfers in the CH4/CH3+, CH4/CH3•, and CH4/CH3- systems: on the electronic nature of the hydrogen (H-, H•, H+) being transferred. II. Analysis of electron-pair interactions from intracule and extracule densities

The nature of the hydrogen transferred in the CH3/CH4+, CH3/CH4·, and CH3/CH4- systems is investigated by analyzing the topology of the contracted intracule and extracule electron-pair densities and their respective Laplacians. The CH3/CH4+, CH3/CH4·, and CH3/CH4- systems are taken as simple models for the study of hydride (H-), hydrogen (H·), and proton (H+) transfer reactions, respectively, under a constrained C-C distance. The study is focused on the comparison of the intracule and extracule densities at the intermediate structures for the three H-transfer reactions, complementing a previous investigation of the same model reactions based on the analysis of one-electron densities. The results obtained by analyzing the contracted electron-pair densities are consistent with those obtained from the analysis of one-electron densities. The electronic nature of the H atom being transferred in the three systems can be differentiated by the topologies of the corresponding intracule and extracule densities. However, the analysis underlies also the difficulties to interpretation of the topologies of contracted electron-pair densities, as different electron-electron interactions may contribute to the same point in the intracule or extracule spaces. In particular, for the systems studied, the contribution of the electron-electron interaction associated to the probability of having two electrons on the H being transferred is not reflected separately neither in the intracule nor in the extracule distributions. Nevertheless, the nature of the H being transferred can still be studied by comparing the importance of the electron-electron interactions associated to the probability of having one electron in C and one in the transferring H. The effects of inclusion of electron correlation are also discussed by means of (HF-CISD//HF) intracule and extracule density difference maps.Key words: hydrogen transfer, electron-pair density, intracule density, extracule density, topological density analyisis.