Intracule function

In addition to D(ri), we have plotted the charge density p(r ) — D(r )/4nri, the electron-electron probability distribution function P(ri2) (defined similarly to D(ri) but with omission of the prefactor 2 ), and the electron-electron density (sometimes referred to as the intracule function). 

Fig. 1. Electron-electron distribution functions for single-configuration He wavefunction (a) radial probability distribution P(ri2) (b) intracule function h(ri2)- In both graphs, the curve with largest maximum is for the closed-shell wavefunction that of intermediate maximum is for the split-shell wavefunction that of smallest maximum is for the wavefunction containing exp( —yri2).

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

Fig. 2. Electron-electron distributions for optimized one, two, and three-configuration He wavefunctions intracule function The curve can be identified by the heights of their maxima larger maxima correspond to fewer configurations.

We recently presented a correlation method based on the Wigner intracule, in which correlation energies are calculated directly from a Hartree-Fock waveftmction. We now describe a self-consistent form of this approach which we term the Hartree-Fock-Wigner method. The efficacy of the new scheme is demonstrated using a simple weight function to reproduce the correlation energies of the first- and second-row atoms with a mean absolute deviation of 2.5 m h. 

We have recently introduced the Wigner intracule (2), a two-electron phase-space distribution. The Wigner intracule, W ( , v), is related to the probability of finding two electrons separated by a distance u and moving with relative momentum v. This reduced function provides a means to interpret the complexity of the wavefunction without removing all of the explicit multi-body information contained therein, as is the case in the one-electron density. 

A. J. Thakkar, Extracules, intracules, correlation holes, potentials, coefficients and all that, in Density Matrices and Density Functionals, R. M. Erdahl and V. H. Smith, Jr., eds. (Reidel, Dordrecht, Holland, 1987), pp. 553-581. 

This interesting field was initiated by Bader [158]. Topological analysis provides the means for a concise description of multivariate functions. For functions that describe physical observables, the number and location of critical points, where the gradient vanishes, and their mutual relationship are often directly related to the properties of the system under study. The application of topological analysis to the one electron density is even more productive, furnishing rigorous quantum-mechanical definitions of and bonds in molecules. Cioslowski has extended this analysis to the study of the electron-electron interactions, based on the analysis of the intracule and extracula densities [159,160]. 

The contributions of the correlated and uncorrelated components of the electron-pair density to atomic and molecular intracule I and extracule E densities and their Laplacian functions were analysed at the HF and Cl level. The correlated components of the / and E densities, and their associated Laplacian functions, reveal the short-ranged nature and high isotropy of Fermi and Coulomb correlation in atoms and molecules. In general, it has been found that the uncorrelated / and E have the same topological structure as the parent functions I and E functions. 

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