Comparison of the localization of an electron as determined by the two-particle distribution function and by the single-particle sharing index

A comparison of the measure of the delocalization of a particle based on the two-particle distribution function and that based on the single-particle density matrix is made using a simple set of wave functions which span states ranging from single determinant ground and doubly excited states through states mimicking correlated states and which include the singly excited state for electrons and for bosons replacing electrons in H2. The comparison further includes an analysis of the application of the measures to a classical ideal gas and a compressible fluid. It is found that the values of the integrated atom-atom measures agree for a range of wave functions involving combinations of the two single determinant (and equivalent Bose) wave functions but disagree for a different range of these wave functions and for the singly excited wave functions. Aside from the single determinant (and equivalent Bose) wave functions, the two sets of point-point measures that underlie the integrated measures all differ. For the sets of wave functions considered, the values of the measures are identical for electrons and bosons. When applied to a closed classical ideal gas and to a closed compressible fluid, the delocalization measure based on the two-particle distribution has a residual long range term, whereas the sharing index in the classical limit gives a completely localized particle. In general, the two measures describe different aspects of the behavior of the particles. The measures based on the two-particle distribution function give only two-particle properties and the single-particle density, and the sharing quantities give only single-particle properties. The latter includes, however, the quantitative measures of the delocalization of a single particle, the point-point sharing index and the sharing amplitude.