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In order to use the sharing amplitude <ζ;ζ′> to investigate the behavior of electrons in bonding the fixed point ζ′ should be located in a valence region of the molecule and not in the regions of the core electrons of the atoms. A procedure has been developed which determines not only the spatial region of valence electrons in an atom but also the inner shell structure of that atom. By integrating the indices of the sharing index I(ζ;ζ′) over two distinct volumes, ζ over the volume of a sphere less than or equal to R and ζ′ over the remaining volume, a measure Δ(R) of the delocalization of the electrons between the regions is found. Δ(R) has a rich shell-like structure: below are two examples, the first beryllium, the second zinc. The color codings used in the figures are: black is the total Δ(R), red is the s contribution, green is the p contribution, and blue is the d contribution to Δ(R).
be delocalization

zn delocalization

Features to note are: the values of Δ(R) at the maxima are close to 0.25 times the number of electrons in the shell which are active in being shared (delocalized) across the sphere of radius R; the regions in which the core electrons and in which the valence electrons reside are clearly delineated; and the p contributions to Δ(R) for beryllium (these being due to correlation) are not insignificant. For further information see:

R. L. Fulton, "On the Delocalization of Electrons in Atoms and Molecules", J. Phys. Chem. A, 108, 11691 (2004)

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