As mentioned in Kimballs_atoms we make life a bit easier by fusing the inner, non-overlapping tetrahedral cloud constructions of Kimball atoms to energy and charge equivalent concentrical spheres. This provokes exchange corrections by overlap of clouds. The overlapping volumes can become quite complicated threedimensional shapes. The simpler of them are lens shaped with different curvature front and aft if the overlapping spheres have not the same radius. It is not difficult to integrate the electrostatic interactions over these space elements. But, in multielectron atoms there are a very large number of different overlap lenses. Their exact interaction energies makes Kimball calculations cumbersome. Therefore, already G.F. Neumark, loc.cit. p.39ff, has proposed to sum electrostatic interactions approximating the lenses with equivalent volume and content of spheres. This recipe is used in Perego's Kimball.exe. Here, I go even further: I ignore the overlap and correct for exchange energy by proper parametrization using a little Density Functional program FDA for atoms, you may download, just adequate for the purpose.|
This procedure is very similar to the "good" quantum chemists resort of using pseudopotential representations of atomic cores in order to reduce the number of electronic interaction integrals to be evaluated in ab initio QC. Good pseudopotentials are available in all QC packages and indispensible for any larger size computation e.g. in the field of transition metal complexes. Optimally, they can reduce a calculation to not much more than the treatment of
|valence electrons. For heavier atoms, the cores may contain more than 99% of the total energy of an atom or molecule and are little influenced by chemical transformations. Hopefully, the errorlimits of pseudopotentials are often unimportant, when the chemical properties are obtained by an energy difference.|
A word about exchange corrections is appropriate here: An overlap lens of two more than singly occupied spheres contains densities not compatible with Pauli's exclusion principle. If this comes from two doubly occupied spheres, we have two electrons each with the same spin, two up and two down. They are not allowed to be in the same volume. Hence, we have to subtract half of the interaction energy within the lens and also of the lens with all other clouds and nuclei. I.e. we only allow density from one up/down pair in that volume. The Kimball energy minimization then adjusts the radii of those clouds slightly to restore a correct kinetic energy, changed by the loss of this excluded space. This procedure within the Kimball model is identical to the quantum mechanical treatment of exchange. In QC we just do not talk about lenses and other simple spatial contructs but integrate complex wavefunctions in the external potential with those reshaped by an exchange operator (over all space!). To get an idea what this means for the K-model, look at the frontispiece of this Kimball site, c-propene with a central exchange hole formed by a "triplelens" of three doubly occupied clouds, compared to c-propane with no exchange and C-C clouds outside of the bond direction. For details, see spherical Li-atom.