Some history

The foregoing makes a good story, but is not entirely true.
G.E. Kimball had a PhD student Victor A. Lewinson at Columbia before G.F. Neumark. Apart from his thesis there exists a paper [1]. In a way this is a precursor of the "Kimball model". Lewinson and Kimball tried

[1] V. A. Lewinson & G.E. Kimball, J.Chem.Phys. 19(1951)690-693
to apply the "cellular method" by E. Wigner & F. Seitz (1933, successful in the theory of metals) to a computation of the hydrogen molecule. It is documented, that this is not really successful. But the Wigner-Seitz' cell, a polyhedron with the symmetry of the Bravais lattice it is in, might be the archetypal idea of a Kimball sphere.

  More history

Another, more cynical, context for Kimball's proposal:
In his paper "The Kimball Free-Cloud Model: A Failed Innovation in Chemical Education?", W.B. Jensen, [2], poses the question, whether G.E. Kimball just wanted to find out, how bad an approximation was still good enough to produce the common 10-30% error of all theoretical molecular work up to the second half of the 20th century. This might be true and ended Kimball's frustration with a positive result: Yes, even such a simple approximation of the molecular electronic structure gives results as good as all those "Complete neglect of ..." truncated Hamiltonian calculations with the meagre computational resources available at the time, see citations [12] in the main page.

[2] J.Chem.Educ. 2014,91(8),1106-1124
However, Kimball acted on that result: He left the field, as many other "Quantum Chemists" have done, to do corporate Operations Research for the rest of his life.
Today, we have adequate computational tools to redo Kimball's work. It turns out, that his Ansatz and using parameterization now gives results with a quality of small basis sets - 6-31G(d). In the time since 1950 computers have been developed to be 100 million times more powerful. Even a Smart Handy can now easily do work which was barely possible with a Cray-1 at 1990!
Of course, that made approximations like Kimball's spheres obsolete for theoretical scientific work. However, there is much demand for simpler approaches which can be served by rethinking Kimball's model, as done in these pages.