In 1982 Robert Brown published the separable exact (non-Muffin Tin) extension of the then well-known Korringa-Kohn-Rostoker (KKR) multiple scattering band theory as his Ph.D. dissertation. This theory had been the target of extensive research on the part of many theorists for ten or fifteen years.
It was of great interest because for many crystalline (regular) materials, variations in electron potentials and densities are significant along bonding directions and in interstitial regions of the crystal. However, the existing (KKR) theory only admitted potentials that were truncated at the surface of the largest sphere that can be inscribed in the Wigner-Seitz cells of the crystal, the so-called Muffin Tin sphere. Outside this sphere the potential was set to ``zero'' (where the overall potentials were adjusted so this zero was the average of the interstitial potential outside of this sphere). Non-Muffin Tin corrections to results obtained with KKR or APW (augmented plane wave) or many other Muffin Tin-limited methods of evaluating band structure and crystal electronic wavefunctions could only be added a posteriori by means of e.g. perturbation theory and could not be made self-consistent.
This latter consideration was very important. Self-consistency (where the wavefunctions produced by solving the Schrödinger equation for the crystal potential result in electron densities which in turn result in the crystal potential used to obtain the wavefunctions) is an obviously desirable property of any purportedly exact solution to the electronic structure problem of a crystal, but it is also one that was essentially impossible to obtain at the time with Muffin Tin methods.
Self-consistency has become even more important as ``Local Density Functional'' methods for determining the important contributions to the overall energies and band structure made by electron correlations of various sorts have been developed over the intervening years. These methods correct the bare electrostatic crystal potential with density dependent terms that account to some extent for the natural repulsion of electrons and their fermi spin exchange properties. Without these corrections crystal band structure and electron density profiles can actually be totally incorrect and lead to incorrect conclusions about the physical properties of the material (whether it is a conductor, semiconductor, or insulator, for example).
Computing technology has at last developed to the point where it is possible to consider writing a completely self-contained package for evaluating band structure, crystal wavefunctions and densities, energies, and more in a fully self-consistent field, local density functional implementation of our exact multiple scattering band theory, and we have recently begun a project to do so. Our goal is to develop the code to the point were we can do a precise computation of the band structure of ``interesting'' materials (for example, Gallium Arsenide) used in the semiconductor industry on an ordinary desktop computer or small compute cluster. By implementing a theory that is formally exact before adding the local density functional contributions, we will also be in an excellent position to critically evaluate the various candidate density functionals proposed for use in these materials or to develop our own. One very interesting idea that we wish to explore is whether one can use neural networks as general nonlinear function approximators to determine ``the'' (near optimal) local density functional if only in a neural representation.