I shall present work which is a collaboration with Tom Hollins and Stewart Clark from Durham University and Keith Refson from STFC. In this talk my intention is to rethink what we believe we know about the Hartree-Fock (HF) approximation in the theory of electronic structure. I shall present and discuss the results of a series of calculations. Conceptually, these calculations are easy to understand: we calculate the HF ground state Slater determinant (Phi) for a system and then we calculate the ground state single-particle density of Phi. Then, we use a simple algorithm to "invert" the HF single-particle density in order to obtain the local single-particle potential, whose ground state Slater determinant (different from Phi) has the same density as Phi. Finally, we study the band structure of this local potential for the various systems of interest. We have applied this method to a number of representative systems where either the HF approximation or the common, local, density functional theory (DFT) approximations (or both) fail to give a qualitatively correct band structure. For example, HF predicts every system to be insulating, including simple metals, that are predicted to be semi-metallic. Typically, the lack of screening is thought to be cause of this anomaly. Semi-conductors are also predicted to have a large gap. On the other hand, common approximations (LDA-GGA) in DFT are accurate for metals but predict too small band gaps for semi-conductors and also give too small or zero band gaps for some transition-metal oxides that are antiferromagnetic insulators. We will see that the method gives a reasonably accurate description for all these systems, despite the fact that the underlying theoretical model is the HF approximation and no correlation effects are included in the calculations. (Defining correlation as whatever is added beyond a HF calculation.) In fact, we exploit the benefit of treating exchange accurately, but with a local single-particle potential, rather than a non-local one.