Simple two-center tight-binding models have often been used in electronic-structure calculations, although uncertainty about the foundations of the tight-binding method has severely limited progress. This uncertainty has been much reduced following the recent successes of the non-self-consistent Harris scheme but several important questions still remain. In particular, the accuracy of the (almost ubiquitous) two-center approximation is still in doubt. It has often been argued that although this is a poor approximation when applied crudely, it can be justified within the more sophisticated context of chemical-pseudopotential theory. The argument looks sensible, but clear quantitative tests only became possible with the advent of the Harris scheme