The « muffin-tin » approximation works well enough for the relatively close-packed structures typical of metals. In « bonded » structures, however, where each atom may have no more than four neighbours, the « valleys » and « hills » of the interstitial potential play a fundamental part in the electronic structure. Some of our recent work at Bristol has been concerned with this difficulty. In most of the band structure « methods », deviations from the simple muffin-tin potential can be represented by the Fourier components of the interstitial potential, together with contributions from non-sphericity of the muffin-tin wells. But a study of the extreme case of polyethylene shows that this procedure fails when the interstitial region contains barriers through which the valence electrons can scarcely penetrate. In the diamond lattice, these effects could be taken care of by the cellular method, using a tetrahedral cell. A more practical procedure is to fill the major part of the interstitial void with a spherical « anti-well », which behaves like another type of atom in a KKR or APW calculation. But in layer or chain structures, such a graphite or polyethylene, it seems essential to introduce new basis functions with appropriate boundary conditions on plane or cylindrical potential barriers. The significance of these ideas for theories of chemical bonding is obvious