  We describe the screened Korringa-Kohn-Rostoker (KKR) method and the
third-generation linear muffin-tin orbital (LMTO) method for solving the
single-particle Schroedinger equation for a MT potential. The simple and
popular formalism which previously resulted from the atomic-spheres
approximation (ASA) now holds in general, that is, it includes downfolding and
the combined correction. Downfolding to few-orbital, possibly short-ranged,
low-energy, and possibly orthonormal Hamiltonians now works exceedingly well,
as is demonstrated for a high-temperature superconductor. First-principles sp3
and sp3d5 TB Hamiltonians for the valence and lowest conduction bands of
silicon are derived. Finally, we prove that the new method treats overlap of
the potential wells correctly to leading order and we demonstrate how this can
be exploited to get rid of the empty spheres in the diamond structure.
