We review the simple linear muffin-tin orbital method in the atomic-spheres
approximation and a tight-binding representation (TB-LMTO-ASA method), and show
how it can be generalized to an accurate and robust Nth order muffin-tin
orbital (NMTO) method without increasing the size of the basis set and without
complicating the formalism. On the contrary, downfolding is now more efficient
and the formalism is simpler and closer to that of screened multiple-scattering
theory. The NMTO method allows one to solve the single-electron Schroedinger
equation for a MT-potential -in which the MT-wells may overlap- using basis
sets which are arbitrarily minimal. The substantial increase in accuracy over
the LMTO-ASA method is achieved by substitution of the energy-dependent partial
waves by so-called kinked partial waves, which have tails attached to them, and
by using these kinked partial waves at N+1 arbitrary energies to construct the
set of NMTOs. For N=1 and the two energies chosen infinitesimally close, the
NMTOs are simply the 3rd-generation LMTOs. Increasing N, widens the energy
window, inside which accurate results are obtained, and increases the range of
the orbitals, but it does not increase the size of the basis set and therefore
does not change the number of bands obtained. The price for reducing the size
of the basis set through downfolding, is a reduction in the number of bands
accounted for and -unless N is increased- a narrowing of the energy window
inside which these bands are accurate. A method for obtaining orthonormal NMTO
sets is given and several applications are presented.Comment: 85 pages, Latex2e, Springer style, to be published in: Lecture notes
in Physics, edited by H. Dreysse, (Springer Verlag