Mattheiss' prescription is an effort to solve for the potential of a crystal in an approximate manner using the spherically averaged components of the individual constituent atomic potentials. In general, the code takes the wavefunctions for the individual atoms and calculates the potential associated with that atom type. The crystal is then built up by treating each atom as a hard sphere of potential and packing them in the appropriate lattice positions. The potentials are then redefined through spherical averaging to reduce the description of the electronic structure at any point to that of a central field problem. The final step in defining the crystal potential is to determine a zero level value for the potential. This is accomplished with Monte Carlo sampling of the gaps between the hard spheres to find an average potential for the interstitial regions. This average potential value, which typically lies in the conduction/valence band of the crystal, is then defined to be the zero level for the potential. All atomic potentials are cut off at the zero level and their values redefined in reference to this new zero value, thus creating the muffin-tin aspect of the potential.
The final potential of the muffin-tin is the sum of three parts: Hartree,
Exchange, and Madelung potentials. The Hartree potential is generically
named. It is the base potential created from the potentials calculated
via the input wave functions. The exchange potential is a correction
added to the Hartree potential to account for exchange terms of the crystal.
The average exchange potential is calculated using Slater's approximation
that while technically different, the various exchange terms have the same
general character and so can be approximated by results taken from the
theory of a free-electron gas.
(See the input paramater, ALPHA).
The Madelung potential is another corrective term. This term is calculated
only if the valence of all atoms is not zero. In that situation,
the correction is calculated by an Ewald split technigue summing the spherically
symmetric potential resulting from a point charge at each atomic position.
The main output of the muffin-tin potential program is, of course, the
muffin-tin. There are two main further results the program can be
used for: calculating the phase shifts and atomic EXAFS matrix elements
for the lattice. Each of these results is called via a subroutine
to the main program. The choice of calling either subroutine is an
input parameter to the program. The phase shifts are computed by
integration of the schrodinger equation for the calculated muffin-tin potentials
of each atom type. The EXAFS matrix elements are computed for the
desired edge using the wave functions calculated by the program.