  The class of the Generalized Coherent Potential Approximations (GCPA) to the
Density Functional Theory (DFT) is introduced within the Multiple Scattering
Theory formalism for dealing with, ordered or disordered, metallic alloys. All
GCPA theories are based on a common ansatz for the kinetic part of the
Hohenberg-Kohn functional and each theory of the class is specified by an
external model concerning the potential reconstruction. The GCPA density
functional consists of marginally coupled local contributions, does not depend
on the details of the charge density and can be exactly rewritten as a function
of the appropriate charge multipole moments associated with each lattice site.
A general procedure based on the integration of the 'qV' laws is described that
allows for the explicit construction the same function. The coarse grained
nature of the GCPA density functional implies great computational advantages
and is connected with the O(N) scalability of GCPA algorithms. Moreover, it is
shown that a convenient truncated series expansion of the GCPA functional leads
to the Charge Excess Functional (CEF) theory [E. Bruno, L. Zingales and Y.
Wang, Phys. Rev. Lett. {\bf 91}, 166401 (2003)] which here is offered in a
generalized version that includes multipolar interactions. CEF and the GCPA
numerical results are compared with status of art LAPW full-potential density
functional calculations for 62, bcc- and fcc-based, ordered CuZn alloys, in all
the range of concentrations. These extensive tests show that the discrepancies
between GCPA and CEF are always within the numerical accuracy of the
calculations, both for the site charges and the total energies. Furthermore,
GCPA and CEF very carefully reproduce the LAPW site charges and the total
energy trends.
