The class of the generalized coherent-potential approximations ͑GCPAs͒ to the density functional theory ͑DFT͒ is introduced within the multiple scattering theory formalism with the aim of 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. Most existing DFT implementations of CPA-based theories belong to the GCPA class. The analysis of the formal properties of the density functional defined by GCPA theories shows that it consists of marginally coupled local contributions. Furthermore, it is shown that the GCPA functional does not depend on the details of the charge density and that it can be exactly rewritten as a function of the appropriate charge multipole moments to be associated with each lattice site. A general procedure based on the integration of the qV laws is described that allows for the explicit construction of the same function. The coarse-grained nature of the GCPA density functional implies a great deal of computational advantages and is connected with the O͑N͒ scalability of GCPA algorithms. Moreover, it is shown that a convenient truncated series expansion of the GCPA functional leads to the charge-excess functional ͑CEF͒ theory ͓E. Bruno et al., Phys. Rev. Lett. 91, 166401 ͑2003͔͒, which here is offered in a generalized version that includes multipolar interactions. CEF and the GCPA numerical results are compared with status of art linearized augmented plane wave ͑LAPW͒ fullpotential density functional calculations for 62 bcc-and fcc-based ordered CuZn alloys, in all the range of concentrations. Two facts clearly emerge from these extensive tests. In the first place, the discrepancies between GCPA and CEF results are always within the numerical accuracy of the calculations, both for the site charges and the total energies. In the second place, the GCPA ͑or the CEF͒ is able to very carefully reproduce the LAPW site charges and a good agreement is obtained also about the total energies.

The technological performances of metallic compounds are largely influenced by atomic ordering. Although there is a general consensus that successful theories of metallic systems should account for the quantum nature of the electronic glue, existing non-perturbative high-temperature treatments are based on effective classical atomic Hamiltonians. We propose a solution for the above paradox and offer a fully quantum mechanical, though approximate, theory that on equal footing deals with both electrons and ions. By taking advantage of a coarse grained formulation of the density functional theory [Bruno et al., Phys. Rev. B 77, 155108 (2008)] we develop a MonteCarlo technique, based on an ab initio Hamiltonian, that allows for the efficient evaluation of finite temperature statistical averages. Calculations of the relevant thermodynamic quantities and of the electronic structures for CuZn and Ni3V support that our theory provides an appropriate description of orderdisorder phase transitions. PACS numbers:Nowadays ground state properties of metallic alloys are routinely calculated by using Density Functional (DF) [1, 2] theory which appropriately incorporates electronic correlations. However, most finite temperature status of art calculations [3] are based on classical Ising models which allow the accurate evaluation of phase equilibria, but are not able to predict the electronic properties. A theory able to cope with both tasks must incorporate quantum mechanics and should be able to explore the very large space of alloy configurations [4] in order to give reliable statistical averages. Unfortunately, quantum simulations as the Car-Parrinello Molecular Dynamics [5] (CPMD) are very hard for the problem at hand and the application of such methods the phase equilibria of metallic alloys seems beyond the capabilities of existing hardware and software. In this Letter we shall outline a new approach to the problem, which, similarly to CPMD is based on the DF theory and the Born-Oppenheimer approximation, but where the relevant quantities shall be obtained as statistical averages (by sampling the space of alloy configurations) rather than as Boltzmann time averages.Although the generalization to n-ary systems is straightforward, in this Letter we focus on binary metallic alloys A c B 1−c . They shall be studied in the (T, c) statistical ensemble defined by the temperature, T , and the mean atomic concentration, c. In order to have a tractable problem, we shall limit ourselves to the solid state and the normal metal regime or, equivalently, to the temperatures between the superconducting and the melting transitions, T SC < T < T M . Furthermore, martensitic and magnetic phenomena shall not be considered.Our first crucial step is to obtain a coarse grained version of the Hohenberg-Kohn DF. If the ions constituting the system are considered frozen on the sites of a simple lattice at their equilibrium positions, R i , then the electrostatic contribution to the DF can be written as a sum of local terms plus some bilinear terms...