## 4 Non-Stoichiometry Studied by First Principle Calculations  
First principle calculations are generally limited to the calculation of perfectly ordered periodic structures which seems, at first, not to be very useful to study the non-stoichiometry and disorder related to atomic mixing. However, several solutions to this problem have been found which are summarized in the following. One of the first attempts to compute thermodynamically properties of FK phases from first principle techniques has been made by [PERSON] _et al._[24]. They proposed to study ordering effects in the \(A15\) structure with Tight-Binding (TB) calculation in the context of the Generalized Perturbation Method (GPM [25]), by coupling with a statistical thermodynamic method like the Monte Carlo simulation or the Cluster Variation Method (CVM [26, 27]). For other FK phases in which local order is known to be less important, such as the \(\sigma\) phase, other approaches have been tested. The first significant work done on the \(\sigma\) phase has been performed on the two systems Cr-Fe [28] and Re-W [29]. The energies associated to the change of local atomic configurations have been parameterized by a set of Effective Cluster Interactions (ECI). 42 configurations have been considered, issued from the \(2^{5}=32\) ordered binary compounds generated by the ordered distribution of the two atoms on the 5 sites, with an additional set of 10 super-structures which allows to determine pair interactions between sites of the same type. The total energy of each configuration was obtained by Density Functional Theory (DFT) using Local Density Approximation (LDA) with the Linear Muffin-Tin Orbital method (LMTO) [30]. Then, the Connolly-Williams Method (CWM [31]) was used to extract the ECI interactions. The ECI was employed to compute the site occupancies and energy terms of the \(\sigma\) phase as a function of temperature and composition, within the tetrahedron approximation of the CVM. Computed site occupancies were in pretty good agreement with the experimental ones, however, they exhibited curious composition dependence at low temperature (see Figure 4). Indeed, Ising-type hamiltonian favors several non-existing single pairs, and affects the preference of single-site and multi-site interactions. In [32], results on Re-W obtained with CWM-CVM has been compared by a simple random distribution of the Bragg-Williams Approximation (BWA [33]), and concluded that the BWA is sufficient to describe the Re-W \(\sigma\) phase without occupancy reversal phenomena at low temperature (see Figure 4). Moreover, the advantages of using only BWA are several: _(i)_ it is well compatible with the Compound Energy Formalism (CEF) used widely in the Calphad method [34]; _(ii)_ it requires only the total energy of each end-member (32 configurations for a binary \(\sigma\) phase) which can be easily obtained by DFT calculations; _(iii)_ no heavy analytic treatment is needed; and _(iv)_ extrapolations to higher-order system are conceivable. Another comparative study of the finite temperature thermodynamic properties was performed on the \(\mu\) Nb-Ni phase [35]. Compared to Cluster Expansion (CE) with CVM, it was shown that BWA used in the CEF was sufficient to describe experimental site occupancies of the \(\mu\) phase (see Figure 5, left). A very good agreement of the Generalized GradientApproximation (GGA) DFT-BWA results [36] with experimental site occupancies was also obtain for another FK phase: the \(\chi\) phase in Mo-Re (Figure 5, right), with also a relative agreement obtained between the computed and experimental phase diagrams of Mo-Re and W-Re systems. Prototype \(\sigma\) Cr-Fe has been by far the most investigated system by many different first-principles calculations. 10 years after the first CWM-CVM study [28], [PERSON] _et al._[38] were able to better reproduce the experimental data by adopting the BWA in the CEF formalism with the 2\({}^{5}\) = 32 end-members calculation in the Korringa-Kohn-Rostoker (KKR) method (see Figure 6).