Exploring the Configurational Space of Binary Alloys: Different Sampling for Different Cell Shapes

In many areas of alloy theory, such as determination of the T=0 ground state structures or calculation of finite- T alloy thermodynamics, one needs to enumerate and evaluate the ∼ 2N configurations σ created by different substitutions of atoms A and B on the N sites of a unit cell. These configurations consist of MICS "inequivalent cell shapes" (ICS's), each having MSSS "same-shape structures" (SSS's). Exhaustive evaluation approaches attempt to compute the physical properties P (σ) of all SSS's belonging to all ICS's. "Inverse band structure" approaches sample the physical properties of all SSS's belonging to a single inequivalent cell shape. We show that the number MICS of ICS's rises only as B Nα, whereas the total number of SSS's scales as A eγN. Thus, one can enumerate the former (i.e., calculate all) and only sample the latter (i.e., calculate but a few). Indeed, we show here that it is possible to span the full configurational space efficiently by sampling all SSS's (using a genetic algorithm) and repeating this by explicit evaluation for all ICS's. This is demonstrated for the problem of ground state search of a generalized cluster expansion for the Au-Pd and Mo-Ta alloys constructed from first-principles total-energy calculations. This approach enables the search of much larger spaces than hitherto possible. This is illustrated here for the 232 alloy configurations relative to the previously possible 220.