Comparison of two cluster-expansion methods for the energetics of Pd-V alloys

The formation energies of substitutional transition-metal alloys are examined by several means. First, two types of direct total-energy calculations are considered, namely, (i) the local-density approximation (LDA), and (ii) a tight-binding (TB) approximation thereof. Second, these directly calculated total energies are used to construct two Ising-like cluster expansions that, if sufficiently accurate, could be used to construct the full statistical mechanics of transition-metal alloys. These are (a) the Connolly-Williams (CW) method, and (b) direct configurational averaging (DCA). Finally, the ability of these two cluster expansions [(a) and (b)] to fit and predict a large number of the underlying directly calculated [(i) and (ii)] total energies is tested, by the average prediction error χ. These tests are performed for a large number of Pd-V alloys, and also, to a more limited extent, for the Pd-Rh, Pd-Ti, and Pt-V systems. We find for Pd-V that (i) direct TB calculations show significant overbinding (too-negative formation energies) relative to the LDA, with average error of χ=112 meV/atom (a typical formation energy of ${\mathrm{Pd}}_{0.50}$ ${\mathrm{V}}_{0.50}$ is ∼-250 meV/atom); (ii) the CW cluster expansion mimics quite well the results of the respective direct calculations, whether LDA (χ=19 meV/atom) or TB (χ=19 meV/atom); (iii) the DCA cluster expansions provides a less accurate depiction of the TB energies on which it is based (χ=65 meV/atom); (iv) the prediction errors for the equimolar random alloys are significantly larger using the DCA than using the CW method. In light of (i) above, it appears that the tight-binding model needs to be refined before it can be used systematically for (either DCA or CW) cluster expansions.