Cluster expansion and the configurational theory of alloys

A rigorous mathematical foundation for the cluster-expansion method is presented. It is shown that the cluster basis developed by Sanchez et al. [Physica A 128, 334 (1984)] is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys. Rev. B 48, 14013 (1993)] corresponds to a multidimensional discrete wavelet transform. For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys. The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach. Several examples aimed at illustrating the main findings and conclusions of this work are given.

Figure 1

Coefficients $J_{\alpha }\left(0\right)$ for eight pairs in the CE expansion of the function $f^{\left(2\right)}\left(\vec{\sigma }\right)=x^2\left(\vec{\sigma }\right)$. The overall fitting error is $\Delta {\left(x^2\right)}_{rms}=0.03$.

Figure 2

Coefficients $J_{\alpha }\left(0\right)$ for the first 50 pairs in the cluster expansion of the function $f^{\left(2\right)}\left(\vec{\sigma }\right)=x^2\left(\vec{\sigma }\right)$. The fitting has been carried out using 200 pairs, in addition to the empty and point clusters, with the condition that the sum of the squares of $J_{\alpha }\left(0\right)$ be minimum. $\Delta {\left(x^2\right)}_{rms}=0$.

Figure 3

Energies of formation for 68 Fe-Co compounds obtained by first-principles total-energy calculations in Ref. 13 (open circles), compared to the energies obtained using the VBCE with nn pairs (filled triangles). $X_{\text{Fe}}$ and $X_{\text{Co}}$ are the atomic concentrations of Fe and Co, respectively. The root-mean-square error is $\Delta E_{rms}=0.27\text{mRy}/\text{atom}$.

Figure 4

Coefficient $J_0\left(x\right)$ for the VBCE of Fe-Co alloys using nearest-neighbor pairs. $J_0\left(x\right)$ is the energy of the random alloy.

Figure 5

Nearest-neighbor pair interaction $J_{\left\{2,1\right\}}\left(x\right)$ as a function of $x$ for the VBCE of Fe- Co alloys.

Figure 6

Energies of formation for 68 Fe-Co compounds obtained by first-principles total-energy calculations (circles) compared to the energies obtained using the VBCE (filled triangles) with 60 pairs, 20 triangles and 10 tetrahedra. The filled circles are five test compound energies not included in the fitting and the empty squares are the energies predicted by the VBCE.

Figure 7

Pair interactions $J_{\left\{2,n\right\}}\left(x\right)$ as a function of $x$ for 60 pairs.

Figure 8

Three-body interactions $J_{\left\{3,n\right\}}\left(x\right)$ as a function of $x$ for 20 triangles.

Figure 9

Four-body interactions $J_{\left\{4,n\right\}}\left(x\right)$ as a function of $x$ for 10 tetrahedra.