Renormalized interactions in truncated cluster expansions

It is shown that truncating the cluster expansion of the energy of alloys gives rise to renormalized effective cluster interactions that are explicit functions of the configurational variables. Such a dependence of the renormalized cluster interactions is in addition to their dependence on the volume of the alloy and on other structural parameters. The physical picture that emerges is different from the commonly used representation of the configurational energy by means of a generalized Ising-like model, which follows from the assumption that the contributions of the effective cluster interactions can be neglected beyond a relatively small cluster size. The physical picture is one in which the sum of the effective interactions contributes over long distances but the expected “near-sightedness” of the energy is preserved by the renormalized interactions. Furthermore, the cluster expansion is implemented by simultaneously fitting the volume- and configuration-dependent energy function to the zero-pressure values of the energies of formation, volumes, bulk moduli, and pressure derivatives of the bulk modulus of a set of ordered compounds. As an example of this formulation of the cluster expansion, we apply the methodology to the Cu-Au system for different types of cell-internal and cell-external relaxations.

Figure 1

Parameters for the Murnaghan's equation of state for the unrelaxed structures: (a) energies of formation, (b) volumes per atom, (c) bulk moduli, and (d) pressure derivatives of the bulk modulus.

Figure 2

Difference between the energies of formation at zero pressure of (a) the cell-internal relaxed structures minus that of the unrelaxed structures and (b) the fully relaxed structures minus that of the unrelaxed structures.

Figure 3

Generalized Ising-model approximation. (a) Contour plots for constant cross-validation scores; (b) energies of formation of the training structures (black symbols) compared to the energies predicted by the Ising-like model (red symbols); and (c) ECIs for pairs (red), triangles (green), and tetrahedra (blue). Not shown are the ECIs for the empty cluster (38.4 meV), the point cluster (17.7 meV), and the nearest-neighbor pairs (33.9 meV). (d) Energies of formation for the test structures (black symbols) compared to the energies predicted by the Ising-like model (red symbols).

Figure 4

Results of the CE with volume-dependent interactions. Corrections due to truncation of the CE are not included. The minimum value of the cross-validation score for the energy of formation of 0.74 meV is found for $n_2=13,n_3=8$ and $n_4=3$. Black and red symbols are, respectively, the calculated values shown in Fig. 1 and those obtained with the CE.

Figure 5

Cluster expansion with volume-dependent and renormalized interactions due to truncation. The minimum value of the cross validation score for the energy of formation of 0.87 meV is found for $n_2=6,n_3=4$, and $n_4=0$. Black and red symbols are, respectively, the calculated values shown in Fig. 1 and those obtained with the CE.

Figure 6

(a) Pair length distribution due to cell-internal relaxations for a structure in the training set, for first (red), second (blue), and third (green) nearest neighbors in the unrelaxed structure. The pair distances are normalized to the first neighbor distance in the unrelaxed structure at the same volume as the relaxed structure. (b) Percent maximum change in the first neighbor bond lengths, relative to the unrelaxed structure, for all 265 structures in the training set.

Figure 7

Volume dependence of the coefficients $K_n\left[v\right]$ of the Taylor expansion to first order in the correlation functions, normalized by the degeneracy ${\omega }_n$ of the corresponding cluster. The clusters are labeled following Table 3.

Figure 8

Effective cluster interactions for pair clusters used with the method $J\left[v\right]$ described in Sec. 7 2 and Table 1. The coordinate of the clusters are given in Table 3. The symbols correspond to the ECIs in the state of order of the compounds used in the training set, while the solid line gives the ECIs in the random state as a function of concentration.

Figure 9

Effective cluster interactions for nonpair clusters used with the method $J\left[v\right]$ described in Sec. 7 2 and Table 1. The coordinate of the clusters are given in Table 3. The symbols correspond to the ECIs in the state of order of the compounds used in the training set, while the solid line gives the ECIs in the random state as a function of concentration.

Figure 10

Volume dependence of the first-order coefficients $K_n\left[v\right]$ of the Taylor expansion of the energy in the method $J[v,z]$. The coefficients are normalized by the degeneracy ${\omega }_n$ of the corresponding cluster, and the clusters are labeled following Table 4.

Figure 11

Volume dependence of the second-order coefficients $K_{n,m}\left[v\right]$ of the Taylor expansion of the energy in the method $J[v,z]$ for all pairs used in the approximation. The coefficients are normalized by the degeneracies $\sqrt{{\omega }_n{\omega }_m}$ of the corresponding clusters and the clusters are labeled following Table 4.

Figure 12

Effective cluster interactions for pair clusters used with the method $J[v,z]$ described in Sec. 7 2 and Table 1. The coordinate of the clusters are given in Table 4.

Figure 13

Effective cluster interaction for nonpair clusters for the method $J[v,z]$ described in Sec. 7 2 and Table 1. The coordinate of the clusters are given in Table 4.