Weighted multibody expansions for computing stable structures of multiatom systems

The effect of structural relaxations in alloys is described using a multibody energy expansion formalism. $N$-body potentials in the multibody expansion are computed from energies of isolated clusters, which, in turn, are calculated from empirical potentials or self-consistent quantum mechanical calculations. Convergence characteristics of multibody expansions (MBEs) are improved by weighting energies obtained from various truncations of many-body expansions in a method called weighted MBE (WMBE). It is shown that multibody expansions of many-atom systems can be efficiently constructed using interpolation of isolated cluster energies from databases. In contrast to the method of cluster expansion, WMBE focuses on positional degrees of freedom and, hence, explicitly handles structural relaxations during computations of stable atom clusters and periodic or amorphous phase structures.

Figure 1

Increase in the total number of clusters involved in a multibody expansion for a 32-atom system with the order of expansion.

Figure 2

(Color online) Left: the space of all possible three-atom clusters within an upper and lower cutoff cluster size. This space represents a convex hull in three dimensions (3D). Right: use of symmetries (in the case where all three atoms are of one type—e.g., Pt-Pt-Pt clusters) can further reduce the space. The simplices used to perform local linear interpolation of energies are also shown. In 3D, the simplex is a tetrahedron.

Figure 3

The plot of energy versus interatomic distance for a two-atom Pt cluster. The location of nodal points in this one-dimensional configurational space and the upper cutoff used for calculations are indicated.

Figure 4

(Color online) In the case of a five-atom cluster, the locations of the fourth and fifth atoms can be fixed with respect to the plane formed by atoms 1-2-3 using the cluster specifiers $[R_{12},R_{23},R_{13},R_{14},R_{24},R_{34},R_{15},R_{25},R_{35}]$. However, these specifiers do not completely represent the cluster. The dependent variable in this case is $R_{45}$ which can take one of possible two values based on the location of atom 5 either above or below the plane formed by atoms 1-2-3.

Figure 5

(Color online) Energy surface $\left[E^{*}(X_1,X_2,X_3)\right]$ for three-atom Pt clusters whose atoms are positioned at the vertices of a right-angled triangle with line joining atoms $X_2$ and $X_3$ forming the hypotenuse. (a) shows computed platinum three-atom cluster energies, while (b) shows the extension of energies beyond the cutoff using energies of smaller clusters [$E^{*}(X_i,X_j)$ and $E^{*}\left(X_i\right)$].

Figure 6

Convergence of the many-body energy expansion of an eight-atom fcc platinum cluster requires at least a seventh-order expansion to reasonably capture the true energy.

Figure 7

Comparison of true energies of the a $4\times 1\times 1$ (16-atom) fcc platinum cluster with that predicted by multibody expansion. Weights in the multibody expansion were computed using three energies at lattice parameters of 7.2, 7.4, and $7.6\text{bohrs}$.

Figure 8

Comparison of the true energies of a $4\times 1\times 1$ (16-atom) fcc platinum cluster with MBE results for an extrapolatory case of a fcc lattice where the face-centered atom in the $x\text{−}y$ plane and $y\text{−}z$ plane in the fcc basis are translated by $(-0.1,-0.1,0)$ and (0, 0.1, 0.1), respectively, in crystal coordinates.

Figure 9

Comparison of the true energies obtained for a $2\times 2\times 1$ (16-atom) fcc platinum cluster with energies computed using second- and third-order WMBEs.

Figure 10

Comparison of the true energies obtained for a $2\times 2\times 1$ (16-atom) fcc platinum cluster with energies computed using third-, fourth-, and fifth-order WMBEs for an extrapolatory case of a fcc lattice with the face-centered atom in the $x\text{−}y$ plane of the fcc basis is translated by $(-0.1,-0.1,0)$ in crystal coordinates.

Figure 11

Decrease in the $l^2$ norm error in energies with increasing order of multibody expansion for the extrapolatory case in Fig. 10.

Figure 12

(Color online) The three-body cluster configurational space for platinum colored by the cohesive energies $[E_c(X_1,X_2,X_3)=E^{*}(X_1,X_2,X_3)-{\sum }_{i=1}^3{E}^{*}\left(X_i\right)]$ computed from ab initio simulations.

Figure 13

(Color online) Comparison of the variation of energies with lattice parameter for a periodic fcc pt lattice with WMBE calculations involving cluster energy interpolation.

Figure 14

(Color online) Comparison of the variation of energies with lattice parameter for a $2\times 2\times 1$ supercell of $\alpha$-alumina (space group $R\overline{3}c$) using third- and fourth-order WMBEs. The true energies and the energies used for computing MBE coefficients are indicated.

Figure 15

(Color online) The three-body cluster configurational space for (a) Cu-Cu-Au and (b) Cu-Au-Au colored by the cohesive energies (as defined in Fig. 12) computed from ab initio simulations. The two spaces have different geometries due to different underlying symmetries.

Figure 16

(Color online) Comparison of the variation of energies with lattice parameter for a periodic fcc $\mathrm{Cu}{\mathrm{Au}}_3$ lattice with WMBE calculations involving cluster energy interpolation.