Multibody expansion of particle interactions: How many-body is a particular element in a cluster

The particle-particle interaction potential of an $N\text{-atom}$ cluster is expanded in $n\text{-body}$ contributions. The expansion allows us to determine the magnitude of each one of the $n\text{-body}$ terms, and consequently quantifies how $n\text{-body}$ a potential really is. This way we obtain bounds for the relative error due to truncation, a feature that could be applicable in several contexts like the search of minimal energy cluster conformations, to obtain adequate seeds for further ab initio refinement, or to speed up molecular dynamics computations. We develop the formalism, and test the procedure numerically for the Lennard-Jones, Murrel-Motram, Gupta, and Sutton-Chen potentials. The contributions of the $n\text{-body}$ terms for Ag, Al, Au, Co, Cu, Fe, Ir, Ni, Pb, Pd, Pt, and Rh clusters are computed up to $n=9$; they show that the importance and magnitude of the $n>2$ interaction terms depend on the particular element. The relevance of the $n\text{-body}$ corrections as a function of cluster size is also explored for $N\le 50$, and for a linear chain of $N\le 1000$ atoms.

Figure 1

Comparison between the Lennard-Jones potential and the expansion obtained with our algorithm. Only the two-body term is nonzero.

Figure 2

Comparison between the two- and three-body expansion terms for the Murrell-Mottram potential as obtained with our algorithm. For the case of the three-body expansion we plot $u_3(r,r,r)$. For the four-body expansion we plot $u_4(r,r,r,r,r,r)$. The parameters we use are for Li.

Figure 3

Two-, three-, and four-body expansion terms of the Gupta potential obtained with our algorithm for $N=4$, and the comparison between the sum of the terms and the full Gupta potential. We use Al parameters.

Figure 4

Two-, three-, and four-body expansion terms of the Sutton-Chen potential obtained with our algorithm for $N=4$, and the comparison between the sum of the terms and the full Sutton-Chen potential. We use Al parameters.

Figure 5

Contribution of the $n\text{-body}$ expansion terms of the Gupta potential with Al parameters. The data are obtained from $M={10}^5$ clusters.

Figure 6

Contribution of the $n\text{-body}$ expansion terms of the Sutton-Chen potential with Al parameters. The data are obtained from $M={10}^5$ clusters.

Figure 7

Contribution of the $n\text{-body}$ expansion terms of the nine-body Gupta potential for different elements. The data is obtained from $M={10}^5$ different seeds.

Figure 8

Contribution of the $n\text{-body}$ expansion for a nine-atom Sutton-Chen potential for different elements, obtained from $M={10}^5$ different seeds.

Figure 9

Energy error when we use the two-body term to approximate the Gupta potential, using $M={10}^5$ clusters of size $N$ generated at random.

Figure 10

Energy error when we use the two-body term to approximate the Sutton-Chen potential, for different elements, using $M={10}^5$ clusters of size $N$ generated at random.

Figure 11

Energy error when we use the two-body term to approximate the Gupta potential, for different elements, using $M={10}^4$ clusters of size $N=50$ randomly generated. The blue area corresponds to the value of ${\varepsilon }_2$ for that given element.

Figure 12

Energy error when we use the two-body term to approximate the Sutton-Chen potential, for different elements, using $M={10}^4$ clusters of size $N=50$ randomly generated. The blue area corresponds to the value of ${\varepsilon }_2$ for that given element.

Figure 13

Relative contribution $|U_{n-i}|/|\text{Min}\left(U\right)|$, relative to the global minima, of the $(N-i)\text{-body}$ potential term for stable clusters of size $N$, using the Gupta potential. The distribution are given by the filled areas, and the averages by the lines.

Figure 14

The relative error ${\sum }_{i=2}^k|U_i-U|/U$ in the truncation of the $n\text{-body}$ potential at a given order $k$ for a linear chain of $N$ Gupta atoms placed at their equilibrium positions, derived from the two-body potential.