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

Abstract

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 $ng2$ 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 $N\ensuremath{\le}50$, and for a linear chain of $N\ensuremath{\le}1000$ atoms.