On the Global Minimum of the Classical Potential Energy for Clusters Bound by Many-Body Forces

This note establishes, first of all, the monotonic increase with N of the average K-body energy of classical N-body ground state configurations with \(N\ge K\) $N\ge K$ monomers that interact solely through a permutation-symmetric K-body potential, for any fixed integer \(K\ge 2\) $K\ge 2$ . For the special case \(K=2\) $K=2$ this result had previously been proved, and used successfully as a test criterion for optimality of computer-generated lists of putative ground states of N-body clusters for various types of pairwise interactions. Second, related monotonicity results are established for N-monomer ground state configurations whose monomers interact through additive mixtures of certain types of k-meric potentials, \(k\in \{1,\ldots ,K\}\) $k\in \{1,\dots ,K\}$ , with \(K\ge 2\) $K\ge 2$ fixed and \(N\ge K\) $N\ge K$ . All the monotonicity results furnish simple necessary conditions for optimality that any pertinent list of computer-generated putative global minimum energies for N-monomer clusters has to satisfy. As an application, databases of N-body cluster energies computed with an additive mix of the dimeric Lennard–Jones and trimeric Axilrod–Teller interactions are inspected. We also address how many local minima satisfy the upper bound inferred from the monotonicity conditions, both from a theoretical and from an empirical perspective.