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 $N\geq K$ monomers that interact solely through a permutation-symmetric $K$-body potential, for any fixed integer $K\geq 2$. For the special case $K=2$ 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, $k\in\{1,...,K\}$, with $K\geq 2$ fixed and $N\geq K$. 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.