The cluster is a real FCC lattice fragment (nearest-neighbor distance r₀ = 2.884 Å, gold's bulk value) cut to a sphere of the chosen radius. Its cohesion is computed two different ways, live, for every atom:
Pairwise (Lennard-Jones):
E_i = Σ_j 4ε[(σ/r_ij)^12 − (σ/r_ij)^6]
Many-body (Gupta / 2nd-moment tight-binding):
E_i = Σ_j A·e^(−p(r_ij/r₀−1))
− √( Σ_j ξ²·e^(−2q(r_ij/r₀−1)) )
The Gupta parameters (A = 0.2061 eV, ξ = 1.790 eV, p = 10.229, q = 4.036) are the standard Cleri–Rosato fit for gold — the square-root "band" term encodes the delocalized-electron bonding that pairwise potentials cannot represent. ε is not a free fudge factor: it is solved once so both models agree exactly for a perfectly bulk-coordinated atom, so any remaining disagreement you see is purely the many-body effect, not a mismatch in the models' overall energy scale.
Because the square root suppresses cohesion sub-linearly as an atom gains neighbors, the two models predict identical bonding only at full bulk coordination — everywhere an atom's neighbor count is reduced (a surface, edge or corner atom) they disagree, and the disagreement grows the smaller and more surface-dominated the cluster is. This is a real, load-bearing fact in nanocluster physics, not a philosophical metaphor: pairwise potentials systematically get metal surface and cluster energies wrong for exactly this reason.
- Radius slider — grows or shrinks the cluster (bigger N → more atoms buried in bulk-like coordination → the two models converge).
- 2nd shell checkbox — extends the interaction cutoff. Where to cut off a nanoscale model's interaction range is itself an unavoidable modeling choice, and changing it visibly shifts every number below.
- Color mode — paints each atom by its own local energy under either model, or by how much they disagree.
The point this sim is built to make concrete: at the nanoscale, "the" cohesive energy of a material is not a single fixed number waiting to be measured — it depends on how many atoms are present, which atoms are on the surface, and which physical model you are willing to trust. Bulk thermodynamic constants are a large-N limit that a nanocluster has, in general, not yet reached — an ontological point (properties genuinely differ by scale) with a direct epistemological consequence (no single simple model is correct at every size).