The cluster is a real 2D hexagonal (triangular) lattice fragment — the two-dimensional analog of a close-packed metal surface, nearest-neighbor spacing a = r₀ = 2.884 Å (gold's bulk value) — cut to a disk 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 2D-lattice atom (6 nearest neighbors, plus 6 second-shell neighbors at a√3 when that shell is included) — so any remaining disagreement you see is purely the many-body effect.
Note on this 2D adaptation: the original 3D version of this simulator uses an FCC lattice (bulk coordination 12). A flat projection of that lattice would give the wrong bulk coordination and silently miscalibrate ε. This 2D engine instead builds a genuine 2D triangular lattice with its own correct coordination shells (6, then 6 more) and re-derives ε from those — the same physical argument, honestly re-derived for two dimensions rather than borrowed from three.
Because the square root suppresses cohesion sub-linearly as an atom gains neighbors, the two models predict identical bonding only at full lattice coordination — everywhere an atom's neighbor count is reduced (an edge or corner atom) they disagree, and the disagreement grows the smaller and more edge-dominated the cluster is. This is a real, load-bearing fact in nanocluster physics: 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). The radial profile panel below the cluster plots every atom's LJ and Gupta energy against its distance from the center, so convergence is visible directly.
- 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.
- Drag to pan the cluster view, scroll/pinch to zoom, Auto-spin rotates the lattice continuously so edge vs. interior atoms are easy to compare from every angle.
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.