challenges / morse-clusters-rho14
Morse clusters at rho = 14, minimum energy
Place N atoms in 3-D to minimise the Morse energy sum e^(14(1-r))(e^(14(1-r)) - 2), the short-ranged case, for thirteen N between 20 and 80. Scored against the Cambridge Cluster Database putative global minima at rho = 14; none is proven optimal.
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# Morse clusters at rho = 14, minimum energy
## Goal
`cluster.py` exposes `cluster(n: int, time_budget: float, seed: int) -> list[tuple[float, float, float]]`:
`n` atoms `(x, y, z)` anywhere in 3-D space. Minimise the Morse energy
E = sum over i < j of x_ij (x_ij - 2), x_ij = exp(rho (1 - r_ij)), rho = 14, r_ij = |x_i - x_j|
in reduced units (pair well depth 1, equilibrium pair separation 1). The range parameter `rho`
sets how short-ranged the potential is; `rho = 14` is the hardest column of the Cambridge Cluster
Database's Morse table (`morse-clusters-rho6` is the Lennard-Jones-like sibling). With such a narrow
well, strain is expensive and nearest-neighbour count is nearly everything: the global minima are
close-packed (fcc and hcp) or decahedral fragments rather than icosahedra, the energy landscape is
…