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Energy landscapes for the self-assembly of supramolecular polyhedra

We develop a mathematical model for the energy landscape of polyhedral supramolecular cages recently synthesized by self-assembly [Sun et al., Science 2010]. Our model includes two essential features of the experiment: (i) geometry of the organic ligands and metallic ions; and (ii) combinatorics. The molecular geometry is used to introduce an energy that favors square-planar vertices (modeling $\mathrm{Pd}^{2+}$ ions) and bent edges with one of two preferred opening angles (modeling boomerang-shaped ligands of two types). The combinatorics of the model involve $2$-colorings of edges of polyhedra with $4$-valent vertices. The set of such $2$-colorings, quotiented by the octahedral symmetry group, has a natural graph structure, and is called the combinatorial configuration space. The energy landscape of our model is the energy of each state in the combinatorial configuration space. The challenge in the computation of the energy landscape is a combinatorial explosion in the number of $2$-colorings of edges. We describe sampling methods based on the symmetries of the configurations and connectivity of the configuration graph. When the two preferred opening angles encompass the geometrically ideal angle, the energy landscape exhibits a very low-energy minimum for the most symmetric configuration at equal mixing of the two angles, even when the average opening angle does not match the ideal angle.