Landau theory and self-assembly of spherical nanoclusters and nanoparticles with octahedral symmetry

Spherical nanoclusters and nanoparticles are rising materials whose functional design provides many useful applications ranging from catalysis, molecular sensing, and gas storage to drug targeting and delivery. Here, we develop phenomenological crystallization theory of such spherical structures with octahedral symmetries $O$ and ${O}_h$. Within the developed theory, we propose a method based on constructing irreducible octahedral density functions and allows us to predict the positions of structural units in the spherical nanoobjects. The proposed theory explains the structures of the simplest known metal nanoclusters, some metal-organic polyhedra and membrane protein polyhedral nanoparticles, and also predicts more complex chiral spherical structures and achiral assemblies characterized by the geometry of semiregular polyhedra. A relationship between the constructed irreducible octahedral functions and spherical lattices, obtained by mapping a plane hexagonal order onto a spherical surface through an octahedron net, is discussed as well.