Icosahedral hybrid orbitals

A set of twelve equivalent icosahedral hybrid orbitals pointing from the centre to the corners of a regular icosahedron has been obtained. Such hybrids can be used to explain the geometry of twelve-coordinate complexes of a rare-earth atom. Using group theoretical considerations, it is shown that these hybrids can be constructed by linear combination of one s, three p, five d and three f-orbitals. Bearing in mind that the twelve hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the sp3d5f3 hybrid pointing along the positive z-axis. In order to obtain elegant mathematical expressions, the x, y and z axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron.