Coordination polyhedra in a crystal structure are considered to be the structural units in a modified rigid-ion model. In the polyhedral model the elastic constants of coordination polyhedra are used as input parameters to calculate the single-crystal elastic constants of corundum and chrysoberyl. The average deviation of the calculated elastic constants from the measured elastic constants is within 6% for corundum and within 1% for chrysoberyl.

The corundum crystal structure is composed entirely of AlO6 octahedra, which are also major polyhedral constituents of the chrysoberyl crystal structure. The nonelectrostatic contribution of the bulk compressibility of an ideal (regular) AlO6 octahedron is found to be 10.5¿0.2 Mbar, an invariant in these two crystal structures. The polyhedral elastic constants and the associated perturbational parameters are not universal, however, as demonstrated in the two crystal structures studied.