Bond valence at mixed occupancy sites. I. Regular polyhedra

Bond valence sum calculations at mixed occupancy sites show the occurrence of systematic errors leading to apparent violations of the Valence Sum Rule (bond valence theory) in regular and unstrained bonding environments. The systematic deviation of the bond valence from the expected value is observed in the long-range structure, and is discussed from geometric and algebraic viewpoints. In the valence-length diagram, such a deviation arises from discrepancies between the intersection points of the long-range bond valences and the theoretical bond valences with the valence-length curves of involved cations. Three factors cause systematic errors in the bond valences: difference in atomic valences, bond valence parameters Ri (the length of a bond of unit valence) and bond valence parameters bi (the bond softness) between the involved cations over the same crystallographic site. One important consequence strictly related to the systematic errors is that they lead to erroneous bond strain values for mixed occupancy sites indicating underbonding or overbonding that actually does not exist.