Determination of the Bonding and Valence Distribution in Inorganic Solids by the Maximum Entropy Method

Abstract

The distribution of valence among the bonds in the bond graph of an inorganic compound is used to calculate an 'entropy'. We show that the distribution of valence that maximizes this entropy (ME) is similar, but not identical, to that obtained using the equalvalence rule (EVR) proposed by Brown [Acta Cryst. (1977), B33, 1305-1310. Since the ME solutions are maximally non-committal with regard to missing information, they give better predictions of the observed valence distributions than the EVR solutions when lattice constraints or electronic anisotropies are present, but worse predictions when these effects are absent. Since valences calculated using ME are necessarily positive, they give significantly better predictions in cases where EVR predicts a negative bond valence. In the absence of electronic distortions the observed bond graph is either the graph with the highest maximum entropy or it has an entropy within 1% of this value. Since the entropy depends on the oxidation states of the atoms, compounds with the same stoichiometry and cation coordination numbers but different atomic valences may adopt different bond graphs and hence different structures.

Comprehensive definition of oxidation state (IUPAC Recommendations 2016)

“…A bond graph represents the infinite periodic network of an extended solid [9,10]. It is constructed on a stoichiometric formula of the network's repetitive unit, with atom symbols connected by a line for each instance of the atom's bonding connectivity.…”

“…An example is AuNCa 3 perovskite [16], where neglecting its metallic character suggests Au 3− anions for which there is no support in theory. 10 Such an OS is merely a practical value for redox balancing, like setting OS = 0 for all metals in intermetallic phases of metallic character (not the semiconducting Zintl phases), or for those inner atoms in metal clusters that are solely bonded to other metal atoms.…”

Oxidation State, A Long‐Standing Issue!

“…This algorithm is tailored to bond graphs.Abond graph represents the infinite periodic network of an extended solid. [25,26] It is constructed on astoichiometric formula of the networks repetitive unit, with atom symbols distributed such that astraight line is drawn for each instance of an atoms bonding connectivity.E ach line carries its own specific bond order.T oo btain the oxidation state,asum is calculated at each atom, of the orders of its bonds weighted by their ionic sign at that atom. Such an "ionized bond order sum", iBOS,t hen equals the atoms oxidation state.F igure 6 explains this on the AuORb 3 perovskite-type structure [27] with bond orders according to the 8 + N rule at Rb,8ÀNrule at O, and the 12ÀN rule at Au.…”

Toward a comprehensive definition of oxidation state (IUPAC Technical Report)

“…A bond graph is a finite representation of the infinite periodic network of an extended solid [50,51]. In a scheme of atoms that corresponds to the stoichiometric formula of the crystalline phase, the bond graph features a line for each instance of bonding connectivity between nearest-neighbor atoms in the structure.…”