Theoretical Prediction of Bond-Valence Networks. II. Comparison of the Graph-Matrix and Resonance-Bond Approaches

Abstract

This paper provides a common framework for the bond-valence and resonance-bond-number methods, both of which explain the principal variations in inorganic bond lengths from the sum of radii, as arising from the connectivity of the structure, and therefore may apply graph information in conjunction with the Valence-Sum Rule. Under these constraints, possible predictions are limited to specific ranges of (M -N + 1) parameters, where M and N are the size and order of a multigraph describing the crystal motif. Further restrictions on these parameters may arise from non-crystallographic graph symmetries. Convenient graphtheoretical calculation schemes are described for both approaches. As it is possible to identify the best possible prediction within the limits described, which is that most closely corresponding to the experimental result, we have a means of making a direct comparison of the effectiveness of the various methods proposed, as well as being able to evaluate them against a statistically based prediction. The resonance-bond-number method proves to be the better predictor in most cases. Examples analysed in this way comprise KVO3 (potassium metavanadate), ot-Ga203 (gallium oxide), TeI4 [tellurium(IV) iodide], Li2SiO3 (lithium metasilicate), Li2GeO3 (lithium metagermanate) and CaCrF5 (calcium chromium fluoride).

Mean bond-length variations in crystals for ions bonded to oxygen

“…The prediction of bond lengths in solids has been proposed by drawing a parallel between crystal structures and electrical networks, and by solving the equivalent of Kirchhoff's circuit laws for the network of chemical bonds to obtain a priori bond valences (Mackay & Finney, 1973;Brown, 1977Brown, , 1981Brown, , 1987Rutherford, 1990Rutherford, , 1998O'Keeffe, 1990). These equivalent rules for chemical-bond networks are called the valence-sum rule and the equal-valence rule by Brown (1977Brown ( , 2002Brown ( , 2016.…”

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

“…For such graphs ME, which ensures all bond valences are positive, produces a valence distribution closer to that observed. However, there are other methods, in particular the RB method, which are able to give predictions that agree well with observation, but avoid the problem of negative valences (Rutherford, 1998).…”

“…1, and Table 1 compares the EVR and ME predictions with the observations. For comparison, the RB valences given by Rutherford (1998) are also shown. The rootmean-square (r.m.s.)…”

“…The presence of lattice constraints is further indicated by the EVR prediction of a negative K--O(2) bond valence which, according to (2), is incompatible with an observed bond. Rutherford (1990) considers the calculation of a negative bond valence as a weakness of the EVR which places too much emphasis on equalizing valences around strongly bonded cations at the expense of those that are weakly bonded and he has recently explored other methods of estimating the bond valence that avoid this problem (Rutherford, 1998). In particular, he explores the Resonance-Bond model (RB) of Boisen et aL (1988), as well as a modification of the EVR constrained to give positive valences.…”

Does mathematical crystallography still have a role in the XXI century?

“…The concept of coordination number itself has been the object of critical considerations (Hoppe, 1970) and generalizations (Hoppe, 1979;O'Keeffe, 1979). To judge up to what distance an atom still coordinates with its neighbours, or the closely related question of accounting for the bond-length distribution in irregular coordination polyhedra, different methods have been proposed, such as bond valence (see a review in Urusov, 1995), resonance bond number (Boisen et al, 1988;Rutherford, 1991Rutherford, , 1998a and charge distribution (Hoppe et al, 1989;Nespolo et al, 1999Nespolo et al, , 2001.…”