Bond valence constraint

We used bond valence rule (BVR) as a constraint for CTR fitting. We note BVR is a robust constraint, which not only makes a faster convergence of differential evolution optimization, but also ensures a fitted model bearing a structure of physical sense. A structural model with extreme short bonds (thus physically impossible) will be essentially ruled out if the model is fitted under BVR constraint. The theory of BVR has been dedicated in detail in this paper. Briefly, a stable ionic structure always features all constituent atoms bearing bond valence saturation, i.e. the valence charge of an ion has to be compensated by its adjacent counter ions in the nearest polyhedra shell. While the valence charge of each ion is already known in a crystalline structure or can be solved under specific solution chemistry using thermodynamic speciation calculations, the bond valence of the covalent bonds formed between the targeted ion can its neighboring counter ions can be calculated using a simple empirical equation as

bond valence = exp((Ro-R)/B)

where the B is an empirical factor (usually 0.37), R0 is the fitted bond valence parameter representing the nominal length of a bond of unit valence, depends on the sizes of the atoms forming the bond. R0 values for different ions have been tabulated in this cif file, and R is the observed bond length. With this equation, we can easily check the bond valence states (undersaturated, oversaturated, or saturated) of all ions in our studied structure. It should be realized that a optimized structure model is likely a reasonable model if the model could pass the bond valence check, whereas a model failing the bond valence check has a high chance of being a wrong model. However, a model passing the bond valence check is not necessarily a correct model, especially if the model fit is poor with many misfitted places. In this situation, you should recheck the model you have constructed and look for any possible missing parts that could lead to the misfits, e.g. absence of interfacial water structure, assigning the reactive sites incorrectly, presence of multiple reactive sites. Try a new model that considers extra features. You may need to repeat this process several times until you find a model, which not only fits your data well but also passes the bond valence check.