The bond-valence model (BVM) posits an inverse relationship between bond valence (essentially bond order) and bond length, typically described by either exponential or power-law equations. To assess the value of these forms for describing a wider range of bond lengths than found in crystals, we first assume that the bond critical point density (ρb, reported in e3) is at least roughly proportional to bond valence. We then calculate ρb -distance curves for several diatomic pairs using electronic structure calculations (CCSD/aug-cc-pVQZ) and Atoms-In-Molecules (AIM) analysis. The shapes of these curves cannot be completely described by the standard exponential and power-law forms, but are well described by a three-parameter hybrid of the exponential and power-law forms. The ρb -distance curves for covalent bonds tend to exhibit exponential behavior, while metallic bonds exhibit power-law behavior, and ionic bonds tend to exhibit a combination of the two. We next use a suite of both experimental and calculated (B3LYP/Def2-TZVP) molecular structures of oxo-molecules, for which we could infer X-O bond valences of ~1 or ~2 v.u., combined with some crystal structure data, to estimate the curvature of the bond valence-length relationship in the high-valence region. Consistent with the results for the ρb -distance curves, the standard forms of the bond valence-length equation become inadequate to describe high-valence bonds as they become more ionic. However, some of these systems demonstrate even higher curvature changes than our three-parameter hybrid form can manage. Therefore, we introduce a four-parameter hybrid form, and discuss possible reasons for the severe curvature. Although the addition of more parameters to the bond valence-length equation comes at a cost in terms of model simplicity and ease of optimization, they will be necessary to make the BVM useful for molecular systems and transition states.

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