Introduction Although the bond valence model is mostly used for validating newly determined structures, it is capable of predicting many of the properties of those chemical structures that can be described by localized bonds In the bond valence model, the
valence of an atom, V, is defined as the number of electrons the atom uses for bonding. This is equal to the number of electrons in its valence shell if all the valence shell electrons are used for bonding. If they are not, the remainder will form non-bonding electron pairs, usually known as
lone pairs. The
valence of a bond, S, is defined as the number of electron pairs forming the bond. In general this is not an integral number. Since each of the terminal atoms contributes equal numbers of electrons to the bond, the bond valence is also equal to the number of valence electrons that each atom contributes. Further, since within each atom, the negatively charged valence shell is linked to the positively charged core by an
electrostatic flux that is equal to the charge on the valence shell, it follows that the bond valence is also equal to the electrostatic flux that links the core to the electrons forming the bond. The bond valence is thus equal to three different quantities: the number of electrons each atom contributes to the bond, the number of electron pairs that form the bond, and the electrostatic flux linking each core to the bonding electron pair.
The valence sum rule It follows from these definitions, that the valence of an atom is equal to the sum of the valences of all the bonds it forms. This is known as the
valence sum rule, Eq. 1, which is central to the bond valence model. : V=\sum(S_j) (Eq. 1) A bond is formed when the valence shells of two atoms overlap. It is apparent that the closer two atoms approach each other, the larger the overlap region and the more electrons are associated with the bond. We therefore expect a correlation between the bond valence and the bond length and find empirically that for most bonds it can be described by Eq. 2: : S=\exp((R_o-R)/b) (Eq. 2) where S is the valence and R is the length of the bond, and R_o and b are parameters that are empirically determined for each bond type. For many bond types (but not all), b is found to be close to 0.37 Å. A list of bond valence parameters for different bond types (i.e., for different pairs of cation and anion in given oxidation states) can be found at the web site. It is this empirical relation that links the formal theorems of the bond valence model to the real world and allows the bond valence model to be used to predict the real structure, geometry, and properties of a compound. If the structure of a compound is known, the empirical bond valence - bond length correlation of Eq. 2 can be used to estimate the bond valences from their observed bond lengths. Eq. 1 can then be used to check that the structure is chemically valid; any deviation between the atomic valence and the bond valence sum needs to be accounted for.
The distortion theorem Eq. 2 is used to derive the distortion theorem which states that the more the individual bond lengths in a coordination sphere deviate from their average, the more the average bond length increases provided the valence sum is kept constant. Alternatively if the average bond length is kept constant, the more the bond valence sum increases
The valence matching rule If the structure is not known, the average bond valence, Sa can be calculated from the atomic valence, V, if the
coordination number, N, of the atom is known using Eq. 3. : S_a=V/N (Eq. 3) If the coordination number is not known, a typical coordination number for the atom can be used instead. Some atoms, such as sulfur(VI), are only found with one coordination number with oxygen, in this case 4, but others, such as sodium, are found with a range of coordination numbers, though most lie close to the average, which for sodium is 6.2. In the absence of any better information, the average coordination number observed with oxygen is a convenient approximation, and when this number is used in Eq. 3, the resulting average bond valence is known as the
bonding strength of the atom. Since the bonding strength of an atom is the valence expected for a bond formed by that atom, it follows that the most stable bonds will be formed between atoms with the same bonding strengths. In practice some tolerance is allowed, but bonds are rarely formed if the ratio of the bonding strengths of the two atoms exceeds two, a condition expressed by the inequality shown in Eq. 4. This is known and the
valence matching rule. Compounds that do not satisfy Eq. 4 are difficult, if not impossible, to prepare, and chemical reactions tend to favour the compounds that provide the best valence match. For example, the aqueous solubility of a compound depends on whether its ions are better matched to water than they are to each other.
The ionic model The bond valence model can be reduced to the traditional ionic model if certain conditions are satisfied. These conditions require that atoms be divided into cations and anions in such a way that (a) the electronegativity of every anion is equal to, or greater than, the electronegativity of any of the cations, (b) that the structure is electroneutral when the ions carry charges equal to their valence, and (c) that all the bonds have a cation at one end and an anion at the other. If these conditions are satisfied, as they are in many ionic and covalent compounds, the electrons forming a bond can all be formally assigned to the anion. The anion thus acquires a formal negative charge and the cation a formal positive charge, which is the picture on which the ionic model is based. The electrostatic flux that links the cation core to its bonding electrons now links the cation core to the anion. In this picture, a cation and anion are bonded to each other if they are linked by electrostatic flux, with the flux being equal to the valence of the bond.
Strengths and limitations of the model The bond valence model is an extension of the electron counting rules and its strength lies in its simplicity and robustness. Unlike most models of chemical bonding, it does not require a prior knowledge of the atomic positions and so can be used to construct chemically plausible structures given only the composition. The empirical parameters of the model are tabulated and are readily transferable between bonds of the same type. The concepts used are familiar to chemists and provide ready insight into the chemical restraints acting on the structure. The bond valence model uses mostly classical physics, and with little more than a pocket calculator, it gives quantitative predictions of bond lengths and places limits on what structures can be formed. However, like all models, the bond valence model has its limitations. It is restricted to compounds with localized bonds; it does not, in general, apply to metals or aromatic compounds where the electrons are delocalized. It cannot in principle predict electron density distributions or energies since these require the solution of the Schoedinger equation using the long-range Coulomb potential which is incompatible with the concept of a localized bond. ==History==