# (p.261) Appendix I Determining Bond Valence Parameters

# (p.261) Appendix I Determining Bond Valence Parameters

# A1.1 Introduction

Pauling (1947) first suggested using eqn 3.1 to associate bond valences (bond numbers or bond strengths) with interatomic distances in metals and alloys. Byström and Wilhelmi (1951) then applied the same relation to oxides. Subsequently others fitted bond-valence–bond-length correlations for the particular bond types they were studying, but Donnay and Allman (1970) first proposed a general method for determining the bond-valence–bond-length parameters for any bond type using eqn 3.2 with a linear extrapolation to $s=0$ at a finite value of *R*_{ij}. Equation 3.2 was later adopted by Brown and Shannon (1973) to produce a systematic listing of bond strength (or valence) parameters for bonds formed between oxygen and the atoms in the first three rows on the Periodic Table, at the same time suggesting some of the uses to which they could be put. Since then there have been many determinations of bond valence parameters for both eqns 3.1 and 3.2 (e.g. Brown 1981; Brown and Altermatt 1985; Brese and O’Keeffe 1991) as well as suggestions for other relationships (Section A1.3). A listing of published parameters for use in eqn 3.1 can be found at Brown (2015).

While the bond valence measures the Coulomb attraction that forms a bond, the correlation between the bond-valence and bond-length measures the core repulsion that determines the equilibrium length. Equation 3.1 is based on the exponential Born–Meyer (1932) repulsion potential and eqn 3.2 on the earlier power law of Born and Landé (1918). Both equations contain two fitted parameters, which is the minimum number needed to give a reasonable fit over the limited ranges of bond lengths normally found between a given pair of atoms. Both expressions give equally good fits, though eqn 3.1 is generally preferred because of its more robust mathematical properties and the universal value of the parameter
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*b* for many bond types. Neither equation describes the relationship particularly well over an extended range of bond lengths as can be seen from Fig. 5.1, which illustrates the correlation for H–O bonds over the total range of observed bond valences from zero to 1.0 vu. The two-parameter equations underestimate the valence for shorter bonds and overestimate it for longer bonds. This is not a problem for most bond types where the observed bond lengths cover a limit range. The exceptions are hydrogen bonds, bonds formed by anions with partially stereoactive lone pairs (Section 7.3) and alkali metals with small coordination numbers.

# A1.2 Determination of Bond Valence Parameters

## A1.2.1 Methods of Determining Bond Valences

Regardless of which expression one uses to describe the correlation between bond valences and bond lengths, they all contain empirical *bond valence parameters* that depend on the bond type: namely the chemical elements of the two atoms that define the bond and their valences (oxidation states). These parameters must be determined from observed structures because the repulsion is a quantum effect that is not easily calculated. Fitting these parameters to observed bond lengths ensures that they represent a distillation of the experimental information stored in the crystallographic databases (Section 12.8).

To determine the bond valence parameters, the observed bond lengths must be assigned their proper bond valences. This can be done in a number of different ways. The most fundamental method is to calculate the ionic flux of the bond in an observed structure (Section 2.3.4). This calculation is not trivial and software is not readily available. It is rarely used, but the points shown in Figs 3.1 and 3.2 were obtained in this way (Preiser *et al.* 1999). In some simple structures such as NaCl (18189) the bond valences can be determined by inspection. More generally, bond valences can be calculated using the network eqns 2.5 and 2.8, but these can only be used if there are no electronic (Chapter 7) or steric (Chapter 11) strains. This excludes any structure containing hydrogen bonds whose asymmetry is the result of the repulsion between the terminal anions (Chapter 5). If one can use any of these methods to calculate the bond valences, one can plot bond valence against bond length directly and find the equation that gives the best fit.

It is normally necessary to use a more indirect method. The most common procedure is to use a trial set of bond valence parameters to calculate an initial valence for each bond in a target set of structures. The bond valence parameters are then refined by minimizing the difference between the atomic valence and its bond valence sum (expression A1.1):

The techniques for performing this refinement are described in Sections of A1.2.2, A1.2.3 and A1.2.4.

## (p.263) A1.2.2 Selection of the Target Set of Observed Atomic Environments

Because experimental results are subject to experimental uncertainties, it is necessary to use a target set of structures and find the bond valence parameters that provide the best overall fit between the theoretical and observed bond valences. A careful selection of the atomic environments to be included in this set is essential, since the quality of the resulting parameters depends on the quality of the structures from which they are derived.

The size of the target set is the first choice that has to be made. The set must contain at least as many atomic environments as there are parameters to be determined, but using a set this small ensures that any error in the observed bond distances is transferred to the resulting bond valence parameters. The more environments that can be included in the target set, the more likely that the statistical averaging will eliminate the errors of measurement, but this will happen only if the environments included in the set accurately reflect the structural chemistry. Using a biased selection or including poorly determined structures can only result in less reliable bond valence parameters.

There are several points to consider. It is simpler if all the bonds in the target set are of the same type, e.g. they are all Ca–O bonds, since then there are only two parameters to refine. It is possible to include atoms that form bonds of more than one type, but for every additional bond type present in the environment of the atom, the more parameters there are that need to be refined. In principle, the refinement should be carried out around both the cation and anion, but few anions are bonded to only one type of cation. It is, however, good practice wherever possible to check that the bond valence parameters refined using the cations also give good valence sums around the anions.

The structures that are included in the target set must be screened for quality. Some filters are obvious and easy to apply: one should include only well-ordered structures with small values of the crystallographic agreement index, *R* (e.g. $R<0.02$). In order to obtain a reliable value for *b* it is important to include atomic environments with as many different coordination numbers as possible, and to ensure that all coordination numbers are weighted equally. Structures with steric strain should be excluded, but such exclusion is not straightforward because steric strain is usually recognized by the failure of the valence sum rule (rule 3.1). The final set of refined bond valence parameters should give good agreement between the bond valence sums and the atomic valences for most of the environments in the target set, but there will usually be a small number where the agreement is poor. It is tempting is to delete these from the target set, but this is poor practice and may bias the results. An environment should be removed from the target set only if a close examination reveals a valid reason for its exclusion. Possible causes include a poor or ambiguous structure determination, or an unstable compound possibly recognized by an anomalously small coordination number. A structure may be excluded if it contains steric strain recognized by the failure of the valence sum rule, but before it can be removed the strain must be confirmed by checking
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that the strain indicates a mechanically stable arrangement of stresses, i.e. that the compressive and tensile stresses represent an equilibrium structure (Chapter 11). If no reason can be found for excluding a particular atomic environment, it should be kept in the target set since it has a story to tell.

A good size for the target set is between 30 and 50 atomic environments. This number is small enough that each environment can be individually examined to ensure quality without bias, and increasing the number of structures does not necessarily improve the quality of the parameters. The quality of the bond valence parameters can be no better than the quality of the target set used to determine them.

The coordination number around most atoms is not usually contentious, but for some coordination spheres, particularly around alkali metals, it may not be obvious how many of the longer bonds should be included (Section 3.3). Brown and Altermatt (1985) and Brese and O’Keeffe (1991) chose to limit the coordination sphere to bonds with a valence greater than 4% of the cation valence and a similar limit has been adopted by most other workers. Adams (2001) examined the effects of including more bonds in each environment and showed that the larger the assumed coordination number, the smaller the value of *R*_{0} and the larger the value of *b*, though once bonds longer than 500 pm were added the values of *R*_{0} and *b* did not change. It is important to use bond valence parameters calculated using the same definition of coordination number. For example, when calculating valence maps of glass structures Adams found it necessary to use bond cut-off distances of 500 to 600 pm to avoid artificial discontinuities in the map (Section 10.5). His ‘soft bond’ parameters (Adams 2015) are not necessarily suitable for normal crystal structure analysis.

## A1.2.3 Refinement

Once the target set as been chosen, there are various methods of refinement. These have been reviewed by Gagné and Hawthorne (2015). They include least squares or other techniques to minimize the expression A1.1. The principal difficulty with this method is that unless the average bond valence is close to 1.0 vu, *R*_{0} and *b* are strongly coupled. One method that gets around this problem is to refine *R*_{s} and *b* in eqn A1.2, which moves the pivot point of the refinements close to the mean bond length and so decouples *R*_{0} and *b*:

Here, *V*_{i} is the valence of the central atom and <*N*_{i}> is the typical coordination number given in Table 2.1. The value of *R*_{0} can then be calculated from eqn A1.3.

If one is calculating bond valence parameters for a hard cation with oxygen or fluorine it may be satisfactory to assume that $b=37\text{}\mathrm{p}\mathrm{m}$, in which case the value
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of *R*_{0} that leads to the valence sum being exactly equal to the atomic valence is given by eqn A1.4:

Calculating *R*_{0} in this way for all the environments of atom *i* in the target set will give a number of estimates of the value of *R*_{0} that can be averaged. However, this method should not be used if either atom *i* or *j* is soft unless one already has a reliable value for *b* (Section 3.5). Sidey (2012) analysed eqn A1.4 and suggested in an alternative expression that gives the geometric mean of the different values of *R*_{0}.

A graphical method of determining *R*_{0} and *b* is based on rewriting eqn 3.1 in the form of eqn A1.5 (Krivovichev and Brown 2001; Sidey 2009):

where *V*_{i} is the valence of the central atom that forms *N* equal bonds each having a length *R*_{N}. If the bonds are not all equal the distortion theorem (rule 3.2) predicts that the average bond length will be larger than *R*_{N}, but if the difference, Δ*R*, is not large the average can be corrected using eqn 3.5. A plot of *R*_{N} against ln(*V*_{i}/*N*), should result in the points lying on a straight line with an intercept of *R*_{0} and slope of *b.*

Any of these methods will have difficulty if the target set contains atoms with only one coordination number, but if the bonds have different lengths, *b* can be estimated by examining the valence sums around the anions. Alternatively, for a distorted environment one could use eqn A1.6 derived from eqn 3.5:

where σ is the standard deviation of the bond lengths from their average, and ΔR is the difference between the average bond length and *R*_{N}. In this approach it is essential that the bond valence sum in each environment be exactly equal to the atomic valence so some normalization may be necessary. Equation A1.6 is only valid in the limit of small distortions and could be derived from the slope of the line in Fig. 3.3 at $\mathrm{\sigma}=0$. If no distorted environment is known, then *b* is indeterminate and any value may be used.

Because *R*_{0} and *b* are strongly coupled, *R*_{0} can change significantly if the value of *b* is changed. *R*_{0} can be determined with much greater precision than *b* which typically cannot be determined to closer than ±5 or 10 pm. The two parameters should be refined together, or if refined separately, the value of *R*_{0} should be determined only after the value of *b* has been fixed. The same argument applies to *R*_{0} and *N* in eqn 3.2

Brown and Altermatt (1985) produced an extensive table of bond valence parameters, mostly of bonds to oxygen, and these are the ones used in this book.
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Brese and O’Keeffe (1991) extended the list to many other anions by assuming that $\mathrm{b}=37\text{}\mathrm{p}\mathrm{m}$ and showing that *R*_{0} for a cation–*X* bond (*R*_{0X}) is related to the value for the bond between the same cation and oxygen (*R*_{0O}) by eqn A1.7:

where *a* and *c* are constants they have tabulate for each of the 11 anions (*X*) they examined. Using eqn A1.7, they were able to calculate values of *R*_{0} for 969 different bond types. While these values are not as accurate as those found by direct fitting, they are satisfactory for most purposes.

## A1.2.4 Special Cases

Underlying this discussion is the assumption that the bond valence parameters depend only on the element and valence of the atoms. Equation 3.1 takes account of differences in the size and coordination number of the atoms. However, there are some exceptions. In recent studies See *et al.* (1998) and Shield *et al.* (2000) show that to obtain correct bond valence sums around transition metals with nitrogen ligands it is necessary to use different values of *R*_{0} according to the coordination number of the nitrogen atom, an effect that is presumably linked to the complex nature of the bonding in late transition metals. The spin state and presence of certain ligands can also affect the values of *R*_{0} for a transition metals. The difference between high and low spin has been explored by Harris *et al.* (2005), (Section 12.3.3), while See and Kozina (2013) have examined variations in the trans effect around d^{8} and low spin d^{6} transition metals (Section 7.4.3).

Ambient conditions such as temperature and pressure can influence the value of *R*_{0}. Corrections for these effects are described in Section 8.8.

The bonds formed by hydrogen need special treatment as described in Section 5.8. No satisfactory analytical expression has been found that relates bond valence to bond length over the full range of observed distances. Using eqn 3.1, Alig *et al.* (1994) fitted H–O bonds with the parameters ${R}_{0}=91.4\text{}\mathrm{p}\mathrm{m}$ and $b=40.4\text{}\mathrm{p}\mathrm{m}$ determined from the bond valence sums around hydrogen but the valence sums around oxygen atoms were not checked. Their parameters do not agree with the curve shown in Fig. 5.1. Sidey (2011) has proposed a three-parameter expression, A1.8, relating the H–O bond distance to the valence over the whole of its range:

The consolidated list of bond valence parameters (Brown 2015) provides three sets of values for *(R*_{0}*,b)*, respectively (90.7, 28 pm), for H–O distances less than 105 pm, (56.9, 94 pm) for H–O distances between 105 and 170 pm, and (99, 59 pm) for H–O distances greater than 170 pm, but in general, it is best to assign valences to H–O bonds using the graphical methods discussed in Section 5.8.

# (p.267) A1.3 Other Expressions

A number of alternative algebraic forms of the relationship between bond valence and bond length have been proposed. Some are motivated by an attempt to use atom-based parameters or to provide some physical justification for the relationship, while others address the failure of eqns 3.1 and 3.2 to give correct valences at the extreme ends of the ranges. They are generally more complex and involve more than two parameters per bond type. Some make direct use of ionic radii which come in a number of different scales.

Ziołkowski (1985) derived an eqn (A1.9) for the bond valence based on notional free ionic radii, *r*_{a} and *r*_{c}, extrapolated to zero coordination:

Here *a, b, c* and *d* are universal constants that, however, depend on the way in which the radii are defined. Since the free ion radii in some scales can be negative (which is unphysical), Ziołkowski defines a second set in which hydrogen is assumed to have zero radius.

Naskar *et al.* (1997) were interested in using bond valences to determine oxidation states around transition metal cations, particularly those with zero or negative formal oxidation states. Since these numbers cannot, in principle, be reached by the standard equations, they proposed to create a fictional positive oxidation state by arbitrarily adding 4.0 vu to the actual oxidation state. They proposed to write the valence sum rule in the form of eqn A1.10:

where *N*_{i} is the coordination number and $a\phantom{\rule{thickmathspace}{0ex}}(=20)$ and *R*_{1} are fitted parameters. One may question the underlying assumptions of this equation and its usefulness, given that different parameters are needed depending on whether the cation or the anion is chosen as the central atom.

Valach (1999) has proposed the use of the five-parameter eqn A1.11 based on a Taylor expansion of the quantum stabilization energy:

The values of α_{ν} are not determined from the theory but, like other bond valence parameters, are fitted to observed bond lengths in the manner described in Section A1.2. The parameters that Valach reports for Cu–O and Cu–N bonds give zero valence at finite-bond lengths, but the valences calculated for very short bonds are probably too low.

Mohri (2000) has proposed eqn A1.12 based qualitatively on the notion that the charge density in the interatomic region will be roughly uniform:

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where *R*^{0} is the bond length observed at a valence of *s*_{0}, and λ is the sum of the core radii (Pauling’s cationic radii) of the two atoms. The parameters of eqn A1.12 can be directly related to the softness parameters, *b* and *N*, of eqns 3.1 and 3.2 by using the value of *R*^{0} For ${s}_{0}=1$:

Using the parameters of eqn A1.12, Mohri found that 75% of the values of *b* lie within 5 pm of 37 pm and that the values of *N* lie within about 10% of the values reported by Brown and Shannon (1973). In a subsequent paper, Mohri (2003) looked for a quantum mechanical quantity that gives the classical bond orders for covalent, i.e. non-polar, bonds, one that also obeys the valence sum rule and that correlates with the bond length. Later, Mohri (2005, 2006) applied the covalent bond order model to the hydrogen bond, *X*-*D*-H ⋯*A*-*Y*.

Sidey (2014) has suggested the expression:

and finds that for Mn–O bonds with $m=0.617\text{}\AA (\text{sic})$ and $1=1.14\text{}\AA $ this equation gives good bond valence sums around manganese regardless of its valence state.

Gagné and Hawthorne (2015) have tested 26 different two- and three-parameter expression against over 30 000 different cation environments involving bonds between 128 different cations and oxygen. They concluded that several expressions give satisfactory bond valences, including eqn 3.1, which they recommend. They refined *R*_{0} and *b* for 135 different bond types, but for hard atoms these provide only a marginal improvement over those in which *b* is fixed at 37 pm. This may not be sufficient to outweigh the convenience of working with the fixed value of *b*.