(p.223) Appendices
(p.223) Appendices
Bond valence parameters
A1.1 Introduction
Pauling (1947) first suggested using eqn (3.1) to associate bond valences (bond numbers or bond strengths) to interatomic distances in metals and alloys. Byström and Wilhelmi (1951) then applied the same relation to oxides. Subsequently Zachariasen (1954) and Zachariasen and Hettinger (1959) published graphical correlations for U–O bonds. Clark et al. (1969) determined bond valences for Si–O bonds by expressing the bond distance as a third degree power series in S with four fitted parameters, though Perloff (1970) found a simple linear relation worked for Cr–O and Mo–O bonds. Donnay and Allman (1970) used eqn (3.2) with a linear extrapolation to S = 0 at a finite value of R to locate hydrogen bonds in minerals. Equation (3.2) was also adopted by Brown and Shannon (1973) to produce a systematic listing of bond valence parameters. Since then there have been many determinations of bond valence parameters for both eqns (3.1) and (3.2) (Brown 1981) as well as more limited listings of parameters for some of the other relations discussed in Section A1.3 below. A listing of published parameters for use in eqn (3.1) can be found at http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/bond_valence_parm/ and a shorter selected list of bond valence parameters is given in Table A1.1
The bond length–bond valence relationship is a measure of the repulsion between ions. 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. Both give equally good fits, though eqn (3.1) is generally preferred because of its more robust mathematical properties and the approximate constancy of the parameter B. Neither equation describes the relationship particularly well over an extended range of bond lengths, but these are encountered in only a few cases such as the O–H bonds discussed in Chapter 7.
A1.2 Determination of bond valence parameters
This section discusses the determination of the bond valence parameters used in eqns (3.1) and (3.2) though the principles can be applied to the other expressions discussed in Section A1.3. Since there is no exact theoretical derivation of the correlation between bond valence and bond length, the bond valence (p.225) parameters, R _{0}, and B or N, must be determined empirically using experimentally determined bond lengths.
Table A1.1 Selected bond valence parameters for eqn (3.1)
Cation |
Anion |
R _{0} (pm) |
B (pm) |
---|---|---|---|
Ag^{+} |
I^{–} |
238 |
37 |
Al^{3+} |
O^{2–} |
162 |
37 |
As^{5+} |
F^{–} |
162 |
37 |
B^{3+} |
O^{2–} |
137.1 |
37 |
Ba^{2+} |
O^{2–} |
228.5 |
37 |
C^{4+} |
O^{2–} |
139 |
37 |
Ca^{2+} |
F^{–} |
184.2 |
37 |
Ca^{2+} |
O^{2–} |
196.7 |
37 |
Cd^{2+} |
Br^{–} |
249 |
37 |
Cd^{2+} |
Cl^{–} |
237 |
37 |
Cl^{7+} |
O^{2–} |
163.2 |
37 |
Co^{2+} |
N^{3–} (two coordinate) |
170 |
37 |
Co^{2+} |
N^{3–} (three coordinate) |
177 |
37 |
Cr^{3+} |
F^{–} |
165.7 |
37 |
Cr^{3+} |
O^{2–} |
172.4 |
37 |
Cr^{6+} |
O^{2–} |
179.4 |
37 |
Cs+ |
Cl^{–} |
279.1 |
37 |
Cs+ |
O^{2–} |
241.7 |
37 |
Cu+ |
O^{2–} |
161 |
37 |
Cu^{2+} |
O^{2–} |
167.9 |
37 |
Cu^{3+} |
O^{2–} |
174 |
37 |
Er^{3+} |
S^{2–} |
246 |
37 |
Eu^{3+} |
O^{2–} |
207.4 |
37 |
Fe^{2+} |
O^{2–} |
173.4 |
37 |
Fe^{3+} |
O^{2–} |
175.9 |
37 |
Ga^{3+} |
O^{2–} |
173.0 |
37 |
Ge^{4+} |
O^{2–} |
174.8 |
37 |
Hg^{2+} |
Cl^{–} |
228 |
37 |
Hg^{2+} |
F^{–} |
217 |
37 |
I^{5+} |
O^{2–} |
200.3 |
37 |
K^{+} |
Cl^{–} |
251.9 |
37 |
K^{+} |
O^{2–} |
213.7 |
37 |
La^{3+} |
O^{2–} |
217.2 |
37 |
Mg^{2+} |
O^{2–} |
169.3 |
37 |
Mn^{3+} |
O^{2–} |
176.0 |
37 |
Mo^{6+} |
O^{2–} |
190.7 |
37 |
N^{5+} |
O^{2–} |
136.1 |
37 |
NH_{4} ^{+} |
O^{2–} |
222.6 |
37 |
NH_{4} ^{+} |
F^{–} |
212.9 |
37 |
NH_{4} ^{+} |
Cl^{–} |
261.9 |
37 |
Na^{+} |
Cl^{–} |
215 |
37 |
Na^{+} |
F^{–} |
167.7 |
37 |
Na^{+} |
O^{2–} |
180.3 |
37 |
Nb^{5+} |
O^{2–} |
191.1 |
37 |
Ni^{2+} |
O^{2–} |
165.4 |
37 |
P^{5+} |
O^{2–} |
161.7 |
37 |
Pb^{2+} |
S^{2–} |
254.1 |
37 |
Pd^{2+} |
Cl^{–} |
253 |
37 |
Rb^{+} |
O^{2–} |
226.3 |
37 |
Ru^{5+} |
O^{2–} |
190 |
37 |
S^{4+} |
O^{2–} |
164.4 |
37 |
S^{6+} |
O^{2–} |
162.4 |
37 |
Sb^{5+} |
O^{2–} |
194.7 |
37 |
Sc^{3+} |
O^{2–} |
184.9 |
37 |
Si^{4+} |
O^{2–} |
162.4 |
37 |
Sn^{2+} |
O^{2–} |
194 |
37 |
Sr^{2+} |
O^{2–} |
211.8 |
37 |
Te^{4+} |
Cl^{–} |
237 |
37 |
Ti^{4+} |
O^{2–} |
181.5 |
37 |
T1^{+} |
O^{2–} |
212.4 |
37 |
Tm^{3+} |
O^{2–} |
200.0 |
37 |
V^{5+} |
O^{2–} |
180.3 |
37 |
Y^{3+} |
O^{2–} |
201.9 |
37 |
Yb^{3+} |
O^{2–} |
196.5 |
37 |
Zn^{2+} |
O^{2–} |
170.4 |
37 |
Zn^{2+} |
S^{2–} |
209 |
37 |
Zn^{2+} |
Te^{2–} |
245 |
37 |
Mostly taken from Brown and Altermatt (1985) and Brese and O’Keeffe (1991). (A complete set is available at http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/bond_valence_parm/)
(p.226) In principle the bond valence parameters could be obtained by comparing the experimental bond valences, S, determined using an initial set of bond valence parameters, against the theoretical bond valences, s, calculated using the network eqns (3.3) and (3.4). These initial values could then be refined to minimize the differences given by the expression (A1.1):
This method has not so far been used but should work well providing that only structures with unstrained bonds are used.
The normal procedure is to refine the bond valence parameters by minimizing the difference between the atomic valence and the sum of the bond valences (p.227) around cations with only one kind of ligand (expression (A1.2)):
Since there are two bond valence parameters to be determined, at least two cation environments are needed. In practice it is best to use at least 20–30 environments if these are available. Even so, there are a number of hazards. If all the cations in the sample have the same coordination number and an average bond valence of 1.0 vu, B necessarily refines to infinity (N refines to 0), but even when different coordination numbers are present there is a strong correlation between B (or N) and R _{0}. The ambiguities caused by this correlation can be resolved by calculating the valence sums around the anions as well as the cations, but this is difficult to do in a systematic way since the anions usually bond to more than one type of cation. A further uncertainty is introduced by the choice of the maximum bonding distance since including longer distances in the coordination sphere systematically increases B and decreases R _{0} (Adams 2001). In the list prepared by Brown and Altermatt (1985), and by inference the list based on it prepared by Brese and O’Keeffe (1991), all the bonds were assumed to have valences greater than 0.04 times the cation valence.
While R _{0} can be determined with much greater precision than B (or N), the correlation between the parameters means that making changes in B (or N), even changes that lie well within the range of experimental uncertainty, requires a corresponding, but significant, change in R _{0}. One must be careful, therefore, to make sure that the value of R _{0} used is the one appropriate to the value of B (or N) chosen.
B (or N) can only be determined with a precision of around 10 per cent even if care is taken (Tytko 1999), but the observation that B lies between 32 and 42 pm for many bonds means that it is convenient to fix its value at 37 pm for all bond types (Brown and Altermatt 1985).^{1} Any error that this assumption introduces is usually negligible for most bonds to O^{2–} provided that the appropriate value of R _{0} is used. Using the same value of B for all bond types makes the determination of R _{0} simpler since only one parameter now needs to be fitted. Combining eqn (3.1) with the valence sum rule and rearranging yields eqn (A1.3):
where the summation is over all the bonds in a single coordination sphere, i. Since all the terms on the right-hand side of this equation are known, it is easy to calculate R_{0i}, the value of R _{0} that will give a valence sum exactly equal to V for (p.228) the ith coordination sphere. This procedure gives separate values of R _{0i} for each coordination sphere examined but, if the valence sum rule and the value of B chosen are valid, the values of R _{0i} should all be similar. Some variation is expected as a result of experimental uncertainty and the presence of cation environments that suffer from lattice-induced strain (Chapter 12), but it is easy to check whether R _{0i} varies systematically with coordination number which would be an indicator that the wrong value of B has been assumed. If the variations in the individual values of R _{0i} are acceptable, the best value for R _{0} is taken as the average (eqn (A1.4)):
The advantage of this method is that it can be easily used with any number of cation coordination spheres whose bond distances are available. Even one coordination sphere is sufficient to give a trial value though the more that are used the more confidence one can have in the value of R _{0}. One needs to exercise a little care if only a few coordination spheres are known, since the oxidation state may be unstable except in the presence of strained bonds which could lead to a false value of R _{0}. There are a number of potential pitfalls in determining bond valence parameters. For example, the inclusion of poorly determined structures in the sample tends to increase the value (and uncertainty) of B with a corresponding decrease in R _{0}. A critique of these problems has been given by Tytko (1999).
The above procedure was used by Brown and Altermatt (1985) to produce an extensive table of bond valence parameters, mostly of bonds to O^{2–}. They based their refinement on cation environments in ordered structures with crystallographic agreement indices (R) of less than 0.1 reported in the Inorganic Crystal Structure Database (Bergerhoff et al. 1983). A routine for calculating R _{0i} is available in the program VALENCE (Brown 1996) which can also be used for calculating bond valences from bond lengths and vice versa. In this book, the bond valence parameters of Brown and Altermatt (1985) are used where available.
Brese and O’Keeffe (1991) extended the table of Brown and Altermatt to many other anions by showing that R _{0} for a bond between any cation and an anion X (R _{0X}) is related to the value for the bond between the same cation and O^{2–} (R _{0O}) by eqn (A1.5):
where a and b are constants which they tabulate for each of the 11 anions (X) they examined. Using eqn (A1.5), 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. In some cases Brese and O’Keeffe ignore the variation of R _{0} with oxidation state but for many (p.229) cations this does not introduce a large error, though for some of the softer ions like Cu, R _{0} is quite sensitive to oxidation state. For the best results, the influence of the oxidation state on R _{0} should not be ignored.
In a second paper O’Keeffe and Brese (1991) showed that R _{0} could be approximately determined from atomic parameters using eqn (A1.6):
where r _{i} and r _{j} are the atomic radii and c _{i} and c _{j} are the electronegativities of the terminal atoms i and j respectively. While eqn (A1.6) extends the range of bond valence parameters to cover virtually all bond types, the parameters it gives are not accurate enough for most of the uses described in this book. These parameters should be used with caution in quantitative work.
More recently Liu and Thorp (1993) and others have addressed the problem of determining the values of R _{0} in cases where the ligand is only present in mixed ligand environments, so that it is impossible to determine R _{0i} in the manner described above. Most of these studies have been made on transition-metal complexes extracted from the Cambridge Structural Database (Allen et al. 1979). Since these complexes usually have a mixture of ligating atoms (typically O, N, or S), Liu and Thorp refined the values of R _{0O}, R _{0N}, and R _{0S} simultaneously against the available bond lengths, the sample varying in size from 13 (for Ni^{3+}) to 116 (for Fe^{2+}). Because the values for O^{2–} and N^{3–} are very similar, there is a strong correlation between the values of R _{0O} and R _{0N}. On the other hand, Liu and Thorp derive different values for cations with different atomic valences and even list different parameters for the vanadyl bond (V^{4+} = 0) and normal V^{4+}–O bonds. Where comparison is possible the values of R _{0} obtained by Liu and Thorp generally differ by less than 3 pm from those of Brown and Altermatt (1985).
In a series of papers Palenik and his coworkers (Palenik 1997 a, b, c; Kanowitz and Palenik 1998; Wood and Palenik 1998, 1999 a, b; Wood et al. 2000) have determined bond valence parameters for transition metals. Some of these have been chosen to be independent of oxidation state in an attempt to provide values of R _{0} that can be used when the oxidation state of the cation is not known. While these parameters are not as accurate as those that take the oxidation state into account, they can be used to make an approximate determination of the oxidation state, after which the correct value of R _{0} can be substituted.
In recent studies See et al. (1998) and Shields et al. (2000) suggest that R _{0} sometimes depends on factors other than the oxidation states of the cation and anion. To obtain correct bond valence sums around transition metals with nitrogen ligands, it is necessary to use different values of R _{0} depending on the coordination number of N^{3–} as discussed in Section 9.2.
As the temperature of a crystal increases, it expands and the length of its bonds also increases though the bond valences do not change. In order to (p.230) calculate the correct bond valence for structures determined at high temperatures, one needs to correct the value of R _{0} for temperature using eqn (9.21) (Brown et al. 1997).
The bonds formed by H^{+} need special treatment as described in Section 7.8. No satisfactory analytical expression has been found that relates bond valence to bond length though some suggested expressions are given in Section A1.3. Using eqn (3.1) Alig et al. (1994) fitted H–O bonds with the parameters R _{0} = 91.4 pm and B = 40.4 pm determined from the bond valence sums around H^{+}. Although these values give good valence sums around the H^{+} ion, the valence sums around O^{2–} were not checked and Alig et al.’s parameters do not agree with the curve shown in Fig. 7.1. The table given at http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/bond_valence_parm/ works around the problem by providing three sets of values for R _{0} and B, respectively 90.7 and 28 pm for H–O distances less than 105 pm, 56.9, and 94 pm for H–O distances between 105 and 170 pm, and 99 and 59 pm for H–O distances greater than 170 pm. In general, it is best to assign valences to H–O bonds using the graphical methods discussed in Section 7.8.
A1.3 Other bond valence expressions
While eqns (3.1) and (3.2) have proved satisfactory for most purposes, several other expressions have been proposed. Most do not address the failure of eqns (3.1) and (3.2) to give correct valences at extreme distances but are motivated by an attempt to use atom-based parameters or to provide some physical justification for the relationship. As a result they are often more complex, sometimes involving more than two parameters per bond. Some make direct use of ionic radii which, however, come in many flavours. In general, it is not clear that these other formulations provide significantly better fits to the valence sum rule than the eqns (3.1) and (3.2), even though they may give more insight into the underlying physics.
Ziołkowski (1985) derived an equation (A1.7) 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 which, however, depend on the way in which the radii are defined. Since the true free ion radii are often negative (which is unphysical), Ziołkowski defines a second set in which hydrogen is assumed to have zero radius.
Ziołkowski’s equation can be simplified to the form given in eqn (A1.8),
since, for a given bond type, e and f are constants which can be treated as empirical parameters and fitted to observed bond lengths. This curve is steeper (p.231) than eqn (3.1) for short bonds, f representing the bond distance at which S becomes infinite. Subtracting a third constant, g, from the left-hand side of eqn (A1.8) gives eqn (A1.9),
and ensures that S = 0 when R = f + e/c. The length of a bond of 1.00 vu is then R = f + e/(1 + g). Brown (1987 b) proposed that the H–O bond valence given by the thin line in Fig. 7.1 could be approximated using eqn (A1.9) with e = 41 pm, f = 60pm and g = 0.16 pm. Bargar et al. (1997 c) recommend eqn (A1.8) with e = 24.1 pm and f = 67.7 pm. Both these sets of parameters are better than eqns (3.1) and (3.2) for O–H bonds, though neither is perfect. Brown’s parameters give a marginally better fit, but at the expense of a third fitted parameter.
Naskar et al. (1997) were interested in using bond valences to determine oxidation states around transition-metal cations, particularly those with negative or zero 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 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 (= 20) and R _{1} are fitted parameters. One may question the underlying assumptions of this equation, and the bond valences determined in this way are different in kind from those determined by the more traditional methods since different values are obtained depending on whether one uses the cation or the anion as the central atom. However, the expression, being empirical, must be judged on how well it discriminates between the oxidation states and this still needs to be demonstrated.
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 $aL_{v} 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 electron density in the interatomic region will be roughly uniform.
(p.232) where R ^{0} is the bond length observed at a valence of S _{0}, and $LD 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 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 finds that 75 per cent of the values of B lie within 5 pm of 37 pm and that the values of N lie within about 10 per cent of the values reported by Brown and Shannon (1973).
Notes:
(1) Recent work suggests that B may be significantly larger than 37 pm for bonds between soft and hard ions (Adams 2001).