Influence of bond softness on bond valence

In order to illustrate the consequences of the approximation of a universal parameter b, it may be helpful to rewrite the monotonic correlation between bond length RA-X and bond valence sA-X   into a pseudopotential of the bond-valence deviation  sA-X .

where sideal represents the bond-valence contribution from a single bond that would give rise to a total valence sum V(A) = Videal(A).

In Fig. 1 it is presumed that a monovalent central cation A+ is symmetrically coordinated by four anions X z-. Assuming that only nearest neighbours contribute to the valence sum the ideal bond-valence contribution from a single bond would thus be
s(ideal,A-X) = ¼. Any deviation of the bond length RA-X   from its ideal value Rideal (corresponding to sideal) leads to an increase of |sA-X | = |sA-X  - sideal|. The concavity of the relation between s and R ensures that the increase of |sA-X| is steeper in the R < Rideal branch. For a given coordination type the postulation of a fixed value of b in this picture would mean that the shape of the pseudopotential becomes identical for all monovalent cations (cf. broken line in Fig. 1), irrespective of the type of bonding or the polarizability of the interacting particles. Thus, applications of the bond-valence concept to any field that might be affected by the shape of the bond-length bond-valence pseudopotential should not rely on bond-valence parameter sets with a fixed parameter b. It appears more natural to incorporate the effects of the polarizability by the determination of a suitable value of b instead of treating it as a correction term to the atomic radii [as had been performed implicitly in (4)].
Figure 1 (from ADAMS, Acta Cryst. B 57, 278-87 (2001).
Pseudopotential representation of the correlation between bond-length R and bond valence s. Rideal is the bond distance that  leads to a bond valence of sideal = 0.25  v.u. (the bond valence for a monovalent cation in a symmetrical tetrahedral  coordination). Full lines refer to bond-valence parameters of Ag-O and Ag-I with freely refined b (Radaev et al., 1994;  Trömel, 1994); the broken line displays the universal shape for pseudopotentials that employ a fixed b = 0.37  Å.

Adjusting the value of b to the softness of a bond at first requires an independent measure of the softness of a bond. Following Parr & Pearson (1983) individual atoms, ions or radicals may be characterized by their `electronic chemical potential'  and their `absolute hardness' . The exact definition of these quantities as


where N equals the number of electrons and  represents the potential of the nucleus and external influences, may appear quite abstract, but for neutral particles approximate values for  and  are experimentally accessible from the relation of these quantities to ionization energy IE and electron affinity EA

In this approximation || becomes identical to Mulliken's definition of the absolute electronegativity abs. It should be noted that for a system in equilibrium abs attains a constant site-independent value, whereas the value of  varies locally with a global average value of

The `absolute softness'   is thereby defined as the reciprocal value of hardness . The softness of a cation M z+ may be calculated in the same manner using the ionization energy of M z+ as IE (= "(z+1)th ionization energy") and replacing EA by the ionization energy of M(z - 1)+ (Pearson, 1985).

As electron affinities of anions are generally inaccessible (and their meaningfulness in the determination of bond softnesses appears questionable), a similar extension to anions is not viable. According to Pearson (1988) the values of IE and EA for the neutral elements may serve as a rough approximation for the anions. In the first part of the paper we will follow this guideline. Later it will be outlined how an empirical correlation between the anion radius and the anion softness may be utilized to obtain a more precise estimate of the anion softness. To derive a measure for the softness of the A-X bond, the softnesses of the interacting species A+ and X z- need to be combined.

In an earlier investigation on a possible connection between bond softness and bond valence, Urusov (1995) had argued that the softness of a bond should increase with the sum of the softnesses of the interacting particles. Contrasting to this assumption, the empirical HSAB (hard and soft acids and bases) concept (Pearson, 1963; Parr & Pearson, 1983; Nalewajski, 1993) suggests that reactions will occur most readily between species that match each other in hardness or softness. If the formation of strong bonds between anions and cations of equal softness is the fundamental reason for this empirical rule, then it appears straightforward to conclude that the interatomic potentials for these bonds should be steeper (and thus correspond to a smaller value of b) than those for the weaker bonds between particles of mismatched softnesses.

The diagrams in Fig. 2 compare the significance of the two suggested correlations based on those literature bond-valence parameters (for halide and chalcogenide compounds) that did not follow the assumption of a universal bond-valence parameter b. Obviously there is no strong correlation to be expected, since the comparison includes data from different sources (some of them were rather old and therefore based on the considerably smaller number of structures available at that time) using slightly different conventions for the selection of `well determined' structures and the choice of the counterions that contribute to the bond-valence sum. In some cases it is also not evident whether the published b parameters were the results of free refinements or biased, e.g. by whatever the authors assumed to be `chemical knowledge'. Some of the bond-valence parameters are intended to apply to atoms for a range of formal valences, but each valence state corresponds to a different softness. The approximative conversion formula suggested by Burdett & Hawthorne (1993) for parameter sets that employ the power law ansatz of (2) systematically produces lower values of b. The converted data have therefore been scaled by a constant factor of 1.25, so that the average value of b for the literature parameter sets from (1) and (2) becomes equal. Moreover, the exponent N of the power law parameters was mostly given as an integer number in the literature, which leads to noticeable rounding errors. Despite all these drawbacks Fig. 2 reveals that the difference of the softnesses anion - cation should be loosely related to b in the sense predicted by the HSAB concept, whereas there seems to be no discernible correlation between b and the sum of softnesses anioncation. From these data it is impossible to decide whether the slopes for the two branches of the correlation between b and the softness difference ( anion  > cation or anion  < cation) differ, since the case anion anion cation occurs only for a low number of cation-anion pairs. The apparent shift of the minimum in the correlation to positive softness differences may serve as an indication that the rough estimate of the anion softness (by assuming equal softnesses for neutral atoms and anions) tends to overestimate the anion softness.
Figure 2 (from ADAMS, Acta Cryst. B 57, 278-87 (2001).  
Comparison of the dependence of the bond-valence parameters b on the difference between the softnesses of anions (halides and chalcogenides) and cations (left-hand side) or on the sum of these softnesses as predicted by Urusov (1995) (right-hand side). Large symbols refer to b values from various literature compilations (Radaev et al., 1994; Trömel,1994; Brown, 1981, 2000), while small symbols refer to b values converted from parameters of the power law ansatz of (2) as described in the text. In harmony with the HSAB concept, bonds between atoms with different softnesses tend to be weaker and therefore correspond to larger values of the parameter b (solid line: fourth-order polynomial fit to all data; broken lines: 99% confidence interval). However, no correlation at all is discernible in the right-hand side diagram between b and the softness sum for the same set of data.

These findings do not preclude that a different combination of the individual softnesses might lead to a more significant correlation. In this context it may be noteworthy that Mohri (2000) recently derived an alternative formulation of the correlation between the bond distance and bond valence from a molecular orbital viewpoint. The conversion of his parameters into the conventional Brown-Altermatt formula leads to values of b that are generally lower (ca 2/3 of literature data), but exhibit the same rough correlation to the softness difference.

The low values of b found by Pauling (1947, 1960) in his early investigations of the analogous relationships between bond length and `bond order' for bonds between the same type of atoms (b = 0.30  Å for metals, b =  0.26  Å for C-C bonds) may be tentatively interpreted as a further line of evidence that bonds between particles of equal softness are characterized by a low value of b.