Bonded atoms

If two neutral atoms A and B are far apart, they each will have spherical symmetry. As they approach each other, the protons in the nucleus of atom A will exert an attractive force on the electron density of atom B and this electron density will move towards atom A, and vice versa for the electron density of atom A. This delocalized electron-density may be represented as: (1) occupying molecular orbitals; or as (2) electron density dispersed along the A–B axis between the two atoms. Both models describe the occurrence of electron density between the two atoms, and this delocalized electron density may be described pragmatically as the chemical bond that is the binding force for both molecules and crystals.

Point-charge models

An arrangement of point charges is not stable (Earnshaw, Reference Earnshaw1842). Thus Earnshaw’s theorem prevents the existence of a stable completely ionic crystal. If such an arrangement were stable, a small perturbation of a charged particle from its equilibrium position would induce a restoring force on that particle, requiring the divergence of the electric field to be negative. Such a condition contradicts Gauss’s law which requires that the divergence of an electric field is zero in free space. This issue was finally resolved only with the advent of quantum theory which proposed the currently accepted modal of the atom in which electrons occupy orbitals around the nucleus. Where atoms approach each other, the valence electrons delocalize, the spherical symmetry of the atom is broken, the local charge of each atom (ion) no longer has spherical symmetry and cannot be considered as a point charge, and Earnshaw’s theorem no longer applies.

A priori bond strengths: definition and calculation

The term ‘bond strength’ is used widely to refer to bond strengths as evaluated by Pauling (Reference Pauling1929, Reference Pauling1960). In order to distinguish between the calculation of bond strengths as done here and as done by Pauling (Reference Pauling1929, Reference Pauling1960), I use the term ‘Pauling bond-strengths’. “A priori bond-strengths are calculated from the bond topology (structure connectivity) of a structure and the charges of all the ions in that structure using all aspects of the conservation law of the electric field.” I use the term a priori to emphasize that no Euclidian metrics are used in the calculation. The method of calculating a priori bond strengths follows that of Gagné et al. (Reference Gagné, Mercier and Hawthorne2018) for bond valences.

Table 4 shows the bond-strength table for C2/m CaM ²⁺Si₂O₆ pyroxenes in which the bond strengths are represented by the variables a to h with the constraints that the sums of the bond strengths exident from the cations and the anions are equal to the magnitudes of the formal charges of the cations and the anions. There are eight variables in Table 3 and six simultaneous equations. The solution of such systems of simultaneous equations is governed by the Rouché-Capelli theorem (Shafarevich and Remizov, Reference Shafarevich and Remizov2013; page 58): “A system of linear equations with n variables has a solution if and only if the rank of its coefficient matrix A is equal to the rank of its augmented matrix [A|b]” where, for the applications described here, [b] is the column matrix involving the magnitudes of the formal charges of the ions in the structure. The system of simultaneous equations derived from Table 4 does not have a solution according to the above condition; what else can we do?

Table 4.

A priori bond-strength table for C2/m pyroxenes CaM ²⁺Si₂O₆

The electric field is a vector field as a result of the positive and negative charges that are involved. Moreover, the graph of the crystal structure is a bipartite graph and any path through the graph of the structure involves the positive and negative bond-strengths associated with the edges of that path. As noted above, electric charge is subject to a conservation law as required by Noether’s first theorem and the gauge invariance of the electromagnetic field results in the conservation of electric charge. Thus if a path though the structure starts and ends at crystallographically identical vertices, the sum of the bond strengths traversed by that path is zero. Three such paths are shown in the C2/m CaM ²⁺Si₂O₆ pyroxene structure depicted in Fig. 4. Such paths provide additional equations for those from the bond-strength table (Table 4) to give a set of simultaneous equations that can be solved (Table 5). We may write these equations in matrix form (Table 6) and solve by Gaussian reduction. This may be done in several mathematical packages. Appendix 1 lists the input and results of this calculation using MATLAB^©.

Figure 4.

Crystal structure of C2/m CaM ²⁺Si₂O₆ pyroxenes showing three paths (labelled P1, P2 and P3) that start and end on crystallographically equivalent ions.

Table 5.

Bond-strength sums around each ion in C2/m pyroxenes CaM ²⁺Si₂O₆ and along bond paths starting and finishing on symmetrically equivalent ions

Table 6.

Matrix equation involving a priori bond-strengths around each ion and along bond paths in C2/m pyroxenes CaM ²⁺Si₂O₆

The solution for the a priori bond strengths is also given in Table 7 and the variation in the calculated Pauling and a priori bond strengths as a function of observed bond lengths is shown in Fig. 5. The a priori bond strengths are a non-linear function of the observed bond lengths despite the fact that the observed bond lengths were not used to calculate the bond strengths. This relation indicates that, unlike the Pauling model, the bond topology of the structure has a major effect on the variation of individual bond lengths within each coordination polyhedron. Note that the topological symmetry of the structure differs from the crystallographic symmetry. Thus in the C2/m CaM ²⁺Si₂O₆ pyroxene structure, there are three pairs of bonds: Ca–O3, Ca–O3′, ^[6]Mg–O1, ^[6]Mg–O1′ and Si–O3, Si–O3′ in which the bonds are identical with regard to bond topology but are not crystallographically identical. Thus the bond strengths for Ca–O3, Ca–O3′ are the same but the corresponding bond lengths are not equal, and the same for the pairs ^[6]Mg–O1, ^[6]Mg–O1′ and Si–O3, Si–O3′. Here, I take their a priori bond strengths and compare them with the mean of the crystallographic bond valences. The corresponding variation in the bond valences in diopside shows a 1:1 correlation with the a priori bond strengths (Fig. 6a) (R ² = 0.988) in contrast to the variation of Pauling bond strengths (Fig. 6b). One would not expect an exact 1:1 correlation of the a priori bond strengths and the corresponding bond-valences as the former are dependent solely on the charges of the ions and the bond topology of the structure whereas the latter are dependent on the charges plus other electronic properties of the ions and the structure. However, an unexpected relation emerges from this comparison: the electrostatic interactions of the constituent ions dominate the variation in bond strengths, and in turn bond lengths.

Figure 5.

Variation of a priori bond-strengths and Pauling bond-strengths as a function of bond-length for diopside (bond-length data from Clark et al., Reference Clark, Appleman and Papike1969).

Figure 6.

(a) Bond valence (bond-valence parameters from Gagné and Hawthorne, Reference Hawthorne2015), and (b) Pauling bond-strength as a function of a priori bond-strength for diopside.

Table 7.

Bond strengths from Pauling’s first rule, a priori values from the text, the corresponding bond lengths (Å) in diopside¹, and the calculated bond-valences² (vu)

¹ From Clark et al. (Reference Clark, Appleman and Papike1969); ² bond-valence parameters from Gagné and Hawthorne (Reference Hawthorne2015).

A priori bond strength: rule 1

The electric field in a crystal is a vector field; bond strengths from cations to anions are positive and bond strengths from anions to cations are negative. The incident bond strengths at all ion sites must equal the formal charges at those sites. Bond-strengths along non-degenerate paths between symmetrically equivalent ions in the structure must sum to zero. This leads to rule 1, the a priori bond-strength rule: “A priori bond strengths may be calculated for all bonds in a structure by constructing a bond-strength table that includes all bond strengths as unknown variables. The corresponding charge-conservation matrix can be solved for all the unknown bond strengths.”

Improvements and advances with the a priori bond-strength rule

1. (1) The resultant a priori bond strengths vary non-linearly with bond length. However, this feature emerges from the calculation, it is not part of the original model. Bond lengths are not a part of the bond-strength calculation; a priori bond strengths depend solely on the bond topology of the structure and the charges of the constituent ions.

2. (2) The theory of a priori bond strengths contrasts with bond-valence theory which uses empirically calculated ‘bond-valence parameters’ (see Gagné and Hawthorne, Reference Hawthorne2015) which collectively incorporate all ion properties (charge plus individual-ion properties such as the stereoactive lone-pair effects, etc.) and the observed bond lengths in a large number of crystal structures.

3. (3) A priori bond strengths are calculated in a different type of space than bond valences. A priori bond strengths are calculated in a set-theoretic space in which the distance metric involves the number of edges between vertices of a graph whereas bond valences are calculated in 3D (dimensional) Euclidian space. Topological symmetry is different from 3D-Euclidian geometrical symmetry: topologically identical atoms and bonds are not necessarily geometrically equivalent, and this may give real structures some additional degrees of freedom. It is these degrees of freedom that give many structures the ability to absorb strain which will also affect individual bond lengths.

4. (4) Although electrostatic interactions affect bond strengths (and, in turn, bond lengths) much more than has hitherto been realised, comparison of a priori bond strengths with bond valences allows the electrostatic effects to be separated from the remaining ion effects plus strain. This was done to a small degree by Gagné and Hawthorne (Reference Gagné and Hawthorne2020) but the potential of this approach is virtually untouched.

Bond-strength theory and bond-valence theory

Bond-valence theory evolved from Pauling’s ideas on bond strength, primarily by the observation that deviations from ideality of Pauling’s second rule correlate with variations in the associated bond lengths (e.g. Zachariasen, Reference Zachariasen1954; Baur, Reference Baur1974). In particular, Brown and Shannon (Reference Brown and Shannon1973) developed quantitative relations between bond lengths and the bond strengths, later denoted as bond valences to distinguish them from Pauling bond-strengths. Brown (Reference Brown, O’Keeffe and Navrotsky1981, Reference Brown2002, Reference Brown2016) went on to develop bond-valence theory from the initial bond-valence relations of Brown and Shannon (Reference Brown and Shannon1973). Bond-valence theory is based on three principal axioms: (1) the valence-sum rule; (2) the loop rule; and (3) the valence-matching principle.

Ion radii and ionic radii

The radii of ions have nothing to do with the ionic model of chemical bonding. In order to disassociate the radii used here from the traditional ionic model, I will use the term ‘ion radii’ for the radii given by Hawthorne and Gagné (Reference Hawthorne and Gagné2024) and the term ‘ionic radii’ where referring to previous sets of radii and arguments that used the ionic model.

Soon after solution of the crystal structure of halite (Bragg, Reference Bragg1913), Bragg (Reference Bragg1920) derived a set of ionic radii and showed that interatomic distances in crystals could be reproduced from the sum of the radii of the bonded atoms. Landé (Reference Landé1920) was the first to propose that anions (in lithium halogenides) are in mutual contact and this idea was used extensively over the next 50 years. Hüttig (Reference Hüttig1920) proposed that the coordination numbers adopted by cations in molecular complexes are determined by radius-ratio considerations, and Goldschmidt (Reference Goldschmidt1926, Reference Goldschmidt1927) used the arguments of Hüttig (Reference Hüttig1920) to predict coordination numbers for a wide range of cations in crystals. A consensus gradually developed that anions are (generally) larger than cations.

The sizes of atoms

The size of an atom depends on: (1) its oxidation state; (2) whether it is non-bonded (i.e. isolated) or whether it is bonded to other atoms; and (3) if bonded, several other factors involving its electronic structure and environment. A series of different radii for Na, Cl, Si and O are shown in Fig. 7 in order to give a sense of the relative magnitudes of these factors. Non-bonded atoms are drawn touching each other to more easily gauge differences in radii. Comparison of Figs 7a and 7b illustrate the diffuse nature of the valence electron-density in both Na⁰ and Cl⁰. The bonded radii (Fig. 7c,d) are smaller than the non-bonded ion radii (Fig. 7a,b). The difference between the neutral non-bonded radius for Si⁰ (Fig. 7a) and the empirical ionic radius for ^[4]Si⁴⁺ (Fig. 7c) emphasizes the diffuse nature of the valence electron-density in isolated neutral atoms.

Figure 7.

Comparison of the sizes (radii) of atoms: (a) calculated neutral non-bonded-atom radii; (b) calculated non-bonded-ion radii; (c) empirical ionic radii; (d) experimental bonded radii; values for (a) and (b) from Rahm et al. (Reference Rahm, Hoffmann and Ashcroft2017), values for (c) from Shannon (Reference Shannon1976), values for (d) from Gibbs et al. (Reference Gibbs, Ross, Cox, Rosso, Iverson and Spackman2013).

The radius of O^2–

The dominance of O^2– over other anions [e.g. (OH)^–, F^–, Cl^–, etc.] in dielectric minerals has led to a focus on the radius of O^2–. Most ionic radii were derived by subtracting a radius for O^2– from observed interatomic distances. Various experimentally based values have been used for the radius of O^2– but Shannon (Reference Shannon1976) used radii for O^2– that are dependent on the coordination number of O^2–. I shall review the experimental evidence for this coordination-number dependency of the radius of O^2– as this is a critically important issue in understanding the radii of ions.

Shannon and Prewitt (Reference Shannon and Prewitt1969) showed a correlation between <Si⁴⁺–O^2–> and the coordination of O^2– (for 4 data points) and developed coordination-dependent ionic radii for O^2–. Figure 8 shows the correlation of Brown and Gibbs (Reference Brown and Gibbs1969) between <Si⁴⁺–O^2–> and the mean anion-coordination number of the (Si⁴⁺O₄) tetrahedra in 46 silicate structures (but omitting several Na-silicates). Shannon (Reference Shannon1976) used this correlation to justify using radii for O^2– that vary as a function of coordination number from ^[2]1.35 Å to ^[8]1.42 Å.

Figure 8.

Variation in distance as a function of the mean coordination number of O^2– in each (SiO₄) tetrahedron; modified from Brown and Gibbs (1969).

Gagné and Hawthorne (Reference Gagné and Hawthorne2017b) examined the variation in mean bond length as a function of (1) anion-coordination number, (2) the electronegativity of the nearest-neighbour cations, (3) bond-length distortion, (4) the ionization energy of the nearest-neighbour cations, and (5) the differences in bond topology, for 55 well-ordered ion configurations. Variation in <Si⁴⁺–O^2–> distance as a function of the constituent-anion coordination number for 334 tetrahedra (Fig. 9) shows negligible effect (R ² = 0.056) of constituent-anion coordination number on variation in <Si⁴⁺–O^2–>. Figure 10 shows the variation of 49 <Si⁴⁺–O^2–> distances with an O^2– coordination of [4]. The range is twice that of the data of Brown and Gibbs (Reference Brown and Gibbs1969) for which [2] ≤ O^2– ≤ [4], the sum of the Shannon (Reference Shannon1976) radii: 0.26 + 1.38 = 1.64 Å is outside the range of values given by Brown and Gibbs (Reference Brown and Gibbs1969) and does not correspond to the average value of the data in Fig. 10: 1.631 Å. We must conclude that any effect of variation in constituent-anion coordination number on variation in <Si⁴⁺–O^2–> distances is not apparent in the data presently available. Moreover, these conclusions are unlikely to be changed by acquisition of additional data. A single value for the radius of O^2– is effective for (SiO₄) tetrahedra and for the 54 other ions tested by Gagné and Hawthorne (Reference Gagné and Hawthorne2017b).

Figure 9.

Mean ^[4]Si–O distance versus mean coordination number of the bonded oxygen atoms for SiO₄ coordination polyhedra; after Gagné and Hawthorne (Reference Gagné and Hawthorne2017b).

Figure 10.

Distribution of mean ^[4]Si–O distances for structures with a mean coordination number for O^2– of [4]. The range of mean Si–O values taken from the trend line on the graph of Brown and Gibbs (Reference Brown and Gibbs1969), and the sum of the ^[4]Si⁴⁺ and ^[4]O^2– radii from Shannon (Reference Shannon1976) are shown; reproduced from figure 7, Gagné and Hawthorne (Reference Gagné and Hawthorne2017b), under the Creative Commons CC-BY license.

Derivation of ion radii

Ion radii were derived by Hawthorne and Gagné (Reference Hawthorne and Gagné2024) from characteristic (grand mean) bond lengths for ordered crystal structures reported by Gagné (Reference Gagné2018) and Gagné and Hawthorne (Reference Gagné and Hawthorne2016, Reference Gagné and Hawthorne2018a, Reference Gagné and Hawthorne2018b, Reference Gagné and Hawthorne2020). These involve: (1) 135 ions bonded to oxygen in 459 configurations (on the basis of coordination number) using 177,143 bond lengths extracted from 30,805 ordered coordination polyhedra from 9210 crystal structures; and (2) 76 ions bonded to nitrogen in 137 configurations using 4048 bond lengths extracted from 875 ordered coordination polyhedra from 434 crystal structures. A single radius for O^2– was subtracted from the characteristic bond lengths for every ion configuration examined in the above-cited references. What value of ion radius for O^2– was used?

For mineral structures, particularly rock-forming minerals, many equations have been derived relating mean bond length and aggregate-ion radius. Most of these equations involve a small number of ion configurations: ^[4]Al³⁺, ^[4]Si⁴⁺, ^[6]Mg²⁺, ^[6]Fe²⁺, ^[6]Mn²⁺, ^[6]Al³⁺, ^[6]Fe³⁺, ^[6]Ti⁴⁺ coordinated by O^2–. We wished to develop ion radii that still work for these equations as the numerous existing relations between mean bond length and mean empirical ionic-radius could then still be used. We subtracted the Shannon radii for this small set of ions from their corresponding characteristic bond lengths (Appendix 2) to get a mean radius for O^2–: 1.366 Å that was subtracted from the characteristic bond lengths for all ion configurations to give the corresponding cation radii.

Characteristic bond lengths for other anions

Appendix 2 also lists the anion radii for O^2–, (OH)^–, F^–, Cl^– and N^3– (taken from Hawthorne and Gagné, Reference Hawthorne and Gagné2024). The differences Δ = r ^anion – r ^O2– are also listed, and the characteristic bond lengths involving these anions may be derived by adding Δ to the corresponding characteristic bond lengths for O^2–.

Mean bond length as a proxy variable for ion radius

Consider the relation between mean bond length D^AB of the ions A^{n +} and B^{m –} and ion radii r^A and r^B:

\begin{equation*}{D^{AB}} = {r^A} + {r^B}\end{equation*}

The ion radius of O^2–, r^B, is fixed (see above). Let us replace it by some constant K:

\begin{equation*}{D^{AB}} = {r^A} + \rm{K}\end{equation*}

Rearranging this equation: r ^A = D ^AB – K. Taking the first derivative with respect to D^AB:

\begin{equation*}{{d{r^A}} \mathord{\left/ {\vphantom {{d{r^A}} {d{D^{AB}}}}} \right. } {d{D^{AB}}}} = {{d{D^{AB}}} \mathord{\left/ {\vphantom {{d{D^{AB}}} {d{D^{AB}}}}} \right. } {d{D^{AB}}}} - {{d\rm{K}} \mathord{\left/ {\vphantom {{dK} {d{D^{AB}}}}} \right. } {d{D^{AB}}}} = 1 - 0 = 1\end{equation*}

Thus, r^A scales as D^AB, and D^AB is thus a ‘proxy variable’ for r^A. Published values of ion radii (and ionic radii) do not represent the radii of ions in crystals.

Uses of ion radii

There are two types of use for ion radii: (1) those which compare the radii of cations with the radii of anions; (2) those which compare the radii of different cations or the radii of different anions. Methods belonging to type 1 use the relative sizes of cation and anion radii to predict local arrangements. As is apparent from the above discussion, I am not aware of experimental derivation of the relative sizes (radii) of cations and anions directly from experimental electron-density maps and on the same scale as the recent derivations of ionic (Shannon, Reference Shannon1976) and ion (Hawthorne and Gagné, Reference Hawthorne and Gagné2024) radii.

Pauling’s first rule states the following: “A coordinated polyhedron of anions is formed about each cation, the cation-anion distance being determined by the radius sum and the ligancy of the cation by the radius ratio [my bold font]”. Pauling’s first rule is the paradigm of type 1 methods as it attempts to predict coordination number from the radius ratio of the constituent cation and anion. Gagné (Reference Gagné2018) and Gagné and Hawthorne (Reference Gagné and Hawthorne2016, Reference Gagné and Hawthorne2018a, Reference Gagné and Hawthorne2018b, Reference Gagné and Hawthorne2020) provide data on cation-coordination numbers with regard to O^2– for almost all atoms of the periodic table and this provides a good test for Pauling’s first rule. The geometric underpinnings of this rule are shown in Fig. 11 for octahedral and cubic coordination and the values of the radius ratio for spheres (spherical ions) which fit into holosymmetric polyhedra with the spheres just touching each other. If the radius ratio is close to that of the octahedron, this coordination is predicted, and likewise for the cube. If the radius ratio is intermediate between the two values (as denoted by the region shaded mauve in Fig. 11), then the ions may adopt either coordination. There are two issues concerning this model. (1) All cations can assume only two coordinations: [6] and [8] in Fig. 11. (2) Figure 11 omits the coordination number [7] (as did Pauling’s model) which is common for medium-sized divalent cations, e.g. Ca²⁺ which has 211, 287 and 519 polyhedra for [6]-, [7]- and [8]-coordination in the data of Gagné and Hawthorne (Reference Gagné and Hawthorne2016), violating condition 1 above. Figure 12 shows the range of coordination numbers for all cations when bonded to O^2– as a function of Lewis acidity (Gagné and Hawthorne, Reference Gagné and Hawthorne2017a) of the cation. Cations in accord with the radius-ratio rule have only two coordination numbers and fall within the yellow region of Fig. 12 and are only ∼20% of the data in the figure; they are far outnumbered by the cations that do not accord with Pauling’s first rule in this regard. Either: (1) the geometrical model underlying Pauling’s first rule is wrong; (2) cation and anion radii vary with chemical composition and structure type, or (3) both, indicating that ion radii cannot be used in such a predictive manner.

Figure 11.

A sketch illustrating the geometrical basis of Pauling’s first rule. The mauve region denotes where R_cation/R_anion is ∼equal distances from the ideal values for octahedral and cubic coordination.

Figure 12.

Variation in range of coordination number as a function of Lewis acidity for 135 cations; the yellow-shaded area denotes the maximum extent of data according to Pauling’s radius-ratio rule. Modified from Gibbs et al. (Reference Gibbs, Hawthorne and Brown2022).

Type 2 methods use the relative sizes (radii) of cations and of anions but they do not rely on the radius ratio of cations and anions. Above, I showed that mean bond length is a proxy variable for ion radius and hence we are replacing what we do not know (the true value of the cation radius) with a proxy variable that we do know very accurately: mean bond length. In crystal structures of solid solutions between two or more ions at a particular site in a structure, relations between mean constituent ion radius and mean bond length can be used to derive site occupancies for two ions with similar scattering factors, e.g. Si⁴⁺ and Al³⁺, and for more than two ions, e.g. Mg²⁺, Al³⁺ and Fe³⁺, where two ions have similar scattering factors and a third has a significantly different scattering factor. Figure 13a shows linear relations between mean bond length, <M(1)–O>, and constituent-cation radius, ^{M (1)}r, for ordered olivine structures (red circles) and for ordered calcium pyroxenes (green circles). Figure 13b shows linear relations between mean bond length, <M(1)–O>, and characteristic mean bond length, <<M ²⁺–O>>, for ordered olivine structures (red circles) and for ordered calcium-pyroxene structures (green circles): the plots in Figs 13a and 13b are identical. This means that we may dispense with ion radii and replace them by ‘characteristic mean bond lengths’ with no loss of accuracy and utility, and with the knowledge that we are dealing with real quantities: mean bond lengths.

Figure 13.

<M(1)–O> in olivines (red circles): M ²⁺₂SiO₄, where M(1) = Ni, Mg, Co, Fe, Mn, Ca; and Ca-dominant clinopyroxenes (green circles): CaM ²⁺Si₂O₆, where M(1) = Mg, Fe, Mn. (a) <M(1)–O> versus ^{M (1)}r; and (b) <M(1)–O> versus <<^[6]M ²⁺–O^2–>> (characteristic distances for inorganic structures).

Ion radius: rule 2

Ion radii derived from experimental bond lengths do not represent the radii of ions in crystals as we cannot objectively divide bond lengths into the radii of the constituent ions. This leads to rule 2, the ion-radius rule: “Ratios of ion radii have no physical meaning whereas sums of ion radii can be used in crystal chemistry (e.g. correlating site occupancies with observed mean bond lengths)”.

Theoretical bonded radii

The electron density in a crystal can be calculated quantum-mechanically by imposing periodicity on its wave functions. The calculated electron-density shows a series of stationary points, known as saddle points, at which the electron density is at a minimum with respect to some directions and at a maximum with respect to other directions (Runtz et al., Reference Runtz, Bader and Messer1977). Saddle points occur on or near lines joining the nuclei of pairs of atoms that are (thought to be) bonded to each other. A gradient path is defined as any line of steepest descent that terminates at a saddle point. The two gradient paths which originate at the same saddle point and end at each of two nuclei define a ‘bond path’, and the included saddle point is called a ‘bond critical point’ (Bader, Reference Bader2009). According to Bader (Reference Bader2009), a bond path is not a chemical bond, it is an indicator of chemical bonding.

Gibbs et al. (Reference Gibbs, Boisen, Beverly and Rosso2001, Reference Gibbs, Ross, Cox, Rosso, Iverson and Spackman2013, Reference Foroutan-Nejad, Shahbazian and Marek2014) used this approach to show that: (1) calculated bonded radii of both individual cations (Fig. 14a) and anions (Fig. 14b) are not fixed but vary with interatomic distance.Footnote ¹ There are large differences between the calculated bonded radii and the ionic radii of Shannon (Reference Shannon1976). This is illustrated in Fig. 15 which shows the experimental electron-density map for coesite, SiO₂, from Gibbs et al. (Reference Gibbs, Ross, Cox, Rosso, Iverson and Spackman2013) with the bond-critical points marked and the ion sizes based on the radius of Shannon (Reference Shannon1976) for ^[2]O^2– and the radius of O^2– based on the position of the bond-critical points. The radii values are 1.35 Å and 0.89 Å with a difference of 0.46 Å. Ion and ionic radii are inverse proxy variables for characteristic bond length and hence we do not expect them to concur with real ion radii (which we do not know). Is the O^2– radius in Fig. 15 the true radius of O^2– ? This we do not know as the bond-critical points shown in Fig. 15 are not calculated bond-critical points but are marked on the figure at the minimum values of experimental electron density along chemical bonds or bond-critical paths depending on your point of view.

Figure 14.

(a) Variation in calculated bonded radii for second- (red), third- (green) and fourth- (yellow) row cations bonded to O^2– as a function of experimental <M–O> bond-lengths; (b) variation in calculated bonded radii for O^2– bonded to second- (red), third- (green) and fourth- (yellow) row cations; data for silicate and oxide structures (modified from Gibbs et al., Reference Gibbs, Ross, Cox, Rosso, Iverson and Spackman2013).

Figure 15.

Experimental electron-density section in coesite through two Si atoms and one bridging O atom (from Gibbs et al., Reference Gibbs, Ross, Cox, Rosso, Iverson and Spackman2013). The bond-critical points are marked as small red circles, the O atom as defined by the bond-critical points is shown as a yellow circle and the O atom, as defined by the Shannon radius of 1.35 Å, is shown by the pale green area bounded by the thick black circle.

The surprising feature of Fig. 14 is the range in size of O^2– (Fig. 14b): 0.65 < r ^O2– < 1.4 Å. This result is in stark contrast to the results of crystal-chemical analysis above: O^2– has a fixed radius of unknown value. Moreover, it is difficult to reconcile the common occurrence of close packing of ions in crystal structures with the calculated range of r ^O2–: 0.65–1.40 Å. QTAIM (Quantum Theory of Atoms In Molecules, Bader, Reference Bader1990) is the basis of this approach. However, QTAIM has engendered considerable controversy and alternative interpretation (e.g. Poater et al., Reference Poater, Sol and Bickelhaupt2006; Foroutan-Nejad et al., Reference Foroutan-Nejad, Shahbazian and Marek2014; Shahbazian, Reference Shahbazian2018; Jabłoński, Reference Jabłoński2019, Reference Jabłoński2023). In view of these controversies, the predictions of QTAIM must be considered uncertain. The sizes of ions need to be measured from experimental electron densities in crystals in order to resolve these radically different models.

Interatomic distances and coordination numbers

Gagné and Hawthorne (Reference Gagné and Hawthorne2016, Reference Gagné and Hawthorne2018a, Reference Gagné and Hawthorne2018b, Reference Gagné and Hawthorne2020) and Gagné (Reference Gagné2018, Reference Gagné2021) described the distribution of bond lengths to O^2– and N^3– in ∼9650 ordered crystal-structures refined since 1975 and listed in the Inorganic Crystal Structure Database (ICSD). From this work, we may derive the ‘characteristic bond length’ (the grand mean bond length) for each ion configuration; these are listed in Appendix 2. We may also derive the ‘characteristic coordination number’ (the weighted grand mean coordination number of a cation in all its ion configurations); these are listed in Appendix 3 together with the corresponding Lewis acidities and the coordination numbers for each specific cation, referred to as ‘individual coordination numbers’.

The Lewis acidity of cations

Gagne and Hawthorne (Reference Gagné and Hawthorne2017a) derived the Lewis acidity for most cations of the periodic table as the ion charge divided by the characteristic coordination number; I shall define these values as ‘characteristic Lewis acidities’. These values are particularly useful where the details of a structure are not known and the characteristic Lewis acidity gives the most probable Lewis acidity of a structure that corresponds to the most probable cation coordination from a statistical perspective. However, the characteristic Lewis-acidity of a cation involves the characteristic coordination number. Any individual structure may adopt a different coordination number (and hence a different Lewis acidity) within the observed range of coordination numbers for that cation (Appendix 3).

Lewis basicity of anions

In general, bond strengths involving simple anions, e.g. O^2–, have a dispersion too large to define a useful Lewis basicity. Thus in MgO, the O^2– anions are [6]-coordinated by Mg²⁺ and have a Lewis basicity of 0.33 e whereas (Cr⁶⁺O₃)_∞ (Stephens and Cruickshank, Reference Stephens and Cruickshank1970) has one [1]-coordinated O^2– with a Lewis basicity of 2 e. This range of bond strengths (0.33–2.00 e) means that we cannot define a useful Lewis basicity for O^2–. However, the situation for oxyanions is different. In a complex oxyanion such as (SO₄)^2– (Fig. 16a), the S⁶⁺ cation provides 1.5 e to each coordinating O^2– ion, each of which needs an additional 0.5 e from the other bonded cations. If the coordination number of O^2– is [n], then the average strength of the bonds to O^2– (exclusive of the S⁶⁺–O^2– bond) is 0.5/(n–1) e; where n = 2, 3, 4 or 5, the mean bond strengths to O^2– are 0.50, 0.25, 0.17 or 0.11 e, respectively. These values are quite tightly constrained (0.50–0.11 e) compared to those for O^2– (0.33–2.00) e and we may calculate a useful Lewis basicity. Appendix 3 lists characteristic Lewis acidities for cations and Appendix 4 lists Lewis basicities for common complex anions.

Figure 16.

Bond-strength matching for (a) Na₂SO₄ and (b) Na₄SiO₄. Characteristic values of Lewis acidity are taken from Gagné and Hawthorne (Reference Gagné and Hawthorne2017a).

Bond-strength matching: rule 3

Where two ions form a bond, the strength of the bond from the cation to the anion is controlled by the Lewis acidity of that cation, and the strength of the bond from the anion to the cation is controlled by the Lewis basicity of that anion. As the bond from the cation to the anion is the same bond as that from the anion to the cation, the Lewis acidity and the Lewis basicity of the constituent ions must be approximately the same for that bond to form. This leads to rule 3, the bond-strength-matching rule: “Stable structures will form where the Lewis acidity of the cation closely matches the Lewis basicity of the anion”.

Rule 3, the bond-strength-matching rule, is the most important and powerful idea in crystal chemistry (Hawthorne, Reference Hawthorne2012, Reference Hawthorne2015). Where structures are known, the availability of bond lengths and site occupancies allows us to interpret details of that known structure. Where a structure is not known, bond-strength matching allows us to test the stability of possible compounds (in terms of whether they can exist or not), which moves us from a posteriore to a priori analysis.

I will give two simple examples (taken from Hawthorne, Reference Hawthorne1994) to illustrate this rule. Consider the composition Na₂SO₄. The Lewis basicity of the (SO₄) group is 0.17 e and the Lewis acidity of Na is 0.17 e (Appendices 3 and 4). The Lewis basicity of the anion matches the Lewis acidity of the cation (Fig. 16a), the bond-strength-matching rule is satisfied, and thénardite (Hawthorne and Ferguson, Reference Hawthorne and Ferguson1975), Na₂SO₄, is stable. Consider the composition Na₄SiO₄. The Lewis basicity of the (SiO₄) group is 0.33 e and the Lewis acidity of Na is 0.17 e (Appendices 3 and 4). The Lewis basicity of the anion does not match the Lewis acidity of the cation (Fig. 16b), the bond-strength-matching rule is not satisfied, and Na₄SiO₄ is not a mineral (and is very unstable as a synthetic structure).

Ways in which bond-strength matching may be optimized

There are ways in which the bond-strength-matching rule may be accommodated: (1) exploitation of differences in topological and crystallographic symmetry; (2) unusual bond lengths; (3) unusual coordination numbers; (4) hydration of coordination polyhedra; (5) hydroxylation of oxyanions; and (6) polymerization and hydroxylation of coordination polyhedra. Below, I consider these mechanisms in more detail and give examples.

Hydration of coordination polyhedra: Lewis acidity

In the atom arrangement in Fig. 17a, the anion receives one bond of strength v e from the cation M. In Fig. 17b, the donor O^2– ion of the (H₂O) group receives a bond strength of v e from the cation; the bond-strength requirements of the O^2– ion are satisfied by two O^2––H⁺ bonds of strength (1 – v/2) e. Each H⁺ ion forms a hydrogen bond with an S anion to satisfy its own bond-strength requirements, and the S anion thus receives a bond strength of v/2 e, one half (Fig. 17b) of what it received where it bonded directly to the M cation (Fig. 17a). The (H₂O) group is called a ‘transformer (H₂O)’ as it transforms one bond into two bonds of half the bond strength (Hawthorne and Schindler, Reference Hawthorne and Schindler2008). In Fig. 17c, two cations bond to an (H₂O) group which bonds to two anions. The O^2– ion receives a bond strength of 2v e from the cations, and local electroneutrality at this O^2– ion is satisfied by two O–H bonds of strength (1 – v) e. Each H⁺ ion forms a hydrogen bond with a neighbouring acceptor anion, which receives the same bond strength (v e, Fig. 17c) as if it were bonded directly to one M cation (Fig. 17a). The (H₂O) group does not affect the strength of the bond from the M cation to the anion S and is a ‘non-transformer (H₂O)’. If (H₂O) is involved in a hydrogen-bond network, the (H₂O) group usually accepts two hydrogen bonds from an adjacent (H₂O) group and donates two hydrogen bonds to an adjacent (H₂O) group; these (H₂O) groups are ‘non-transformer’ and just propagate bonds to more distant anions, as is the case where the (H₂O) group is bonded to two cations (Fig. 17c). Consider the atomic arrangement in Fig. 17d in which the (H₂O) accepting the bond from the M cation hydrogen bonds to two adjacent transformer (H₂O) groups which then hydrogen bond to the anion S with a bond-strength one quarter that of the initial M–(H₂O) bond: v/4 e (Fig. 17d). As we shall see below, transformer and non-transformer (H₂O) groups play a major role in greatly enlarging the number of possible minerals.

Figure 17.

The bond-strength structure around (H₂O) as a function of local bond-topology: (a) a cation, M (green) bonded to an anion, S (violet) with bond strength v e; (b) a cation bonded to an (H₂O) group (O = yellow, H = red) with a bond strength of v e; the H⁺ ions hydrogen-bond to the anions S with bond strengths of v/2 e per bond; (c) two cations bonded to an (H₂O) group with bond strengths of v e per bond; the H⁺ ions hydrogen-bond to the anions S with bond strength of v e per bond; (d) a cation bonded to an (H₂O) group with a bond strength of v e; the H⁺ ions hydrogen-bond to other (H₂O) groups which hydrogen bond to the anions S with bond strengths of v/4 e per bond. Modified from Hawthorne et al. (Reference Hawthorne, Hughes, Cooper and Kampf2022).

Hydroxylation of oxyanions: Lewis basicity

Where a H⁺ ion is attached to a ligand of an oxyanion, this produces an acid oxyanion, e.g. (BO₃)^3– + H⁺ → [BO₂(OH)]^2–. This process also perturbs the bond strengths in the oxyanion. The ideal bond-strengths in the (BO₃)^3– group are 3^–/3 = 1 e. Taking the coordination number of O^2– as [4], each anion needs an additional three bonds of strength 0.333 e to satisfy its local bond-strength requirements, i.e. produce local electroneutrality. Where a single ligand of the (BO₃)^3– group is hydroxylated, an additional bond is formed at one of the O^2– anions, converting it into an (OH)^– group. The Lewis acidity of H⁺ when bonded to a donor O^2– is 0.87 e (Gagné and Hawthorne, Reference Gagné and Hawthorne2017a); the acid anion receives a net bond-strength of 1.0 + 0.87 = 1.87 e and requires an additional bond of strength 0.13 e. The remaining two O^2– ions each requires 3 bonds of strength 0.333 e which means that the [BO₂(OH)]^2– needs 7 bonds of strength 2.13/7 = 0.304 e. The [BO(OH)₂]^2– needs 3 + 1 × 2 = 5 bonds of strength 1.26/5 = 0.252 e. These results are summarized in Fig. 18a,b,c and similar results are shown for (^[4]BΦ₄), Φ = O^2–, (OH)^– in Fig. 18d,e,f. Figure 19 shows the effect that the coordination number of the central cation has on the Lewis basicity of the oxyanion. The triangular coordination of the (BΦ₃) causes a much more pronounced decrease in Lewis basicity with increasing hydroxylation than is the case for the (BΦ₄) tetrahedra.

Figure 18.

Successive hydroxylation of (a–c) (BΦ₃) groups and (d–f) (BΦ₄) groups with changes in chemical composition and Lewis basicity.

Figure 19.

Lewis basicity in (BΦ₃) and (BΦ₄) groups as a function of successive hydroxylation.

Polymerization and hydroxylation of coordination polyhedra

Detailed investigation of post-mining secondary minerals in the Uravan Mineral Belt of roll-front uranium-vanadium deposits in western Colorado and eastern Utah, pioneered by Tony Kampf, has resulted in the discovery of many new vanadate minerals of the pascoite family (Table 8). These minerals involve decavanadate polyanions, the simplest form of which is the [V⁵⁺₁₀O₂₈]^6– group (Fig. 20). Inspection of Table 8 shows that [V⁵⁺₁₀O₂₈]^6– is by far the most common form of decavanadate in these minerals but more complex (sensu lato) forms also occur. The general form of the chemical formulae consists of a decavanadate polyanion (shown in bold in Table 8) which is the ‘structural unit’ (Hawthorne, Reference Hawthorne1983, Reference Hawthorne2014), and lower-valence cations plus (H₂O) groups that form the ‘interstitial complex’. Details of the [V⁵⁺₁₀O₂₈]^6– decavanadate group are shown in Fig. 20. As indicated in Fig. 20, there is considerable variation in coordination number for the O^2– ions of the decavanadate group, and this variation plays a key role in the adaptability of the decavanadate group to a wide variety of interstitial constituents.

Figure 20.

The [V₁₀O₂₈]^6– decavanadate polyanion. V atoms = black circles, [1]-coordinated O atoms = red circles, [2]-coordinated O atoms = blue circles, [3]-coordinated O atoms = green circles, [6]-coordinated O atoms = yellow circles, V–O_vanadyl bonds = thick black line, V–O_trans bonds = thin black line, V–O_equatorial bonds = grey shaded line.

Table 8.

Minerals of the pascoite family

The decavanadate group consists of ten (VO₆) octahedra that share edges to form a closely packed polyanion of maximum symmetry 2/m2/m2/m. The [V⁵⁺₁₀O₂₈]^6– polyanion shown in Fig. 20 may be decorated by H⁺ ions and modified by replacement of (usually one) V⁵⁺ ion by a different ion (e.g. V⁴⁺, Ti⁴⁺), but the undecorated [V⁵⁺₁₀O₂₈]^6– polyanion is by far the most common structural unit in the pascoite-family of minerals (Table 8). Due to the compact nature of the group, there is wide variation in the coordination numbers of the various O^2– ions (Fig. 20). In the centre of the group, there are two [6]-coordinated O^2– ions (shown as yellow circles in Fig. 20); they are completely surrounded by V⁵⁺ ions and cannot bond to any ions external to the decavanadate group. In addition, there are four [3]-coordinated O^2– ions (shown in green), fourteen [2]-coordinated O^2– ions (shown in blue), and eight [1]-coordinated O^2– ions (shown in brown) (Fig. 20). The colour scheme of Fig. 20 indicates only the topological commonalities of the different O^2– ions in the different decavanadate groups.

Binary structure representation

We can divide a mineral into two parts: for example, thénardite (Hawthorne and Ferguson, Reference Hawthorne and Ferguson1975), Na₂SO₄, is divided into a ‘structural unit’, (SO₄)^2–, and an ‘interstitial complex’, Na⁺. Using the bond-strength-matching rule as we did above involves direct one-on-one ion interactions. When considering a more complicated mineral such as pascoite (Hughes et al., Reference Hughes, Schindler and Francis2005), Ca₃(H₂O)₁₇[V⁵⁺₁₀O₂₈], direct one-to-one ion interactions are less straightforward to define. We may overcome this problem as shown in Fig. 21. We define the Lewis basicity of the [V⁵⁺₁₀O₂₈]^6– structural unit as the average strength of an anion–cation bond with the interstitial complex and we define the Lewis acidity of the Ca₃(H₂O)₁₇ interstitial complex as the average strength of a cation–anion bond with the structural unit. This is a mean-field equivalent of the bond-strength-matching rule and is designated as the rule of correspondence of Lewis acidity–Lewis basicity (Hawthorne and Schindler, Reference Hawthorne and Schindler2008). Details on how to calculate Lewis basicities and Lewis acidities for complicated oxide and oxysalt minerals are given by Hawthorne and Schindler (Reference Hawthorne and Schindler2008) and specific details for decavanadates are given by Hawthorne (Reference Hawthorne2025). Table 9 lists the Lewis basicities for the various varieties of the decavanadate ion and Table 10 lists the Lewis acidities for the interstitial complexes of Table 8. The Lewis basicities for the decavanadates vary over the range 0.054 to 0.135 e and Lewis acidities for the interstitial complexes vary over the range 0.054 to 0.500 e.

Figure 21.

The rule of correspondence of Lewis acidity–Lewis basicity applied to the structure of pascoite; modified from Hawthorne (Reference Hawthorne2025).

Table 9.

Lewis basicities for decavanadate polyanions

Table 10.

Range in Lewis acidity for hydrated interstitial cations

What check do we have on these values? They are calculated theoretically but validation requires experimental verification. Decavanadates and other vanadate species are soluble in aqueous solution, forming protonic acids at near-ambient temperatures, and the stable species are very sensitive to pH (Fig. 22). According to the rule of correspondence of Lewis acidity-basicity (see above), the Lewis basicity of the decavanadate polyanions should correlate with the pH of the aqueous solution in which they have their maximum stability. The Lewis basicities calculated for the polyanions [H₃V⁵⁺₁₀O₂₈]^3–, [H₂V⁵⁺₁₀O₂₈]^4–, [H₁V⁵⁺₁₀O₂₈]^5– and [V⁵⁺₁₀O₂₈]^6– are linearly correlated with the pH values of the aqueous solutions at the maximum stabilities of these polyanions (Fig. 23), suggesting that these calculations are valid.

Figure 22.

Concentration of aqueous vanadate species as a function of pH at 0.200 M, with the concentrations of the decavanadate species shown in red. Reproduced from figure 2, Aureliano et al. (Reference Aureliano, Gumerova, Sciortino, Garribba, McLauchlan, Rompel and Crans2022), under the Creative Commons CC-BY license.

Figure 23.

Lewis acidity versus pH at maximum stability for decavanadate species in aqueous solution.

Figure 24 shows the variation in mean coordination number for simple cations as a function of Lewis acidity (red circles) and the Lewis basicity for the vanadate polyanions (green circles at bottom right). The dashed black vertical line AB shows the limit of the rule of correspondence between Lewis basicity and Lewis acidity. The monovalent cations Cs⁺ to Tl⁺ plus (NH₄)⁺ have Lewis acidities in the same range as the Lewis basicities of the vanadate polyanions and hence satisfy the rule of correspondence of Lewis acidity-basicity. The Lewis acidity for Na⁺ lies slightly outside the range of Lewis basicity for the decavanadate polyanions. However, for simple cations, the Lewis acidity of a cation reflects the grand-mean bond length, and a cation can decrease its effective Lewis acidity by increasing its coordination number above its average value. Na⁺ has a characteristic coordination-number of 6.31 (Appendix 3) and a Lewis acidity of 1/6.31 = 0.158 e. Increasing its coordination number to [8] will change the Lewis acidity to 1/8 = 0.125 e in accord with the rule of correspondence of Lewis acidity-basicity. Thus Na⁺ is the most common interstitial cation in decavanadate minerals and has coordination numbers ≥ [8]. It is apparent from Table 9 that protonated vanadate polyanions have the lowest Lewis basicities, and synthetic protonated vanadates such as Cs₄(H₂O)₄[H₂V₁₀O₂₈] (Rigotti et al., Reference Rigotti, Rivero and Castellano1987) and Rb₄Na(H₂O)₁₀[HV₁₀O₂₈] (Yakubovich et al., Reference Yakubovich, Steele, Yakovleva and Dimitrova2015) contain the least-acid cations, i.e. Cs⁺ and Rb⁺.

Figure 24.

Mean observed coordination number for 91 cations as a function of their Lewis acidity (red circles). The Lewis acidity for NH₄⁺, 0.109 vu (from Hawthorne et al., Reference Hawthorne, Hughes, Cooper and Kampf2022 from the results of García-Rodríguez et al., Reference García-Rodríguez, Rute-Pérez, Piñero and González-Silgo2000) overlaps that of K⁺, 0.110 vu. The green circles indicate the Lewis basicities of the decavanadate units listed in Table 9. The broken black line denotes the maximum value of the green circles.

Inspection of Table 8 shows that Ca²⁺, Mg²⁺ and Al³⁺ are common interstitial cations in decavanadate minerals and yet according to Fig. 24, their Lewis acidities lie far outside the region conforming to the rule of correspondence of Lewis acidity-basicity. How can this happen? It happens because of the presence of transformer (H₂O) in these minerals. The effect of transformer (H₂O) on the acidities of the interstitial cations is shown in Fig. 25. The ranges of Lewis acidities of hydrated interstitial cations ^[6]M ⁺ to ^[6]M ³⁺ (from Table 10) are shown by the yellow (coordinated by single transformer (H₂O) groups) and green (coordinated by double transformer (H₂O) groups) boxes at the top of Fig. 25. The dashed line CD shows the limit of the rule of correspondence between Lewis basicity and Lewis acidity for interstitial cations coordinated single- and double-transformer (H₂O) groups. It is apparent that this mechanism greatly expands the compositional range for decavanadate minerals, and will do so for all other oxysalt minerals.

Figure 25.

Ranges in Lewis acidity of complex interstitial cations with one layer of coordinating transformer (H₂O) groups (yellow boxes) and complex interstitial cations with two layers of coordinating transformer (H₂O) groups (green boxes). The green circles indicate the Lewis basicities of the decavanadate units listed in Table 9. The broken black lines denote the maximum Lewis acidity of simple cations that can combine with (H₂O) to form complex cations that can combine with decavanadate in accord with the rule of correspondence of Lewis acidity–Lewis basicity.

Rule 1: The a priori bond-strength rule

The electric field in a crystal is a vector field; bond strengths from cations to anions are positive and bond strengths from anions to cations are negative. The incident bond strengths at all ion sites must equal the formal charges at those sites. Bond strengths along non-degenerate paths between symmetrically equivalent ions in the structure must sum to zero. This leads to rule 1, the a priori bond-strength rule: “A priori bond strengths may be calculated for all bonds in a structure by constructing a bond-strength table that includes all bond-strengths as unknown variables. The corresponding charge-conservation matrix can be solved for all the unknown bond-strengths.”

Rule 2: The ion-radius rule

Ion radii derived from experimental bond lengths do not represent the radii of ions in crystals as we cannot objectively divide bond lengths into the radii of the constituent ions. This leads to rule 2, the ion-radius rule: “ratios of ion radii have no physical meaning whereas sums of ion radii can be used in crystal chemistry (e.g. correlating site occupancies with observed mean bond-lengths).”

Rule 3: The bond-strength-matching rule

The characteristic Lewis acidity of a cation may be defined as its characteristic bond strength, which is equal to its charge/characteristic-coordination-number. The Lewis basicity of an anion can be defined as the characteristic strength of the bonds formed by the anion. This leads to rule 3, the bond-strength-matching rule: “Stable structures will form where the Lewis acidity of the cation closely matches the Lewis basicity of the anion”.

These rules are particularly important for Mineralogy as most minerals show some sort of disorder: e.g. long-range isovalent and coupled heterovalent solid-solution, order/disorder of ions over different sites in a structure, short-range (local) coupled heterovalent substitutions, and as a result, quantum-chemistry methods cannot be used on most minerals. Quantitative calculations on such materials require mean-field methods, i.e. the rules of Crystal Chemistry.