1 THE BOND VALENCE MODEL AS A TOOL FOR TEACHING INORGANIC CHEMISTRY: THE IONIC MODEL REVISITED I.D.Brown Brockhouse Institute for Materials Research McMaster University Hamilton, Ontario, Canada L8S 4M1
[email protected] KEYWORDS: Inorganic Chemistry, Bonding Theory, Materials Science, Solid State Chemistry, Theoretical Chemistry, Water/Water Chemistry
Abstract The ionic model is shown to give a good description of most inorganic materials, such as salts, ceramics and minerals, regardless of the covalent or ionic character of their bonds. The virtue of the model is its ability to treat chemical bonding using simple electrostatic theory, all the quantum mechanical effects being contained in a short range potential which is treated empirically. By exploiting the properties of the electrostatic field, a rigorous but simple and intuitive bond model is developed (the bond valence model).
This paper shows how the model can be used to explore the structural, chemical and physical properties of inorganic compounds, including their stability and solubility. The on-line version of the paper explores further application, showing how the model can be used to understand hydrogen bonding, the factors that determine coordination number and the unusual chemistry and physics of the new high tech inorganic solids,
The simplicity of the model, the insights it provides without the need for extensive computation, and its wide range of applicability make it particularly useful in teaching as well as research.
2 Contents1
1. Introduction 2. The electrostatic bond model 3. Bonding strength and bonding strength matching 4. Aqueous solubility 5. Bond valences and bond lengths OL-6. Coordination numbers OL-7. Strained bonds OL-7.1. Electronically driven strains OL-7.2. Geometrically strained structures OL-7.3. Steric strains: Hydrogen bonds OL-7.4. Lattice induced strains OL-8. Applications of structure related properties OL-8.1. Ferroelectricity OL-8.2. Superconductivity OL-8.3. Ionic conductivity 9. Epilogue 10. Acknowledgements Appendix Solution of the Network Equations
1
The full version of this article is available only in JCE On-line.. Sections labelled OL- only appear in the on-line
version, other sections appear in whole or in part in the printed version. Tables, figures and equations have the same numbers in both versions resulting in some figures and equations not appearing in the printed version. References are numbered differently in the two versions.
3 1. Introduction The concepts of atoms and bonds were introduced into chemistry in the early nineteenth century to provide a way of understanding and organising the many compounds that were being discovered, particularly those containing carbon. According to this model, a compound was seen as a collection of atoms connected by bonds to form a molecular network. At this time the nature of neither the atoms nor the bonds was known, but the model was so successful that chemists soon accepted their reality. The development of atomic physics and x-ray diffraction in the early twentieth century confirmed the existence of atoms as well as their geometric arrangements in organic molecules, but the nature of the chemical bond has remained elusive and continues to be the subject of much discussion.
In contrast to organic compounds, inorganic compounds form extended crystalline arrays of atoms that do not follow the established rules of organic chemistry. In 1918, to account for this different behaviour, Born and Landé (1) proposed the Ionic Model, in which the crystals were assumed to be held together by the Coulomb attraction between atoms carrying opposite charges. They postulated that the total energy of the crystal could be calculated as the sum taken over all pairs of atoms of interatomic potentials of the form
Uelectrostatic + Ushortrange
(1)
where the first term is the electrostatic potential that holds the crystal together and the second term is a short range potential that describes the repulsion that keeps the atoms apart. The electrostatic term was calculated by Madelung for NaCl (2) using classical electrostatics. The repulsive term was, in the absence of a proper theoretical model, approximated by an inverse power law with empirically fitted parameters. While this model had some success in predicting the properties of simple solids like NaCl, the calculations were too difficult to perform for more complex structures.
In 1929 Pauling developed a more descriptive approach to the ionic model in his empirical `Principles
4 Determining the Structure of Complex Ionic Solids' (3). Pauling's rules state that positively charged ions (cations) surround themselves with negatively charges ions (anions) in such a way as to preserve local charge neutrality. This latter condition he expressed in terms of the electrostatic valence principle in which the charge, V, of a cation is divided between the N bonds that it forms to give each bond an electrostatic bond strength of V/N. He showed that the sum of the bond strengths received by each anion was approximately equal to the negative charge on the anion. Bragg pointed out that this principle can be represented by a picture of lines of electrostatic field linking the cations to the anions (4). Because the energy is a minimum when the lines of field are as short as possible, each cation will surround itself with anions and vice versa.
Bragg's picture was never developed further, but many newly determined structures, particularly those of minerals, were interpreted in terms of Pauling's rules which, in the last few decades, have been developed into the bond valence model which is the subject of this paper.
More recently the introduction of computers has put life back into the original ionic model. Madelung summations can now be rapidly performed and good empirical short-range potentials can be found by quantum mechanical calculation or by fitting the parameters to observed physical properties. By minimising the energy, the ionic model can, in experienced hands, predict the positions of atoms in complex crystals to within a few pm and can predict many of a crystal's physical properties (5). However, powerful as this form of the ionic model is, details of the calculation are lost inside the computer and the model does not provide the kind of intuitive insights that are required of a good teaching model.
These insights can, however, be recovered if the model is developed in terms of the electrostatic field rather than the energy. Such a derivation, presented in Section 2, yields a simple model of chemical bonding that is identical to the empirically developed bond valence model (6,7,8). The remaining sections of the paper show how this model can be used to understand many chemical and physical properties of inorganic compounds, particularly the interesting and unusual properties of the new high-tech ceramics used in fuel cells and other
5 electrical devices. Further, the model is simple and intuitive and permits predictions to be made without recourse to a computer.
2. The electrostatic bond model The particular virtue of the ionic model is that the long range forces are described by a classical electrostatic potential while all the properties that depend directly on quantum mechanics are contained a short range potential, Ushortrange, that is fitted empirically. Therefore for compounds that satisfy the assumptions of the model, quantum mechanical calculations are not needed. These assumptions limit the scope of the model but not as much as is generally supposed. They are:
1. Atoms can be treated as carrying a net charge, either positive (cations) or negative (anions). The charge on the anions is the charge necessary to complete the electron octet in the valence shell. The charge on the cations is then determined by the requirement of electroneutrality. In many cases this will leave the cation also with a filled shell configuration.
2. There is no bonding between two cations or between two anions. That is, all the bonds have a cation at one end and an anion at the other. This means that all the bonds are between an electron donor and an electron acceptor, and the assumption effectively restricts the model to inorganic compounds. The assumption is, however, satisfied by some organic compounds (e.g., organic salts) and, on the other hand, there are some inorganic compounds (e.g., carbonyl complexes) where the model does not apply.
3. The electron density of the atoms is spherically symmetric so the electric field in the region outside the atom is the same as if the atom were replaced by a point charge equal to the net charge on the atom. However, the model can be adapted to situations where the electron density is not spherically symmetric by adding point multipoles to the point charges.
6 These assumptions are obeyed by most inorganic salts, minerals and ceramics, but they are not obeyed by most organic and metallic compounds. Note that there is nothing in the assumptions that excludes covalent bonds per se, since most covalent bonding does not destroy the (approximate) spherical symmetry of the electron density. Covalent effects, together with any differences between formal and actual ionic charges, are included in the fitted short range potential, Ushortrange.
Figure 1. A section through the Coulomb filed of the rutile form of TiO2. Each Ti4+ ion bonds to six O2- ions but only half of these bonds appear in this section.
Fig. 1 shows the electrostatic field in a section through a crystal of rutile (TiO2). Each Ti4+ ion is represented by a charge of +4 electrons and each O2- ion by a charge of -2 electrons. Since lines of field cannot cross, they must connect nearest neighbour ions as pointed out by Bragg (4). As Fig. 1 shows, this field, which includes all the long-range electrostatic effects, naturally partitions itself into localised bond-like regions, a phenomenon which provides a theoretical justification for drawing localised bonds between neighbouring cations and anions and which provides a unique definition of the coordination number. It should be emphasised that the localised bond-like regions are a complete representation of the Coulomb field and a knowledge of their properties is sufficient to determine all the long-range electrostatic effects.
7 A property that uniquely characterises each bond, i, is the electrostatic flux (the number of lines of field),
0i, linking the ions.
According to Gauss' theorem, the total flux emanating from a charge, V, is equal to the
charge (eq 2).
(i 0i = V
(2)
The ionic charge and flux are measured in valence units, v.u., which formally correspond to units of electronic charge, though the actual charges may be smaller. Eq 2 necessarily holds for sums around both the anions and the cations.
The bonds can be treated as capacitors, since a charged capacitor consists of two equal and opposite charges, q, connected by an electrostatic flux, 0, which is equal to q. By replacing each bond in a structure by an equivalent capacitor, Ci, the chemical bond network can be converted to an equivalent electric circuit as shown in Fig. 2 for the network illustrated in Fig. 1.
By the law of conservation of energy, the sum of the electric potential differences around any loop in such a network is zero and, since the potential across a capacitor is given by qi/Ci and qi is equal to 0i, it follows that
(loop0i/Ci = 0
(3)
each term in the summation being taken as positive or negative depending on the direction in which the bond is traversed.
Eqs 2 and 3 constitute the Kirchhoff equations for the equivalent circuit shown in Fig. 2 and they allow
8 the values of 0i to be calculated providing one knows the values of the capacitances, Ci, and the bond connectivity. Unfortunately, values of Ci depend on the positions of the atoms and can only be determined a priori from a quantum mechanical calculation. The problem of determining the capacitances is taken up again in Section 5, but Sections 3 and 4 describe some of the chemistry that can be derived using eq 2 alone. Since eq 2 does not require a knowledge of the atomic coordinates, it can be used to derive relationships that apply to all states of matter including amorphous solids, solutions and melts, as shown in the next two sections.
3. Bonding Strength and Bonding Strength Matching By recognising that the flux tends to be distributed uniformly among the various bonds, one can use eq 2 to estimate how much flux each bond will have. If one knows the coordination number, N, of one of the ions forming the bond, this estimate is provided by eq 42.
0 = V/N i
(4)
Since cations have characteristic coordination numbers, this estimate can often be made even if the actual coordination number is not known. For example, S6+ is always 4-coordinated by O2-, so that S6+-O2- bonds always have an average flux of 1.5 v.u. When coordinated to O2-, Na+ is found with coordination numbers ranging from 3 to 12, but most frequently with a coordination number close to the average of 6.4 (10). One would expect then to find that the flux of Na-O bonds lies between 0.33 and 0.08 v.u., with the most likely value of 0.16 v.u. obtained by substituting the average coordination number into eq 4. Average coordination numbers found around a number of cations are listed in
2
The factors affecting coordination number are discussed in Section OL-6.
9 Table 1 . The corresponding fluxes are shown Cation on the Bonding left hand Strengths side of Fig. 3. The Table 1 Cation
Ionic charge
Coordination number
Bonding strength (v.u.)
Cs+
1
9.2
0.11
K+
1
7.9
0.13
Na+
1
6.4
0.16
H+ (weak bond)
0.18
H2O
0.18
Ba2+
2
10.24
0.20
Li+
1
4.87
0.20
Sr2+
2
8.57
0.23
Ca2+
2
7.31
0.27
Mg2+
2
5.98
0.33
La3+
3
8.5
0.35
Zn2+
2
4.98
0.40
Be2+
2
3.99
0.50
Al3+
3
5.27
0.57
H+ (strong bond)
0.82
B3+
3
3.46
0.87
Si4_
4
4.02
1.00
P5+
5
4.01
1.25
C4+
4
2.96
1.35
S6+
6
4.00
1.50
N5+
5
3.00
1.67
Cl7+
7
4.00
1.75
Average observed coordination numbers to O2- are mostly taken from (10). The bonding strength of H+ is discussed in Section OL-7.3, and of H2O in Section 5.
10 flux calculated using eq 4 is called the cation bonding strength and represents the flux one would expect for bonds formed by the cation.
Cation bonding strengths are a measure of Lewis acid strength and correlate with electronegativity (Fig. 4). It is remarkable that a quantity defined in terms of the formal charge and average coordination number found in solids (the bonding strength) should correlate so well with a property determined from the heats of formation of binary compounds (Pauling electronegativity) but the reason is clear. Both are determined by the size and charge of the electron core. Both increase as the number of valence electrons increases and the size of the electron core decreases.
One can define an anion bonding strength in a similar way. Cl- has 6 cation neighbours in NaCl and 8 in CsCl leading to an estimated average coordination number of around 7. O2- is rarely bonded to fewer than 2 or more than 6 cations with the average close to 4. Thus the bonding strength of Cl- is 1/7 = 0.14 v.u. and of O2- is 2/4 = 0.50 v.u. Bonding strengths of complex oxyanions can be assigned by assuming a coordination number of 4 for O2- . The bonding strength of SO42- is calculated by recognising that O2- forms one bond of 1.5 v.u. with S6+, leaving 0.5 v.u. available for the remaining three bonds, i.e., the external bonds will have an average flux of 0.5/3 = 0.17 v.u. Alternatively, one can recognise that each of the four O2- ions will form three external bonds to give a total of 12 external bonds which must share the net charge of -2 on the complex SO42- ion, giving each bond a flux of 2/12 = 0.17 v.u. The bonding strength of PO43- can be shown in a similar way to be 3/12 = 0.25 v.u. Bonding strengths of a number of anions are shown on the right hand side of Fig. 3 and are listed in Table 2. Anion bonding strengths are a measure of the Lewis base strength of the anion.
11 Table 2 Anion
Anion Bonding Strengths Ionic charge
Coordination number
Anion bonding strength (v.u.)
I-
-1
ClO4-
-1
Br-
-1
NO3-
-1
3x3= 9
0.11
CdCl64-
-4
6 x 6 = 36
0.11
Cl-
-1
7
0.14
H2PO4-
-1
6
0.17
HCO3-
-1
6
0.17
SO42-
-2
H2O
0
HPO42-
-2
CO32-
-2
F-
-1
PO43-
-3
3 x 4 = 12
0.25
SiO44-
-4
3 x 4 = 12
0.33
BO33-
-3
3x3= 9
0.33
OH-
-1
3
0.33
O2-
-2
4
0.50
0.08 3 x 4 = 12
0.08 0.10
3 x 4 = 12
0.17 0.18
9 3x3= 9
0.22 0.22
4
0.25
Anion bonding strengths are calculated using eq 4 with the coordination numbers of O2- and F- assumed to be 4. For the higher halogens the bonding strength is taken from the maxima in Fig. 5. For the treatment of protonated anions see footnote 3. The anion bonding strength of H2O is discussed in Sections 5 and OL-7.3.
12 Since the anion and cation bonding strengths are estimates of the flux that links the cation to the anion, it follows that the most favourable conditions for bonding will occur when both the cation and the anion have similar bonding strengths. This is expressed by the bonding strength matching rule which states that:
The most stable bonds will be formed between cations and anions with similar bonding strengths.
A glance at Fig. 3 immediately shows which cations and anions will readily form bonds. Mg2+ and SiO44- both form bonds with fluxes of 0.33 v.u. making Mg2SiO4 the stable mineral, forsterite, which is believed to form a large portion of the earth's upper mantle. PO43- readily bonds to Ca2+ to form minerals such as apatite which is a major constituent in bone and teeth, but PO43- is poorly matched with K+. Even though K3PO4 is a stoichiometrically allowed compound, it is unstable because of the poor match between the cation and anion bonding strengths (0.12 v.u. and 0.25 v.u. respectively). K3PO4 is, in fact, deliquescent and readily absorbs water from the air to form H2PO4- ions as shown in Section 4.
The most stable phosphate of potassium, the one with
the best bonding strength match, is KH2PO4 (anion bonding strength 0.17 v.u.), whose interesting physical properties are discussed in Sections OL-7.3 and OL-8.1. A compound with a poor valence match, such as Si3(PO4)4 (cation and anion bonding strengths of 1.00 and 0.25 v.u. respectively), would be unstable and difficult if not impossible to prepare because of the poor match, even though it obeys the electroneutrality rule. Hawthorne (12) has used the matching rule to show what kind of crystal will form in a cooling silicate magma. Condensed silicates (SiO4 tetrahedra sharing corners to form chains and frameworks) have low anion bonding strengths and will form when weakly bonding cations such as alkali metals are present, but orthosilicates (isolated SiO4 tetrahedra), which have a larger bonding strength (0.33 v.u.) will be formed in the presence of more strongly bonding cations such as transition metals.
4. Solubility The process by which a solid is dissolved in water is a chemical reaction with an equation of the following form:
13 AX + nH2O ----> A(H2O)n+ + X-
(5)
The bonding strength matching rule can be used to decide whether the left or right hand side of this equation is the most stable, but first it is necessary to know the cation and anion bonding strengths of water. To calculate these requires, in turn, an understanding of the unique bonding properties of hydrogen which bonds asymmetrically between two O2- ions, an arrangement called a hydrogen bond. The asymmetry, which is fully discussed in Section OL-7.3, is a consequence of the repulsion between the two terminal O2- ions which causes the H-O bonds to be stretched. This strain can be removed if the H+ ion is displaced towards one of the O2- ions to form one short bond of 0.82 v.u. and one long bond of 0.18 v.u. (I) (13).
An isolated water molecule contains O-H bonds of valence 1.00 v.u. (II), but in the liquid, each molecule has four neighbours with which it forms hydrogen bonds (III). In liquid water, therefore, the water molecule behaves as both a cation and an anion. The H+ ion acts as a cation with a bonding strength of 0.18 v.u. and the O2ion acts as an anion with a bonding strength (based on 4-coordinated O) also of 0.18 v.u. (IV).
14 From the matching rule it is clear that water molecules will readily bond to each other since they have equal cation and anion bonding strengths, but they can also bond to anions with a bonding strength between 0.10 and 0.25 v.u. and cations with a bonding strength between 0.10 and 0.36 v.u. (V). Compounds whose ions have bonding strengths lying in these ranges can dissolve in water since the water molecule will be able to bond to them and equation (5) can proceed to the right. For example, Na+ will tend to surround itself with 6 water molecules, forming bonds (0.16 v.u.) that are only a little weaker than the hydrogen bonds between the water molecules themselves (0.18 v.u., VI). Mg2+ will surround itself with six water molecules but, since the Mg-O bond must have a flux of 0.33 v.u., each O2- ion will only bond to only one Mg2+ ion (VII). The O-H bonds of the coordinated water molecule will have fluxes of (2 - 0.33)/2 = 0.83 v.u. and the hydrogen atoms will form bonds of 1 - 0.83 = 0.17 v.u. with other water molecules or anions. Because the flux of the Mg-O bond (0.33 v.u.) is almost twice that of the bonds between water molecules, the Mg-O bond is stronger than those in the surrounding water and Mg(H2O)62+ is a stable complex ion even in aqueous solutions.
We can identify four situations that might occur when an inorganic compound is placed in water. If the cation and anion are well matched and both have large bonding strengths, e.g., Mg2+ and SiO44- (see Fig. 3), then the solid will be insoluble since the match between the two ions is better than the match between either of them and water.
If the two ions are well matched but have bonding strengths that are small so that each is also well matched to water, e.g., Na+ and Cl- then the compound will be soluble but will also readily recrystallize when the
15 water is removed.
More interesting are compounds in which a weak anion such as SO42- is found with a strong cation such as Mg2+. As discussed above, water can readily bond to Mg2+ to give the hydrated cation Mg(H2O)62+ with a bonding strength of 0.17 v.u., just right for forming bonds with both water and SO42- Anhydrous MgSO4 readily dissolves in water, even to the extent of picking up water from the atmosphere, but the solid that recrystallizes from aqueous solution is the well matched heptahydrate, epsomite, whose formula can be written Mg(H2O)6SO4.H2O, the seventh water molecule occupying an otherwise empty cavity between the ions. Compounds of strongly bonding cations with weakly bonding anions are normally found as hydrates and their anhydrous forms are hygroscopic.
The situation of a weakly bonding cation, e.g., Na+, and a strongly bonding anion, e.g., CO32- (bonding strength 0.22 v.u.), is somewhat different, but again water is able to mediate the bonding. In this case the CO32ion binds a hydrogen atom from the water in order to lower its anion bonding strength to 0.17 v.u. to match that of the surrounding water according to eq 6
CO32- + H2O ---> HCO3- + OH-
(6)
Attaching a hydrogen atom reduces the charge on the carbonate ion but also reduces the number of external bonds it can form. The addition of H+ effectively satisfies the bonding requirements of one of the O2- ions3, leaving the HCO3- ion with a charge of -1 and a coordination number of 6, three bonds being formed by each of the remaining two O2- ions. The anion bonding strength is therefore 1/6 = 0.17 v.u. which gives a better match than CO32-
3
In practice the H+ ion in (CO3H)- donates a hydrogen bond to a water molecule or other anion and the O2- ion
to which H+ is attached accepts a hydrogen bond or bond of equivalent strength. These two bonds effectively cancel each other and the net effect is the same as if the OH group formed no bonds except that to C.
16 (bonding strength 0.22 v.u.) to Na+ (bonding strength 0.16 v.u.) One therefore expects that, because of the poor match, Na2CO3 will dissolve in water and that the carbonate anion will convert to bicarbonate (hydrogen carbonate) with the excess OH- ions leaving the solution alkaline. In neutral pH the salt which crystallizes is NaHCO3 (baking soda). The original Na2CO3 (washing soda) will only crystallize if the solution is made sufficiently alkaline. The same arguments would lead one to expect CaCO3 to be only sparingly soluble, since both ions have moderately large bonding strengths (0.27 v.u. and 0.22 v.u.) and are reasonable matched.
Fig. 5 shows the free energies of solvation of the alkali metal and alkaline earth halides as a function of the cation bonding strength. Positive free energies correspond to insoluble compounds and large negative free energies to hygroscopic or deliquescent compounds. Only F-, with its relatively large anion bonding strength (0.25 v.u.), forms insoluble salts, the most insoluble being formed with Ca2+ which provides the best match. Both MgF2 and BaF2 are more soluble than CaF2. This is not what one would expect from the periodic table sequence (Mg, Ca, Sr, Ba) but it is readily predicted by the bond valence model.
Cl- and Br-, whose anion bonding strengths (0.14 and 0.10 v.u.
respectively) lie in the range of good hydrogen bond acceptors, form only soluble salts, those with poor valence matches, such as BeCl2 and MgBr2, being hygroscopic or deliquescent. I-, having a very small bonding strength (0.083 v.u.), does not form a good match with any of the alkali metals or alkaline earths, but the relative insolubility of CsI suggests that I- should form an insoluble salt with an even weaker cation. Tetramethylammonium iodide with a cation bonding strength of 0.06 v.u. is only slightly soluble.
5. Bond Flux and Bond Lengths
17 Bond lengths are determined by the size of the bond flux and the mutual repulsion between the atoms, the larger the flux the more closely the atoms can be brought together. The repulsion is dependent on the way the electron density is distributed in the atoms and can only be properly determined using quantum mechanics. However, as pointed out in Section 2, the ionic model avoids the explicit use of quantum mechanics by treating this repulsion empirically, the effect being hidden in the unknown bond capacitances of eq 3. It is no simple matter to calculate these capacitances even when the positions of the atoms are known, so an alternative approach is needed.
In the years following Pauling's paper (3), it was noticed that there was in inverse relationship between bond length and bond flux. This correlation, shown for Ca-O bonds in Fig 6 and represented by eq 7, is now well established for virtually all the bonds encountered in inorganic compounds.
s = exp((ro-r)/B)
(7)
where r is the bond length and s is the bond flux, also known as the bond valence4. The parameters, ro and B, are chosen to ensure that the valences of the bonds formed by an atom sum to the formal ionic charge according to eq 8, known as the valence sum rule,
(isi = V
4
(8)
The bond valence model was developed before it was realised that the bond flux was the same as the bond
valence. Bond valence was originally defined in terms of the bond length using eq 7 and the term `bond valence’ is retained here for the quantity derived experimentally from bond lengths, or theoretically from the bond connectivity using eqs 8 and 9, to distinguish it from the bond flux calculated from the Coulomb field.
18 V is the charge on the ion as defined in Section 2 and is therefore a positive number for cations and a negative number for anions. In the bond valence model it is called the atomic valence. The sign of the bond valence, s, is Table 3
Selected bond valence parameters (ro in eq 7) in pm mostly taken from (14). Al3+-F- 154.5 Al3+-O2- 162.0 As3+-O2- 178.9 As5+-O2- 176.7 B3+-O2- 137.1 Ba2+-O2- 228.5 Be2+-O2- 138.1 C4+-C4- 154.0
C4+-N3- 144.2
Ca2+-O2- 196.7
Ca2+-F- 184.2
Cr3+-F- 165.7
Cr6+-O2- 179.4
Cu2+-O2-167.9 Fe2+-O2- 173.4 K+-O2- 213.2 La3+-O2- 217.2 Li+-O2- 146.6 N5+-O2- 143.2 Na+-O2- 180.3 P5+-O2- 161.7 Rb+-O2- 226.3 S6+-O2- 162.4 Se6+-O2- 178.8 Si+-O2- 164.0 Ti4+-O2- 181.5
Fe3+-O2- 175.9
C4+-O2- 139.0
19 Zn2+-Cl- 202.7
Zn2+-O2- 170.4
determined by the assumption that bonds are directed from the anion to the cation. For the majority of observed structures it is possible to find values for the bond valence parameters of eq 7 that satisfy eq 8. The parameter B, which gives the slope of the curve shown in Fig. 6, is found to be close to 37 pm for almost all bonds, but ro, which represents the notional length of a bond of unit valence, depends on the sizes of the two atoms that form the bond. These values have been determined for most bond types (14,15). A selected list of values of ro are given in Table 35.
New structure determinations are often checked by calculating bond valences using eq 7, since violations of eq 8 usually indicate a problem with the structural model. More interestingly, eq 8 can be used to distinguish between the different
oxidation states of
a cation even in
metallorganic
complexes and
enzymes (16). It can
also be used, as
discussed in
Section OL-7.4, to
analyse the strained
bonds that cause
some inorganic solids
to have unusual
physical and
chemical properties.
5
Since eq 7 is only valid over the small distance ranges normally found for a given bond, it does not describe the
full O-H bond correlation shown in Fig. 17. Therefore no value is given for ro(O-H) in Table 3 and Fig. 17 should be used instead.
20 It is possible to represent an inorganic compound by an array of atoms linked by a network of bonds. The network may be finite as in a molecule such as P4O10 (VIII) but in most inorganic compounds, such as NaCl (IX), it is infinite. Whether finite or infinite, for large class of compounds the valences of the bonds determined from eq 7 are found to obey eq 9 as well as eq 8.
(loopsi = 0
(9)
The summation in eq 9 is over any set of bonds that form a closed loop in the bond network, the sign of s depending on whether the bond is traversed from cation to anion or anion to cation. Eq 9 is known as the equal valence rule because it leads to the most uniform distribution of valence between the bonds (17).
Eqs 8 and 9
taken together are known as the network equations and give a unique solution for si for every bond in the network. Together the network equations correspond to the statement that each atom shares its valence as equally as possible between the bonds that it forms. Methods of solving these equations are given in the Appendix.
While eq 8 is obeyed by most inorganic solids, there are some important compounds that do not obey eq 9. These cases are examined in Section OL-7 and correspond to structures in which various additional constraints result in an unequal distribution of the valence. It is therefore convenient to differentiate between the experimental bond valence, determined from the bond lengths using eq 7 and represented by the symbol `S’, and the theoretical bond valence, being the solution of eqs 8 and 9 and represented by the symbol `s’. Compounds in which these two quantities are the same, on average, within 0.05 v.u. are defined as having unstrained structures.
In unstrained structures, the bond flux, 0, of the Coulomb field, calculated as described in Section 2, is equal to the experimental bond valence, S, (hence also to the theoretical bond valence, s) (9), so that eqs 2 and 3 lead to the same solution as eqs 8 and 9. Given that the flux, 0, is equal to the bond valence, s, and that eq 2 has the same form as eq 8, eq 3 must be the same as eq 9, which requires that the capacitances of eq 3 cancel and therefore must be equal. Thus the observed tendency for the valence to distribute itself uniformly among the
21 bonds (expressed by eq 9)
means that, at
equilibrium, all the bond
capacitances are equal. Eq
8 is thus a
statement of the laws of
conservation of charge (eq
2) and eq 9 a
statement of the law of
conservation of energy (eq 3) in unstrained equilibrium
structures.
Like the example of NaCl (IX), most inorganic compounds form structures whose bond network is infinite. In crystalline solids, the structure consists of a very large number of identical small units indefinitely repeated like the patterns on wallpaper. Because these small units are identical, the problem of solving eqs 8 and 9 for a crystal is greatly simplified since it is only necessary to consider the basic unit of the repeating pattern which often is just the formula unit. For example, the repeating unit for NaCl crystals is just the formula unit `NaCl' and the infinite network can be mapped into a graph containing a single Na+ ion and a single Cl- ion linked by 6 bonds (X). Unlike the more familiar bond graph used in organic chemistry, this diagram does not indicate a hextuple bond, but rather represents six separate bonds connecting each Na+ ion to six different Cl- ions and vice versa. A more interesting and less trivial example is CaCrF5 (XI). The bond valences shown here are those calculated using eqs 8 and 9. The bond lengths calculated from these valences correspond quite closely to those observed as shown in Fig. 7 (18).
The bond valence model can be used for structure prediction. For many ternary compounds it is possible
22 to draw the correct bond graph knowing only the formula unit. Bonds are drawn between the cations and anions, starting with those with largest bonding strength, in such a way that the cations have coordination numbers close to their observed average. Where there is a choice, the most symmetric bond graph is chosen as this distributes the valence most uniformly among the bonds. The network equations can then be used to determine the ideal valences of the bonds and from these the ideal bond lengths can be calculated using eq 7. In favourable cases, the 3-dimensional structure can be found by searching for the highest symmetry space group that is able to accommodate the bond graph (19).
The procedure is illustrated with the bond graph of LaAlO3 shown in Fig. 8. The atoms of the formula unit are first written down, then the bonds from the cation with the highest bonding strength (Al3+) are drawn. Al3+ can adopt coordination numbers ranging from 4 to 6, but only 6-coordination leads to a symmetric graph. Since the average observed coordination number for La3+ is 8.5 (Table 1), La3+ best preserves the symmetry by forming nine bonds. The equivalence of the bonds around each cation makes it easy to determine the bond valences, hence the bond lengths, as shown in Fig. 8. The highest symmetry space group into which this graph
c, yields the structure shown in Fig. 9. can be mapped, R3
OL-6. Coordination numbers In Section 3 the bonding strength of a cation was defined as the formal ionic charge (valence) divided by the coordination number. The method of determining the formal ionic charge was described in Section 2 but the coordination numbers used were experimental values and no attempt was made to explain why different cations have different coordination numbers. It is now time to address this question.
23 Clearly a maximum limit is set by how many anions (for the sake of argument we will assume that they are all O2- ions) can fit around the cation 10). This can be determined by comparing the O-O distance expected in a given coordination sphere with the minimum distance allowed between two O atoms in contact. If the expected O-O distance is smaller than the contact distance, the coordination number cannot exist. Calculating the O-O distance expected in a given coordination sphere is straightforward if the bond lengths are known and, using eq 7, bond lengths can be found from the predicted bond valences even for coordination numbers that are not found in nature. However, determining the distance that brings two O2ions into contact is not so easy because the O2- ions behave like soft spheres. The stronger the bonds formed by the cations, the closer the O2- ions can be pulled together. The O-O distances for a variety of cation coordination numbers are shown in Fig. 10 as a function of the effective valence pulling the O2- ions together. The number of sides of the polygon gives the coordination number. The large blue polygons represent coordination environments that are known, the small yellow polygons, those that are not known. Polygons referring to ions of the same elements are linked by lines. The effective valence is the valence of the cation-O bond projected onto the line joining the two O2- ions and represents the valence that is available to bring the two O2- ions together. The heavy line is drawn so as to divide the observed coordination numbers from the unobserved and represents the O-O contact distance as a function of the effective valence.
As an example, 3-coordinate N5+ is known in the NO3- ion but the 4-coordinate NO43- ion is not known. Using eq 7 the N-O distance is predicted to be 124 pm (for bond valence = 5/3) in the planar NO3- ion and 135 pm (for bond valence = 5/4) in the tetrahedral NO43- ion. For both ions the O-O distance is predicted to be similar, 215 pm for NO3- and 220 pm for NO43-, but the bonds in NO3- are much stronger than those in NO43- (20). O-O distances of 215 pm are possible only if the two O2- ions are attached to the same cation with bonds stronger than
24 about 1.5 v.u. but the weaker bonds that would be found in NO43- are not strong enough to bring the O2- ions even as close as 220 pm. Because the N-O bonds in NO3- are particularly strong, the O-O distances are exceptionally short. A more typical O-O distance is 266 pm found around 6-coordinated Al3+ where the Al-O bonds have valences of 3/6 = 0.5 v.u. Al3+ is known in 4-, 5- and 6but not 7- or 8-coordination since 260 pm is the closest that Al3+ can bring two O2- ions together (Fig. 10).
There is however a second factor that determines cation coordination numbers and this is the bonding strength of the counterion. If the bonding strength of the anion is larger than that of the cation, the cation will tend to form bonds stronger than its bonding strength and its coordination number will be correspondingly reduced to satisfy the valence sum rule. Anions with weaker bonding strengths, on the other hand, will tend to increase the
co
ordination number of the cation. Consider a large weak cation such as Cs+. If
it
surrounds itself by the maximum possible number of 14 O2- ions, the bonds it
fo
rms will have valences of only 1/14 = 0.07 v.u. which is smaller than the
bo
nding strength (0.08 v.u.) of the weakest inorganic anion, ClO4-, shown in
Ta
ble 2 and Fig. 3. Since ClO4- will tend to form bonds with a valence of 0.08
v.
u. it will favour a Cs+ coordination number of only 12. Stronger anions will
te
nd to lower the coordination number even further. The frequency with which
va
rious coordination numbers are found around Cs+ are shown at the top of Fig.
11
(10). They range all the way from 3 to 14 but most commonly lie between 6 and 12 with an average close to 9. For Cs+, the coordination number is therefore primarily determined by the bonding strengths of available anions, not by the maximum number of O2- ions that can fit around it.
25 The opposite effect is seen around very strong cations such as Cl7+, S6+ and N5+. The histogram at the bottom of Fig. 11 shows that N5+ is found with essentially only one coordination number. The reason is that all the anions listed in Table 2 have smaller bonding strengths than N5+ (1.67 v.u.). Only the anion with the largest bonding strength, O2-, is able to form bonds to N5+ and even then the match is very poor, the bonding strength of N5+ being more than 3 times that of O2-. The small bonding strength of the anion forces the coordination number of N5+ to be as large as possible, thus bringing the O2- ions into the closest possible contact resulting in the NO3ion as discussed above. Because the N5+ and O2- ions are so poorly matched, NO3- is relatively unstable and, like the equally poorly matched ClO4- ion, it makes a good oxidising agent.
Zn2+ is a medium strength cation and its coordination number, whose frequency is shown in the middle of Fig. 11, is determined by both its size and the bonding strength of the anion. O-O contacts restrict the coordination number of Zn2+ to a maximum of 6, in which state it will form bonds with a valence of 2/6 = 0.33 v.u. Zn2+ is expected to show its maximum coordination of 6 when it bonds to anions with bonding strengths less than 0.33 v.u. but smaller coordination numbers when it bonds to stronger anions. The strongest available anion, O2-, has a bonding strength of 0.5 v.u. corresponding to a cation coordination number of 4 which is why ZnO adopts a tetrahedrally coordinated structure related to that of diamond rather than the NaCl structure in which it would be 6-coordinate. As shown in Fig. 12, Zn2+ forms most of its compounds with anions having bonding strengths between 0.25 v.u. and 0.5 v.u. and is rarely found with coordination numbers less than 4. The line drawn in Fig. 12 shows the predicted coordination number as a function of the anion bonding strength and the area of the circles is proportional to the number of observed occurrences. With anions having a bonding strength less than 0.33 v.u., Zn2+ is usually 6-coordinate, but with stronger anions the coordination number of Zn2+ is correspondingly reduced (10).
26 Zn2+ is a relatively soft Lewis acid (cation) in the Pearson classification of hard-soft acids and bases (HSAB) (21,22,23). This means that its electron shell can be polarised by its environment, allowing Zn2+ to adopt a range of coordination numbers. By contrast, Mg2+, which is similar in size and charge to Zn2+, is hard. Its electron density cannot so easily respond to the requirements of the ligands and 80% of Mg2+ ions are found with a coordination number of 66. Thus, unlike ZnO, MgO adopts the NaCl structure.
OL-7. Strained bonds In Section 5 unstrained structures were defined as those in which the experimental bond valences derived from the observed bond lengths are the same as the theoretical bond valences derived from the network equations 8 and 9. Unstrained structures are, therefore, ones in which the bond valence, or bond flux, is most uniformly distributed among the bonds since this is the condition implicit in the network equations. This does not necessarily mean that all the bonds have the same length, as the example of CaCrF5 shows (XI and Fig. 7), but the predicted valences correspond to an equilibrium structure and hence to the lowest possible energy. One characteristic of unstrained structures is that they tend to adopt the most symmetrical bond graph consistent with the given coordination numbers.
There are, however, a number of structures where the theoretical and experimental bond valences are significantly different. If we assume that the network equations normally predict the equilibrium geometry, these structures must contain some perturbing influence that shifts this equilibrium. The result is that the bonds are strained, that is, they are either shorter or longer than predicted by the network equations.
An examination of these structures reveals that there are two principal causes for these strains, intrinsic electronic distortions and geometric effects. The first arises from an asymmetry in the electronic structure of an
6
Mg2+ is occasionally found in 4 and 8 coordination, the latter being stabilized by stronger chelating cations
which pull the O2- ions closer together.
27 ion, e.g., a stereoactive non-bonding electron pair or transition metal with empty d orbitals. Geometric effects include anion-anion repulsion and incommensuration between different parts of the structure, resulting from the inability to fit bonds of the ideal length into the available 3-dimensional space without stretching or compressing them. In some crystals, such as BaTiO3, both electronic and geometric effects work together to produce the strain. If the strains are large, the structure may even adopt a bond network of lower symmetry in which the theoretical valences result in lower strain. These effects are discussed and illustrated in more detail in the following subsections.
OL-7.1. Electronically Driven Strains One of the assumptions made in Section 2 is that the ions are all spherically symmetric. However, this is not always true. Some ions, such as the lone pair cations S4+ and Pb2+, have low lying electronic states with significantly asymmetric electron densities. To give a proper description of the Coulomb field around such cations, point multipoles must be added to the ionic point charges. Adding such multipoles causes a redistribution of the fluxes in the bonds formed by the cation and hence a change in the bond lengths. The multipole field has a short range and only directly affects the first coordination sphere around the ion. Long range effects continue to be transmitted by the monopole field which is, however, altered by the distortion that appears in the cation environment. Thus eqs 2 and 3 (or equivalently eqs 8 and 9) still apply except in the immediate environment of the asymmetric cation where eq 9 will fail since the bond flux (or valence) is not uniformly distributed between the bonds. Eq 3 still holds but cannot be used because the unknown bond capacitances are no longer equal. Two examples illustrate this effect.
The first concerns cations whose valence shell contains electronpairs that are non-bonding, usually referred to as lone pairs. If the lone pair occupies an s orbital, it is spherically symmetric and the bonding around the atom will not be distorted, but if it occupies an orbital with p character it can be represented by a point dipole or, according to the VSEPR (Valence Shell
28 Electron Pair Repulsion) model (24,25,26), the lone pair blocks a site that would normally be occupied by a bonding electron pair. In this case a cation such as S4+ in the sulfite anion SO32- (Fig. 13a) will form three S-O bonds that point to three of the vertices of a tetrahedron, the lone pair (or the negative pole of the point dipole) pointing in the direction of the fourth (empty) vertex. In solids, particularly around heavy cations such as Sb3+, there are often additional weak (secondary) bonds close to the direction of the lone pair (Fig. 13b). However, in some compounds of Tl+ and Pb2+, such as PbS which has the NaCl structure, the secondary bonds are as strong as the primary ones, the lone pair occupies an s orbital, and the electronic structure and the bonding around the cation are both spherically symmetric (Fig. 13c).
The structural chemistry of thallium (I) is particularly interesting. In TlNO3, Tl+ is symmetrically bonded, behaving very much like the alkali metal Rb+ (Fig. 14a), but in Tl3BO3, the non-bonding electron pair is fully stereoactive and, like S4+ in SO32-, Tl+ forms three bonds at three of the vertices of a tetrahedron (Fig. 14b). Tl+ thus has two electronic configurations of similar energy, one in which the lone pair occupies an s orbital and one in which it occupies an orbital with p character. Quite small changes in the environment are sufficient to cause the ion to switch from one configuration to the other making it a classic soft cation (21,22,23). Which state is the more stable in a particular compound is readily predicted using the bonding strength matching principle (Section 3). When Tl+ bonds to an anion with a large bonding strength such as BO33- (bonding strength = 3/9 = 0.33 v.u.), it adopts a conformation in which it forms matching bonds of 0.33 v.u. Since Tl+ can only form three such bonds, its coordination sphere has plenty of space for the lone pair and a distorted environment is expected (Fig. 14b). In cases where the anion has a small bonding strength, e.g., NO3- (bonding strength = 1/9 = 0.11 v.u.), Tl+ adopts a conformation which allows it to form bonds of 0.11 v.u. In this case it will form nine bonds, filling the coordination shell. The lone pair is forced into the s orbital and the structure is symmetric (Fig. 14a). Anions of intermediate strength cause the lone pair to be partially stereoactive with a coordination number lying between 3 and 9. One consequence of this electronic rearrangement is that the cation bonding strength of the soft Tl+ ion
29 ranges from 0.11 to 0.33 v.u. and Tl+ can form stable compounds with a much wider range of anions than the similarly sized but hard cations K+ and Rb+. Even though the alkali metals are known with low coordination numbers, the resulting compounds are unstable and absorb water from the air as discussed in Section 4.
The d0 transition metal cations, Sc3+, Ti4+, V5+ and Cr6+ constitute another system where the electronic conformation leads to large distortions in the coordination environment. When the oxidation state of the cation is small (e.g., Sc3+), the empty d orbitals of the cation are sufficiently high in energy that they have a negligible effect on the bonding, but as the oxidation state becomes larger and the d orbitals are less screened from the nuclear charge, their energy is lowered to the point where it becomes comparable to the energy of the filled p orbitals of the ligands. The p and d states overlap and can mix, leading to an easily polarizable electronic structure around the transition metal. The stable state can be predicted using an extension of the VSEPR model. Gillespie and Robinson (27) show that strongly bonded negatively charged ligands polarise the cation core electrons away from the bond. This has the effect of destabilising any bonds that are trans to an existing bond. As centrosymmetric arrangements of ligands necessarily contain trans pairs of bonds, they are less stable than non-centrosymmetric arrangements. Consequently tetrahedrally coordinated d0 transition metals are found in stable undistorted environments, but regular octahedrally coordinated environments are unstable and the cations move off-center in an attempt to avoid forming trans pairs of bonds. Thus Ti4+ is often found shifted from the center of the octahedron formed by its ligands, as for example in BaTiO3 (Fig. 15) discussed further in Section OL-7.4. By contrast, the hard cation, Sn4+, which is similar in size to Ti4+ but retains a spherically symmetric electron density, is found in octahedral environments that are rarely distorted.
30 The cation V5+, the element to the right of Ti4+ in the periodic table, is more readily polarised and its octahedral environments distort spontaneously to give typically one very short V-O bond, often referred to as a vanadyl bond. Because of this distortion, ZnV2O6, for example, crystallises with a structure whose bond network reinforces this distortion (Fig. 16b) while ZnSb2O6, which contains the isoelectronic but harder spherical Sb5+ ion, adopts the more symmetric network shown in Fig. 16a. The distortion is so much greater for Cr6+, the next ion to the right in the periodic table, that octahedral coordination, which is expected on the basis of its size (see Fig. 10) is completely destabilised and only tetrahedral coordination is found. These examples illustrate how the progressive increase in the ability of the ion to be polarised leads first to a simple distortion of the environment of an octahedrally coordinated cation, then to the adoption of a less symmetric bond graph and finally to a change in the coordination number.
OL-7.2. Geometrically Strained Structures Knowing the topology of the bond network allows one to calculated the expected bond lengths using eqs 8, 9 and 7, but there is no guarantee that these distances can be maintained when the structure is mapped into 3dimensional space. Sometimes the bonds must be squeezed or stretched to make them fit. Such structures are said to be geometrically strained because the bonds in such compounds are forced to deviate from their ideal lengths.
There are two ways in which geometric strain can arise. If the strain is caused by the repulsion between nonbonded atoms it is called steric strain, but if it arises from the need for all parts of the structure to adopt the same crystallographic translations it is called lattice induced strain. Steric strain is often found in overcrowded organic molecules, but lattice induced strain can only occur in crystals with an infinitely extended bond network. It is therefore mostly confined to inorganic crystals where, as discussed in Sections OL-7.4 and OL-8, it is responsible for many
31 unusual physical and chemical properties.
OL-7.3 Steric Strain: Hydrogen bonds In Section OL-6 it was shown that coordination numbers are limited by the number of anions that can be fitted around a cation. But suppose that the anions are held in place in a crystal in such a way that they provide a cavity that is too large for the cation. If the cation were at the center of the cavity, the bonds would all be stretched, and the experimental bond valences calculated from these bond lengths would be smaller than expected. Consequently, the sum of the bond valences around the cation will fall short of the cation valence. However, if the cation is displaced from the center of the cavity, some bonds will become shorter and others longer though keeping the average bond length constant. An examination of the bond valence - bond length correlation shown in Fig. 6 shows that the valence of the shorter bond will increase more rapidly than valence of the longer bond decreases, so that the sum of the bond valences will be increased. If the cation moves far enough off-center, the bond valence sum can be made large enough to equal the atomic valence of the cation.
This result can be stated in the form of a useful theorem, the distortion theorem, which states that:
If the average bond length is held fixed, distorting the environment of an atom will tend to increase the sum of the bond valences.
An alternative but equivalent statement is:
If the bond valence sum is held fixed, distorting the environment of an atom will tend to increase the average bond length.
The best example of a sterically strained bond is the
32 hydrogen bond, an arrangement in which a hydrogen ion forms an asymmetric link between two anions, typically O2-, to create a relatively weak bridge that is of particular importance in biological chemistry, being one of the essential components that makes life possible. This unusual behaviour of hydrogen arises entirely because the distance between two O2- ions in contact is too large to form two equal O-H bonds of valence 0.5 v.u. The length of an O-H bond of 0.5 v.u. is 115 pm, and the formation of two such bonds by H+ requires the O2- ions to be no further than 230 pm apart (see the letter H in Fig. 10). However, the shortest O-O contact possible in this case is 244 pm. In order to exist, the O-H bonds must be stretched to 122 pm corresponding to a valence (Fig. 17) of 0.45 v.u. and a valence sum around H+ of only 0.90 v.u. The distortion theorem predicts that this sum can be increased to 1.00 v.u. if the H+ ion moves off center so that one bond has a valence of around 0.82 v.u. (length 97 pm) and the other a valence of 0.18 v.u. (length 181 pm) (13). The resulting combination of one strong and one weak O-H bond (generally represented as O-H...O) is called a hydrogen bond. The repulsion between the O2- ions ensures that the O-H...O angle is close to 180(. This is the normal hydrogen bond formed, for example, in water or ice.
The asymmetry of the hydrogen bond causes water to have a cation bonding strength (through H+) of 0.18 v.u. and an anion bonding strength (through O2-) of 0.18 v.u. as discussed in Section 4. If it were not for this asymmetry, water would be linked into a silica-like solid held together with bonds of 0.5 v.u., causing it to be a mineral with a high melting point. The steric effects that result in the asymmetry of the hydrogen bond are responsible for the weakness of the bonding between water molecules, and consequently for the chemistry of water that makes life possible. The asymmetry in S-H...S bonds is even greater so that bonding between H2S molecules is weak and H2S remains a gas at room temperature.
Structures that form hydrogen bonds often adopt bond networks that themselves favour the distortion and hence minimise the deviation from the equal valence rule (eq 9). In [Mg(H2O)6]2CdCl6 (28) the steric strain
33 around H+ and the arrangement of the bonds in the bond graph shown in Fig. 18 both lead to a distorted environment around the water hydrogen atoms. The valence of the Mg-O bonds is 2/6 = 0.33 v.u., so the O-H bonds of the water must have valences of 0.83 (to ensure a valence sum of 2.00 around O2-), and application of the valence sum rule to the H+ ions shows that the weaker H...Cl bond must have a valence of only 0.17 v.u., a value which provides a nearly ideal distortion for the hydrogen bonds. This is an example of the symbiotic effects of different distortion mechanisms. In this case the crystal adopts a bond graph whose theoretical bond valences lead to bond lengths that exactly match the steric distortion expected around H+.
The ideal conformation of the hydrogen bond described above is the one that has the lowest energy, but other conformations are possible providing that the energy required to make the distortion larger or smaller is recovered somewhere else in the structure. Hydrogen bonds both stronger and weaker than ideal are known.
Weaker (more asymmetric) hydrogen bonds can be formed when one of the anions has a small anion bonding strength. Such bonds are found in Li(H2O)3ClO4 (29) where the anion bonding strength of the ClO4- ion is only 0.083 v.u., a value which ensures that the H...O bond is relatively weak and hence long (204 pm, Fig. 19). Since the combined length of the O-H and H...O bonds now exceeds the minimum O...O distance, the hydrogen bond is free to bend and in this compound the O-H...O angle is reduced to 164(. This allows a third O2- ion to enter the coordination sphere at a distance of 262 pm forming an angle at H+ of only 106(. For these weak hydrogen bonds there is a good correlation between the bond length and the bond angle at H+ as is expected with the geometry determined by the repulsion between the O2- ions (13).
34 Stronger (more symmetric) hydrogen bonds can be formed when both anions are strongly bonding, e.g., the H2PO4- ion (bonding strength = 0.22 v.u.) in KH2PO4 (30) (Fig. 20). The observed structure is a compromise between one in which the environment of P5+ is regular but the hydrogen bond is symmetric and strained, and one in which the hydrogen bond has its ideal distortion but the environment of P5+ is consequently distorted and strained. The energy required to make the hydrogen bond more symmetric, therefore, comes from the energy saved by making the phosphate ion less distorted.
OL-7.4 Lattice Induced Strain A crystal consists of identical groups of atoms, each group comprising one or more formula units, repeated at regular intervals in all three directions of space rather like 3-dimensional wallpaper. The adjacent repeating units in a crystal must also fit together so that, like the patterns on the wallpaper, the
35 structure flows seamlessly in all directions7. This means that each part of the structure must conform to the same repeat distances. BaTiO3, for example, adopts the perovskite structure which is made up of an alternation of BaO layers and TiO2 layers (Fig. 21). The size of the repeating unit of the BaO layer is equal to 2 times the length of the Ba-O bond and the size of the repeating unit of the TiO2 layer is twice the length of the Ti-O bond. Since the ideal lengths of the Ba-O and Ti-O bonds calculated using the bond valence model are 295 pm and 196 pm, the repeat distances in the two layers are expected to be different (417 and 393 pm respectively). However, if the two layers are to coexist in the crystal, the repeat distances must be the same. Thus the Ba-O bonds must be compressed and the Ti-O bonds stretched until both layers have the same repeat distance (405 pm). Consequently the valences calculated from the observed Ba-O bond lengths will be too large and those calculated from the Ti-O bond lengths too small. This strain is readily recognised by the failure of the experimental bond valences to obey the valence sum rule, eq 8. If the difference between the bond valence sums and the atomic valences averaged over all the atoms in the formula unit exceeds about 0.2 v.u. the structure is unstable and, if it is to exist, it must find some way to relax. According to the distortion theorem described in Section OL-7.3, the valence sum around Ti4+ can be increased without shortening the average bond length if the Ti4+ ion moves away from the center of its surrounding octahedron of O2- ions as shown in Fig. 15. This distortion is observed in crystals of BaTiO3 at room temperature and is reinforced by the tendency of the electronic structure of Ti4+ to favour a distorted environment as discussed in Section OL-7.1. This is another example of the symbiotic effect of two different distortion mechanisms: in this case the electronic structure helps to stabilise a lattice induced distortion and vice versa.
7
Students will find an excellent and enjoyable introduction to the constraints imposed by symmetry using the
program Escher Web Sketch available at http://www-sphys.unil.ch.
36 La2CuO4 is built on the same principles as BaTiO3, but with each CuO2 layer (ideal spacing 417 pm) separated by two LaO layers (ideal spacing 351 pm) (Fig. 22) (31). In this case the LaO layers need to be stretched and the CuO2 layer compressed to give an average spacing of 388 pm. A strain of this magnitude (10%) is unstable and the structure therefore relaxes by absorbing oxygen. It does this spontaneously if prepared in air at high temperatures to give the non-stoichiometric compound La2CuO4.15. The extra O2ions help to fill the voids between the two LaO layers allowing La3+ to form additional bonds and so to increase the sum of its bond valences. Adding O2- has a second beneficial effect, that of increasing the oxidation state of Cu from +2 to +2.3, thereby increasing the valence of the Cu-O bonds which shrinks the size of the CuO2 layer. The addition of interstitial oxygen thus relieves the strain in two complementary ways: it reduces the tensile strain in the LaO layers by creating more La-O bonds, and reduces the compressive strain in the CuO2 layers by increasing the valence of Cu and thus shrinking the Cu-O bonds. This example shows how the steric requirements of 3-dimensional geometry can affect the chemistry of a solid. The formula is not only non-stoichiometric but copper is found in an unusual oxidation state.
OL-8. Applications of structure related properties OL-8.1 Ferroelectricity Ferroelectricity is the electrical equivalent of ferromagnetism. A ferroelectric crystal possesses an electric dipole moment which can be reversed by the application of an electric field, just as a ferromagnet possesses a magnetic dipole moment which can be reversed by application of a magnetic field. The electric dipole is caused by the center of the positive electric charge carried by the cations being slightly displaced from the center of the negative charge carried by the anions. There are quite a number of crystals (polar crystals) which have an electric dipole moment, but in only a small number can the direction of polarisation be reversed by an
37 electric field. Among the better known ferroelectric crystals are BaTiO3 and KH2PO4.
As mentioned in Section OL-7.4, the Ti4+ ion in BaTiO3 is spontaneously displaced from the center of its octahedron of O2- ions by the combined effect of a lattice induced strain and electronic instability. At different temperatures the Ti4+ moves towards a face, an edge or a corner of its coordination octahedron, but there is nothing to say which face, edge or corner since, in principle, they are all equivalent. However, if one Ti4+ ion moves towards a particular O2- neighbour, the corresponding Ti-O bond will be shortened and will have a higher valence. This in turn will reduce the valence of the Ti-O bond on the other side of the O2- ion forcing the next Ti4+ to move off-center in the same direction (Fig. 15). The displacement is thus transmitted to all Ti4+ atoms throughout the crystal in a cooperative manner, resulting in a relative displacement between the centers of positive and negative charge and leading to a polar crystal. However, a small applied electric field is sufficient to reverse the direction of the displacement of the Ti4+ ions allowing the polarity to be reversed. One consequence is that BaTiO3 has an extremely high dielectric constant making it useful in capacitors and microwave circuits and, since the polarisation can be switched, it can be used in computer memories. Ferroelectric memories based on analogues of BaTiO3 are likely to be the basis of the new technology of smart cards.
The hydrogen bond is responsible for ferroelectricity in KH2PO4. Since H+ bridges between two identical PO43- ions, it can move off-center equally well towards either PO43- group, but once the first H+ ion has made its selection, the PO43- group is distorted in a way that determines how the next H+ ion will move (Fig. 20 top). As in BaTiO3 the distortion is transmitted to all parts of the crystal and results in the center of positive charge being displaced from the center of negative charge. Application of an electric field can flip the H+ ions to the other PO43- group causing the dipole moment to reverse (Fig. 20 bottom).
OL-8.2 Superconductivity Although the mechanism of superconductivity in metals at temperatures below 10 K is reasonably well understood, this is not true of superconductivity in ceramic materials which can occur at temperatures well above
38 100 K. Most of these so-called `high temperature' ceramic superconductors contain CuO2 layers such as those found in La2CuO4, and, although the mechanism of superconductivity is not understood, one prerequisite is that the oxidation state of Cu must be close to +2.3. This unusual oxidation state occurs quite naturally in La2CuO4 where the lattice induced strain causes the crystals to absorb oxygen to form the compound La2CuO4.15 as described in Section OL-7.4. Lattice induced strain is important in achieving the correct oxidation state for Cu in other superconducting cuprates, usually in conjunction with the controlled introduction of impurities and vacancies, each of which introduce their own strains.
Although ceramic superconductors have only been discovered relatively recently, they are of interest because many superconduct at the temperature of liquid nitrogen, a temperature that is relatively easily achieved. They are already finding uses in electronic applications and they are likely to become increasingly important in the future.
OL-8.3 Ionic conductors The search for better batteries and fuel cells has led to a growing interest in ionic conductors. Electric batteries generate currents by dissolving ions from the metal electrodes and allowing them to diffuse under a chemical potential gradient through an electronically insulating liquid barrier to the opposite electrode. In doing so, they carry an electric charge and so build up an electric potential difference between the electrodes of the battery. Fuel cells work in a similar way, but instead of obtaining the ions by dissolving the electrodes, the ions are supplied externally, usually in the form of hydrogen and oxygen gas. The oxygen diffuses through the anode and moves as O2- ions through the insulating barrier to the cathode where it reacts with the hydrogen to form water, in the process carrying negative charge between the electrodes. In this way, the electrodes are not dissolved and the fuel cell does not get used up. If such cells are to be run efficiently at high temperatures, it is necessary to replace the liquid barrier by a solid, but it must be a solid through which the ions can diffuse. For a fuel cell one requires a ceramic solid in which there are channels along which H+, Li+, Na+, O2- or other appropriate ion can easily diffuse. Structures with lattice induced strain, e.g., compounds similar to La2CuO4, are
39 particularly suitable for ionic conduction because of the extra space available in the stretched layers. Thus the same property that allows O2- to diffuse into the stretched LaO layers of La2CuO4 to form La2CuO4.15 also allows O2- to diffuse through the crystal under the influence of a chemical potential gradient caused when the O2- ions react with hydrogen at the cathode to form water.
9. Epilogue This paper presents an alternative view of the ionic model based on the properties of the electric field rather than the energy. Although this approach has limitations, for example it does not directly give the energy of the system, it provides many insights that the traditional ionic model does not. In either form, the ionic model is restricted to acid-base compounds, i.e., salts, ceramics and minerals, but within these systems it describes all bonds, ionic or covalent, equally well. Unlike other chemical models, the ionic model as developed here leads directly to the concept of a chemical bond whose flux obeys the valence sum rule (eq 2 or 8). The need for quantum mechanical solutions is avoided by treating the short-range forces empirically via the correlation between bond flux and bond length (eq 7).
This model differs in many important ways from the more traditional approaches to describing chemical bonding in inorganic compounds. Lewis structures are frequently used in teaching, but the difficulty is to know how much weight to give to each of the many possible Lewis structures that can be drawn. This is well illustrated by a recent discussion of how one should draw Lewis structures for the sulfate ion, SO42- (32). What are the criteria for preferring one Lewis structure over another? The large number of Lewis diagrams that can be drawn, and the lack of clear criteria for choosing between them can be confusing for the neophyte chemist. Although one can argue that Lewis structures that obey the octet rule give a better physical description, Lewis structures in which all the atoms have zero formal charge necessarily obey the valence sum rule (eq 8) if one assigns a valence of 1.00 to each electron pair bond. Averaging all such structures gives a quantity known as the resonant bond number which measures the number of electron pairs in the bond and should therefore be a measure of covalence. But averaging all equivalent structures leads to the most symmetric distribution of resonant bond number thereby
40 ensuring that the resonant bond number also satisfies the equal valence rule (eq 9). Since resonant bond numbers satisfy the network equations they must be identical to the fluxes or valences of the bond valence model. It is not trivial to calculate resonant bond numbers for crystals since, in theory, one needs an infinite number of Lewis structures spanning every atom in the crystal, but the result has been verified for complex crystalline silicate minerals with the aid of some simplifying assumptions and a lot of computer time (33).
If the resonant bond number is a measure of the number of electron pairs (the covalence) in a bond, and if it is identical to the bond valence, it follows that the bond valence itself is a measure of covalence - the larger the bond valence, the more covalent the bond (34). When describing structure and structure-related properties, it is therefore unnecessary, and often unhelpful, to worry about whether a bond is ionic or covalent. The simple theorems of network theory give the same quantitative description of both ionic and covalent chemical bonding because they describe how the bonding power (whether covalent or ionic) of an atom is distributed to give the most symmetric bonding arrangement. Such descriptions do not, of course, make any statement about how the electrons are distributed between the bonded atoms (35) but, as this paper has shown, they can be used to understand a great deal of chemistry without any need to take electron distributions into account.
Atomic orbitals have only been briefly mentioned in this paper. While a basis set of atomic orbitals is needed to calculate the energy and electron distribution in a molecule or crystal, the usefulness of an orbital description for inorganic compounds is limited. Atoms usually adopt the most symmetric arrangement of neighbours, and the frequency with which 4- and 6-coordination are found has less to do with the availability of s, p or d orbitals than with the fact that the tetrahedron and octahedron are among the few geometries that allow all the bonds to be equivalent. Only for the electronically distorted atoms discussed in Section OL-7.1 is it useful to invoke orbitals (36).
The simple model presented here has many applications in both solution and solid state inorganic chemistry. The examples discussed here show how the model can account for many of the chemical and physical
41 properties of inorganic compounds, including examples that violate the traditional rules of chemistry.
10. Acknowledgements Credit is due to many of my colleagues who, through discussions over morning coffee, have helped to shape this model. Their work is acknowledged in the citations. Drs R.J.Gillespie and D.A.Humphreys read through early drafts of this paper and, with the anonymous referees, provided most helpful and constructive comments. The Natural Science and Engineering Research Council of Canada provided the funds that made the research possible.
42
APPENDIX I
Solution of Network Equations
Methods for solving the network equations have been described by Brown (37), O'Keeffe (38) and Rutherford (39). Here the example of CaCrF5 is worked through in detail.
The symmetry of the bond graph can be used to simplify the calculation by recognising that two bonds joining the same pair of atoms, and symmetry equivalent bonds, must have the same valences. This reduces the number of independent bonds to six, whose valences are given symbolic values a, b, c, d, e and f as shown in Fig. A1. The valence sum equations (eq 8) around Ca, Cr, F1 and F2 are then respectively: 2a + 4b + c = 2
(A1)
2d + 2e +2f = 3
(A2)
a+d =1
(A3)
2b + e = 1
(A4)
The sum equation around F3 is redundant since it is determined by the condition of charge neutrality. Two loop equations (eq 9) are needed to solve for six unknowns. They are a-b+e-d=0
(A5)
a-c+f-d=0
(A6)
One then proceeds to eliminate the variables one at a time. From A3 and A4 one gets d=1-a e = 1 - 2b Substituting these into the remaining four equations gives 2a + 4b + c = 2 (unchanged)
(A1')
43 2a + 4b - 2f = 1
(A2')
2a - 3b = 0
(A5')
2a - c + f = 1
(A6')
From A5' one gets 2a = 3b Substituting this into the remaining three equations gives 7b + c = 2
(A1")
7b - 2f = 1
(A2")
3b - c + f = 1
(A6")
From A1" and A2" one gets c = 2 -7b f = (7b - 1)/2 Substituting this into A6" gives b = 7/27 = 0.26 hence substituting back one finds c = 5/27 = 0.18 f = 11/27 = 0.41 a = 21/54 = 0.39 e = 13/27 = 0.48 d = 33/54 = 0.61 Alternatively the equations can be solved using standard matrix methods.
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