R. JI FRIESEN, editor univer.ity of Waterloo Waterloo, Ontorio, Conoda
Bond Valence Theory: Part I, A New Twist to an Old Description of Chemical Bonding
I. D. Brown Institute for Materials Research McMaster University Hamilton, Ontario L8S 4Ml In the early nineteenth century, the theory of valence was introduced into chemistry to explain the fact that elements combined chemically in simple proportions. The concept proved so fruitful that by mid-century it had evolved into a theory descrihing the arrangement of chemical linkages hetween atoms, the valence of an atom being equal to the numher of such linkages (honds) it could form. By the end of the century, it had reached the point where i t was able to predict the relative spatial arrangement of the honds so that it was possihle to descrihe in a qualitative manner the three-dimensional geometry of molecules. This theory was the more remarkable because i t was based entirely on circumstantial evidence; namely, the chemical properties of the materials produced during reactions. There was no direct evidence for even the existence of atoms and only a vague idea of what their size might he. As late as the first decade of the present century, chemists were still careful to point out that the chemical theories of atoms and molecules were only hypotheses, very useful and compelling hypotheses hut supported by no direct observation. The discovery of X-ray diffraction by crystals in 1912 changed the picture, providing not only direct evidence for the existence of atoms hut allowing their size and, more importantly, their arrangements in solids to be measured. In many cases the structures derived from the valence theory were strikingly confirmed, particularly for those compounds that formed discrete molecules. But for inorganic compounds, the structural predictions of valence theory were found to he less valid. For these compounds a new theory, the ionic or electrostatic theory, soon superseded the old valence theory. Over the vears since 1912. manv further theories have been developed to descrihe the natlre of chemical honding. Some of them have been simde and intuitive, others r i ~ o r ous and complex, hut very fkw of them have been able t o make quantitative predictions of the honding in complex solids. During the course of these developments, the old valence theory has been largely abandoned. One reason for this was the discovery that ionic crystals, such as NaC1, did not form discrete diatomic molecules with one Na and one C1 atom, hut a giant molecule containing around loz0 atoms, each bonded to six atoms of the opposite kind. However, such a "molecule" can he described by the valence theory if we do not insist that each valence forms a single bond. The valence represents the honding power of an atom, but the number of honds formed is determined by I 0 0 / Journal of Chemical Education
the geometrical constraints of packing atoms together. Thus different honds will have different strengths or valences associated with them. The Na atom (valence 1) in NaCl forms six honds, each with a valence of Mth of a valence unit (v.u.) to the C1 atoms. The carbon-carbon double hond in ethylene is a bond with a strength of 2.0 v.u., and the C-C hond in benzene has a strength of 1.5 V.U. The idea of honds of non-integral valence was introduced by Pauling in 1929, and has been found by geologists to he much more useful in descrihing the complex bonding in minerals than either the covalent or the ionic models. In order to introduce partial bond valences, one can modifv the old valence theorv hv treatine the valence of an atom as its bonding power raihe; than t h i number of bonds that it forms: the latrer is ralled the coordination numher. Where the valence and coordination number are the same, as in many organic compounds, each bond will have a strength of 1 v.u., hut in inorganic compounds, the two numbers are usuallv different. and the valence of each atom is distributed between the honds rhat i t forms. It follows from classical valence theorv that it is always ~ossible to associate a valence with each bond in such a way that the sum of the hond valences around each atom is equal to its atomic valence. One can regard the valence of an atom as being the number of electrons that it has available for forming bonds. Each bond is formed by an equal contribution of electrons from the two atoms that form i t so that the hond valence is the numher of electron pairs that are associated with a bond. But how can one have a bond that contains only Mth of a pair of electrons? Each electron occupies an orbital that extends over a considerable volume around an atom and behaves exactly as if its charge were diffused over the whole orbital. If all the electron density of a pair of electrons is concentrated in the region between two atoms, it will tend to attract both nuclei, but if it is in the region hehind the atoms,
CI
Na
a
Figure 1. The pair of valence electrons in NaCl surround the CI atom, but only 1/6 of the pair lies in the binding region between any two atoms.
2.0 1.5 .
11
'
'
" I
$ 1.0
3 z
0-0 F-F
W
0.5
0 W LL
LL W
1
1.0
20 30 ANGSTROMS Figure 2. T b relation between bond valence (uenical)and bond length (horizontal) lor 1) H-0. 2) Li-0. Be-0. 6-0. C-0. N-0. 3) Na-0. Mg-0, A I 4 . Si-0. P-0, S-0. 4) K-0. Ca-0. Si-0. T i ( l V t 0 . V ( V t 0 , CW)--0, and 5) M n ( l l t 0 and F e ( l l l k 0 bonds. it will tend to pull the nuclei apart. Since most orbitals will lie partly in the binding and partly in the antibinding region, only a part of the electron pair will contribute to the bond. In NaCl both bonding electrons will lie close to the C1 atom and symmetrically around i t so that only %th of the electron pair will be in the region between the C1 atom and any one of its six Na neighbors (Fig. 1). In the last twenty years, with the development of more accurate methods of structure determination, it has become apparent that there is a correlation between the length of a bond and its valence. The more valence there is in a bond, the tighter the atoms will be bonded together. On the other hand, the electrons in the core are not involved in bonding and tend to repel a t close distances. The shorter the bond, the greater the repulsive force between the atoms and the larcer the bonding force has to be to balance it. The relationship hetween i o n d length and bond valence is shown for certain bonds in Fieure 2. From the observed bond lengths, one can therefore calculate the bond valences. In oxides and fluorides, these calculated bond valences are found to add up to within 4% of the atomic valences in most inorganic solids. One can also work the calculation the other way. If one can successfully predict the bond valences, one can calculate what the hond lengths should be. In general, there are several ways of assigning valence to the bonds that ensure that the hond valence sums are eaual to the atomic valence. but it's often possible to write down a plausible arrangement by inspection. Some examdes will be eiven in Part 11 of this article. The valence OF a bond obviously depends on the number of bonds formed bv a eiven atom: that is. the coordination number of an atom. ~ h f o r ethe structure of a solid can be predicted, the coordination numbers must be known. These ari determined by the tendency of atoms to bond as closely as possible in a solid. In general, atoms will tend to surround themselves with atoms to which they can bond, but in most cases a limit will be reached when atoms that do not bond with each other (e.g., two oxygen atoms) come into contact. Thus we find the coordination around lithium and sodium is restricted to six oxygen atoms, but around the larger potassium and rubidium atoms it is possible to place seven, eight, or more oxygen atoms. Beryllium, which bas a valence of two and thus binds oxygen more closdy than lithium, can coordinate with only four oxygen atoms while four-valent carbon is only able to
0
2.0
3.0
X - X DISTANCE
Figure 3. The minimum 0-0 and F-F given effective valence (venicsl).
distences (horizontal)allowed for a
V Figure 4. Calculation of the effective valence for 0 4 repulsions.
coordinate to three oxygen atoms (e.g., C0s2- group). AS the valence of the honds between the central atom and the oxygen increases, the non-bonding oxygen atoms can be brought closer together, so that in the Nos- group with N-0 bonds of 1.67 v.u., the oxygen atoms are only 2.2 A apart compared to a typical separation of 3.0 A when surrounding sodium. The theory is a quantitative theory. I t can be seen from Figure 2 that a number of bond types have the same curve; for example, Li-0, Be-0, B-0, C-0 and N-0 honds all have the same length for a given valence. The same is also true of honds to atoms of the second and third rows of the periodic table. Similar curves can be plotted for the non-bonding contacts between oxygen or fluorine atoms, as shown in Figure 3. In this case, the closest possible 0-0 or F-F approach is plotted against an effective valence; that is, the bond valence available for bringing the non-bonded atoms closer together. This will be the component of the bond valence that lies along the 0-0 or F-F direction; namely, the M-0 bond valence (s) multiplied by the cosine of the angle (ol) between M-0 and 0-0 (see Fig. 4). The more strongly M is bonded to 0 , the closer the 0 atoms can be brought to each other.
Volume 53. Number 2 February 1976 / 101
