The bond valence model

let s get a bit more chemical and imagine the formation of an H2 molecule from two separated hydrogen atoms, Ha and Hb, initially an infinite distance apart. Electron 1 is associated with nucleus A, electron 2 with nucleus B, and the terms in the electronic Hamiltonian / Ab rA2 and rgi are all negligible when the nuclei are at infinite separation. Thus the electronic Schrodinger equation becomes 

I am going to leave you to prove for yourself that the wavefunction corresponding to this infinite-distance H2 problem is a product of two hydrogen atom wavefunc-tions. Physically, you might have expected this the two atoms are independent and so the electronic wavefunctions multiply to give the molecular electronic wavefunction. 

If I write possible atomic orbitals for hydrogen atom A as Xa aQd possible atomic orbitals for hydrogen atom B as Yb the molecular electronic wavefunction will be 

The x s can individually be Is, 2s, 2p. atomic orbitals. The lowest-energy solution will be when the x s correspond to Is orbitals on each of the two hydrogen atoms, the next-highest-energy solution will be when one of the x s is a Is, the other a 2s atomic orbital, and so on. Possible solutions of the electronic problem, with the two H atoms at infinity, are shown in Table 4.1. 

In this table, eis is the energy of a hydrogenic Is orbital, S2S the energy of a hydrogenic 2s orbital. Before we worry about comparison with experiment, there are a couple of loose ends that have to be tidied up. 

In practice, the valence bond picture has probably exerted more influence on how chemists actually think than the HMO picture. However most early applications were primarily qualitative in nature. This qualitative VB picture can be summarized under die name of resonance theory [10]. The basic concept is that in general the more ways one has of arranging the spin pairing in the VB wave function, the more stable the molecule is likely to be. Thus, VB theory predicts that phenanthrene with 14 carbon atoms and 5 Kekule structures should be more stable than anthracene with 14 carbon atoms but just 4 Kekule structures, in complete accord with the experimental evidence. It also predicts that benzenoid hydrocarbons with no Kekule structures should be unstable and highly reactive, and in fact no such compounds are knowa Extensions of this qualitative picture appear, for example, in Clar s ideas of resonant sextets [11], which seem to be very powerful in rationalizing much of the chemistry of benzenoid aromatic hydrocarbons. The early ascendancy of HMO theory was thus largely based on the ease with which it could be used for quantitative computations rather than on any inherent superiority of its fundamental assumptions. 

---

We 11 expand our picture of bonding by introducing two approaches that grew out of the idea that electrons can be described as waves—the valence bond and molecular orbital models In particular one aspect of the valence bond model called orbital hybridization, will be emphasized... 

We 11 begin our discussion of hydrocarbons by introducing two additional theories of covalent bonding the valence bond model and the molecular orbital model... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The structural features of methane ethane and propane are summarrzed rn Ergure 2 7 All of the carbon atoms have four bonds all of the bonds are srngle bonds and the bond angles are close to tetrahedral In the next sectron we 11 see how to adapt the valence bond model to accommodate the observed structures... 

We saw in the last chapter how covalent bonds between atoms are described, and we looked at the valence bond model, which uses hybrid orbitals to account for the observed shapes of organic molecules. Before going on to a systematic study of organic chemistry, however, we still need to review a few fundamental topics. In particular, we need to look more closely at how electrons are distributed in covalent bonds and at some of the consequences that arise when the electrons in a bond are not shared equally between atoms. 

Until about 20 years ago, the valence bond model discussed in Chapter 7 was widely used to explain electronic structure and bonding in complex ions. It assumed that lone pairs of electrons were contributed by ligands to form covalent bonds with metal atoms. This model had two major deficiencies. It could not easily explain the magnetic properties of complex ions. 

The concept of a bond has precise meaning only in terms of a given model or theory. In the Lewis model a bond is defined as a shared electron pair. In the valence bond model it is defined as a bonding orbital formed by the overlap of two atomic orbitals. In the AIM theory a bonding interaction is one in which the atoms are connected by a bond path and share an interatomic surface. 

For example, students develop an elementary understanding of bonding from the Lewis model. Then they refine it through the valence bond model and finally molecular orbital theory. Some exercises challenge students to refine models further—and to develop new ones. Students will see how current chemical knowledge is based on the authority—and the fallibility—of modern experimental techniques. 

The valence bond model of covalent bonding is easy to visualize and leads to a satisfactory description for most molecules. It does, however, have some problems. Perhaps the most serious flaw in the valence bond model is that it sometimes leads to an incorrect electronic description. For this reason, another bonding description called molecular orbital (MO) theory is often used. The molecular orbital model is more complex than the valence bond model, particularly for larger molecules, but sometimes gives a more satisfactory accounting of chemical and physical properties. 

Now that we ve looked at bonding in the H2 molecule, let s move up a level in complexity by looking at the bonding in several second-row diatomic molecules— N2,02, and F2. The valence bond model developed in Section 7.10 predicts that the nitrogen atoms in N2 are triply bonded and have one lone pair each, that the oxygen atoms in 02 are doubly bonded and have two lone pairs each, and that the fluorine atoms in F2 are singly bonded and have three lone pairs each ... 

The valence bond model constructs hybrid orbitals which contain various fractions of the character of the pure component orbitals. These hybrid orbitals are constructed such that they possess the correct spatial characteristics for the formation of bonds. The bonding is treated in terms of localised two-electron two-centre interactions between atoms. As applied to first-row transition metals, the valence bond approach considers that the 45, 4p and 3d orbitals are all available for bonding. To obtain an octahedral complex, two 3d, the 45 and the three 4p metal orbitals are mixed to give six spatially-equivalent directed cfisp3 hybrid orbitals, which are oriented with electron density along the principal Cartesian axes (Fig. 1-9). 

The sequence of energy levels obtained from a simple molecular orbital analysis of an octahedral complex is presented in Fig. 1-12. The central portion of this diagram, with the t2g and e levels, closely resembles that derived from the crystal field model, although some differences are now apparent. The t2g level is now seen to be non-bonding, whilst the antibonding nature of the e levels (with respect to the metal-ligand interaction) is stressed. If the calculations can be performed to a sufficiently high level that the numerical results can be believed, they provide a complete description of the molecule. Such a description does not possess the benefit of the simplicity of the valence bond model. 

To test the generality of the jr-distortivity phenomenon and of the Valence Bond model for delocalization, it is of interest to apply the o-it partition to conjugated molecules other -than hydrocarbons, e.g. containing nitrogen, silicon or phosphorus atoms that we have kept in a constrained planar geometry. The total distortion energies, as well as their a and it components are displayed in Table 3, as calculated at the 6-31G/it-CI level. 

I. D. Brown, The Chemical Bond in Inorganic Chemistry The Valence Bond Model, Oxford University Press, New York, 2002. 

Thus, the simplest wavefunction describing say three bonds using the Valence Bond model in addition to an orthogonal core is given by... 

---