Bonding in H2 The Valence Bond Model

Valence bond picture of bonding in H2 as illustrated by electrostatic potential maps. The Is orbitals of two hydrogen atoms overlap to give an orbital that contains both electrons of an H2 molecule. 

A bond in which the orbitals overlap along a line connecting the atoms (the inter-nuclear axis) is called a sigma ( r) bond. The electron distribution in a ct bond is cylindri-cally symmetrical were we to slice through a ct bond perpendicular to the intemuclear axis, its cross section would appear as a circle. Another way to see the shape of the electron distribution is to view the molecule end-on. 

Orbitals overlap along a line connecting the two atoms 

Circular electron distribution when viewing down the H—H bond 

We will use the valence bond approach extensively in our discussion of organic molecules and expand on it shortly. First though, let s introduce the molecular orbital method to see how it uses the Is orbitals of two hydrogen atoms to generate the orbitals of an H2 molecule. 

Recall from Section 1.1 that electron waves in atoms are characterized by their wave function, which is the same as an orbital. For an electron in the most stable state of a hydrogen atom, for example, this state is defined by the Is wave function and is often called the 1 orbital. The valence bond model bases the connection between two atoms on the overlap between half-filled orbitals of the two atoms. The molecular orbital model assembles a set of molecular orbitals by combining the atomic orbitals of all of the atoms in the molecule. 

For a molecule as simple as H2, valence bond and molecular orbital theory produce very similar pictures. The next two sections describe these two approaches. 

In-phase overlap of two Is orbitals gives new orbital encompassing both hydrogen atoms. 

---

As chemists, much of our intuition concerning chemical bonds is built on simple models introduced in undergraduate chemistry courses. The detailed examination of the H2 molecule via the valence bond and molecular orbital approaches forms the basis of our thinking about bonding when confronted with new systems. Let us examine this model system in further detail to explore the electronic states that arise by occupying two orbitals (derived from the two Is orbitals on the two hydrogen atoms) with two electrons. 

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 ... 

Historically, molecular orbital theory was preceded by an alternative and successful description of the bonding in H2. In 1927, W. Heitler and F. London proposed the valence bond theory, in which each electron resides in an atomic orbital. In other words, in this model, the identity of the atomic orbital is preserved. There are two ways in which the two electrons in H2 can be accommodated in the pair of Is atomic orbitals ... 

Interatomic distance is calculated by mathematical modelling of the electron exchange that constitutes a covalent bond. Such a calculation was first performed by Heitler and London using Is atomic wave functions to simulate the bonding in H2. To model the more general case of homonuclear diatomic molecules the interacting atoms in their valence states are described by monopositive atomic cores and two valence electrons with constant wave functions (3.36). 

The application of Walter Heftier and Fritz London s valence bond theory was the first description of the binding forces in the H2 molecule, the simplest neutral molecule. Linus Pauling and John Slater later extended the principles to larger molecules. The key element in their proposal was the synthesis of a bonding wavefunction resulting from a combination of atomic orbitals that link the two atoms in a bond. It was hugely important that this localized approach concurred with the Lewis dot model. For the simplest neutral molecule, H2, the Hamiltonian operator may be written... 

Further evidence for the validity of the frontier orbital approach derives from its success in predicting the shift (increase or decrease) in naked cluster IP upon the chemisorption of small reactant molecules. For all metal clusters examined thus far, H2 chemisorption induces an increase in cluster IP. ° This follows directly from interactions (1) and (2), since the creation of the two new metal-hydride bonding orbitals effectively removes two electrons from the cluster valence orbital manifold. Thus with resjiect to the metal cluster, H2 chemisorption can be viewed as an oxidative addition process. If a one-electron (Aufbau filling) approximation is assumed as above, the Fermi level of the cluster is shifted toward lower energy, that is, there is an increase in IP. As the cluster grows larger, the shift in IP diminishes. This is simply a manifestation of cluster-size-dependent variations in the valence orbital density of states, and is again consistent with the frontier orbital model. 

FIGURE 6.37 The electron density for the if/g and ifil wave functions in the simple valence bond model for H2. (a) The electron density pg for if/g and Pu for calculated analytically as described in the text, (b) Three-dimensional isosurface of the electron density for the ipg wave function, as calculated numerically by Generalized Valence Bond Theory (GVB). 

The valence bond and molecular orbital theories differ in how they use the orbitals of two hydrogen atoms to describe the orbital that contains the electron pair in H2. Both theories assume that electron waves behave much like more familiar waves, such as sound and light waves. One property of waves that is important here is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other ( in phase ) destructive interference occurs when they oppose each other ( out of phase ) (Figure 1.15). In the valence bond model constructive interference between two electron waves is seen as the basis for the shared electron-pair bond. In the molecular orbital model, the wave functions of molecules are derived by combining wave functions of atoms. 

In the quantum mechanical model, electrons are simply one type of wave, and we can picture the formation of chemical bonds as an example of constructive interference between electron waves. But for electron waves from different atoms to interact, they must first occupy the same region of space. Another way of saying this is that the atoms must be positioned so that the valence orbitals from one atom must overlap those of the other atom if a bond is to form. This idea forms the basis for the valence bond model of chemical bonding, in which all bonds are seen as the result of overlap between atomic orbitals. Let s examine this idea of orbital overlap by looking at the simplest possible molecule, H2. 

The molecular orbital (MO) is the basic concept in contemporary quantum chemistry. " It is used to describe the electronic structure of molecular systems in almost all models, ranging from simple Hiickel theory to the most advanced multiconfigurational treatments. Only in valence bond (VB) theory is it not used. Here, polarized atomic orbitals are instead the basic feature. One might ask why MOs have become the key concept in molecular electronic structure theory. There are several reasons, but the most important is most likely the computational advantages of MO theory compared to the alternative VB approach. The first quantum mechanical calculation on a molecule was the Heitler-London study of H2 and this was the start of VB theory. It was found, however, that this approach led to complex structures of the wave funetion when applied to many-electron systems and the mainstream of quantum ehemistry was to take another route, based on the success of the central-field model for atoms introduced by by Hartree in 1928 and developed into what we today know as the Hartree-Foek (HF) method, by Fock, Slater, and co-workers (see Ref. 5 for a review of the HF method for atoms). It was found in these calculations of atomic orbitals that a surprisingly accurate description of the electronic structure could be achieved by assuming that the electrons move independently of each other in the mean field created by the electron cloud. Some correlation was introduced between electrons with... 

---