9.7 Molecular Orbitals

Valence-bond theory and hybrid orbitals allow us to move in a straight-forward way from Lewis dot structures to rationalizing the observed geometries of molecules in terms of atomic orbitals. For example, we can use this theory to understand why methane has the formula CH4, how the carbon and hydrogen atomic orbitals are used to form electron-pair bonds, and why the arrangement of the CH bonds about the central carbon is tetrahedral. This model, however, does not explain all aspects of bonding. It is not successful, for example, in describing the excited states of molecules, which we must understand in order to explain how molecules absorb light, giving them color.

Some aspects of bonding are better explained by another model called molecular orbital theory. In Chapters 6 and 7 we saw that electrons in atoms exist in allowed energy states, which we call atomic orbitals. In a similar way, molecular orbital theory describes the electrons in molecules as existing in allowed energy states called molecular orbitals.

Molecular orbitals have many of the same characteristics as atomic orbitals. For example, molecular orbitals hold a maximum of two electrons (with opposite spins), they have definite energies, and their electron-density distributions can be visualized by using contour representations, as we did when we discussed atomic orbitals. Molecular orbitals are associated with the entire molecule, however, not with a single atom.

The Hydrogen Molecule

To get a sense of the approach taken in molecular orbital theory, consider the hydrogen molecule, H2. Whenever two atomic orbitals overlap, two molecular orbitals form. Thus, the overlap of the 1s orbitals of two hydrogen atoms to form H2 produces two molecular orbitals (Figure 9.32).

FIGURE 9.32 The combination of two H 1s atomic orbitals forms two molecular orbitals of H2. In the bonding molecular orbital, 1s, the atomic orbitals combine constructively, leading to a buildup of electron density between the nuclei. In the antibonding molecular orbital, *1s, the orbitals combine destructively in the bonding region: Note that the *1s orbital has a node between the two nuclei.

The lower-energy molecular orbital of H2 concentrates electron density between the two hydrogen nuclei and is called the bonding molecular orbital. This sausage-shaped molecular orbital results from summing the two atomic orbitals so that the atomic orbital wave functions enhance each other in the bond region. Because an electron in this molecular orbital is strongly attracted to both nuclei, the electron is more stable (at lower energy) than it is in the 1s orbital of the hydrogen atom. Because it concentrates electron density between the nuclei, the bonding molecular orbital holds the atoms together in a covalent bond.

The higher-energy molecular orbital in Figure 9.32 has very little electron density between the nuclei and is called the antibonding molecular orbital. Instead of enhancing each other in the region between the nuclei, the atomic orbitals cancel each other in this region, and the greatest electron density is on opposite sides of the nuclei. Thus, this molecular orbital excludes electrons from the very region in which a bond must be formed. An electron in this molecular orbital is actually repelled from the bonding region and is therefore less stable (at higher energy) than it is in the 1s orbital of a hydrogen atom.

The electron density in both the bonding and the antibonding mo-lecular orbitals of H2 is centered about the internuclear axis, an imaginary line passing through the two nuclei. Molecular orbitals of this type are called sigma () molecular orbitals. The bonding sigma molecular orbital of H2 is labeled 1s, the subscript indicating that the molecular orbital is formed from two 1s orbitals. The antibonding sigma orbital of H2 is labeled *1s (read "sigma-star-one-s"), the asterisk denoting that the orbital is antibonding.

The interaction between two 1s orbitals to form 1s and *1s molecular orbitals can be represented by an energy-level diagram (also called a molecular orbital diagram), like those in Figure 9.33. Such diagrams show the interacting atomic orbitals in the left and right columns and the molecular orbitals in the middle column. Note that the bonding molecular orbital, 1s, is lower in energy than the atomic 1s orbitals, whereas the antibonding orbital, *1s, is higher in energy than the 1s orbitals. Like atomic orbitals, each molecular orbital can accommodate two electrons with their spins paired (Pauli exclusion principle). (For more information, see Section 6.7)

FIGURE 9.33 Energy-level diagram for (a) the H2 molecule, and (b) the hypothetical He2 molecule.

The molecular orbital diagram of the H2 molecule is shown in Figure 9.33(a). Each H atom has one electron, so there are two electrons in H2. These two electrons occupy the lower-energy bonding (1s) molecular orbital with their spins paired. Electrons occupying a bonding molecular orbital are called bonding electrons. Because the 1s orbital is lower in energy than the isolated 1s orbitals, the H2 molecule is more stable than the two separate H atoms.

In contrast, the hypothetical He2 molecule requires four electrons to fill its molecular orbitals, as in Figure 9.33(b). Because only two electrons can be put in the 1s orbital, the other two must be placed in the *1s. The energy decrease from the two electrons in the bonding molecular orbital is offset by the energy increase from the two electrons in the antibonding orbital. (In fact, antibonding molecular orbitals are slightly more unfavorable than bonding orbitals are favorable. Thus, whenever there is an equal number of electrons in bonding and antibonding orbitals, the energy is slightly higher than it is for the isolated atoms and no bond is formed.) Hence, He2 is not a stable molecule. Molecular orbital theory correctly predicts that hydrogen forms diatomic molecules but helium does not.

Bond Order

In molecular orbital theory the stability of a covalent bond is related to its bond order, defined as follows:

That is, the bond order is half the difference between the number of bonding electrons and the number of antibonding electrons. We take half the difference because we are used to thinking of bonds in terms of pairs of electrons. A bond order of 1 represents a single bond, a bond order of 2 represents a double bond, and a bond order of 3 represents a triple bond. Because molecular orbital theory also treats molecules with an odd number of electrons, bond orders of are possible.

Because H2 has two bonding electrons and no antibonding ones [Figure 9.33(a)], it has a bond order of (2 - 0) = 1. Because He2 has two bonding electrons and two antibonding ones [Figure 9.33(b)], it has a bond order of (2 - 2) = 0. A bond order of 0 means that no bond exists.

Sample Exercise 9.9

What is the bond order of the He2+ ion? Would you expect this ion to be stable relative to the separated He atom and He+ ion?

SOLUTION The energy-level diagram for this system is shown in Figure 9.34. The He2+ ion has a total of three electrons. Two are placed in the bonding orbital, the third in the antibonding orbital. Thus, the bond order is

Because the bond order is greater than 0, the He2+ molecular ion is predicted to be stable relative to the separated He and He+. Formation of He2+ in the gas phase has been demonstrated in laboratory experiments.

Practice Exercise

FIGURE 9.34 Energy-level diagram for the He2+ ion.

Determine the bond order of the H2- ion. Answer: