Just as we treated the bonding in H2 by using molecular orbital theory, we can consider the molecular orbital description of other diatomic molecules. Here we will restrict our discussion to homonuclear diatomic molecules (those composed of two identical atoms) of elements in the second row of the periodic table. As we will see, the procedure for determining the distribution of electrons in these molecules closely follows the one we used for H2.
Second-row atoms have more than one atomic orbital. We will therefore consider some rules that relate to the formation of molecular orbitals, and the way we place electrons in them:
1. The number of molecular orbitals formed equals the number of atomic orbitals combined.
2. Atomic orbitals combine most effectively with other atomic orbitals of similar energy.
3. The effectiveness with which two atomic orbitals combine is proportional to their overlap with one another; that is, as the overlap increases, the bonding orbital is lowered in energy, and the antibonding orbital is raised in energy.
4. Each molecular orbital can accommodate, at most, two electrons, with their spins paired (Pauli exclusion principle).
5. When molecular orbitals have the same energy, one electron enters each orbital (with the same spin) before spin pairing occurs (Hund's rule).
Lithium, the first element of the second period, has a 1s22s1 electron configuration. When lithium metal is heated above its boiling point (1347°C), Li2 molecules are found in the vapor phase. The Lewis structure for Li2 indicates a LiLi single bond. We will now address the description of the bonding in Li2 in terms of molecular orbitals.
Because the 1s and 2s orbitals of Li are so different in energy, we may assume that the 1s orbital on one Li atom interacts only with the 1s orbital on the other atom (rule 2). Likewise, the 2s orbitals interact only with each other. The resulting energy-level digram is shown in Figure 9.35. Notice that we are combining a total of four atomic orbitals to produce four molecular orbitals (rule 1).
FIGURE 9.35 Energy-level diagram for the Li2 molecule.
The 1s orbitals of Li combine to form 1s and *1s bonding and antibonding orbitals, as they did for H2. The 2s orbitals interact with one another in exactly the same way, producing bonding (2s) and antibonding (*2s) molecular orbitals. Because the 2s orbitals of Li extend farther from the nucleus than do the 1s, the 2s orbitals overlap more effectively. As a result, the energy separation between the 2s and *2s orbitals is greater than that for the 1s-based molecular orbitals. Note that, despite being an antibonding orbital, the *1s molecular orbital is lower in energy than the 2s bonding orbital. The 1s orbitals of Li are so much lower in energy that their antibonding orbital is still well below the 2s-based bonding orbital.
Each Li atom has three electrons, so six electrons must be placed in the molecular orbitals of Li2. As shown in Figure 9.35, these occupy the 1s, *1s, and 2s molecular orbitals, each with two electrons. There are four electrons in bonding orbitals and two in antibonding orbitals, so the bond order equals (4 - 2) = 1. The molecule has a single bond, in accord with its Lewis structure.
Because both the 1s and *1s molecular orbitals of Li2 are completely filled, the 1s orbitals contribute almost nothing to the bonding. The single bond in Li2 is due essentially to the interaction of the valence 2s orbitals on the Li atoms. This example illustrates the general rule that core electrons usually do not contribute significantly to bonding in molecule formation. The rule is equivalent to our use of only the valence electrons when drawing Lewis structures. Thus, we need not consider further the 1s orbitals while discussing the other second-row diatomic molecules.
The molecular orbital description of Be2 follows readily from the energy-level diagram for Li2. Each Be atom has four electrons (1s22s2), so we must place eight electrons in molecular orbitals. Thus, we completely fill the 1s, *1s, 2s, and *2s molecular orbitals. We have an equal number of bonding and antibonding electrons, so the bond order equals 0. Consistent with this analysis, Be2 does not exist.
Before we can consider the remaining second-row molecules, we must look at the molecular orbitals that result from combining 2p atomic orbitals. The interaction between p orbitals is shown in Figure 9.36, where we have arbitrarily chosen the internuclear axis to be the z axis. The 2pz orbitals face each other in a "head-to-head" fashion. Just as we did for the s orbitals, we can combine the 2pz orbitals in two ways. One combination concentrates electron density between the nuclei and is therefore a bonding molecular orbital. The other combination excludes electron density from the bonding region; it is an antibonding molecular orbital. In each of these molecular orbitals, the electron density lies along the line through the nuclei. Hence, they are molecular orbitals: 2p and *2p.
FIGURE 9.36 Contour representations of the molecular orbitals formed by the 2p orbitals on two atoms. Each time we combine two atomic orbitals, we obtain two molecular orbitals: one bonding and one antibonding. In (a) the p orbitals overlap "head-to-head" to form and * molecular orbitals. In (b) and (c) they overlap "sideways" to form and * molecular orbitals.
The other 2p orbitals overlap in a sideways fashion and thus concentrate electron density on opposite sides of the line through the nuclei. Molecular orbitals of this type are called pi () molecular orbitals. We get one bonding molecular orbital by combining the 2px atomic orbitals and another one from the 2py atomic orbitals. These two 2p molecular orbitals have the same energy; they are degenerate. Likewise, we get two degenerate *2p antibonding molecular orbitals.
The 2pz orbitals on two atoms point directly at one another. Hence, the overlap of two 2pz orbitals is greater than that for two 2px or 2py orbitals. From rule 3 we therefore expect the 2p molecular orbital to be lower in energy (more stable) than the 2p molecular orbitals. Similarly, the *2p molecular orbital should be higher in energy (less stable) than the *2p molecular orbitals.
We have thus far independently considered the molecular orbitals that result from s orbitals (Figure 9.32) and from p orbitals (Figure 9.36). We can combine these results to construct an energy-level diagram (Figure 9.37) for homonuclear diatomic molecules of the elements boron through neon, all of which have valence 2s and 2p atomic orbitals. Several features of the diagram are notable:
1. As we expect for many-electron atoms, the 2s atomic orbitals are lower in energy than the 2p atomic orbitals. (For more information, see Section 6.7) Consequently, both of the molecular orbitals that result from the 2s orbitals, the bonding 2s and antibonding *2s, are lower in energy than the lowest-energy molecular orbital that is derived from the 2p atomic orbitals.
2. As previously discussed, the overlap of the two 2pz orbitals is greater than that of the two 2px or 2py orbitals. As a result, the bonding 2p molecular orbital is lower in energy than the 2p molecular orbitals, and the antibonding *2p molecular orbital is higher in energy than the *2p molecular orbitals.
3. Both the 2p and *2p molecular orbitals are doubly degenerate; that is, there are two degenerate molecular orbitals of each type.
Before we can add electrons to the energy-level diagram in Figure 9.37, there is one more effect that we must consider. We have constructed the diagram by assuming that there is no interaction between the 2s orbitals on one atom and the 2p orbitals on the other. In fact, such interactions can and do take place; recall Figure 9.12(b), which showed the overlap of a 1s orbital on an H atom with a 3p orbital on a Cl atom. These interactions affect the energies of the 2s and 2p molecular orbitals in such a way that these molecular orbitals move farther apart in energy, the 2s falling and the 2p rising in energy (Figure 9.38). These 2s-2p interactions are strong enough that the energetic ordering of the molecular orbitals can be altered: For B2, C2, and N2, the 2p molecular orbital is above the 2p molecular orbitals in energy. For O2, F2, and Ne2, the 2p molecular orbital is below the 2p molecular orbitals.
FIGURE 9.37 General energy-level diagram for molecular orbitals of second-row homonuclear diatomic molecules. The diagram assumes no interaction between the 2s atomic orbital on one atom and the 2p atomic orbitals on that same atom.
Given the energy ordering of the molecular orbitals, it is a simple matter to determine the electron configurations for the second-row diatomic molecules B2 through Ne2. For example, a boron atom has three electrons (remember that we are ignoring the inner-shell 1s electrons). Thus, for B2 we must place six electrons in molecular orbitals. Four of these fully occupy the 2s and *2s molecular orbitals, leading to no net bonding. The last two electrons are put in the 2p bonding molecular orbitals; one electron is put in each 2p orbital with the same spin. Therefore, B2 has a bond order of 1. Each time we move to the right in the second row, two more electrons must be placed in the diagram. For example, on moving to C2, we have two more electrons than in B2, and these electrons also are placed in the 2p molecular orbitals, completely filling them. The electron configurations and bond orders for the diatomic molecules B2 through Ne2 are given in Figure 9.39.
FIGURE 9.38 When the 2s and 2p orbitals interact, the 2s molecular orbital falls in energy and the 2p molecular orbital rises in energy. For O2, F2, and Ne2, the interaction is small, and the 2p molecular orbital remains below the 2p molecular orbitals, as in Figure 9.37. For B2, C2, and N2 the 2s-2p interaction is great enough that the 2p molecular orbital rises above the 2p molecular orbitals, as shown on the right.
FIGURE 9.39 Molecular orbital electron configurations and some experimental data for several second-row diatomic molecules.
It is interesting to compare the electronic structures of these diatomic molecules with their observable properties. The behavior of a substance in a magnetic field provides an important insight into the arrangements of its electrons. Molecules with one or more unpaired electrons are attracted into a magnetic field. The more unpaired electrons in a species, the stronger the force of attraction. This type of magnetic behavior is called paramagnetism.
Substances with no unpaired electrons are weakly repelled from a magnetic field. This property is called diamagnetism. Diamagnetism is a much weaker effect than paramagnetism. A straightforward method for measuring the magnetic properties of a substance, illustrated in Figure 9.40, involves weighing the substance in the presence and absence of a magnetic field. If the substance is paramagnetic, it will appear to weigh more in the magnetic field; if it is diamagnetic, it will appear to weigh less. The magnetic behaviors observed for the diatomic molecules of the second-row elements agree with the electron configurations shown in Figure 9.39.
FIGURE 9.40 Experiment for determining the magnetic properties of a sample. (a) The sample is first weighed in the absence of a magnetic field. (b) When a field is applied, a diamagnetic sample tends to move out of the field and thus appears to have a lower mass. (c) A paramagnetic sample is drawn into the field and thus appears to gain mass. Paramagnetism is a much stronger effect than is diamagnetism.
The electron configurations can also be related to the bond distances and bond-dissociation energies of the molecules. As bond orders increase, bond distances decrease and bond-dissociation energies increase. Notice the short bond distance and high bond-dissociation energy of N2, whose bond order is 3. The N2 molecule does not react readily with other substances to form nitrogen compounds. The high bond order of the molecule helps explain its exceptional stability. We should also note that molecules with the same bond orders do not have the same bond distances and bond-dissociation energies. Bond order is only one factor influencing these properties. Other factors, including the nuclear charges and the extent of orbital overlap, also contribute.
Bonding in the dioxygen molecule, O2, is especially interesting. Its Lewis structure shows a double bond and complete pairing of electrons:
The short OO bond distance (1.21 Å) and the relatively high bond-dissociation energy (495 kJ/mol) of the molecule are in agreement with the presence of a double bond. However, the molecule is found to contain two unpaired electrons. The paramagnetism of O2 is demonstrated in Figure 9.41. Although the Lewis structure fails to account for the paramagnetism of O2, molecular orbital theory correctly predicts that there are two unpaired electrons in the *2p orbital of the molecule (Figure 9.39). The molecular orbital description also correctly indicates a bond order of 2.
Going from O2 to F2, we add two more electrons, completely filling the *2p molecular orbitals. Thus, F2 is expected to be diamagnetic and have a FF single bond, in accord with its Lewis structure. Finally, we see that the addition of two more electrons, to make Ne2, leads to the filling of all the bonding and antibonding molecular orbitals; therefore, the bond order of Ne2 is zero, and the molecule is not expected to exist.
Predict the following properties of O2+: (a) number of unpaired electrons; (b) bond order; (c) bond-dissociation energy and bond length.
SOLUTION (a) The O2+ ion has one electron less than O2. The electron removed from O2 to form O2+ is one of the two unpaired * electrons (see Figure 9.39). Therefore, O2+ should have just one unpaired electron left.
(b) The molecule has eight bonding electrons (the same number as O2) and three antibonding ones (one less than O2). Thus, its bond order is
(c) The bond order of O2+ is between that for O2 (bond order 2) and N2 (bond order 3). Thus, the bond-dissociation energy and bond length should be about midway between those for O2 and N2, approximately 720 kJ/mol and 1.15 Å, respectively. The observed bond-dissociation energy and bond length of the ion are 625 kJ/mol and 1.123 Å, respectively.
Predict the magnetic properties and bond order of (a) the peroxide ion, O22-; (b) the acetylide ion, C22-Answers: (a) diamagnetic, 1; (b) diamagnetic, 3