8.5 Magnetism & MO Theory

2s-2p interactions in  MO theory

An important factor influences MO energy order. We know that AOS of similar energy interact to form MOs. We assume that s and p orbitals are so different in energy that they don’t mix. This is true for O, F and Ne. With O₂ we really see the power of MO theory

Now consider the example of O₂. Experimentally, O₂ is known to paramagnetic. According to VBT, O₂ should look like this:

Figure 8.79 O₂ Molecule

Ref: commons.wikimedia.org/

As you can see, all the electrons are paired. Therefore VBT predicts that O₂ should be diamagnetic.

Now let’s examine how the electrons are arranged according to MOT. In MOT, unlike VBT, it involves the creation of bonding and anti-bonding MOs. MOs are basically the supposition of the wavefunctions of atomic orbitals. In O₂ the 2s AOs of each oxygen atom constructively and destructively overlap with each other while their 2p AOs also constructively and destructively overlap with each other The resulting MOs for O₂ looks like this:

Figure 8.80 MO diagram for O₂ Molecule

Ref: commons.wikimedia.org/

Now that we got the MOs, all we have to do is fill them with electrons using the same method that we use for AOs. By doing that we get 2 unpaired electrons. Therefore MOT correctly predicts that O2 should be paramagnetic, unlike VBT which predicts that O_­2 is diamagnetic.

Figure 8.81 Paramagnetic O₂ Molecule

Ref: commons.wikimedia.org/

Figure 8.82 MO diagram for Paramagnetic O₂ Molecule

Ref: commons.wikimedia.org/

3Sigma and pi orbitals

The molecules we have considered thus far are composed of atoms that have no more than four electrons each; our molecular orbitals have therefore been derived from s-type atomic orbitals only. If we wish to apply our model to molecules involving larger atoms, we must take a close look at the way in which p-type orbitals interact as well. Although two atomic p orbitals will be expected to split into bonding and antibonding orbitals just as before, it turns out that the extent of this splitting, and thus the relative energies of the resulting molecular orbitals, depend very much on the nature of the particular p orbital that is involved.

The importance of direction

You will recall that there are three possible p orbitals for any value of the principal quantum number. You should also recall that p orbitals are not spherical like s orbitals, but are elongated, and thus possess definite directional properties. The three p orbitals correspond to the three directions of Cartesian space, and are frequently designated p_x, p_y, and p_z, to indicate the axis along which the orbital is aligned. Of course, in the free atom, where no coordinate system is defined, all directions are equivalent, and so are the p orbitals. But when the atom is near another atom, the electric field due to that other atom acts as a point of reference that defines a set of directions. The line of centers between the two nuclei is conventionally taken as the x axis. If this direction is represented horizontally on a sheet of paper, then the y axis is in the vertical direction and the z axis would be normal to the page.

These directional differences lead to the formation of two different classes of molecular orbitals. The above figure shows how two p_x atomic orbitals interact. In many ways the resulting molecular orbitals are similar to what we got when s atomic orbitals combined; the bonding orbital has a large electron density in the region between the two nuclei, and thus corresponds to the lower potential energy. In the out-of-phase combination, most of the electron density is away from the internuclear region, and as before, there is a surface exactly halfway between the nuclei that corresponds to zero electron density. This is clearly an antibonding orbital— again, in general shape, very much like the kind we saw in hydrogen and similar molecules. Like the ones derived from s-atomic orbitals, these molecular orbitals are σ (sigma) orbitals.

Sigma orbitals are cylindrically symmetric with respect to the line of centers of the nuclei; this means that if you could look down this line of centers, the electron density would be the same in all directions.

Figure 8.85 Pi bonding and antibonding MOs

When we examine the results of the in- and out-of-phase combination of p_y and p_z orbitals, we get the bonding and antibonding pairs that we would expect, but the resulting molecular orbitals have a different symmetry: rather than being rotationally symmetric about the line of centers, these orbitals extend in both perpendicular directions from this line of centers. Orbitals having this more complicated symmetry are called π (pi) orbitals. There are two of them, π_y and π_z differing only in orientation, but otherwise completely equivalent.

The different geometric properties of the π and σ orbitals causes the latter orbitals to split more than the π orbitals, so that the σ* antibonding orbital always has the highest energy. The σ bonding orbital can be either higher or lower than the π bonding orbitals, depending on the particular atom.

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MO splitting patterns for second-row diatomics

If we combine the splitting schemes for the 2s and 2p orbitals, we can predict bond order in all of the diatomic molecules and ions composed of elements in the first complete row of the periodic table. Remember that only the valence orbitals of the atoms need be considered; as we saw in the cases of lithium hydride and dilithium, the inner orbitals remain tightly bound and retain their localized atomic character.

Notice that the relative energies of the 2p-derived σ and π bonding molecular orbitals are reversed in O₂ and F₂. This is attributed to interactions between the 2s orbital each atom with the 2p_x orbital of the other, an effect similar to hybridizaton. However, the order in which these two orbitals are filled has no effect on the predicted bond orders, so there is ordinarily no need to know which molecules follow which scheme.

Figure 8.86 Pi –Pi interactions in MOs

MO’s in dicarbon: Carbon has four outer-shell electrons, two 2s and two 2p. For two carbon atoms, we therefore have a total of eight electrons, which can be accommodated in the first four molecular orbitals. The lowest two are the 2s-derived bonding and antibonding pair, so the “first” four electrons make no net contribution to bonding. The other four electrons go into the pair of pi bonding orbitals, and there are no more electrons for the antibonding orbitals— so we would expect the dicarbon molecule to be stable, and it is. (But being extremely reactive, it is known only in the gas phase.)

Figure 8.87 MO diagram of C₂                            

You will recall that one pair of electrons shared between two atoms constitutes a “single” chemical bond; this is Lewis’ original definitio>n of the covalent bond. In C₂ there are two paris of electrons in the π bonding orbitals, so we have what amounts to a double bond here; in other words, the bond order in dicarbon is two.

More on dicarbon.

MO’s in dioxygen

Figure 8.88 MO diagram of O₂

The electron configuration of oxygen is 1s²2s²2p⁴. In O₂, therefore, we need to accommodate twelve valence electrons (six from each oxygen atom) in molecular orbitals. As you can see from the diagram, this places two electrons in antibonding orbitals. Each of these electrons occupies a separate π* orbital because this leads to less electron-electron repulsion.

The bond energy of molecular oxygen is 498 kJ/mole. This is smaller than the 945 kJ bond energy of N₂— not surprising, considering that oxygen has two electrons in an antibonding orbital, compared to nitrogen’s one.

Although this paramagnetic ground-state form of O₂ (also known as triplet oxygen) is the energetically favored, and therefor is the common form of this element, another variety, in which the two electrons are paired up in a single pi antibonding orbital, is also well known. This singlet oxygen, as it is called, has a bond energy of only 402 kJ/mole. The lower value reflects the action of electrostatic repulsion between the two electrons in the same orbital.

Singlet oxygen, being less stable, does not exist under normal conditions. It can be formed by the action of light and in certain chemical reactions, and it has an interesting and unique chemistry of its own.

Figure 8.89 Paramagnetic behavior of Liquid Oxygen

The two unpaired electrons of the dioxygen molecule give this substance an unusual and distinctive property: O₂ is paramagnetic. The paramagnetism of oxygen can readily be demonstrated by pouring liquid O₂ between the poles of a strong permanent magnet; the liquid stream is trapped by the field and fills up the space between the poles.

Since molecular oxygen contains two electrons in an antibonding orbital, it might be possible to make the molecule more stable by removing one of these electrons, thus increasing the ratio of bonding to antibonding electrons in the molecule. Just as we would expect, and in accord with our model, O₂⁺ has a bond energy higher than that of neutral dioxygen; removing the one electron actually gives us a more stable molecule. This constitutes a very good test of our model of bonding and antibonding orbitals. In the same way, adding an electron to O₂ results in a weakening of the bond, as evidenced by the lower bond energy of O₂^–. The bond energy in this ion is not known, but the length of the bond is greater, and this is indicative of a lower bond energy. These two dioxygen ions, by the way, are highly reactive and can be observed only in the gas phase.

References

Introductory tutorials from various sources

The Chemogenesis WebBook site has a nice page on MO theory

This page from Imperial College (London) has MO diagrams for molecules such as ethane, ethylene, and water.

Wikipedia article on molecular orbitals

Visualizations

Viewer applet for the MO’s of the hydrogen molecule ion

Atom-in-a-box (Mac only) – Orbital viewer (Windows only)

More advanced stuff (probably best avoided by first-year students!)

A more advanced introduction to MO theory (J Harvey, U of Bristol)

This one is even more advanced, and introduces the mathematical methods used to calculate molecular properties

What you should be able to do

1. Make sure you thoroughly understand the following essential ideas which have been presented above.
2. In what fundamental way does the molecular orbital model differ from the other models of chemical bonding that have been described in these lessons?
3. Explain how bonding and antibonding orbitals arise from atomic orbitals, and how they differ physically.
4. Describe the essential difference between a sigma and a pi molecular orbital.
5. Define bond order, and state its significance.
6. Construct a “molecular orbital diagram” of the kind shown in this lesson for a simple diatomic molecule, and indicate whether the molecule or its positive and negative ions should be stable.

*** Adapted from Chem I Virtual textbook  http://www.chem1.com/acad/webtext/virtualtextbook.html.