Molecular Orbital Theory | Research & Encyclopedia Articles

Molecular Orbital Theory

Atoms are the fundamental building blocks of all matter. This is the basis of chemistry. But if they are the bricks, then what is the mortar? What holds the atoms together to form molecules? The answer is "bonds" which, of course, doesn't really explain anything because it raises the next question--what is a bond? It is the nature of science that the answer to one question invariably leads to another question.

A simple thought experiment can be used to determine the basic nature of bonds. Electrostatics tells us that oppositely charged particles attract and that similarly charged particles repel. This means that nuclei will repel each other, as will electrons. Simply trying to push two nuclei together will not allow for the formation of a molecule. But attractions do exist between electrons and nuclei and this tends to favor molecule formation. Here's what happens when two hydrogen atoms approach to within bonding distance. Each atom has a single electron that is attracted to its own nucleus but also is attracted to the nucleus of the other atom. There are four such interactions--each of the two electrons are attracted to two nuclei. These four interactions overcome the electron-electron and nuclear-nuclear repulsions as they develop, allowing a molecule to form. In this case, a bond is a pair of shared electrons.

While this is qualitatively the picture of bonding, it is by no means complete. For example, if pure electrostatic interaction was all that was necessary then the more electrons and protons an atom had, the stronger it would bond.Metals, such as gold, would have so many bonding interactions that they would never melt or be malleable. The simplistic view of bonding outlined above is represented by Lewis structures, for example, and a great deal of chemistry can be done without ever pursuing the exact nature of the chemical bond any further. Most synthetic chemists don't need to know the details.

Various models have been put forward to explain molecular interactions. The two most dominant are the "valence bond approach" and "molecular orbital theory." They are very similar in the way that they describe molecules and come to roughly the same conclusion but from slightly different angles. In grossly simplified terms, the valence bond approach fills the atoms with electrons and then interacts the orbitals, whereas molecular orbital theory interacts the orbitals and then fills them with electrons.

Orbitals are a mathematical construct used to explain the location of electrons around atoms and their physical properties. No one has ever seen an orbital. But they are completely consistent with all that we know about atoms and the way that electrons behave in atoms. This may sound like metaphysics--and it is--but it is important to remember that orbitals are a consequence of the quantum mechanics and the equations that underlay it. This is important because representing orbitals by equations means that adding together equations should give new combinations or orbitals. This is exactly what molecular orbital theory does. By treating the molecule as a single unit and all of the atoms in the molecule as part of that unit, a new set of orbitals for the whole molecule are constructed out of linear combinations of atomic orbitals (LCAO).

Linear combinations of atomic orbitals have the property that orbitals are conserved. Thus, if two orbitals are combined, then the product is two orbitals. If three orbitals are combined, then three new orbitals are created. The two new orbitals created from the original orbitals, designated Ψ and Ψ, are (Ψ+Ψ) and (Ψ- Ψ), i.e., the sum and difference of the original orbitals. This is what is meant by the term "linear combination".

Orbitals, though, are really electron density distributions and the symbol Ψ represents the solution to the wave function. The combination (Ψ+Ψ) is a "piling up" of electron density. This is a simplistic view, but useful as this combination is bonding. It results in an increase in the electron density between the atomic nuclei. But if (Ψ+Ψ) is an increase in electron density, then (Ψ- Ψ) must represent the removal of electron density from between the nuclei. And if an increase in electron density is bonding, then what is a decrease? The answer is that (Ψ-Ψ) represents an anti-bonding orbital--a combination that negates the bonding interaction. The energy of a bonding orbital is decreased relative to the atomic orbital and the antibonding orbital is increased. The energy differences are equal and opposite in sign so that energy is conserved in the system. The energy of the two atomic orbitals is the same as the combination of the bonding and anti-bonding orbitals.

Consider the interaction of two hydrogen atoms. The molecular combination has a bonding and anti-bonding pair of orbitals generated from the 1s atomic orbital on each atom (Figure 1). Each atom also has a single electron. In forming H, these electrons fill the molecular orbitals by the Aufbau principle--from the lowest energy orbital up. The result in H is that both electrons reside in the bonding combination. The resulting molecule is stabilized relative to the individual atoms. On the other hand, consider trying to form the dihelium molecule, He. In this case, the 1s orbital also would form a bonding and anti-bonding pair. But with each atom contributing two electrons, the result would be the filling of both the bonding and anti-bonding combinations. The energy required to put the electrons into the anti-bonding orbital is slightly more (due the overlap integral) than the energy that is obtained from the electrons in the bonding orbital. This is the reason that He doesn't form. It is energetically unfavorable relative to the separate atoms. Indeed, this is the reason that none of the noble gases occurs as a diatomic molecule. It would cost energy and there is no profit in it.

Now, consider the hydrogen molecule again. What are the consequences of combining a hydrogen atom with a hydrogen ion--H with H+ ? In this case, the bonding orbital would only have one electron in it, but this enough to cause the molecule to form. The result is H+ , which is a real, experimentally observable molecule containing a half bond between the hydrogens. The energy of this bond is about one half that of the hydrogen-hydrogen interaction and the bond length is 106 pm versus the 74.2 pm of , as would be expected for a weaker interaction. Similarly, it is possible to combine H- with H to give H- . Again, there is a half bond between the atoms, but in this case it is composed of two electrons in the bonding orbital and one in the anti-bonding. The bond order is defined as one half of the difference between the number of bonding and anti-bonding electrons.

Consider a slightly more complicated molecule, dilithium. It does exist, although it is unlikely to ever be used to power a starship, such as Star Trek's Enterprise. In fact, it is a rather ordinary molecule. In this case, each lithium atom contributes two orbitals to the molecule, a 1s and a 2s orbital. And much as you would expect, these result in bonding and anti-bonding combinations. The 1s orbitals overlap with each other and the 2s orbitals do likewise. The 1s orbital on one atom doesn't interact with the 2s on the other due to the relative energy differences. The 2s orbital is much higher in energy and inaccessible to the 1s orbital. Dilithium, therefore, has two bonding combinations and two anti- bonding combinations and with three electrons contributed from each atom, the net result is a single bond (i.e. one half of four minus two).

So far, as described, molecular orbital theory hasn't really differentiated itself from the Lewis model. Bonds are still the result of electron pairs residing in bonding molecular orbitals. It explains why simply adding more electrons and protons doesn't result in massive bonding. The anti-bonding orbitals insure that the bond order never gets too high. But this can be explained in a Lewis approach using lone pairs. What does molecular orbital theory tell us that makes chemists believe that it is a good model for bonds? Consider the p-orbitals. Unlike the s- orbitals, which are spherically symmetric, the p- orbitals have direction. They orient along the x, y, and z axes of a Cartesian coordinate system. If a line joining the two nuclei is defined as the z- axis, then it is fairly easy to see that the P orbitals will be pointing at each other. The result is a bonding and anti-bonding combination, usually designated and * , respectively, to distinguish them from the and * orbitals that are generated from the s-orbitals. The means that the orbital is spherically symmetric around the bond axis, while the asterisk is used to designate anti-bonding orbitals.

But what about the interaction of the P and P orbitals? Their interaction is much weaker because of the distance separating them, but they do overlap with their opposite partner to give a bonding and anti-bonding combination. These combinations are designated and * where the symbol means that they have a nodal plane that includes the bond axis. Viewed down the bond axis, -bonds look like s- orbitals and -bonds look like p-orbitals.

Since P and P are equivalent, they have the same energy, but they don't interact with each other because of their spatial orientation. The result is that there are two sets of and * orbitals created with the same energy. Furthermore, since the p-orbital overlap isn't as strong as the s-orbital overlap, the resulting difference in energy between the bonding and anti-bonding orbital is not as large. The result is that is filled after , but * is filled before * . That is, the filling order for a molecule such as oxygen is: , * , , * , , 2(), 2(pi;* ), * . With each oxygen contributing eight electrons, the highest occupied molecular orbital (HOMO) is the anti-bonding interaction and since there are two degenerate orbitals, each orbital gets one electron. This results in oxygen being a "diradical" and paramagnetic, both of which properties can be confirmed experimentally. This result is explained only by molecular orbital theory.

The HOMOs of oxygen could also be designated as SOMOs or singly occupied molecular orbitals. The next orbital up in energy, which is left unoccupied, is the LUMO or lowest unoccupied molecular orbital. Collectively, these are termed the "frontier orbitals", because it is at the frontier of the molecular orbitals that chemistry happens. For example, it is these orbitals that are involved when oxygen binds to hemoglobin or when it reacts with combustible materials.

Molecular orbital theory is the best explanation of molecular bonding that chemists have. In its simplest form, it provides qualitative explanations for the formation and reactivity of molecules. It can also be used with rigorous calculations to provide the energy level diagrams for molecules that qualitatively explain the results of photoelectron spectroscopy. In addition, it can also lead to daunting calculations as each atom contributes its orbitals to the whole molecule. While this explains such phenomena as the hyperfine and superhyperfine coupling observed for the magnetic resonance spectroscopies, it does destroy the intuitive picture of one bond/two electrons that was originally used to describe bonding. In the end, though, the advantages of molecular orbital theory vastly outweigh the disadvantages.

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