Goldilocks and the Three Energy Levels:

An Introduction to Frontier Molecular Orbital Theory

Bob Hanson

St. Olaf College

October 20, 2000 Draft

This is a guide to the presentations I made in Chemistry 247B, Organic Chemistry, on October 18 and 20, 2000. The subject is Frontier Molecular Orbital Theory, developed by Kenishi Fukui, cowinner of the Nobel Prize in Chemistry in 1981. Much of this material is hard to find or has never been put together quite in this way before, at least as far as I am aware. It is an extension of what I call Data-Driven Chemistry, which I have used in Chemistry 123, Atomic and Molecular Structure. Here I present it more with a bent toward organic chemistry.

Atomic Energy Wells. The story begins with the lowly hydrogen atom. For the hydrogen atom (just a proton and an electron for our purposes), the solution of the Schroedinger Equation indicates an infinite number of possible energy states, all with negative total energy. The picture I would like you to consider is one of an energy well, where zero energy is at the top. Energy is required to rise from state to state. These states get closer and closer in energy as the electron gets higher and higher on the energy scale and (on average) further and further from the nucleus.

Ultimately (at E=0), the system becomes a continuum, where any energy is possible. At this point we have essentially a free electron flying through space at a speed determined by its overall kinetic energy. We say that at this point the atom is ionized, and the energy required to do this is called the the ionization energy:

Atoms Other Than Hydrogen. The premise is that the s, p, d, f orbital solutions to the hydrogen atom wave equation are useful in talking about molecules with more than one electron. That is, we take the known solution to the Schroedinger Equation for the hydrogen atom and extend it through approximate methods to molecules having more than one electron while still preserving the idea of orbitals.

If we do this, than we are likely to say that the energy required to remove an electron from atoms other than hydrogen has to do with the orbitals from which the electrons arise. Basically, the picture I want to introduce extends the hydrogen atom well to other atoms. Realize that this is somewhat of a stretch. But it is amazingly effective. The figure to consider is that of a whole set of atomic energy levels, from hydrogen through sodium (pdf).

Take a close look at this figure. It is set to scale, showing the ionization energy for each atom as a vertical bar dropping down from the continuum. Superimposed upon this is a diagram representing the progressive deepening of the atomic energy level systems of the elements. The periodic ionization energies of the elements show up in that throughout a row of the periodic table, as the nuclear charge increases, electrons are more tightly held, and all orbitals decrease in energy. This makes it generally harder to remove an electron from atoms as we go across the periodic table from left to right.

However, there are exceptions, and these are seen as the result of electron-electron repulsion. For example, in going from nitrogen to oxygen there is a big decrease in the amount of energy required to remove an electron rather than an increase. This anomaly is attributable to the cost of electron pairing. The difference between the ionization of a nitrogen atom and the ionization of an oxygen atom is primarily in that the electron coming out of the oxygen is pair while the one in nitrogen is not. In essence, the electron being removed from oxygen gets an extra boost from its paired partner not available to the electron in nitrogen. The same is true of fluorine and neon, and their ionization energies continue in line with oxygen's.

OK, the point here is that as one level fills and simultaneously drops in energy, another one drops in to replace it. Thus, the properties of atoms going down a period of the table tend to be similar. Sodium is similar in properties to lithium; phosphorus is similar to nitrogen. This is simply the natural consequence of having an infinite supply of empty orbitals that are continually lowering in energy while filling as one goes down the line from atom 1 to atom 110 of the periodic table.

Now for Goldilocks. We can classify the electrons in filled orbitals of atoms into two basic camps: those that are too low in energy to be effectively removed or shared upon reacting with other atoms, and those that are high enough to do so. Sometimes we talk of core and valence electrons, and here we are extending this a bit, arguing that even some of the valence electrons may be too deep for effective reaction. We can think of there being somewhat of a cutoff in energy that, for all practical purposes separates these two camps.

Likewise, the unoccupied orbitals of atoms (those high energy electronic states close to the continuum) can be divided into two camps. Those that are too high in energy can't be effectively used in interactions with other atoms, because as electrons start to interact with these orbitals in making a bond, those external electrons also are repulsed by the electrons already in the atom. The cost of that repulsion can't be overcome by interaction with these high-energy orbitals, and we can essentially ignore them for all practical purposes. On the other hand, vacant orbitals that are lower in energy hold significant promise for reaction. Below some cutoff in energy, they offer a significant stabilization to external electrons that just might overcome the intrinsic repulsion of electrons in the system.

Frontier Molecular Orbital Theory. What Fukui discovered was that these ideas could be extended to molecules as well as atoms. Thus, when two hydrogen atoms react, for example, two electrons from the Goldilocks region are stabilized and drop out of it.

A critical point to realize is that in all such interactions to form a bond, one orbital increases in energy, becoming antibonding (sigma-star) and one decreases in energy, becoming bonding (sigma). In the case of H₂, the stabilization is so great that those electrons are no longer particularly reactive. Fukui referred to the orbitals involved in bonding as Frontier Molecular Orbitals. In particular, he showed that it is largely the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) that are involved in chemical reactions.

Photoelectron Spectroscopy. It is possible to probe the electronic orbital structure of molecules by sending high-energy radiation into them. Only certain energies are allowed, and what can be measured is the amount of energy required to fully ionize the molecule. Electrons ejected from a molecule due to the absorption of light are called photoelectrons, and they can be measured in the technique of photoelectron spectroscopy. In this technique, light of increasingly strong energy is passed through a sample in a strong magnetic field, and any ejected electrons are detected. The result is a spectrum (set of lines) that indicate at what energies of incident light electrons were thrown out of the molecule.

In the accompanying figure, the data for the first-row hydrides, CH₄, NH₃, H₂O, HF, as well as neon are shown. This is a very special and informative group, having the same number of protons and electrons overall, but from one to the next more and more protons are in the centerof the molecule (N has one more proton than C, O one more than N, etc.) and fewer and fewer are on the periphery as H atoms. I've turned the data on its side, with increasing required energy indicated down the left side of the graph. (The symbols next to the lines are spectroscopists' assignments for the peaks in the spectrum.).

Do you see the picture I'm seeing here? Just as for atoms, we see the effect of increasingly positive central nuclear charge. Most importantly, as we go across the data from left to right we see the emergence of lone pairs in the frontier region.

An interpretation of the data in terms of orbitals is given in the next figure. The vertical scale is drawn using the numbers from the photoelectron spectroscopy data. Note that methane, which is not particularly reactive, has no orbitals in the frontier region. The lone pairs of nitrogen and oxygen both are in the region, leading to these compounds' nucleophilic nature. However, notice that the lone pairs in HF and neon are simply too deep to get involved.

In terms of the accepting of electrons (electrophilicity), the antibonding orbitals of both methane and ammonia are too high to be utilized, but the antibonding orbitals in water and HF are quite usable. Note that as a nucleophile approaches either of these two molecules, electrons in the nucleophile interact with the antibonding orbital of water or HF. This weakens the bond to H, and the proton is transferred. Interestingly, water can behave both as a nucleophile and an electrophile, depending upon the context.

Conclusions. The goal of this brief introduction is to get you thinking about the structure and energy of molecules in relation to reactions. You will find that these concepts are very easily transferred to discussions of alkenes, aromaticity, and resonance stabilization of intermediates in organic reactions. Be on the lookout for opportunities to use the ideas presented here. If a substance is an electrophile, can you imagine the antibonding orbital that might get involved? If a substance is a nucleophile, which orbital contains the electrons that are doing the work? The goal is not to have a detailed quantitative picture. Frontier molecular orbital theory gives us the opportunity to focus in on only the orbitals and electrons that are most influential in determining the course of a reaction.