Molecular Orbital theory (MO) is the most important quantum mechanical theory for describing bonding in molecules. It is an approximate theory (as any theory that utilizes "orbitals"), but it is a very good approximation of the bonding.
The MO perspective on electrons in molecules is very different from that of a localized bonding picture such as valence bond (VB) theory. In VB we describe particular bonds as coming from the overlap of orbitals on atomic centers. In MO this idea is not completely gone, but now rather than just looking at individual bonds, MO describes the whole molecule as one big system. The orbitals from MO theory are spread out over the entire molecule rather than being associated with a bond between only two atoms. Each MO can have a particular shape such that some orbitals have greater electron density in one place or another, but in the end the orbitals now "belong" to the molecule rather than any particular bond.
For diatomic molecules (which we look at a lot), the VB picture and the MO picture are very similar. This is because the whole molecule is simply two atoms bonded together. The difference become more apparent when we look at MO in larger molecules.
Why in the world would you ever do a molecular orbital calculation? There are many reasons, but one of the most common is to understand the geometry of a molecule. The calculation can answer the following questions: What are the bond lengths? What are the bond angles? What are the bond strengths?
The MO calculation is a quantum mechanical calculation similar to those discussed for atomic systems. The calculation yields a set of orbitals (wavefunctions) and energies for the molecule. The electrons in the system (the molecule) can then be put into these orbitals starting from the lowest energy orbitals moving up to the highest. The wavefunctions (and their energies) for the electrons are calculated for a specific fixed position of the atomic nuclei (this is called the Born-Openheimer approximation). In this way the energy for the molecule can be calculated for a given geometry and compared to the energy of a different geometry. This is the key idea. The To find the lowest energy geometry, the energy for the molecule is calculated for all possible positions of the atoms. This meas that calculation is essentially repeated over and over and over again until the position of minimum energy is found. This lowest energy configuration is considered the ground state of the molecule. From this geometry we can get the lengths, angles, and strengths of all the bonds in the molecule. This is very hard to visualize for a large molecule as there are many degrees of freedom. However, we can easily imagine this for a diatomic system such as H2. For they hydrogen molecule there is only one thing we can change, the distance between the two nuclei. Therefore, one can perform an MO orbital calculation for different distances between the two nuclei and plot the ground state energy as of this distance. Such a plot is shown below.
You have seen this plot before when we introduced the idea of covalent bonding. Only now we can actually calculate quantitatively the energy of the molecule as a function of the distance between the two nuclei. Thus this calculation for this diatomic molecule would yield the information that the minimum energy is at a separation of 74 pm. We would say the bondlength of this molecule is 74 pm. Moreover, the difference in energy between the molecule at this distance (-436) and the atoms at infinite separation (0) is 436 kJ mol-1. We would say the bond energy for this diatomic bond is 436 kJ mol-1
This diagram is essentially a comparison between the energy of the atoms and the energy of the molecule. In MO theory we are often comparing the energy of the atomic orbitals (AO) to the energy of the molecular orbitals (MO). While this comparison could be done at any distance, we are alway making this comparison between the atom (at infinite distance) and the molecule at it lowest energy position (the bond length distance). When the MO energy is lower than the AO energy it is important to realize that this comparison is at a particular distance (the minimum energy distance).
The simplest molecule to understand is molecular hydrogen, H2, because it is a combination of the simplest atom.
What do we get from the quantum mechanical solution to the hydrogen molecule using MO theory? Just as with quantum mechanical calculations for atomic system, the MO calculation yields orbitals (wavefunctions) and energies. The electrons can then be placed into those orbitals starting at the lowest energy orbitals with two electrons going into in each orbital. A key difference is that now we must choose a geometry at which to perform the MO calculation. For H2 the geometry is defined by the separation between the two hydrogen atoms.
To examine the "bonding" we can compare the molecular orbitals (MO) to the atomic orbitals (AO). We need to make this comparison at a particular distance, since the MOs depend on the atomic separation. Generally we will look at the MOs at a separation that minimizes their energy (the bond length). We can represent this comparison by the following diagram.
Here we show the MOs in the middle and the AOs on each side of the diagram. Energy is lowest at the bottom of the diagram and increases moving up. This particular diagram shows the orbitals for both the hydrogen atom and the hydrogen molecule. The AOs are the 1s orbitals from the hydrogen atom. The MOs, in the middle of the diagram, shows the two lowest energy MOs. Each hydrogen atom has one electron. For the atomic system we put one electron into each hydrogen (one atom is depicted on the left side and the other on the right side). This can be compared to the molecular system by placing these same electrons into the MOs. There are two total electrons, so we put these same two electrons into the MOs. Now both electrons can go into the lowest energy MO (the one label \(\sigma\)). This orbital is lower in energy than the AO orbitals. This means the energy of the molecule (the two electrons in the sigma orbital) is lower in energy than the separated atoms (one electrons in each 1s AO). The molecule is more stable than the atoms. Thus we have a chemical bond between these two atoms (they are lower in energy together than apart).
Moreover, the MO calculation is quantitative, and we know not only that it is lower, but how much lower and at precisely what distance! Not all the MOs are lower in energy than the AOs. You can see the one labeled \(\sigma\) is lower while the one labeled \(\sigma *\) is higher in energy. MOs that are lower in energy than their corresponding AOs we call "bonding". MOs that are higher in energy than their corresponding AO's we call "anti-bonding". If the MOs happen to have an identical energy (or very similar energy) to the AOs we label them as "non-bonding".
We can understand the nature of bonding orbitals by examining how they are formed from the atomic orbitals as the atoms are brought together. Bonding orbitals are formed when atomic orbitals combine in ways that lead to predominantly constructive interference. The key feature of bonding orbitals is that the molecular orbitals have a lower energy than the corresponding atomic orbitals. Thus, the molecule (the atoms separated by at a particular small distance) has a lower energy than the separated atoms (atoms separated by a large distance).
Another characteristic of bonding orbitals is that the electron density is found between the atoms. This leads to our idea that covalent bonding is "shared" electrons. The electrons have a high probability of being between the nuclei in the molecule.
σ-bonding is due to the end-to-end overlap of orbitals having constructive interference (in phase). All σ-bonding is "on axis" meaning the electron density is centered directly between the two bonding nuclei. Below are two figures showing σ-bond formation. The first figure shows two s-orbitals overlapping to give the σ-bond. The second figure shows the end-to-end overlap of two p-orbitals to give the σ-bond.
In diatomic systems, the MO look remarkably like the bonds we think about with VB theory since the whole molecule is only two atoms (and the bond between them). Thus we use the same labels of "sigma" and "pi" bonds. In larger molecules, the orbitals will have complicated shapes but the same ideas will hold. Lower energy molecular orbitals compared to separate atomic orbitals with higher energy favors bonding.
Similarly, we can understand anti-bonding orbitals by looking at other ways the atomic orbitals can combine. Anti-Bonding orbitals are essentially the "opposite" of bonding orbitals. They are formed when atomic orbitals combine in ways that lead to predominantly destructive interference. The key feature of anti-bonding orbitals is that the molecular orbitals have a higher energy then the corresponding atomic orbitals. Thus the molecule (the atoms separated by a particular small distance) has a higher energy than the separated atoms (atoms separated by a large distance) and the atoms would prefer to be in the lower energy atomic state.
Another characteristic of anti-bonding orbitals is a "node" or place of zero electron density between the atoms. The higher in energy the MO, the more nodes it will have. In general, anti-bonding orbitals have higher energy and more nodes than their bonding counter parts.
We denote anti-bonding orbitals with a * symbol. Thus if someone is talking about π* orbital they are referring to an anti-bonding orbital. Below is a diagram showing the formation of an s-to-s anti-bonding orbital. Note that the orbital has a nodal plane between the two nuclei.
So anytime two atomic orbitals combine to give a lower-energy bonding orbital, an analogous higher energy anti-bonding orbital is also formed. Below is a figure depicting this simple bond/anti-bond molecular orbital diagram that we had for hydrogen. The diagram also shows that electrons (in this case) completely fill the bonding orbital and leave the anti-bonding orbital empty.
A common question is why would "anti-bonding" orbitals exist. In some ways this is a complicated question, but in others it is quite straightforward. When combining atomic orbitals there will always be two ways in which we can add up the atomic orbitals (remember they are just mathematical functions). They can add such that there is constructive interference. This leads to bonding orbitals. Or they can add up such that there is destructive interference. This leads to anti-bonding orbitals. When we are making MOs for molecules with lots of atoms, there are lots of AOs. This means that there is only one orbital in which all the AO add up constructively. However, the lowest energy MOs (the bonding ones) will have more constructive interference and the higher energy MOs (the anti-bonding ones) will have more destructive interference.
For a molecule, we can calculate the bond order to characterize the bonding in the molecule. The bond order is equal to half of the sum of the number of electrons in bonding orbitals minus the number of electrons in anti-bonding orbitals. For diatomics, the bond order will tell us if we have single, double, or triple bonds. For example, below is, again, our MO diagram for hydrogen.
Hydrogen has two electrons in a bonding orbital and zero electrons in anti-bonding orbitals. Therefore the bond order is one. [ B.O = 0.5(2-0) = 1].
For larger molecules, the idea is more difficult to translate as MO deals with the whole molecule rather than individual bonds. For example, below is a picture of the molecular orbitals of methane.
You can see the atomic orbitals from carbon (2s and 2p) and four 1s orbital from hydrogen. The MOs are in the middle. Methane has four σ MOs and four σ* MOs. All of the electrons are in the bonding orbitals. The bond order in methane is then = 0.5(8-0) = 4. This is not because there is a quadruple bond. Instead, this is because MO theory looks at the whole molecule. We would describe methane as having four C-H single bonds. Thus the bond order is four. However, you have to remember that MO looks at the whole molecule not local bonds. This is closer to "reality" as far as QM, but it is confusing conceptually. That is why as chemists we use MO to find geometries and energies, but we talk about bonds and local bonding theories.
Non-Bonding MOs are orbitals that have essentially the same energy in the molecule as they do in the atomic system. Therefore, they don't contribute to the bond order.
Below is a MO diagram for hydrogen fluoride. It shows the 1s orbital for H as well as the valence orbitals for F. The bonding MO is formed from a combination of the 1s on H and a 2p orbital on F. However there are several MOs (in the middle) that are the same as AO on F. You can see there is one that is identical to the 2s orbital and two other that are the same as the 2p on F. In the diagram, these orbitals are filled, but they don't contribute to the bond order. We can think of them as essentially the lone pairs we would get from the Lewis dot structure.
We can build up MO diagram for homonuclear diatomics from our combination of AOs just like we did for the hydrogen molecule. For each diatomic the diagram will be qualitatively the same. However, it is important to realize that the actually MO calculation will yield specific energies and bond lengths that are unique for each diatomic. While this is true, the general ordering of the MOs and their relative energies compared to the AOs is consistant for the 2nd row homonuclear diatomics. Thus if we know this diagram, we can answer a variety of questions regarding these molecules by simply placing the appropriate number of electrons into the diagram.
The following two diagrams are the MO diagrams for the 2nd row diatomic molecules. There are two diagrams because the order of the energy levels is slightly different for O2 and F2 than for the rest of the diatomic molecules. Note the changing in the order of the σ and π bonding orbitals from the 2p atomic orbitals.
If you'd like, here is a full-size pdf of these 2 MO diagrams..
As MO for larger molecules tend to be more complicated (the electrons are spread out all over the big molecule). At this point we generally no longer focus on a comparison between the MOs and the AOs but instead just take the MOs as a set of orbitals for the whole molecule. There are many such orbitals (an infinite number), but we will always focus simply on the electrons that are most important for the chemistry. These are: the highest occupied molecule orbital or HOMO and the lowest un-occupied molecular orbital or LUMO. The HOMO is the highest energy MO that has any electrons in it. The LUMO is the next highest energy orbital (it will be empty). The LUMO is the lowest energy place to put or excite an electron.
The energy difference between the HOMO and LUMO or HOMO-LUMO gap is generally the lowest energy electronic excitation that is possible in a molecule. The energy of the HOMO-LUMO gap can tell us about what wavelengths the compound can absorb. Or alternatively, measuring the wavelengths a compound absorbs in the lab can be used as a measure of the HOMO-LUMO gap.
Finally, for most molecules (that are not extremely symmetric) there are no degenerate molecular orbitals (MOs with the same energy). Thus the MOs are a series of single orbitals of increasing energy. Since most stable molecules have an even number of electrons (closed shell), nearly all molecules will have all their electrons paired in molecular orbitals.
Paramagnetic compounds (and atoms) are attracted to magnetic fields while diamagnetic compounds (and atoms) are repelled from magnetic fields.
Paramagnetic compounds have unpaired electrons while in diamagnetic compounds the electrons all have paired spins.
Very few individual atoms are paramagnetic since this requires having a half-filled MO. In contrast, nearly all molecules are diamagnetic (O2 is a notable exception). That is, they essentially have all paired electrons in MOs.
See this video of a frog floating in a magnetic field. Why does the frog float? The frog is repelled by the magnetic field since all the molecules that make up the frog (or at least the vast majority of them) are diamagnetic. In addition, the magnet is really strong such that the repulsion of the diamagnetic compounds in the frog is strong enough to overcome gravity.
Levitating Frog VideoSometimes chemists will combine ideas from VB theory with the MO to describe the bonding in a molecule. This is because VB theory is nice to describe bonds (it is a local theory looking at individual bonds). In contrast, MO theory looks at the entire molecule rather than individual bonds. However, MO theory better captures ideas of delocalized bonds. To deal with this, chemists will often describe the σ bonds in molecule with VB and the delocalized π bonds with a more molecular orbital description. This is particularly true when we would like to capture the idea of resonance structures.
The short video examines this idea for benzene.
Combining VB Theory and MO Theory