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Molecular Orbitals
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3.6 Molecular Orbitals
Almost done! Up until now in this chapter, you have seen how we came from plum pudding models to atomic orbitals. At the end of this chapter, we will bring in one last concept called molecular orbitals.
As you have hopefully learned from previous sections, atomic orbitals are essentially functions that describe where electrons are the most likely to be found in an atom. However, what (most) scientists care about are not atoms, but rather molecules which are made up of a bunch of atoms. Thus, it is not surprising that the concept of “orbitals” needs to be extended to the context of a molecule.
But solving the Schrödinger equation to get the atomic orbital of a single H atom is already a non-trivial task, how are we supposed to do that to a molecule? Fortunately, scientists came up with an idea which allows us to obtain a decent guess of what molecular orbitals look like, but involving minimal effort. Motivated by the principle of superposition for waves, the molecular orbitals can be thought of as a superposition of atomic orbitals. This is known as the linear combination of atomic orbitals (LCAO).
Let's start with something very simple: an H_{2} molecule. This molecule is obviously formed by two hydrogen atoms, which we name H_{a} and H_{b}. These two atoms each have their 1s orbitals, which are denoted as \chi_{a} and \chi_{b}. When these two atoms approach each other, the electron clouds surrounding them overlap and merge together, and thus the electron probability distribution needs be described by new functions (molecular orbitals). The possible LCAOs are:
\begin{equation}
\psi_{+}=N(\chi_{a}+\chi_{b})\hspace{30pt}\psi_{-}=N(\chi_{a}-\chi_{b})
\end{equation}where N is the normalisation factor. An intuitive way to understand this is to think of the wave functions as actual waves: if you recall from physics courses in the past, waves can interfere with each other either constructively or destructively. In this case, we can understand \psi_{+} as a result of constructive interference between \chi_{a} and \chi_{b}, and \psi_{-} as a result of destructive interference (see the figure below for visualization).
Moreover, just like how each atomic orbital has its corresponding energy, here the two molecular orbitals we created also have specific energies. The molecular orbital that arises from constructive interference (\psi_{+}) has the energy E_{1}, which is lower than the energies of the two atomic orbitals. On the other hand, \psi_{-}, which comes from destructive interference, has the energy E_{2}, which is higher than the energies of the two atomic orbitals. By filling both electrons into \psi_{+}, the total energy of the molecule is lowered compared to the two individual atoms. This is where bonding comes from! This is why we give \psi_{+} is called the bonding orbital, and \psi_{-} the antibonding orbital. In summary, this is all shown in the molecular orbital diagram below.
Just like how transitions between different energy levels in a hydrogen atom, transitions between molecular energy levels also correspond to absorbing and emitting light. A famous example is \beta-carotene, which is the pigment that gives carrots their orange colour. The most prominent energy transition of this molecule corresponds to the absorption of blue light, which is why this molecule is orange in colour!
Try it yourself!
Shown above is the molecular structure of \beta-carotene. Given that its most prominent energy transition corresponds to an energy gap of 2.050 * 10^{-19} J, verify that this corresponds to blue light. (hint: use E=\frac{hc}{\lambda}!)
Let's make one final return to the H_{2} molecule. What do you expect its emission spectrum to look like? Following the logic shown above, you may expect to see one single emission line that corresponds to the energy E_{2}-E_{1}. However, take a look at Table 1 of this paper: the actual emission spectrum of H_{2} observed by telescopes is actually much more complex!
Why? Chapter 4 awaits you!