10.1 Molecular Solids

The hydrogen molecule is the most basic example in quantum mechanics of how atoms can combine into molecules in order to share electrons. So, the question suggests itself whether, if hydrogen molecules are brought close together in a solid, will the atoms start sharing their electrons not just with one other atom, but with all surrounding atoms? The answer under normal conditions is no. Metals do that, but hydrogen under normal conditions does not. Hydrogen atoms are very happy when combined in pairs, and have no desire to reach out to further atoms and weaken the strong bond they have already created. Normally hydrogen is a gas, not a metal.

However, if you cool hydrogen way down to 20 K, it will eventually condense into a liquid, and if you cool it down even further to 14 K, it will then freeze into a solid. That solid still consists of hydrogen molecules, so it is called a molecular solid. (Note that solidified noble gases, say frozen neon, are called molecular solids too, even though they are made up of atoms rather than molecules.)

The forces that glue the hydrogen molecules together in the liquid and solid phases are called Van der Waals forces, and more specifically, they are called London forces. (Van der Waals forces are often understood to be all intermolecular forces, not just London forces.) London forces are also the only forces that can glue noble gas atoms together. These forces are weak.

It is exactly because these forces are so weak that hydrogen must be cooled down so much to condense it into liquid and finally freeze it. At the time of this writing, that is a significant issue in the hydrogen economy. Unless you go to very unusual temperatures and pressures, hydrogen is a very thin gas, hence extremely bulky.

Helium is even worse; it must be cooled down to 4 K to condense it into a liquid, and under normal pressure it will not freeze into a solid at all. These two, helium and hydrogen are the worst elements of them all, and the reason is that their atoms are so small. Van der Waals forces increase with size.

To explain why the London forces occur is easy; there are in fact two explanations that can be given. There is a simple, logical, and convincing explanation that can easily be found on the web, and that is also completely wrong. And there is a weird quantum explanation that is also correct, {A.33}.

If you are the audience that this book is primarily intended for, you may already know the London forces under the guise of the Lennard-Jones potential. London forces produce an attractive potential between atoms that is proportional to 1$\raisebox{.5pt}{$/$}$$d^6$ where $d$ is a scaled distance between the molecules. So the Lennard-Jones potential is taken to be

\begin{displaymath}
V_{\rm LJ} = C\left(d^{-12} - d^{-6}\right)
\end{displaymath} (10.1)

where $C$ is a constant. The second term represents the London forces.

The first term in the Lennard-Jones potential is there to model the fact that when the atoms get close enough, they rapidly start repelling instead of attracting each other. (See section 5.10 for more details.) The power 12 is computationally convenient, since it makes the first term just the square of the second one. However, theoretically it is not very justifiable. A theoretically more reasonable repulsion would be one of the form $\bar{C}e^{-d/c}$$\raisebox{.5pt}{$/$}$$d^n$, with $\bar{C}$, $c$, and $n$ suitable constants, since that reflects the fact that the strength of the electron wave functions ramps up exponentially when you get closer to an atom. But practically, the Lennard-Jones potential works very well; the details of the first term make no big difference as long as the potential ramps up quickly.

It may be noted that at very large distances, the London force takes the Casimir-Polder form 1$\raisebox{.5pt}{$/$}$$d^7$ rather than 1/$d^6$. Charged particles do not really interact directly as a Coulomb potential assumes, but through photons that move at the speed of light. At large separations, the time lag makes a difference, [26]. The separation at which this happens can be ballparked through dimensional arguments. The frequency of a typical photon corresponding to transitions between energy states is given by $\hbar\omega$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E$ with $E$ the energy difference between the states. The frequency for light to bounce back and forwards between the molecules is given by $c$$\raisebox{.5pt}{$/$}$$d$, with $c$ the speed of light. It follows that the frequency for light to bounce back and forward is no longer large compared to $\omega$ when $Ed$$\raisebox{.5pt}{$/$}$${\hbar}c$ becomes order one. For hydrogen, $E$ is about 10 eV and ${\hbar}c$ is about 200 eV nm. That makes the typical separation at which the 1$\raisebox{.5pt}{$/$}$$d^6$ relation breaks down about 20 nm, or 200 Å.

Molecular solids may be held together by other Van der Waals forces besides London forces. Many molecules have an charge distribution that is inherently asymmetrical. If one side is more negative and the other more positive, the molecule is said to have a “dipole strength.” The molecules can arrange themselves so that the negative sides of the molecules are close to the positive sides of neighboring molecules and vice versa, producing attraction. (Even if there is no net dipole strength, there will be some electrostatic interaction if the molecules are very close and are not spherically symmetric like noble gas atoms are.)

Chemguide [[1]] notes: “Surprisingly dipole-dipole attractions are fairly minor compared with dispersion [London] forces, and their effect can only really be seen if you compare two molecules with the same number of electrons and the same size.” One reason is that thermal motion tends to kill off the dipole attractions by messing up the alignment between molecules. But note that the dipole forces act on top of the London ones, so everything else being the same, the molecules with a dipole strength will be bound together more strongly.

When more than one molecular species is around, species with inherent dipoles can induce dipoles in other molecules that normally do not have them.

Another way molecules can be kept together in a solid is by what are called “hydrogen bonds.” In a sense, they too are dipole-dipole forces. In this case, the molecular dipole is created when the electrons are pulled away from hydrogen atoms. This leaves a partially uncovered nucleus, since an hydrogen atom does not have any other electrons to shield it. Since it allows neighboring molecules to get very close to a nucleus, hydrogen bonds can be strong. They remain a lot weaker than a typical chemical bond, though.


Key Points
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Even neutral molecules that do not want to create other bonds can be glued together by various “Van der Waals forces.”

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These forces are weak, though hydrogen bonds are much less so.

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The London type Van Der Waals forces affects all molecules, even noble gas atoms.

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London forces can be modeled using the Lennard-Jones potential.

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London forces are one of these weird quantum effects. Molecules with inherent dipole strength feature a more classically understandable version of such forces.