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

$$V_{\rm LJ} = C\left(d^{-12} - d^{-6}\right)$$ (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 to 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.