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In the classic freshman physics text Electricity and Magnetism by Purcell, one of his first examples is the computation of the electrical binding energy of a $\ce{NaCl}$ crystal. He models the crystal in terms of perfect ionic bonding and sums up the energy $q_1q_2/r$ pairwise, computing the $r$ values in three dimensions. (Actually he just gives the first few terms explicitly and then says he carried out the computation numerically on a computer.) The result is $U=-0.8738Ne^2/a$, where $N$ is the number of ions, $a$ is the edge length of the cubic lattice, and the units are cgs, so that the Coulomb constant equals 1.

I thought this was a nice example, but it seemed unfortunate that the actual result could only be found through a fairly complicated numerical computation. I thought it would be nice if I could come up with a similar but simpler example for use with my freshman E&M classes, using a linear molecule with charges $\pm e$ arranged like $\cdots +-+-+-\cdots$. A relatively simple calculation gives an energy of $$-\frac{Ne^2}{4a}\sum_{k=1}^\infty \frac{1}{k(k-1/2)}.$$

I imagine that the sum can be done in closed form, but anyway it's clearly finite and positive, so that the result is a finite binding energy that has the right sign to keep the system bound.

Is there any molecule that actually has this structure, or something close to it? Googling and asking a chemist in the hall resulted in a variety of examples that are not exactly what I had in mind:

  • There are various 3-ion molecules such as $\ce{BeF2}$.

  • You can have long carbon molecules, with hydrogens hanging off of them, but they're not exactly linear, and in any case the carbons are all the same element, so there is no ionic bonding.

  • Acetylene is linear.

Does nature provide anything like the $+-+-$ chain that I'm describing? Do they not occur because they're too unstable with respect to crumpling up? Could they exist on a substrate (the kind of thing people build using scanning electron microscope probes) or as linear impurities in a bulk solid?

For my purposes, I don't think it really matters whether the bonding is perfect ionic bonding, as long as there is some segregation of charge.

Related:

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    $\begingroup$ I believe I read many years ago in a solid state physics textbook that the calculated speed of sound in a 1D crystal is imaginary. If there is no context I am forgetting, wouldn't that suggest they cannot exist? $\endgroup$ – Nicolau Saker Neto Dec 3 '18 at 22:09
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    $\begingroup$ While bonding in molecule may have more ionic then covalent component, they are rather treated as covalent - any simple ionic model shouldn't be useful. $\endgroup$ – Mithoron Dec 4 '18 at 1:44
  • $\begingroup$ My main criticism is on line with the comment of @Mithoron. Treating a long chain with alternating (partially) oppositely charged fragments as kept in place by electrostatic force isn't the proper treatment. Perhaps one can estimate a coulombic term but then we enter in the computational and quantum chemistry field, truly. $\endgroup$ – Alchimista Dec 4 '18 at 12:23
  • $\begingroup$ @Alchimista: By Earnshaw's theorem, no system can be in a stable, static equilibrium due to electrical forces. That isn't what Purcell is claiming. He goes into a lengthy discussion of this. $\endgroup$ – Ben Crowell Dec 4 '18 at 14:29
  • $\begingroup$ Therefore I say. It is somewhat the Peierl instability in a charged version. It is not clear what you want calculate. $\endgroup$ – Alchimista Dec 4 '18 at 16:49
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An example of a linear structure is given by cycopentadienyl lithium, whose structure is explored in this question. There is some covalent interaction in this polymeric structure, as in most organolithium compounds, but the bonding between lithium and cyclopentadienyl groups is highly polar.

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Check for polyoxymethylene (POM).

enter image description here

It has partial positive charge on carbons and partial negative on oxygens, and it's "long" enough to serve as plausible example for sum with large maximum k. But I should mention that there are no perfectly linear polymers which are like string of beads.

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    $\begingroup$ Not true how about indefinitely long polyynes & cumulenes? bit.ly/2Sp81yb I'd imagine any deviations from linearity would be pretty minor! ; ) $\endgroup$ – ManRow Dec 3 '18 at 21:54
  • $\begingroup$ @ManRow: I'm having trouble parsing your comment. What are you saying is not true? By the way, it's not necessary to give shortened bit.ly links; you can just give a link straight to Wikipedia, which makes it easier for people to understand what the link is going to be about. The link re linear acetylenic carbon is interesting, but it isn't ionic bonding, since every atom in the chain is a carbon...? $\endgroup$ – Ben Crowell Dec 3 '18 at 21:59
  • $\begingroup$ Hi! I was commenting on what Kelly Shepphard said about linear polymers "like a string of beads" not existing, and just provided some counterexamples : ) $\endgroup$ – ManRow Dec 3 '18 at 22:02
  • $\begingroup$ ManRow, they are still not perfectly linear (and I've said about perfect linearity), and definitely have impurities (as any other polymer). Besides, usually they exist only as rigid part of molecule whis is linear copolymer, dendric structure or, at least, crosslinked polymer. I even bet that such "strings of beads" would crosslink spontaneously. $\endgroup$ – Kelly Shepphard Dec 4 '18 at 9:39
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    $\begingroup$ Then it's not a conterexample for now. But maybe one day humanity will master some new reactions and ways to support perfectly linear long structures - this day will be the day when your example becomes counterexample :) For now, it's like example that polymers can't be perfectly linear - even if parts of them are. $\endgroup$ – Kelly Shepphard Dec 4 '18 at 12:25

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