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Crystal-Cloudy, Crystal-Clear

Roald Hoffmann

What Crystal Structures Tell

Crystallographic studies are usually done very carefully, although sufficient mistakes of a certain basic type (so-called space-group assignment) are made to provoke one distinguished crystallographer to publish a dreaded article per year replete with his colleague's errors. The crystal structures are carried out by people perhaps more aware of systematic and random error (that "standard deviation") than almost any subculture of our science. So why do I malign their labors?

Actually I don't malign them. I am a voracious consumer of crystal structures—I value them, I treasure them. It's just that the deviations I need and love are not the standard deviations they provide. I want "chemical" estimates of uncertainty: They give me experimental variances.

The standard deviation of a geometrical parameter in a crystallographic structure determination—be it a unit-cell coordinate, a distance, a bond angle—is the square root of a variance, the latter usually denoted σ2 owing to a multitude of experimental uncertainties.

These are accounted for usually quite well by the careful experimentalist, though I wonder today—when the variances are spewed out in an inkling by a computer—whether they are really given as much serious consideration as to origins as they were decades ago.

In my years of molecular voyeurism, I saw the crystal structures get better, and the standard deviations sink, for good data sets to the 0.002 angstrom (Å) [1 Å = 10–8 centimeters] level for organic-molecule distances. But it was also clear to me that ± 0.002Å made no chemical sense. For I saw in some of the structures (technically those with several molecules in the asymmetric unit) molecules that were chemically identical yet whose relevant matching distances or angles differed by much more than the listed standard deviations, because of the intermolecular forces at play. In fact after a while I began to look for these cases. Or to get an estimate of a chemical standard deviation I focused on the part of the crystal structure that was least interesting—that phenyl group in a triphenylphosphine ligand (hundreds of them, and they should be very much alike). And I saw big deviations.

So that's the reason for my flippant π and e factor. What I also record here is a missed opportunity on my part—the information was on hand, and I just quipped. Antonio Martín and A. Guy Orpen of the University of Bristol in the United Kingdom did what needed to be done (and didn't get their idea from me, either). In what is certain to be a classic paper, in 1996 they showed from an examination of thousands of metal-complex structures that crystal-packing forces cause standard deviations of the order of 0.01 to 0.02Å in ligand distances. So, as it turns out, my π was too conservative!

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