On the presence of multiple molecules in the crystal asymmetric unit (Z′ > 1)

The presence of multiple molecules in the asymmetric unit (Z′) has been noted from the earliest days of crystallography, when such an occurrence might even have prevented structure solution and refinement, till today when the phenomenon is being commented upon regularly¹ and, as some claim, can even be engineered.² In the end, however, the fact that some crystals take more than one symmetry independent molecule is still something of an enigma. Is it a matter of no consequence whatsoever, which occurs randomly, or is there a deeper underlying reason why Z′ > 1?

Perhaps high Z′ structures appear mysterious simply because there is still not a critical mass of relevant literature on their phenomenological analysis.³ However, a few authors have attempted to study this difficult problem. Brock has surveyed alcohols and phenols extensively and has given an elegant explanation as to why these compounds have a higher proportion of Z′ > 1 crystal structures when compared with the global sampling of organic molecular crystals.⁴ According to her, the tendency of these hydroxy compounds (ROH) to form cooperative hydrogen bonded chains of the O–H⋯O–H type is countered by the steric demands of the R-groups, and high Z′ is one of the outcomes (the other is crystallisation in a high symmetry system). In general, there seems to be a consensus that when packing problems make it difficult to achieve a structure with Z′ = 1, a higher Z′ is a good alternative option. The idea of interaction frustration has also echoes in the work of Nangia, Steed and Clegg among others.^3,5–9 These authors have also elaborated other interesting themes: that structures with high Z′ have a loose packing but good interactions;⁵ that crystals which are grown from the melt or by sublimation have a significantly higher proportion of Z′ > 1 structures;⁶ that high Z′ structures can be described as modulations;^5,10 that high Z′ is obtained when the molecule has a large number of equi-energetic conformations, these conformations co-existing in the crystal.¹¹ Pseudosymmetry is certainly implicated in some cases and the pseudo-elements of symmetry may be either global or local. In the latter category, we noted a very unusual subset more than 15 years ago of P1̄ crystals which have Z′ = 2, the two symmetry independent molecules being related by a local pseudo-centre of inversion.¹² Why does the crystal take this pseudo-centre, an “extra”, almost “wasted” symmetry element?

All these observations are undoubtedly interesting, and possibly important, but what struck me as the most unusual fact about high Z′ structures, an observation that seemed almost counter-intuitive, is that the proportion of Z′ > 1 structures relative to all crystal structures has remained practically invariant over the decades. I must admit that I had always felt (and I daresay that this might be true of others, too) that with the advent of CCD diffractometers and high throughput crystallography, the proportion of high Z′ structures should be steadily increasing. Reality is different. During the period 1970–2006, during which time the CSD has become nearly 43 times larger, the proportion of Z′ = 1, Z′ < 1 and Z′ > 1 organic structures has remained virtually constant (±1%) at 73, 16 and 11%.¹³ Does this constancy of numbers say anything about the origin of this phenomenon? Is there a basic, universal reason why high Z′ structures are obtained?

I would like to suggest that the secrets of high Z′ structures will be most easily revealed through a study of polymorphic systems, wherein a high Z′ structure may be most easily compared with a lower Z′ structure (ideally one with Z′ = 1) of the same chemical substance. We noted that at least in two cases, pentafluorophenol and trans-1,4-bis(phenylethynyl)cyclohexane-1,4-diol, there are two structures with different Z′ values.¹⁴ The structure with the lower Z′ is the more stable structure; the one with the higher Z′ value is obtained under what essentially amount to kinetic conditions: cryo-crystallography for the phenol and melt cooling for the diol. These observations are suggestive. Steed put forth the idea a few years ago that a high Z′ structure is a “fossil relic” of a more stable form.¹ In this way, it is not difficult to associate the higher Z′ polymorphs of these two compounds with kinetic modifications. Indeed, we showed that the discrete O–H⋯O–H⋯O–H trimer synthon in the high Z′ polymorph of pentafluorophenol is an “incomplete” version of the infinite O–H⋯O–H chain in the low Z′ polymorph. Similarly, the high Z′ polymorph of the diol contains a large number of molecular conformations that are essentially “frozen” into the crystal, and these anneal out in the lower Z′ high temperature modification.

It is worthwhile to reflect that all the reasons which have been put forward in the past for the adoption of high Z′ structures (packing difficulties and inconsistencies, modulation, pseudosymmetry, equi-energetic conformations, better interactions) are simply different ways of saying the same thing. A high Z′ structure is a crystal “on the way”. Many of Brock's observations on alcohols and phenols¹⁵ follow from the idea that molecules which form stable clusters in solution because of strong hydrogen bonding have a higher tendency to form Z′ > 1 crystals because these clusters are carried forward more or less unaltered into the crystal. Possibly, O–H⋯O hydrogen bonding in alcohols and phenols is so strong and directional that it becomes difficult to observe a better packed, high temperature, Z′ = 1 polymorph with poorer hydrogen bonds. However, this is certainly not the case for the weaker C–H⋯O interactions and polymorphic sets of high and low Z′ structures are being reported regularly for C–H⋯O bonded solids.^5,9 A very interesting example (4,4-diphenylcyclohexa-2,5-dien-1-one) was published recently by Nangia and Kruger.¹¹ This almost schizophrenic compound has 4 polymorphs which contain a total of no less than 19 distinct molecular conformations. Significantly, the lower Z′ structures are more stable but the higher Z′ structures have better C–H⋯O hydrogen bonds. Similarly, in α-D-glucofuranose-1,2:3,5-bis(p-tolyl)boronate, the higher Z′ polymorph has the looser packing but better hydrogen bonds.⁵ This dichotomy—better directional interactions versus better packing—is becoming a recurring theme in crystal engineering. It is both expected and normal, and it has to do with the differences between kinetic and thermodynamic crystallisation.

Finally, I return to the constancy of frequency of high Z′ structures and I will go out on a limb with a possible explanation as to why the number of organic Z′ > 1 structures has remained stubbornly fixed over the decades. Let us assume that Z′ > 1 structures occur because of “incomplete crystallization”. The proportion of crystals wherein the crystallisation is “incomplete”, in the sense of being frozen in some high energy kinetic form, should depend on the difference between ΔG^‡_{thermodynamic} and ΔG^‡_kinetic of potentially polymorphic substances. Given the fact that intermolecular interactions in most molecular crystals generally occur in a small energy window (say 0.5 to 8 kcal mol⁻¹), these energy differences will be modest. Accordingly, the probability of appearance of a high Z′ structure will depend largely on the crystallisation temperature. Since this too generally occurs within a small window (say 10 to 30 °C), the proportion of high Z′ structures will always be roughly the same whether a thousand crystal structures have been determined, or a million. In those cases where the kinetic and thermodynamic forms are one and the same, Z′ > 1 is not possible. Since such monomorphic substances are numerous, the overall proportion of Z′ > 1 structures is rather small, at 11%.

The idea that a higher Z′ polymorph is a manifestation of incomplete or interrupted crystallisation is attractive, but more proof is needed for this conjecture. The most interesting aspect, to my mind, of all this is that high Z′ structures may teach us something about the mechanism of crystallisation. At a higher level, one may pose the question: must a Z′ = 1 structure be the most stable packing for an organic compound if directional interactions are disregarded? This is like asking if a single crystal must be the most stable form of a condensed phase. Like the latter, the answer to my question might not be found in the domain of chemistry but rather in mathematics.

References

1. J. W. Steed, CrystEngComm, 2003, 5, 169.
2. K. M. Anderson, K. Afarinkia, H. Yu, A. E. Goeta and J. W. Steed, Cryst. Growth Des., 2006, 6, 2109.
3. N. J. Babu and A. Nangia, Cryst. Growth Des., 2006, 6, 1995 , and the references cited therein.
4. C. P. Brock and L. L. Duncan, Chem. Mater., 1994, 6, 1307.
5. S. K. Chandran and A. Nangia, CrystEngComm, 2006, 8, 581.
6. B. Sarma, S. Roy and A. Nangia, Chem. Commun., 2006 10.1039/b610323e.
7. A. M. Todd, K. M. Anderson, P. Byrne, A. E. Goeta and J. W. Steed, Cryst. Growth Des., 2006, 6, 1750.
8. K. M. Anderson, A. E. Goeta, K. S. B. Hancock and J. W. Steed, Chem. Commun., 2006, 2138.
9. W. Clegg and G. S. Nichol, Cryst. Growth Des., 2006, 6, 451.
10. X. Hao, M. A. Siegler, S. Parkin and C. P. Brock, Cryst. Growth Des., 2005, 5, 2225.
11. S. Roy, R. Banerjee, A. Nangia and G. J. Kruger, Chem.–Eur. J., 2006, 12, 3777.
12. G. R. Desiraju, J. C. Calabrese and R. L. Harlow, Acta Crystallogr., Sect. B: Struct. Sci., 1991, 47, 77.
13. The reader will note that a very small proportion of Z′ = 1 structures have Z′ = 0.5 + 0.5, in other words there are two symmetry independent molecules situated on distinct inversion centres in space groups like P1̄ and P2₁/c. These structures are phenomenologically identical to Z′ = 2 structures in which the molecules occupy general positions. However, there is no easy way in which the number of these structures (Z′ = 0.5 + 0.5) can be counted.
14. D. Das, R. Banerjee, R. Mondal, J. A. K. Howard, R. Boese and G. R. Desiraju, Chem. Commun., 2006, 555.
15. X. Hao, S. Parkin and C. P. Brock, Acta Crystallogr., Sect. B: Struct. Sci., 2005, 61, 689; H.-J. Lehmler, L. W. Robertson, S. Parkin and C. P. Brock, Acta Crystallogr., Sect. B: Struct. Sci., 2002, 58, 140; L. L. Duncan, B. O. Patrick and C. P. Brock, Acta Crystallogr., Sect. B: Struct. Sci., 2002, 58, 502.
16. There are cases where the higher Z′ polymorph in a pair of crystals has the lower energy. Whether further calculations will reverse the energy order, or whether there is a yet undiscovered polymorph with Z′ = 1 and an even lower energy are still open questions.

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