Acc. Chem. Res. 1994, 27, 309—314
`Are Crystal Structures Predictable?
`Angelo Gavezzotti*
`Dipartimento di Chimica Física ed Elettrochimica, Universita di Milano, Milano, Italy
`Received. May 16, 1994
`
`309
`
`“No”: by just writing down this concise statement,
`in what would be the first one-word paper in the
`chemical literature, one could safely summarize the
`an honorarium from the
`present state of affairs, earn
`American Chemical Society, and do a reasonably good
`In the main-
`service to his or her own
`reputation.
`stream of academic tradition, one could then concede
`a conditional “yes”, thus making a
`a “maybe”, or even
`good point for discussion; and then, in the mainstream
`of publication policy tradition, proceed eventually to
`have his or her papers rejected by referees taking the
`opposite stand.
`Fortunately, there is a rhetorical way out of this
`predicament, known to medieval philosophers as
`in plain words it means, when you
`amplificatio:
`cannot provide an answer, just rephrase and expand
`the statement of the question. To this very old trick
`we will resort in this paper.
`In fact, the title question
`is a bit too straightforward and simple-minded; such
`broad terms as “crystal structure” and “prediction”
`detail. There are several
`need be defined in more
`levels of desirable a priori information on a solid; they
`will be described by posing a number of typical, more
`restricted questions, in order of increasing complexity.
`Organic substances only will be considered.
`It is assumed that it need not be explained to the
`reader why control or prediction of the structure of a
`solid, at a molecular level,
`there are
`is desirable;
`several self-evident justifications, on both theoretical
`and practical grounds, for striving to understand the
`basic factors that dictate the arrangement of molecules
`in space when they recognize each other at a short
`distance and eventually coagulate in a rigid configura-
`tion. While the present knowledge of intramolecular
`valence can be considered satisfactory, that of inter-
`molecular “valence” is rudimentary; and the perspec-
`tive of being able to design molecular solids with
`predetermined physical properties, which depend on
`structure, is appealing (an understatement) to applied
`chemists in the fields of pigments,1 pharmaceuticals,2
`magnets,3 conductors,4 and photosensitive5 or opto-
`electronic6 materials. So one has here a big theoretical
`challenge going hand in hand with big business.
`In the early days of X-ray crystallography, guessing
`at the crystal structure by minimizing intermolecular
`repulsions was considered a viable method of solving
`the phase problem, when cell dimensions and diffrac-
`tion intensities were available. From such a perspec-
`tive, knowledge of the cell volume implied that inter-
`molecular attractions had been satisfied, and that only
`
`Angelo Gavezzotti (bom in 1944) studied chemistry at the University of Milano
`and graduated in 1968 with a thesis in X-ray crystallography. He has worked in the
`fields of theoretical and structural chemistry with supervision and friendly advice from
`M, Simonetta. He spent research terms in Orsay, France (1973), and in Ann Arbor,
`Ml (1977-1978), with L. S. Bartell. He is presently professor of physical chemistry
`at the University of Milano, and his research interests focus on the structural chemistry
`of organic crystals. He served a three-year term as Coeditor of Acta Crystallographies.
`0001-4842/94/0127-0309$04.50/0
`
`mutual avoidance between rigid objects had to be
`accomplished, either by rough (but surprisingly ef-
`ficient) mechanical devices73 or by computer sieving.7b
`These procedures were suddenly made obsolete, and
`dismissed, by the advent of direct methods. Crystal
`structure prediction resurfaced only in very recent
`times, and with a much more ambitious connotation;
`the new problem is to consider an organic compound
`formula has been written on
`for which a structural
`paper, but whose synthesis (presumably expensive in
`terms of materials or human resources) has not yet
`In keeping with the rhetorical
`been accomplished.
`profile of this paper, typical questions on its future as
`a solid will now be posed.
`1. Will this compound crystallize at all? Thermo-
`dynamics holds that any substance must crystallize,
`provided it
`is pure and the temperature is low (or
`pressure is high) enough. But organic chemistry
`thrives in mild temperature-pressure regimes, prone
`elusive dictates of kinetics. Dis-
`to the much more
`solution always works in the proper solvent while
`crystal growth from solution is problematic; melting
`at higher temperatures than
`nearly always occurs
`freezing; a crystal is more readily destroyed than built.
`The organic solid state ranges from waxes
`or glasses
`twinned crystals, to pow-
`to disordered, strained, or
`ders, and eventually, to well-shaped single crystals.
`Chemists often come to grips with tough problems in
`the control of solidification, crystal growth, and crystal
`morphology, mainly due to the perverse kinetic control
`of nucleation; and this is a well-developed research
`field of its own.8
`For example, sexithienyl, a compound of great
`importance in nonlinear optics, has a high melting
`point, yet no single crystals of this substance could
`be grown, in spite of considerable effort. A reasonable
`and stable crystal structure has been predicted9 by
`calculations based on empirical potentials. Recently,
`a Rietveld analysis of powder specimens (the best that
`* New address: Dipartimento di Chimica Strutturale e Stereochimica
`Inorgánica, Universitá di Milano, Milano, Italy.
`(1) Klebe, G.; Graser, F.; Hadicke, E.; Berndt, J. Acta Crystallogr.
`1989, B45, 69-77.
`(2) Haleblian, J.; McCrone, W. J. Pharm. Sci. 1969, 58, 911-929.
`(3) Miller, J. S.; Epstein, A. J.; Reiff, W. M. Acc. Chem. Res. 1988,21,
`114-120.
`(4) Hunig, S.; Erk, P. Adv. Mater. 1991, 3, 225-236.
`(5) For the structural problems connected with epitaxy and photo-
`conductivity of small molecules on polymers, see: Scaringe, R. P.; Perez,
`S. J. Phys. Chem. 1987, 91, 2394-2403 and references therein.
`(6) Chemla, D. S.; Zyss, J. Nonlinear Optical Properties of Organic
`Molecules and Crystals; Academic Press: Orlando, 1987.
`(7) (a) Kitaigorodski, A. I. Organic Chemical Crystallography; Con-
`sultants Bureau: New York, 1961 (the structure-seeking apparatus). For
`a review of Kitaigorodski’s work, see also: Struchkov, Y. T.; Fedin, E. I.
`Acta Chem. Hung. 1998,130,159-172.
`(b) Rabinovich, D.; Schmidt, G.
`M. J. Nature (London) 1966, 211, 1391—1393.
`(8) Hulliger, J. Angew. Chem., lnt. Ed. Engl. 1994, 33, 143-162.
`(9) Gavezzotti, A.; Filippini, G. Synth. Met. 1991, 40, 257-266.
`© 1994 American Chemical Society
`
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`

`310 Acc. Chem. Res., Vol. 27, No. 10, 1994
`
`Gavezzotti
`
`a
`
`pounds17 (acids, alcohols, and amides). Correlations
`found which allow an estimate of sublimation
`were
`enthalpies from molecular parameters like the number
`of valence electrons (Z) or
`the van der Waals surface
`(S). For example, in non-hydrogen-bonded oxohydro-
`carbons,
`
`(a) The main motif in the predicted crystal structure of sexithienyl (ref 9; P2\/a, Z = 2).
`(b) The same for the structure
`Figure 1.
`from a Rietveld refinement of powder data (ref 10; P2i/c, Z = 4). The two structures differ mainly in the interplanar angle between
`neighbor molecules (49° vs 67°), better shown in the side views on the right.
`could be obtained) has been published.10 While the
`agreement between the main features of the predicted
`and experimental crystal structures is pleasing (Fig-
`the riddle of the lack of sexithienyl single
`ure
`1),
`crystals is still unanswered.
`Is this crystal high-melting? The melting tem-
`2.
`perature (Tm) is high for high melting enthalpy or for
`low melting entropy. The entropic factor implies that
`disordered crystals, or crystals whose liquids are
`heavily associated (e.g., by hydrogen bonding), have
`higher Tm’s. Correlations between Tm and crystal
`cohesion should therefore be taken with caution.
`A very old rule of thumb states that more symmetric
`molecules form higher-melting crystals;11 this idea has
`been analyzed12 using ortho-, meta-, and para-disub-
`stituted benzenes (XCeFLtY, X and Y being any sub-
`stituents). A survey of their TVs shows that para
`isomers are the highest-melting ones, with very few
`for only 18 out of 238 para-meta and
`exceptions;
`para-ortho couples, the para isomer melts at a lower
`temperature. However, the definition of molecular
`symmetry in this context is really elusive and merges
`uncomfortably with that of molecular shape. The rule
`of thumb stays such, and cannot be given a sound
`theoretical or structural foundation. Tm is still one of
`the most difficult crystal properties to predict.
`3. What is the lattice energy (heat of sublimation)!
`Extensive statistical studies have been conducted on
`relationships between molecular and crystal proper-
`ties for non-hydrogen-bonding compounds containing
`C,  , N, O, S, and Cl atoms,13-16 as well as for the
`families of hydrogen-bonding com-
`most
`common
`(10) Porzio, W.; Destri, S.; Mascherpa, M.; Bruckner, S. Acta Polym.
`1993, 44, 266-272.
`(1Í) An early statement
`is by Hückel: Hückel, W. Theoretische
`Grundlage der Organischen Chemie; Akademische Verlagsgesellschaft:
`Leipzig, 1931; Vol. II, pp 185-186.
`(12) Gavezzotti, A. To be published.
`(13) Gavezzotti, A. J. Am. Chem. Soc. 1989, 111, 1835-1843.
`(14) Gavezzotti, A. J. Phys. Chem. 1991, 95, 8948—8955.
`(15) Gavezzotti, A.; Filippini, G. Acta Crystallogr. 1992, B48, 537-
`545.
`(16) Gavezzotti, A.; Filippini, G. Acta Chim. Hung. 1993, 130, 205—
`220.
`
` Hs = 0.201 Z + 9.4 kcaVmol
`AHS = 0.077S(A2) + 8.9 kcaVmol
`Standard deviations of these linear regressions are
`comparable to experimental uncertainties of measure-
`in this respect,
`truly predictive
`least
`ments;18 at
`correlations between molecular and crystal properties
`In some cases, errors
`in experi-
`can be established.
`mental AHs’s have been detected by redeterminations
`prompted by large deviations from the correlation.14
`Needless to say, the total lattice energy as such carries
`no information on the geometrical structure of the
`crystal.
`4. Will the crystal structure be non-centrosymmetric!
`This is a simple but vital
`for some
`requirement
`practical applications of crystal chemistry.19 Crystal
`centrosymmetry is often a matter of debate, and it is
`sometimes one of the refinable parameters in X-ray
`crystal structure analysis, rather than a stringent a
`priori condition.20 One sees here a wide gap between
`the high (sometimes too high) resolution of diffraction
`experiments, where a single non-centrosymmetrically
`arranged atom in a large molecule would make a total
`difference, and the coarse view of the applied chemist.
`No one, except a neutron diffractionist, would consider
`non-centrosymmetric a hypothetical P2i crystal struc-
`ture of monodeuteriobenzene.
`(17) Gavezzotti, A.; Filippini, G. J. Phys. Chem. 1994,98, 4831-4837.
`(18) For a review of available sublimation enthalpies of organic
`compounds, see: Chickos, J. S. In Molecular Structure and Energetics;
`Liebman, J. F., Greenberg, A., Eds.; VCH: New York, 1987; Vol. 2.
`(19) Paul, I. C.; Curtin, D. Y. Chem. Rev. 1981, 81, 525-541.
`(20) For a PI reassigned as Pi, see: Marsh, R. E. Acta Crystallogr.
`1990, C46, 1356—1357. See also: Marsh, R. E. Acta Crystallogr. 1994,
`A50, 450, 455. The author is so assiduous in this kind of exercise that
`papers so reconsidered are commonly said to have been “marshed”.
`
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`

`Are Crystal Structures Predictable?
`a
`
`Figure 2. Arrangement of molecules in (a) the X-ray crystal
`structure of 1,3,5-triamino-2,4,6-trinitrobenzene (ref 24; PI, Z
`= 2) and (b) the simulated crystal structure (ref 26; PI, Z = 2).
`Oxygen atoms in one nitro group are filled in.
`The opinion that molecules with a high dipole
`tend to crystallize in a head-to-tail cen-
`moment
`trosymmetric fashion is untenable, as has been dem-
`onstrated by a detailed analysis:21 the dipole repre-
`sentation of a charge distribution applies at
`large
`distances from it, while neighbor molecules in crystals
`see each other at distances comparable to molecular
`dimensions. On the other hand, the carboxylic acid
`group nearly always forces crystal centrosymmetry by
`forming cyclic dimers.17,22 As is often the case, we only
`know how to produce the effect we do not want.
`A crystal grown out of a solution containing only one
`enantiomer will perforce be non-centrosymmetric, but
`the spontaneous
`nothing can be said a priori on
`resolution of racemic solutions by crystallization. The
`relative stability of resolved and racemic crystals has
`been analyzed,23 but there are at present no really
`predictive concepts on this fascinating subject, which
`may be related to the chirality of the chemistry of life.
`Quite often, non-centrosymmetric molecular layers
`are readily formed, but they cannot be prevented from
`assuming an apparently very favorable centro sym-
`metric arrangement in the crystal. For example, the
`crystal structure of 1,3,5-triamino-2,4,6-trinitroben-
`zene has been assigned to a centrosymmetric space
`group (PI) by X-ray analysis,24 while the material
`displays a second harmonic generation propensity,25
`a property of non-centrosymmetric structures. Plau-
`sible non-centrosymmetric structures, with lattice
`energies quite comparable to that of the X-ray one,
`have been generated (Figure 2); the discussion of the
`(21) Whitesell, J. K.; Davis, R. E.; Saunders, L. L; Wilson, R. J.;
`Feagins, J. P. J. Am. Chem. Soc. 1991,113, 3267-3270.
`(22) Leiserowitz, L. Acto Crystallogr. 1976, B32, 775—802.
`(23) Brock, C. P.; Schweizer, W. B.; Dunitz, J. D. J. Am. Chem. Soc.
`1991, 113, 9811-9820.
`(24) Cady,  . H.; Larson, A. C. Acta Crystallogr. 1965,18, 485-496.
`(25) Ledoux, I.; Zyss, J.; Siegel, J. S.; Brienne, J.; Lehn, J. M. Chem.
`Phys. Lett. 1990, 172, 440-444.
`
`Acc. Chem. Res., Vol. 27, No. 10, 1994 311
`results26 has a lot of academic ifs and buts, perhaps
`than to the advance-
`contributing to confusion more
`ment of knowledge. The formation of non-centrosym-
`metric domains seems, however,
`the most
`likely
`explanation of the unusual properties of this crystal.
`5. Will some parts of the molecule take up a
`predictable orientation in the crystal? Use of the
`information contained in the Cambridge Structural
`Database27 has led to a number of statistical studies
`on the geometry of hydrogen bonding, of halogen-
`halogen interactions, and of other preferred approach
`paths between chemically recognizable molecular moi-
`eties. The reader is referred to an excellent review28
`on the subject.
`Much work (and speculation) has been devoted29 to
`interactions between aromatic rings,
`the so-called  - 
`driving to stacking, against the “electrostatic” attrac-
`tions between rim hydrogen atoms and core carbon
`atoms, driving to T-shaped arrangements; preference
`for the latter is often assumed, quoting as
`a key
`example the benzene crystal, which in fact does
`contain also almost stacked neighbor molecules. A
`paper30 in which the distribution of phenyl group
`orientations in hydrocarbon crystals has been exam-
`ined, with peaks for both parallel and T-shaped
`arrangements, and a non-negligible population in
`between, has not been considered too seriously. Rules
`for the prediction of the appearance of herringbone
`stacked motifs in condensed aromatics have,
`versus
`apparently, been derived.31
`In crystals of monofunctional carboxylic acids and
`amides, virtually no exceptions to the formation of
`cyclic dimers for the former and of single N—H~0=C
`hydrogen bonds in the latter were found.17 Hydrogen
`bond formation has undoubtedly a very high priority
`in the construction of a crystal structure, but mol-
`ecules with several acceptor and/or donor groups quite
`often crystallize in different polymorphic forms with
`different hydrogen-bonding networks.32
`To conclude this section, one could say that some
`broad trends in the dependence of crystal packing from
`the presence of certain substituents or fragments have
`been identified; but this “substituent effect” in crystal
`chemistry stands on a shaky pedestal, since interac-
`tions in crystals of complex molecules are diverse and
`diffuse, and relying on local effects is always danger-
`ous.
`6. What can be the space group and the number of
`molecules in the asymmetric unit? The very concept
`of “space group” needs a little revision for crystal
`chemistry purposes. The presence or absence of a
`center of symmetry may be questionable;20 the same
`applies to every symmetry element. To the eyes of
`is not
`an X-ray crystallographer, a glide plane is or
`present according to an extinction pattern, but the
`borderline between extinct and very weak reflections
`can sometimes be a matter of subjective judgement
`(parasitic diffraction phenomena also contribute).
`Minor molecular displacements may destroy some
`(26) Filippini, G.; Gavezzotti, A. Submitted.
`(27) Allen, F. H.; Kennard, O.; Taylor, R. Acc. Chem. Res. 1983, 16,
`146-153.
`(28) Desiraju, G. R. Crystal Engineering·, Elsevier: Amsterdam, 1989.
`(29) See, e.g.: Dahl, T. Acto Chem. Scand. 1994, 48, 95—106 and
`vpfprdnrAQ tnprpin
`(30) Gavezzotti,’A. Chem. Phys. Lett. 1989, 161, 67-72.
`(31) Desiraju, G. R.; Gavezzotti, A. Acto Crystallogr. 1989, B45, 473-
`482.
`(32) Sulfa drugs provide striking examples: see, e.g.: Bar, I.; Bern-
`stein, J. J. Pharm. Sci. 1985, 74, 255—263.
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`312 Acc. Chem. Res., Vol. 27, No. 10, 1994
`symmetry element and bring about a change in space
`group (to the overdetailed eyes of the X-ray analyst),
`without really affecting the properties of the solid. For
`the crystal chemist, the prediction of the space group
`may be a whimsical exercise, if what counts is just a
`broad understanding of how molecules arrange them-
`selves in space. Besides, in a molecular crystal (here
`meaning one in which distinguishable chemical enti-
`ties appear, for which forces within the entity are
`considerably stronger than forces between entities) a
`distinction must be made between intramolecular, or
`point-group, symmetry and the true “intermolecular”
`symmetry, when the asymmetric unit is less than one
`molecule.33
`Overall, crystal symmetry has two facets. On one
`side, in a milestone mathematical development, it was
`the combinations of symmetry
`demonstrated that
`than 230
`elements give rise to no fewer and no more
`independent three-dimensional space groups. On the
`other side, crystal symmetry has to do with the mutual
`recognition of molecules to form a stable solid, a
`fascinating and essentially chemical problem that
`It
`requires an evaluation of intermolecular forces.
`should be clear that no necessary relationship holds
`between these two views; 230 space groups exist, but
`molecules cannot freely choose among them. Far from
`it, there are rather strict conditions that can be met
`only by a limited number of combinations of very few
`symmetry elements; for organic compounds, these are
`axis, and the
`the inversion center, the 2-fold screw
`glide plane, plus the ubiquitous translation (some-
`times disguised as centering), itself a respectable, if
`often forgotten, symmetry operator. Thus, the choice
`of the space group for organic crystals is usually
`restricted to those including the above combinations:
`PI, PI, P2i, P2i/c, C2/c, P2i2i2i, Pbca. The well-
`known statistics on space group populations34 for
`organic compounds confirms this, as Kitaigorodski
`pointed out decades ago.35
`Some crystals reach a stable (or at least a lasting
`metastable) state with more
`than one molecule in the
`asymmetric unit. Statistics on the Cambridge Data-
`at 8.3%,36 but this is
`base have these occurrences
`presumably an underestimation, since the Database
`is socially biased: structures with several molecules
`in the asymmetric unit pose a small supplementary
`technical problem in final space group assignment and
`structure refinement and were often in the past (and
`probably still are) put aside by busy crystallographers
`as unsavory members of their waiting lists. Once
`again, the reader is reminded of the discussion on the
`presence or absence of a symmetry operator, in this
`case the one that could provide a relationship between
`the partners of the plurimolecular asymmetric unit.
`the formation of
`Some basic rules that preside over
`intra- and intermolecular hydrogen bonding have been
`identified.37 In addition, it turns out that molecules
`which form very stable clusters in the liquid by
`likely to form plurimo-
`hydrogen bonding are more
`lecular asymmetric units, since these clusters are
`intact into the crystal, and perfect sym-
`carried over
`(33) See the discussion in the following: Scaringe, R. P. In Electron
`Crystallography of Organic Molecules; Fryer, J. R., Dorset, D. L., Eds.;
`Kluwer: Dordrecht, 1991, especially pp 92-94.
`(34) Baur, W. H.; Kassner, D. Acta Crystallogr. 1992, B48, 356-369.
`(35) See ref 7a, introductory chapters.
`(36) Padmaja, N.; Ramakumar, S.; Viswamitra,  . A. Acta Crystallogr.
`1990, A46, 725-730.
`(37) Etter, M. C. Acc. Chem. Res. 1990, 23, 120—126.
`
`Gavezzotti
`metry within them is energetically irrelevant, or even
`slightly unfavorable: 40% of the alcohol crystals in
`the Cambridge Database have more than one molecule
`in the asymmetric unit.17 For non-hydrogen-bonded
`crystals a similar explanation may be proposed,
`although no simple rules based on chemical reasoning
`can be put forward for preaggregation in the liquid
`state.
`the cell parameters? The cell volume
`7. What are
`per molecule is rather easily estimated from molecular
`volume, after the Kitaigorodski
`idea of a constant
`packing coefficient;35 hence, the crystal density too can
`be roughly estimated (see refs 15 and 17 for average
`packing coefficients of different chemical classes). If
`space is to be efficiently used in a condensed phase,
`there must be broad correlations between molecular
`for example, if Ds is the
`dimensions and cell edges:
`shortest molecular dimension, Cs the shortest cell
`edge, Dh the longest molecular dimension, and Ch the
`longest cell edge, the following restrictions apply38 (Á):
`Ds - 2 < Cs < Ds + 5
`
`3
`
`Ch > Dh -
`Cell dimensions are indeed a bad identifier of a crystal
`structure, since their choice is not always unique.
`Distances between molecular centers of mass may be
`more useful; of course, some of these coincide with the
`length of screw or glide translations and, hence, are
`equal to one-half the cell parameters along unique
`crystallographic axes. These distances are the main
`quantitative descriptors of crystal geometry and are
`dictated solely by the strength and directionality of
`this level, therefore, not
`intermolecular forces. At
`much can be predicted with decent accuracy unless
`quantitative intermolecular potentials are available.
`The systematic calibration of a set of potential
`energy parameters for organic crystals containing H,
`C, N, O, S, and Cl atoms, with17 or without39 hydrogen
`bonds, has been (painstakingly) accomplished. The
`reader will be spared the details of, and the endless
`the methods employed in such work;
`disputes on,
`space forbids also a quotation of the many alternative
`force fields available in the literature.40 Suffice it to
`say that these parameters are as few as possible, and
`that with them one can safely calculate lattice energies
`(since experimental heats of sublimation18 are repro-
`duced), trusting that lattice dynamics is not grossly
`misrepresented (since reasonable lattice vibration
`frequencies are calculated41 for observed crystal struc-
`tures). The functional form includes one exponential
`inverse sixth power
`term in interatomic
`and one
`distances, so that computing times are not inflated by
`slowly-converging summations. These potentials (Table
`1) have been tailored for the explicit task of performing
`large scale searches of crystal potential surfaces, or,
`in fewer words, for crystal structure prediction.
`8. Are crystal structures predictable? Of course, the
`final question is whether it
`is possible to predict ab
`initio the complete structure of any organic crystal,
`(38) Gavezzotti, A. J. Am. Chem. Soc. 1991, 113, 4622-4629.
`(39) Filippini, G.; Gavezzotti, A. Acta Crystallogr. 1993, B49, 868-
`880.
`(40) See refs 17 and 39 for perspective and discussion; see also:
`Pertsin, A. J.; Kitaigorodski, A. I. The Atom—Atom Potential Method;
`Springer-Verlag: Berlin, 1987.
`(41) The lattice-dynamical procedure is described in the following:
`Filippini, G.; Gramaccioli, C. M. Acta Crystallogr. 1986, B42, 605-609.
`
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`

`Are Crystal Structures Predictable?
`Table 1. Atom-Atom Potential Parameters:
`E = A exp(-BRy) - CRy~e
`interaction
`Aa
`Bb
`Cc
`H-H
`5774
`0.010
`4.01
`26.1
`3.36
`H-C
`4.10
`28 870
`0.049
`3.29
`113
`H-N
`0.094
`2.99
`4.52
`54 560
`120
`H-0
`0.121
`2.80
`4.82
`70 610
`105
`H-S
`0.110
`4.03
`279
`64 190
`3.35
`H-Cl
`0.120
`4.09
`3.30
`70 020
`279
`C-C
`0.093
`3.89
`3.47
`54 050
`578
`C-N
`117 470
`0.201
`3.86
`667
`3.50
`C-0
`3.74
`93 950
`0.161
`641
`3.61
`c-s
`126 460
`0.217
`1504
`3.41
`3.96
`C-Cl
`3.52
`93 370
`0.160
`923
`3.83
`N-N
`3.65
`87 300
`0.150
`3.70
`691
`N-0
`0.110
`64 190
`3.86
`364
`3.50
`0-0
`0.080
`3.74
`46 680
`319
`3.61
`o-s
`0.189
`3.72
`3.63
`906
`110 160
`O-Cl
`0.139
`3.72
`3.63
`80 855
`665
`s-s
`3.52
`259 960
`2571
`0.445
`3.83
`Cl-Cl
`140 050
`0.240
`3.83
`3.52
`1385
`HB-0 (amides)
`4.0
`3 607 810
`238
`7.78
`1.80
`HB-0 (acids)
`6 313 670
`8.75
`205
`7.0
`1.60
`HB-0 (alcohols)
`4 509 750
`298
`7.78
`5.0
`1.80
`HB-N (-N-H-N)
`7 215 600
`476
`7.78
`8.0
`1.80
`HB-N (-NHz-N)
`1 803 920
`7.37
`2.0
`1.90
`165
`a Kcal/mol. 6 A / c Kcal/imol'A 6). d Potential well depth (kcal/
`mol). 6 Distance at the minimum (A). From refs 17 and 39.
`
`space group, cell parameters, and atomic positions,
`in X-ray single-crystal
`much in the same
`style as
`structure analysis. The answer
`here is definitely “no”.
`Undoubtedly, no true prediction in the above sense
`can be accomplished without calculating the crystal
`potential energy, but one fundamental point is the
`choice of the best coordinates for the energy space.
`Intramolecular structure can be described by just a
`few (mostly torsional) conformational parameters, full
`relaxation of intramolecular vibrational degrees of
`freedom being pointless, since coupling with intermo-
`lecular vibrations is negligible. The location of mol-
`ecules in the cell is described by three translational
`and three rotational rigid-body coordinates (restric-
`tions apply for some point-group symmetries). Rather
`than using space groups and cell parameters, it
`is
`from the constituents of
`convenient38 to start
`more
`spacial symmetry, that is, the four basic symmetry
`operators (inversion center, screw, glide, and transla-
`tion); molecular clusters are built under their action,
`and their energies are calculated by empirical poten-
`In this procedure, molecular conformation must
`tials.
`be assumed as fixed, and the fact that polymorphs may
`exist with different molecular conformations is one
`addition to an already uncomfortably long list of
`difficulties. Anyway, a number of promising clusters
`are selected and are translated in space or coupled
`with other operators until a full three-dimensionally
`periodic crystal structure is reached.42 One advantage
`of this procedure is that, say, a two-molecular cluster
`a center of symmetry can be used to try both PI
`over
`and P2i/c. The most questionable feature is that there
`is no guarantee that a stable cluster will actually
`appear in the crystal, whose stability is determined
`by the overall features of its three-dimensional struc-
`ture.
`
`(42) Gavezzotti, A. PROMET: A Program for the Generation of
`Possible Crystal Structures from the Molecular Structure of Organic
`Compounds, and Space Group Symmetry: A Primer, University of
`Milano, 1993 (available from the author upon request). Using this
`program is an excellent way of learning the basics of space group
`symmetry.
`
`Acc. Chem. Res., Vol. 27, No. 10, 1994 313
`Methods38’43 which involve an examination of a great
`many possible crystal structures, using strategical
`shortcuts and sequential sieves, may be called “static”.
`Their success
`in full prediction has been modest, but
`encouraging; they should be helpful when auxiliary
`information—from spectroscopy, powder or partial
`single-crystal diffraction, or structural correlation to
`similar compounds—is available. The construction of
`stable aggregates is made much easier when the
`consideration of predominant hydrogen-bonding
`schemes is possible.44
`A “dynamic” approach uses Monte Carlo or molecu-
`lar dynamics calculations.45-47 The starting point is
`a collection of molecules in random orientations, and
`the predicted equilibrium state is the result of averag-
`a large configurational space, or of evolution
`ing over
`in time after solution of the classical equations of
`In both cases, molecular interactions must
`motion.
`be calculated by empirical potentials, which retain
`their pivotal role in the whole procedure. Computing
`times increase steeply with the number of molecules
`in the statistical sample and put a severe
`strain even
`Ideally, this approach al-
`on present-day machines.
`lows the simulation of the complete phase behavior
`of the substance, as a function of temperature and
`pressure. Although its scope and promise are
`cer-
`tainly wider than those of the static approach, only
`the reproduction of the crystal structure of benzene48
`and a few other organic molecules46 has been achieved
`so far, and a definite proof that such methods can give
`an unequivocal solution to the problem of crystal
`structure prediction has not been produced. The
`author of the present paper would be more
`than happy
`if this statement could be falsified in the near
`future.
`The computer software described and used in ref 46
`(presumably the best available at the moment) is now
`being commercialized by a profit company (module
`“Polymorph” of the CERIUS package, by Molecular
`Simulations).
`Polymorphism
`Does the blame for the present, hardly satisfactory
`situation lie with technicalities? Is it just a matter of
`better path-finding algorithms and faster computers,
`or are there other basic obstacles to crystal structure
`prediction by calculations? There are. All computa-
`tions and experiments demonstrate that many crystal
`structures for the same compound have quite similar
`lattice energies, or heats of sublimation. The AHS of
`a medium-size organic molecule is 20-50 kcal/mol;
`typically V3 of AHS;
`heats of melting (Aífm) are
`enthalpy differences between crystalline phases49
`in all evidence, just a fraction of AHm, or
`must be,
`something like 1-5 kcal/mole:
`the range of
`just
`experimental uncertainties of AHs’s on which empiri-
`calibrated. Besides, crystalline
`cal potentials are
`(43) Holden, J. R.; Du, Z.; Ammon, H. L. J. Comput. Chem. 1993,14,
`422-437.
`(44) See, for example: Zerkowski, J. A.; Whitesides, G. M. J. Am.
`Chem. Soc. 1994, 116, 4298-4304.
`(45) Linert, W.; Renz, F. J. Chem. Inf. Comput. Set. 1993, 33, 776-
`781.
`(46) Karfunkel, H. R.; Gdanitz, J. R. J. Comput. Chem. 1992, 13,
`1171-1183.
`(47) Perlstein, J. J. Am. Chem. Soc. 1994, 116, 455—470.
`(48) Gdanitz, R. J. Chem. Phys. Lett. 1992, 190, 391-396.
`(49) A vast literature deals with differences in solution enthalpies and
`phase transformation enthalpies of polymorphic pharmaceuticals; typical
`results are from a few down to fractions of a kcal/mol. For one example,
`see: Kojima, H.; Kiwada, H.; Kato, Y. Chem. Pharm. Bull. 1982, 30,
`1824—1830. See also ref 32 and references therein.
`
`Merck Exhibit 2164, Page 5
`Mylan Pharmaceuticals Inc. v. Merck Sharp & Dohme Corp.
`IPR2020-00040
`
`

`

`314 Acc. Chem. Res., Vol. 27, No. 10, 1994
`Table 2. Organization and Tools of an
`experimental
`synthetic chemistry
`commercially available compounds
`
`(re)crystallization
`studies of nucleation
`studies of morphology
`systematic search for polymorphs
`standardized studies of dissolution
`
`X-ray crystallography
`single crystal
`powder and Rietveld
`
`thermochemistry
`differential scanning calorimetry
`thermogravimetry
`vapor pressure measurements
`solid-state NMR
`
`Gavezzotti
`
`Ideal Department of Organic Solid State Chemistry
`theoretical
`Cambridge Structural Database
`First-principles calculations on sample systems
`
`empirical methods
`crystal potentials
`
`lattice dynamics
`search of crystal potential hypersurfaces
`Monte Carlo methods
`molecular dynamics
`
`ultramicroscopy (SEM/TEM, AFM)°
`“ The acronyms refer to surface visualization techniques: scanning and tunneling electron microscopy; atomic force microscopy.
`Concluding Remarks
`phases with higher enthalpy are usually less