摘要:
The errors in lattice energy minimisation studies: sensitivity to experimental variations in the molecular structure of paracetamol Theresa Beyer and Sarah L. Price* Centre for Theoretical and Computational Chemistry, Department of Chemistry, University College London, 20 Gordon Street, London, UK WC1H 0AJ. E-mail: s.l.price@ucl.ac.uk Received 11th August 2000, Accepted 9th November 2000 Published on the Web 12th December 2000 Fourteen experimental crystal structure determinations of paracetamol, covering two polymorphic forms and a range of temperatures, show the usual minor variations in the molecular structure, which reØect the limitations of X-ray crystallography and dynamical motion of the methyl group. Minima in the intermolecular lattice energy have been found starting from these experimental structures, keeping the molecular structure rigid and using the same distributed multipole-based model intermolecular potential in all calculations.These have been compared and contrasted with the lattice energy minima obtained using an ab initio optimised molecular model. Genuine crystal structure prediction requires the use of non-crystallographic molecular models, and so in this study we illustrate the accuracy that could be achieved in such a prediction study. The small variations in the experimental molecular structure produce variations of a few percent in the cell lengths and a few kJ mol21 of the lattice energy in the rigid-molecule crystal structure minima. This produces considerable uncertainty in the calculated energy difference between the polymorphs.The structural and lattice energy differences between the experimental crystal structures and the corresponding lattice energy minima vary sufÆciently with temperature and determination to show that the uncertainties in the molecular structure and neglect of temperature dominate this difference (of around 3% in the cell lengths). Thus, this study suggests that the level of signiÆcance for reÆning model potentials by comparing lattice energy minima and experimental crystal structures is a few % in the cell lengths. This is comparable with previous estimates based on the thermal expansion of this type of organic crystal structure between 0 K and room temperature. Introduction Most modelling of organic crystals is done by static lattice energy minimisation, which is now such a widely and easily used technique,1,2 that its limitations warrant more frequent discussion.Many sets of model intermolecular potentials3±7 have been parameterised by optimising the agreement of the lattice energy minima with the experimental crystal structures and available heats of sublimation. These model potentials have then been used for studying other properties of the crystals,1 such as elastic constants,8 or in Molecular Dynamics or Monte Carlo studies of the solid9,10 and liquid phases.10,11 When the potential parameters are being extrapolated to model another molecule, by assuming that the atom±atom parameters are transferable, then the model potential could be validated by its ability to reproduce the experimental crystal structure of that molecule in a lattice energy minimisation.In recent years, lattice energy minimisation has played a major role in the development of computational methods to predict molecular crystal structures. Indeed, there have been several successes in predicting the crystal structures of simple organic molecules,12 including in blind tests,13,14 by various methods of searching for the global minimum in the static lattice energy. It is generally assumed that the main cause for the differences between the experimental crystal structure and the lattice energy minimum is the approximations in the model intermolecular potential. However, there are many other approximations involved in comparing the two structures.It is necessary to assess the typical magnitude of the errors produced by the other approximations, as when this level of agreement between the minimum and the experimental structure is reached, further improvements to the model potential by empirical optimisation are not possible. Early work on empirical potential Ætting to crystal structures had a DOI: 10.1039/b006604o This journal is # The Royal Society of Chemistry Paper target accuracy of 1% in the lattice constants, 1� in cell angles, 2� in the molecular rotation and 0.1 A in the molecular translation.15 However, this accuracy has rarely been attained. More recent work considers a crystal structure to be satisfactorily modelled when the lattice energy minimum differs from the experimental structure by about the thermal expansion of organic crystals between 0 K and room temperature.7,16 This difference in structure between the formal temperature of the simulation and the temperature of most experimental determinations is typically a few % in the cell lengths.17 The aim of this paper is to evaluate how much variation in the temperature and accuracy of the experimental molecular structure affect the calculated lattice energy minima.Although we can only use one particular molecule in this study, it provides a concrete example of the `level of signiÆcance' of the accuracy with which crystal structures can be reproduced by lattice energy minimisation.This complements the estimates that differences of 2 kcal mol2118 or 3±4 kcal mol211 between the minimised lattice energy and the experimental heat of sublimation are not a cause for concern, based on the theoretical approximations, as well as experimental errors. One major approximation in lattice energy minimisation is in the model used for the molecule, i.e. the intra-molecular force Æeld. The difÆculties in obtaining a sufÆciently accurately balanced inter- and intra-molecular force Æeld, as well as the increased computational expense, mean that most lattice energy minimisation studies are performed using rigid molecules. This raises the issue as to which molecular structure is used, from which temperature and type of determination, and the accuracy required.Even for molecules that most chemists would consider rigid, the molecular structure in the crystal will be slightly affected by the environment, which will differ between polymorphs. Although the extent of these variations has been studied19 for organic molecular crystals, CrystEngComm, 2000, 34, 1±8 1their effect on the corresponding lattice energy minima has not. Furthermore, for genuine crystal structure predictions, it is necessary to obtain the rigid molecular model from a noncrystallographic source, and ab initio structure optimisation is a common choice amongst the many possible computational methods.13 These methods estimate the gas phase structure of the molecule, and so this approximation will affect how accurately the experimental crystal structure can be predicted.It has already been suggested for organometallic salts that polymorph prediction may be more severely hampered by inaccuracies in molecular structures than by inadequacies in the intermolecular potentials.20 Hence, we compare here the lattice energy minima obtained using an ab initio optimised molecular structure with those using the experimental molecular structures, to estimate the additional errors expected in predicting crystal structures using ab initio geometries. Other approximations involved in lattice energy minimisation are the complete neglect of explicit temperature effects and zero-point motion, i.e. the lattice energy minimum would be the 0 K classical structure for an accurate intermolecular potential.Hence, the lattice energy minimum structure corresponds to making the harmonic lattice approximation, as it is the anharmonicity in the potential that results in thermal expansion. Here, we determine the lattice energy minima derived from a large set of crystal structures for the same molecule, N-(4- hydroxyphenyl)acetamide (Scheme 1), covering two polymorphic forms. We compare these minima, which use the rigid experimental molecular structures, with those obtained using an ab initio optimised molecular structure, as might be used in a crystaructure prediction study. The aim is to assess the variation in the predicted lattice energies and crystal structures with the experimental determination, as an estimate of the uncertainties involved for similar molecules where only one experimental crystal structure is available.A second aim is to illustrate how accurately structures and relative stabilities of polymorphs could be predicted by lattice minimisation methods, when there are no experimental structures available. The molecule (Scheme 1) is better known as paracetamol or acetaminophen. It is a popular over-the-counter pharmaceutical. It has become the most widely accepted antipyretic (fever suppressant) and analgesic (pain killer) in the world, used by over 30 million people in the UK every year with an annual production of about 3 billion tablets.21 Although the more stable monoclinic polymorph I is currently marketed, it cannot be compressed directly into the pure form,22 requiring binders to be added for tableting.The metastable orthorhombic form II is considerably more compressible,23 which has pharmaceutical advantages in the tabletting process, and so both forms have been extensively investigated providing an unusual range of crystal structure determinations for this study. There are reports24,25 of another polymorph, which is too unstable to have been characterized. Method We have investigated both neutron and X-ray determinations of the crystal structure of paracetamol over a wide range of temperatures. The earliest X-ray determinations are by Haisa et al. for form II (HXACAN)26 in 1974, and for form I (HXACAN01)27 in 1976. Further X-ray determinations are by Scheme 1 Molecular structure of paracetamol.2 CrystEngComm, 2000, 34, 1±8 Fig. 1 An overlay of experimental and ab initio molecular structures of paracetamol. Blue: the lowest temperature (20 K) neutron determination of form I (CCW_20 K). Green: the lowest temperature (123 K) determination (X-ray) of form II (CSFII_LT). Red: the ab initio optimised structure. Click image or here to access a 3D representation. Naumov et al. (HXACAN04)28 in 1998, and more recently by Frampton et al.23 at 123 K (CSFI_LT) and at 298 K (CSFI_RT) for form I and for form II at 123 K (CSFII_LT) and at 298 K (CSFII_RT). Form I was determined by single crystal pulsed neutron diffraction at 20 K (CCW_20 K), 50 K (CCW_50 K), 80 K (CCW_80 K), 150 K (CCW_150 K), 200 K (CCW_200 K), 250 K (CCW_250 K) and at 330 K (CCW_330 K) by Wilson.29 For each of these 14 crystal structures, a lattice energy minimisation was performed, using both the corresponding experimental molecular structure, and a model gas phase structure.For all experimental structures determined by X-ray diffraction the bonds to hydrogen were adjusted to a standard length30 of 1.08 A for C±H, 1.02 A for O±H and 1.01 A for N± H along the experimental bond direction. For neutron data, the original hydrogen bond lengths were taken. A molecular model was generated without reference to an experimental crystal structure by geometry optimisation of a high quality ab initio SCF/6±31G** wavefunction, obtained using the program suite CADPAC.31 The crystal structure models using this ab initio molecular structure are referred to as 'prediction' models, since it represents the closest possible prediction of the crystal structure from the chemical diagram using rigid molecule lattice energy minimisation.The model intermolecular potential consisted of an ab initiobased distributed multipole model for the electrostatic contribution and an empirical isotropic atom±atom repulsion±dispersion potential. The electrostatic model includes all terms in the atom±atom multipole series up to R25, using atomic multipoles up to hexadecapole, which have been obtained by a Distributed Multipole Analysis (DMA)32 of the MP2/6±31G** wavefunction of the isolated molecular structure.Hence the electrostatic model reØects changes in the charge distribution with changes in the molecular structure. The electrostatic contributions to the lattice energy were summed over entire molecules to a 15 A centre of mass cut-off, except for the charge±charge, charge±dipole and dipole±dipole contributions that were evaluated by Ewald summation. All other contributions to the intermolecular potential were represented by an empirical repulsion±dispersion potential of the form U~ Uik~ i[1,k[2 X(AiiAkk)1=2 exp ({(BiizBkk)Rik=2){(CiiCkk)1=2 (1) R6ik i[1,k[2 X where atom i in molecule 1 is of type i, and atom k in molecule 2 is of type k. The parameters for the C, HC, O and N atoms were taken from the empirically Ætted potentials of Williams et al.33,34 The additional parameters used for both polar hydrogens (H1 and H4) were obtained by Ætting to many organic crystal structures involving polar hydrogens HN in N±Table 1 Differences between the experimental and the ab initio optimised molecular structure of paracetamola Form I, neutron Form I, X-ray Form II, X-ray Ab initio HXACAN CSFII LT CSFII RT HXACAN01 HXACAN04 CSFI RT CSFI LT CCW, 20 K CCW, 50 K CCW, 80 K CCW, 150 K CCW, 200 K CCW, 250 K CCW, 300 K values Lengths /A D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) D (%) 20.1 20.2 1.6 1.8 2.0 1.4 1.5 1.2 1.2 0.5 0.0 0.5 0.7 0.3 1.381 C2C3 20.1 20.6 20.4 1.2 0.7 1.0 0.4 0.6 0.6 0.4 0.0 0.6 0.1 0.4 1.385 C5C6 20.1 0.8 0.5 0.8 0.7 0.9 0.7 1.2 1.3 1.6 1.7 1.6 1.8 2.0 1.354 C4O1 21.6 20.8 20.9 21.0 21.2 21.0 21.1 21.1 21.5 21.0 21.4 21.7 21.1 21.3 1.360 N1C7 2.1 2.4 2.3 2.7 3.1 3.3 3.1 3.2 2.8 3.2 2.8 3.1 3.2 2.1 1.198 C7O2 20.1 20.5 20.6 20.7 20.7 20.4 20.4 20.7 20.5 20.3 0.8 1.2 0.7 0.3 1.515 C7C8 21.2 20.1 20.9 210.6 211.8 212.8 213.2 213.0 28.5 211.2 0.2 1.2 0.5 0.7 1.069 C2H2 20.7 28.3 29.7 210.3 28.9 213.5 210.7 212.7 0.5 1.0 1.5 1.0 0.7 0.2 1.074 C3H3 21.3 22.5 22.1 26.1 213.0 24.7 212.7 2.3 3.6 3.9 3.3 4.9 4.7 4.9 0.942 O1H1 20.1 20.3 20.4 28.4 210.7 210.4 211.8 213.7 210.8 210.8 0.2 0.1 0.4 0.4 1.077 C5H5 21.9 22.5 21.3 21.9 21.2 29.4 211.6 29.1 210.9 213.7 27.7 212.3 0.5 0.3 1.077 C6H6 21.5 27.2 28.2 26.9 29.0 213.4 26.7 220.1 0.1 1.7 1.0 2.2 1.2 1.3 0.993 N1H4 214.7 213.3 211.5 210.2 26.6 21.4 20.9 210.5 214.2 213.4 219.0 211.4 26.8 214.8 1.084 C8H7 25.8 22.8 24.2 21.5 21.1 20.1 29.5 210.3 213.1 212.5 211.1 28.9 218.8 0.3 1.08 C8H8 29.6 25.0 23.9 24.0 20.3 212.2 211.1 213.5 21.1 211.6 212.1 211.1 1.086 Angles/� 0.0 0.3 C8H9 21.6 20.8 20.8 20.8 21.6 20.8 20.8 20.8 20.8 21.6 21.6 21.6 0.0 0.0 123 O1C4C5 21.7 21.7 21.7 21.7 21.7 21.7 21.7 20.8 20.8 20.9 20.9 21.7 21.7 0.0 118 C6C1N1 20.8 21.6 20.8 20.8 20.8 20.8 20.8 20.8 20.8 20.8 20.8 0.8 0.0 0.8 129 C1N1C7 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1.7 1.7 121 O2C7C8 20.9 22.7 21.8 22.7 20.9 1.8 1.8 0.9 1.8 0.9 0.9 0.9 0.0 0.0 111 H1O1C4 20.9 20.9 20.9 22.7 20.9 23.5 20.9 0.9 0.0 0.0 0.9 0.9 0.9 0.0 113 C7C8H7 21.9 2.8 0.9 0.0 2.8 1.9 1.9 1.9 2.8 1.9 0.9 0.9 0.9 2.8 108 C7C8H8 21.8 20.9 21.8 0.9 0.9 0.0 0.0 0.0 0.9 0.0 0.0 0.0 0.0 0.9 109 C7C8H9 Overall/A 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.918 O1C8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 6.506 O1O2 3 CrystEngComm, 2000, 34, 1±8 Torsions/� 217 217 217 218 219 217 218 219 21 223 17 16 17 22 0 H1O1C4C5 2164 2164 2163 2180 2159 163 163 162 163 162 163 163 161 158 180 H1O1C4C3 21 2 2 0 1 0 2 1 23 23 4 3 1 21 0 O1C4C5H5 26 1 2 24 23 23 23 23 23 123 22 22 2 0 O1C4C3H3 216 218 215 216 219 16 16 15 15 15 15 15 13 15 12 C6C1N1H4 223 211 210 214 212 214 21 11 214 229 233 17 6 0 22 N1C7C8H7 2153 2147 2146 2143 2141 2139 2138 2106 2106 2103 2146 2120 2100 2159 2154 N1C7C8H8 94 104 104 105 105 106 139 147 137 110 120 147 84 88 N1C7C8H9 104 aFor the bond lengths, angles and for the overall dimension, % errors with D~100(expt2ab initio)/ab initio and for the torsions the absolute values in degrees are listed. Parameters for which the comparison of the ab initio form with all experimental structures yields Dv1% or torsional differences of dv3� are omitted (atom labels are shown in Scheme 1).Fig.2 Variation of the experimental and lattice minimised cell lengths of form I with temperature of determination. Constant offsets of 7.05 A in a, 9.1 A in b and 11.5 A in c have been used. HNºN and N±HNºOC hydrogen bonds.16 Hence the model potential corresponds to the FIT potential scheme,16 reÆned so that the effect of electron correlation on the molecular charge density is derived from a correlated wavefunction rather than by scaling the SCF multipoles.The repulsion±dispersion contributions to the lattice energy were summed out to 15 A . The lattice energies of all available experimental crystal structures were calculated and minimised using the program DMAREL,35 in a later version in which the space group symmetry was constrained during the minimisation. Thus the analytically calculated forces and torques on the rigid molecules, and the strains on the unit cell parameters, are relaxed using a pseudo-Newton±Raphson technique. The Hessian of the second derivatives at each lattice energy minimum was examined to conÆrm that it was a true minimum (i.e. all the eigenvalues are positive).The `prediction' ab initio molecular model was incorporated into each experimental crystal structure so that the centres of mass of the prediction and experimental molecular structure coincided, and their axes systems were parallel. The molecular axis system was deÆned with x along C4 to C1, and the xy plane being parallel to the plane containing C1, C4 and C2, so essentially the phenyl ring was located in the same position. The minimum obtained by this process (of minimising the experimental crystal structure using the same rigid molecular structure as would have been used in the search procedure) is the closest that a crystal structure prediction method based on rigid lattice energy minimisation could 'predict' the experimental crystal structure.Results Analysis of the molecular structures The variation in the molecular structures between the different determinations of the monoclinic form was known to be small for the non-hydrogenic atoms. An overall similarity quantiÆcation,36 based on superimposing the conformations and coordination environment of the non-hydrogenic atoms onto the low temperature structure CCW_20 K, gave a root-meansquare deviation from 0.003 up to 0.012 A for the conformations of the molecules, and from 0.013 up to 0.12 A for the superposition of a co-ordination group of 10 molecules, with the higher errors corresponding to higher temperatures. The ab initio molecular structure is contrasted with the experimentally determined molecular structures from each crystal structure in Table 1.This presents the bond lengths and angles of the experimental and ab initio optimised molecular structures, for all bond lengths and angles where any experimental structure deviates from the ab initio structure by more than 1% and all torsion angles where there is a difference of more than 3�. These deviations can be visualised in Fig. 1, where the ab initio optimised structure is overlaid with the lowest temperature experimental structures for each form. The differences in the 4 CrystEngComm, 2000, 34, 1±8 molecular framework (the bond lengths and angles not involving hydrogen atoms) are mainly too small to be in Table 1. Thus, there is no signiÆcant difference in the overall dimensions of the frame, as measured by the separation of the hydroxyl oxygen and the amide carbonyl (O1ºO2) or methyl carbon (O1ºC8) atoms.The X-ray bond lengths to hydrogen are all quite severely underestimated (Table 1), due to the systematic error resulting from the X-ray determination locating electron density,37 which is displaced into the bond for hydrogen nuclei. This systematic error18 is accounted for in the lattice energy calculations, where the hydrogen atoms are repositioned by elongating the bonds to standard neutron values. The ab initio N±H and O±H bond lengths are somewhat shorter than both these standard values and the neutron determinations in paracetamol. Most of the crystal structures have the hydroxyl group tilted out of the plane of the aromatic ring by around 17� in form I and 22� in form II (with the exception of CSFII_RT), whereas it is coplanar in the ab initio structure.This suggests that these are genuine distortions of the molecular structure by the hydrogen bonding in the crystal, which appear to be similar in the two forms. In both polymorphs the hydrogen bonds have the same graph sets38 and quite a degree of similarity, although the sequence of the different types of hydrogen bond (N(H)ºO(H) or O(H)ºOC) is different and there is a considerable difference in the angles between the planes of the hydrogen bonded molecules. The intramolecular energy differences between the molecules in the gas and crystalline phases, due to the rotation of the hydroxyl group, are estimated to be less than 1.6 kJ mol21.This is the difference in the total MP2 energy between the ab initio optimised molecular structure with a torsion angle of 0� and the same structure with the hydroxyl proton repositioned to give a torsion angle of 20�. This estimate is consistent with the nature of the atoms involved and the observation that torsion angles associated with an intramolecular energy of more than 1 kcal mol21 appear19 to be very unusual in crystal structures. The errors in this estimate of the intramolecular energy difference arise from the ab initio method and the assumption that all other geometric parameters are constant. Although the ab initio optimised structure is, as expected, the most stable at the SCF level at which it was optimised (by 16 kJ mol21 relative to CCW_20 K), the inclusion of electron correlation makes it less stable by 0.8 kJmol21.This is another example39 of an experimental crystal structure providing a more accurate estimate of the gas phase structure than SCF optimisation, by the criterion of the variation principle. We also note that the justiÆed concerns20 about calculating the relative stability of conformers using ab initio methods on molecules of Æxed geometry40 are well illustrated by paracetamol. The energies of the experimental molecular structures, relative to the most stable (CCW_20 K), range fromTable 2 Experimental crystal structures of paracetamol and the changes in parameters on lattice energy minimisation Reduced cell parameters/A (error in min D(%))a Hydrogen bonds Initial Method Origin.Space Min lattice energy/kJ mol21 RMS % lattice energy/ kJ mol21 O(H)ºO/A N(H)ºO/A a b c b/� Cell vol /A 3 (R factor) group T/K Error FORM I 2118.3 2111.1 P21/a 20 CCW_20 K 2.646 (8.2) 2.895 (0.9) 2.2 741.16 (4.5) 98.06 (2.0) 11.546 (0.1) 9.166 (1.7) 7.073 (3.2) Neutron 2118.6 2111.6 1/a P2 2.647 (8.0) 2.896 (0.8) 2.1 743.54 (4.2) 97.97 (2.1) 11.572 (0.1) 9.173 (1.6) 7.073 (3.0) Neutron 50 CCW_50 K 2119.6 2113.0 P21/a 80 CCW_80 K 2.654 (7.4) 2.897 (0.7) 1.9 744.23 (3.7) 97.90 (2.3) 11.574 (0.1) 9.173 (1.5) 7.077 (2.7) Neutron 2118.4 2111.2 11.620 (20.3) 97.82 (2.2) 1/n P2 2.656 (8.4) 2.907 (0.5) 2.0 753.94 (3.2) 9.232 (0.5) 7.094 (3.5) X-Ray 123 CSFI_LT 2119.7 2113.7 P21/a 150 CCW_150 K 2.652 (7.3) 2.906 (0.0) 1.6 753.61 (2.1) 97.78 (2.6) 11.628 (0.3) 9.240 (0.0) 7.079 (2.4) Neutron 2118.3 2110.6 11.657 (20.7) 97.67 (2.4) 1/n P2 2.656 (8.6) 2.913 (0.3) 2.0 759.09 (2.3) 9.262 (0.3) 7.094 (3.4) X-Ray HXACAN04 150 (0.055) 2.912 (20.1) 2.66 (6.7) 2119.9 2114.1 P21/a 200 CCW_200 K 1.7 759.77 (1.3) 97.65 (2.8) 11.664 (0.0) 9.278 (0.0) 7.084 (2.1) Neutron 2.916 (20.5) 2.659 (6.7) 2119.6 2113.8 9.315 (20.5) 11.676 (0.0) 1/a P2 1.7 763.89 (0.7) 97.58 (2.9) 7.086 (2.0) Neutron 250 CCW_250K 2.934 (20.5) 2.663 (7.8) 2118.3 2110.9 9.400 (20.4) 11.721 (20.7) 97.12 (2.8) P21/a HXACAN01 295 1.9 776.27 (0.9) 7.100 (2.7) X-Ray (0.072) 2.934 (20.6) 2.666 (7.4) 2119.4 2112.9 9.382 (20.7) 11.704 (21.1) 97.36 (2.8) P21/n 298 CSFI_RT 1.9 773.88 (0.5) 7.106 (3.0) X-Ray 2.931 (21.2) 2.66 (6.1) 2120.4 770.51 (21.0) 2115.2 9.370 (22.7) 11.706 (1.1) 1/a P2 2.1 97.46 (3.6) 7.085 (1.6) Neutron 330 CCW_330 K 2110.1 2.877 2.912 776.99 99.97 12.119 8.944 7.278 Prediction model 5 Cr2000, 34, 1±8 FORM II 2117.4 2111.9 2.708 (6.9) 2.942 (0.3) 2.2 1458.08 (6.1) 90 (0.0) 17.166 (1.8) 11.777 (0.8) 7.212 (3.4) X-Ray 123 CSFII_LT Pbca 2.967 (20.5) 2.726 (6.9) 2116.2 2112.0 1.1 1497.98 (3.0) 90 (0.0) 17.164 (1.6) 11.805 (1.2) 7.393 (0.2) X-Ray 295 HXACAN Pcab (0.077) 2112.2 2108.2 7.405 (20.1) 11.831 (0.6) 2.727 (6.9) 2.975 (1.8) 1.5 1502.99 (2.9) 90 (0.0) 17.156 (2.5) X-Ray 298 CSFII_RT Pbca 2106.5 2.949 2.924 1539.25 90 17.263 12.076 7.384 Prediction model aD(%)~100(min2expt)/expt.Fig.3 Overlay of the experimental (123 K) and prediction model (blue) crystal structures. (a) Form I in P21/n, using CSFI_LT (click image or here to access a 3D representation which compares the minimised crystal structures); (b) Form II using CSFII_LT in Pabc. In the reduced cell setting the layers are in the bc plane (bvc), and this diagram is viewed down a (click image or here to access a 3D representation which compares the minimised crystal structures). 1.1 kJ mol21 for CCW_50 K to over 100 kJ mol21 for CCW_300 K. This is consistent with the great sensitivity of Æxed geometry calculations to slight errors in the experimental bondlengths and angles, and the uncertainties in the positioning of the hydrogen atoms increasing with thermal motion.There is also considerable variation in the methyl group geometry. The considerable deviations from a tetrahedral methyl group observed in HXACAN01 and CSFI_RT are primarily an artefact of the inaccuracy of X-ray located protons.37 For the more accurately located methyl protons, the variation in torsion angles (Table 1) is consistent shown with their large thermal ellipsoids. As Wilson deduced from the thermal parameter analysis,29 there is considerable libration of this group with a residual zero-point root mean squared librational amplitude of 11�, and a barrier height for rotation of 2.2 kJ mol21. Thus, the substantial libration of the methyl group, above 80 K, results in an apparent shortening of the C8± H bondlengths in the neutron structures.Analysis of the experimental temperature effects The temperature dependence of the lattice constants of the monoclinic form I of paracetamol are shown in Fig. 2 and Table 2 (throughout this paper, the reduced cell parameters are used. These correspond to the P21/n setting for form I and the Pabc setting for form II). There is considerable variation in the three thermal expansion coefÆcients. There is a small expansion coefÆcient of a#161025 deg21 along a, which is approxi- 6 CrystEngComm, 2000, 34, 1±8 mately the direction of the hydrogen bonds [Fig. 3(a)]. There is a moderate expansion along c of a#461025 deg21, corresponding to changes in the relative tilt of the molecules in the zig±zag layer and the protrusion of the methyl groups.By far the largest expansion of a#961025 deg21 is in the b direction along which the pleated sheets are stacked. The more limited data on form II show quite small expansions in the bc plane of the hydrogen bonded sheets, with one side being essentially temperature independent, but considerable expansion of a which measures the separation of the sheets [Fig. 3(b)]. Analysis of the lattice energy minima All the lattice energy minima are quite close to their corresponding experimental crystal structures (Table 2). The maximum % change in a cell length is 3.6%, and the r.m.s. error over the three cell lengths is less than 2.3%. The lowest r.m.s. % error for both forms can be found for structure determinations in the region around 150vTv298 K for form I and T~295 K for form II, i.e.approximately room temperature. This probably reØects the fact that the repulsion±dispersion parameters were Ætted to room temperature crystal structures. The variations in the minimised cell lengths with experimental structure reØect the changes in the molecular structures with temperature only, as the simulation is effectively 0 K. As Fig. 2 shows, this results in a signiÆcant but erratic variation in the cell lengths of the lattice energy minima with the temperature of the experimental determination. In particular, the b and c parameters of form I can be overestimated or underestimated by lattice energy minimisation, depending on the crystal structure used.A more detailed examination of the lattice energy minima (Table 2) shows that only the a parameter of form I is consistently predicted to have signiÆcant deviation from experiment, albeit only being over-estimated by 1.6 to 3.5%. The intermolecular N(H)ºO(H) hydrogen bond lengths are well modelled (21.2vDv1.8%), but the difference between the experimental and calculated O(±H)ºO(C) intermolecular bonds is much higher (6.1vDv8.6%). This is consistent with these hydrogen bonds having a signiÆcant component along a for form I and in the bc plane of form II. It has been observed that this FIT potential gave a comparable elongation of the COºH±O bonds found in carboxylic acids.41 Hence, the use of distinct repulsion parameters for the two polar hydrogens, HN± N and HO±O, may lead to an improvement in the model potential.In the prediction model, when the ab initio optimised structure is pasted into any of the experimental structures for a given form, the lattice energy minima are identical. This implies that the crystal structures for each form correspond to the same basin of attraction for the minimisation, as using the same intermolecular potential and molecular model produces the same lattice energy minimum from each crystal structure. The prediction model minima differ from the experimental lattice energy minima in having a larger cell volume. In the monoclinic form I, the zig-zag sheets have expanded along the nonhydrogen bonded direction c, which reduces the protrusion of the methyl groups so that they stack with a smaller b [Fig.3(a)]. The sheets of the orthorhombic form are expanded [Fig. 3(b)]. These changes are nevertheless quite small and consistent with the enforced methyl conformation and planarity of the hydroxyl group. The O(±H)ºO distance is about 0.2 A larger in the prediction model than in the experimental reproductions. The calculations agree with the available energetic information on this system, i.e. that the monoclinic form I is the more stable. The various experimental structures give lattice energy minima estimates that vary with the experimental molecular model, from between 2118 and 2120 kJ mol21 for form I, and between 2112 and 2117.5 kJ mol21 for form II. In both cases,the prediction model gives a signiÆcantly less stable intermolecular lattice energy, by about 10 kJ mol21. The prediction model estimates that the orthorhombic form II is 3.6 kJ mol21 less stable than the monoclinic form I, whereas the experimental estimates would be 7.2 kJ mol21 using the 298 K crystal structures, or 1 kJ mol21 for the 123 K crystal structures.We note from Table 2 that the initial lattice energies show a somewhat larger range of values than the minimised lattice energies, and also that the polymorphic energy difference, for comparable determinations, such as the 123 or 295 K sets, predicts the wrong order of stability. This emphasises that lattice energy estimates derived from minimised crystal structures are more reliable than those derived from the initial lattice energies at the experimental structures.This is because small errors in the structure or repulsive potential can signiÆcantly destabilise the lattice since the repulsive wall is exponential. Conclusions The comparison of the eleven experimental molecular structures within form I and three within form II show that the paracetamol molecule in its crystalline phases is as well represented by a rigid ab initio optimised structure as can be reasonably expected for a typical organic molecule. There are variations with crystal structure determination and temperature, but these are mainly in the positions of the hydrogen atoms and reØect environmental distortions, dynamical motions and the limitations of X-ray crystallography37 which are much as would be expected, and are only qutiÆed through the extensive neutron studies.However, lattice energy minimisation with the same model intermolecular potential overestimates the b cell length by 1.7% in CCW_20 K and underestimates it by 2.7% in CCW_330 K. Hence a variation of over 4% in the minimised b cell length of form I comes from changing the experimental determination and therefore the molecular structure. The c cell length similarly can be under- or over-estimated using the same model potential, though with smaller but more erratic variations. The more limited data for form II also suggest that the differences between the lattice energy minimum and corresponding experimental structure are too small and erratic to reØect inadequacies in the model potential.Thus, the comparison of the paracetamol lattice energy minima suggests that differences of around 3% between the minimised and experimental structure may arise from approximations other than inadequacies in the model potential. The cell parameters of the experimental structures for form I vary by about 0.5% for a, 2.3% for b and 1.3% for c between 20 K and room temperature. These thermal expansions are approximately similar to the larger differences between each experimental structure and the corresponding lattice energy minimum for b and c. Thus this comparison of lattice energy minima of paracetamol produces the same conclusion as the `typical thermal expansion' argument, namely that a few % in the cell parameters is satisfactory agreement between lattice energy minima and experimental structures for neutral organic molecules.The reduced cell parameter a of the monoclinic structure is consistently overestimated, (though only by 1.6 to 3.2%), so that it could be interpreted as revealing a deÆciency in the model potential, probably in the COºH±O interactions. However, this conclusion could only be reached from having a range of determinations over two polymorphs, and because it is reasonable for hydrogens bonded to oxygen to have different repulsion±dispersion parameters from those bonded to nitrogen. The results from comparing any one lattice-energy minimum and the corresponding crystal structure would suggest that the potential was adequate within the limitations of lattice energy minimisation.Thus, although the model potential used in this study is far from deÆnitively accurate, and the theory of intermolecular forces suggests improvements in the repulsion±dispersion and neglected contributions, such improvements cannot be made through empirical Ætting using lattice energy minimisation. The accuracy with which lattice energy minimisation can predict crystal structures using an ab initio molecular structure is qualitatively excellent, but the crystal structures are not reproduced quantitatively as well as if the actual experimental molecular structures had been used. In the case of paracetamol, the only systematic difference between the experimental and the ab initio structures that is likely to be a genuine effect of the crystal packing is a small out-of-plane rotation of the hydroxyl group.Thus, this is a system where the rigid molecule approximation in a genuine crystal structure prediction should be fairly good. This is apparently the case, as the prediction model gives cell lengths that are generally within the range of the experimental determinations and reproductions, and only slightly outside this range in the directions most affected by the intramolecular rotation of the hydroxyl group. The differences in the rigid methyl group model appear to 6 have a fairly minor but observable effect on the crystal structure modelling of paracetamol.This is similar to the effect of positioning of the methyl groups in theophylline, where a full 60� rotation of the methyl group left the lattice energy remarkably unaffected.42 In contrast, two molecular structures for tetrolic acid,41 differing mainly in the placing of the undetermined methyl hydrogen atoms, had differences of 3% in the r.m.s. cell length errors, and several kJ mol21 in the lattice energy, giving signiÆcant quantitative but not qualitative differences. Hence, it seems that the uncertainties associated in using a rigid model for the methyl hydrogen atoms may be quite small, but this is very dependent on the crystal packing. This is fortunate, as it would be very demanding to model the dynamical motion of methyl groups43 beyond correcting for the apparent bond shrinkage caused by librational motion.Recent rigid-body lattice energy calculations on PF 2 salts of metal complexes showed that correcting the P±F bondlength for librational motion had a signiÆcant effect on calculated lattice energies.20 The standard methodology adopted here, of correcting X-ray determined proton positions to give standard bondlengths, does provide a rigid-body correction for the foreshortening caused by the librational motion. However, the neutron molecular structures were not corrected for this effect, and the results indicate that this might have had a small effect on the lattice energy minima for the higher temperature structures. Although the changes in the molecular and crystal structures are not atypical for the range of temperatures and types of determination considered, the sensitivity of calculated lattice energies to the positions of the hydrogen atoms does result in some variation in the predicted lattice energies.The differences of order of 1 kcal mol21 (i.e. 2 kJ mol21 for form I, 5 kJ mol21 for form II) are small relative to the total lattice energy. Even the lattice energy estimates using the gas phase molecular structure are not unreasonable compared with typical experimental errors in determining of heats of sublimation and the other errors implicit in comparing the two quantities. Unfortunately these differences are highly signiÆcant compared with the estimated energy difference between the two polymorphs.Thus, even for good quality experimental crystal structures of rigid molecules, discrepancies in lattice energy minima of a few percent in the cell dimensions and a kcal mol21 or so in the lattice energy could arise from variations in the experimental molecular structure. Thus static minimisation estimates of the energy differences between polymorphs, based on experimental molecular and crystal structures, have associated errors that are generally comparable to the actual energy difference. 7 CrystEngComm, 2000, 34, 1±8Crystal structure prediction calculations, based on a gas phase ab initio optimised molecular structure will have additional errors, due to changes in the molecular structure. Nevertheless, these paracetamol prediction model results are certainly sufÆciently close to any of the experimental crystal structures to be useful.The structures could be used as a starting point for reÆnement of crystal structures from powder data, and the relative lattice energies are sufÆciently close that both structures would be predicted as energetically feasible. Thus, we are currently attempting a crystal structure prediction search on paracetamol, to seek insight into its polymorphism. Acknowledgement We thank Dr C. C. Wilson (CLRC Rutherford Appleton Laboratories) and Professor C. S. Frampton (Roche Discovery) for providing the unpublished atomic coordinates of many of the crystal structures, the Cambridge Crystallographic Data Centre for a partial studentship for T.B., and the EPSRC for Ænancial support. References 1 A. J. Pertsin and A. I. Kitaigorodsky, The Atom±Atom Potential Method. Applications to Organic Molecular Solids, Springer± Verlag, 1987. 2 Theoretical Aspects and Computer Modeling of the Molecular Solid State, ed. A. Gavezzotti, Wiley, Chichester, UK, 1997. 3 A. T. Hagler, E. Huler and S. Lifson, J. Am. Chem. Soc., 1974, 96, 5319. 4 S. Lifson, A. T. Hagler and P. Dauber, J. Am. Chem. Soc., 1979, 101, 5111. 5 G. Filippini and A. Gavezzotti, Acta Crystallogr., Sect. B, 1993, 49, 868. 6 A. Gavezzotti and G. Filippini, J. Phys. Chem., 1994, 98, 4831. 7 D. E. Williams, J. Mol. Struct., 1999, 486, 321. 8 G. M. Day, S. LPrice and M. Leslie, Crystal Growth and Design, 2001, 1, 13.9 D. C. Sorescu, B. M. Rice and D. L. Thompson, J. Phys. Chem. B, 1997, 101, 798. 10 P. M. Rodger, A. J. Stone and D. J. Tildesley, Mol. Phys., 1988, 63, 173. 11 H. Sun, J. Phys. Chem. B, 1998, 102, 7338. 12 P. Verwer and F. J. J. Leusen, Reviews in Computational Chemistry, ed. K. B. Lipkowitz and D. B. Boyd, New York, 1998, vol. 12, p. 327. 13 J. P. M. Lommerse, W. D. S. Motherwell, H. L. Ammon, J. D. Dunitz, A. Gavezzotti, D. W. M. Hofman, F. J. J. Leusen, W. T. M. Mooij, S. L. Price, B. Schweizer, M. U. Schmidt, B. P. van Eijck and D. E. Williams, Acta Crystallogr., Sect. B, 2000, 56, 697. 14 B. S. Potter, R. A. Palmer, R. Withnall, B. Z. Chowdhry and S. L. Price, J. Mol. Struct., 1999, 486, 349. 15 L.-Y. Hsu and D.E. Williams, Acta Crystallogr., Sect. A, 1980, 36, 277. 8 CrystEngComm, 2000, 34, 1±8 16 D. S. Coombes, S. L. Price, D. J. Willock and M. Leslie, J. Phys. Chem., 1996, 100, 7352. 17 S. L. Price, in Computer Modelling in Inorganic Crystallography, ed. C. R. A. Catlow, Academic Press, San Diego, 1997, p. 269. 18 A. Gavezzotti, in Theoretical Aspects and Computer Modelling of the Molecular Solid State, ed. A. Gavezzotti, 1997, Wiley, Chichester, p. 97. 19 F. H. Allen, S. E. Harris and R. Taylor, J. Comput.-Aided Mol. Design, 1996, 10, 247. 20 J. Breu, H. Domel and P.-O. Norrby, Eur. J. Inorg. Chem., 2000, 2409. 21 40 years of Paracetamol. Advertising Feature, Pharm. J., 1996, 256. 22 E. Joiris, P. Di Martino, C. Berneron, A.-M. Guyot-Hermann and J.-C. Guyot, Pharm. Res., 1998, 15, 1122. 23 G. Nichols and C. S. Frampton, J. Pharm. Sci., 1998, 87, 684. 24 A. Bu» rger, Acta Pharm. Technol., 1982, 28, 1. 25 P. Di Martino, P. ConØant, M. Drache, J.-P. Huvenne and A.- M. Guyot-Hermann, J. Therm. Anal., 1997, 48, 447. 26 M. Haisa, S. Kashino and H. Maeda, Acta Crystallogr., Sect. B, 1974, 30, 2510. 27 M. Haisa, S. Kashino, R. Kawai and H. Maeda, Acta Crystallogr., Sect. B, 1976, 32, 1283. 28 D. Y. Naumov, M. A. Vasilchenko and J. A. K. Howard, Acta Crystallogr., Sect. C, 1998, 54, 653. 29 C. C. Wilson, Chem. Phys. Lett., 1997, 280, 531. 30 F. H. Allen, O. Kennard, D. G. Watson, L. Brammer and A. G. R. Orpen, J. Chem. Soc., Perkin Trans. 2, 1987, S1. 31 R. D. Amos, I. L. Alberts, J. S. Andrews, S. M. Colwell, N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, N. Koga, K. E. Laidig, P. E. Maslen, C. W. Murray, J. E. Rice, J. Sanz, E. D. Simandiras, A. J. Stone and M. D. Sul, in CADPAC6: The Cambridge Analytic Derivatives Package, Cambridge University, 1995. 32 S. L. Mayo, B. D. Olafson and W. A. Goddard III, J. Phys. Chem., 1990, 94, 8897. 33 S. R. Cox, L.-Y. Hsu and D. E. Williams, Acta Crystallogr., Sect. A, 1981, 37, 293. 34 D. E. Williams and S. R. Cox, Acta Crystallogr., Sect. B, 1984, 40, 404. 35 D. J. Willock, S. L. Price, M. Leslie and C. R. Catlow, J. Comput. Chem., 1995, 16, 628. 36 J. P. Lommerse, Int. J. Appl. Crystallogr., in preparation. 37 J. C. Speakman, in Molecular Structure by Diffraction Methods, ed. G. A. Sim and L. E. Sutton, The Chemical Society, London, 1973, vol. 1, p. 203. 38 M. C. Etter, Acc. Chem. Res., 1990, 23, 120. 39 C. B. Aakeroy, M. Nieuwenhuyzen and S. L. Price, J. Am. Chem. Soc., 1998, 120, 8986. 40 D. Buttar, M. H. Charlton, R. Docherty and J. Starbuck, J. Chem. Soc., Perkin Trans. 2, 1998, 763. 41 T. Beyer and S. L. Price, J. Phys. Chem. B, 2000, 104, 2647. 42 E. D. Smith, R. B. Hammond, M. J. Jones, K. J. Roberts, J. B. O. Mitchell, S. L. Price, R. K. Harris, D. C. Apperley, J. C. Cherryman and R. Docherty, J. Phys. Chem. B, submitted. 43 M. R. Johnson, M. Prager, H. Grimm, M. A. Neumann, G. J. Kearley and C. C. Wilson, Chem. Phys., 1999, 244, 49.
ISSN:1466-8033
DOI:10.1039/b006604o
出版商:RSC
年代:2000
数据来源: RSC