An Experimental and Computational Study of 1-Phenylethylammo

Peter W. Cains,§ Martin Vickers,† Derek A. Tocher,† Alastair J. Florence,| and. Sarah L. Price*,†. Department of Chemistry, UniVersity College ...
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J. Phys. Chem. B 2007, 111, 5326-5336

Toward the Computational Design of Diastereomeric Resolving Agents: An Experimental and Computational Study of 1-Phenylethylammonium-2-phenylacetate Derivatives Panagiotis G. Karamertzanis,† Parathy R. Anandamanoharan,‡ Phillipe Fernandes,| Peter W. Cains,§ Martin Vickers,† Derek A. Tocher,† Alastair J. Florence,| and Sarah L. Price*,† Department of Chemistry, UniVersity College London, 20 Gordon Street, London, WC1H 0AJ, U.K., Department of Chemical Engineering, UniVersity College London, Torrington Place, London, WC1E 7JE, U.K., AVantium Technologies BV, Zekeringstraat 29, 1014 BV Amsterdam, The Netherlands, and Institute of Pharmacy & Biomedical Sciences, UniVersity of Strathclyde, 27 Taylor Street, Glasgow, Lanark, G4 0NR, U.K. ReceiVed: December 12, 2006; In Final Form: February 27, 2007

The crystal structures, including two new polymorphs, of three diastereomerically related salt pairs formed by (R)-1-phenylethylammonium (1) with (S&R)-2-phenylpropanoate (2), (S&R)-2-phenylbutyrate (3), and (S&R)-mandelate (4) ions were characterized by low-temperature single crystal or powder X-ray diffraction. Thermal, solubility, and solution calorimetry measurements were used to determine the relative stabilities of the salt pairs and polymorphs. These were qualitatively predicted by lattice energy calculations combining realistic models for the dominant intermolecular electrostatic interactions and ab initio calculations for the ions’ conformational energies due to the distortion of their geometries by the crystal packing forces. Crystal structure prediction studies were also performed for the highly polymorphic diastereomeric salt pair (R)-1phenylethylammonium-(S&R)-2-phenylbutyrate (1-3) in an attempt to predict the separation efficiency without relying on experimental information. This joint experimental and computational investigation provides a stringent test for the reliability of lattice modeling approaches to explain the origins of chiral resolution via diastereomer formation (Pasteurian resolution). The further developments required for the computational screening of single-enantiomer resolving agents to achieve optimal chiral separation are discussed.

1. Introduction The world market for optically pure products has recently experienced a marked increase1 that has stimulated the development of novel methods for asymmetric synthesis and chiral separation. The revenues generated in 2003 by the application of chiral technology for pharmaceutical and agrochemical intermediates was estimated2 to be eight billion dollars, with an annual growth of around 11%. To a large extent, this was the response to drug regulatory guidelines3 that delineated the development of stereoisomeric drugs, following several cases of enantiomers exhibiting different pharma- or toxicological properties, such as the high-profile case of thalidomide.4 Manufacturing racemates is usually more economical than the synthesis of enantiomers,5 especially if their separation can be efficiently achieved with high yield and optical purity and the unwanted enantiomer utilized or recycled in some manner.1 A wide variety of chiral separation techniques are available, which exploit either the enantioselective behavior of biological systems or the differences in the molecular recognition of enantiomers by auxiliary chiral agents. In the “classical” resolution method (Pasteurian resolution),6 the enantiomers are converted to a diastereomeric salt pair by reaction (usually acid-base) with a single-enantiomer resolving agent, e.g., * Corresponding author. E-mail: [email protected]. † Department of Chemistry. ‡ Department of Chemical Engineering. § Avantium Technologies BV. | Institute of Pharmacy & Biomedical Sciences.

(R)-B + (S&R)-A-H+ f (R)-BH+(S)-A- + (R)-BH+(R)-Aand separated by precipitation of the least soluble salt. At present, there is no systematic methodology to choose the optimal resolving agent that has to be found on a trial and error basis.7 For example, a recent study of the resolution efficiency of 1,4-benzodioxane-2-carboxylic acid by (S)-1-arylethylamines revealed no systematic variations in the physicochemical properties of the diastereomeric salts with the functional groups present.8 Hence, there would be considerable industrial benefit to a computational method of designing the optimal resolving agent and process conditions. The most important determinant for an efficient resolution is the solubility ratio cRS/cRR of the two diastereomers. If we assume that the difference in pKa values between the base B and the acid A-H+ is large and that the free energy difference of the two diastereomers in solution is negligible, i.e. the ions are fully dissociated and solvated, then it can be shown9 that the solubility ratio is related to the diastereomers’ free energy difference:

GRS - GRR ) HRS - HRR - T(SRS - SRR) ) 2RT ln(cRS/cRR) (1) Under these simplifying assumptions, the solubility ratio is linked to the solid-state properties and does not depend on the solution chemistry. Approximate estimates of the diastereomers’ stability difference have been made by visually examining the crystal structures of the less and more soluble diastereomer for

10.1021/jp068530q CCC: $37.00 © 2007 American Chemical Society Published on Web 04/19/2007

1-Phenylethylammonium-2-phenylacetate Derivatives CHART 1: Diastereomeric Salt Pairs Formed by (R)-2Phenylethylammonium (1) with (S&R)-2-Phenylpropanoate (2), (S&R)-2-Phenylbutyrate (3), and (S&R)-Mandelate (4)a

a Torsion angles likely to be significantly affected by the packing forces are indicated.

the presence or lack of stabilizing hydrogen-bond motifs,10,11 CH‚‚‚π interactions12 and the overall packing efficiency.10 Although such qualitative analyses can recognize trends when there are marked differences in the dominant intermolecular interactions, they cannot quantify the stability difference. On the other hand, the computational prediction of the free energy difference is very demanding, as the zero-point energy, entropy, and temperature-dependent contributions to the enthalpy need to be accurately estimated, accounting for molecular flexibility and thermal expansion.13 However, a systematic study of the resolution efficiency of 11 diastereomeric salt pairs of ephedrine with phenyl-substituted cyclic phosphoric acids showed that the solubility ratio correlates well with the enthalpy difference between the diastereomeric salt pairs;9 i.e., the entropies of the two diastereomers are approximately equal. If we further assume that the zero-point energy and specific heats of the two diastereomers are similar, then, to a first approximation, the solubility ratio can be estimated by comparing the diastereomers’ static lattice energies. The static lattice energy, which includes the intermolecular contributions and the intramolecular energy penalties for the deformation of the ions’ conformation in the solid state, is amenable to theoretical predictions. Early computational studies9,14,15 showed that the models for the intermolecular forces and the ion conformational energies were not sufficiently accurate for reliable relative lattice energy estimates, and in some cases, the experimental crystal structures did not correspond to a minimum on the lattice energy surface as modeled with the available force fields. In order to provide the experimental information necessary for assessing the ability of current computational models to predict the diastereomers’ relative lattice energy and hence design resolving agents, we have performed an interdisciplinary study on a series of three polymorphic, chemically related diastereomeric salt pairs. These are formed by the optically pure base (R)-1-phenylethylamine (1) with the two enantiomers of 2-phenylpropanoic acid (2), 2-phenylbutyric acid (3), and mandelic acid (4) (see Chart 1). The crystal structures, including two new polymorphs, have been determined by single crystal or powder X-ray diffraction. Transitions to high-temperature polymorphs were studied with variable temperature powder X-ray diffraction. The thermodynamic relative stabilities (in terms of enthalpy and Gibbs free energy ordering) of the diastereomeric forms were established with a set of complementary techniques, including solubility, solution calorimetry, and thermal measurements and found to be consistent with the reported separability behavior16 of the diastereomeric salt pairs. The structural and thermodynamic data were then used to test the ability of a state-of-the-art computational model based on

J. Phys. Chem. B, Vol. 111, No. 19, 2007 5327 lattice energy minimization to reproduce the crystal structure and experimental relative stabilities. The lattice energy was computed by combining a realistic model for the dominant intermolecular electrostatic interactions with ab initio intramolecular energies. The ultimate aim of computational studies is the prediction of the stability difference of the diastereomeric salt pair without relying on experimental information, which also requires the ab initio prediction of the diastereomers’ crystal structures. The prediction of the crystal structure of chiral salts is a challenging problem15 for two reasons. First, the number of possible packing arrangements that need to be considered is significantly larger compared with typical crystal structure prediction studies for rigid, non-ionic systems. This is because the smallest asymmetric unit in a salt crystal structure comprises two crystallographically independent ions. In addition, chiral ions generally contain single bonds around which rotation, in response to the crystal packing forces, is energetically feasible, and hence, the ions’ conformations should also be varied within the search. Second, accuracy in the calculated energies is vital for the quantitative prediction of the solubility ratio, as the latter depends exponentially on their relative thermodynamic stability (see eq 1). Because the ions are flexible, their predicted energy difference is particularly sensitive to the balance between the intra- and intermolecular forces. In a preliminary study, we explored the lattice energy landscape of the diastereomeric salt pair R-(1)R-(2), R-(1)S-(2) by using a set of rigid-body ion conformations and located all known polymorphs in the vicinity of the corresponding global energy minima.17 However, the lattice energy was very sensitive to even subtle changes in the ion conformations arising from the crystal packing forces. Subsequently, we refined the rigidbody predictions by explicitly modeling the ions’ deformation (using quantum mechanical calculations) under the intermolecular forces within the lattice, with the recently developed flexible-torsion lattice energy minimization algorithm DMAflex.18 This approach improved the stability ranking of the known forms and led to good agreement between the predicted lattice energy difference and experimental relative enthalpy of the diastereomers.18 In this paper, we report results on the crystal structure prediction of the conformationally more challenging system R-(1)R-(3), R-(1)S-(3) by using this methodology to refine the hypothetical crystal structures generated with a novel flexible-molecule search algorithm.19 The ability to predict the structures of this diastereomeric salt pair is of particular interest because of the large number of polymorphs discovered in this study. 2. Method 2.1. Experimental. 2.1.1. Crystallography. Powder samples for systems R-(1)S-(3) and R-(1)S-(4) were characterized by powder X-ray diffraction at room temperature and found to correspond to the literature structures (CSD refcodes PEAPEA10 and PIVGEG, respectively), against which they were refined. In particular, powder data for R-(1)S-(3) were collected on a Siemens D500 theta-2theta Bragg-Brentano diffractometer with a copper anode at 30 mA, 40 kV and a quartz pre-sample monochromator to eliminate all but KR1. The diffraction pattern was recorded from 2 to 45° 2θ at 0.05° increments with a count time of 40 s per step using a scintillation counter detector. For R-(1)S-(4), data were collected on a Stoe StadiP transmission diffractometer with similar radiation and tube power and a 0.5 mm sample capillary. The KR1 monochromator used was germanium, and the scan was from 3 to 30° 2θ at 0.1° increments and 140 s per step with a linear position sensitive detector (PSD). Binning at 4 channels per point gave a data

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TABLE 1: Experimentally Determined Enantiomorphous Crystal Structures for the (R)-1-Phenylethylammonium, (S&R)-2-Phenylacetate Derivativesa reduced cell structure/ CSD reference

symmetryb

a (Å)

b (Å)

R (deg)

c (Å)

β (deg)

γ (deg)

density (g cm-3)

RSc RR Ic RR II (NMACEP01)

P21 P212121 P212121

(R)-1-Phenylethylammonium, (S&R)-2-Phenylpropanoate (1-2) 6.539 11.008 12.160 116.0 90.00 90.00 1.146 5.761 15.376 16.824 90.00 90.00 90.00 1.209 5.941 15.469 17.501 90.00 90.00 90.00 1.121

RS (PEAPEA10)

P41

(R)-1-Phenylethylammonium, (S&R)-2-Phenylbutyrate (1-3) 6.408 16.642 16.642 90.00 90.00 90.00 1.068

RR Ic RR IId RR IIId

P212121 P212121 P21

RS (PIVGEG)

P1, Z′ ) 4

6.398

RR Ic RR II (PEAMAN01)

P212121 P21

6.849 6.801

5.757 6.062 11.882

15.433 16.779 5.976

17.571 16.888 13.075

90.00 90.00 113.51

90.00 90.00 90.00

90.00 90.00 90.00

1.214 1.103 1.113

(R)-1-Phenylethylammonium, (S&R)-Mandelate (1-4) 14.807 16.109 75.33 82.97 81.03 1.250 8.325 8.322

25.441 12.885

90.00 91.74

90.00 90.00

90.00 90.00

1.252 1.245

experimental details SXRD, 150 K, R-factor 4.07% SXRD, 150 K, R-factor 3.58% SXRD, RT, R-factor 3.15% SXRD, RT, R-factor 5.6%, positions of some hydrogen atoms not determined SXRD, 150 K, R-factor 3.98% PXRD, 295 K, Rwp 2.9% PXRD, 295 K, Rwp 3.0% PXRD, 122 K, R-factor 4.49% positions of some hydrogen atoms not determined SXRD, 150 K, R-factor 5.88% SXRD, 122 K, R-factor 4.4%

a Structures determined in this work are shown in bold; for literature structures, the determinations with the lowest temperature/R-factor are given. b Space group and number of crystallographically independent ion pairs if greater than one. c Re-determined at lower temperature by SXRD in this work, leading to smaller R-factor compared to CSD structures (PMACEP01, NMACEP02, PBUPEA, and PEAMAN, respectively). d Determined from powder X-ray data in this work.

step of 0.02° 2θ. The unit cells for both measurements were refined against the published structure models using Rietica.20 Single-crystal structures were determined by mounting on a glass fiber, and all geometric and intensity data were taken from the sample on a Bruker SMART APEX CCD diffractometer using graphite-monochromated Mo-KR radiation (λ ) 0.71073 Å) at 150 ( 2 K. Data reduction and integration were carried out with SAINT+21 and absorption corrections applied using the program SADABS.22 Structures were solved by direct methods and developed using alternating cycles of least-squares refinement and difference Fourier synthesis. All non-hydrogen atoms were refined anisotropically; hydrogen atoms were refined isotropically. Structure solution and refinement used the SHELXTL PLUS V6.12 program package.23 Variable temperature powder X-ray diffraction data were collected from polycrystalline samples held in 0.7 mm borosilicate glass capillaries. Capillaries were mounted on a BrukerAXS D8 instrument fitted with a curved germanium 111 primary monochromator (Cu KR1, λ ) 1.54056 Å), Vantec PSD, and transmission capillary geometry. Samples were heated in the range of 295 and 370 K using an Oxford Cryosystems Crystostream 700 Series Plus device. Data were collected using the following settings: 50 kV, 40 mA, and 0.016° 2θ step size. Step times for structure identification and variable temperature scans were typically 1 s per step, with variable count time schemes implemented for data collections for structure determinations. Structure solution was carried out using the simulated annealing global optimization procedure described previously24 and implemented in the DASH structure solution package.25 Rietveld refinements were carried out using TOPAS.26 2.1.2. Crystal Growth and Determination of Structures. For all crystallization experiments, we used (R)-1-phenylethylamine (99+% purity) and the six enantiomer acids (97-99%) obtained from Lancaster Synthesis and Alfa Aesar, respectively, without further purification. All solvents were HPLC grade. Single crystals of diffraction quality were produced for R-(1)S-(2), R-(1)R-(2) I, R-(1)R-(3) I, and R-(1)R-(4) I by solvent evaporation of the corresponding enantiomorphous salt from ethanol. Their structures, which corresponded to the published

determinations,27-29 were re-determined by low-temperature X-ray diffraction. In the case of the highly soluble R-(1)S-(3) and R-(1)S-(4), the same procedure led to microcrystalline material. PXRD measurements yielded a sufficiently good quality pattern in reasonable agreement with the literature structures30,31 (the unit cells were refined to P41, a ) 16.659 Å, c ) 6.416 Å for R-(1)S-(3) and P1, a ) 6.446 Å, b ) 14.976 Å, c ) 16.330 Å, R ) 75.06°, β ) 82.96°, γ ) 80.64°, for R-(1)S-(4) both at room temperature). The polymorph R-(1)R-(4) II was produced by separation experiments in ethanol16 and confirmed to correspond to the literature structure29 by unit cell determination (single-crystal X-ray diffraction). Due to the large solubility difference of the two diastereomers, the product of separation experiments16 was predominantly the RR diastereomer for the majority of starting ratios of S/R mandelate. The unit cell determination of the precipitates revealed that the polymorphic outcome was dependent on the starting ratio as R-(1)R-(4) II was separated for ratios R/S less than 4/1 and R-(1)R-(4) I for larger ratios. The new polymorph R-(1)R-(3) II was produced as a fine polycrystalline powder by solvent evaporation from ethanol solution with a starting ratio of 2:1 R-2-phenylbutyric acid:R1-phenylethylamine, and its crystal structure was solved from powder X-ray diffraction data.32 Variable temperature powder X-ray measurements showed that the endothermic events observed upon heating forms R-(1)R-(3) I and R-(1)R-(3) II (see Supporting Information) are due to phase transitions leading to the same high-temperature polymorph, R-(1)R-(3) III, that was also solved from powder X-ray diffraction data.33 These diastereomeric salt crystal structures are summarized in Table 1. Our X-ray experiments did not reveal the presence of any cocrystals with neutral acid molecules coexisting in the unit cell with the two ions, although they have been reported for system 1-4.34,35 Moreover, the use of a single base enantiomer prevented the crystallization of double (both enantiomers of the 1-phenylethylammonium cation in the unit cell)36 and racemic37,38 salts known to exist for systems 1-2 and 1-4. A thorough polymorph screen may produce additional enantiomorphous or double salt structures (both enantiomers of the

1-Phenylethylammonium-2-phenylacetate Derivatives

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anion in the unit cell) or other solid forms (e.g., solvates or cocrystals with neutral acid molecules) but remained outside the scope of the present investigation. 2.1.3. Solution Calorimetry, Solubility, and Thermal Measurements. Solution calorimetry was used to determine the enthalpy change associated with the transition from the crystallized solid to a dilute aqueous solution (nominally infinite dilution of less than 1 mg L-1). HPLC data show complete dissociation of the ions, and hence, the differences in the enthalpies of formation of the lattices can be determined by the differences in the enthalpy changes measured on dissolution, because each salt pair will dissociate to a common base (R-1phenylethylamine) and the corresponding enantiomer acid. Solution calorimetry results are reported only for systems 1-2 and 1-4. For 1-3, the polymorphic purity of the sample could not be determined, and hence, the results were considered unreliable. The enthalpies and entropies of dissolution of salt 1-2 in ethanol have also been determined by fitting the equation39

2 ln x ) -

∆Hdiss + ∆Sdiss/R RT

(2)

where x denotes the salt mole fraction, to solubility data in the temperature range 283-322 K.16 The fit of the data was very good (R ∼ 0.998). The solubility data obtained for the other two salt pairs were not suitable for fitting. Combined differential scanning calorimetry and thermogravimetric analysis were carried out for systems 1-3 and 1-4 using a Netszch Jupiter STA449C instrument. Aluminum sample pans (10 µL) (open) were used, and samples were heated at a rate of 10 K min-1 from room temperature to above the melting point of the sample under a flow of pure nitrogen (20 mL min-1). No thermal analysis was performed for the thoroughly characterized27,38 1-2 system. 2.2. Computational. 2.2.1. Calculation of Lattice Energies of the Experimental Structures. The ability to predict relative stabilities from the crystal structures of all experimentally determined enantiomorphous crystals (Table 1) for the three diastereomeric salt pairs was studied by rigid-body lattice energy minimization using the crystal structure modeling program DMAREL.40 The ion conformations had the flexible torsion angles θ identified in Chart 1 constrained to their experimental values to reflect the effect of the packing forces on the intramolecular degrees of freedom. The rest of the ion geometry was determined by MP2/6-31G(d,p) ab initio constrained optimization using the electronic structure program GAUSSIAN.41 The lattice energy was computed as the sum of the intermolecular contributions and the ion conformational energies. The latter were expressed as the energy difference between the θ-constrained and global, unconstrained ab initio minima. The lattice energies ignore the zero-point and temperaturedependent contributions to the energy and are an approximation to the enthalpy of formation at an indeterminate temperature, because of the use of a repulsion-dispersion potential that has been empirically fitted42 to experimental organic crystal structures and heats of sublimation. This isotropic atom-atom intermolecular exp-6 repulsion-dispersion potential used the C, N, O, and HC (hydrogen connected to carbon) parameters of Williams et al.43-45 and a HN,O (hydrogen connected to nitrogen or oxygen) parameter added to this force field by fitting to polar, hydrogen-bonded organic crystal structures in conjunction with a distributed multipole model.42 The electrostatic interactions were modeled with a set of distributed multipole moments up to hexadecapole computed at the atomic nuclei with a distributed

multipole analysis46 of the MP2/6-31G(d,p) ion charge densities. The lattice energy contributions from the slowly convergent charge-charge, charge-dipole, and dipole-dipole interactions were evaluated by Ewald summation;47 the repulsion-dispersion interactions and those involving higher multipole moments (decaying as R-5 or faster) were computed in direct space up to a 15 Å cutoff distance. The geometric reproduction of the crystal structures was used to confirm that the accuracy of the intermolecular potential was sufficient and that the flexible degrees of freedom had been correctly identified (the mean RMS discrepancy for the nonhydrogen atoms between the experimental and ab initio constrained-optimized ion conformations was 0.05 Å). In all cases, the root-mean-square error in the lattice lengths was a few percentage points, and the root-mean-square deviation in the overlay of all non-hydrogenic atoms in a 15-ion coordination sphere48 was smaller than 0.35 Å. The validity of the computational model was further confirmed by the excellent reproduction of all known racemic, double salt, and acid crystal structures retrieved from the Cambridge Structural Database (CSD)49 involving the base and acid moieties under investigation without other molecules present. 2.2.2. Crystal Structure Prediction for the Diastereomeric Salt Pair R-(1)R-(3), R-(1)S-(3). Crystal structure prediction for 1-3 was performed by a comprehensive, two-stage search methodology for packing arrangements of low lattice energy. The approximate lattice energy surfaces for both diastereomers were first explored for low-energy minima with a recently developed search algorithm,19,50 which utilizes a computationally efficient, isotropic intermolecular potential based on conformationally dependent atomic charges and an empirical repulsion-dispersion potential.51 The distortions in the rotation of the phenyl, carboxylate, ethyl, and ammonium groups (torsion angles θ in Chart 1) from their in Vacuo values due to the packing forces were explicitly considered during both the generation and minimization of hypothetical crystal structures.19 The searches were limited to the five most common chiral space groups (P212121, P21, P1, C2, and P21212) plus the tetragonal space group P41, so that the experimentally determined RS structure could be found. In the second stage, approximately the 50 most stable structures for each diastereomer were refined with the algorithm DMAflex,18 allowing the same set of flexible torsion angles θ (Chart 1) to vary under the forces of the crystal lattice, now modeled using conformationally dependent, atomic multipole moments. The atomic multipoles and ion intermolecular energies were repeatedly computed directly by quantum mechanical calculations during the lattice energy minimization. This algorithm has been successful in describing the fine balance between the inter- and intramolecular interactions for a wide range of crystals containing flexible molecules and in predicting the crystal structure and relative stability of the conformationally less challenging diastereomeric salt pair R-(1)R-(2), R-(1)S-(2).18 The ab initio intramolecular energies were computed at the HF/ 6-31G(d,p) level, and the intermolecular potential was identical to the one used for the rigid-body minimizations (i.e., distributed multipoles computed using single-point MP2/6-31G(d,p) calculations). However, the cutoff distance for both repulsiondispersion and higher multipole interactions was increased to 60 Å to avoid numerical instabilities from molecules coming in and out of the cutoff region in these flexible-torsion minimizations. Moreover, the electrostatic model was constructed with a recently reported, modified multipole analysis52 to avoid discontinuities in the electrostatic component of the

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TABLE 2: Summary of Thermal Measurements system

onset (K)

peak (K)

∆Ha (kJ mol-1)

remarks

(R)-1-Phenylethylammonium, (S&R)-2-Phenylpropanoate (1-2)b 437.0 42.5 melting 377.9 4.7 transformation to RR II 422.0 39.8 melting

RS RR I RR II RS RR I RR II RR III

405.8 378.6 367.9 429.4-434.7

RSd RR I RR II

383.1 452.5 445.4

(R)-1-Phenylethylammonium, (S&R)-2-Phenylbutyrate (1-3)b 408.1 44.4 melting 382.7 6.0 transformation to RR III 373.0 1.1 transformation to RR III 435.6-438.8 40-44 melting, decomposition with >10% mass lossc (R)-1-Phenylethylammonium, (S&R)-Mandelate (1-4)b 389.7 23.0 melting 463.9 39 melting, decomposition with 9.2% mass lossc 453.6 36 melting, decomposition with 9.6% mass lossc

a All enthalpies reported correspond to endothermic events. b For system 1-2, thermal analysis data were taken from ref 38; detailed thermal measurements (DSC and variable temperature XPRD) for systems 1-3 and 1-4 can be found in the Supporting Information. c Significant mass loss shows that the heat of fusion cannot be used in quantitative comparisons with theoretical predictions due to sample decomposition. The melting temperature and heat of fusion for R-(1)-R-(3) III shows significant variation depending on the quantity of the sample used and rate of heating leading to variable degrees of mass loss and decomposition. d A minor endothermic event (∆H ) -0.8 kJ mol-1) is also observed with onset at 353 K (peak 358 K). Variable temperature PXRD showed that this event is not accompanied by appreciable changes in the structure of the system.

TABLE 3: Thermodynamic Data from Solution Calorimetrya at Room Temperature and Solubility Measurements solution calorimetry (water, room temperature)

solubility measurements (ethanol, 283-322 K)b

∆Hdiss ∆[∆Hdiss]c ∆Hdiss ∆[∆Hdiss]c ∆Sdiss system (kJ mol-1) (kJ mol-1) (kJ mol-1) (kJ mol-1) (J mol-1 K-1) (R)-1-Phenylethylammonium, (S&R)-2-Phenylpropanoate (1-2) RS 7.1 -3.9 55.1 -10.0 133.2 RR I 11.0 65.1 178.3 RS RR I

(R)-1-Phenylethylammonium, (S&R)-Mandelate (1-4) 0.5 -14.9 15.4

a No measurements are reported for 1-phenylammonium,2-phenylbutyrate (1-3) because of the presence of multiple polymorphs in the solid sample used in the calorimetry measurements. b The enthalpy and entropy of dissolution were approximately determined by fitting eq 2 in the temperature range 283-322 K (solubility data from ref 16) by assuming unit activity coefficients and full dissociation of the salts in ethanol solution, as observed by HPLC.16 They are double the values quoted previously16 that were calculated for non-dissociated molecular entities. c Defined as ∆[∆Hdiss] ) ∆Hdiss,RS - ∆Hdiss,RR ) HRR - HRS.

lattice energy, arising from contributions to multipole moments being switched to different nuclei as the ions’ conformations change. The required CPU time per refinement is approximately 3 days on a modern workstation and involves, on average, 70 ab initio calculations to compute the intramolecular energy and charge density for each ion. The total cost, which also includes the preliminary searches that involved approximately 120 000 lattice energy minimizations for each diastereomer and the construction of the ab initio intramolecular potentials grids for the two ions,19 is estimated to exceed the equivalent of approximately two CPU years on a single workstation. This investigation used distributed computing facilities. 3. Results 3.1. Experimental Relative Stability. The results of the experimental measurements on the relative stability of the three diastereomeric salt pair crystal structures are summarized in Tables 2 and 3. Heats of fusion and solution calorimetry measurements are used to determine enthalpy differences, and

solubility ratio measurements are typically used to establish differences in the Gibbs free energy between the diastereomers’ crystal structures. In order to derive an approximate stability order of the diastereomers and their polymorphs that allows a direct, qualitative comparison with static lattice energy calculations, we will assume that the free energy and enthalpy order is the same and independent of temperature; i.e., the zero-point and thermal contributions are equal for all crystal structures. We will critically examine the experimental data to assess the validity of the assumptions and hence the reliability of the computed static lattice energy differences to predict the resolution efficiency. 3.1.1. (R)-1-Phenylethylammonium, (S&R)-2-Phenylpropanoate (1-2). The RS diastereomer is known to exist in one form;27,53 the RR diastereomer has two enantiotropically related polymorphs, with RR I27,53 being more stable than RR II53 at low temperatures. The heat of the endothermic transition38 RR I f RR II is -4.7 kJ mol-1 (Table 2). The solution calorimetry measurements (Table 3) show that the RR I diastereomer has 3.9 kJ mol-1 lower enthalpy of formation than RS at room temperature. If we assume that the disorder in the diastereomers’ melts is sufficient for the entropy difference to be negligible and that the entropy difference between the two solids is independent of temperature, we can estimate the entropy difference SRS - SRRI ≈ -∆Hf,RS/Tf,RS + ∆Hf,RRII/Tf,RRII - ∆Ht/ Tt, where Tt and ∆Ht are the temperature and enthalpy difference for the transition RR I f RR II. The estimated entropy difference SRS - SRRI ≈ 9.5 J K-1 mol-1 agrees reasonably well with the value of 19 J K-1 mol-1 obtained by using the free energy difference at 25 °C estimated from solubility data16,17 and the room-temperature enthalpy difference obtained by solution calorimetry given the approximations involved. This shows that entropy stabilizes the RS diastereomer by several kiloJoules per mole at room temperature compared with the RR I polymorph. Hence, the marginally lower solubility16 of the RS diastereomer for temperatures above 283 K is due to the entropic contributions. Fitting eq 2 to solubility measurements16 in the temperature range 283-323 K gives somewhat larger enthalpy (HRS - HRRI ) 10 kJ mol-1) and entropy differences (SRS - SRRI ) 45 J K-1 mol-1), although the entropy and enthalpy ordering is consistent with the other measurements. These results demonstrate that the interplay between the enthalpic and entropic contributions to the Gibbs free energy may alter the stability order of the diastereomers at different

1-Phenylethylammonium-2-phenylacetate Derivatives temperatures. However, as present computational models are not yet able to capture this effect, we can only examine whether lattice energy calculations qualitatively reproduce the approximate experimental stability order RS ∼ RR I > RR II, which correlates well with the poor resolution efficiency of 1-2.16 3.1.2. (R)-1-Phenylethylammonium, (S&R)-2-Phenylbutyrate (1-3). The second system has one RS and three RR diastereomeric salt structures. The endothermic transitions RR I f RR III (onset 378.6 K, ∆H ) -6.0 kJ mol-1) and RR II f R III (onset 367.9 K, ∆H ) -1.1 kJ mol-1) suggest54,55 that RR III is enantiotropically related to both RR I and RR II (see Supporting Information). RR I is probably the thermodynamically most stable at low temperatures given that the transition RR I f RR III is significantly more endothermic than RR II f RR III. At the transition temperature, the Gibbs free energy difference between the interconverting forms vanishes, and hence, the transition entropies (by using the transition enthalpies and transition onset temperatures in Table 2) SRRIII - SRRI and SRRIII - SRRII are calculated to be equal to 16 and 3 J K-1 mol-1, respectively. If we assume that entropy differences are independent of temperature, we can estimate that at room temperature the entropy stabilizes form RR II by approximately 4 kJ mol-1. Although the entropy contributions are smaller at lower temperatures, this suggests that the predicted relative lattice energies will probably overestimate the actual free energy differences by several kiloJoules per mole. The diastereomer RS is the least stable form given its significantly lower melting point and higher solubility (the solubility ratio cRS/cRR is 4.4 and 5.9 at 286 and 323 K, respectively).16 From the solubility ratio, we can estimate that the free energy difference (eq 1) GRS - GRRI at 323 K will be approximately 10 kJ mol-1. The predicted lattice energy difference is expected to be larger than that, as the less dense tetragonal RS structure will be stabilized by larger entropic and zero-point contributions. Hence, the relative stability of the experimentally determined forms at low temperatures is RS , RR III < RR II < RR I. Notwithstanding the uncertainties in the application of the Burger-Ramberger rules54,55 in deriving this stability order, the RR diastereomeric structures are unequivocally more stable than the only known tetragonal RS structure, which is consistent with the excellent resolution efficiency. 3.1.3. (R)-1-Phenylethylammonium, (S&R)-Mandelate (1-4). The third system considered in this work exhibits two RR polymorphs. If we ignore the decomposition during their heating, we can infer that they are monotropically related (RR I < RR II), as the lower melting polymorph has lower enthalpy of fusion. The RR polymorphs are significantly more stable than the only known RS structure, as they have significantly higher melting points and heats of fusion. This is consistent with the significantly higher enthalpy of dissolution of RR I compared with RS (and hence lower enthalpy of formation HRRI - HRS ) -14.9 kJ mol-1 at room temperature; see Table 3). From the solubility ratio16 of cRS/cRRI 4.55 at 318 K, we estimate that GRRI - GRS ) -8 kJ mol-1, which again shows that the free energy difference is smaller than the enthalpy difference. If we assume that the entropy difference of the RR and RS melts is negligible and that the entropy difference between the RS and RR I crystal structures is invariant with temperature, we can estimate from the enthalpies of fusion (ignoring the effect of mass loss due to decomposition, which is of similar magnitude) that SRS - SRRI ) ∆Hf,RRI/Tf,RRI - ∆Hf,RS/Tf,RS ) 26 J K-1 mol-1. This suggests

J. Phys. Chem. B, Vol. 111, No. 19, 2007 5331

Figure 1. (R)-1-Phenylethylammonium-(R)-2-phenylbutyrate form I (R-(1)R-(3) I) shown parallel to the c crystallographic axis. The hydrogen-bonded ladder shown comprises R34 (10) rings sharing one N-H‚‚‚O edge and is present in all experimental and almost all putative structures. The ladders are always parallel to the unique and shortest cell axis in the case of monoclinic and orthorhombic crystal structures, respectively, which varies between 5.9 and 6.8 Å.

that at 318 K the entropy contribution to the Gibbs free energy difference (GRRI - GRS) is +8 kJ mol-1, which is consistent with the solution calorimetry and solubility measurements given the assumptions involved. Hence, the qualitative experimental stability order is RS , RR I < RR II and is consistent with the efficient resolution of the two diastereomers.16 3.2. Comparison of Crystal Structures. In all enantiomorphous forms of 1-phenylethylammonium (1) with 2-phenylpropanoate (2) and 2-phenylbutyrate (3), the hydrogen-bonding motif is similar. Each ammonium group of the cation is hydrogen-bonded to three anion carboxylate groups, with one of the oxygen atoms accepting two hydrogen bonds and the other only one. In all these forms, there are parallel hydrogenbonding ladders that are related by translational, 2-fold, or 4-fold symmetry and comprise R34(10) rings that share one N-H‚‚‚O edge (Figure 1). However, the structures have differences in the relative orientation of the phenyl rings of the hydrogenbonded ions belonging to the same ladder and in the packing of the ladders. Thus, the intermolecular interaction between the π electrons and the aliphatic substituents differs significantly between the structures. There are also marked differences in the conformations of both ions, confirming that the rotation of the phenyl, ammonium, and carboxylate groups is determined by the fine balance of inter- and intramolecular forces in the lattice. Finally, there are pronounced differences in the ethyl rotation (θ5) of the 2-phenylbutyrate anion in the four 1-3 diastereomeric crystal structures. The forms R-(1)R-(3) II and R-(1)R-(3) III have the most subtle structural differences as the anion conformation and the relative orientation of the phenyl groups of the hydrogen-bonded ions with respect to the ladder are similar, with only the packing of the ladders being different. This is consistent with the small size of the endotherm (-1.1 kJ mol-1) for the transformation (Table 2). This contrasts with the significantly larger endotherm associated with the R-(1)R-(3) I f R-(1)R-(3) III transition (-6.0 kJ mol-1), which corresponds to changes in both the anion’s conformation and the relative orientation of the phenyl rings of the ions that belong to the same ladder. Changes in the relative orientation of the hydrogen-bonded ions in the same ladder are also observed27,38 for the endothermic transition R-(1)R-(2) I f R-(1)R-(2) II (-4.7 kJ mol-1). The (R)-1-phenylethylammonium-(R)-mandelate (1-4) polymorphs exhibit the same hydrogen-bonding ladders with identical orientation of the ions with respect to the ladder. However

5332 J. Phys. Chem. B, Vol. 111, No. 19, 2007

Karamertzanis et al.

TABLE 4: Reproduction Accuracy and Predicted Lattice Energies of the Experimentally Determined Diastereomeric Salt Pairs Shown in Chart 1 intramolecular energyb (kJ mol-1) structure

lattice energya (kJ mol-1)

RS RR I RR II

-648.48 -646.79 -639.05

RS RR I RR II RR III

-606.16 -648.97 -643.08 -628.09

RS

-599.55

RR I RR II

-593.87 -584.65

cation

anion

quality of reproductionc RMS error cell lengths (%)

(R)-1-Phenylethylammonium, (S&R)-2-Phenylpropanoate 0.80 3.55 1.70 0.30 3.71 2.94 2.29 5.86 3.50

density (% error)

RMS15 (Å)

1.126 (-1.75%) 1.158 (-1.78%) 1.103 (-1.61%)

0.173 0.211 0.287

1.027 (-3.81%) 1.158 (-4.61%) 1.107 (+0.30%) 1.109 (-0.30%)

0.253 0.307 0.225 0.344

1.215 (-2.76%)

0.186

1.224 (-2.22%) 1.206 (-3.15%)

0.182 0.159

(1-2)d

(R)-1-Phenylethylammonium, (S&R)-2-Phenylbutyrate (1-3)e 0.08 10.09 2.97 0.24 6.56 3.48 0.80 6.34 2.82 1.97 4.94 4.89 (R)-1-Phenylethylammonium, (S&R)-Mandelate (1-4)f 3.35 74.86 1.28 3.52 65.87 0.75 71.38 0.01 12.65 8.37 85.14 1.72 6.38 85.55 1.46

a Lattice energy in kJ mol-1 per ion pair. Repulsion-dispersion and higher multipole interactions were summed up to a 15 Å cutoff in atomic and ion’s centers of mass separation, respectively. b Expressed as the difference between the energy of the constrained optimized ion conformation minus the energy of the global conformational minimum at the MP2/6-31G(d,p) level of theory. c Quality of reproduction assessed via the RMS error in the lattice lengths and RMS discrepancy of a 15 ion cluster.48 The density of the energy minimized crystal and the % error are also given. d Reference 17. e Missing hydrogen atoms for RS (PEAPEA10) were added with SHELX.63 These hydrogen atoms and those in the PXRD determined structures RR II, RR III were ab initio optimized for the isolated ions. f Most anion conformations in the crystal appear to have large intramolecular energy because the hydroxyl hydrogen atom is hydrogen-bonded to carboxylate oxygen atoms of neighboring molecules and hence are significantly less stable than the global conformational minimum that exhibits an intramolecular hydrogen bond. For RS, two anions and two cations have missing hydrogen atoms that were added with SHELX.63 All of these hydrogen atoms were ab initio optimized for the ions in isolation apart from one hydroxyl hydrogen that was constrained to satisfy plausible intermolecular hydrogen-bond geometry.

the additional hydroxyl donor forms another hydrogen bond with the carboxylate oxygen that is acting as a single acceptor in systems 1-2 and 1-3. The hydrogen-bonding pattern is more elaborate in the case of the triclinic, Z′ ) 4 R-(1)S-(4) structure. Although not all hydrogen atoms have been experimentally determined, a visual examination of the packing arrangement suggests that there are bifurcated hydrogen bonds to ammonium hydrogen atoms and a combination of intra- and intermolecular hydrogen bonding between the hydroxyl hydrogen and the carboxylate oxygen atoms. Despite the complexity of the packing arrangement and the low symmetry, the density of R-(1)S-(4) is almost equal to the density of the R-(1)R-(4) polymorphs. 3.3. Computed Relative Stability from the Experimental Crystal Structures. The results of the rigid-body lattice energy minimization of all crystal structures are shown in Table 4. For system 1-2, the predicted lattice energy order is RR II < RR I ∼ RS, and for the 1-3 system, the predicted order is RS , RR III < RR II < RR I. These stability orders are in good agreement with experimental evidence, although the stability differences are severely overestimated. For example, for the polymorph pairs R-(1)R-(2) I-II, R-(1)R-(3) I-III, and R-(1)R-(3) II-III, the static lattice differences are 7.7, 20.9, and 15.0 kJ mol-1 compared with the experimental enthalpy differences of 4.7, 6.0, and 1.1 kJ mol-1, respectively. The latter also include the zero-point and temperature-dependent contributions to the enthalpy, and so, it is consistent that the discrepancies are larger for the differences involving the high-temperature polymorph R-(1)R-(3) III. For system 1-4, the predicted stability order of the RR polymorphs is RR I > RR II, which is in good agreement with experimental evidence. However, the cruder estimate of the lattice energy of the RS diastereomer is in poor agreement with experiment (RR I > RR II . RS). The RS lattice energy is very sensitive to the positioning of the two hydroxyl hydrogen atoms that have not been experimentally determined.56 The lattice

energy in Table 4 corresponds to a plausible structure where the missing hydroxyl hydrogen atom of one of the anions was assumed to form an intramolecular hydrogen bond, as there are no available acceptors in neighboring ions. The low RS lattice energy is mainly because the intramolecular energy of the mandelate ion that was assumed to exhibit intramolecular hydrogen bonding is approximately 50 kJ mol-1 lower than the energy of the mandelate ions involved in only intermolecular hydrogen bonds in the RS and both RR polymorphs. It is encouraging that the overall geometric reproduction of the crystal structure is satisfactory with 1.28% and 0.19 Å rootmean-square error in the lattice lengths and 15 ion coordination sphere, respectively, which suggests that the assumed positions of the missing hydrogen atoms are reasonably correct. However, the accurate prediction of the experimental stability order is much more demanding of the balance of the inter- and intramolecular energy model. The conformational stabilization due to the intramolecular hydrogen bond is computed at the electronic structure level and hence accurately includes both electrostatic and induction contributions to the hydrogen bond, but the induction energy is not explicitly accounted for in the intermolecular lattice energy. Although the empirical fitting of the repulsion-dispersion model potential to neutral hydrogenbonded molecules will have absorbed some intermolecular induction effects, it will not be appropriate for the stronger electrostatic fields in salts. This problem in computing energy differences57 between crystal structures, where an intermolecular hydrogen bond is replaced by an intramolecular bond, probably explains the artificial stabilization of the RS structure compared with the RR polymorphs for the 1-4 system. Fortunately, the hydrogen-bonding motif for all other experimental structures is intermolecular and similar, and here the error in neglecting induction is expected to approximately cancel. 3.4. Crystal Structure Prediction for (R)-1-Phenylethylammonium, (S&R)-2-Phenylbutyrate (1-3). The genuine prediction of crystal structures cannot assume the knowledge

1-Phenylethylammonium-2-phenylacetate Derivatives

J. Phys. Chem. B, Vol. 111, No. 19, 2007 5333

TABLE 5: Predicted Crystal Structures for the Diastereomeric Salt Pair (R)-1-Phenylethylamine, (S&R)-2-Phenylbutyrate (1-3) cell geometrya

ion conformations anion

rank, space group

U + ∑ ∆Eb (kJ mol-1)

∑ ∆E (kJ mol-1)

RR I, P212121 experimental RR II, P212121 experimental RR III, P21 experimental

-658.96

5.46

-654.94

7.98

-641.81

9.46

1, P212121 2, P212121 3, P212121 4, P212121 5c, P21212 6, P21 7, P212121 8, P21 9, P212121 10, P212121

-653.46 -651.93 -648.4 -647.10 -646.28 -646.01 -645.47 -645.32 -644.65 -643.73

10.03 8.56 8.70 12.12 9.41 10.09 7.14 10.64 11.58 11.37

57.61 76.59 56.86 49.68 67.20 66.93 57.30 65.99 57.91 29.63

RS, P41 experimental

-619.01

1, P212121 2, P212121 3, P212121 4, P21 5, P212121 6, P212121 7, P212121 8, P212121 9, P212121 10, P212121

-654.96 -654.30 -649.05 -648.97 -647.47 -647.20 -647.18 -646.78 -645.67 -644.27

cation V (Å3)

conventional cell a (Å), b (Å), c (Å), β (°)

407.37 390.32 427.45 429.44 426.20 425.67

6.058, 15.831, 16.990, 5.757, 15.433, 17.571, 6.223, 16.428, 16.698, 6.062, 16.779, 16.888, 11.921, 6.253, 12.355, 112.24 11.882, 5.976, 13.077, 113.51

-157.04 64.94 -72.20 -62.61 -66.61 72.15 -67.26 73.18 -56.83 -168.72

414.62 420.42 419.42 443.64 395.50 430.22 425.90 425.33 413.15 430.66

6.585, 12.891, 19.538, 6.435, 15.927, 16.409, 6.100,13.968, 19.691, 6.332, 13.472, 20.836, 5.682, 15.917, 17.493, 8.666, 6.554, 15.357, 99.42 6.277, 15.950, 17.017, 8.690, 6.669, 14.724, 94.54 5.945, 16.127, 17.237, 5.982, 16.817, 17.125, -

10.35

RS-diastereomer, experimentally determined 70.79 -63.32 -76.72 99.51 61.20 69.96 -68.15 -76.20 93.88 66.87

456.48 443.68

16.439, -, 6.757, 16.642, -, 6.408, -

12.36 8.72 1.29 0.47 11.30 13.89 8.26 12.58 7.14 14.39

59.29 78.63 72.54 73.30 47.30 48.40 84.65 38.34 66.12 51.81

405.15 410.09 449.20 425.86 435.31 406.21 415.40 429.47 420.45 424.67

5.990, 15.778, 17.148, 5.860, 14.628, 19.135, 6.371, 16.693, 16.895, 11.674, 6.603, 11.729, 109.61 6.370, 13.333, 20.502, 6.701, 15.183, 15.971, 5.904, 15.697, 17.931, 6.081, 11.586, 24.385, 6.813, 122.748, 19.364, 6.330, 16.118. 16.649, -

θ1 (°)

θ2 (°)

θ3 (°)

θ4 (°)

θ5 (°)

RR-diastereomer, experimentally determined 74.38 -56.46 62.95 -64.28 -65.74 68.60 -56.76 68.15 -70.55 -64.11 62.45 -78.70 75.25 -89.04 -172.65 63.94 -56.76 79.18 -88.22 -174.67 58.61 -80.37 76.16 -86.28 -167.51 53.41 -56.76 75.37 -84.69 -170.47 RR-diastereomer, putative -89.32 51.21 -61.41 -58.16 40.59 -32.80 -72.15 43.48 -61.41 -91.84 46.28 -71.26 -42.62 67.53 -76.05 -66.19 30.80 -21.54 -70.54 44.13 -66.48 -67.78 29.09 -20.82 -88.88 65.53 -64.91 -64.38 44.15 -72.03

RS-diastereomer, putative -76.23 -20.00 61.29 -51.37 -60.84 37.86 -64.47 -44.39 90.65 -59.34 -49.39 76.78 -75.97 -67.11 74.20 -84.54 -54.34 115.89 -82.84 -59.97 54.87 -68.51 -48.17 107.68 -68.66 -98.16 98.67 -92.50 -50.21 95.77

-56.80 68.09 168.91 170.86 65.02 161.95 63.28 165.18 167.54 69.92

a Cell volume per ion pair and cell lengths and angles not constrained by space group symmetry. b Lattice energy (in kJ mol-1 of ion pair) expressed as the sum of inter- and intramolecular energies. To avoid numerical instabilities, repulsion-dispersion and higher multipole interactions were summed up to an increased 60 Å cutoff in atomic and ion’s centers of mass separation, respectively. The distributed multipole models were constructed with a modified multipole analysis.52 c The only structure that does not exhibit the R43(10) hydrogen-bond motif illustrated in Figure 1 but comprises R42(8) and R65(16) rings.

of the experimental torsion angles, as done in the evaluation of lattice energies in Table 4. Hence, in order to evaluate the structure prediction methodology, all experimentally determined R-(1)S-(3) and R-(1)R-(3) crystal structures were also lattice energy minimized by allowing the distortion of the flexible torsion angles in Chart 1 due to the packing forces with the DMAflex algorithm. The reproduction accuracy (Table 5) is excellent with the flexible torsion angles being optimized to within a few degrees of their experimental values. The apparent exceptions are the rotations of the ammonium group θ2 for the two structures determined from powder X-ray diffraction data R-(1)R-(3) II and R-(1)R-(3) III, although this is probably due to the limited accuracy with which H-atom positions are determined in these experimental structures. The ammonium proton positions were fixed during the global optimization procedure by setting θ2 equal to the value found in the singlecrystal X-ray determined polymorph R-(1)R-(3) I. The subsequent Rietveld refinements on each structure did allow H-atoms to vary; however, their shifts were subject to slack constraints to restrict their movement within chemically sensible ranges. The weak scattering contribution from H-atoms means that the final Rwp value is relatively insensitive to H-atom position; hence, equally good fits to the data are likely for a range of θ2 values. The ammonium rotation changes significantly during the flexible-ion refinement leading to almost linear N-H‚‚‚O hydrogen bonds in the direction of the oxygen lone pairs, which

is physically reasonable. Although the lattice energies are lowered by over 10 kJ mol-1 (cf. Tables 4 and 5) due to the relaxation of the ion geometries (plus the increased cutoff and the use of a modified multipole analysis), the relative stability order is retained and remains in good agreement with experiment. The lattice energies for the 100 most stable crystal structures found in the search for 1-3 (50 for each diastereomer) are plotted against their energies after refinement in Figure 2, and the 20 most stable putative structures for each diastereomer obtained after the refinement are given in Table 5. It is encouraging that the preliminary search located all known crystal structures and that the refinement consistently improves the rank of the experimentally determined structures. For example, the search for the RR diastereomer identified polymorphs RR I and RR II as the 2nd and 26th most stable packing arrangements, and their rank improved to 1st and 2nd, respectively, after refinement. However, the structures RR III and RS were not within the top 50 structures generated in the search and even after the refinement are high in energy (17.15 and 35.95 kJ mol-1 above the corresponding global energy minima; see Table 5). The reranking observed during refinement is mainly attributed to the use of an accurate electrostatic model based on conformationally dependent multipole moments for the dominant electrostatic interactions (compared with point charges in the search) and, second, on the use of a different repulsion-dispersion potential

5334 J. Phys. Chem. B, Vol. 111, No. 19, 2007

Figure 2. Re-ranking due to refinement of the 50 most stable search crystal structures for each 1-3 diastereomer. Open and full symbols denote hypothetical and experimental crystal structures, respectively. Some of the search crystal structures that led to the same minimum after refinement are indicated with the horizontal broken lines. Note that the RR III and RS salts were found in the search but are not shown due to their high energies of -592.96 and -577.33 kJ mol-1, respectively. Their refined lattice energies remained high at -641.81 and -619.01 kJ mol-1, respectively.

(FIT42 versus Williams51). A more subtle difference is that in the preliminary search the rigid degrees of freedom were frozen to their gas-phase values,19 and in the refinement, they are indirectly allowed to adapt as at each optimization step the ion conformations are ab initio optimized with the flexible torsion angles (θi, i ) 1, ... , 5 in Chart 1) being constrained.18 There is sufficient correlation between the lattice energy before and after refinement (Figure 2) that the refinement of additional search structures is not likely to produce new global minima. Some search crystal structures lead to the same minimum after refinement, showing that separate minima on the lattice energy surface coalesce into the same basin with improvement in the realism of the intermolecular potential. The hydrogen-bond motif in 19 out of the 20 putative structures shown in Table 5 is identical to the one encountered in all experimentally determined structures for systems 1-2 and 1-3 (see Figure 1). As in the experimental crystal structures, the ladders are parallel to the unique and shortest cell axis in the case of monoclinic and orthorhombic structures, respectively. The length of the ladders varies from 5.9 to 6.8 Å before they are translationally repeated (Figure 1). The predictions show a variety of possible relative orientations of the hydrogen-bonded ions with respect to the ladder as well as a diverse set of possibilities for the packing of the ladders for both diastereomers. Despite the large number of energetically possible packing arrangements, the predictions are clearly successful in identifying the dominant hydrogen-bond motif and in providing insights for the propensity of polymorphism, as the lack of strong directional forces between adjacent ladders may facilitate structural rearrangements during heating and crystal growth. The search produced several putative RR structures that are more stable than RR III and many RS structures that are significantly more stable than the only experimentally observed RS structure. The search for the RR diastereomer was successful in locating the thermodynamically stable form as the global minimum, but the known RS structure is particularly unstable compared with several of the putative structures in the P212121 and P21 space groups, although there are no energetically competitive structures with the tetragonal symmetry of the experimental form. The known RS structure is the least dense among all experimental and 10 most stable putative structures for both diastereomers, which suggests that its thermodynamic

Karamertzanis et al. stability has been underestimated by neglecting the entropic contributions. The flexible torsion angles in the putative structures vary considerably, although the total intramolecular energy for the two ions does not exceed 15 kJ mol-1 in any of the experimental or hypothetical crystal structures. This further emphasizes the need to explicitly model the deformation of the ion conformations under the packing forces. Although the anion conformations in the experimental structures are close to the two lowest energy conformational minima19 (C-C-C-C(Ph) ) θ5 ≈ -60 or 180° for the R enantiomer), there are several putative crystal structures with the ethyl group rotation in a third configuration (C-C-C-C(Ph) ) θ5 ≈ +60° for the R enantiomer). For these structures, the increased intramolecular energy is compensated by more favorable intermolecular interactions. However, a search of the CSD for 2-phenylbutyrate fragments found none with the C-C-C-C(Ph) ≈ +60° configuration, which may imply that there is a kinetic barrier to transformation to this conformer during crystallization. 4. Discussion This interdisciplinary study has shown that, despite the similarities in the hydrogen-bonding motifs in the crystal structures of three chemically related diastereomeric salt pairs, the thermodynamic stability (and hence separability) and tendency to polymorphism varies considerably. The experimental structural and thermodynamic data provide a stringent test for developing a computational methodology for the prediction of the energy difference of a diastereomeric salt pair with or without the knowledge of their crystal structure. The crystal structures and qualitative relative stability of the chemically related diastereomer pairs have been reproduced well by lattice energy calculations using a distributed multipole model for the dominant electrostatic interactions and ab initio estimates of the ion conformational energies. The only exception is the overestimation of the thermodynamic stability of the R-(1)-S(4) crystal structure compared with the R-(1)-R(4) polymorphs, due to the different treatment of the intra- and intermolecular hydrogen-bonding energies by the computational model. Hence, the intermolecular potential should be extended to account for the effect of polarization of the ion’s charge density in the crystalline environment, in order to correctly rank structures where the type of hydrogen-bonding motif can vary. Ignoring this exception, the computational model appears able to even predict the subtle stability ordering of the polymorphs of the same diastereomer. Thus, the effect of the uncertainties in the intermolecular potential and ion conformational energies generally appears to be small compared with the stability difference of efficiently resolving diastereomeric salt pairs. Despite the considerable progress in obtaining qualitative agreement with experimental relative stabilities, further methodological advances are necessary to quantitatively predict the solubility ratio of diastereomeric salt pairs. The solubility ratio depends exponentially on the diastereomers’ Gibbs free energy difference, which include the relative zero-point and thermal contributions that have been ignored in this study. The presence of strong hydrogen bonds in the ionic systems lowers the enthalpy but at the same time imposes a relatively precise geometry, which may disfavor compactness and lead to entropic stabilization.7 It is the interplay of entropic vs enthalpic stabilization (e.g., for system 1-4, GRRI - GRS ) -8.0 kJ mol-1, and HRRI - HRS ) -15.0 kJ mol-1 at approximately room temperature) that determines the solubility ratio of the diastereomers as a function of temperature and explains the overes-

1-Phenylethylammonium-2-phenylacetate Derivatives timation of their relative stability when we only consider their static lattice energies. For example, the Gibbs free energy difference between the tetragonal R-(1)S-(3) and the more densely packed R-(1)R-(3) I diastereomers (GRS - GRRI ) 10 kJ mol-1 at 323 K) is significantly smaller than the predicted lattice energy difference (39.95 kJ mol-1). The quantitative prediction of the relative Gibbs free energy as a function of temperature will require the consideration of the vibrational contributions, while maintaining the accuracy in the modeling of the intra- and intermolecular forces, and will be the subject of a future study. The theoretical identification of the optimal resolving agent for a given racemic mixture should not rely on any experimental information and hence should also involve the ab initio prediction of the crystal structures of the diastereomeric salt pair.15 Our results for the diastereomeric salt pair 1-3 show that the development of a search strategy that strikes a balance between computational efficiency and model realism requires more research. The search located all known R-(1)S-(3) and R-(1)R-(3) crystal structures. However, only forms RR I and RR II were sufficiently low in energy to be within the large set of structures considered for refinement; forms RS and RR III were only modeled further because they had been experimentally observed. The interest in developing an extensive yet accurate method to explore complex lattice energy surfaces is likely to grow in the future as crystal structure prediction expands into multicomponent systems.15,17 The predictions suggest that the known R-(1)S-(3) form may be a kinetically trapped, metastable structure as there are hypothetical structures that are up to 36 kJ mol-1 more stable and 10% denser. It is enticing to postulate that the known structure was the first to crystallize and is metastable according to the Ostwald’s rule of stages.58 Thus, our calculations warn that a more stable R-(1)S-(3) polymorph could crystallize during a thorough polymorph screening59 that was beyond the scope of the present investigation. The growth of a thermodynamically more stable RS form may limit the lattice energy difference of the most stable RR and RS diastereomers to a few kiloJoules per mole. This illustrates the importance of developing a methodology that will reliably predict the relative stability of diastereomers, as the discovery of an unexpected, thermodynamically more stable polymorph60,61 may considerably change the separability behavior in an industrial process. In contrast, the thermodynamically stable forms of both 1-2 diastereomers correspond to the global minima in the corresponding searches with the same model.17,18 Hence, although the discovery of additional 1-2 polymorphs cannot be ruled out, they are likely to be metastable. Although the growth of metastable polymorphs will probably have a smaller impact on the resolution efficiency than thermodynamically stable ones, the conditions of the resolution process may affect the polymorphic outcome and, hence, indirectly determine the solubility ratio and separability of the two diastereomers. This is corroborated by the polymorphic outcome of the separation experiments16 for (R/S) mandelic acid and (R)-1-phenylethylamine (1-4) being dependent on the starting ratio of the two acid enantiomers. Theoretical calculations predict that the lattice energy of the low melting polymorph R-(1)R-(4) II is approximately 10 kJ mol-1 lower than R-(1)R-(4) I. Their predicted stability difference appears sufficiently large for the polymorphic outcome to have an important effect on the observed separability behavior,16 which warrants a more detailed investigation.

J. Phys. Chem. B, Vol. 111, No. 19, 2007 5335 The prediction of the resolution efficiency from lattice calculations is based on the assumption that a successful resolution is mainly dependent on the solubility ratio of the diastereomeric salt pair. This provides a reasonable basis for the theoretical screening of resolving agents to eliminate unsuitable candidates by performing lattice calculations on the diastereomers’ crystal structures (see eq 1). However, one has to recognize that the validity of this approach requires (inter alia) that the free energy difference of the two diastereomers in solution vanishes, despite their solubility difference. Moreover, the resolution efficiency depends on the complete diastereomeric salt pair-solvent ternary phase diagram and not only the position where the solubility curve crosses the solventdiastereomer axes. Our separability measurements for system 1-2 showed16 that the direction of enrichment changes when the starting diastereomer ratio passes through a eutonic point composition, although the resolution efficiency will be less affected when the ternary phase diagram is very disymmetric. Finally, the crystallization outcome may not always be thermodynamically controlled and may also involve the growth of unexpected forms, such as solvates, double salts, or solid solutions, which will also alter the resolution efficiency.62 Despite the challenges in predicting all factors that influence the separation behavior, the development of computational methods that will reliably compute the Gibbs free energy difference of diastereomeric salt pairs can provide a valuable guide to experimental investigations, by short listing resolving agents whose performance is likely to be satisfactory. 5. Conclusions This study demonstrated that the stability difference of the diastereomeric salt pairs of three 1-phenylethylammonium-2phenylacetate derivatives is related to their resolution efficiency and varies considerably, despite the similarity in their molecular structure and hydrogen-bonding motifs. The computed lattice energy differences were found in qualitative agreement with experimental evidence (with the exception of R-(1)S-(4) that could have been foreseen from the intramolecular hydrogen bond in the crystal structure). This suggests that the accuracy of the intermolecular potential and conformational energies are approaching the level required to predict the diastereomers’ relative stability, although further methodological developments will be required to include the effect of zero-point and thermal contributions to the free energy. The experimental measurements presented in this study provide a test bed for the necessary developments in computational modeling. The advances in the prediction of the crystal structure and relative stability of diastereomeric salt pairs demonstrated in this work renders the theoretical screening of resolving agents a foreseeable possibility of significant industrial importance.7 Computational models have the potential not only to guide experimental investigations, by focusing on promising candidates that lead to a large stability difference of the resulting diastereomers, but may also provide insights on the inherent danger of polymorphism and its implications on the resolution efficiency. Acknowledgment. This work was part of the CPOSS project (http://www.cposs.org.uk) funded by the Basic Technology Programme of the Research Councils U.K., and used UCL research computing infrastructure. Dr. Ashley Hulme is thanked for assistance with the experimental work. Supporting Information Available: DSC measurements for systems 1-3 and 1-4, discussion of the structural relationship

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