2887
J. Phys. Chem. 1092, 86, 2867-2873
Pressure Dependence of Mlxlng Enantiomeric Liquids: 1,P-Dichloropropane‘ Wllllam L. Jorgensen*Zand Bernard Blgot3 Department of Chemistry, Purdue UnhwsiYy, West Lafayeite, Indiana 47907 (Received: March 18, 1982)
Monte Carlo statistical mechanics simulations have been carried out for a pure enantiomer and the racemate of liquid 1,2-dichloropropane (DCP) at 25 “C and 1and 5ooo atm. The intermolecularinteractions were described by Coulomb and Lennard-Jones terms in the TIPS format. The internal rotation about the Cl-C2 bond in the monomers w a included by using a rotational potential function derived from experimental data on related systems. The effects of pressure on the structure and thermodynamics of the liquids are analyzed with the aid of various distribution functions. No statistically signifcant shifts in the conformer populations are apparent in going from 1to 5000 atm. However, the intermolecular interactions are more favorable at the higher pressure and the radial distribution functions indicate greater structure. Furthermore, computed heats and volumes of mixing, conformationalresulta, and neighbor analyses were all considered in searching for chiral discrimination in the liquids. No definitive evidence could be found for such an effect at either pressure. The necessity and methodology for umbrella sampling involving the intramolecular rotational potential in the simulations are also discussed.
Introduction Racemic liquids usually crystallize as a racemic compound in which the unit cells contain equal numbers of opposite enantiomers or as a conglomerate which is a mixture of separate crystals consisting of the two pure enantiomers. The occurrence of the latter permitted Pasteur’s mechanical resolution of (&)-sodiumammonium tartrate in 1848.4~~These alternatives demonstrate a preference for heterochiral (d-l) or homochiral (d-d) interactions in the solids, though conglomerate formation is energetically disfavored by an intrinsic entropic factor of RT In 2. Indeed, heats of mixing for the formation of racemic compounds from the solid enantiomers have been determined and range from +0.3 to -2.3 kcal/mol.6 Furthermore, the densities of solid pure enantiomers and racemic compounds are known to vary by as much as 8% .5 Chiral discrimination has also beep investigated in the liquid state. For example, differences in heats of mixing for some enantiomeric dicarboxylic acids in CHC1, are -30 to -50 cal/mol.s More pronounced effects (200 to 350 cal/mol) have recently been found by Arnett and Zingg for the formation of diasteriomeric ion pairs between chiral amines and mandelic acid in several solvents? In contrast, the mixing of enantiomeric liquids to form a racemate is generally considered to be essentially ideal under standard condition^.^ This is supported by the few available heats of mixing for enantiomeric liquids (2-octanol, 2-b-nitrophenyl)butane, fenchone, and a-methylbenzylamine)which are 0-3 cal/m01.~-~ Our theoretical studies of organic liquids have evolved to the point where it seemed that we might be able to meaningfully treat as intricate a problem as the mixing of (1)Quantum and Statistical Mechanical Studies of Liquids. 22. (2) Alfred P. Sloan Foundation Fellow, 1979-81; Camille and Henry Dreyfus Foundation Teacher-Scholar, 1978-83. (3)On leave from the Universite Pierre et Marie Curie and Ecole N o d e Sup6rieure de Saint Cloud, France; CNRS-NSF Fellow, 19%-1. (4)Pasteur, L.C. R. Acad. Sci. 1848,26,535.Ann. Chin. Phys. 1860,
28,M. (5) For a recent, thorough review on the properties and separation of chiral compounds,see: Jacques, J.; Collet, A.; Wilen, S. H. ‘Enantiomers, Racemates and Resolutions”; Wiley-Interscience: New York, 1981. (6)Leclercq, M.; Collet, A.; Jacques, J. Tetrahedron 1976,32, 821. (7)(a) Guette, J. P.; Boucherot, D.; Horeau, A. Tetrahedron Lett. 1973,465; (b) Atik, Z.; Ewing, M. B.; McGlashan, M. L. J. Phys. Chem. 1981,86,3300. J. Chem. Thermodyn. Submitted for publication. (8)Horeau, A.; Guette, J. P. Tetrahedron, 1974,30, 1923. (9)Arnett, E. M.; Zingg, S. P. J . Am. Chem. SOC.1981,103, 1221. 0022-3654/82/2086-2867$01.25/0
T A B L E I:
TIPS Parameters for D C P site
CH, primary C1 secondary C1
10-4~2
795 7 29 680 525 525
cz
4
2400 1825 1150 2600 2600
0.0 0.25 0.265 -0.25 -0.265
Units are for A Z((kcal AI2)/mol), Cz ((kcal A 6 ) / m o l ) , and q (electrons). e z in eq 1 is 332.17752 (kcal A ) / m o l .
enantiomeric liquids. The present paper describes the outcome of our first effort along these lines. A key computational issue is the statistical accuracy obtainable for the heats and volumes of mixing. Such work requires intermolecular potential functions which yield good thermodynamic and structural results for organic liquids, the ability to perform the simulations in the isothermal, isobaric (NPT) ensemble, and procedures for efficiently sampling over internal rotational degrees of freedom which may involve substantial barriers between conformers. These topics have all been addressed in a series of papers from our laboratory which have included the development of transferable intermolecular potential functions (TIPS) for water, alkanes, alcohols, ethers, and alkyl chlorides and their use in Monte Carlo statistical mechanics simulations of the corresponding Condensed-phaseeffects on conformational equilibria have been studied for liquid methano1,’l ethano1,l’ n-butane,12J4 1,2-dichloroethane (DCE),12methyl ethyl ether,13diethyl ether,’, and tetrahydrofuran.16 In addition, efficient sampling procedures for internal rotational degrees of freedom have been devised by using “umbrella” sampling with “chopped” barr i e r ~ Many . ~ ~ of the simulations have also been run in the NPT ensemble. This has demonstrated that the TIPS yield liquid densities within 6% of experiment including pressures up to 15000 atm for liquid n-butane and methano1.14J5 (10)Jorgensen, W.L. J. Am. Chem. SOC.1981,103,335. (11)Jorgensen, W.L. J . Am. Chem. SOC.1981,103,341,345. (12)Jorgensen, W. L. J . Am. Chem. SOC.1981,103,677.Jorgensen, W. L.; Binning, R. C.; Bigot, B. Zbid. 1981,103,4393. (13)Jorgensen, W. L.; Ibrahim, M. J. Am. Chem. SOC.1981,103,3976. (14)Jorgensen, W.L.J. Am. Chem. SOC.1981,103,4721. (15)Jorgensen, W. L.;Ibrahim, M. J.Am. Chem. SOC.1982,104,373. (16)Chandrasekhar, J.; Jorgensen, W. L. To be submitted. (17)Bigot, B.; Jorgensen, W. L. J. Chem. Phys. 1981,75, 1944.
0 1982 American Chemical Society
2888
The Journal of Physical Chemistty, Vol. 86, No. 15, 1982
Jorgensen and Bigot
Cl RClTflTIONflL ENERGY
FUNCTION
J R ?,?-dichloropropane
- 1
H
* ;aLw4e
0
-I
60
120
180
240
300
360
PHI
60
-.d's
'80 )
;a,:he
331
Figure 1. Conformers of (R)-1,2dichloropropane (DCP).
As reported here, 1,2-dichloropropane (DCP) has been chosen for the first simulations of a chiral liquid and of the mixing of enantiomers to form a racemic liquid. It also seemed promising to study the pressure dependence of the mixing, since higher density as in a solid should exentuate any preference for segregation in the liquid racemate. Consequently, simulations at both 1 atm and 5000 atm were carried out. So, besides the search for chiral discrimination, further insights into the effects of pressure on the structure, properties, and conformational equilibria for organic liquids have been obtained.
Statistical Mechanics Calculations Intermolecular Potential Functions. A principal element in the Monte Carlo simulations is the intermolecular potential functions describing the interactions between monomers in the fluids. Since the TIPS were found to yield good thermodynamic and conformational results for liquid DCE,12they have been used for the simulations of DCP as well. In this case there are five interaction sites for each monomer centered on the carbon and chlorine atoms with the alkyl hydrogens implicit.'O The sites interact intermolecularly via Coulomb and Lennard-Jones terms (eq 1). The A and C parameters for alkyl groups
have remained unchanged since the original report as summarized in Table I.O ' The C value for chlorine is also unchanged;l2 however, the A value has been reduced by about 7% to fine tune the computed densities for liquid alkyl chlorides. The charge for primary chlorine is -0.25e as beforel2and the charge for secondary chlorine is -0.265e. The latter value was derived from the ratio of experimental dipole moments for isopropyl chloride (2.17 D)18and ethyl chloride (2.05 D)lStimes the charge for primary chlorine (-0.25e). Then, for neutrality to be maintained, the charges on the CH2 and CH groups in DCP are taken as +0.25e and +0.265e. As usual, standard geometries were employed for the monomers that are consistent with diffraction data for 1,2-dichloroalkanes:19r(CC) = 1.530 A, r(CC1) = 1.785 A, LCCC = LCCCl = 109.47O. Intramolecular Potential Function. Although the bond lengths and angles in the monomers are not varied in the liquid simulations, the principal torsional degree of freedom about the Cl-C2 bond is included. This permits the interconversion of the three conformers illustrated in (18) Nelson, R. D.; Lide, D. R.; Maryott, A. Natl. Bur. Stand. ( U S . ) , Circ. 1967, No. IO. (19) Kveseth, A. Acta Chem. Scand., Ser. A 1975, 29, 307.
Figure 2. Intramolecular rotational potential function for R-DCP from eq 2.
Figure 1 for the R-(+) enantiomer.20 The gauche* (g*) conformer has a dihedral angle of 60" between the chlorines with the chlorine on C1 and the methyl group anti. The chlorines are anti in the trans (t) conformer, while the chlorine on C1 is between the other chlorine and the methyl group in the gauche (8) conformer. There have been a few experimental and theoretical studies dealing with the rotational potential surface. It should be noted that the R and S enantiomers have rotational potentials that are mirror images of each other, e.g., for an S molecule the 60' conformer is the gauche looking down the Cl-C2 bond as in Figure 1. After some earlier experimental work with invalid assumptions,21an electron diffraction study gave a gauche-trans energy difference of 1.1-1.3 kcal/mol without distinguishing the gauche and gauche* conformers and a trans-gauche barrier height around 5 kcal/mo1.2z Other gas-phase studies with IR, Raman,23and NMRZ4spectroscopy specifically considered the gauche and gauche* conformations and yielded gauche-trans and gauche*-trans energy differences of 1.9 and 1.2 kcal/mol, respectively. The theoretical studies have been based on molecular mechanics calculation^.^"^^ The results for the g-t and g*-t energy differences are 1.2-1.7 and 1.0-1.3 kcal/mol; however, no barrier heights were reported. For 1,2-dichloroethane derivatives, the most precise comparison between theory and experiment can be made for DCE itself. Since Allinger's force field gives the best results in this case,25we used his MM2 programz8 to calculate the entire rotational potential surface for DCP. The g-t and g*-t energy differences are 1.16 and 1.12 kcal/mol as found previouslyz5and the t-g, t-g*, and g-g* barrier heights are computed to be 7.27,4.57, and 6.03 kcal/mol, respectively. By comparison with MM2 calculations for 2-chlorobutane,29the t-g and g-g* barriers appear too high since they imply, for example, that a chlorine eclipsing a hydrogen or methyl is 1-2 kcal/mol less stable than a methyl eclipsing the same groups. This is at variance with what is expected for these interactions by considering experi(20) Fickett, W.; Garner, H. K.; Lucas, H. J. J . Am. Chem. SOC.1951, 73, 5063. (21) Oriani, R. A.; Smyth, C. P. J. Chem. Phys. 1949, 17, 1174. Morino, Y.; Miyagawa, I.; Haga, T. Ibid. 1951, 19, 791. (22) Wood,W. W.; Fickett, W.; Kirkwood, J. G. J. Chem. Phys. 1952, 20, 561. (23) Mizushima, S. "Structures of Molecules"; Academic Press: New York, 1954. Pure Appl. Chem. 1963, 7 , 1. (24) Dempster, A. B. J.Mol. Struct. 1974,23, 193. (25) Meyer, A. Y.;Allinger, N. L. Tetrahedran 1975, 31, 1971. (26) Meyer, A. Y. J. Mol. Struct. 1977, 40, 127. (27) Abraham, R. J.; Parry,K. J. Chem. SOC.B 1970, 539. (28) Allinger, N. L.; Yuh, Y. H. QCPE 1981, 13, 395. (29) Allinger, N. L., personal communication. For 2-chlorobutane, the t-g, t-g*, and g-g* barriers are computed to be 4.7, 3.2, and 5.0 kcal/mol.
The Journal of phvsical Chemistry, Vol. 86, No. 15, 1982 2889
Pressure Dependence of Mixing Enantiomeric Liquid
mental data on related conformations of reference systems such as propane, chloroethane, n-butane, and l-chloropropane.30 From analyses of these component interactions, it is possible to make the following predictions for DCP: hE(g*-t) = 1.1, hE(g-t) = 1.2, and the t-g, t-g*, and g-g* barriers should be ca. 5.3,4.9, and 4.2 kcal/mol. These values are consistent with the electron diffraction results22and the molecular mechanics predictions for the relative energies of the three conformers. To obtain a continuous potential function, these data were fit to a Fourier series with a least-squares program which yielded eq 2. The upper and lower signs on the sin terms are for V(4) = 2.997 + 0.413 COS 4 - 0.343 COS 24 + 2.237 cos 34 =F 0.184 sin 4 f 0.160 sin 24 (2) the R and S enantiomers, respectively. The function is illustrated in Figure 2 and yields the following data: bE(g*-t) = 1.12, hE(g-t) = 1.15, and the t-g, t-g*, and g-g* barriers are 5.52, 4.93, and 4.15 kcal/mol. Monte Carlo Simulations and Umbrella Sampling. Monte Carlo simulations were carried out for the pure R and racemic DCP at 25 "C and with applied pressures 1 and 5000 atm. Cubic samples of 216 monomers were employed in each case with periodic boundary conditions and Metropolis sampling. Complete descriptions of the formalism of Monte Carlo simultations in the NPT ensemble can be found in earlier work.13J4J1 New configurations were generated by randomly selecting a monomer, translating it in all three Cartesian directions, rotating it about one randomly selected Cartesian axis, and performing the internal rotation. Volume moves were attempted on each 1000th configuration and involved scaling all the intermolecular separations. Acceptance rates of ca. 40% for new configurations were maintained by using ranges of f0.12 A, f12", f20°, and f700 A3 at 1 atm for the translations, total rotations, internal rotations, and volume moves and fO.10 A, f8O, f15", and f350 A3 at 5000 atm. Spherical cutoffs at 12 8,were used in evaluating the intermolecular potential functions which include interactions with a molecule's ca. 50 nearest neighbors. An important and novel point in the present simulations concerns the sampling for the internal rotation. For high rotational barriers, complete sampling of the configuration space may be impeded to the point where conformational equilibrium is not attained. We have found barriers above ca. 3.5 kcal/mol to be problematic in this regard for liquid simulations under normal PT conditions and of normal length. Thus, the ca. 5 kcal/mol rotational barriers for DCP would seriously inhibit convergence of the calculations. Fortunately, the problem is efficiently overcome with an "umbrella sampling" p r o c e d ~ r e . ' ~ ~ ~ ~ Briefly, the average of a property e can be represented by eq 3 where ( )w indicates an average obtained from
1(e/w)w exp(-PH) d X dV -
( e ) = l l ( l / w ) w exp(-PH) d X dV
(em),/
( l / w ) w (3)
sampling over the non-Boltzmann distribution, w(X) exp(-pH), and the integrations are over all geometric configurations (X)and volumes (V) for the system. The enthalpy (H) is given by the sum over all intermolecular interactions and internal rotational energies (V,) for the (30)Lowe, J. P. h o g . Phys. Org. Chem. 1968,6,1. (31)Owicki, J. C.;Scheraga, H. A. J. Am. Chem. SOC.1977,99,7403. (32)(a) Valleau, J. P.; Torrie, G. In 'Statistical Mechanics", Part A, Berne, B. J., Ed.; Plenum Press: New York, 1977, p 169 (b) Rebertus, D.W.; Berne, B. J.; Chandler, D. J. Chem. Phys. 1979,70,3395.
monomers plus a P V term (eq 4). This permits the actual N
H=
a , .
GAUCHE
4
Figure 4. Total intermolecular bonding energy distributions for monomers in liquid DCP. Units for the ordinate are mole fraction per kcai/moi.
are shown in Figure 4. Increasing the pressure to 5000 atm makes the intermolecular interactions significantly more favorable and lowers the energy (Table 11). As discussed previo~sly,'~ Bridgman found that the internal energy for many organic liquids reaches a minimum near 4000-9000 atm at 25 0C.34 Compression beyond 1 atm yields more interactions near the bottoms of the potential energy wells until a limit is reached when the short-range repulsive regions are intruded. The separate distributions in Figure 4 show that the gauche and gauche* monomers are more bound than the trans. Of course, if this were not the case, then the shift to lower trans population in the liquid vs. the gas would not occur (Figure 3). The difference is substantial with the gauche and gauche* monomers bound by about 2 kcal/mol more than the trans. Presumably, this is due to the more favorable electrostatic interactions possible for the gauche and gauche* monomers with their neighbors. For example, the positively charged CH2 and CH groups in a gauche or gauche* monomer can both have unobstructed interactions with a chlorine in a neighbor, while Coulombically repulsive chlorine-chlorine intermolecular interactions reduce the favorability of the corresponding interaction with a trans monomer. The distributions of dimerization energies that the monomers in the liquid experience are shown in Figure 5. The shoulders at low energy are assigned to the particularly attractive interactions with near neighbors, while the spikes near 0 kcal/mol are due to the weak interactions with the many distant molecules in the bulk. As before, interactions with gauche and gauche* monomers are shown to be more favorable than with trans conformers. In addition, the interactions with near neighbors are clearly shown to be more attractive at the higher pressure. Integrating the shoulders can also provide estimates for the number of near neighbors. An integration limit of -0.6 kcal/mol yields coordination numbers of 10-12. Structure. Radial distribution functions (rdfs) represent the deviations for distributions of atoms in a liquid from
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The Journal of phvsical Chemistry, Vol. 86, No. 15, 1982 I
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CH-CL
RADIAL DISTRIBUTION FUNCTIONS I
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Jorgensen and Bigot
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GRUCHEx
TRRNS
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1- I
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