Moses K. Kaloustian
Americon Universitv, of Beirut Berur, Lebonon
I I
The Electrostatic Dimension in Conformational Analysis
The past few years have witnessed a proliferation of conformational studies of saturated heterocycles and acyclic systems bearing more than one heterosubstituent (I3). The quantitative treatment of such systems differs from that-of analogous alicycles and acyclic systems with one or no polar substituents, since in polar systems dipolar interactions and solvation effects eain in imwrtance.' Whereas, in theory, the relative energies of two conformers can be evaluated by quantum mechanical principles (4-5), in practice the calculations, especially for larger molecules, turn out to he verv complex. More commonly, one resorts to the classical approaih according to which molecules are treated as mechanical systems subject to certain boundary conditions. The conformational energy, E,2 of a given conformer of a polar molecule in a solvent S a t a given temperature must now be expressed as the sum of six terms,3 instead of four, (6-8), each of which relates energy to geometrical parameters
E
-
E,
+ Ee + E, + E" + E, + E,
(1)
In eqn. (1) E p is the sum of bond-compressing and stretching energies, Ele is the sum of energies due to deformations of bond angles (Baeyer strain), ET is the sum of torsional energies (Pitzer strain), Ev is the sum of energies due to nonbonded interactions (van der Waals interactions), E D is the sum of energies due to intramolecular dipolar interactions, and E is the energy of solvation. The forces associated with EF, E e , ET, and Ev are due to short-range interactions; the energy terms are for nearest neighbors. In contrast, intramolecular dipolar interactions, responsihle for ED, are relatively long-range. The functions re~resentinethe various terms are derived empirically from thermodynamics and spectroscopy. In the classical method. a covalent bond is treated as a harmonic spring and its energy is governed by Hooke's law. EF (in kcal/mole) is then expressed as
where kij is the stretching force constant (in dyne/cm) of the bond between atoms i and j, and Axil is the change in bond length (in A), due to compression or stretching, relative to the corresponding "normal" bond length. The expression for E,e (in kcal/mole) has a quadratic form similar to that for EF ,
where k i j ~is the bending force constant (in erg/radian2) for the angle defined by contiguous atoms i, j, and k; ARl,r is the deformation of the internuclear angle (in degrees) from the "normal" value. Bending and stretching force constants can be derived from Raman and infrared data.4 E t for a given bond m-n, on the other hand, is a sinusoidal function and may he formulated as a Fourier series or more simply as
where Eo is the torsional energy barrier along bond m-n, n is the periodicity, and Arp is the deviation of the dihedral angle defined by contiguous atoms o, m, n and p (o and p are monovalent atoms of Groups I or W). ET for the entire molecule then, is the sum of El's for all m-n bonds where m and n are atoms bonded to more then one atom. The two best, known formulations for Eu.,,,, the interaction energy between a pair of nonhonded atoms o and p, are Lennard Jones' "6,12" potential (eqn. (5)) (9) and Buckingham's "six-exp" potential (eqn. (6))(10)
+ Brnp-'2
15)
+ C(exp -Dr.,)
(6)
E ,,,v = -Ar,,-6
E
=
-Ar,,*
where A, B, C, and D are positive constants, and rorl is the distance (in A) hetween atoms o and p. In eqns. (5) and (6) the f i s t term describes the attraction between atoms o and p (London dispersion force), whereas the second term represents the repulsion between them a t a distance roD. The former is dominant a t large distances, the latter, a t short internuclear distances. In other) empirical equations \for EuO,",the attraction usually retains the r-6 dependence but the repulsive term varies betweenr-8 and r-'2 (11). Nonbonded interactions between different pairs of atoms may also be described by Hill's semi-empirical relationship (eqn. ( I ) )(12) in terms of two parameters: a, which is a function of distance, and a function of energy E , , >= + 868 X id expl-a/O.0736)] ( 7 )
+
rj*); r,j is the distance between where a = ril/(ri* atoms i and j, rj* and rj* are the van der Waals radii of atoms i and j, respectively, and 0i.j is an energy parameter characteristic of the atom pair i,j.5 In eqn. (I), Ev is the sum of all E,'s and represents the total energy due to nonbonded interactions between all pairs of nearest neighhor atoms. The classical method of calculating conformational ene m was introduced hv Westheimer (6). Later. Hendrickson (13-16) improved k and calculated.the conformational energies of C5-Clo cycloalkanes and on the basis of these 'For a detailed survey of solvation effects see Coetzee, J. F., and Ritchie, C. D., (Editors), "Solute-Solvent Interactions," Marcel Dekker,New York, 1969. 2The energy term E is not a thermodynamic function; it represents the potential energy of the system and the terms in eqn. (1) contributing to E may he thought of as also contributing to U,the total internal. energy, to H, the enthalpy or heat content, and therefore also to G,the Gibbs free energy. 3The discussion here is restricted to molecules with polar covalent bonds, and in principle is applicable to ground states and transition states. Molecules in which ion-ion and ion-dipole interactions come into play and cases in which hydrogen bonding and special resonance effects exist cannot be accomadated in the present treatment. 'For a listing of typical force constants see Herzberg, G., "Molecular Spectra and Molecular Structure," D. van Nostrand, Princeton, N. J., 1960, Vol. 2, p. 193. 5Typical values for both parameters are listed in Ref. (71, p. 452.
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calculations determined the preferred molecular geometries. Wiberg (17-18) adapted the method to the process of energy minimization and examined the Cs, Cm, C12 cycloalkanes and the cyclohexane chair s boat equilibrium. Force-field calculations have also been carried out by Allineer and coworkers (19-24) on a laree number of organic mzecules, e.g., alkines, 'cycloalk&es, carhonyl ;omwunds. alcohols. sulfoxides. nitriles, and silanes. Classical cal&lations have been extended by Schleyer (25-27) to bridgehead carbonium ions and polycycloalkanes with two-carbon bridges, by Ouellette (28, 29) to group N organometallic compounds, by Favini and coworkers (30-32) to cycloalkenes and cycloalkadienes, and by Boyd's group to bicyclo[n.m.o.] alkanes. An extensive application has been made by Lifson's school (34-41) to alkanes, cycloalkanes, amides, and nonconjugated olefins. With a judicious choice of parameters, Lifson and his collaborators were able to determine molecular conformations, enthalpies, molecular and lattice vibrational energies, heats of suhlimation. unit-cell narameters. and heats of hvdroeenation. In another study, Dunitz and coworkers (42j in& meted the X-rav diffraction data of 1.1.5.5-tetramethvlcvclodecane-8-carboxylic acid on the basis of a random mixture of two conformations-the geometries of which were determined by Westheimer-type calculations. Calculations based on classical mechanics have also been performed bv Altona and Sundaralineam 143) on substituted norbornaies and camphanes, by keublein and coworkers (44. 45) on 2.3-dihalohutanes and 3.4-dihalohexanes, and by ' ~ e i p(46) 'on tetrahydrofuran and 1,2,4-trioxapentane. These are but a few examples from a rapidly growing number of theoretical force-field calculations. The first four terms of eqn. (1) provide a reasonably good approximation of the gas phase conformational energy of molecules with a limited number of or no heteroatoms. However, for solutions of molecules having several heteroatoms, one must also include in the calculations En and Es, the last two terms of eqn. (1).The evaluation of these two terms is based on the classical theory of electrostatics or dielectrics (uide infra). In conformational analysis, one need evaluate only the difference in the energies of conformers AE6 AE
=
AE,
+ AE,, + AE, + AEV + AE,, + AE,
(8)
Thus, the conformational energy difference can be arhitrarily subdivided into a structural contribution and an electrostatic one. In principle, AED and AEs can have opposite signs; hence, their sum, AE,~,,t,,,t,ti,, may reinforce or offset the contribution of AE8trueturoito AE. AE may also he expressed as AE
=
AE,.,
+ AE,
(10)
whelie AE,,.
=
AE?
+ AE,, + AET + AE, + AE,
E D , for a molecule containing several dipolar bonds, is the sum of all the intramolecular dipolar interactions, i.e., of all the Ed's (cf. eqn. (11)). It represents the minimum work done in bringing all the imaginary point dipoles, associated with all the diuolar bonds in the molecule. from infinity to the correspoiding orientations in the moiecular framework. En. like E d . mav be stabilizing or destabilizing depending& the r2ativk orientations of the individual bond point dipoles. In the gas phase, where the dielectric constant is approximately unity, ED, for a molecule with dipolar bonds, is a well-defined quantity and cannot be neglected. For example, consider the diequatorial (e,e) and diaxial (a,a) conformers of trans-1,4-dihalocyclohex778
/ Journal of Chemical Education
Figure 1. The diequalorial /e.eJ and diaxial la,a) confarmallanal equilibrium of trans-1.4-dihalocyclohexane.
Figure 2. The conformational equilibrium between the all-anti (a,a,a/ and anti-gauche-anti(a.g,a/ farms of dimethoxyethane. ane (Fig. 1). Neither of the two conformers has an overall dipole moment, but each form has two antiparallel C-X dipoles. In accordance with eqn. (11)) the e,e form is destabilized by a relatively large ED term, whereas the a,a conformer is stabilized by a small E D term. The difference in the two E D terms, AEo, would be an appreciable quantitv.? .. and must he taken into account in calculating AE. As another example, consider gaseous 1,2-dimethoxyethane for which the predominant conformers are here assumed to be all-anti-(a,a,a) and anti-gauche-anti (a,g,a) (Fig. 2) (47). The a,a,a conformer has no overall dipole moment, since the C-O-C dipoles are antiparallel to one another; i t is nevertheless stabilized, relative to the all-anti form of n-hexane say, by a relatively large ED term. On the other hand, models show that the two C-O-C dipoles of the a,g,a conformer have parallel components, and according to eqn. (11) the ED term for this conformer will be destabilizing. If either one of the previous compounds is immersed in a solvent, the effective dielectric constant ( 6 ) between the interacting dipoles increases, and the magnitude of Ed ( = ED in either case) diminishes to Edle. The surrounding solvent molecules "insulate" one dipole from another, and cause a diminution of the interaction between the dipoles. While the dipolar energy between two dipoles, Ed, as in the e,e and a,a (Fig. I), a,a,a and a,g,a forms (Fig. 2), diminishes with increasing dielectric constant of the solvent, it does not necessarily become negligible in media of high polarity. Nevertheless, the magnitudes of the individual Ed's. and hence of En d e ~ e n don the nature of the medium surrounding rhe interact& dipoles. The conceptual distinction between El, and E , is extremely important and is one that has not been fully appreciated in the past (48). Whereas in the gas phase ED is not negligible (uide supra), E is zero. Even in solvents of low dielectric constant, e.g. CC4, the value of Es can be-
S I n calculating AE, differences in zero-point energies, contributions of higher vibrational states, and of APdV are assumed to be negligible; cf the discussion by J. Reisse in Ref. (I), p. 219. Hawever, to be able to compare AE with AGO meaningfully, one must take into account any differences in entropy of mixing and/or symmetry. 'In the absence of reliable values of the effective dielectric constant between the two dipoles, one cannot arrive at a meaningful AED term. If one assumes e = 1, AEo can be as large as 5 kcall mole in favorof the a,a conformer.
come significant. As the polarity of the medium increases, E D reaches a minimum non-zero value whereas - E s reaches an algebraic maximum. That is to say, as the medium becomes increasingly polar, a polar molecule (with a positive E D term) immersed in the medium gains in stability. This is due to a smaller contribution from ED, as a result of diminished dipolar interactions, and due to a larger (negative) contribution from E.5, as a result of enhanced solvation. A molecule with a negative ED term would he destabilized hy a smaller ED, (relative to the situation in the gas phase), but could still he stabilized (negative ES term) by salvation. It was noted earlier that in the gas phase E D is not uegligible hut E s is almost zero. It follows that AEn, for a system of equilibrating conformers or diastereomers, is a non-negligible term but AEs is clearly zero. Therefore, in the gas phase AE,~,,t,,,t,t,, " OED, and AE = AE,, = AEstrueturo~ AED. In other words, the conformational energy difference in the gas phase is equal to the sum of energies due to structural contributions (stretching, hending, torsional, and van der Waals) plus those due to intramolecular dipolar interactions. For the (e,e) t (a,a) conformational equilibrium of trans-1,4-dihalocyclohexane (Fig. 1) in the gas phase, AGO = AE = AEst,,,t,,,~ + AEn.B If one assumes the conformational enerev of C1 to be6.46 kcal/mole (49), AGO = (2 x 0.46) + A&. aEDalmost certainlv offsets the structural contribution" and the a,a conforme; must predominate in the gas phase (55). A parallel reasoning would enable the determination of the position of the configurational equilibrium for diastereomeric 1.4-dichlorocyclohexanes. For dilute solutions of polar substances in inert solvents of low or medium dielectric constant, both AEs and AED contribute significantly to AEe~ectro8tatic. That is to say, for equilibria in solvents such as carbon tetrachloride, chloroform, or methylene chloride, one must take into account a new contrihution, AEs, due to differential solvation of the equilibrating forms. Thus, for each of the two chair forms of the conformationally mobile 2-bromocyclohexanone (Fig. 3), one can calculate the En term due to the interaction of the C=O and C-X dipoles (48); AED would then represent the difference in such interactions, and when added to AE,trucrura~would give AE,,,. When 2-bromocyclohexanone is dissolved in a solvent, e.g., carbon tetrachloride, AEn would change, inasmuch as the intramolecular dipolar interactions depend on the effective dielectric constant of the medium. In addition, each conformer would be solvated to a different extent, since each would have a different overall dipole moment. The relative stabilization due to solvation of each form, would lead to a difference in conformational free energy, and this contribution, AEs (eqn. 14)), (see below) should be included in the expression for AE (eqn. (8)). Therefore, for dilute solutions of equilibrating polar molecules in inert solvents of low or medium dielectric constant, the conformational energy is the sum of the structural term (stretching, bendinp, torsion, non-bonded interactions) and the elec&ostatic term (dipolar interactions and solvation). In high dielectric solvents the value of AED would be finite but small, even though theoretically
+
lim AE, = 0
,+
-
in real media, dielectric constants are considerably less than infinity. Consequently, as the dielectric constant of the medium increases, the absolute contribution of AED to AE tends towards a uon-zero minimum, whereas that of AEs approaches a finite maximum. For equilibrating systems in highly polar media, AE is still equal to the sum of AE,t,,,t,,,~ and AE,~,,t,,,t,tre But, it is the solvation term AE;, rather than the dipolar term AED, that
Figure 3. Conformational equilibrium of 2-bromocyclahexanone.
anti
gauche
Figure 4. Canformational equilibrium of 1.2-difluoroethane.
trans
cis
Figure 5. Contigurational equilibrium of diastereoisomeric 2-isopropyl-5fluoro-1.3-diaxanes.
contributes significantly to AEPreetrostotre;the reverse was true for the gas phase (uide supra). The following examples illustrate the principles discussed above. Consider the conformational equilibrium in gaseous 1,2-difluoroethane (Fig. 4). In the gas phase (t = 1) the gauche form is favored by 0.6 kcal/mole (51), perhaps hecause in the gauche form F . . . F attractive London dispersion forces (i.e., negative E v ) (50) dominate over the dipolar repulsion (positive E D ) between the C-F dipoles. In the liquid phase ( 6 > 1) however, the difference in conformatioual energy increases to 2.6 kcal/mole. The enhanced stabilization of the gauche conformer is, in part, due to a smaller positive E D in the liquid versus vapor; to a larger extent i t is due to a much more negative E.7 term (liquid versus vapor). This follows from eqn. (13), while keeping in mind that the gauche form has the higher calculated dipole moment of 3.2 D (the anti form has no net dipole moment). Another striking example is provided by the configurational equilibria in 2-isopropyl-5-fluoro-1,3-dioxanes (Fig. 5) (52). The cis diastereomer has a calculated dipole moment of 3.09 D and is disfavored by a larger E D term relative to the trans comoound. The latter has a calculated dipole moment of only 1.07 D and presumably has a smaller En term because of the quasi-antiparallel arranpements of the two dipoles. In the gas the trans isomer is predicted (52) and is observed (53) to predominate. In solution. however. i t is the cis diastereomer which is more stable: AGO = + o . ~ ~ ( c c I ~ 0.62 ) , (diethyl ether), and 1.22 kcallmole (CHnCN). The dramatic reversal in stahility is a &nsequenci of the "better" solvation of the more polar cis form-the overwhelming stabilization of the cis form due to salvation offsets the disfavoring effects of E,t,,,tulal and En and causes a reversal of the equilibrium in the gas phase. According to Bijttcher (54), the interaction energy Ed between two unpolarizahle point dipoles ,,I and ,,2 in vacuum is given by
Ed = - 3 ( p , . r ) ( p 2 . r \ / r i
+ p,.@Jr3
Volume 51, Number 12, December 1974
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/
779
where r is the vector joining the midpoints of the interacting dipoles ,,I and For the simple case where p l = 112, and +I, $ 2 and r are in the same direction, E d = -2 pZ/r3; when and are antiparallel to each other and both are parallel to r, Ed = +2p2/r3. On the other hand, when p 1 and p z are antiparallel and both perpendicular to r, E d = -p2/$; for the case ,a and',,^ parallel to each other and both perpendicular to r, Ed = +p2/$. According to Ahraham (55), Ed (in kcal/mole) for two antiparallel dipoles of equal magnitude is Ed = 14.41p2(3cd8- l)/r3
(12)
where r is the distance (in A) between the dipoles, p is the magnitude of either dipole (in Dehyes), and 0 is the angle between the p's and r. It is evident from these considerations that En. for a system of two-point dipoles, may he negative (stabilizing) or positive (destabilizing) depending on the relative orientations of the indi~idualdi~oies. For two interacting polurizable point dipoles yl and p z of wlarizabilities a, and a?. resnectivelv. one must remember that due & mutual pofarization; the moments are changed in direction and mamitude. For the simple case where p1 = p 2 , 01 = a 2 , a n d p l , pz, and r have ihe same direction, E d = -2 p 2 / P ( 1 - 2 a / r 3 ) ; when ,,I and pz are antiparallel and both perpendicular to r, E d = - p 2 / $(l - a/r3) (56). The last term of eqn. (I), Es, the energy of the dipolar field for a spherical polarizable molecule, (57), is equivalent to the work done on the system in bringing the molecule from an infinite distance outside the dielectric medium to the center of a spherical cavity inside the dielectric. It corresponds to the work done to increase the dipole moment, to polarize the dielectric, and to surround the diwle with the dielectric: in other words. E.; renresents the stabilization energy of dipolar molecul& d"e io solvation. Unlike En which mav be newtive or ~ositive. . E .q is alwavs negative (stabilizing)
-
E,
1 = - s f ~ ~ / ( l
-f
~ )
(13)
where f = l/a3(2r - 2 ) / ( 2 ~+ 1)and y = a 3 ( n $ - l ) / ( n o 2 + 2). In the preceding equations, p is the dipole moment of the solute molecule, a is the molecular radius, c is the dielectric constant of the medium and no is the refractive index of the solute. The value of the molecular radius can he obtained from 4 r a 3 / 3 = M/Np where M is the molecular weight, N is Avogadm's number, and p is the density of the solute. According to eqn. (131, for a given solvent, the greater the dipole moment of a molecule, the greater the stabilization due to solvation. In the gas phase, ESis negligible since c 1and E.? = 0. But, as the dielectric constant of the medium increases, e.g., in polar solvents acetone or dimethylformamide, the absolute value of EFtends towards a maximum; since, according to I'Hbpital's rule,s
--
-. +
lim E,
=
y!
- ~ ' ~ p a X-2Co_
Ahraham, (51, 58) has derived a useful expression for for two dipolar (or quadmpolar) rotamers or equilibrating diastereomers A and B
.AE.q
According to this rule, f o r two functions f(e) and F ( e )
lim f'(e)/F'(c)
