A Tale of Two Carenes: Intrinsic Optical Activity ... - ACS Publications

Article pubs.acs.org/JPCA

A Tale of Two Carenes: Intrinsic Optical Activity and Large-Amplitude Nuclear Displacement Priyanka Lahiri, Kenneth B. Wiberg, and Patrick H. Vaccaro* Department of Chemistry, Yale University, P.O. Box 208107, New Haven, Connecticut 06520-8107, United States S Supporting Information *

ABSTRACT: The specific rotation for two isomeric members of the terpene family, (S)(+)-2-carene and (S)-(+)-3-carene, has been investigated under complementary solvated and isolated conditions, where the latter vapor-phase work has been performed at excitation wavelengths of 355 and 633 nm by means of ultrasensitive cavity ring-down polarimetry (CRDP). Linear-response computations of dispersive optical activity built upon analogous density-functional (B3LYP/aug-cc-pVTZ) and coupled-cluster (CCSD/aug-cc-pVDZ) levels of theory have been enlisted to unravel the structural and electronic origins of observed behavior. The six-membered portion of the bicyclic skeleton in the nominally rigid 3-carene system is predicted to be near-planar in nature, with calculated and measured rotatory powers for the isolated (gas-phase) species shown to be in excellent agreement. In contrast, the inherent flexibility of 2-carene gives rise to two quasidegenerate conformations that are interconnected by a large-amplitude ringpuckering motion and exhibit antagonistic chiroptical properties. Various approaches to simulate the intrinsic response evoked from a thermally equilibrated ensemble of gaseous (S)-(+)-2-carene molecules have been considered, including implicit averaging over independent conformers and explicit (albeit restricted) averaging over nuclear degrees of freedom. A polarizable continuum model for implicit solvation was found to describe solvent-dependent trends reasonably well in the case of (S)-(+)-2-carene, but failed to reproduce the specific-rotation patterns emerging from polarimetric studies of (S)-(+)-3-carene.

I. INTRODUCTION Recent years have witnessed a veritable renaissance in the application of chiroptical probes to problems of molecular structure and dynamics,1 with numerous experimental and theoretical efforts highlighting the ability to determine the absolute stereochemical configuration and to extract ancillary conformational information for diverse chemical species. To a large extent, this resurgence of interest has been stimulated by the advent of robust and efficient quantum-chemical paradigms for reliably predicting the spectral response (magnitude and sign) evoked from a chiral molecule; 2 however, the complications incurred by extrinsic environmental perturbations (e.g., solvation processes) and intrinsic nuclear couplings (e.g., vibrational motion as well as large-amplitude displacements) still present formidable challenges.3 The present work strives to elucidate the provenance of dispersive electronic optical activity (circular birefringence or CB)4,5 by examining the wavelength-dependent specific rotation (optical rotatory dispersion or ORD) for two isomeric members of the terpene family under complementary solution-phase and vapor-phase conditions, where the latter isolated-molecule studies have exploited the ultrasensitive techniques of cavity ring-down polarimetry (CRDP).6,7 Figure 1 depicts the two molecules of interest for the present work, (a) (1S,6R)-3,7,7-trimethylbicyclo[4.1.0]hept-2-ene and (b) (1S,6R)-3,7,7-trimethylbicyclo[4.1.0]hept-3-ene, which will be referred to as (S)-(+)-2-carene and (S)-(+)-3-carene in the ensuing discussion. These species clearly are structural isomers of one another that differ primarily by position of the olefinic © 2012 American Chemical Society

Figure 1. Structures of the targeted carenes. The minimum-energy structures of (a) (S)-(+)-2-carene and (b) (S)-(+)-3-carene, as predicted by the B3LYP/apVTZ level of theory, are depicted, with the numbering scheme used to distinguish the carbon atoms composing the bicyclic ring system being superimposed.

moiety within the six-membered portion of the bicyclic ring systema modification in chromophore topology that might be expected to affect the nature and location of excited electronic states only slightly. Nevertheless, previous studies performed in the condensed phase have documented the substantial changes in optical activity imbued by this modest alteration in skeletal bonding patterns, with the highest specificrotation values reported for sodium D-line (589.3 nm) Received: April 5, 2012 Revised: August 9, 2012 Published: August 13, 2012 9516

dx.doi.org/10.1021/jp303270d | J. Phys. Chem. A 2012, 116, 9516−9533

The Journal of Physical Chemistry A

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excitation of neat (S)-(+)-2-carene8 and neat (S)-(+)-3-carene9 being +97.7 deg dm−1 (g/mL)−1 and +17.7deg dm−1 (g/mL)−1, respectively. The molecular geometries in Figure 1 display subtle differences in puckering of the carbon skeleton accompanied by reorientation of the methyl groups to minimize steric hindrance. Such structural preferences have been the subject of numerous experimental and theoretical efforts that, oftentimes, have led to contradictory conclusions. An early investigation of Kerr constants in 3-carene suggested an equilibrium to exist between boat and inverted-boat conformations,10 with the former dominating under ambient conditions. This assertion was supported, in part, by the product distribution generated from the sensitized photo-oxygenation of 3-carene,11 which was believed to stem from a rapidly interconverting 1:1 mixture of the aforementioned conformers; however, the interference patterns observed in gas-phase measurements of electron diffraction could be simulated well by considering only the boat motif.12 While the temperature dependence of chemical shifts observed for the methyl protons of 3-carene was interpreted in terms of a conformational equilibrium in which the dominant boat form was stabilized by an estimated 6.2 kJ mol−1 relative to its inverted-boat counterpart,13 analogous proton-NMR studies performed at higher magnetic-field strengths argued that the six-membered portion of the bicyclic ring is essentially planar in nature.14 Conflicting assertions also have emerged from theoretical analyses of 3-carene. While early molecular-mechanics simulations by Favini and co-workers15 reported a stable planar arrangement for the six-fold carbon ring, geometry optimizations performed at minimal Hartree−Fock (HF/STO-3G) levels of theory suggested the inverted-boat conformer to reside 10.7 kJ mol−1 lower in energy than the boat form,16 a prediction at odds with prior experimental findings. A subsequent MMP2 molecular-mechanics study performed in conjunction with measurements of NMR proton−proton coupling constants lent further support to the existence of a planar ring motif.17 In the most recent computational investigations of 3-carene, Stephens et al.18 exploited an extensive Monte Carlo search algorithm based upon the MM94 force field to locate a single, globally stable conformation, the validity of which was confirmed by examining the potential surface as a function of the C 3C 4C 5 C 6 dihedral angle (cf., Figure 1). This coordinate was found to be essentially nil (0°) at the minimum-energy configuration, in keeping with the expectation for a planar six-membered carbon skeleton. The refined structure obtained from this work subsequently was employed for density-functional calculations of chiroptical properties by means of B3LYP/6-31G* and B3LYP/aug-cc-pVDZ linearresponse methods, leading to an estimated specific-rotation value of +31.9 deg dm−1 (g/mL)−1 for the sodium D-line excitation of isolated (solvent-free) (S)-(+)-3-carene molecules. In contrast to 3-carene, few experimental and theoretical studies of conformational dynamics have been reported for 2carene. Two stable forms adopting pseudochair and pseudoboat motifs have been proposed for this system,13 with chemicalshift measurements and chemical-reactivity arguments suggesting the former to be more stable than the latter. Molecularmechanics simulations15 and minimal (HF/STO-3G) geometry-optimization procedures19 supported this conjecture, predicting the attendant energy difference (between conformers) to be 8.8 and 4.9 kJ mol−1, respectively. The ensuing

analyses build upon such prior efforts to elucidate the structural preferences of (S)-(+)-2-carene and (S)-(+)-3-carene, which will be shown to manifest themselves prominently through distinct chiroptical properties.

II. EXPERIMENTAL PROCEDURES AND COMPUTATIONAL METHODS Requisite samples of (S)-(+)-2-carene and (S)-(+)-3-carene were obtained from a commercial source (Sigma-Aldrich) with specified chemical purities of 97% and 99%, respectively, and used without further treatment. Corresponding percentages of enantiomeric excess (% ee) were estimated to be >92% and >97% on the basis of solvated and neat specific-rotation measurements performed at the sodium D-line (589.3 nm). Materials employed for vapor-phase studies were sealed in glass vessels having grease-free (Teflon) vacuum stopcocks and subjected to at least three freeze−pump−thaw cycles so as to minimize potential contamination from entrained (atmospheric) gases. Measurements performed on isolated (vapor-phase) species exploited the basic CRDP instrumentation and analysis procedures reported elsewhere,6,7 with the most salient details being discussed briefly here. Requisite visible and ultraviolet light was generated by pumping a high-resolution pulsed dye laser (Lambda Physik FL3002E; 0.035 cm−1 bandwidth) with the second harmonic of a Nd:YAG system (Spectra-Physics GCR-4-20; 20 pps repetition rate; ∼10 ns pulse duration). While the fundamental dye-laser output was used for 633 nm work, studies performed at 355 nm necessitated the additional step of frequency doubling (Inrad Autotracker III; BBO crystal) followed by isolation of the emerging ultraviolet radiation. The resulting linearly polarized 633/355 nm beam propagated through a variable attenuator before entering a Keplerian telescope where it was spatially filtered and refocused to match the mode structure of a ring-down cavity having a total length of L = 170.7284(35) cm, where the value in parentheses denotes the one standard-deviation uncertainty for the final two significant digits. The mode-matched excitation light passed though a circular polarizer (consisting of a tandem calcite prism and λ/4retarder) prior to entering the CRDP linear resonator assembly through the planar rear surface of one cavity mirror. The concave mirrors employed for 355 and 633 nm studies (Los Gatos Research; 6 m radius of curvature) had specified intensity reflectivities of ≥99.95% and ≥99.99%, respectively. Two intracavity quarter-wave plates of composite zero-order construction (Alpine Research Laboratories; optically contacted with antireflection coatings exhibiting [α]Tλ,gas, an assertion diametrically 9526



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extracted for 355 and 633 nm excitation are plotted against the Onsager dielectric-constant function:55,56 f (ε) =

ε−1 2ε + 1

(8)

which spans the range 0 ≤ f(ε) ≤ 0.5 as the surrounding medium varies from being completely nonpolar (ε = 1) to being infinitely polar (ε ≫ 1). In particular, this quantity stems from the progenitor of all implicit solvation models known as Onsager reaction-field theory,55 where it mediates the solvent− solute stabilization energy engendered by the lowest-order multipole moment (i.e., the permanent electric dipole moment) of the neutral solute molecule (vide infra). Excluding the anomalous behavior of chloroform that has been attributed to dominant quadrupolar interactions,32,57 a strong linear relationship is found between [α]Tλ and f(ε) for the remaining solvents, with extrapolation to the origin (ε = 1) suggesting the attendant vapor-phase specific rotations to be 77.5 and 14.1 deg dm−1 (g/mL)−1 at 355 and 633 nm, respectively. These predictions differ markedly from the present CRDP measurements of intrinsic optical activity, reinforcing prior assertions that a serious incongruity exists between dispersive chiroptical studies performed under solvated and isolated conditions.7 Table 9 contains a subset of the optical-activity data acquired for (S)-(+)-2-carene in five different solvents, with Figure 8 depicting associated ORD profiles. In contrast to 3-carene, solvation now causes a modest increase in the magnitude of rotatory power. When compared to their vapor-phase counterparts (cf., Table 5 or 6), solution-phase [α]Tλ parameters extrapolated for 355 nm (633 nm) excitation rise by +26.3% (+16.8%) and +10.6% (+4.2%) in the presence of cyclohexane and acetonitrile media, thus making the latter, once again, the best mimic for isolated-molecule measurements. Also of note is the overall tendency for specific rotation to decrease in proportion to solvent polarity; however, both ethanol and chloroform appear to deviate from this pattern. The inherent nonrigidity of the bicyclic framework in 2-carene can be expected to complicate interpretation of these results, leading to patterns that reflect the modification of chiroptical response by the surrounding environment as well as the differential stabilization experienced by individual conformers.37 Some insight can be gleaned by considering the nature of solvation processes, which often are dominated by electrostatic effects built upon the lowest-order multipole moments of the entrained solute. The most rudimentary implementation of Onsager reaction-field theory predicts a solute−solvent interaction energy that scales as Esol ∝ −f(ε)μ2,55,56 where μ denotes the magnitude of the permanent electric dipole moment for the solute. Consequently, species possessing larger values of μ will be relaxed to a greater extent than their lesspolar analogues, with the overall degree of stabilization increasing in proportion to the polarity of the solvent as encoded in the dielectric-constant function f(ε). While the magnitude of the permanent electric dipole moment estimated for isolated 3-carene molecules is rather small (0.169 D from B3LYP/apVTZ), this quantity grows to 0.364 and 0.358 D for the (A) and (B) forms of 2-carene, respectively, and rises further to 0.374 D for the transition-state configuration. The differences in μ among these 2-carene structures are slight; however, the preferential stabilization of polar species in high-dielectric surroundings might be expected to increase the energy separation between conformers and (potentially) decrease the accompanying barrier to isomer-

Figure 8. ORD of solvated (S)-(+)-2-carene. The specific-rotation values measured at discrete excitation wavelengths (closed symbols and solid curves) for (S)-(+)-2-carene are shown in the top panel for a variety of solvents. The accompanying vapor-phase data at 355 and 633 nm (isolated symbols) highlight the role of environmental perturbations. The bottom panel depicts ORD profiles predicted by a thermal conformer-averaging analyses (T = 300 K) based on structures optimized at the B3LYP/apVTZ level of theory in the presence of implicit (PCM) acetonitrile solvation. These results follow from B3LYP/apVTZ linear-response calculations, with the energy (ΔEη) and free-energy (ΔGη) estimates for individual conformers being obtained from three computational methods: B3LYP/apVTZ, G3B3, and G3.

ization. Such assertions are mostly in accord with the computational results compiled in Table 10, which highlight the solvent-dependent energies of stationary points optimized at the B3LYP/apVTZ level of theory in the presence of a PCM treatment for solvation. In particular, the B3LYP/apVTZ, G3B3, and G3 approaches concur in predicting the relative energy of 2-carene(B) to climb upon moving from a weakly polar cyclohexane environment to a strongly polar acetonitrile medium. Table 11 contains wavelength-resolved values of specific rotation computed for the three stationary points of groundstate (S)-(+)-2-carene molecules entrained in the five solvents targeted by the present study. These results follow from application of the B3LYP/apVTZ linear-response formalism to geometries optimized at the same level of density-functional theory, with both analyses being performed in the presence of a PCM treatment for solvation. As such, the tabulated quantities embody changes effected in both structural and chiroptical properties. The putative role of the latter was accessed by 9527



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Table 10. Predicted Relative Energy for Stationary Points of Solvated (S)-(+)-2-Carenea relative energy (cm−1)

population fraction

source

solvent

conformer (B)

transition state

conformer (A)

conformer (B)

ΔEη [B3LYP/apVTZ]

acetonitrile ethanol chloroform dibutyl ether cyclohexane

57.79 56.74 44.80 38.88 32.18

449.00 449.02 449.20 449.24 449.22

0.569 0.568 0.554 0.546 0.539

0.431 0.432 0.446 0.454 0.461

ΔGη [G3B3]


80.33 79.01 63.87 56.62 48.50

731.51 731.07 725.36 723.39 721.63

0.595 0.594 0.576 0.567 0.558

0.405 0.406 0.424 0.433 0.442

ΔGη [G3]


109.74 109.08 100.96 96.13 90.86

··· ··· ··· ··· ···

0.629 0.628 0.619 0.613 0.607

0.371 0.372 0.381 0.387 0.393

a Relative energies for the (A) and (B) conformers of solvated 2-carene molecules, as well as for the associated transition-state (TS) configuration, are tabulated for five different solvents. The predicted energy (ΔEη) and free-energy (ΔGη) parameters follow from B3LYP/apVTZ, G3B3, and G3 calculations performed on geometries optimized at the B3LYP/apVTZ level of theory, with all analyses incorporating implicit PCM solvation. The listed fractional populations of 2-carene(A) and 2-carene(B) are evaluated for thermal equilibrium at T = 300 K.

Table 11. Predicted Specific Rotation for Stationary Points of Solvated (S)-(+)-2-Carenea specific optical rotation [deg dm−1 (g/mL)−1] species

wavelength (nm)

acetonitrile

ethanol

chloroform

dibutyl ether

cyclohexane

(S)-2-carene A

355.00 365.02 435.83 546.07 589.30 633.00

−231.29 −208.39 −117.80 −64.86 −53.94 −45.60

−234.53 −211.30 −119.40 −65.72 −54.65 −46.20

−262.08 −235.89 −132.56 −72.59 −60.29 −50.91

−268.30 −241.30 −135.08 −73.77 −61.23 −51.68

−281.76 −253.28 −141.39 −77.02 −63.89 −53.90

(S)-2-carene TS

355.00 365.02 435.83 546.07 589.30 633.00

739.23 663.85 362.54 190.65 156.51 130.88

738.49 663.17 362.16 190.45 156.33 130.73

746.34 669.84 364.81 191.45 157.08 131.31

731.54 656.69 358.02 188.03 154.30 129.00

731.04 656.11 357.37 187.56 153.90 128.64

(S)-2-carene B

355.00 365.02 435.83 546.07 589.30 633.00

1223.26 1102.40 613.89 328.66 271.03 227.50

1229.02 1107.49 616.48 329.94 272.07 228.36

1254.54 1129.93 627.55 335.31 276.39 231.92

1238.59 1115.68 620.01 331.45 273.25 229.30

1247.06 1123.07 623.49 333.09 274.56 230.37

a The wavelength-resolved specific rotation computed for the solvated (A) and (B) conformers of (S)-(+)-2-carene, as well as for the attendant transition state (TS), are presented for five solvents. These results follow from application of B3LYP/apVTZ linear-response methods to structures refined at the same level of density-functional theory, with all analyses incorporating a PCM treatment of implicit solvation.

repeating calculations of dispersive optical activity without the PCM ansatz while still retaining solvent-optimized configurations of the nuclear framework. This procedure led to [α]Tλ parameters (for the unrelaxed “gas-phase” species) that increased in magnitude by ≤10%, again contradicting the behavior anticipated from the Lorentz local-field correction of eq 7. At a given excitation wavelength, the magnitude of specific rotation computed for each stationary point of (S)-(+)-2-carene

tends to decrease with increasing dielectric constant of the surroundings; however, clear deviations from this pattern are evident for chloroform and ethanol. The most favorable comparison between B3LYP/apVTZ solution-phase and vapor-phase predictions of dispersive optical activity are found in the case of acetonitrile, where solvated [α]λT parameters for 2-carene(A) differ from their isolated-molecule counterparts (cf., Table 3) by near-negligible amounts. In contrast, analogous acetonitrile results for 2-carene(B) and 29528



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activity performed at the B3LYP/apVTZ and CCSD/apVDZ (MVG) levels of theory were found to be in excellent agreement with each other and to reproduce isolated-molecule measurements of specific rotation in a quantitative fashion. As such, the intrinsic chiroptical properties of (S)-(+)-3-carene would appear to be described well by purely electronic effects, with cumulative contributions arising from nuclear degrees of freedom (e.g., small-amplitude vibrational motion) ostensibly being of negligible consequence. Two nearly degenerate conformers, which interconvert along a large-amplitude ring-puckering coordinate that presents a potential barrier of finite (400−800 cm−1) height, have been identified for (S)-(+)-2-carene. These species are predicted to have antagonistic chiroptical properties, with the wavelengthresolved (positive) rotatory power for 2-carene(B) being four times larger in magnitude than the corresponding (negative) quantity for 2-carene(A) while the transition-state configuration (which supports a quasiplanar six-membered ring akin to that in 3-carene) displays an intermediate (albeit positive) behavior. Despite the uniformity attained among theoretical calculations in (S)-(+)-3-carene, coupled-cluster (MVG) estimates of dispersive optical activity for all stationary points in (S)(+)-2-carene are almost a factor of 2 smaller than their densityfunctional counterparts at each targeted wavelength. Various methods to account for the inherent structural flexibility of 2-carene have been considered, including Boltzmann-weighted averaging over properties of independent conformers and restricted vibrational averaging over torsional displacement of the bicyclic ring system. A conformer-averaging procedure, whereby relative free-energies estimated for each equilibrium configuration from composite G3B3 analyses (ΔGη) are combined with corresponding B3LYP/apVTZ eq ), affords the best calculations of specific rotation ([α]λ,η simulation of observed isolated-molecule behavior, with use of analogous CCSD/apVDZ estimates for [α]eq λ,η significantly reducing the quality of theoretical results. The overall agreement achieved between the conformer-averaging and vibration-averaging treatments confirms the reliability of the former scheme (despite the rudimentary nature of its underlying assumptions)40 and reinforces the important role played by the large-amplitude, ring-puckering degree of freedom. The specific rotation of (S)-(+)-3-carene drops markedly upon solvation, with acetonitrile offering the best mimic for isolated-molecule behavior. While dispersive optical-activity measurements in the vapor phase are reproduced well by B3LYP/apVTZ and CCSD/apVDZ calculations, attempts to describe solution-phase results by incorporating a PCM ansatz into the linear-response framework of density functional theory proved to be less successful. Such analyses were found to shift solution-phase ORD curves uniformly in the wrong direction (toward larger magnitudes) relative to their vapor-phase counterpart; however, the overall growth in observed specific rotation with decreasing solvent polarity was reproduced successfully, including the deviation of a chloroform medium from this basic trend. In contrast to (S)-(+)-3-carene, (S)-(+)-2-carene undergoes a modest increase in rotatory power when introduced into the solution phase, with acetonitrile, once again, found to best duplicate the intrinsic (vapor-phase) response. The observed specific rotation tends to increase with decreasing solvent polarity; however, both ethanol and chloroform deviate from this general pattern. The PCM treatment (used to calculate

carene(TS) deviate by 15% or more. Given the discussion above, such observations can be attributed primarily to the structural relaxation brought about by solute−solvent interactions. The condensed-phase specific rotation exhibited by a thermally equilibrated ensemble of (S)-(+)-2-carene molecules has been simulated under the conformer-averaging ansatz of eq 3, with requisite fractional populations fη(T) following from B3LYP/apVTZ, G3B3, and G3 calculations performed in the presence of implicit (PCM) solvation (cf., Table 10). In the case of acetonitrile, excellent agreement between experiment and theory is obtained by using B3LYP/apVTZ estimates of ΔEη, which give a deviation of +2.2% for sodium D-line excitation. Analogous treatments based on more robust G3B3 and G3 predictions of ΔGη increase the magnitude of discrepancies at 589.3 nm to −7.9% and −20.8%, respectively. Similar levels of accord are found for the other solvents and the tendency for [α]Tλ at a fixed wavelength to decrease in proportion to the polarity of the surroundings is reproduced uniformly. The solvent dependence of dispersive optical activity in (S)(+)-2-carene is summarized by the top panels of Figure 8, which plot [α]Tλ parameters extrapolated for 355 and 633 nm excitation against the Onsager dielectric-constant function, f(ε). In contrast to the kindred analysis for (S)-(+)-3-carene, the graphs now display an inverse (negative) slope with the points for chloroform and ethanol departing markedly from those of other media. The latter behavior reflects the putative action of nondipolar effects32,57 as well as the influence of other processes (e.g., specific solute−solvent couplings) that mediate conformational and chiroptical properties. A linear relationship is obtained among the acetonitrile, di-n-butyl ether, and cyclohexane results, with the projected specific rotations for the nominally isolated solute molecule (ε = 1) being 468.8 and 84.8 deg dm−1(g/mL)−1 at 355 and 633 nm, respectively. Once again, CRDP measurements of intrinsic rotatory power depart appreciably from these estimates (by +36.8% at 355 nm and +26.2% at 633 nm), displaying characteristics more reminiscent of those obtained in strongly polar environments.

IV. SUMMARY AND CONCLUSIONS A synergistic experimental and computational investigation of dispersive optical activity has been performed for two isomeric members of the terpene family, (S)-(+)-2-carene and (S)(+)-3-carene, under ambient (room-temperature) conditions. The specific rotation of isolated molecules in the vapor phase was measured at 355 and 633 nm by exploiting the ultrasensitive methods of cavity ring-down polarimetry.6,7 Complementary condensed-phase work was conducted at discrete excitation wavelengths spanning the visible and ultraviolet regions of the spectrum by using solvent media selected to display a wide range of chemical and physical properties. Linear-response calculations based on the densityfunctional (B3LYP/apVTZ) and coupled-cluster (CCSD/ apVDZ) frameworks were enlisted to unravel the provenance of observed chiroptical behavior, with a polarizable continuum model (PCM) for implicit solvation being used to explore the putative roles of solute−solvent interactions. In the case of (S)-(+)-3-carene, quantum-chemical calculations confirmed the presence of a single, low-lying conformer where the six-membered portion of the bicyclic ring is held in a near-planar configuration by a nominally rigid molecular framework. Linear-response predictions of dispersive optical 9529



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ΔGη and [α]eq λ,η for the conformer-averaging procedure) appears to afford a reasonable description for solute−solvent interactions in the case of (S)-(+)-2-carene, reflecting, in part, differential stabilization of conformers that exhibit antagonistic chiroptical properties. Nevertheless, analyses of continuumdielectric scaling based on a rudimentary implementation of Onsager reaction-field theory reinforce previous assertions that a serious incongruity exists between specific-rotation values extracted from solvated and isolated environments,7 with the latter routinely displaying characteristics reminiscent of those encountered in nonintuitive solvents of high polarity. The transposition of single/double carbon−carbon bonds between the isomers of carene does alter electronic structure slightly, with TDDFT(B3LYP)/apVTZ and EOM-CCSD/ apVDZ calculations showing the lowest-lying excited singlet state in 3-carene to reside at 5.30 and 6.00 eV, respectively, while the analogous feature in 2-carene(A) is predicted to be roughly 6% lower in energy (cf., Tables S6 and S7 in the Supporting Information). Nevertheless, for the nonresonant excitation wavelengths examined during the present study, the dispersive chiroptical response is not dominated by a particular (electronic) eigenstate, but instead reflects a subtle superposition of contributions arising from the entire manifold of excited states. As such, the marked differences observed in the rotatory powers of the targeted carene species might be attributable to factors other than those of purely electronic origin. Since the excellent agreement found between theory and experiment for the specific rotation of isolated (S)-(+)-3-carene molecules would tend to discount effects arising from smallamplitude vibrations, the pronounced ring-puckering motion that mediates interconversion between the quasidegenerate (A) and (B) conformers of (S)-(+)-2-carene becomes a prime suspect. Indeed, inspection of geometries predicted for the two equilibrium configurations (cf., Figure 3) reveals the sixmembered portion of the bicyclic-ring system to be twisted helically in opposite directions, thereby affording a viable explanation for the antagonistic chiroptical properties exhibited by these isomeric forms. Further computational and experimental efforts designed to unravel the dependence of electronic optical activity on nuclear degrees of freedom are underway, with special emphasis being directed toward molecular frameworks that are subject to large-amplitude displacements.

internal coordinates, with the corresponding determinant of g being denoted by g  ∥g∥. Straightforward expansion and simplification of eq A1 yields59 T̂ =T̂′ + V̂ ′ ⎧⎛ jk ⎞⎛ ⎞ ⎪ ∂g ⎟⎜ ∂ ln g ⎟ ⎨⎜ ⎜ ⎜ ⎟ ∂q ∂qk ⎟⎠ j,k j,k ⎪ ⎩⎝ j ⎠⎝ ⎤⎫ ⎡⎛ 2 ⎞ ⎞ ⎛ 1 ⎜ ∂ ln g ⎟⎛⎜ ∂ ln g ⎞⎟⎥⎪ jk ⎢⎜ ∂ ln g ⎟ ⎬ +g ⎜ + ⎜ ⎢ ∂q ∂q ⎟ 4 ⎝ ∂qj ⎟⎠⎜⎝ ∂qk ⎟⎠⎥⎦⎪ ⎣⎝ j k ⎠ ⎭

=−

ℏ2 −1/4 g 2

∑ j,k

∂ 1/2 jk ∂ −1/4 g g g ∂qj ∂qk

∑

∂ jk ∂ ℏ2 g + 8 ∂qj ∂qk

3N − 3

∑

where the symmetric nature of the metric tensor, gjk=gkj, has been exploited to arrive at the final expression. The second term in eq A2 represents an effective potential (or pseudopotential), V̂ ′, that arises from the kinetic energy operator. Although this quantity can be incorporated into the actual potential energy, V̂  V(q), its magnitude, which reflects the variation of elements gjk and determinant g with respect to internal coordinates, often is negligibly small in comparison to that of V̂ .60 Under such circumstances, the kinetic energy operator becomes T̂ ≈ T̂ ′ = −

ℏ2 2

3N − 3

∑ j,k

⎧ ⎫ ⎛ jk ⎞ ⎪ jk ∂ 2 ∂g ⎟ ∂ ⎪ ⎨g ⎬ + ⎜⎜ ⎟ ⎪ ∂qj∂qk ⎝ ∂qj ⎠ ∂qk ⎪ ⎩ ⎭

(A3)

where the second term in the summation can be discounted if gjk displays only a weak dependence on qj. Likewise, the mixed partial derivatives of the first term would vanish if the transformation between Cartesian and curvilinear coordinates leads to a diagonal metric tensor, gjk = gjj δjk (where δjk denotes the canonical Kronecker delta symbol), thus allowing (gjj)−1 to be associated with an effective mass factor for displacement along qj. By construction, the kinetic energy operator of eq A3 embodies both rotational and vibrational degrees of freedom, with g capable of being partitioned into two blocks (representing pure-rotational and pure-vibrational contributions) that are coupled by off-diagonal terms.46,60 The functional form of gjk for pure-rotational motion and for rotation−vibration interaction depends on how the body-fixed axis system is attached to the molecular framework; however, this is not the case for pure-vibrational elements and for the metric-tensor determinant.46,59 Consequently, any convenient choice of axis embedding can be used to evaluate the latter quantities.

APPENDIX A For evaluation of the kinetic energy associated with the torsional isomerization of 2-carene, it is assumed that the center-of-mass motion for a free molecule comprised of N atoms has been separated, leaving 3N − 3 internal degrees of freedom to partition among rotations and vibrations. As first elaborated by Podolsky,58 the position representation for the quantum-mechanical operator of kinetic energy, T̂ , can be recast in terms of curvilinear internal coordinates, {qj; j = 1, 2, ..., 3N − 3}, by means of T̂ = −

3N − 3

(A2)

■

3N − 3

ℏ2 2

■

ASSOCIATED CONTENT

S Supporting Information *

Structure for (S)-(+)-3-carene (Table S1); structure for (S)(+)-2-carene(A) (Table S2); structure for (S)-(+)-2-carene(B) (Table S3); structure for (S)-(+)-2-carene(TS) (Table S4); origin dependence for coupled-cluster predictions of specific rotation (Table S5); density-functional predictions for excited electronic states (Table S6); coupled-cluster predictions for excited electronic states (Table S7); and temperature dependence of chiroptical response in (S)-(+)-2-carene (Figure S1). This material is available free of charge via the Internet at http://pubs.acs.org.

(A1)

which is appropriate for an accompanying rovibrational wave function, ψ(q), normalized with respect to the volume element dq = dq1dq2···dq3N−3. The quantity gjk signifies an element of the contravariant mass-weighted metric tensor, g, that reflects the conversion between rectangular-Cartesian and curvilinear9530



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: (203) 4323975. Fax: (203) 432-6144. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was performed under the auspices of grants CHE0809856 and CHE-1112239 awarded by the Chemical Structures, Dynamics, and Mechanisms Program in the Directorate for Mathematical and Physical Sciences of the United States National Science Foundation. The authors wish to thank Prof. T. Daniel Crawford (Virginia Tech) for technical assistance with the Psi3.4 quantum-chemistry package. Computational resources utilized in this work were supported in part by the Yale University Faculty of Arts and Sciences High Performance Computing Center and by the National Science Foundation under grant CNS-0821132, which partially funded the acquisition of requisite computer facilities.

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