Intrinsic Optical Activity and Large-Amplitude Displacement

Article pubs.acs.org/JPCA

Intrinsic Optical Activity and Large-Amplitude Displacement: Conformational Flexibility in (R)‑Glycidyl Methyl Ether 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 dispersive optical activity of (R)-(−)-glycidyl methyl ether (R-GME) has been interrogated under ambient vapor-phase and solution-phase conditions, with quantum-chemical analyses built on density functional (B3LYP and CAM-B3LYP) and coupled-cluster (CCSD) implementations of linear-response theory exploited to interpret experimental findings. Inherent flexibility of the heavy atom skeleton leads to nine lowlying structural isomers that possess distinct chiroptical and physicochemical properties, as evinced by marked changes in the magnitude and the sign of rotatory powers observed in various media. These species are interconverted by independent motion along two largeamplitude torsional coordinates and are stabilized differentially by interaction with the surroundings, thereby reapportioning their relative contributions to the collective response evoked from a thermally equilibrated ensemble. The intrinsic behavior exhibited by isolated (vapor-phase) R-GME molecules was calculated through use of both conformeraveraging and restricted vibrational-averaging procedures, the former affording moderately good agreement with measurements of optical rotatory dispersion (ORD) and the latter providing strong evidence for sizable effects arising from vibrational degrees of freedom. A similar conformer-averaging ansatz based on the polarizable-continuum model (PCM) for implicit solvation was deployed to elucidate R-GME specific-rotation parameters acquired for six dilute solutions. This approach gave reasonable predictions for sodium D-line (589.3 nm) experiments performed in the extremes of solvent polarity represented by cyclohexane and acetonitrile but failed to reproduce the overall shape of ORD profiles and suggested more complex processes might be involved in the case of an aromatic medium.

I. INTRODUCTION The dispersive interaction of light with an isotropic ensemble of chiral molecules,1 which most commonly manifests itself as a rotation in the emerging plane of polarization by equal magnitude yet opposite sign for the two members of an enantiomeric pair,2 has been studied for over two-hundred years and remains a routine method for the relative discrimination of stereoisomers.3 Ongoing advances in quantum chemistry have enabled this phenomenon, as well as other forms of natural optical activity, to be predicted from first-principles,4 thereby transforming chiroptical spectroscopy into a robust tool for determining absolute stereochemical configuration and elucidating conformational dynamics.5 Although computational limitations usually restrict such analyses to isolated (gas-phase) species, the vast majority of experiments are performed in the condensed phase, where observed behavior may be influenced markedly by environmental perturbations (e.g., solute−solvent coupling) that remain poorly understood.6−8 Ultrasensitive polarimetric schemes, as exemplified by the techniques of cavity ringdown polarimetry (CRDP),9 have been developed to probe dispersive chiroptical properties directly in rarefied media, leading to benchmark measurements of vapor-phase rotatory power that reflect the electronic/structural provenance of the observed response as well as its mediation by the surroundings. The present work extends our ongoing investigations of © XXXX American Chemical Society

intrinsic optical activity to (R)-(−)-glycidyl methyl ether (RGME), a small chiral system in which two large-amplitude nuclear degrees of freedom conspire to yield a potential-surface topography characterized by numerous low-lying minima. In particular, values of specific optical rotation, [α]Tλ [in deg dm−1 (g/mL)−1], have been acquired under complementary isolated and solvated conditions at ambient temperatures (T) and at discrete excitation wavelengths (λ) spanning the visible and ultraviolet portions of the spectrum, with high-level ab initio calculations serving to guide the interpretation of experimental findings. The structure of GME follows from that of the chiral epoxide methyloxirane (MeOX) or propylene oxide upon replacing a methyl hydrogen atom by a methoxy group (−OCH3). In particular, MeOX, as well as other substituted oxiranes including epichlorohydrin (EPH), epifluorohydrin (EPF), and 1,2-epoxybutane (EPB), have been studied extensively by gasphase polarimetry10 and quantum-chemical calculations11−20 to elucidate the origins of intrinsic chiroptical response and to unravel the marked influence of solvation. Though MeOX is nominally rigid save for the equivalent configurations obtained by 3-fold symmetric rotation of the −CH3 appendage, the Received: May 30, 2015 Revised: June 27, 2015

A

DOI: 10.1021/acs.jpca.5b05177 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

Measurements of intrinsic (vapor-phase) rotatory power were performed at the discrete wavelengths of 355 and 633 nm by employing instrumentation and procedures described elsewhere for the modulated mode of CRDP operation.9,26,28,29 In brief, requisite excitation light was obtained by pumping a high-resolution pulsed dye laser (Lambda Physik FL3002E; ∼0.035 cm−1 etalon-narrowed bandwidth) by the second harmonic of a Nd:YAG system (Spectra-Physics GCR-4-20; 20 pps repetition rate; ∼10 ns pulse duration). The fundamental dye output was used directly for 633 nm experiments whereas their 355 nm counterparts necessitated frequency doubling (Inrad Autotracker III; BBO crystal) followed by isolation of the ultraviolet radiation. The resulting 355/633 nm beam was variably attenuated, spatially filtered, and circularly polarized prior to entering a high-finesse linear cavity of total length L = 170.656(64) cm (one standard deviation uncertainty on last two significant digits) formed by two concave mirrors (Los Gatos Research; 6 m radius of curvature) having specified reflectivities of R ≥ 99.95% for 355 nm or R ≥ 99.99% for 633 nm. Evacuated by a liquidnitrogen-baffled diffusion pump (≤10−6 Torr base pressure), this apparatus contained an intracavity quarter-wave retardation plate (Alpine Research Optics; optically contacted, composite zero-order construction with ΔEη whereas the latter tends to have ΔEη′ < ΔEη. Although the vibrationless (υ = 0) levels for higher-lying species display fractional changes in stability (ΔE′η/ΔEη) that deviate only slightly from unity, the energy separations among the three lowest-lying forms of R-GME decrease by ∼26.8% under the coupled-cluster harmonic force field and increase by at least a factor of 2 in the case of analogous density functional treatments. Such disparities among energy metrics imply different population distributions for a thermally equilibrated ensemble of molecules, which will have repercussions for subsequent predictions of the collective chiroptical response. The most rudimentary treatment of dispersive optical activity in R-GME relies on an independent-conformer assumption (vide inf ra) that requires the wavelength-dependent specific rotation for each contributing isomer to be evaluated at the corresponding equilibrium configuration of the nuclear frameE


Article

The Journal of Physical Chemistry A Table 3. Predicted 355, 589.3, and 633 nm Responses for Isolated R-GME Isomersa (a) 355 nm Response B3LYP/apVTZ geometry isomer

LGB3LYP

LGCB3LYP

LGCCSD

I II III IV V VI VII VIII IX

−28.13 6.45 −19.11 314.98 −267.82 −192.61 −369.22 650.59 1001.02

−54.29 28.45 −27.51 236.41 −294.97 −127.74 −412.14 599.28 917.17

−51.06 −0.16 −55.73 220.55 −320.74 −175.47 −388.63 569.9 860.56

B3LYP

CB3LYP

CCSD/apVDZ geometry VGCCSD

−149.22 −194.08 −87.43 128.99 −780.27 −440.3 −790.93 720.18 1196.77 (b) 589.3 nm Response

MVGCCSD

LGCCSD

VGCCSD

MVGCCSD

−24.49 −8.97 1.99 228.72 −363.59 −236.09 −399.62 594.94 865

−52.09 −1.52 −78.45 223.09 −333.14 −137.58 −391.23 571.21 891.83

−145.30 −208.94 −135.47 136.72 −810.70 −390.04 −798.74 734.40 1259.89

−22.96 −10.16 −20.04 233.91 −379.48 −202.41 −398.92 599.46 899.24

B3LYP/apVTZ geometry isomer

LG

CCSD

CCSD/apVDZ geometry MVG

CCSD

CCSD

VGCCSD

MVGCCSD

−29.74 −6.15 −31.21 54.34 −129.90 −38.61 −154.88 180.05 289.41

−143.95 −209.18 −129.28 −36.77 −581.25 −248.85 −564.10 330.32 666.24

−21.01 −10.11 −13.44 57.73 −144.56 −58.71 −158.37 187.65 293.69

LG

LG


−25.42 −1.28 −12.19 75.18 −122.52 −52.06 −160.79 206.09 324.56

−28.80 5.51 −13.41 58.76 −118.42 −36.24 −158.40 190.79 295.58

isomer

LGB3LYP

LGCB3LYP

LGCCSD

VGCCSD

MVGCCSD

LGCCSD

VGCCSD

MVGCCSD


−22.67 −1.26 −10.81 63.2 −106.92 −44.20 −140.06 176.86 278.72

−25.35 4.54 −11.78 49.53 −102.92 −30.87 −137.37 163.79 253.85

−25.21 −4.25 −19.86 45.88 −108.94 −43.05 −132.32 154.22 240.94

−143.19 −192.79 −94.1 −51.66 −537.26 −263.61 −527.47 285.17 575.56

−18.46 −7.68 −4.68 48.07 −120.58 −59.39 −136.17 159.93 243.79

−26.24 −5.58 −27.15 45.72 −112.74 −32.86 −134.52 154.49 248.66

−141.12 −207.83 −127.37 −48.56 −556.55 −237.74 −537.41 295.89 613.10

−18.78 −9.05 −11.93 48.63 −125.34 −50.10 −137.59 160.95 252.45

−28.59 −4.66 −22.79 54.46 −125.49 −50.49 −152.4 179.73 280.35

VG

CCSD

−145.41 −20.69 −193.69 −8.58 −94.49 −5.06 −42.71 57.02 −555.72 −139.04 −273.75 −69.53 −548.12 −156.82 311.7 186.46 615.31 283.55 (c) 633 nm Response

LG

B3LYP/apVTZ geometry

CCSD/apVDZ geometry

a Isolated-molecule values of specific rotation computed for the nine low-lying isomers of R-GME ([α]eq λ,η) are tabulated for three discrete wavelengths: (a) 355 nm, (b) 589.3 nm, and (c) 633 nm. Independent results are shown for fully optimized B3LYP/apVTZ and CCSD/apVDZ equilibrium configurations, with attendant chiroptical analyses being performed by length-gauge implementations of B3LYP/apVTZ (LGB3LYP), CAM-B3LYP/apVTZ (LGCB3LYP ), and CCSD/apVDZ (LGCCSD) linear-response theory, as well as through the velocity-gauge (VGCCSD) and modified-velocity gauge (MVGCCSD) variants of the latter coupled-cluster scheme.

rotation for wavelength λ in a thermally equilibrated ensemble at temperature T to be formulated as: [α]Tλ =

∑ [α]eqλ ,η fη (T ) = ∑ [α]eqλ ,η η

η

e−ΔEη / kBT ϑ(T )

species8,10,16,17,29,47−50 and has proven to be surprisingly robust,50 especially when the underlying potential-energy surface topography presents substantial barriers between adjacent minima. Table 4 contains a subset of dispersive optical activity predictions obtained for a T = 300 K thermally equilibrated ensemble of isolated R-GME molecules by implementing the are conformer-averaging ansatz of eq 2. Estimates of [α]300K λ shown for five discrete wavelengths of interest, with analogous results for other choices of λ being deposited in the repository of Supporting Information. As depicted graphically in Figure 3, for a given energy metric (ΔEη or ΔGη) the ORD profiles emerging from the CAM-B3LYP (LGCB3LYP) linear-response framework are shifted uniformly downward (toward more negative rotatory powers) relative to their B3LYP (LGB3LYP) counterparts, thereby bringing them into better accord with the corresponding CCSD (MVGCCSD) curves. Composite calculations based upon G3 and CBS-APNO treatments are in

(2)

where kB is the Boltzmann constant and the quantity ϑ(T) = ∑ηe−ΔEη/kBT ensures overall normalization of the temperature-dependent population fractions such that ∑η fη(T) = 1. This approach can be implemented in a myriad of ways, reflecting the different schemes available for computing requisite specific-rotation parameters ([α]eq λ,η) and conformer stabilities, with free-energy (ΔGη) values that incorporate both enthalpic and entropic considerations often supplanting their internal-energy (ΔEη) counterparts for studies performed under ambient conditions. The independent-conformer or conformer-averaging ansatz of eq 2 has become the de facto standard for chiroptical analyses performed on flexible F


Article

The Journal of Physical Chemistry A Table 4. Predicted Optical Activity for Isolated R-GMEa specific optical rotation [deg dm−1 (g/mL)−1]

calculation type method

response

355 nm

435.83 nm

546.07 nm

589.3 nm

633 nm

B3LYP ΔEη B3LYP ΔEη B3LYP ΔEη

LGB3LYP LGCB3LYP MVGCCSD

9.059 −4.086 −7.736

−8.068 −12.069 −15.448

−9.716 −10.879 −13.364

−9.162 −9.927 −12.122

−8.481 −8.990 −10.936

B3LYP ΔE′η B3LYP ΔEη′ B3LYP ΔEη′


10.836 −2.565 −5.145

−7.038 −11.192 −13.982

−9.121 −10.371 −12.525

−8.663 −9.501 −11.420

−8.057 −8.627 −10.340

G3 ΔGη G3 ΔGη G3 ΔGη


1.273 −7.949 −12.888

−10.981 −13.219 −17.320

−10.813 −11.124 −14.016

−9.965 −10.047 −12.580

−9.086 −9.033 −11.267

G4 ΔGη G4 ΔGη G4 ΔGη


12.510 3.236 −3.022

−3.735 −6.136 −11.096

−6.324 −6.768 −10.196

−6.140 −6.340 −9.330

−5.790 −5.844 −8.471

CBS ΔGη CBS ΔGη CBS ΔGη


1.339 −7.557 −13.380

−10.755 −12.863 −17.473

−10.613 −10.866 −14.053

−9.783 −9.821 −12.601

−8.921 −8.834 −11.278

CCSD ΔEη CCSD ΔEη′

MVGCCSD MVGCCSD

−42.781 −36.603

−36.946 −33.202

−26.315 −24.073

−23.091 −21.194

−20.336 −18.712

−5.212 6.314

−5.103 4.803

2D torsion (ZPE) 2D torsion (full)

LGB3LYP LGB3LYP

18.622 53.666

Intrinsic specific rotations computed for a T = 300 K thermally equilibrated ensemble of isolated R-GME molecules are listed for five wavelengths. Most of the tabulated values reflect conformer-averaging analyses built on B3LYP/apVTZ (B3LYP) and CCSD/apVDZ (CCSD) equilibrium geometries, with the resulting internal energies (ΔEη and ΔEη′, the latter having zero-point offsets) being supplemented by free energy metrics (ΔGη) from composite G3, G4, and CBS-APNO calculations. Chiroptical properties were deduced from B3LYP/apVTZ (LGB3LYP), CAM-B3LYP/ apVTZ (LGCB3LYP), and CCSD/apVDZ (MVGCCSD) linear-response theory. The final entries, which stem from two-dimensional treatments of torsional motion, give a restricted vibrational-averaging prediction for dispersive optical activity [2D Torsion (Full)] and a conformer-averaging estimate based on zero-point expectation values [2D Torsion (ZPE)]. a

increase to −91.7% (+91.3%) and −91.3% (+87.8%) at 355 nm (633 nm) upon use of G3 and CBS-APNO free-energy estimates, respectively, the best agreement is realized by combining G4 predictions of ΔGη with LGB3LYP calculations eq to find differences of −18.5% at 355 nm and +21.9% at of [α]λ,η 633 nm. The analogous performance obtained for G4-based LGCB3LYP and MVGCCSD analyses at long wavelengths (633 nm) are +23.0% and +78.3%. The use of coupled-cluster chiroptical predictions (MVGCCSD) in conformer-averaging analyses based on either B3LYP/apVTZ internal-energy metrics (ΔEη and ΔE′η) or composite free-energy estimates (ΔGη) generates ORD profiles that are displaced markedly toward more negative rotatory powers. This situation is exacerbated by introducing the relative stabilities of conformers (ΔEη and ΔEη′ ) suggested by CCSD/ apVDZ geometry optimizations, with the bottom panel of Figure 3 illustrating the distinct wavelength-resolved forms assumed by the resulting values of specific rotation. Although this disparate behavior could reflect a breakdown of the independent-conformer assumption or inadequacies inherent to the underlying quantum-chemical methodology, it also might suggest the need to include effects beyond those arising from purely electronic contributions (vide inf ra). A similar situation exists for MeOX, where quantitative agreement between CRDP measurements and high-level calculations of intrinsic optical

excellent agreement with one another but depart significantly from those of G4 which consistently give more positive rotatory powers. The specific-rotation parameters computed for CCSDoptimized geometries are substantially larger in magnitude than those determined for B3LYP structures, displaying MVGCCSD ratios (i.e., for CCSD ΔEη/B3LYP ΔEη) of 5.53, 1.90, and 1.86 at 355, 589.3, and 633 nm, respectively. Introduction of zeropoint vibrational energy corrections (to obtain ΔE′η) causes the absolute optical rotation to decrease consistently by ∼10% throughout the visible portion of the spectrum; however, the DFT harmonic force field appears to imbue greater variability in the near-ultraviolet region. As illustrated in the top panel of Figure 3, all conformeraveraging procedures built upon the LGB3LYP linear-response framework correctly predict the sign of specific rotation measured for ambient R-GME vapor at 355 and 633 nm; however, the positive sign of this quantity in the near-ultraviolet (355 nm) is not reproduced uniformly by LGCB3LYP (middle panel of Figure 3) and MVGCCSD (bottom panel of Figure 3) treatments. The B3LYP/apVTZ internal-energy metric (ΔEη) in conjunction with LGB3LYP chiroptical parameters ([α]eq λ,n) underestimates the 355 nm rotatory power by 41.0% and overestimates the corresponding 633 nm value by 78.6%, with incorporation of vibrational zero-point corrections (ΔEη′ ) modestly improving results to give percentage deviations of −20.9% and +69.6%. Although deviations from experiment G


Article


requisite eigenvalues Ev and eigenfunctions ψv(Q), where v ≡ {v1, v2, ..., vNvib} and Q ≡ {Q1, Q2, ..., QNvib} represent the sets of quantum numbers and displacement coordinates specifying the Nvib total vibrational degrees of freedom available to the system. In particular, the distribution of probability density defined by ψv(Q) nominally extends over the entire conformational space but becomes highly localized (at least for a subset of v) proximate to the potential-energy minimum ascribed to the equilibrium configuration of a given isomer. Provided that the functional dependence of the wavelengthresolved specific rotation on molecular geometry is known, [α(Q)]λ, the characteristic response produced by a single vibrational eigenstate can be cast in terms of a quantummechanical expectation value: ⟨[α]λ ⟩v = ⟨ψv|[α(Q)]λ |ψv ⟩ =

∫Q ψv*(Q)[α(Q)]λ ψv(Q) dQ

(3)

where it has been assumed that ⟨ψv′|ψv⟩ = δv′v with the Kronecker delta symbol δv′v denoting unity if v′ = v and zero otherwise. The collective chiroptical properties for a thermally equilibrated ensemble of molecules follows from an averaging procedure akin to that in eq 2: [α]Tλ =

∑ ⟨[α]λ ⟩v fv (T ) = ∑ ⟨[α]λ ⟩v v

v

e−Ev / kBT q(T )

(4)

where the summation extends over all thermally accessible vibrational levels and the partition function q(T) = ∑ve−Ev/kBT, which defines the mean number of quantum states occupied at absolute temperature T, ensures overall normalization of the Boltzmann-weighted population fractions such that ∑v f v(T) = 1. The full vibrational-averaging ansatz embodied in eqs 3 and 4 usually represents a formidable task; however, it can be performed readily in reduced dimensionality provided that a salient subset of nuclear motions can be identified.19,28 A judicious choice for a restricted two-dimensional treatment of R-GME would appear to be the dual torsional degrees of freedom highlighted in Figure 2 that mediate transformations among the nine low-lying conformers. The ensuing analyses have adopted ∠O1C2C3O2 ≡ θ and ∠C2C3O2C4 ≡ ϕ as viable surrogates for describing large-amplitude isomerization pathways, enabling partial B3LYP/apVTZ geometry optimizations to be performed for fixed values of these angles as selected by making independent 10° increments over the range of −180° to +180°. The resulting array of 36 × 36 discrete energies was subjected to a periodic cubic-spline interpolation procedure to give the attendant potential-energy surface, V(θ,ϕ), which is depicted in the leftmost portion of Figure 4 in the form of a topographical map. The locations of stable isomers are marked and labeled by the previously introduced Roman-numeral indices (I−IX) whereas the superimposed patterns of concentric contour lines reflect the spread in probability density for the corresponding “zero-point” vibrational wavefunctions (vide inf ra). These local minima are separated from one another by barriers of different heights and shapes that ultimately govern the propensities for species to interconvert. For each partially relaxed configuration of the R-GME molecular framework, the B3LYP/apVTZ linear-response formalism (LGB3LYP) was deployed to predict rotatory powers at three wavelengths, 355, 589.3, and 633 nm. Continuous

Figure 3. ORD predictions for isolated R-GME. Dispersive optical activity measured for isolated R-GME molecules (circular symbols and error bars) is compared with calculations based on LGB3LYP (top panel), LGCB3LYP (middle panel), and MVGCCSD (bottom panel) implementations of linear-response theory. Most predictions stem from conformer-averaging procedures employing various energy metrics as described by the legends of the middle and bottom panels, with the top panel also displaying results of a restricted (twodimensional) vibrational-averaging ansatz [2D torsion (full)] and of a conformer analysis utilizing chiroptical properties averaged over nuclear zero-point motion [2D torsion (ZPE)].

activity only can be realized by explicitly taking nuclear degrees of freedom into consideration.14,19 d. Intrinsic Response by Vibrational Averaging. The influence of vibrational motion on dispersive optical activity cannot be discounted a priori, with several studies demonstrating the notable effects incurred even by zero-point displacements of otherwise rigid molecular frameworks.51−53 This especially is true in the case of small intrinsic optical activities (e.g., MeOX),14,19,20 which can allow vibrationally mediated processes to dominate over their usually much larger electronic counterparts. Quantitative descriptions of such phenomena are encumbered further by the presence of multiple conformers, each of which possesses its own manifold of vibrational states that couple strongly as barriers to interconversion are surmounted. Rigorous analyses require solution of the multidimensional nuclear Schrödinger equation to obtain H


Article


Figure 4. Restricted torsional analysis of R-GME. Two-dimensional relative potential-energy surface, V(θ,ϕ) (left panel), and 589.3 nm specificrotation function, [α(θ,ϕ)]D (right panel), for R-GME are shown in the form of topographical maps. These results follow from partial B3LYP/ apVTZ geometry optimizations performed at discrete (yet independent) values of the large-amplitude coordinates θ and ϕ, with optical activities being computed through use of the kindred LGB3LYP linear-response formalism. The location of each low-lying conformer is indicated and distinguished by the appropriate Roman-numeral label, and adjacent contours lines superimposed on V(θ,ϕ) reflect the spread of probability density for the attendant “zero-point” vibrational wavefunction.

because they typically are much smaller in magnitude than actual nuclear potential-energy terms.54 By neglecting overall rotation of the molecular framework and restricting attention to the two large-amplitude torsional displacements of interest, q1 ≡ θ and q2 ≡ ϕ, eq 5 can be simplified dramatically, with the three requisite elements of the metric tensor, gθθ, gθϕ = gϕθ, and gϕϕ, following directly from the array of B3LYP/apVTZ partialgeometry optimizations used to compute V̂ ≡ V(θ,ϕ) by exploiting the Gmat (ver. 2.1) software package of Muñoz-Caro and co-workers.55 To determine torsional eigenvalues and eigenfunctions for RGME, the two-dimensional isomerization Hamiltonian, Ĥ = T̂ + V̂ , was cast in a Fourier-product basis composed of complex exponential functions:

[α(θ,ϕ)]λ functions were generated by performing a periodic cubic-spline interpolation of these individual data points, with the rightmost portion of Figure 4 highlighting the contour plot (or topographical map) deduced for the case of sodium D-line (589.3 nm) excitation. The computed specific-rotation surfaces vary dramatically in magnitude and sign over the conformational space, tending to attain the largest absolute values in regions proximate to the highest-lying isomeric structures. The nuclear kinetic-energy operator, T̂ , employed for the present analyses of R-GME dynamics builds upon the basic formalism developed during our previous studies of 2-carene.28 Assuming the center-of-mass motion for an N-atom molecule has been separated, the general form of T̂ can be cast in terms of the remaining 3N − 3 curvilinear coordinates, {q1, q2, ..., q3N−3}, needed to describe rotational and vibrational degrees of freedom: ℏ2 T̂ ≈ − 2

3N − 3

ℏ2 2

3N − 3

=−

∑ j,k

∑ j,k

Φk , l (θ ,ϕ) = Φk (θ ) Φl (ϕ) =

∂ jk ∂ g ∂qj ∂qk ⎡ ⎛ jk ⎞ ⎤ 2 ∂g ⎟ ∂ ⎥ ⎢g jk ∂ + ⎜⎜ ⎟ ⎥ ⎢ ∂q ∂q ⎝ ∂qj ⎠ qk ⎦ j k ⎣

e ikθ e ilϕ π π

(6)

which are orthonormal over the full range of θ and ϕ coordinates: ⟨Φk ′ , l ′|Φk , l ⟩ = δk ′ , kδl ′ , l

(7)

The spans of integer indices k = 0, ±1, ±2, ..., nθ and l = 0, ±1, ±2, ..., ±nϕ were selected to ensure convergence of features residing below the 1000 cm−1 cutoff used for subsequent optical-activity analyses. Numerical diagonalization of the resulting (2nθ + 1)(2nϕ + 1)-dimensional square matrix Ĥ generated the requisite spectrum of energy eigenvalues and eigenvectors, the latter expressed in terms of expansion

(5)

where gjk denotes an element of the contravariant massweighted metric tensor, g, that specifies the conversion between rectangular-Cartesian and curvilinear-internal coordinates. In formulating this expression, pseudopotential contributions that scale in proportion to the determinant of g have been ignored I


Article

The Journal of Physical Chemistry A coefficients for the Φk,l(θ,ϕ) functions. The specific-rotation contributions for each vibrational state of interest were computed by means of eq 3, with the wavelength-resolved rotatory power, [α(θ,ϕ)]λ, also being cast in the Fourierproduct basis to facilitate evaluation of integrals over the conformational space.

for isomerization dynamics. The LGB3LYP rotatory powers estimated for R-GME(I) by isolated-conformer (Table 3) and zero-point-averaged (Table 5) calculations are very similar; however, higher-lying forms show more pronounced differences, including wavelength-dependent changes in sign for species II, III, VII, and VIII. Such behavior is reflected in the collective chiroptical response predicted at T = 300 K by replacing [α]eq λ,η in eq 2 with the analogous ⟨[α]λ⟩υη=0 quantity from Table 5, which causes the 355 nm specific rotation to nearly double in magnitude relative to that obtained from B3LYP/apVTZ independent-conformer analyses whereas those for 589.3 and 633 nm excitation are nearly halved. Indeed, as depicted in the top panel of Figure 3, this hybrid approach affords the best agreement with CRDP measurements of intrinsic (vapor-phase) optical activity, yielding percentage deviations of +21.3% at 355 nm and +7.43% at 633 nm. Proper implementation of the vibrational-averaging ansatz demands consideration of R-GME torsional eigenstates beyond those ascribed to the zero point of each conformer, with an enumerated list of features residing below 250 cm−1 relegated to the repository of Supporting Information. The present chiroptical analyses have taken into account all levels spanning the 0 ≤ ΔEv ≤ 1000 cm−1 range, where ΔEv is defined relative to the minimum energy emerging from diagonalization of the model Hamiltonian and the fractional population at T = 300 K of the highest-lying state included in the summation of eq 4 amounts to 0.83% of that for its lowest-lying counterpart [i.e., f v(T=300 K) factors of 2.85 × 10−4 vs 3.44 × 10−2]. Predicted rotatory powers for three discrete wavelengths are compiled in Table 4 and depicted in the top panel of Figure 3, revealing a nearly 6-fold increase in the magnitude of response at 355 nm (relative to the independent-conformer treatment based on B3LYP/apVTZ geometries and LGB3LYP response) as well as an inversion of signs at both 589.3 and 633 nm. When compared to vapor-phase measurements of intrinsic optical activity, the vibrationally averaged ORD profile seems to be shifted uniformly upward (toward more positive values of [α]Tλ ) leading to wavelength-resolved absolute deviations between theory and experiment in excess of 200%. Although such glaring discrepancies could be construed as evidence for shortcomings inherent to the two-dimensional description of nuclear motion, they also might imply that vibrational effects in isolated R-GME molecules are substantial (as found in MeOX)14,19 with purely electronic contributions to chiroptical

Table 5. Isomer Zero-Point Levels from Two-Dimensional Torsional Analysisa two-dimensional torsional wavefunction

zero-point averaged specific optical rotation [deg dm−1 (g/ml)−1]

isomer level

energy (cm−1)

355 nm

I (υI = 0) II (υII = 0) III (υIII = 0) IV (υIV = 0) V (υV = 0) VI (υVI = 0) VII (υVII = 0) VIII (υVIII = 0) IX (υIX = 0)

0 10.528 43.628 196.393 322.746 403.090 543.721 560.166 823.703

−29.995 17.463 2.851 309.436 −256.370 −105.534 663.768 −387.224 989.97

589.3 nm −25.691 2.033 −5.619 73.819 −119.331 −53.429 209.794 −164.652 321.163

633 nm −22.889 1.574 −5.193 62.057 −104.214 −45.402 180.006 −143.285 275.809

a

Relative energies extracted from a two-dimensional treatment of isomerization dynamics are listed for vibrational states assigned to the “zero-point” level of each low-lying conformer in isolated R-GME molecules. The attendant estimates of dispersive optical activity at three wavelengths follow from explicit averaging of B3LYP/apVTZ (LGB3LYP) linear-response calculations over the corresponding “zeropoint” wavefunction.

Before examining results of the vibrational-averaging ansatz defined in eq 4, it proves useful to consider the effects of nuclear displacement on individual conformers. Table 5 summarizes the relative energies and optical activities computed for two-dimensional torsional states, ψv(θ,ϕ), nominally ascribed to the vibrationless (υη = 0) levels for each low-lying isomer of R-GME, with the contour lines superimposed on the topographical potential-energy surface in the leftmost portion of Figure 4 illustrating the attendant distributions of probability density. The listed zero-point energy shifts are in excellent accord with their full-dimensionality B3LYP/apVTZ harmonic counterparts (ΔEη′ ) in Table 2 (absolute deviations of 300 cm−1 and ΔG > 150 cm−1) isolating it from the next lowest-lying conformation.

The solvent-optimized geometries for each R-GME conformer were subject to PCM-based analyses of chiroptical properties performed at both the B3LYP/apVTZ (LGB3LYP) and the CAM-B3LYP/apVTZ (LGCB3LYP) levels of linearresponse theory. Tables 8 and 9 contain [α]eq λ,η values predicted in cyclohexane and acetonitrile media for 355, 589.3, and 633 nm excitation, with analogous results for other wavelengths and solvents being compiled in the Supporting Information. Also tabulated are estimates for the phase dependence of optical activity, Δαλ,η, which give the fractional change in rotatory power accompanying “solvation”: Δαλ , η =

[α]eq,sol − [α]eq,gas λ,η λ,η [α]eq,gas λ,η

(8)

where [α]eq,gas and [α]eq,gas denote specific-rotation parameters λ,η λ,η evaluated for a given isomer (η) and wavelength (λ) in the presence (solvated) and absence (gaseous) of PCM solvation. In particular, this quantity permits solvent-induced effects to be separated into contributions arising from changes in nuclear geometry and electronic response, with a large value of |Δαλ,η| suggesting the latter to be of importance. L


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The Journal of Physical Chemistry A Table 11. Predicted Optical Activity for R-GME in Acetonitrilea specific optical rotation [deg dm−1 (g/mL)−1]

calculation type method

response

355 nm

435.83 nm

546.07 nm

589.3 nm

633 nm

B3LYP ΔEη B3LYP ΔEη

LGB3LYP LGCB3LYP

12.595 −4.924

−8.237 −14.648

−10.778 −13.240

−10.255 −12.085

−9.552 −10.954

G3 ΔGη G3 ΔGη

LGB3LYP LGCB3LYP

16.612 3.301

−3.552 −8.085

−7.153 −8.714

−7.028 −8.139

−6.683 −7.495

G4 ΔGη G4 ΔGη

LGB3LYP LGCB3LYP

21.484 7.709

−0.659 −5.422

−5.436 −7.119

−5.579 −6.790

−5.443 −6.339

CBS ΔGη CBS ΔGη

LGB3LYP LGCB3LYP

16.363 2.755

−3.775 −8.462

−7.314 −8.957

−7.170 −8.349

−6.808 −7.677

a

Specific-rotation values computed for a T = 300 K thermally equilibrated ensemble of R-GME molecules entrained in an implicit acetonitrile medium are given at five discrete excitation wavelengths. These conformer-averaged predictions follow from equilibrium configurations optimized by density-functional (B3LYP/apVTZ ) methods in the presence of a PCM treatment for solvation, with the resulting values of internal energy (ΔEη) being supplemented by free-energy metrics (ΔGη) deduced from composite calculations (G3, G4, and CBS-APNO). Chiroptical properties were estimated from the length-gauge implementations of both B3LYP/apVTZ (LGB3LYP) and CAM-B3LYP/apVTZ (LGCB3LYP) linear-response theory.

For a given incident wavelength, the difference between rotatory powers estimated for an individual R-GME conformer by PCM-solvated LGB3LYP and LGCB3LYP methods are nearly identical to those found in analyses of the isolated molecule (cf. Table 3). Although the Δαλ,η parameters defined in eq 8 typically show the influence of solvation upon electronic response to be ≪10% in magnitude, substantially larger effects exist for isomers I and II, both of which are expected to contribute heavily to the collective behavior observed under ambient thermal conditions. Comparison of solvent-free calculations (Table 3) performed at the sodium D-line (589.3 nm) with their solvated counterparts (Tables 8 and 9) usually reveals only modest changes in the optical activity predicted for each species of interest, with acetonitrile presenting more pronounced deviations than cyclohexane due presumably to greater variations in equilibrium configurations incurred by the polar medium. In contrast, the ORD profile for R-GME(II) seems to be shifted upward (toward more positive rotatory powers) by solvation such that the slightly negative B3LYP vapor-phase values of [α]eq analyses at λ,II suggested by LG 589.3 and 633 nm are inverted in sign and increased dramatically in size by either C6H12 or CH3CN PCM treatments. This is in keeping with the |Δαλ,η| metrics computed for isomer II, which are almost 100-fold larger than those obtained for other forms of R-GME. The relative energies (ΔEη and ΔGη) and optical activities eq ) obtained from PCM calculations were combined ([α]λ,η through the conformer-averaging ansatz of eq 2 to determine the collective behavior expected for a thermally equilibrated ensemble of implicitly solvated R-GME molecules. Tables 10 and 11 contain a subset of predictions obtained for cyclohexane and acetonitrile at T = 300 K, with similar findings for other wavelengths and solvents being relegated to the repository of Supporting Information. As depicted graphically in Figure 5, the ORD profiles emerging from LGCB3LYP linear-response theory are displaced uniformly downward (toward more negative rotatory powers) relative to their LGB3LYP counterparts for each choice of energy metric, leading to a near-constant difference between LGB3LYP and LGCB3LYP specific-rotation values at a given wavelength. Although the ΔGη parameters emerging from G3 and CBS-APNO composite calculations yield dispersion curves that essentially overlap, the analogous

Figure 5. ORD predictions for solvated R-GME. Wavelength dependence of dispersive optical activity predicted for R-GME is compared with experimental measurements performed in dilute cyclohexane (top panel) and acetonitrile (bottom panel) solutions. Calculations stem from a conformer-averaging procedure implemented with PCM treatments of implicit solvation by combining various estimates of relative conformer energies (distinguished by different symbols as described in legends) with either LGB3LYP (solid curves) or LGCB3LYP (dashed curves) linear-response calculations of chiroptical properties.

G4 results are shifted strongly upward (toward more positive rotatory powers). At long wavelengths in either C6H12 or CH3CN solutions, the best agreement between PCM theory M


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Figure 6. Comparison of predictions for R-GME optical activity. Fractional deviation between theoretical and experimental values of specific optical rotation at 589.3 nm is displayed for the six solvents of interest. Predictions stem from T = 300 K conformer-averaging analyses that employed implicit PCM treatments of solvation. Successive bars shown for each medium distinguish the level of agreement attained by utilizing internal-energy (ΔEη) or free-energy (ΔGη) metrics derived from electronic-structure (B3LYP/apVTZ) or composite (G3, G4, and CBS-APNO) calculations combined with B3LYP/apVTZ (LGB3LYP) and CAM-B3LYP/apVTZ (LGCB3LYP) linear-response predictions of chiroptical properties.

specific rotation in the vapor phase were performed at 355 and 633 nm whereas analogous solution-phase studies made use of six solvents probed at discrete visible and ultraviolet wavelengths. Interpretation of experimental results is encumbered by the extreme flexibility of R-GME, which leads to nine low-lying conformers that possess distinct physicochemical and chiroptical properties. These species are distinguished by different orientations of the −CH2OCH3 appendage on the epoxide ring, as specified by two large-amplitude torsional coordinates, ∠O 1 C 2 C 3 O 2 ≡ θ and ∠C 2 C 3 O 2 C 4 ≡ ϕ. For each isomer (η = I−IX) contributing to vapor-phase behavior, internal energies without (ΔEη) and with (ΔEη′ ) harmonic zeropoint vibrational correction were computed at geometries fully optimized by density-functional (B3LYP/apVTZ) and coupledcluster (CCSD/apVDZ) methods, with composite calculations (G3, G4, and CBS-APNO) affording estimates of relative freeenergy values (ΔGη). Requisite linear-response predictions of wavelength-resolved optical activity for the isolated equilibrium eq ) exploited the length-gauge configurations of interest ([α]λ,η formulations of B3LYP/apVTZ (LGB3LYP) and CAM-B3LYP/ apVTZ (LGCB3LYP), as well as the modified velocity-gauge treatment of CCSD/apVDZ (MVGCCSD). The collective response evoked from a T = 300 K thermally equilibrated ensemble of isolated R-GME molecules was computed initially by means of a conformer-averaging ansatz, where the predicted specific rotation for each participating eq ) was weighted by Boltzmann population isomer ([α]λ,η fractions derived from various energy metrics (ΔEη, ΔEη′ , or ΔGη). Such treatments uniformly gave ORD profiles that were displaced downward (toward more negative rotatory powers) relative to vapor-phase measurements, with the best agreement between experiment and theory being realized by using G4 values of free energy in conjunction with LGB3LYP estimates of conformer properties. To assess the putative roles of nuclear motion, a restricted (two-dimensional) vibrational-averaging procedure was implemented by combining the eigenvalues and

and polarimetric measurements is realized by combining B3LYP/apVTZ estimates of ΔEη with LGCB3LYP predictions of chiroptical properties; however, the G3 and CBS-APNO analyses offer slightly better performance for near-ultraviolet portions of the cyclohexane data. A graphical summary of R-GME solution-phase results is provided in Figure 6, where the fractional deviation between computed and measured specific-rotation values at 589.3 nm is shown for all six solvents targeted by the current work. In particular, the eight quantum-chemical analyses depicted for each medium stem from conformer-averaging procedures implemented by combining four distinct sets of isomer energies (B3LYP/apVTZ, G3, G4, and CBS-APNO) with two different linear-response methods (LGB3LYP and LGCB3LYP). Excluding chloroform, where the small magnitude of optical rotation leads to more erratic behavior, theoretical ORD profiles are displaced uniformly upward (toward more positive rotatory powers) relative to their experimental counterparts. The LGB3LYP predictions based on B3LYP (ΔEη), G3 (ΔGη), G4 (ΔGη), and CBS-APNO (ΔGη) energy metrics yield root-mean-square deviations between calculated and observed chiroptical properties of 5.41, 6.37, 9.10, and 6.35 deg dm−1 (g/mL)−1, respectively, with alternative use of LGCB3LYP parameters reducing these errors to 4.56, 5.85, 8.37, and 5.77 deg dm−1 (g/mL)−1. The consistently poor performance realized for toluene might suggest a more systematic cause, in keeping with kindred studies of MeOX in benzene which have implicated a solute-imprinted chiral solvation shell that dominates over the chiroptical response of the lone solute molecule.25

IV. SUMMARY AND CONCLUSIONS The dispersive optical activity of a small nonrigid chiral epoxide, (R)-(−)-glycidyl methyl ether (R-GME), has been probed at ambient temperatures under complementary isolated and solvated conditions. Isolated-molecule measurements of N


Article


energies and structural parameters for R-GME isomers in PCM solvents; (Table S7) predicted chiroptical response for solvated R-GME isomers; (Table S8) predicted optical activity for solvated R-GME. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.jpca.5b05177.

eigenstates deduced from a relaxed B3LYP/apVTZ potentialenergy surface, V(θ,ϕ), with analogous LGB3LYP specificrotation functions, [α(θ,ϕ)]λ, to allow for evaluation of stateresolved contributions to optical activity. This approach led to a substantial positive displacement of the predicted ORD curve, strongly suggesting that vibrational effects dominate over their electronic counterparts in the dispersive chiroptical behavior of vapor-phase R-GME, as reported previously for the case of methyoxirane (MeOX).14,19 A more detailed investigation of such phenomena, including two-dimensional analyses of potential-energy and optical-activity topography performed at potent coupled-cluster levels of theory, will be the subject of a forthcoming publication; however, further studies clearly are needed to assess the influence that small-amplitude (vibrational) and large-amplitude (dynamical) nuclear degrees of freedom exert upon intrinsic chiroptical behavior. Solute−solvent coupling perturbs the dispersive optical activity of R-GME profoundly, affecting both the magnitude and the sign of the wavelength-resolved response measured in the six solvents examined during the current study. Interpretation of these findings has relied on a conformeraveraging ansatz in which the polarizable-continuum model (PCM) for implicit solvation was deployed to compute relativeenergy metrics (ΔEη and ΔGη) and specific-rotation parameters ([α]eq λ,η) for isomer configurations optimized at the B3LYP/ apVTZ level of theory. Although semiquantitative agreement was obtained for sodium D-line (589.3 nm) excitation in the extreme cases of solvent polarity represented by cyclohexane and acetonitrile, the PCM treatment failed to reproduce the overall shape of attendant ORD profiles successfully, thereby indicating that effects beyond those arising from purely electrostatic interactions might need to be considered. Inspection of the 589.3 nm rotatory powers predicted for all solvents of interest showed the best accord with polarimetric measurements to be realized by combining B3LYP/apVTZ internal-energy estimates (ΔEη) with CAM-B3LYP/apVTZ linear-response calculations (LGCB3LYP), where the resulting root-mean-square deviation from experiments of 4.56 deg dm−1 (g/mL)−1 (taken across all solvents) increased modestly to 5.41 deg dm−1 (g/mL)−1 upon use of the alternative LGB3LYP formalism. The notably poorer performance found for an aromatic toluene medium hints that a different mechanism might need to be evoked to fully explain the observed chiroptical behavior observed in this low-polarity medium, including putative formation and participation of an imprinted chiral solvation shell as has been implicated for MeOX entrained in benzene.25 Further elaboration of this assertion, as well as a comprehensive understanding of solventinduced changes in the intrinsic (vapor-phase) optical activity of R-GME, must await more sophisticated analyses that incorporate explicit coupling among solute and solvent degrees of freedom.

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AUTHOR INFORMATION

Corresponding Author

*P. H. Vaccaro. Electronic mail address: patrick.vaccaro@yale. edu. Telephone/fax number: (203) 432-3975/(203) 432-6144. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Dr. James R. Cheeseman (Gaussian, Inc.) and Prof. Marco Caricato (University of Kansas) for insightful discussions on chiroptical properties. The work described in this paper was completed under the auspices of grant CHE1112239 awarded by the Chemical Structures, Dynamics, and Mechanisms (CSDM-A) Program in the Directorate for Mathematical and Physical Sciences of the United States National Science Foundation, the continuing support of which is gratefully acknowledged. Computational efforts 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 acquisition of requisite computer facilities.

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REFERENCES

(1) Barron, L. D. Molecular Light Scattering and Optical Activity, 2nd ed.; Cambridge University Press: Cambridge, U.K., 2004. (2) Vaccaro, P. H. Optical Rotation and Intrinsic Optical Activity. In Comprehensive Chiroptical Spectroscopy; Berova, N., et al., Eds.; John Wiley & Sons, Inc.: Hoboken, NJ, 2012; Vol. 1, Chapter 11, pp 275− 323. (3) Eliel, E. L.; Wilen, S. H.; Doyle, M. P. Basic Organic Stereochemistry; John Wiley and Sons, Inc.: New York, 2001. (4) Pecul, M.; Ruud, K. The Ab Initio Calculation of Optical Rotation and Electronic Circular Dichroism. Adv. Quantum Chem. 2005, 50, 185−212. Crawford, T. D. Ab Initio Calculation of Molecular Chiroptical Properties. Theor. Chem. Acc. 2006, 115, 227−245. Crawford, T. D.; Tam, M. C.; Abrams, M. L. The Current State of Ab Initio Calculations of Optical Rotation and Electronic Circular Dichroism Spectroscopy. J. Phys. Chem. A 2007, 111, 12057−12068. Autschbach, J. Computing Chiroptical Properties with First-Principles Theoretical Methods: Background and Illustrative Examples. Chirality 2009, 21, E116−E152. (5) Polavarapu, P. L. Optical Rotation: Recent Advances in Determining the Absolute Configuration. Chirality 2002, 14, 768− 781. Polavarapu, P. L. Erratum: Optical Rotation: Recent Advances in Determining the Absolute Configuration. Chirality 2002, 15, 284−285. Polavarapu, P. L. Renaissance in Chiroptical Spectroscopic Methods for Molecular Structure Determination. Chem. Rec. 2007, 7, 125−136. Polavarapu, P. L. Why is it Important to Simultaneously Use More Than One Chiroptical Spectroscopic Method for Determining the Structures of Chiral Molecules. Chirality 2008, 20, 664−672. (6) Kumata, Y.; Furukawa, J.; Fueno, T. Effect of Solvent on the Optical Rotation of Propylene Oxide. Bull. Chem. Soc. Jpn. 1970, 43, 3920−3921. (7) Wiberg, K. B.; Wang, Y.-G.; Wilson, S. M.; Vaccaro, P. H.; Jorgensen, W. L.; Crawford, T. D.; Abrams, M. L.; Cheeseman, J. R.; Luderer, M. Optical Rotatory Dispersion of 2,3-Hexadiene and 2,3Pentadiene. J. Phys. Chem. A 2008, 112, 2415−2422.

ASSOCIATED CONTENT

S Supporting Information *

Aside from full reference citations (section S1), the following compilations of supporting material have been included: (Table S1) regression parameters from least-squares analyses of ORD profiles; (Table S2) optimized geometries for isolated R-GME conformers; (Table S3) predicted chiroptical response for isolated R-GME conformers; (Table S4) predicted optical activity for isolated R-GME; (Table S5) low-lying levels from restricted vibrational analysis of isolated R-GME; (Table S6) O


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Intrinsic Optical Activity and Large-Amplitude Displacement

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