Observation and calculation of vibrational circular birefringence: A new form of vibrational optical activity

Authors


  • Contribution to the Special Thematic Project “Advances in Chiroptical Methods”

Abstract

We report the first mid-infrared observation of vibrational circular birefringence (VCB) arising from individual chiral molecules. VCB can also be called vibrational optical rotatory dispersion (VORD) and is the Kramers-Kronig transform of vibrational circular dichroism (VCD). The method of measurement involves a simple change in the optical set-up and electronic processing of a VCD spectrometer such that the VCB spectrum appears at twice the polarization modulation frequency as a pseudo vibrational linear dichroism (VLD) spectrum. VCB spectra are also calculated with density function theory (DFT) for the first time using a commercially available program for rotational strengths where the calculated intensities are convolved with the real, dispersive part of a normalized complex Lorentzian lineshape rather than the imaginary, absorptive part, normally used for IR and VCD intensity calculations. Comparison of themeasured and calculated VCB, VCD, and IR spectra of (+)-R-limonene and (−)-S-α-pinene show close agreement and confirm the validity of the new VCB measurements. Chirality 21:E277–E286, 2009. © 2009 Wiley-Liss, Inc.

INTRODUCTION

The first observation of optical activity dates back to the publication by Arago in 1811, who observed colors in light transmitted along the optic axis of a quartz crystal positioned between two crossed polarizers.1 Arago did not explain the origin of the colors, but extensive studies of polarization phenomena by Biot published in the following year did explain the colors as arising from the rotation of the plane of polarization of the incident light beam by the quartz crystal.2 In addition, he found that the rotation was greatest for the color violet and decreased in magnitude through blue, green, to yellow, and was the smallest for red. He, thus, simultaneously discovered the phenomena of optical rotation (OR) at a single wavelength, and optical rotatory dispersion (ORD) for different degrees of OR as a function of wavelength. In 1818, Biot published an inverse square law for the wavelength dependence of ORD,3 which was subsequently improved by Drude in 1902,4 and is known today as Drude's Law.5, 6 In 1836, Biot and Melloni first described optical rotation in the infrared region of the spectrum from quartz crystal.7 In the present article, nearly two centuries after the first observations of OR, and over 170 yr following the first observation of infrared OR, we report, and confirm theoretically, the first direct observation of mid-infrared ORD for vibrational transitions in a chiral substance.

Vibrational optical activity (VOA)6, 8–11 is comprised of two principal areas, infrared vibrational circular dichroism (VCD)12 and vibrational Raman optical activity (ROA),13, 14 based on infrared absorption spectroscopy and Raman scattering spectroscopy, respectively. Both arise from a differential response with respect to left (LCP) and right circularly polarized (RCP) radiation for a chiral molecule undergoing a vibrational transition. VCD is observed in the mid-IR and near-IR spectral regions,15 whereas ROA is observed in the visible region of the spectrum, except for a recent report using near-IR laser excitation.16

There are a number of different forms of VOA.17 There are four forms of circular polarization (CP) ROA depending on whether CP intensity differences in Raman scattering are taken with respect to the incident laser radiation, the scattered Raman radiation, or both either in-phase (RR minus LL) or out-of-phase (RL minus LR). Each form of CP ROA has various relative advantages and disadvantages, and in general, particularly as resonance is approached, each contains a unique combination of invariants that contribute to the observed intensity.18 There is only one form of CP-VCD, namely the difference in absorbance for LCP and RCP infrared radiation.

Less well known, and still unobserved experimentally, are four analogous forms of linear polarization (LP) ROA.17, 19 These are comprised of Raman intensity differences with respect to LP states of incident and scattered radiation that are at ±45° with respect to a vertical or horizontal axis along the path of the radiation. LP-ROA is predicted to be observable only under conditions of resonance Raman scattering.

In this article, we demonstrate experimentally, theoretically, and computationally, that a form of infrared VOA exists that can be measured as a ±45°-LP difference measurement analogous to that of LP-ROA. It could be called LP-VCD, similar to LP-ROA, but this form of infrared VOA is simply vibrational optical rotatory dispersion (VORD), or alternatively and preferably, vibrational circular birefringence (VCB), the difference in the real part of the infrared refractive index for LCP and RCP radiation. To measure VCB, the VCD spectrometer is set up to measure vibrational linear dichroism (VLD), which for a sample of an isotropic liquid or solution is simply zero, but a second polarizer is added after the sample at 45° from vertical, which cancels stray VLD that may be present, but allows a VLD signal to be measured if the sample rotates plane polarized light through positive or negative angles, the sign of the measured VLD depending on the sign of the rotation. In this way, infrared optical rotation can be measured across the region of vibrational transitions. A normal VCD spectrometer can be converted between VCD and VCB operation in a few minutes provided one has a second IR linear polarizer for placement after the sample.

We report here the first mid-infrared measurements of VCB/VORD within the absorption bands of individual chiral molecules, namely infrared VCB Cotton effects. Earlier reports of near-IR ORD for polymer samples have been published in which hints of Cotton effects further in the infrared region were inferred from the shape of the measured ORD spectrum, but not directly observed.20, 21 In addition, infrared ORD has been reported for liquid crystal samples. These IR-ORD spectra are orders of magnitude larger than VCB arising from individual chiral molecules and were measured without the sensitive instrumentation used for traditional VCD measurement.22

The VCB reported here, in particular, for (+)-R-limonene and (−)-S-α-pinene, are related to their VCD spectra by Kramers-Kronig transforms.6 Thus, VCB spectra contain useful new information if one is interested in the refractive index properties of a chiral molecule or sample. On the other hand, if one has the entire VCD spectrum, the VCB intensity at any spectral location can be determined from the Kramers-Kronig transform. As is well known, the same set of rotational strengths is the source of both the VCD and VCB spectra of a molecule, and in that sense, a VCB spectrum is, relative to its corresponding VCD spectrum, simply an alternative experimental observable of the same theoretically-based information.

The results of this article were presented previously in preliminary form at the First International Conference on Vibrational Optical Activity in August, 2008.23 We note that very recently, VORD in the near-infrared carbon–hydrogen stretching region was reported by Cho and co-workers using a new femtosecond approach to VOA measurement.24

THEORY

The theoretical formulation of infrared VCB is presented here in relation to the corresponding theory of VCD. In the most general sense, the complex index of refraction, ñ(ν), of a material has a real part, n(ν), which is simply called the refractive index or the index of refraction and an imaginary part which is the absorption index k(ν) or loss index of the material. Thus, one can write, as a function of radiation frequency, ν, that

equation image(1)

Here and below, a complex quantity is designated by an overscript tilda. The second equation expresses ñ(ν) using an alternative convention in which a single prime corresponds to real part and a double prime to the imaginary part of the complex quantity. Throughout this article, we use the convention with single and double primes since it will be applied to quantities, both the real and imaginary parts, which do not otherwise have conventional symbols.

The effect on a radiation beam I0( equation image) that is incident on a medium of complex refractive index ñ(equation image) and pathlength l in cm, and radiation wavenumber frequency, equation image = 1/λ in cm−1, is expressed by

equation image(2)

The attenuation of the beam due to absorption is given by substitution of eq. 1 into eq. 2 and keeping the absorptive part

equation image(3)

The refractive index is a unitless, bulk property of the medium dependent on the mass density. By contrast, absorption properties of molecules are typically expressed in terms of molar absorptivity, ϵ( equation image), and defined by the expression

equation image(4)

where C is the concentration of the solution in mol/l, and by analogy to the refractive index, the molar absorptivity can be written as a complex quantity, namely

equation image(5)

The second equality in eq. 4 relates the molar absorptivity to the decadic absorbance, usually written as A( equation image) but here written as A″(equation image) to emphasize that the absorbance, is related to the imaginary part of a complex absorbance function, namely

equation image(6)

which, corresponding to eq. 5, has both a real dispersive part and an imaginary absorptive part. As shown below, eq. 6 represents the measured quantity and can be equated to equation image as equation image/Cl provided that Beer-Lambert's law, specified in eq. 4, is valid. Comparing eqs. 3 and 4, the conversion between refractive index absorption and molar absorptivity is given by

equation image(7)

Infrared (IR) absorption and VCD spectra can be simulated from the quantum mechanical expressions for the dipole and rotational strengths, Di and Ri, respectively, where

equation image(8)

and μi is the vibrational electric dipole transition moment for normal mode i and mi is the corresponding magnetic dipole transition moment. One can define VCD in terms of the molar absorptivity as Δϵ″( equation image) = ϵ″L(equation image) − ϵ″R(equation image), and write the standard equations for IR absorption and VCD spectra in terms of dipole and rotational strength, respectively,

equation image(9)
equation image(10)

where N is Avagadro's number, h is Planck's constant, and c is the speed of light. Here, fmath image( equation image) is the normalized Lorentzian lineshape function for mode i applied to each dipole or rotational strength and summed over the normal modes of the desired range of vibrational frequencies to simulate the measured spectrum.

equation image(11)

The Lorentzian lineshape function can also be written in complex form as

equation image(12)

where the imaginary part is the normalized Lorentzian absorption lineshape function eq. 11, and the real part is the corresponding normalized Lorentzian dispersive lineshape. The absorption and dispersive lineshapes are plotted in Figure 1 where it can be seen that the dispersive line shape is zero at the resonance frequency and has a minimum and a maximum at the half maximum points of the absorption lineshape.

Figure 1.

Real dispersive (blue trace) and corresponding imaginary absorptive (red trace) parts of the normalized complex Lorentzian line shape function used to convolve the calculated rotational and dipole strengths to produce the calculated VCB, VCD, and IR spectra. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

It can be shown that corresponding absorption and dispersive Lorentzian lineshapes are Kramers-Kronig transforms of each other. It follows that the VCB spectrum associated with a particular VCD spectrum can be constructed from the same set of rotational strengths by simply changing the lineshape function from an absorptive Lorentzian to the corresponding dispersive Lorentzian as

equation image(13)

where fmath image( equation image) is given in eq. 12 as

equation image(14)

Using eq. 7, the VCB spectrum can be expressed as a refractive index difference quantity that depends on the concentration of molecules, and hence, the effective mass density of the medium, as

equation image(15)

MEASUREMENT FORMALISM

The description of the optical-electronic configuration for the measurement of VCB spectra is described below in terms of Stokes vectors and Mueller matrices.6, 17, 25 We begin with the description of VCD measurement and show how this configuration forms the basis for the measurement of VCB spectra.

A Stokes vector is a four-component vector with entries from top to bottom representing the total intensity, the excess intensity for vertically polarized minus horizontally polarized light, followed by the corresponding intensity difference for ±45° linearly polarized light, and last the excess intensity for right minus left circularly polarized light. The Stokes vector for unpolarized light with a total intensity I0( equation image) is given by

equation image(16)

A Mueller matrix is a 4 × 4 matrix that transforms one Stokes vector into another as the result of the light beam passing through or interacting with an optical element. If the Mueller matrix of a vertical polarizer, MP(0°), is applied to the Stokes vector in eq. 16 as

equation image(17)

one obtains the Stokes vector, S1( equation image), of a vertically polarized light beam reduced in intensity by one-half. For a pure polarized state of light, the sum of the bottom three Stokes vector elements equals the top element.

VCD Measurement

The minimum optical set up for VCD measurement is source (S), vertical linear polarizer (P), photoelastic modulator (PEM), sample (X), and detector (D). The Stokes vector for each stage in the optical train is obtained by applying the Mueller matrix for that stage to the previous Stokes vector. The Mueller matrix for the PEM can be written in terms of Bessel functions through second order in the PEM modulation frequency as

equation image(18)

This matrix describes the action of the PEM on a polarized light beam. The two off-diagonal terms, 2J1math image( equation image)], are associated with conversion of vertical or horizontal linear polarization to circular polarization at the PEM modulation frequency. The diagonal terms, J0math image(equation image)] and 2J2math image(equation image)], describe the effect of the PEM on the non-PEM modulated intensity and on the intensity modulated between vertical and horizontal polarized radiation at twice the PEM frequency, respectively. The terms 2J1math image(equation image)] are associated with VCD measurement while the terms 2J2math image(equation image)] are associated with the measurement of VLD, and in the present application, VCB. The Stokes vector emerging from the modulator is given by

equation image(19)

The next optical element is the sample of an isotropic chiral medium or solution of chiral molecules. The Mueller matrix of such a sample is given by

equation image(20)

The overall intensity is diminished by the absorbance pre-factor, 10A(equation image). The effect of CD on the transmitted intensity, given by CD(equation image) = ln(10)ΔA″(equation image)/2, is to modulate the total intensity at the PEM frequency, and the effect of CB, equal to CB(equation image) = 2πΔn′(equation image)lequation image, does not involve the total intensity, but coverts vertical-horizontal light to ±45° polarized light by the continuous rotation of the plane of polarization of the light as the beam passes through the sample. The CD and CB of the sample in units of absorbance and refractive index are defined, respectively, as

equation image(21)

The Stokes vector after the sample is given by

equation image(22)

The term J0math image( equation image)]CB(equation image) appearing in the third component element of the Stokes vector has been dropped since it only contributes constant (non-modulated) intensity and is very small relative to 1 (unity) of the total intensity expressed by first component of the Stokes vector. If the intensity of the beam is now measured by a detector, the final intensity is simply the total-intensity component of the final Stokes vector since the light beam and all polarization information terminates at the detector. This is accomplished formally by the Mueller matrix for the detector, assumed to have no linear or circular polarization bias, is simply the row vector

equation image(23)

which when multiplied into a Stokes column vector produces a single element scalar intensity. The total intensity at the detector is, therefore, given by

equation image(24)

Here, the first term with pre-factor 10−A″(equation image) is the DC-term, IDC(0,equation image), whereas the second one, IACM,equation image), carrying the VCD intensity is the AC-term. The AC-term is isolated from the DC-term by a lock-in amplifier tuned to the PEM frequency, ωM. The ratio of the AC- and DC-terms, and use of CD(equation image) = ln(10)ΔA″(equation image)/2, allows isolation of the VCD spectrum as

equation image(25)

where ln(10)/2 = 1.1513. A calibration measurement to determine the first-order Bessel function J1math image( equation image)], needed to isolate the VCD spectrum, ΔA″(equation image), can be carried out using a multiple-wave plate and polarizer as described previously.26–28

VCB Measurement

The measurement of VCB can be achieved simply by inserting a polarizer oriented at 45° immediately after the sample in the VCD set up. The Mueller matrix for this polarizer is

equation image(26)

The Stokes vector of the light after the polarizer is obtained by applying MP(45°) to the Stokes vector after the sample, S3( equation image), given in eq. 22.

equation image(27)

The effect of this second polarizer is to mix and equalize the first the third components of S3( equation image) to yield a Stokes vector S4(equation image) with a new total intensity component. The intensity at the detector is simply this total intensity. Including the Stokes–Mueller expression for the total optical train, we have

equation image(28)

The VCB spectrum can be measured by isolating the second AC-term with a lock-in amplifier tuned to twice the PEM frequency, 2ωM. Using CB( equation image) = 2πΔn(equation image)lequation image, one obtains after dividing by the DC term

equation image(29)

Note that it is possible with two lock-amplifiers, one tuned to ωM and the other to 2ωM, to measure sequentially, or even simultaneously, VCD and VCB with the same optical set up.

Comparing eq. 29 with eq. 25, it is possible to express VCB as a difference in absorbance since it is measured as a pseudo VLD measurement at twice the PEM frequency. Using eq. 7, we can write

equation image(30)

Now eq. 29 can be written as

equation image(31)

Equations 25 and 30 show that VCD and VCB are measured by closely related quantities, ΔA″( equation image) and ΔA′(equation image), respectively. The two spectra are both Kramers-Kronig transforms of one another and the real and imaginary parts of the complex vibrational optical activity absorbance spectrum, ΔÃ(equation image), given by

equation image(32)

The connection to molar absorptivity, given in eqs. 10 and 13, can be obtained by assuming Beer-Lambert's law and dividing the terms in eq. 32 by the concentration and pathlength, namely

equation image(33)

The VCB spectrum requires a calibration spectrum to determine the magnitude and shape of the second-order Bessel function, J2math image( equation image)]. This can be obtained simply by removing the sample and turning the polarizer first to the vertical position and then to the horizontal position. The detector signal for these two cases (upper sign vertical, lower sign horizontal) is given by

equation image(34)

The average of the two DC terms (upper and lower signs) can be defined as

equation image(35)

and the ratio of I±AC( equation image) to IDC,ave(equation image)

equation image(36)

is the desired calibration function for eq. 29 or eq. 31.

EXPERIMENTAL

Neat liquid samples of (−)-S-α-pinene, (+)-R-α-pinene, (+)-R-limonene, and (−)-S-limonene were obtained from Sigma-Aldrich Chemical Company and used without further purification. Samples were placed in a fixed pathlength IR cell with BaF2 windows, and IR, VCD, and VCB spectra were recorded with a modified Dual-PEM ChiralIR™ FT-VCD spectrometer (BioTools, Jupiter, FL) using a resolution of 4 cm−1 and a collection time of 20 min. For VCD, the optimum retardation of the two ZnSe photoelastic modulators (PEMs) was set at 1400 cm−1. Any baseline offsets in the VCD or VCB spectra were eliminated by subtracting the spectra of the enantiomer and dividing by two. VCD spectra were measured with the dual-PEM option by subtracting in real time the VCD spectra associated with each of the two PEMs as previously described.28 VCB spectra cannot be corrected in the same way with dual-PEM dynamic subtraction, and for the VCB experiments the first PEM was set to 1200 cm−1 and the second PEM was inactive.

Calculations of the optimized geometry, vibrational frequencies and IR, VCD, and VCB intensities were carried out on a dual-processor Pentium IV PC using density functional theory (DFT), and three different combinations of basis and functionals obtained from the program Gaussian 03W (Gaussian, Wallingford, CT).29 We first used the minimal recommended, and commonly used, 6-31G(d) basis set and B3LYP functional, for IR, VCD, and VCB calculations of S-α-pinene and R-limonene. To achieve great accuracy, we carried out the same calculations using two different triple zeta basis sets, cc-pVTZ and TZVP, using the B3PW91 functional. The calculated frequencies were uniformly scaled by 0.97 for the 6-31G(d) calculations and by 0.98 for the cc-pVTZ and TZVP calculations. The triple zeta basis sets cc-pVTZ and TZVP required more computational time than the 6-31G(d) basis set for the IR/VCD/VCB calculations. The times required for each molecule were ∼19 h for the cc-pVTZ basis set, 2.5 h for the TZVP, and only ∼40 min for the 6-31G(d) calculations. The basis set TZVP appears, at least for these two molecules, to be an attractive alternative to the much more costly cc-pVTZ basis set if improvement beyond the 6-31G(d) basis set is desired. The calculated intensities were convolved with normalized absorptive or dispersive Lorentzian lineshape functions with half-width at half-maximum of the absorptive lineshape set to 6 cm−1 and summed over all transitions for comparison to the measured spectra.

RESULTS

The observed IR, VCD, VCD noise, VCB, and VCB noise spectra are presented in Figure 2 for (−)-S-α-pinene and in Figure 3 for (+)-R-limonene. The intensities plotted on the left-hand scale are A″( equation image) for the parent IR spectra, ΔA″(equation image) for the VCD spectra and ΔA′(equation image) for VCB spectra. There are no units since absorbance is a unitless quantity. On the right-hand intensity scale, the intensities are plotted as the corresponding molar absorption coefficients, ϵ″(equation image), Δϵ″(equation image), and Δϵ′(equation image) for the IR, VCD, and VCB spectra, respectively. The units for all the molar absorption coefficients are l/mol/cm.

Figure 2.

VCB noise, VCB, VCD noise, VCD, and IR spectra (top to bottom) of (−)-S-α-pinene neat liquid with a sample pathlength of 50 μ. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 3.

VCB noise, VCB, VCD noise, VCD, and IR spectra (top to bottom) of (+)-R-limonene neat liquid with a sample pathlength of 100 μ. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

The comparisons of measured and calculated IR, VCD, and VCB spectra of (−)-S-α-pinene and (+)-R-limonene are given in Figures 4 and 5, respectively. Both measured and calculated spectra are plotted in molar absorption coefficients with the same expansion scale for corresponding measured and calculated spectra. The degree of intensity agreement reflects in part the fact that the spectra were measured at 4 cm−1 resolution whereas the calculated spectra were convolved with Lorentzian absorption line shapes with half-widths of 6 cm−1 at half-maximum (12 cm−1 full width). If Lorentzian halfwidths of less than 6 cm−1 are chosen for comparison to experiment, the calculated spectra become noticeably sharper and more intense on average compared to the measured spectra.

Figure 4.

Comparison of measured VCB, VCD, and IR spectra of (−)-α- pinene neat liquid (red) versus the calculated VCB, VCD, and IR spectra of S-α-pinene for 6-31G(d)/B3LYP (blue), cc-pVTZ/B3PW91 (green), and TZVP/B3PW91 (black). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 5.

Comparison of measured VCB, VCD, and IR spectra of (+)-limonene neat liquid (red) versus the calculated VCB, VCD, and IR spectra of R-limonene for 6-31G(d)/B3LYP (blue), cc-pVTZ/B3PW91 (green), and TZVP/B3PW91 (black). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

The computational results obtained for α-pinene are in agreement with previous detailed analysis of the measured and calculated spectrum of this molecule.30 In particular, we confirm the more accurate calculations are obtained with the triple zeta basis sets than with double zeta basis set 6-31G(d). This is illustrated in Figures 4 and 5 where all three sets of calculated IR, VCD, and VCB spectra are superimposed for both molecules.

DISCUSSION

The measured spectra in Figures 2 and 3 show the expected relationship between VCD and VCB. From Figure 1, it is clear that on the same intensity scale a single VCD band has a maximum twice as far from zero as the two extrema of the corresponding VCB spectrum. It is thus expected that the range of VCD intensities should be approximately twice the range of the intensities corresponding VCB spectrum. In addition, it is possible to see correlations between the major VCD features and the major VCB features by considering the differences in the line shapes of the transitions as shown in Figure 1.

As is well known, VCB spectra are more overlapping between neighboring transitions than are VCD spectra for the same molecular sample. The origin of this effect is the nature of the line shapes associated with each transition. This can be seen in Figure 1 where the intensity of a VCD band has fallen to much smaller values in the wings of the bands compared the higher intensity of VCB in the wings, even though intensity extrema of the VCB are smaller than that of VCD by a factor of two. In the far-from-resonance limit where ( equation imageiequation image)2math image, the frequency dependence of the Lorentzian VCD lines shape decreases as the inverse second power multiplied by γi as γi/π(equation imageiequation image)2 whereas the VCB decreases only to the inverse first power multiplied by unity as 1/π(equation imageiequation image), and hence, VCB persists with significant intensity far from the resonance frequency. The dispersion line shape is essentially Drude's Law for the long wavelength dependence of optical rotation spectra (ORD).5, 6

Because the VCB spectra were measured experimentally as a pseudo linear dichroism spectrum, and because, as Kramers-Kronig transform pairs, VCB and VCD are spectral functions with the same units whether they be degrees, such as rotation and ellipticity angles, unitless absorbances, or molar absorptivities in units of l/mol-cm, VCB can be referenced to the parent IR absorbance spectrum in the same units and in the same way that VCD spectra are referenced. When CD and CB/OR are given in degrees, the origin of the parent spectrum is more difficult to conceptualize, but nonetheless, it can be referenced back from degrees to ellipticity phase shifts of the imaginary part of the index of refraction. Plotting VCB relative to a parent absorbance spectrum provides such a reference. If desired, one could reference the real part of the infrared index of refraction and this is now being carried for certain optical applications where needed as mentioned below for research in negative refraction.

Also, shown in Figures 2 and 3 are the noise spectra for the VCD and VCB spectra. It is clear that VCD features contain a distinct noise advantage for the same time of collection for the same sample. There are three immediate reasons for this. The first is from eq. 24 to eq. 28 where it can be seen the VCD intensities, which can be measured without an analyzing polarizer, enjoy an IR throughput that is twice that of VCB (as well as to VCD measured with the same optical set up). The second is that the value of J2math image( equation image)] at its maximum is smaller by ∼50% compared the maximum of J1math image(equation image)]. Finally, the VCB intensity scale is approximately two times more sensitive than the corresponding VCD scale, and hence, for the same size of spectral display for VCD and VCB, there will be larger visible noise for the VCB spectra. There may also be additional factors more difficult to identify and quantify, such as the relative electronic efficiencies at PEM frequencies ωM and 2ωM.

The comparisons of the three sets of calculated VCB spectra to the measured VCB spectra in Figures 4 and 5 show close agreement in overall shape, magnitude, and sign. The calculations using the two triple zeta basis sets pVTZ/B3PW91 and TZVP/B3PW91 are very close to one another and are in closer agreement to the measured spectra for IR, VCD, and VCB than the spectra calculated at the 6-31G(d)/B3LYP level. In particular, the comparison between calculated and measured VCB/VCD/IR presented here appears to be at the same level of agreement as that of previously published comparisons, including triple zeta basis sets, between calculated and measured IR and VCD spectra for α-pinene.30 It should be noted that the calculations with the TZVP basis set required significantly less time and were closer to that needed for the 6-31G(d) calculations than the other triple zeta basis set tested, cc-pVTZ, where no improvement relative to TZVP was apparent. The agreement between measured and calculated VCB thus confirms the authenticity of the measured VCB spectra and secures these results as the first examples of mid-infrared VCB, or infrared VORD, originating from individual molecules.

Because the VCB spectra reported here were measured using a Fourier transform spectrometer, they share the Fourier multiplex advantage whereby all spectral frequencies are measured simultaneously without a time-bias for the spectral measurement. ORD spectra, or CD spectra, measured in the visible and UV region must typically be scanned over time making it impossible, without step-scanning augmented by pulse-delay methods, to measure the kinetics of any sample being measured unless the timescale of the kinetics is say two orders of magnitude or more slower than the time for a single spectral scan. Thus, FT-VCB, as well as, FT-VCD, offers the possibility of measuring chemical kinetics across the entire spectrum with a time resolution comparable to the shortest time for which the signal-to-noise ratio is adequate to discern spectral changes of interest.

From the computational results presented, it can be concluded that VCB spectra can be calculated given an accurate calculation of the corresponding VCD where accuracy is known by the quality of agreement between measured and calculated VCD spectra. There may arise, however, samples for which the VCB spectrum, as a chiral property of the index of refraction a sample is sought but for which a VCD, and therefore, a VCB, calculation is not possible, perhaps due to the size or complexity of the sample at the molecular level.

One particular example where the VCB spectrum of a complex sample is needed, but may not be available from quantum chemistry calculations, is that of extended solids or polymers that are candidates to exhibit negative refraction by the chiral mechanism. Materials that exhibit negative refraction, known as negative index materials or NIMs, possess many remarkable properties such as perfect lensing and cloaking of macroscopic objects.31, 32 A few years ago, a chiral route to negative refraction was discovered,33 which can occur if the index of refraction of a material is positive but closer to zero than half the absolute magnitude of the CB at that point in the spectrum, namely whenever the following relationship holds

equation image(37)

if this is true, then the refractive index for one sense of circular polarization will be positive while the other sense must be negative. For example, if Δn′( equation image) > 0, then the refractive index for left circularly polarized radiation is positive while that for right circularly polarized radiation is negative, namely

equation image(38)
equation image(39)

To confirm the presence of the chiral mechanism for negative refraction, or to gauge how close to negative refraction a material is, accurate measures of both the real part of the index of refraction, as well as the CB, are needed. In our laboratory, the search for chiral NIMs is currently underway in the infrared region and is of intense interest for a variety of reasons in the area of applied optics.

Finally, we cite another example where direct measurement of the steady-state VCB of a sample may be of interest. Recently, a report of laser coherent measurement of vibrational optical activity was reported from the laboratory of Minhaeng Cho in which VCD and VORD (VCB) spectra were simultaneously measured with femtosecond resolution.24 In this report, the time-based Fourier components of VCD and VCB spectra were scanned by a monochromator after sample and reference femtosecond pulses were combined for detection where the VCD and VCB signals were in quadrature out-of-phase from one another. The sample used was limonene, the same chiral molecule featured here with α-pinene; however, the region of vibrational frequencies of the Cho paper was the near-infrared CH-stretching region, so direct comparison of VCB spectra obtained by these two different methods cannot at present be compared.

CONCLUSIONS

Here, we report the achievement of a number of new advances in the measurement and calculation of natural optical activity. The first is the first report of mid-infrared vibrational optical rotatory dispersion (VORD) originating from individual chiral molecules which we prefer to call vibrational circular birefringence, VCB. Second, VCB (or any form of CB/ORD) has been measured for the first time with a Fourier transform instrument, and hence, a simultaneous measurement of optical rotation across a wide range of frequencies and multiple numbers of resonances has been achieved. Third, we report the first theoretical calculation of VCB and its comparison to the corresponding measured VCB spectrum. The resulting agreement between calculated and measured VCB spectra, comparable in quality to the degree of agreement between calculated and measured IR and VCD spectra for two different chiral molecules, confirms the veracity of both the new Fourier transform polarization modulation method of VCB measurement and the new method of calculating a VCB spectrum, the latter being accomplished by simply changing the underlying line shape function from absorptive to dispersive, each of which are Kramers-Kronig transform pairs of one another, and also, real and imaginary parts of a unified complex line shape function.

The achievement of a simple route to the routine measurement of VCB, as well as an equally simple route to the routine calculation of VCB, opens a new area of natural optical activity in general, and vibrational optical activity in particular. Also, included in this advance is the potential to measure magnetic vibrational optical activity as both magnetic VCD (MVCD), as previously studied extensively for small non-chiral molecules,34, 35 and now MVCB. Important applications of direct measurements of VCB are foreseen to arise for complex chiral materials for which theoretical simulation is beyond the scope of current theoretical calculations, an important example of which is the search for chiral negative index materials. Another area of application of VCB measurement, as well as VCD measurement, is the determination of reference spectra for time-resolved femtosecond VCD and VCB kinetic studies.

Acknowledgements

The authors thank Dr. Shengli Ma for an early version of the VCB plotting program and Professor Tess Freedman for helpful discussions.

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