Thomas Müller, Kenneth B. Wiberg, Patrick H. Vaccaro, James R. Cheeseman, and Michael J. Frisch, "Cavity ring-down polarimetry (CRDP): theoretical and experimental characterization," J. Opt. Soc. Am. B 19, 125-141 (2002)
Detailed theoretical analyses are presented for cavity ring-down polarimetry, a recently developed scheme for probing circular birefringence (nonresonant rotatory dispersion) and circular dichroism (resonant differential absorption) with unprecedented sensitivity. Aside from elucidating the nature of time-resolved signals generated by various modes of operation, the influence of instrumental imperfections on polarimetric response is examined. The unique ability of cavity ring-down polarimetry to interrogate nonresonant optical activity in low-pressure chiral vapors is demonstrated by extracting specific rotation parameters at two complementary excitation wavelengths (355 nm and 633 nm) for gaseous samples of α-pinene, β-pinene, and cis-pinane. The resulting isolated-molecule properties are contrasted with those derived from conventional solution-phase experiments and state-of-the-art ab initio calculations.
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Room-Temperature Specific Rotation ( in deg dm-1 (g/ml)-1)
Gas Phase (Measured)
Gas Phase (Calculated)
Solution Phase (Extrapolated)
Solution Phase (γs Corrected)
(1R, 5R)-(+)-α-Pinene
≥99% (97%ee)
355
191.2 ± 2.8
88.7
165.2
121.2
633
48.1 ± 1.9
33.6
45.5
33.9
(1S, 5S)-(−)-α-Pinene
99% (97%ee)
355
−188.8 ± 2.8
−88.7
−165.2
−121.2
(1S, 5S)-(−)-β-Pinene
≥99% (97%ee)
355
70.5 ± 2.0
−253.6
21.8
16.0
633
−12.1 ± 1.9
−20.7
−17.0
−12.7
(1S)-(−)-cis-Pinane
99%
355
−63.0 ± 5.9
−60.1
−88.4
−64.9
(1R)-(+)-cis-Pinane
99%
633
5.2 ± 1.1
12.8
19.0
14.1
For each of the chiral systems examined during the present investigation, the specific optical rotation parameters measured under room-temperature conditions at excitation wavelengths of 355 nm and 633 nm, and , are tabulated in the canonical units of deg dm-1 (g/ml)-1. In particular, circular-birefringence results were obtained both for isolated gas-phase species (as probed directly through CRDP experiments) and for molecules solvated in a cyclohexane medium (as extrapolated from conventional ORD spectra), with the corresponding ab initio predictions for calculated at the B3LYP/6-311++G(2d, 2p) level of theory. The final column contains “solvent-corrected” specific rotation values derived from experimental quantities by exploiting the Lorentz scaling factor , where the index of refraction for the cyclohexane solvent medium (n355 nm = 1.4451 and n633 nm = 1.4247) was interpolated from published data.48 The chemical and enantiomeric purities of the targeted compounds also are listed, with the latter given as the enantiomeric excess revealed by gas–liquid chromatography.
Table 2
Influence of Retardation Errors on Specific Rotation Measurementsa
Imposed Retardation Error η (rad)
Measured Optical Rotation
(rad/m)
(rad/m)
π/256
53
π/512
106
π/1024
212
The results of simulated measurements designed to extract effective optical rotation values,
are summarized for the stationary
and modulated
modes of operation in an empty CRDP apparatus
that has a retardation error of η imposed upon one of its intracavity quarter-wave plates (see text for details). While
follows from a closed-form analytical expression [Eq. (22)], determination of the corresponding
quantities necessitated the use of numerical least-squares fitting procedures. The relative deviations of
from zero, as well as the tabulated
ratio, highlight the superior immunity to instrumental imperfections afforded by the modulated CRDP scheme.
Room-Temperature Specific Rotation ( in deg dm-1 (g/ml)-1)
Gas Phase (Measured)
Gas Phase (Calculated)
Solution Phase (Extrapolated)
Solution Phase (γs Corrected)
(1R, 5R)-(+)-α-Pinene
≥99% (97%ee)
355
191.2 ± 2.8
88.7
165.2
121.2
633
48.1 ± 1.9
33.6
45.5
33.9
(1S, 5S)-(−)-α-Pinene
99% (97%ee)
355
−188.8 ± 2.8
−88.7
−165.2
−121.2
(1S, 5S)-(−)-β-Pinene
≥99% (97%ee)
355
70.5 ± 2.0
−253.6
21.8
16.0
633
−12.1 ± 1.9
−20.7
−17.0
−12.7
(1S)-(−)-cis-Pinane
99%
355
−63.0 ± 5.9
−60.1
−88.4
−64.9
(1R)-(+)-cis-Pinane
99%
633
5.2 ± 1.1
12.8
19.0
14.1
For each of the chiral systems examined during the present investigation, the specific optical rotation parameters measured under room-temperature conditions at excitation wavelengths of 355 nm and 633 nm, and , are tabulated in the canonical units of deg dm-1 (g/ml)-1. In particular, circular-birefringence results were obtained both for isolated gas-phase species (as probed directly through CRDP experiments) and for molecules solvated in a cyclohexane medium (as extrapolated from conventional ORD spectra), with the corresponding ab initio predictions for calculated at the B3LYP/6-311++G(2d, 2p) level of theory. The final column contains “solvent-corrected” specific rotation values derived from experimental quantities by exploiting the Lorentz scaling factor , where the index of refraction for the cyclohexane solvent medium (n355 nm = 1.4451 and n633 nm = 1.4247) was interpolated from published data.48 The chemical and enantiomeric purities of the targeted compounds also are listed, with the latter given as the enantiomeric excess revealed by gas–liquid chromatography.
Table 2
Influence of Retardation Errors on Specific Rotation Measurementsa
Imposed Retardation Error η (rad)
Measured Optical Rotation
(rad/m)
(rad/m)
π/256
53
π/512
106
π/1024
212
The results of simulated measurements designed to extract effective optical rotation values,
are summarized for the stationary
and modulated
modes of operation in an empty CRDP apparatus
that has a retardation error of η imposed upon one of its intracavity quarter-wave plates (see text for details). While
follows from a closed-form analytical expression [Eq. (22)], determination of the corresponding
quantities necessitated the use of numerical least-squares fitting procedures. The relative deviations of
from zero, as well as the tabulated
ratio, highlight the superior immunity to instrumental imperfections afforded by the modulated CRDP scheme.