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 (${[\alpha ]}_{\mathrm{\lambda}}^{25\xb0\mathrm{C}}$ in deg dm^{-1} (g/ml)^{-1})

Gas Phase (Measured)

Gas Phase (Calculated)

Solution Phase (Extrapolated)

Solution Phase (γ_{s} 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, ${[\alpha ]}_{355\mathrm{nm}}^{25\xb0\mathrm{C}}$ and ${[\alpha ]}_{633\mathrm{nm}}^{25\xb0\mathrm{C}}$, 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 ${[\alpha ]}_{\mathrm{\lambda}}^{25\xb0\mathrm{C}}$ 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 ${\gamma}_{s}=({n}_{\lambda}^{2}+2)/3$, where the index of refraction for the cyclohexane solvent medium (n_{355 nm} = 1.4451 and n_{633 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

The results of simulated measurements designed to extract effective optical rotation values, ${\varphi}_{\mathrm{eff}},$
are summarized for the stationary $(\alpha =0)$
and modulated $(\alpha =\pi /20)$
modes of operation in an empty CRDP apparatus $(\u220a=\mathrm{\Delta}\u220a=\varphi =0)$
that has a retardation error of η imposed upon one of its intracavity quarter-wave plates (see text for details). While ${\varphi}_{\mathrm{eff}}(\alpha =0)$
follows from a closed-form analytical expression [Eq. (22)], determination of the corresponding ${\varphi}_{\mathrm{eff}}(\alpha =\pi /20)$
quantities necessitated the use of numerical least-squares fitting procedures. The relative deviations of ${\varphi}_{\mathrm{eff}}$
from zero, as well as the tabulated ${\varphi}_{\mathrm{eff}}(\alpha =0)/{\varphi}_{\mathrm{eff}}(\alpha =\pi /20)$
ratio, highlight the superior immunity to instrumental imperfections afforded by the modulated CRDP scheme.

Room-Temperature Specific Rotation (${[\alpha ]}_{\mathrm{\lambda}}^{25\xb0\mathrm{C}}$ in deg dm^{-1} (g/ml)^{-1})

Gas Phase (Measured)

Gas Phase (Calculated)

Solution Phase (Extrapolated)

Solution Phase (γ_{s} 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, ${[\alpha ]}_{355\mathrm{nm}}^{25\xb0\mathrm{C}}$ and ${[\alpha ]}_{633\mathrm{nm}}^{25\xb0\mathrm{C}}$, 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 ${[\alpha ]}_{\mathrm{\lambda}}^{25\xb0\mathrm{C}}$ 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 ${\gamma}_{s}=({n}_{\lambda}^{2}+2)/3$, where the index of refraction for the cyclohexane solvent medium (n_{355 nm} = 1.4451 and n_{633 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

The results of simulated measurements designed to extract effective optical rotation values, ${\varphi}_{\mathrm{eff}},$
are summarized for the stationary $(\alpha =0)$
and modulated $(\alpha =\pi /20)$
modes of operation in an empty CRDP apparatus $(\u220a=\mathrm{\Delta}\u220a=\varphi =0)$
that has a retardation error of η imposed upon one of its intracavity quarter-wave plates (see text for details). While ${\varphi}_{\mathrm{eff}}(\alpha =0)$
follows from a closed-form analytical expression [Eq. (22)], determination of the corresponding ${\varphi}_{\mathrm{eff}}(\alpha =\pi /20)$
quantities necessitated the use of numerical least-squares fitting procedures. The relative deviations of ${\varphi}_{\mathrm{eff}}$
from zero, as well as the tabulated ${\varphi}_{\mathrm{eff}}(\alpha =0)/{\varphi}_{\mathrm{eff}}(\alpha =\pi /20)$
ratio, highlight the superior immunity to instrumental imperfections afforded by the modulated CRDP scheme.