Optical Rotation

Linear Optical Rotation

The first contribution to the induced electric dipole moment arising from the time dependence of the magnetic field, \({\omega^{-1}}G'\), gives rise to two different observable properties. In the dispersive region, it determines the optical rotatory power, optical rotation (OR) for short, whereas in the absorptive region it determines the rotational strength observed in electronic circular dichroism (ECD).

Since there are no terms bilinear in the electric and magnetic fields in the Hamiltonian, \(G^{\prime}\), the mixed electric dipole–magnetic dipole polarizability, can be expressed as a linear response function

\[ G^{\prime}_{\alpha\beta}(-\omega;\omega)= -\frac{2\omega}{\hbar} \sum_{n \neq 0} {{\mathcal Im}} \frac{\left< 0 \mid {\hat{\mu}}_{\alpha} \mid n\right>\left<n \mid {\hat{m}}_{\beta} \mid 0\right>}{ \omega_{n}^2-\omega^2} = -{\mathcal Im}\left<\left<\hat{\mu}_{\alpha};\hat{m}_{\beta}\right>\right>_\omega. \]

\(G^{\prime}\) vanishes if the electromagnetic field is static (\(\omega=0\)).

Measurements of optical rotation are almost exclusively carried out on liquid samples (though gas-phase measurements is also possible [MullerWV00] and we will therefore be concerned primarily with the rotational average of the \(G^{\prime}\) tensor, which is described by the quantity \(^{\rm ORD}\beta\) defined as

\[ ^{\rm ORD}\beta = -\frac{1}{3\omega}G^{\prime}_{\alpha\alpha }. \]

Experimental issues

The most common experimental set-up involves the determination of the rotation of plane-polarized light as it passes through a sample in which there is an excess of one enantiomer. The standard optical rotation \(\left[\alpha\right]_D^{25}\), proportional to \(^{\rm ORD}\beta\), is reported for light with a frequency corresponding to the sodium D-line (589.3~nm) at a temperature of \(25^\circ{\rm C}\).

The dispersion of the optical rotation was for a long time also the focus of much experimental attention through optical rotatory dispersion measurements. Even after it became customary to restrict the optical rotation measurements to a single frequency, ORD served as an important tool for determining excitation energies in chiral molecules, although it has now been surpassed by electronic circular dichroism for these purposes.

At the sodium frequency, the optical rotation is rather small for most molecules. However, the individual diagonal elements of the mixed electric dipole-magnetic dipole polarizability may be fairly large in absolute value, often cancelling each other out in the trace. Consequently, the optical rotation is highly sensitive to numerical errors in the tensor components, because small residual errors in the individual tensor components, arising from the solution of the linear response equations, may lead to substantial errors in \(^{\mathrm {ORD}}\beta\). Another consequence of this cancellation is that \(^{\mathrm {ORD}}\beta\) is very sensitive to the choice of molecular geometry, as well as the zero-point vibrational effects [RTAAstrand01], since the small changes introduced by these effects in the molecular charge distribution, and thus on the different diagonal elements of \(G^{\prime}\), can give rather large effects on \(^{\mathrm {ORD}}\beta\).

Units

Unit conversion factors for electric dipole-magnetic dipole polarizability \(G^{\prime}\) :

1 a.u. (\(e^{2} a_0^{3} {\hbar}^{-1}\)) equals \(3.60702\times 10^{-35}\) C\(^2\)m\(^3\)J\(^{-1}\)s\(^{-1}\) (SI) and \(1.08136\times 10^{-27}\) Fr cm G\(^{-1}\) (CGS)

Gauge Origin Dependence

Since the mixed electric dipole-magnetic dipole polarizability involves the magnetic dipole operator, in approximate calculations \(G^{\prime}\) carries an origin dependence. Indeed, the individual tensor elements of \(G^{\prime}\) are origin dependent. The trace of \(G^{\prime}\) must be origin independent, since the optical rotation is an experimental observable. In non-isotropic media, contributions to the optical rotation tensor arise from the mixed electric dipole-electric quadrupole polarizability \(A\),

\[ A_{\alpha,\beta\gamma}\left(-\omega;\omega\right) = -\langle\langle\hat{\mu}_\alpha;\hat{\Theta}_{\beta\gamma}\rangle\rangle_\omega, \]

and it is the combination of the \(G^{\prime}\) and \(A\) contributions that is gauge-origin independent for exact wave functions. The contribution from \(A\) vanishes in isotropic samples, since this mixed electric dipole-electric quadrupole polarizability is traceless.

For approximate variational wave functions, the origin independence of the trace of \(G^{\prime}\) is only achieved in the limit of a complete basis set. One way to overcome this problem is to introduce local gauge origins by using London atomic orbitals [HRB+94].

Conformationally flexible molecules

The sensitivity of the optical rotation is perhaps most clearly illustrated in the case of conformationally flexible molecules that is molecules that have a significant population of multiple stable minima. In this case different molecular conformations can have optical rotations that differ by orders of magnitude and even in sign. A thorough conformational search is therefore mandatory before the absolute sign of the optical rotation (or any birefringence or dichroism for that matter) is determined from theoretical calculations for conformationally flexible molecules. The optical rotation of the flexible molecule can then be determined by Boltzmann averaging over the dominant molecular conformations.

Basis sets for optical rotation

In order to ensure reasonably well-converged results for the optical rotation, basis sets of polarized valence double-zeta quality are required, and sets such as the aug-cc-pVDZ of Woon and Dunning [WmboxDJr94], or the polarized triple-zeta basis of Sadlej [Sad88] have been shown to perform well for calculations of optical rotation. Most importantly, diffuse \(p\) functions in the outer regions of the electron density, which in most cases is described by the electron density of hydrogen atoms, are required to ensure qualitatively correct results [ZH04].

The effects of electron correlation

Since the optical rotation is very sensitive even to small changes in the electron density, electron correlation effects should also be taken into account in order to get accurate results. Due to the fact that chiral molecules in general have very low symmetry, if any symmetry at all, the only viable approach for calculating electron correlation effects in chiral molecules is currently density functional theory [RSD+03, SDCF01]. Methods such as MP2 and CCSD could also be used to describe electron correlation, but due to their non-variational nature it is not ensured that gauge-origin independent results can be obtained in the conventional length gauge formulation even in the limit of a complete basis set. In principle, the use of the dipole velocity gauge will ensure that the calculated results are independent of the gauge origin. For all but the smallest basis sets, the differences between the velocity and the length gauges are not very large, with the length gauge in general performing better. However, in the case of the optical rotation an additional complication arises: whereas \(G^{\prime}\) given by the equation given at the beginning of this section will vanish in the limit of a static field, this is not the case for the dipole-velocity analogue of the optical rotation. It has been suggested that the much slower basis set convergence of the optical rotation compared to other properties calculated using the velocity-gauge formulation can be improved by subtracting the static-limit value of the corresponding response function [PKBmboxSanchezdMeras04], such that

\[ G' \propto \langle\langle \hat{p}_\alpha;\hat{l}_\beta \rangle\rangle_\omega - \langle\langle \hat{p}_\alpha;\hat{l}_\beta \rangle\rangle_0. \]

Since there are no empirical rules relating the stereochemistry of a molecule to the observed sign of the optical rotation, an important area of application for optical rotation calculations would be the combined theoretical and experimental determination of absolute configuration. However, this is a difficult task due in part to the large variations with geometry in the optical rotation of conformationally flexible molecules, and in part because of the small magnitude of \(^{\mathrm {ORD}}\beta\). These factors make the sign of the optical rotation hard to determine with confidence. One way to increase the predictive power of the calculations is to change the frequency of the incident light to shorter wavelengths. The magnitude of the optical rotation increases dramatically as the frequency approaches that of a resonance. Thus, if both theory and experiment could determine the optical rotation at frequencies closer to electronic excitation energies, theoretical calculations could provide a much more reliable proof of the absolute configuration of the molecule [GVZR04].