Expanding the Toolkit for Optical Rotation Calculations: Development of Interpretive Models and Extension to Solid State Systems

Abstract

The measurement of optical rotation (OR) is a long standing approach forcharacterizing chiral systems. The ability to predict optical rotation theoretically without resorting to expensive \textit{ab initio} simulations would greatly aid in absolute configuration assignment of chiral species and would help develop a more chemically intuitive sense of the physical processes that drive this phenomenon. However, it has proven challenging to develop predictive models for OR, not only due to the difficulty of computing the intrinsic OR arising from the structure of a molecule, but also the need to disentangle it from optical activity induced by the surrounding environment. We have worked to unravel these competing effects by developing models to decompose the intrinsic OR into more interpretable structural contributions and by extending quantum mechanical methods to compute OR to solid state systems, allowing for a more thorough analysis of environmental effects. We have worked to extend the \st[] analysis framework, which decomposes the OR intooccupied-virtual orbital pair contributions, to make it more generally applicable. We calculated OR for a small test of molecules in both the modified velocity gauge (MVG) and length gauge (LG) in order to determine how the physical interpretation provided by \st[] analysis is affected by the gauge. We found that the distribution of \st[] contributions was consistent across different gauges. The \st[] values from different gauges also converged at a similar rate to the total OR when summing contributions from largest in magnitude to smallest. Examining the electric and magnetic vectors associated with the largest magnitude transitions, we found that each gauge gave a consistent interpretation of the dominant physical processes driving the OR. We have also worked to remove the unphysical origin dependence from computed \st[]values. To this end, we derived two origin invariant MVG formulations of \st{}: $\tilde{S}^{\text{Avg}}$, which averages \st[] computed with an electric (MVG-E) and magnetic (MVG-M) perturbation \st[] to cancel out their opposite sign, equal magnitude origin dependence, and $\tilde{S}^{\text{Hemi}}$, which eliminates the origin dependence by contracting together "hemi-perturbed" densities formed via a Cholesky decomposition of the response matrix. We tested these methods on two small organic molecules and confirmed that \st{} remained unchanged at a displaced origin. While further work is needed to develop associated electric and magnetic vectors for these origin invariant \st[], they can already be used to detect significant deviation due to the chosen origin. As a first step towards developing quantum mechanical methods to compute the OR ofsolids, we have created a benchmark set of helical chains of diatomic molecules, for which the OR can be computed using existing molecular methods. We probed the effect of cell size, cell spacing, and helical orientation on the computed OR. We found that even for these simple models, convergence to the macroscopic limit was slow with system size and the effect of helical orientation could not be readily reproduced using the semi-empirical Kirkwood model. Our results made clear the need for a periodic, quantum mechanical methods to compute the full OR tensor, including both the electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizability contributions. To aid in the development of highly accurate methods for computing solid state OR, wecompared MVG and origin invariant length gauge [LG(OI)] calculations of the full OR tensor using coupled cluster with single and double excitations (CCSD) and Density Functional Theory (DFT) methods. We found that DFT and CCSD components of the OR tensor were well correlated, though DFT could significantly overestimate these values depending on the chosen functional. CCSD MVG and LG(OI) tensor components were highly correlated, but the extent of correlation showed a high dependence on the inclusion/exclusion of outliers. Our results suggest that some care must still be taken with the choice of gauge, as for some cases CCSD may not provide a sufficient treatment of electron correlation to achieve gauge invariance. We have derived translation invariant forms for the magnetic dipole andelectric quadrupole in order to implement the periodic, full OR tensor into Gaussian. While a periodic form of the magnetic dipole has recently been published, a derivation of the periodic electric quadrupole is reported for the first time in this work. Once the implementation is complete, we aim to distinguish intrinsic and environmental influences on the OR via simulations of molecular crystals.