A Conceptual Shift In Our Understanding of Degenerate Radical Spin Systems: Spin-Rotation Coupling Turned On Its Head

Abstract

For most chemists, Kramers' degeneracy refers to the fact that for any radical system, every potential energy surface is at least doubly degenerate (with spin up and spin down, time-reversed solutions) for all nuclear positions $\mathbf{X}$. That being said, as is well-known to the community of spin chemists, one can experimentally detect a splitting of almost every rotational energy level for a doublet system -- highlighting the fact that nuclear motion breaks the spin degeneracy of such BO electronic states. Thus, as far as predicting experimental spectra, the implications of BO degeneracy are very limited unless one further includes a complete treatment of nuclear-electronic entanglement in a robust fashion; indeed, understanding radical molecules (and the degeneracy of their stationary states) can be extremely non-intuitive within the paradigm of Born-Oppenheimer potential energy surfaces. Now, as an alternative to BO theory, recent theory has suggested characterizing radical potential energy surfaces as functions of both nuclear position $\mathbf{X}$ and nuclear momentum $\mathbf{P}$, an approach which has been shown to recover a host of observables outside of BO theory, e.g., vibrational circular dichroism, Raman optical activity, and lambda doubling. Here, we show that such a technique predicts that different spin states will follow different (nondegenerate) potential energy surfaces and that the differences in these spin-dependent surfaces is quantitatively consistent with experimental spin-rotation couplings -- all without any contradiction with regard to Kramers' degeneracy. Thus, the present finding suggests there is still a great deal to learn about spin-resolved molecular reactivity, demanding a conceptual shift in our understanding of coupled spin-nuclear motion, especially in the context of chiral molecules and materials where spin-separation is known to arise.

Figures (6)

• Figure 1: Lowest two potential energy surfaces under the a) BO approximation; b) the phase space theory, plotted as a function of the canonical nuclear momentum $L^\mathrm{n}$ and colored based on the spin character; c) the same phase space PES, but colored based on their relationship to the total electronic and nuclear eigenstates. Figure not to scale. Intuitively, the spin-rotation splittings arise from the difference in energy between the broken symmetry ground and excited state surfaces labeled as $\Delta E$ in c).
• Figure 2: Spin-rotation coupling splitting of CH$_3$ observed in the rotational spectroscopy. Each peak is labeled based on the type of transition; on the left hand side, $\Delta\nu_i$ and $\Delta\nu_f$ represent the spin-rotation splitting of the initial and final states, respectively. The original spectrum is reproduced from Ref. davis1997jet with permission of AIP publishing. The frequencies between the split levels are plotted relative to the frequency difference between the unsplit transitions (according to an experimental fit). The experimental transitions here are from vibration level $v=0$ to vibrational level $v=1$, but changes in vibrational level are removed by considering the relative frequency.
• Figure S1: Predicted spin-rotation coupling tensor element $\epsilon_{bb}$ of CH$_3$ using various DFT functionals in comparison to the exact solver FCI in the minimal basis (STO-6G).
• Figure S2: Predicted spin-rotation coupling tensor element $\epsilon_{bb}$ of CH$_3$ in various bases with TPSS functional. The black dashed line is the $\epsilon_{bb}$ extracted from experimental spectra davis1997jet.
• Figure S3: Lowest two potential energy surfaces from the phase space theory, plotted as a function of the canonical nuclear momentum $L^\mathrm{n}$ and colored based on their relationship to the total electronic and nuclear eigenstates. Figure not to scale. Intuitively, the spin-rotation splittings for $N=1$ arise from the difference in energy between the broken symmetry ground state surface (with spin aligned with the rotation) at $L^\mathrm{n}=\pm 1.5$ and the excited state surface (with spin antialigned with the rotation) at $L^\mathrm{n}=\pm0.5$, where the energy difference is labeled as $\Delta E$.