Rotational Coupling in Methyl-Tunneling Electron Spin Echo Envelope Modulation

Coherence between tunnel-split states of a methyl quantum rotor can be generated and observed in stimulated and spin-locked echo experiments, if hyperfine coupling of a nearby electron spin to the methyl protons breaks C$$_3$$ 3 symmetry and is of the same order of magnitude as the tunnel splitting. Here, we consider the case of two methyl groups bound to the same sp$$^{3}$$ 3 -hybridized atom, which is important in the context of common nitroxide spin labels. For a simple form of the rotor-rotor coupling Hamiltonian, we provide an approach that allows for density operator computations of this system with 1152 quantum states with moderate computational effort. We find that, in the regime where the ratio between rotor-rotor coupling and rotational barrier is much smaller than unity, three-pulse ESEEM and hyperfine-decoupled ESEEM depend only on the tunnel splitting, but not on this ratio. This finding may simplify the treatment of tunnel-induced electron decoherence in systems where the methyl groups are bound to sp$$^{3}$$ 3 -hybridized atoms.