Spin-orbit coupling and strong electronic correlations in cyclic molecules

In atoms spin-orbit coupling (SOC) cannot raise the angular momentum above a maximum value or lower it below a minimum. Here we show that this need not be the case in materials built from nanoscale structures including multi-nuclear coordination complexes, materials with decorated lattices, or atoms on surfaces. In such cyclic molecules the electronic spin couples to currents running around the molecule. For odd-fold symmetric molecules (e.g., odd membered rings) the SOC is highly analogous to the atomic case; but for even-fold symmetric molecules every angular momentum state can be both raised and lowered. These differences arise because for odd-fold symmetric molecules the maximum and minimum molecular orbital angular momentum states are time reversal conjugates, whereas for even-fold symmetric molecules they are aliases of the same single state. We show, from first principles calculations, that in suitable molecules this molecular SOC is large, compared to the energy differences between frontier molecular orbitals. Finally, we show that, when electronic correlations are strong, molecular SOC can cause highly anisotropic exchange interactions and discuss how this can lead to effective spin models with compass Hamiltonians.