  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.

Strongly Correlated Electrons