Spin-orbit coupling in crystals is known to lead to unusual direction dependent exchange interactions, however understanding of the consequeces of such effects in molecular crystals is incomplete. Here we perform four component relativistic density functional theory computations on the multinuclear molecular crystal Mo3S7(dmit)3 and show that both intra-and inter-molecular spin-orbit coupling are significant. We determine a long-range relativistic single electron Hamiltonian from first principles by constructing Wannier spin-orbitals. We analyse the various contributions through the lens of group theory. Intermolecular spin-orbit couplings like those found here are known to lead to quantum spin-Hall and topological insulator phases on the 2D lattice formed by the tight-binding model predicted for a single layer of Mo3S7(dmit)3.

We show how quasi-one-dimensional correlated insulating states arise at two-thirds filling in organometallic multinuclear coordination complexes described by layered decorated honeycomb lattices. The interplay of spinorbit coupling and electronic correlations leads to pseudospin-one moments arranged in weakly coupled chains with highly anisotropic exchange and a large trigonal splitting. We show that the in-plane exchange coupling is very different from the interlayer coupling; in particular the latter is much larger, despite the underlying hopping integrals being close to isotropic. Surprisingly, the effective dimensionality of the pseudospin model is strongly dependent on the strength of the electronic correlations: With increasing Hubbard U the pseudospin-one model becomes increasingly one dimensional, even though the crystal is almost isotropic. We predict that the trigonal splitting leads to a quantum phase transition from a Haldane phase to a topologically trivial phase as the relative strength of the spin-orbit coupling increases.

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Motivated by recent synthetic and theoretical progress we consider magnetism in crystals of multinuclear organometallic complexes. We calculate the Heisenberg symmetric exchange and the Dzyaloshinskii-Moriya antisymmetric exchange. We show how, in the absence of spin-orbit coupling, the interplay of electronic correlations and quantum interference leads to a quasi-one-dimensional effective spin model in a typical trinuclear complex, Mo 3 S 7 (dmit) 3 , despite its underlying three-dimensional band structure. We show that both intra-and intermolecular spin-orbit coupling can cause an effective Dzyaloshinskii-Moriya interaction. Furthermore, we show that even for an isolated pair of molecules the relative orientation of the molecules controls the nature of the Dzyaloshinskii-Moriya coupling. We show that interference effects also play a crucial role in determining the Dzyaloshinskii-Moriya interaction. Thus, we argue that multinuclear organometallic complexes represent an ideal platform to investigate the effects of Dzyaloshinskii-Moriya interactions on quantum magnets.

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 multinuclear 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.