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Frustrated triangular molecule magnets such as \{Cu$_3$\} are characterizedby two degenerate S=1/2 ground-states with opposite chirality. Recently it hasbeen proposed theoretically [PRL {\bf 101}, 217201 (2008)] and verified by {\itab-initio} calculations [PRB {\bf 82}, 155446 (2010)] that an external electricfield can efficiently couple these two chiral spin states, even in the absenceof spin-orbit interaction (SOI). The SOI is nevertheless important, since itintroduces a splitting in the ground-state manifold via theDzyaloshinskii-Moriya interaction. In this paper we present a theoretical studyof the effect of the SOI on the chiral states within spin density functionaltheory. We employ a recently-introduced Hubbard model approach to elucidate theconnection between the SOI and the Dzyaloshinskii-Moriya interaction. Thisallows us to express the Dzyaloshinskii-Moriya interaction constant $D$ interms of the microscopic Hubbard model parameters, which we calculate fromfirst-principles. The small splitting that we find for the \{Cu$_3$\} chiralstate energies ($\Delta \approx 0.02$ meV) is consistent with experimentalresults. The Hubbard model approach adopted here also yields a better estimateof the isotropic exchange constant than the ones obtained by comparing totalenergies of different spin configurations. The method used here for calculatingthe DM interaction unmasks its simple fundamental origin which is theoff-diagonal spin-orbit interaction between the generally multireference vacuumstate and single-electron excitations out of those states.

First-principle studies of the spin-orbit and the Dzyaloshinskii-Moriya interactions in the \{Cu$_3$\} single-molecule magnet