  Frustrated triangular molecule magnets such as \{Cu$_3$\} are characterized
by two degenerate S=1/2 ground-states with opposite chirality. Recently it has
been proposed theoretically [PRL {\bf 101}, 217201 (2008)] and verified by {\it
ab-initio} calculations [PRB {\bf 82}, 155446 (2010)] that an external electric
field can efficiently couple these two chiral spin states, even in the absence
of spin-orbit interaction (SOI). The SOI is nevertheless important, since it
introduces a splitting in the ground-state manifold via the
Dzyaloshinskii-Moriya interaction. In this paper we present a theoretical study
of the effect of the SOI on the chiral states within spin density functional
theory. We employ a recently-introduced Hubbard model approach to elucidate the
connection between the SOI and the Dzyaloshinskii-Moriya interaction. This
allows us to express the Dzyaloshinskii-Moriya interaction constant $D$ in
terms of the microscopic Hubbard model parameters, which we calculate from
first-principles. The small splitting that we find for the \{Cu$_3$\} chiral
state energies ($\Delta \approx 0.02$ meV) is consistent with experimental
results. The Hubbard model approach adopted here also yields a better estimate
of the isotropic exchange constant than the ones obtained by comparing total
energies of different spin configurations. The method used here for calculating
the DM interaction unmasks its simple fundamental origin which is the
off-diagonal spin-orbit interaction between the generally multireference vacuum
state and single-electron excitations out of those states.
