  We have derived an expression of the Dzyaloshinskii-Moriya (DM) interaction,
where all the three components of the DM vector can be calculated
independently, for a general, non-collinear magnetic configuration. The
formalism is implemented in a real space - linear muffin-tin orbital - atomic
sphere approximation (RS-LMTO-ASA) method. We have chosen the Cr triangular
trimer on Au(111) and Mn triangular trimers on Ag(111) and Au(111) surfaces as
numerical examples. The results show that the DM interaction (module and
direction) is drastically different for collinear and non-collinear states.
Based on the relation between the spin and charge currents flowing in the
system and their coupling to the non-collinear magnetic configuration of the
triangular trimer, we demonstrate that the DM interaction can be significant,
even in the absence of spin-orbit coupling. This is shown to emanate from the
non-collinear magnetic structure, that can induce significant spin and charge
currents even with spin-orbit coupling is ignored.
