  A novel general method of describing the spin-lattice interactions in
magnetic solids was proposed in terms of first principles calculations. The
spin exchange and Dzyaloshinskii-Moriya interactions as well as their
derivatives with respect to atomic displacements can be evaluated efficiently
on the basis of density functional calculations for four ordered spin states.
By taking into consideration the spin-spin interactions, the phonons, and the
coupling between them, we show that the ground state structure of a
representative spin-frustrated spinel, MgCr2 O4, is tetragonally distorted, in
agreement with experiments. However, our calculations find the lowest energy
for the collinear spin ground state, in contrast to previously suggested
non-collinear models.
