  The present paper proposes the direct calculation of the microscopic
contributions to the magneto-electric coupling, using ab initio methods. The
electrostrictive and the Dzyaloshinskii-Moriya contributions were evaluated
individually. For this purpose a specific method was designed, combining DFT
calculations and embedded fragments, explicitely correlated, quantum chemical
calculations. This method allowed us to calculate the evolution of the magnetic
couplings as a function of an applied electric field. We found that in $\rm
YMnO_3$ the Dzyaloshinskii-Moriya contribution to the magneto-electric effect
is three orders of magnitude weaker than the electrostrictive contribution.
Strictive effects are thus dominant in the magnetic exchange evolution under an
applied electric field, and by extension on the magneto-electric effect. These
effects remain however quite small and the modifications of the magnetic
excitations under an applied electric field will be difficult to observe
experimentally. Another important conclusion is that the amplitude of the
magneto-electric effect is very small. Indeed, it can be shown that the linear
magneto-electric tensor is null due to the inter-layer symmetry operations.
