  Aiming at increasing the ferroelectric polarization in AFM-E ortho-\hmo, we
investigate the in-plane strain effects on both the magnetic configuration and
the polarization by means of density functional theory calculations and model
Hamiltonian approaches. Our results show that the net polarization is largely
enhanced under compressive strain, due to an increase of the electronic
contribution to the polarization, whereas the ionic contribution is found to
decrease. We identify the electron-lattice coupling, due to Jahn-Teller (JT)
distortions, and its response to strain to be responsible for the observed
behavior. The JT-induced orbital ordering of occupied Mn-e$_g^1$ electrons in
alternating $3x^2-r^2/3y^2-r^2$ orbital states at equilibrium changes to a
mixture with $x^2-z^2/y^2-z^2$ states under in-plane compressive strain. The
asymmetric hopping of e$_g$ electrons between Mn ions along zig-zag spin chains
(typical of the AFM-E spin configuration) is therefore enhanced under strain,
explaining the large value of the polarization. We reproduce the change in the
orbital ordering pattern in a degenerate double-exchange model supplemented
with electron-phonon interaction. In this picture, the orbital ordering change
is related to a change of the Berry phase of the e$_g$ electrons, which in turn
causes an increase of the polarization, whose origin is purely electronic.
