  Using density functional theory we investigate the lattice instability and
electronic structure of recently discovered ferroelectric metal LiOsO$_3$. We
show that the ferroelectric-like lattice instability is related to the Li-O
distortion modes while the Os-O displacements change the d-p hybridization as
in common ferroelectric insulators. Within the manifold of the d-orbitals, a
dual behavior emerges. The ferroelectric transition is indeed mainly associated
to the nominally empty e$_g$ orbitals which are hybridized with the oxygen p
orbitals, while the t$_{2g}$ orbitals are responsible of the metallic response.
Interestingly, these orbitals are nominally half-filled by three electrons, a
configuration which suffers from strong correlation effects even for moderate
values of the screened Coulomb interaction.
