  The electronic and structural properties of Li$B$O$_3$ ($B$=V, Nb, Ta, Os)
are investigated via first-principles methods. We show that Li$B$O$_3$ belong
to the recently proposed hyperferroelectrics, i.e., they all have unstable
longitudinal optic phonon modes. Especially, the ferroelectric-like instability
in the metal LiOsO$_3$, whose optical dielectric constant goes to infinity, is
a limiting case of hyperferroelectrics. Via an effective Hamiltonian, we
further show that, in contrast to normal proper ferroelectricity, in which the
ferroelectric instability usually comes from long-range coulomb interactions,
the hyperferroelectric instability is due to the structure instability driven
by short-range interactions. This could happen in systems with large ion size
mismatches, which therefore provides a useful guidance in searching for novel
hyperferroelectrics.
