  The oxide pyrochlore Bi$_2$Ti$_2$O$_7$ is in some ways analogous to
perovskite PbTiO$_3$, in that Bi$_2$Ti$_2$O$_7$ has two cations, Bi$^{3+}$ and
Ti$^{4+}$ in oxidation states that are normally associated with a propensity to
off-center. However, unlike PbTiO$_3$, Bi$_2$Ti$_2$O$_7$ is experimentally
observed to remain cubic down to 2 K, while the only observed ionic
displacements are local and incoherent. Here we report first-principles
calculations of the zone-center phonons of the ordered cubic pyrochlore which
reveal several lattice instabilities. An analysis of the structural energetics
suggest that the ordered cubic pyrochlore is unstable with respect to
distortion towards a ferroelectric ground state with a large polarization. Our
results suggest a key role of a frustrated soft polar mode in the dielectric
properties of bismuth pyrochlores.
