  A first-principles study of the vibrational modes of PbTiO_3 in the
ferroelectric tetragonal phase has been performed at all the main symmetry
points of the Brillouin zone (BZ). The calculations use the local-density
approximation and ultrasoft pseudopotentials with a plane-wave basis, and
reproduce well the available experimental information on the modes at the Gamma
point, including the LO-TO splittings. The work was motivated in part by a
previously reported transition to an orthorhombic phase at low temperatures
[(J. Kobayashi, Y. Uesu, and Y. Sakemi, Phys. Rev. B {\bf 28}, 3866 (1983)]. We
show that a linear coupling of orthorhombic strain to one of the modes at Gamma
plays a role in the discussion of the possibility of this phase transition.
However, no mechanical instabilities (soft modes) are found, either at Gamma or
at any of the other high-symmetry points of the BZ.
