  Our first-principles computations show that the ground state of PbTiO$_3$
under hydrostatic pressure transforms discontinuously from $P4mm$ to $R3c$ at 9
GPa. Spontaneous polarization decreases with increasing pressure so that the
$R3c$ phase transforms to the centrosymmetric $R\bar{3}c$ phase at around 30
GPa. The first-order phase transition between tetragonal and rhombohedral phase
is exceptional since there is no evidence for a bridging phase. The essential
feature of the $R3c$ and $R\bar{3}c$ phases is that they allow the oxygen
octahedron to increase its volume $V_B$ at the expense of cuboctahedral volume
$V_A$ around a Pb ion. This is further supported by the fact that neither the
$R3m$ nor $Cm$ phase, which keep the $V_A/V_B$ ratio constant, is a ground
state within the pressure range between 0 and 40 GPa. Thus tetragonal strain is
dominant up to 9 GPa, whereas at higher pressures efficient compression through
oxygen octahedra tilting plays the central role for PbTiO$_3$. Previously
predicted pressure induced colossal enhancement of piezoelectricity in
PbTiO$_3$ corresponds to unstable $Cm$ and $R3m$ phases. This suggests that the
phase instability, in contrast to the polarization rotation, is responsible for
the large piezoelectric properties observed in systems like Pb(Zr,Ti)O$_3$ in
the vicinity of the morphotropic phase boundary.
