  We present a modified bond-valence model of PbTiO$_3$ based on the principles
of bond-valence and bond-valence vector conservation. The relationship between
the bond-valence model and the bond-order potential is derived analytically in
the framework of a tight-binding model. A new energy term, bond-valence vector
energy, is introduced into the atomistic model and the potential parameters are
re-optimized. The new model potential can be applied both to canonical ensemble
($NVT$) and isobaric-isothermal ensemble ($NPT$) molecular dynamics (MD)
simulations. This model reproduces the experimental phase transition in $NVT$
MD simulations and also exhibits the experimental sequence of
temperature-driven and pressure-driven phase transitions in $NPT$ simulations.
We expect that this improved bond-valence model can be applied to a broad range
of inorganic materials.

Materials Science;Computational Physics