Reinterpretation of bond-valence model with bond-order formalism: an improved bond-valence based interatomic potential for PbTiO$_3$

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.