  According to X-ray diffraction (XRD) measurements, PbTiO$_3$ undergoes a
phase transition from a tetragonal ferroelectric phase to a cubic paraelectric
phase at 763 K. However, X-ray absorption fine-structure (XAFS) measurements
indicate that PbTiO$_3$ is locally tetragonal even after the phase transition.
The difference in these results is because XAFS measurements can probe local
features of a structure, while XRD averages over such local features. For both
measurements to be consistent, PbTiO$_3$ is macroscopically cubic but locally
tetragonal after the phase transition. Despite this, most models, such as the
Laundau-Ginsburg-Devonshire theory and effective Hamiltonians, are still unable
to explain this phenomenon. Moreover, these methods involve model parameters
fitted to experimental or theoretical data and do not consider other tetragonal
configurations, such as domain walls, to predict the phase transition. Our
previous study used our novel zentropy approach to predict the phase transition
by considering the tetragonal ferroelectric ground state configuration and the
tetragonal 90{\deg} and 180{\deg} domain wall configurations with their total
energies at 0 K. In the present work, the Helmholtz energies of the three
configurations are obtained from density functional theory calculations through
energy-volume curves and phonon calculations. The predicted phase transition
temperature using the meta-GGA $r^2\text{SCAN}$ and revised multiplicities of
configurations is 716 K, showing good agreement with the experimental value of
763 K.
