  The band convergence strategy, which improves Seebeck coefficient by inducing
multi-valley in bandstructures, has been widely used in thermoelectric
performance (TE) enhancing. However, the phonon-assisted intervalley scattering
effect is neglected and the mode-selection rules remain unclear. In this work,
TE properties for $\alpha$-, $\beta$- and $\gamma$-PbP are intestigated under
the consideration of full mode-, energy- and momentum-resolved electron-phonon
interactions (EPI). The group theory is used to analyze the selection rules for
EPI matrix elements. Our calculations reveal that, the intervalley scattering
contributes non-trivially to the total carrier relaxation time, and the
intervalley scattering can be modulated through crystal symmetry. In addition,
the investigation on the thermal properties reveals that four-phonon scattering
effect dominates the phonon relaxation processes, since the three-phonon
scattering is suppressed due to the significantly large acoustic-optical phonon
bandgap in $\alpha$-, $\beta$- and $\gamma$-PbP. By considering full EPI effect
and high-order phonon scattering processes, the calculated ZT values reach
0.90, 0.24 and 1.25 for $\alpha$-, $\beta$- and $\gamma$-PbP, repectively,
indicating their promising applications in thermoelectric devices.
