Phonon Self-Energy Corrections: To Screen, or Not to Screen

First-principles calculations of phonons are often based on the adiabatic approximation and on Brillouin-zone samplings that might not always be sufficient to capture the subtleties of Kohn anomalies. These shortcomings can be addressed through corrections to the phonon self-energy arising from the low-energy electrons. The exact self-energy involves a product of a bare and a screened electron-phonon vertex [Rev. Mod. Phys. 89, 015003 (2017)]; still, calculations often employ two adiabatically screened vertices, which have been proposed as a reliable approximation for self-energy differences [Phys. Rev. B 82, 165111 (2010)]. We assess the accuracy of both approaches in estimating the phonon spectral functions of model Hamiltonians and the adiabatic low-temperature phonon dispersions of monolayer TaS$_2$ and doped MoS$_2$. We find that the approximate method yields excellent corrections at low computational cost, due to its designed error cancellation to first order, while using a bare vertex could in principle improve these results but is challenging in practice. We offer an alternative strategy based on downfolding to partially screened phonons and interactions [Phys. Rev. B 92, 245108 (2015)]. This is a natural scheme to include electron-electron interactions and tackle phonons in strongly correlated materials and the frequency dependence of the electron-phonon vertex.