Phonon self-energy corrections: To screen, or not to screen

First-principles calculations of phonons are often based on the adiabatic approximation, and 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. A well-founded correction method exists [Phys. Rev. B 82, 165111 (2010)], which only relies on adiabatically screened quantities. However, many-body theory suggests to use one bare electron-phonon vertex in the phonon self-energy [Rev. Mod. Phys. 89, 015003 (2017)] to avoid double counting. We assess the accuracy of both approaches in estimating the low-temperature phonons of monolayer TaS₂ and doped MoS₂. We find that the former yields excellent results at low computational cost due to its designed error cancellation to first order, while the latter becomes exact in the many-body limit but is not accurate in approximate contexts. We offer a third strategy based on downfolding to partially screened phonons and interactions [Phys. Rev. B 92, 245108 (2015)] to keep both advantages. This is the natural scheme to include the electron-electron interaction and tackle phonons in strongly correlated materials and nonadiabatic renormalization of the electron-phonon vertex. This record contains (i) a patch for the PHonon and EPW codes of Quantum ESPRESSO, (ii) the Python scripts and data necessary to create all figures shown in our paper, (iii) a minimal working example of the optimization of quadrupole tensors, and (iv) the Quantum ESPRESSO input files we have used.