In compound semiconductors and insulators, the polar electron-phonon coupling diverges at long range, known as the Fröhlich interaction. Modern first-principles electron-phonon calculations treat the Fröhlich interaction in a semiclassical electrostatic formalism based on density-functional perturbation theory. Here, using many-body $GW$ perturbation theory, we reveal important electron correlation effects in the Fröhlich-type electron-phonon coupling, which are missed by the prevailing approaches and rooted in the fundamentally distinct behavior between quasiparticle self-energy and semilocal exchange-correlation functionals in the long-range limit. Going beyond the electrostatic treatment, we derive and implement the $GW$ self-energy contribution to the long-range polar electron-phonon coupling, and demonstrate its critical role and nontrivial behaviors in properties such as electron linewidth and polaron formation in several prototype semiconductors. Remarkably, calculations on photoemission kink in electron-doped ${\mathrm{TiO}}_2$ achieve excellent agreement with experiment. Our work establishes the many-body generalization of the Fröhlich interaction that is essential for accurate electron-phonon calculations at the full $GW$ level combined with Wannier interpolation techniques.

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• Electron-phonon coupling
• Polarons
• GW method
• Many-body techniques