We present a new first-principles linear-response theory of changes due to perturbations in the quasiparticle self-energy operator within the $GW$ method. This approach, named $GW$ perturbation theory ($GW\mathrm{PT}$), is applied to calculate the electron-phonon ($e\text{−}\text{ph}$) interactions with the full inclusion of the $GW$ nonlocal, energy-dependent self-energy effects, going beyond density-functional perturbation theory. Avoiding limitations of the frozen-phonon technique, $GW\mathrm{PT}$ gives access to $e\text{−}\text{ph}$ matrix elements at the $GW$ level for all phonons and scattering processes, and the computational cost scales linearly with the number of phonon modes (wave vectors and branches) investigated. We demonstrate the capabilities of $GW\mathrm{PT}$ by studying the $e\text{−}\text{ph}$ coupling and superconductivity in ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{BiO}}_3$. We show that many-electron correlations significantly enhance the $e\text{−}\text{ph}$ interactions for states near the Fermi surface, and explain the observed high superconductivity transition temperature of ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{BiO}}_3$ as well as its doping dependence.

• Received 18 July 2018
• Revised 15 February 2019

DOI:https://doi.org/10.1103/PhysRevLett.122.186402