Electron-phonon interactions in the projector-augmented-wave method

The projector-augmented-wave (PAW) method is a powerful approach for improving the accuracy and efficiency of first-principles calculations. Here, we present two formulations of the electron-phonon coupling within the PAW framework. When the phonon perturbation is introduced prior to the PAW transformation, the resulting electron-phonon matrix elements are referred to as all-electron (AE). Conversely, applying the PAW transformation before the phonon perturbation yields pseudo (PS) matrix elements. While the AE matrix elements are universally applicable in second-order perturbation theory, the PS matrix elements provide superior numerical efficiency, robustness and convergence when computing the phonon-induced bandstructure renormalization. We demonstrate the advantages of each approach by calculating key observables, offering insights into their respective strengths for different computational scenarios.