Verification of first-principles codes: comparison of total energies, phonon frequencies, electron-phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo

With the ever-increasing sophistication of codes, the verification of the implementation of advanced theoretical formalisms becomes critical. In particular, cross comparison between different codes provides a strong hint in favor of the correctness of the implementations, and a measure of the (hopefully small) possible numerical differences. We lead a rigorous and careful study of the quantities that enter in the calculation of the zero-point motion renormalization of the direct band gap of diamond due to electron-phonon coupling, starting from the total energy, and going through the computation of phonon frequencies and electron-phonon matrix elements. We rely on two independent implementations : Quantum Espresso + Yambo and ABINIT. We provide the order of magnitude of the numerical discrepancies between the codes, that are present for the different quantities: less than $10^{-5}$ Hartree per atom on the total energy (-5.722 Ha/at), less than 0.07 cm$^{-1}$ on the $Gamma,L,X$ phonon frequencies (555 to 1330 cm$^{-1}$), less than 0.5% on the square of the electron-phonon matrix elements and less than 4 meV on the zero-point motion renormalization of each eigenenergies (44 to 264 meV). Within our approximations, the DFT converged direct band gap renormalization in diamond due to the electron-phonon coupling is -0.409 eV (reduction of the band gap).