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 Ha per atom on the total energy (−5.722 Ha/at), less than 0.07 cm - 1 on the Γ , L , X phonon frequencies (555–1330 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 (44–264 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).