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We compute electron-phonon coupling (EPC) of selected phonon modes ingraphene and graphite using various ab-initio methods. The inclusion ofnon-local exchange-correlation effects within the GW approach stronglyrenormalizes the square EPC of the A$'_1$ {\bf K} mode by almost 80$%$ withrespect to density functional theory in the LDA and GGA approximations. WithinGW, the phonon slope of the A$'_1$ {\bf K} mode is almost two times larger thanin GGA and LDA, in agreement with phonon dispersions from inelastic x-rayscattering and Raman spectroscopy. The hybrid B3LYP functional overestimatesthe EPC at {\bf K} by about 30%. Within the Hartree-Fock approximation, thegraphene structure displays an instability under a distortion following theA$'_1$ phonon at {\bf K}.

Impact of the electron-electron correlation on phonon dispersions: failure of LDA and GGA functionals in graphene and graphite