  We show that the electron-phonon coupling (EPC) in many materials can be
significantly underestimated by the standard density functional theory (DFT) in
the local density approximation (LDA) due to large non-local correlation
effects. We present a simple yet efficient methodology to evaluate the
realistic EPC going beyond LDA by using more advanced and accurate GW and
screened hybrid functional DFT approaches. The corrections we propose explain
the extraordinarily high superconducting temperatures that are observed in two
distinct classes of compounds-the bismuthates and the transition metal
chloronitrides, thus solving a thirty-year-old puzzle. Our work calls for the
critically reevaluation of the EPC of certain phonon modes in many other
materials such as cuprates and iron-based superconductors. The proposed
methodology can be used to design new correlation-enhanced high temperature
superconductors and other functional materials involving electron-phonon
interaction.
