We report ab initio calculations of the electronic and vibrational properties in ${\mathrm{CuPb}}_9{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$, including the electron-phonon coupling strength via strong-coupling Migdal-Eliashberg theory. We verify the presence of appealing flat electronic bands near the Fermi level, a strong hybridization between the Cu $3d$ and O $2p$ states, and soft low-energy phonons, which can suggest high-temperature superconducting behavior. However, the electron-phonon coupling strength appears insufficient to overcome the Coulomb repulsion between an electron pair and thus does not support high-temperature superconductivity in ${\mathrm{CuPb}}_9{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ via the conventional electron-phonon Migdal-Eliashberg mechanism. Even neglecting Coulomb repulsion of the electron pair we find this electron-phonon coupling suggests a superconducting transition temperature of less than 2 K.

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• Electron-phonon coupling
• Electronic structure
• Phonons
• Superconductivity
• Density functional theory
• Eliashberg theory