Electron-phonon coupling is at the origin of conventional superconductivity, enabling the pairing of electrons into Cooper pairs. The electron-phonon matrix elements depend on the electronic eigenstates and, in the standard linear approximation, on the first derivative of the potential felt by the electrons with respect to ionic perturbations. Here, we focus on the derivatives of the potential with a twofold aim: to assess their contribution to the overall coupling and to analyze the limitations of neglecting higher-order derivatives. Several real-space functions are proposed to do the analysis, and are computed for some well-known superconductors. Our results show that, in hydrides, the derivatives of the potential tend to be larger in regions of high electron localization, explaining the success of electronic descriptors previously observed to correlate with the critical temperature. The new functions introduced here are able to tell apart structures with similar types of bonding but very different critical temperatures, such as ${\mathrm{H}}_3\mathrm{S}$ and ${\mathrm{H}}_3\mathrm{Se} Im\overline{3}m$ phases, where electronic descriptors alone fail. Moreover, they provide a method to discriminate promising superconductors from conventional low-$T_c$ materials. Interestingly, our descriptors are capable of easily estimating the impact of higher-order terms in the electron-phonon coupling. In fact, we capture the limitations of the linear approximation expected for PdH, and we predict an even more important nonlinear behavior in other hydrides.

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• Electron-phonon coupling
• Superconductivity
• Hydrides
• First-principles calculations