Strong electron-electron interactions are known to significantly modify the electron-phonon coupling relative to the predictions of density functional theory, but this effect is challenging to calculate with realistic theories of strongly correlated materials. Here we define and calculate a version of the electron-phonon coupling applicable beyond band theory by combining first-principles density functional theory plus dynamical mean-field theory with finite-difference phonon perturbations, presenting results for several representative phonon modes in two materials of interest. In the three-orbital correlated metal ${\mathrm{SrVO}}_3$, we find that intra-V-$t_{2g}$-band correlation significantly increases the coupling of these electrons to a Jahn-Teller phonon mode that splits the degenerate orbital energies, while slightly reducing the coupling associated with a breathing phonon that couples to the charge on each V atom. In the infinite layer cuprate ${\mathrm{CaCuO}}_2$, we find that local correlation within the $d_{x^2-y^2}$ orbital derived band has a modest effect on coupling of near-Fermi-surface electrons to optical breathing modes. In both cases, the interaction correction to the electron-phonon coupling predicted by dynamical mean-field theory has a significant dependence on the electronic frequency, arising from a lattice-distortion dependence of the correlated electron dynamics, showing the inadequacy of the simple picture in which correlations change static local susceptibilities. We also show that the electron-phonon scattering and phonon lifetimes associated with these phonon modes are modified by the electronic correlation. Our findings shed light on the material- and mode-specific role of dynamical electronic correlation in electron-phonon coupling and highlight the importance of developing efficient computational methods for treating electron-phonon coupling in correlated materials.

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• Electron-phonon coupling
• Electronic structure
• Strongly correlated systems
• Transition metal oxides
• Dynamical mean field theory
• First-principles calculations