  Electron-electron ($e$-$e$) and electron-phonon ($e$-ph) interactions are
challenging to describe in correlated materials, where their joint effects
govern unconventional transport, phase transitions, and superconductivity. Here
we combine first-principles $e$-ph calculations with dynamical mean field
theory (DMFT) as a step toward a unified description of $e$-$e$ and $e$-ph
interactions in correlated materials. We compute the $e$-ph self-energy using
the DMFT electron Green's function, and combine it with the $e$-$e$ self-energy
from DMFT to obtain a Green's function including both interactions. This
approach captures the renormalization of quasiparticle dispersion and spectral
weight on equal footing. Using our method, we study the $e$-ph and $e$-$e$
contributions to the resistivity and spectral functions in the correlated metal
Sr$_2$RuO$_4$. In this material, our results show that $e$-$e$ interactions
dominate transport and spectral broadening in the temperature range we study
(50$-$310~K), while $e$-ph interactions are relatively weak and account for
only $\sim$10\% of the experimental resistivity. We also compute effective
scattering rates, and find that the $e$-$e$ interactions result in scattering
several times greater than the Planckian value $k_BT$, whereas $e$-ph
interactions are associated with scattering rates lower than $k_BT$. Our work
demonstrates a first-principles approach to combine electron dynamical
correlations from DMFT with $e$-ph interactions in a consistent way, advancing
quantitative studies of correlated materials.
