Cooperation between Electron-Phonon Coupling and Electronic Interaction in Bilayer Nickelates La_{3}Ni_{2}O_{7}

The recent observation of high-T_{c} superconductivity in the bilayer nickelate La_{3}Ni_{2}O_{7} under pressure has garnered significant interest. While researches have predominantly focused on the role of electron-electron interactions in the superconducting mechanism, the impact of electron-phonon coupling (EPC) has remained elusive and unexplored. In this Letter, we perform first-principles calculations to study the phonon spectrum and electron-phonon coupling within La_{3}Ni_{2}O_{7} under pressure and explore the interplay between EPC and electronic interactions on the superconductivity by employing functional renormalization group (FRG) approach. Our calculations reveal that EPC alone is insufficient to trigger superconductivity in La_{3}Ni_{2}O_{7} under pressure. We identify unique out-of-plane and in-plane breathing phonon modes which selectively couple with the Ni d_{z^{2}} and d_{x^{2}-y^{2}} orbitals, showcasing an orbital-selective EPC. Within the bilayer two-orbital model, it is revealed that solely electronic interactions foster s_{±}-wave pairing characterized by notable frustration in the band space, leading to a relatively low transition temperature. Remarkably, we find that the out-of-plane EPC can act in concert with electronic interactions to promote the interlayer pairing in the d_{z^{2}} orbital, partially releasing the pairing frustration and thus elevating T_{c}. In contrast, the inclusion of in-plane EPC only marginally affects the superconductivity, distinct from the cuprates. Potential experimental implications in La_{3}Ni_{2}O_{7} are also discussed.