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This paper investigates the correlation between electrons and fermionic $^{13}$C and $^{19}$F nuclei using the random-phase approximation (RPA) and $GW$ self-energies. It establishes a link between RPA and second-order perturbation theory for inter-fermionic interactions and highlights the critical role of vertex corrections in mitigating self-interaction errors within $GW$ calculations for nuclear densities. The study provides theoretical and technical guidelines for quantum mechanical treatments of these nuclei.
Vertex corrections are essential for accurate $GW$ calculations of nuclear densities, revealing the limitations of standard self-energy approximations in electron-nuclei correlation.
We present a theoretical and numerical study of the correlation between electrons and the fermionic $^{13}$C and $^{19}$F nuclei. We use the random-phase approximation (RPA) as a valuable tool in obtaining these correlation energies. A special connection between the RPA and second-order perturbation theory for the inter-fermionic interaction is outlined. Subsequently, Green's function based $GW$ self-energies are evaluated for the nuclear densities. The strong influence of self-interaction errors is outlined, and vertex corrections are shown to be strictly necessary to obtain reasonable results. The theoretical and technical requirements for a quantum mechanical treatment of $^{13}$C and $^{19}$F nuclei are also addressed in this work, thereby facilitating further research in this area.
