<i>Ab initio</i> correlated calculations without finite-basis-set error: Numerically precise all-electron random-phase-approximation correlation energies for diatomic molecules

In wave-function-based ab initio quantum mechanical calculations, achieving absolute convergence with respect to the one-electron basis set is a long-standing challenge. In this work, using the random-phase approximation (RPA) electron correlation energy as an example, we show how to compute the basis-error-free RPA correlation energy for diatomic molecules by iteratively solving the Sternheimer equations for first-order wave functions in the prolate spheroidal coordinate system. Our approach provides RPA correlation energies across the periodic table to any desired precision; in practice, the convergence of the absolute RPA energies to the meV-level accuracy can be readily attained. Our method thus provides unprecedented reference numbers that can be used to assess the reliability of the commonly used computational procedures in quantum chemistry. The numerical techniques developed in the present work also have direct implications for the development of basis error-free schemes for the GW method or the ab initio quantum chemistry methods such as second-order M\o{}ller-Plesset perturbation theory.