| Summary: | Over time, many different theories and approaches have been developed to
tackle the many-body problem in quantum chemistry, condensed-matter physics,
and nuclear physics. Here we use the helium atom, a real system rather than a
model, and we use the exact solution of its Schr\"odinger equation as a
benchmark for comparison between methods. We present new results beyond the
random-phase approximation (RPA) from a renormalized RPA (r-RPA) in the
framework of the self-consistent RPA (SCRPA) originally developed in nuclear
physics, and compare them with various other approaches like configuration
interaction (CI), quantum Monte Carlo (QMC), time-dependent density-functional
theory (TDDFT), and the Bethe-Salpeter equation on top of the GW approximation.
Most of the calculations are consistently done on the same footing, e.g. using
the same basis set, in an effort for a most faithful comparison between
methods.
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