  For more than three decades, nearly free electron elemental metals have been
a topic of debate because the computed bandwidths are significantly wider in
the local density approximation to density-functional theory (DFT) than
indicated by angle-resolved photoemission (ARPES) experiments. Here, we
systematically investigate this using first-principles calculations for alkali
and alkaline-earth metals using DFT and various beyond-DFT methods such as
meta-GGA, G$_0$W$_0$, hybrid functionals (YS-PBE0, B3LYP), and LDA+eDMFT. We
find that the static non-local exchange, as partly included in the hybrid
functionals, significantly increase the bandwidths even compared to LDA, while
the G$_0$W$_0$ bands are only slightly narrower than in LDA. The agreement with
the ARPES is best when the local approximation to the self-energy is used in
the LDA+eDMFT method. We infer that even moderately correlated systems with
partially occupied s-orbitals, which were assumed to approximate the uniform
electron gas, are very well described in terms of short-range dynamical
correlations that are only local to an atom.
