  Using multiwavelets, we have obtained total energies and corresponding
atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a
test set of 211 molecules with an unprecedented and guaranteed $\mu$Hartree
accuracy. These quasi-exact references allow us to quantify the accuracy of
standard all-electron basis sets that are believed to be highly accurate for
molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric
atom-centered orbitals (NAOs) and full-potential augmented plane wave (APW)
methods. We show that NAOs are able to achieve the so-called chemical accuracy
(1 kcal/mol) for the typical basis set sizes used in applications, for both
total and atomization energies. For GTOs, a triple-zeta quality basis has mean
errors of ~10kcal/mol in total energies, while chemical accuracy is almost
reached for a quintuple-zeta basis. Due to systematic error cancellations,
atomization energy errors are reduced by almost an order of magnitude, placing
chemical accuracy within reach also for medium to large GTO bases, albeit with
significant outliers. In order to check the accuracy of the computed densities,
we have also investigated the dipole moments, where in general, only the
largest NAO and GTO bases are able to yield errors below 0.01 Debye. The
observed errors are similar across the different functionals considered here.
