  We revisit the two derivations of the PBE correlation functional: The
real-space cut-off of the exchange-correlation hole and the imposition of exact
conditions. These differ in the Lieb-Simon limit, exemplified by the scaling of
neutral atoms to large $N$ and $Z$, in which LDA becomes relatively exact. We
use the leading correction to this limit for neutral atoms to design an
asymptotically corrected correlation GGA as a compromise between these two
constructions which becomes relatively more accurate for atoms with increasing
atomic number. When paired with a similar correction for exchange, this acGGA
satisfies more exact conditions than PBE. Combined with the known
$r_s$-dependence of the gradient expansion for correlation, this correction
accurately reproduces correlation energies of closed shell atoms down to Be. We
test this acGGA for atoms and molecules, finding substantial improvements over
PBE, but also showing that optimal global hybrids of acGGA do not improve upon
PBE0, and are similar to meta-GGA values. We discuss the relevance of these
results to Jacob's ladder of non-empirical density functional construction.
