  The non-relativistic large-$Z$ expansion of the exchange energy of neutral
atoms provides an important input to modern non-empirical density functional
approximations. Recent works report results of fitting the terms beyond the
dominant term, given by the local density approximation (LDA), leading to an
anomalous ZlnZ term that can not be predicted from naive scaling arguments.
Here, we provide much more detailed data analysis of the mostly smooth
asymptotic trend describing the difference between exact and LDA exchange
energy, the nature of oscillations across rows of the periodic table, and the
behavior of the LDA contribution itself. Special emphasis is given to the
successes and difficulties in reproducing the exchange energy and its
asymptotics with existing density functional approximations.
