  The recent non-empirical semi-local exchange functional of Armiento and
K\"ummel, the AK13 [PRL 111, 036402 (2013)] incorporates a number of features
reproduced by higher-order theory. The AK13 potential behaves analogously with
the discontinuous jump associated with the derivative discontinuity at integer
particle numbers. Recent works have established that AK13 gives a qualitatively
improved orbital description compared to other semi-local methods, and
reproduces a band structure closer to higher-order theory. However, its
energies and energetics are inaccurate. The present work further investigates
the deficiency in energetics. In addition to AK13 results, we find that
applying the local-density approximation (LDA) non-self-consistently on the
converged AK13 density gives very reasonable energetics with equilibrium
lattice constants and bulk moduli well described across 14 systems. We also
confirm that the attractive orbital features of AK13 are retained even after
full structural relaxation. Hence, the deficient energetics cannot be a result
of the AK13 orbitals having adversely affected the quality of the electron
density compared to that of usual semi-local functionals; an improved orbital
description and good energetics are not in opposition. We also prove that the
non-self-consistent scheme is equivalent to using a single external-potential
dependent functional in an otherwise consistent KS-DFT scheme. Furthermore, our
results also demonstrate that, while an internally consistent KS functional is
presently missing, non-self-consistent LDA on AK13 orbitals works as a
practical non-empirical computational scheme to predict geometries, bulk
moduli, while retaining the band structure features of AK13 at the
computational cost of semi-local DFT.
