  The Group V and VI transition metals share a common Fermi surface feature of
hole ellipsoids at the N point in the Brillouin zone. In clear contrast to the
other Fermi surface sheets, which are purely of d character, these arise from a
band that has a significant proportion of p character. By performing local
density approximation (LDA), generalized gradient approximation (GGA), strongly
constrained and appropriately normed (SCAN) meta-GGA, and GW approximation
calculations, we find that the p character part of this band (and therefore the
Fermi surface) is particularly sensitive to the exchange-correlation
approximation. LDA and GGA calculations inadequately describe this feature,
predicting N hole ellipsoid sizes that are consistently too large in comparison
to various experimental measurements, whereas quasiparticle self-consistent GW
calculations predict a size that is slightly too small (and non-self-consistent
GW calculations that use an LDA starting point predict a size that is much too
small). Overall, for the metals tested here, SCAN provides the most accurate
Fermi surface predictions, mostly correcting the discrepancies between
measurements and calculations that were observed when LDA calculations were
used. However, none of the tested exchange-correlation approximations succeeds
in simultaneously bringing all of the measurable properties of these metals
into good experimental agreement, particularly where magnetism is concerned.
The SCAN calculations predict antiferromagnetic moments for Cr that are 3 times
larger than the experimental value (1.90 $\mu_B$ compared to 0.62 $\mu_B$).
