  First-principles calculations are used to gauge different levels of
approximation to calculate the magnetocrystalline anisotropy energies (MAE) of
five $L1_0$ FeMe alloys (Me=Co, Cu, Pd, Pt, Au). We find that a second-order
perturbation (2PT) treatment of the spin-orbit interaction (SOI) breaks down
for the alloys containing heavier ions, while it provides a very accurate
description of the MAE behaviour of FeCo, FeCu, and FePd. Moreover, the
robustness of the 2PT approximation is such that in these cases it accounts for
the MAE of highly-non-neutral alloys and, thus, it can be used to predict their
performance when dopants are present or when they are subject to applied gate
bias, which are typical conditions in working magnetoelectric devices. We also
observe that switching of the easy axis direction can be induced in some of
these alloys by addition or removal of, at least, one electron per cell. In all
cases, the details of the bandstructure are responsible for the finally
observed MAE value and, therefore, suggest a limited predicting power of models
based on the expected orbital moment values and bandwidths. Finally, we have
confirmed the importance of various calculation parameters to obtain converged
MAE values, in particular, those related to the accuracy of the Fermi level
determination.
