  We derive a second-order perturbation formula for an electronic system
subject to spin-orbit interactions (SOI). The energy correction originates in
the spin-conserving and the spin-flip transitions. The former are represented
by the orbital angular momentum (OAM) acquired via the SOI. The latter come
from the quantum fluctuation effect. By using our formula, we examine the
relativistic electronic structures of a d orbital chain and L1_0 alloys. The
appearance of OAM in the chain is understood by using a parabolic-bands model
and the exact expressions of the single-particle states. The total energy is
found to be accurately reproduced by the formula. The self-consistent fully
relativistic first-principles calculations based on the density functional
theory are performed for five L1_0 alloys. It is found that the formula
reproduces qualitatively the behavior of their exact magnetocrystalline
anisotropy (MCA) energies. While the MCA of FePt, CoPt, and FePd originates in
the spin-conserving transitions, that in MnAl and MnGa originates in the
spin-flip contributions. For FePt, CoPt, and FePd, the tendency of the MCA
energy with the variation in the lattice constant obeys basically that of the
spin-flip contributions. These results indicate that not only the anisotropy of
OAM, but also that of spin-flip contributions must be taken into account for
the understanding of the MCA of the L1_0 alloys.
