  In this chapter we review the quantitative and qualitative aspects of
describing the properties of magnetic solids on the basis of electronic
Hamiltonian. We show that a spin Hamiltonian approach becomes consistent with
an electronic Hamiltonian approach if the spin lattice and its associated spin
exchange parameters, to be used for the spin Hamiltonian, are determined by the
energy-mapping analysis based on DFT calculations. The preferred spin
orientation (i.e., the magnetic anisotropy) of a magnetic ion is not predicted
by a spin Hamiltonian because it does not include the orbital degree of freedom
explicitly. In contrast, the magnetic anisotropy is readily predicted by
electronic structure theories employing both orbital and spin degrees of
freedom, if one takes into consideration the spin-orbit coupling (SOC). It was
shown that the preferred spin orientation of a magnetic ion can be predicted
and understood in terms of the HOMO-LUMO interactions of the magnetic ion by
taking SOC as perturbation. A spin Hamiltonian gives rise to the spin-half
misconception, namely, the blind belief that spin-half magnetic ions do not
possess magnetic anisotropy that arise from SOC. This misconception is a direct
consequence from the limitedness of a spin Hamiltonian that it lacks the
orbital degree of freedom. We show that the magnetic properties of 5d ion
oxides are better explained by the LS-coupling than by the jj-coupling scheme
of SOC, that the spin-orbital entanglement of 5d ions is not as strong as has
been assumed.
