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In this chapter we review the quantitative and qualitative aspects ofdescribing the properties of magnetic solids on the basis of electronicHamiltonian. We show that a spin Hamiltonian approach becomes consistent withan electronic Hamiltonian approach if the spin lattice and its associated spinexchange parameters, to be used for the spin Hamiltonian, are determined by theenergy-mapping analysis based on DFT calculations. The preferred spinorientation (i.e., the magnetic anisotropy) of a magnetic ion is not predictedby a spin Hamiltonian because it does not include the orbital degree of freedomexplicitly. In contrast, the magnetic anisotropy is readily predicted byelectronic structure theories employing both orbital and spin degrees offreedom, if one takes into consideration the spin-orbit coupling (SOC). It wasshown that the preferred spin orientation of a magnetic ion can be predictedand understood in terms of the HOMO-LUMO interactions of the magnetic ion bytaking SOC as perturbation. A spin Hamiltonian gives rise to the spin-halfmisconception, namely, the blind belief that spin-half magnetic ions do notpossess magnetic anisotropy that arise from SOC. This misconception is a directconsequence from the limitedness of a spin Hamiltonian that it lacks theorbital degree of freedom. We show that the magnetic properties of 5d ionoxides are better explained by the LS-coupling than by the jj-coupling schemeof SOC, that the spin-orbital entanglement of 5d ions is not as strong as hasbeen assumed.

Magnetic Properties from the Viewpoints of Electronic Hamiltonian: Spin Exchange Parameters, Spin Orientation and Spin-Half Misconception