The magnetic energy levels of a given magnetic solid are closely packed in energy because the interactions between magnetic ions are weak. Thus, in describing its magnetic properties, one needs to generate its magnetic energy spectrum by employing an appropriate spin Hamiltonian. In this review article we discuss how to determine and specify a necessary spin Hamiltonian in terms of first principles electronic structure calculations on the basis of energy-mapping analysis and briefly survey important concepts and phenomena that one encounters in reading the current literature on magnetic solids. Our discussion is given on a qualitative level from the perspective of magnetic energy levels and electronic structures. The spin Hamiltonian appropriate for a magnetic system should be based on its spin lattice, i.e., the repeat pattern of its strong magnetic bonds (strong spin exchange paths), which requires one to evaluate its Heisenberg spin exchanges on the basis of energy-mapping analysis. Other weaker energy terms such as Dzyaloshinskii–Moriya (DM) spin exchange and magnetocrystalline anisotropy energies, which a spin Hamiltonian must include in certain cases, can also be evaluated by performing energy-mapping analysis. We show that the spin orientation of a transition-metal magnetic ion can be easily explained by considering its split d-block levels as unperturbed states with the spin–orbit coupling (SOC) as perturbation, that the DM exchange between adjacent spin sites can become comparable in strength to the Heisenberg spin exchange when the two spin sites are not chemically equivalent, and that the DM interaction between rare-earth and transition-metal cations is governed largely by the magnetic orbitals of the rare-earth cation.

The ferroelectric polarization of triangular-lattice antiferromagnets induced by helical spin-spiral order is not explained by any existing model of magnetic-order-driven ferroelectricity. We resolve this problem by developing a general theory for the ferroelectric polarization induced by spin-spiral order and then by evaluating the coefficients needed to specify the general theory on the basis of density functional calculations. Our theory correctly describes the ferroelectricity of triangular-lattice antiferromagnets driven by helical spin-spiral order and incorporates known models of magnetic-order-driven ferroelectricity as special cases.

The nature of the charge order in the charge frustrated compound ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ and its effect on magnetocapacitance were examined on the basis of first-principles electronic structure calculations and Monte Carlo simulations of electrostatic energy. Our work shows that two different types of charge order of almost equal stability (i.e., $\sqrt{3}\times \sqrt{3}$ and chain types) occur in the ${\mathrm{Fe}}_2{\mathrm{O}}_4$ layers of ${\mathrm{LuFe}}_2{\mathrm{O}}_4$, and that the ground state of ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ has a ferrielectric arrangement of the ${\mathrm{Fe}}_2{\mathrm{O}}_4$ layers with $\sqrt{3}\times \sqrt{3}$ charge order. The giant magnetocapacitance effect of ${\mathrm{LuFe}}_2{\mathrm{O}}_4$ at room temperature is accounted for in terms of charge fluctuations arising from the interconversion between the two types of charge order, that becomes hindered by an applied magnetic field.