The origin of the spin-crossover phenomenon in ${\mathrm{LaCoO}}_3$ is discussed based on the electronic state ${}^{5}D$ realized in a $3d^6$ system subjected to an intermediate crystal field. We calculate the splitting of the ${}^{5}D$ state under a crystal field whose symmetry is predominantly cubic but with a trigonal distortion. We confine ourselves to the ${\mathrm{\Gamma }}_5$ state and use a fictitious orbital angular momentum $\mathit{l}$ of magnitude one. Eigenvalues of a Hamiltonian including the spin-orbit interaction $-{\lambda }^{\prime }\mathit{l}·\mathit{S}$ and the trigonal crystal fields $9B_2^0(l_z^2-2/3)-80B_4^0(l_z^2-9/10)$ are rigorously calculated. A singlet ground state with an energy gap to the excited states is realized for certain values of the parameters in the Hamiltonian. The magnetization calculated rigorously using the energy levels subject to a magnetic field accords with published measurements.

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• Magnetic order
• Magnetism
• Spin-orbit coupling