${\mathrm{TiF}}_3$: Almost nonmagnetic state of a spin-$\frac{1}{2}$ system

We have explained the nonmagnetic state of ${\mathrm{TiF}}_3$, down to 10 K, in a strongly-correlated electron approach treating one $d$ electron of the ${\mathrm{Ti}}^{3+}$ ion as fully localized. We have derived the low-energy states, their energies in the meV energy scale, and their eigenfunctions, considering octahedral crystal-field, trigonal distortion, and spin-orbit interactions. The calculated ground state of the ${\mathrm{Ti}}^{3+}$ ion in ${\mathrm{TiF}}_3$ originates from the $e_g^{\pi }$ orbital doublet, contrary to the $a_{1g}$ orbital singlet reported on the basis of first-principles calculations. The ground Kramers-doublet state is almost nonmagnetic due to compensation of the spin and orbital momenta. Our approach reveals the fundamental importance of intra-atomic relativistic spin-orbit coupling for calculations of the magnetism and the electronic structure of $3d$ fluorides/oxides, pointing out that strong electron correlations are predominantly of a local on-site origin.

Figure 1

The calculated extra effect of the spin-orbit (${\lambda }_{\text{SO}}$) interactions on the $t_{2g}$ triplet states, split by the trigonal stretched off-octahedral distortion (measured by a $B_2^0$ CEF parameter of $-40\mathrm{K})$ in the presence of an octahedral crystal field $(B_4^z=+120\mathrm{K})$. ${\lambda }_{\text{SO}}$ of $+220\mathrm{K}$ is relevant to the ${\mathrm{Ti}}^{3+}$ ion. The distortion and SO energy effects are of comparable size. Zero energy is at the two Hund's rules ${}^{2}D$ term energy. The magnetic moment of the lowest Kramers doublet is very small, of $\pm 0.06{\mu }_B$ for ${\lambda }_{\text{SO}}=220\mathrm{K}$, due to the cancellation of the spin moment by the orbital moment.