Negative $g$ Factors, Berry Phases, and Magnetic Properties of Complexes

It is shown that the sign of the product of three Zeeman splitting factors corresponding to the main magnetic axes defines the sign of the Berry phase of a (pseudo) spin in an applied magneic field. Ab initio calculations show that $g_X{g}_Y{g}_Z<0$ is often the case for lanthanide and transition metal complexes, while we prove that it is never achieved in $S$ complexes with dominant second-order magnetic anisotropy. In the case of polynuclear compounds, it is argued that the signs of individual $g_i$, $i=X$, $Y$, $Z$, on each metal site can be extracted from experiment.

Figure 1

The structure of the ${\mathrm{Er}}^{3+}$ complex [24] showing $g_X{g}_Y{g}_Z<0$ in the ground state.

Figure 2

The values of $g_X{g}_Z/g_Y$ for individual Kramers doublets arising from the zero-field splitting of $S=3/2$ (dotted blue line), $S=5/2$ (dashed green line), and $S=7/2$ (solid red line) in the weak spin-orbit coupling limit.

Figure 3

(a) Energies and (b) $g_X{g}_Z/g_Y$ values of the lowest six Kramers doublets in ${\mathrm{CoCl}}_6^{4-}$ as a function of tetragonal elongation or compression of two Co-Cl bonds along one axis. The octahedral geometry corresponds to $R_{\mathrm{Cu}\mathrm{\text{−}}\mathrm{Cl}}=2.60\AA$.

Figure 4

Variations of $g_X{g}_Z/g_Y$ for the ground Kramers doublet of ${\mathrm{CuCl}}_4^{2-}$ as a function of two deformation modes of the ligand environment.

Figure 5

Zeeman splitting of the exchange multiplets of a model binuclear system with (a) the same and (b) opposite signs of the $g$ factors on the two metal sites.