Microscopic origin of isotropic non-Heisenberg behavior in $S=1$ magnetic systems

We have reanalyzed the microscopic origin of the isotropic deviations that are observed from the energy spacings predicted by the Heisenberg–Dirac–van Vleck (HDVV) Hamiltonian. Usually, a biquadratic spin operator is added to the HDVV Hamiltonian to account for such deviations. It is shown here that this operator cannot describe the effect of the excited atomic non-Hund states which brought the most important contribution to the deviations. For systems containing more than two magnetic centers, non-Hund states cause additional interactions that are of the same order of magnitude as the biquadratic exchange and should have significant effects on the macroscopic properties of extended systems.

Figure 1

Two pathways of the fourth-order quasidegenerate perturbation theory interaction between $T_{+}{T}_{-}{T}_0$ and $T_0{T}_{+}{T}_{-}$. The coupling goes through ionic configurations, then a function involving a non-Hund state, and finally again ionic configurations. The local spin functions $T_0$ and $S_0$ are, respectively, distinguished by the signs + and − in upper index of the parentheses.

Figure 2

Left strongly localized orbitals.

Figure 3

Comparison of the model spectra with the $\mathrm{CAS}+\mathrm{DDCI}2$ spectrum. (a) Heisenberg model with $J_{12}^{\mathit{eff}}$, (b) Heisenberg model extended with ${\lambda }_{12}^{\mathit{eff}}$, (e) magnetic Hamiltonian given in Eq. (5) including $J_{12}^{\mathit{eff}}$, ${\lambda }_{12}^{\mathit{eff}}$, and the three-body operator interactions ${\gamma }_{12}^{\mathit{eff}}{\gamma }_{23}^{\mathit{eff}}$, and (f) magnetic Hamiltonian given in Eq. (5) extended with the second neighbor interactions $J_{13}^{\mathit{eff}}$, ${\lambda }_{13}^{\mathit{eff}}$, and ${\gamma }_{12}^{\mathit{eff}}{\gamma }_{13}^{\mathit{eff}}$.