Zero-Field Splitting of S -State Ions. II. Overlap and Covalency Model

A general formulation including both overlap and charge-transfer covalency effects is developed for the parameters D and E in the spin Hamiltonian HS=D(3Sz²-S(S+1))+E(Sₓ²-Sz²) for a ⁶S ion surrounded by six singly charged negative closed-shell ions. The contributions to D and E arise in second order from the spin-orbit interaction and in first order from the spin-spin interaction. These contributions can be broken down into three categories: local, nonlocal, and distant, depending on whether the ligand orbitals are involved not at all, once, or twice. Specific application is made to the case of a Mn²⁺ ion surrounded by six F^- ions (Mn²⁺:ZnF₂). The calculations are performed for the recent crystal-structure parameters given by Baur. We find, on the one hand, that for the spin-spin interaction, the nonlocal terms predominate over the local and distant terms, and are of the same sign. On the other hand, for the spin-orbit interaction, the local and distant terms predominate over the nonlocal terms, and are of opposite sign, though similar in magnitude. The total (overlap only) contribution to D is found to be -10.94×{}10^-4 cm^-1, which is comparable and of opposite sign to the point-charge contribution 24.01×{}10^-4 cm^-1 calculated in our earlier publication on this problem. The sum of the two contributions (point charge and overlap) is found to be 13.07×{}10^-4 cm^-1, very close to the experimental value of +10.5×{}10^-4 cm^-1. The overlap contribution to E is found to be -19.77×{}10^-4 cm^-1, which is rather smaller than the point-charge contribution of -102.32×{}10^-4 cm^-1. The combined result, E=-122.09×10^-04 cm^-1, is also close to the experimental value of -113.5×{}10^-4 cm^-1.