Zero-Field Splitting of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>-State Ions. II. Overlap and Covalency Model

Abstract

A general formulation including both overlap and charge-transfer covalency effects is developed for the parameters $D$ and $E$ in the spin Hamiltonian ${\mathcal{H}}_{S}=D(3{{S}_{z}}^{2}\ensuremath{-}S(S+1))+E({{S}_{x}}^{2}\ensuremath{-}{{S}_{z}}^{2})$ for a $^{6}S$ 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 ${\mathrm{Mn}}^{2+}$ ion surrounded by six ${\mathrm{F}}^{\ensuremath{-}}$ ions (${\mathrm{Mn}}^{2+}$:Zn${\mathrm{F}}_{2}$). 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\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, which is comparable and of opposite sign to the point-charge contribution 24.01\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ calculated in our earlier publication on this problem. The sum of the two contributions (point charge and overlap) is found to be 13.07\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, very close to the experimental value of +10.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$. The overlap contribution to $E$ is found to be -19.77\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, which is rather smaller than the point-charge contribution of -102.32\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$. The combined result, $E=\ensuremath{-}122.09\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}04}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, is also close to the experimental value of -113.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$.