Covalency Effects in KNiF3. III. Theoretical Studies

Abstract

For the ${(\mathrm{N}\mathrm{i}{\mathrm{F}}_{6})}^{4\ensuremath{-}}$ complex in KNi${\mathrm{F}}_{3}$ we have constructed molecular orbitals (MO) which are linear combinations of the ${\mathrm{Ni}}^{2+}$ and ${\mathrm{F}}^{\ensuremath{-}}$ Hartree-Fock atomic orbitals. These LCAO-MO, introduced by Van Vleck, are of the form $\ensuremath{\Psi}={N}^{\ensuremath{-}\frac{1}{2}}(\ensuremath{\phi}\ensuremath{-}\ensuremath{\lambda}\ensuremath{\chi})$ in which $\ensuremath{\phi}$ is the ${\mathrm{Ni}}^{2+}$ $3d$ function and $\ensuremath{\chi}$ a linear combination of the suitable ${\mathrm{F}}^{\ensuremath{-}}$ functions. The orbitals were assumed to be solutions of Schr\"odinger's equation $h\ensuremath{\Psi}=E\ensuremath{\Psi}$, where the Hamiltonian was $h=\ensuremath{-}\frac{\ensuremath{\Delta}}{2}+{V}_{M}+{V}_{L}$. The terms ${V}_{M}$ and ${V}_{L}$ describe the Coulombic and exchange interactions with the metal ion and ligands, respectively. Matrix elements of the form $〈\ensuremath{\Psi}|h|\ensuremath{\Psi}〉$ were evaluated numerically on an IBM 7090. Assuming $\ensuremath{\lambda}$ and the overlap between $\ensuremath{\phi}$ and $\ensuremath{\chi}$ to be small, the energy was minimized and the parameters $\ensuremath{\lambda}$ were determined. For the $2p\ensuremath{\sigma}$ bonding and the $2s$ bonding the calculated values were ${{N}_{e}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\lambda}}_{\ensuremath{\sigma}}=0.383$ and ${{N}_{e}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\lambda}}_{s}=0.109$ which agreed very well with the values ${{N}_{e}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\lambda}}_{p\ensuremath{\sigma}}=0.337$ and ${{N}_{e}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\lambda}}_{s}=0.116$ determined in the nuclear magnetic resonance experiment. The molecular orbitals were used to calculate the cubic crystal field splitting $10Dq=({\ensuremath{\Psi}}_{e}|h|{\ensuremath{\Psi}}_{e})\ensuremath{-}({\ensuremath{\Psi}}_{t}|h|{\ensuremath{\Psi}}_{t})$ which is the promotion energy of an electron from a ${t}_{2g}$ orbital to an ${e}_{g}$ orbital. The calculated value of $10Dq=6350$ ${\mathrm{cm}}^{\ensuremath{-}1}$ agreed quite well with the observed value of $10Dq=7250$ ${\mathrm{cm}}^{\ensuremath{-}1}$ considering the accuracy of the calculation. Furthermore, the reduction of the spin-orbit parameter and the Racah parameter $B$ from their free-ion values are satisfactorily explained by the molecular orbital approach. The physical interpretation of these results is emphasized. In particular, the only contributions to $10Dq$ with the correct sign come from the off-diagonal matrix elements associated with the covalency; the amount of $\ensuremath{\pi}$ electron admixture is shown to be large; one novel physical mechanism partly responsible for the large $\ensuremath{\pi}$ bonding is the crystal field splitting of the ${\mathrm{F}}^{\ensuremath{-}}$ $p\ensuremath{\sigma}$ and $p\ensuremath{\pi}$ levels by the ${\mathrm{Ni}}^{2+}$ ions; expanding the ${\mathrm{Ni}}^{2+}$ radial function is shown to be unnecessary for some purposes and incorrect for the remainder. Details of the calculation are presented and implications of the LCAO-MO model discussed.