Covalency Effects in KNiF3. III. Theoretical Studies

Abstract

For the ${\left(\mathrm{N}\mathrm{i}{\mathrm{F}}_6\right)}^{4-}$ complex in KNi${\mathrm{F}}_3$ we have constructed molecular orbitals (MO) which are linear combinations of the ${\mathrm{Ni}}^{2+}$ and ${\mathrm{F}}^{-}$ Hartree-Fock atomic orbitals. These LCAO-MO, introduced by Van Vleck, are of the form $\Psi =N^{-\frac{1}{2}}(\varphi -\lambda \chi )$ in which $\varphi$ is the ${\mathrm{Ni}}^{2+}$ $3d$ function and $\chi$ a linear combination of the suitable ${\mathrm{F}}^{-}$ functions. The orbitals were assumed to be solutions of Schrödinger's equation $h\Psi =E\Psi$, where the Hamiltonian was $h=-\frac{\Delta }{2}+V_M+V_L$. 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 $〈\Psi \mathrm{}\left|h\right|\mathrm{}\Psi 〉$ were evaluated numerically on an IBM 7090. Assuming $\lambda$ and the overlap between $\varphi$ and $\chi$ to be small, the energy was minimized and the parameters $\lambda$ were determined. For the $2p\sigma$ bonding and the $2s$ bonding the calculated values were ${N_e}^{-\frac{1}{2}}{\lambda }_{\sigma }=0.383\mathrm{}$ and ${N_e}^{-\frac{1}{2}}{\lambda }_s=0.109\mathrm{}$ which agreed very well with the values ${N_e}^{-\frac{1}{2}}{\lambda }_{p\sigma }=0.337\mathrm{}$ and ${N_e}^{-\frac{1}{2}}{\lambda }_s=0.116\mathrm{}$ determined in the nuclear magnetic resonance experiment. The molecular orbitals were used to calculate the cubic crystal field splitting $10Dq=\left({\Psi }_e\mathrm{}\right|h\left|\mathrm{}{\Psi }_e\right)-\left({\Psi }_t\mathrm{}\right|h\left|\mathrm{}{\Psi }_t\right)$ which is the promotion energy of an electron from a $t_{2g}$ orbital to an $e_g$ orbital. The calculated value of $10Dq=6350$ ${\mathrm{cm}}^{-1\mathrm{}}$ agreed quite well with the observed value of $10Dq=7250$ ${\mathrm{cm}}^{-1\mathrm{}}$ 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 $\pi$ electron admixture is shown to be large; one novel physical mechanism partly responsible for the large $\pi$ bonding is the crystal field splitting of the ${\mathrm{F}}^{-}$ $p\sigma$ and $p\pi$ 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.