Theory of transferred hyperfine interaction in $\sigma$-bonding systems: ${\mathrm{Cu}}^{2+}$ ion and ${\mathrm{Cu}}^{2+}$${\mathrm{F}}^{-}$${\mathrm{Cu}}^{2+}$ pair

The theory of transferred hyperfine interaction at the ${\mathrm{F}}^{-}$ site in the $\sigma$-electron systems ${\mathrm{Cu}}^{2+}$ ${\mathrm{F}}^{-}$ and ${\mathrm{Cu}}^{2+}$ ${\mathrm{F}}^{-}$ ${\mathrm{Cu}}^{2+}$ in cubic crystals is developed from the first principles. The transfer of the $\sigma$ electron from the occupied $p$ and $s$ orbits of ${\mathrm{F}}^{-}$ to the empty $d_{z^2}$ orbit on the ${\mathrm{Cu}}^{2+}$ ion is considered and all the integrals of the kinetic energy and Coulomb terms are evaluated using Hartree-Fock wave functions. The spin density in the case of one ${\mathrm{Cu}}^{2+}$ ion is found to be different from that in the ${\mathrm{Cu}}^{2+}$ ion-pair case. The $s$-spin density is found to vary with the interactomic distance as $R^{-5.63\mathrm{}}$ in the pair case. The variation of the $p_{\sigma }$ spin density with interatomic separation is rather involved. As the${\mathrm{Cu}}^{2+}$ ${\mathit{F}}^{-}$ distance is increased from 1.8 Å, the $p_{\sigma }$ spin density first decreases going through a minimum at about $R=2.0\mathrm{}$ Å and is almost independent of distance from $R=2\mathrm{}$ Å to $R=2.1\mathrm{}$ Å. For $R$ larger than about 2.2 Å this spin density decreases monotonically with increasing distance. The isotropic as well as the anisotropic transferred hyperfine coupling are calculated.