Generalizedd-electron interaction matrices-their derivation and impact on existing results

The restriction of the "pure" $d$ orbitals on the Tanabe-Sugano electron-electron interaction matrices has been removed and new matrices have been derived appropriate to $d$-electron transition-metal ions in solids and complexes. We have adopted the group-theoretical technique developed by Racah involving the coefficients of fractional parentage. A general expression useful for obtaining the matrix elements for the complementary states has also been given. The significance of the present study has been emphasized by checking and comparing our results with the recently published results on GaAs:${\mathrm{Cr}}^{2+}$ and GaAs:${\mathrm{Cr}}^{3+}$ and by presenting the examples of Mg${\mathrm{F}}_2$:${\mathrm{Co}}^{2+}$ and Mg${\mathrm{F}}_2$:${\mathrm{Mn}}^{2+}$ where it has been possible to deduce the values of the $d$-electron Coulomb and exchange parameters in conjunction with the fine experimental optical spectra. Other applications such as the removal of accidental degeneracies have been indicated. Important points for further improvements have been discussed.