Configuration interaction and crystal-field splitting of the tetrahedralCr2+ion

The effect of configuration interaction on the crystal-field splitting of the tetrahedral ${\mathrm{Cr}}^{2+}$ ion via crystal field and the residual interaction that occurs in the formalism of the central-field approximation have been discussed within the framework of crystal-field theory using Richardson functions for the radial integrals. A simplified method which can replace the laborious method adopted earlier for the calculation of the effect of configuration interaction via crystal field alone has been indicated. In addition to the well-known mechanism of configuration interaction via crystal field another mechanism involving the combined action of crystal field and the aforesaid residual interaction has been discussed and it is found that the latter mechanism is in no way less important than the former. In the case of tetrahedral ${\mathrm{Cr}}^{2+}$ in CdS it has been shown that via crystal field alone the interaction of $3d^{3}4p$ and $3d^{3}4f$ with the ground $3d^4$ configuration narrows the crystal-field splitting ${\Gamma }_3-{\Gamma }_5$ by about 215 and 90 ${\mathrm{cm}}^{-1\mathrm{}}$, respectively, while the interaction of $3d^{3}4d$ widens the separation by a small amount of about 15 ${\mathrm{cm}}^{-1\mathrm{}}$ only. The residual interaction alone shifts equally both the levels ${\Gamma }_3$ and ${\Gamma }_5$ yielding no effect on the separation ${\Gamma }_3-{\Gamma }_5$. On the other hand, via the combined action of crystal field and residual interaction $3d^4$ interacts significantly with $3d^{3}4d$ and as a result the separation ${\Gamma }_3-{\Gamma }_5$ increases by about 1130 ${\mathrm{cm}}^{-1\mathrm{}}$. The net result of configuration interaction is thus to widen the crystal-field splitting ${\Gamma }_3-{\Gamma }_5$ by about 840 ${\mathrm{cm}}^{-1\mathrm{}}$.