Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2A / Section A / 15

The octahedral splitting ${\Delta }_o$ separates the metal $t_{2g}$ and $e_g$ orbital sets. The Racah parameter $B$ characterizes interelectronic repulsion and hence the spacing of many-electron d-shell terms; it is not another name for ${\Delta }_o$. A Tanabe-Sugano diagram plots term excitation energies $E/B$ against ${\Delta }_o/B$, for specified values of the other repulsion-parameter ratios. Each ordinate is measured relative to whichever term is the ground state at that field strength. The original many-electron treatment diagonalizes the relevant d-shell energy matrices rather than identifying every absorption with a one-electron $t_{2g}\to e_g$ gap.^[ref]

The ground state changes from high-spin $t_{2g}^3{e}_g^2$, ${}^6{A}_{1g}$, to low-spin $t_{2g}^5$, ${}^2{T}_{2g}$. The larger splitting eventually compensates the additional pairing cost. Because all plotted excitation energies are re-referenced to the new ground state, the branches change slope at the crossover. The break is not an infinite physical excitation energy or a discontinuity of the underlying Hamiltonian.

Here ${\Delta }_o/B=20$, so the ground term is ${}^6{A}_{1g}$. There are no spin-allowed d–d bands in the displayed manifold under the spin selection rule: the excited terms shown have doublet or quartet rather than sextet multiplicity. Very weak spin-forbidden absorptions can still gain intensity through spin–orbit coupling, and even spin-allowed d–d transitions would be forbidden by the Laporte rule in ideal centrosymmetric coordination. Thus a strong ordinary d–d spectrum should not be predicted simply by multiplying the splitting by a numerical factor. For scale, the lowest quartet branches read roughly $E/B\simeq 17,23,33$ at this abscissa, corresponding to weak features around $1.4\times 10^4$, $1.8\times 10^4$ and $2.6\times 10^4\mathrm{c}{\mathrm{m}}^{-1}$ if such forbidden intensity is detectable. These are graphical estimates, not precision assignments of every weak line.

Now ${\Delta }_o/B=40$ and the ground term is ${}^2{T}_{2g}$. Spin-conserving transitions to the labelled doublet excited terms are possible. Reading the diagram gives approximatelyThe first two upper terms are coincident or unresolved at this diagram's resolution. These doublet transitions should dominate over the spin-forbidden transitions to sextet or quartet levels. Their intensity still depends on vibronic or other relaxation of the Laporte rule. The plotted curves justify only approximate wavenumbers, not several-digit band positions.