Understanding $d-d$ Transition Spin Rules

The nature of $d-d$ transitions (spin-allowed or spin-forbidden) depends on whether the total spin multiplicity changes during the electronic excitation. The spin selection rule states that transitions are allowed only if $\Delta S = 0$. We analyze the spin states for the given $d^5$ and $d^7$ high-spin configurations in an octahedral field.

High-Spin $d^5$ Configuration Analysis

For a high-spin $d^5$ metal ion in an octahedral complex, the electron configuration is $t_{2g}^3 e_g^2$. All five electrons are unpaired.

• Number of unpaired electrons = 5.
• Total spin quantum number, $S = \frac{\text{Number of unpaired electrons}}{2} = \frac{5}{2}$.
• Spin multiplicity = $2S + 1 = 2(\frac{5}{2}) + 1 = 6$. This corresponds to a sextet state.
• In $d-d$ transitions, an electron moves between $d$ orbitals. For $d^5$ high-spin, the ground state is a sextet ($^6S$). The possible excited states typically do not have the same sextet multiplicity.
• Since the transition involves a change in spin multiplicity ($\Delta S \neq 0$), the $d-d$ transitions are spin-forbidden.

High-Spin $d^7$ Configuration Analysis

For a high-spin $d^7$ metal ion in an octahedral complex, the electron configuration is $t_{2g}^5 e_g^2$. We determine the number of unpaired electrons:

• In the $t_{2g}$ orbitals ($t_{2g}^5$): Two orbitals are paired, one is unpaired ($\uparrow$). So, 1 unpaired electron from $t_{2g}$.
• In the $e_g$ orbitals ($e_g^2$): Both orbitals have one unpaired electron ($\uparrow, \uparrow$). So, 2 unpaired electrons from $e_g$.
• Total number of unpaired electrons = 1 + 2 = 3.
• Total spin quantum number, $S = \frac{3}{2}$.
• Spin multiplicity = $2S + 1 = 2(\frac{3}{2}) + 1 = 4$. This corresponds to a quartet state.
• The ground state term for $t_{2g}^5 e_g^2$ is a quartet state (e.g., $^4A_{2g}$). The excited states arising from $d-d$ transitions (e.g., from $t_{2g}^4 e_g^3$) also result in quartet states.
• Since the transition occurs between states of the same spin multiplicity ($\Delta S = 0$), the $d-d$ transitions are spin-allowed.

Conclusion on Transitions

Based on the analysis:

• $d^5$ high-spin: $d-d$ transitions are spin-forbidden.
• $d^7$ high-spin: $d-d$ transitions are spin-allowed.

Therefore, the correct description is that $d-d$ transitions are spin-forbidden for $d^5$ and spin-allowed for $d^7$.