Crystal Field Splitting Energy (CFSE) and High-Spin/Low-Spin Complexes

Once the d-orbitals of a metal ion have been split by the surrounding ligand field (Sections 5.12-5.13), filling them with the metal's actual d-electrons releases a net energetic stabilization compared with the unsplit, spherical-field reference — this is the crystal field stabilization energy (CFSE). For an octahedral complex, it is calculated directly from the orbital occupation:

where $n_{t_{2g}}$ and $n_{e_g}$ are the number of electrons occupying the $t_{2g}$ and e_g} sets respectively.

For electron counts $d^1$-$d^3$ and $d^8$-$d^{10}$, there is only one possible way to distribute the electrons (Section 5.6), so the CFSE has only one value. But for $d^4$ through $d^7$, two genuinely different distributions are possible, giving two different CFSE values, and which one is actually adopted depends on comparing $\Delta_o$ (set by the ligand) against the metal's intrinsic pairing energy, $\boldsymbol{P}$ (the energy cost of forcing two electrons to share one orbital, arising from their mutual electrostatic repulsion).

If $\Delta_o < P$ (a weak-field ligand), it costs more energy to pair two electrons than to promote one into the higher $e_g$ set instead, so the electrons spread out as much as possible — the high-spin configuration, with the maximum number of unpaired electrons for that electron count. If $\Delta_o > P$ (a strong-field ligand), the reverse is true, and electrons preferentially pair up within the lower $t_{2g}$ set rather than occupy the costlier $e_g$ set — the low-spin configuration, with the minimum number of unpaired electrons.

For $\text{Co}^{3+}$ ($d^6$) in the weak-field, high-spin complex $[\text{CoF}_6]^{3-}$: the configuration is $t_{2g}^4 e_g^2$, so $\text{CFSE} = \left(-0.4(4) + 0.6(2)\right)\Delta_o = (-1.6 + 1.2)\Delta_o = -0.4\,\Delta_o$. In the strong-field, low-spin complex $[\text{Co}(\text{NH}_3)_6]^{3+}$ (or $[\text{Co}(\text{CN})_6]^{3-}$), the configuration is $t_{2g}^6 e_g^0$, so $\text{CFSE} = \left(-0.4(6) + 0.6(0)\right)\Delta_o = -2.4\,\Delta_o$ — a substantially larger orbital stabilization, though this must be weighed against the extra pairing energy cost of forcing three electron pairs into $t_{2g}$ (two more pairs than the single pair present in the high-spin case); by definition of a strong-field ligand, $\Delta_o$ is large enough that the low-spin arrangement still wins out overall. …