2.3 High-Spin and Low-Spin Complexes

High-Spin and Low-Spin Complexes

In crystal field theory, the competition between two energies determines how electrons fill the split d-orbitals of a transition metal complex. High-spin and low-spin states arise from this competition, and they directly control a complex's magnetic behavior, color intensity, and reactivity. Since spin state connects ligand identity, metal properties, and observable spectra, it's one of the most practically important ideas in this unit.

High-Spin vs. Low-Spin Complexes

Electronic Configurations

The distinction comes down to how electrons populate the $t_{2g}$ and $e_g$ sets in an octahedral field.

In a high-spin complex, electrons spread across all five d-orbitals before any pairing occurs, following Hund's rule. This maximizes the number of unpaired electrons and gives a higher total spin quantum number ($S$). For example, a $d^6$ high-spin octahedral complex has the configuration $t_{2g}^4 e_g^2$ with four unpaired electrons.

In a low-spin complex, electrons fill the lower-energy $t_{2g}$ set completely (pairing up as needed) before occupying the higher-energy $e_g$ set. This minimizes unpaired electrons and lowers $S$. The same $d^6$ ion in a low-spin octahedral complex has the configuration $t_{2g}^6 e_g^0$ with zero unpaired electrons.

Determining Spin State

The spin state depends on the balance between two quantities:

• Crystal field splitting energy ($\Delta_o$): the energy gap between the $t_{2g}$ and $e_g$ orbital sets.
• Pairing energy ($P$): the energy cost of forcing two electrons into the same orbital, arising from electron-electron repulsion.

The decision rule is straightforward:

• If $\Delta_o > P$: it costs less energy to pair electrons in $t_{2g}$ than to promote them to $e_g$, so the complex is low-spin.
• If $P > \Delta_o$: it costs less energy to place electrons in $e_g$ than to pair them, so the complex is high-spin.

Predicting Spin State

Crystal Field Splitting Energy and the Spectrochemical Series

The magnitude of $\Delta_o$ depends primarily on the ligands. Strong-field ligands donate electron density in ways that produce a large $t_{2g}$–$e_g$ gap, while weak-field ligands produce a small gap.

The spectrochemical series ranks common ligands by their field strength:

$I^- < Br^- < Cl^- < F^- < OH^- < H_2O < NH_3 < en < NO_2^- < CN^- < CO$

Ligands on the right ($CN^-$, $CO$) tend to force low-spin configurations. Ligands on the left ($I^-$, $Br^-$) tend to give high-spin complexes. Ligands in the middle ($H_2O$, $NH_3$) can go either way depending on the metal.

Pairing Energy

Pairing energy ($P$) is relatively constant for a given metal ion in a given oxidation state. A few trends are worth knowing:

• $P$ increases with higher effective nuclear charge ($Z_{eff}$), because the d-orbitals contract and electron-electron repulsion intensifies.
• 4d and 5d metals have significantly lower pairing energies than 3d metals. Their d-orbitals are spatially larger, so repulsion between paired electrons is reduced. This is a major reason why second- and third-row transition metal complexes are almost always low-spin, regardless of the ligand.

Factors Influencing Spin State

Ligand Field Strength

This is the single most important factor for 3d metals. Strong-field ligands like $CN^-$ and $CO$ produce large $\Delta_o$ values that exceed $P$, favoring low-spin states. Weak-field ligands like $I^-$, $Br^-$, and $Cl^-$ produce small $\Delta_o$ values, leaving $P$ dominant and favoring high-spin states.

Metal Ion Properties

• Oxidation state: Higher oxidation states contract the d-orbitals and draw ligands closer, increasing $\Delta_o$. For instance, $[Fe(H_2O)_6]^{2+}$ ($Fe^{2+}$, $d^6$) is high-spin, but $[Fe(CN)_6]^{3-}$ ($Fe^{3+}$, $d^5$) is low-spin partly because the higher charge on $Fe^{3+}$ boosts $\Delta_o$ (and $CN^-$ is a strong-field ligand).
• Row in the periodic table: As noted above, 4d and 5d metals have larger $\Delta_o$ values and smaller $P$ values than their 3d counterparts. Both effects push toward low-spin.
• d-electron count: The high-spin/low-spin distinction only matters for $d^4$ through $d^7$ configurations in octahedral geometry. For $d^1$–$d^3$, there aren't enough electrons to require pairing in either scenario, so the configuration is the same regardless of field strength. For $d^8$–$d^{10}$, the $t_{2g}$ set is already full and the filling pattern is again fixed.

Geometry and Temperature

• Octahedral vs. tetrahedral: Tetrahedral splitting ($\Delta_t$) is only about $\frac{4}{9}$ of $\Delta_o$ for the same metal-ligand combination. Because $\Delta_t$ is so small, tetrahedral complexes are almost always high-spin.
• Square planar: The splitting pattern is very different from octahedral, and the energy gap between the highest and next-highest orbital is typically large. Square planar geometry strongly favors low-spin states, which is why $d^8$ ions like $Pt^{2+}$, $Pd^{2+}$, and $Ni^{2+}$ (with strong-field ligands) commonly adopt this geometry as diamagnetic, low-spin complexes.
• Temperature: Some complexes sit right at the $\Delta_o \approx P$ boundary and exhibit spin-crossover behavior. Increasing temperature provides thermal energy that can populate the high-spin state. These systems can switch between spin states with changes in temperature, pressure, or even light irradiation.

Properties of Spin States

Magnetic Properties

Measuring the magnetic moment of a complex is one of the most direct experimental ways to determine its spin state.

• High-spin complexes are paramagnetic (attracted to a magnetic field) because of their unpaired electrons.
• Low-spin complexes may be diamagnetic (no unpaired electrons, as in low-spin $d^6$) or weakly paramagnetic (fewer unpaired electrons than the high-spin case).

The spin-only magnetic moment is calculated as:

$\mu_{s.o.} = \sqrt{n(n+2)} \; \mu_B$

where $n$ is the number of unpaired electrons. For example, a high-spin $d^5$ complex ($n = 5$) gives $\mu_{s.o.} = \sqrt{35} \approx 5.92 \; \mu_B$, while a low-spin $d^5$ complex ($n = 1$) gives $\mu_{s.o.} = \sqrt{3} \approx 1.73 \; \mu_B$. That's a huge, easily measurable difference.

Deviations from the spin-only formula occur when there is a significant orbital angular momentum contribution (common for $t_{2g}$-degenerate ground states like $T$ terms) or spin-orbit coupling (especially important for heavier metals).

Spectroscopic Properties

High-spin and low-spin complexes of the same metal ion have different ground-state term symbols, so their d-d absorption spectra differ in both position and intensity.

• Spin-allowed transitions ($\Delta S = 0$) are relatively intense, with molar absorptivities ($\varepsilon$) typically in the range of 10–50 $M^{-1}cm^{-1}$ for Laporte-forbidden d-d bands.
• Spin-forbidden transitions ($\Delta S \neq 0$) are much weaker, often with $\varepsilon < 1 \; M^{-1}cm^{-1}$.

Because a low-spin complex has a different ground-state multiplicity than a high-spin one, the set of spin-allowed excited states changes. For example, in a $d^5$ octahedral complex:

• High-spin ground state is $^6A_{1g}$. All d-d transitions from this sextet state to quartet excited states are spin-forbidden, making the complex very pale (think of the faint pink of $[Mn(H_2O)_6]^{2+}$).
• Low-spin ground state is $^2T_{2g}$. Transitions to other doublet states are spin-allowed, producing noticeably more intense absorption and deeper color.

This pattern generalizes: high-spin complexes with half-filled or fully occupied subshells tend to be weakly colored, while low-spin complexes of the same ion often display stronger, more vivid colors.