3.3 Electronic Absorption Spectra, Tanabe-Sugano Diagrams, and Magnetic Properties

3.3 Electronic Absorption Spectra, Tanabe-Sugano Diagrams, and Magnetic Properties

Quick Answer: The vibrant colors and magnetic behaviors of coordination complexes stem from electronic promotions between split $d$-orbitals and molecular orbitals. Electronic transitions are governed by the Laporte selection rule (parity forbidden, $\Delta l = \pm 1$) and the spin selection rule ($\Delta S = 0$). While $d-d$ transitions in centrosymmetric octahedral complexes are Laporte-forbidden and rely on vibronic coupling (yielding pale colors with $\epsilon < 50\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$), tetrahedral complexes and charge transfer (LMCT/MLCT) processes yield intense absorption ($\epsilon > 10^3 - 10^5$). Magnetism is quantified using the spin-only formula $\mu_s = \sqrt{n(n+2)}\,\mu_B$, with deviations reflecting unquenched orbital angular momentum in complexes with orbitally degenerate ground terms ($T_{1g}, T_{2g}$).

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Selection Rules Governing Electronic Transitions

According to the Beer-Lambert Law ($A = \epsilon b c$), the intensity of an absorption band is quantified by its molar absorption coefficient ($\epsilon$, in $\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$). The magnitude of $\epsilon$ is strictly governed by quantum mechanical selection rules for electric-dipole transitions:

1. The Laporte Selection Rule (Orbital Parity Rule)

In a molecule or complex possessing an inversion center ($i$), transitions are allowed only between states of opposite parity:

$g \leftrightarrow u \quad (\text{Allowed}) \qquad g \leftrightarrow g, \; u \leftrightarrow u \quad (\text{Forbidden})$

where $g$ (gerade, even) denotes symmetry with respect to inversion, and $u$ (ungerade, odd) denotes antisymmetry. In atomic terms, this requires a change in the orbital angular momentum quantum number: $\Delta l = \pm 1$.

Because all five $d$-orbitals have $l = 2$ and possess gerade symmetry, all pure $d \rightarrow d$ transitions are Laporte-forbidden ($\Delta l = 0$, $g \rightarrow g$).

Mechanisms Relaxing the Laporte Rule

1. Vibronic Coupling (Herzberg-Teller Coupling): In octahedral complexes, asymmetric molecular vibrations (such as $u$-symmetry bending or stretching modes) temporarily destroy the center of inversion. This dynamic loss of centrosymmetry mixes odd-parity ligand or metal $p$-orbital character ($u$) into the $d$-orbitals ($g$), granting the transition a small electric-dipole moment. This yields pale colors with modest molar absorptivities: $\epsilon \approx 1 - 50\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$ (e.g., pale pink $[Co(H_2O)_6]^{2+}$).
2. Non-Centrosymmetric Geometries: Tetrahedral complexes ($T_d$) lack an inversion center. In tetrahedral symmetry, both metal $3d$ and $4p$ orbitals transform under the identical symmetry representation ($t_2$), allowing significant direct $p-d$ orbital mixing. Because $p \rightarrow d$ is Laporte-allowed, tetrahedral $d-d$ transitions are much more intense: $\epsilon \approx 100 - 2,000\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$ (e.g., deep intense blue $[CoCl_4]^{2-}$).

2. The Spin Selection Rule

Electric-dipole transitions cannot involve a simultaneous change in the spin state of the electron:

$\Delta S = 0 \quad (\text{Allowed})$

Transitions between states with different spin multiplicities ($\Delta S \neq 0$) are spin-forbidden. They can occur only very weakly through spin-orbit coupling, yielding tiny extinction coefficients: $\epsilon < 1\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$.

Classic Example: High-spin $d^5$ complexes such as $[Mn(H_2O)_6]^{2+}$ have a $^6A_{1g}$ ground state (all five $d$-electrons parallel, $S = 5/2$). Any electronic excitation requires flipping an electron spin to yield an excited state with $S = 3/2$ or $1/2$. Consequently, every possible $d-d$ transition in high-spin $Mn^{2+}$ is both spin-forbidden and Laporte-forbidden, resulting in an almost colorless solution ($\epsilon \approx 0.01 - 0.1\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$).

Absorption Intensity Hierarchy

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Charge Transfer (CT) Spectra: LMCT and MLCT

Charge transfer bands arise when an electron is promoted between molecular orbitals localized primarily on the ligand and those localized on the central metal ion. Because these transitions involve different atomic orbital types across different atoms, they are fully Laporte-allowed and spin-allowed, producing brilliant, intensely colored solutions with extinction coefficients exceeding $10^4\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$.

1. Ligand-to-Metal Charge Transfer (LMCT)

An electron is transferred from a ligand-dominated $\sigma$ or $\pi$ bonding orbital into an empty or partially filled metal $d$-orbital. LMCT is favored when:

• The central metal is in a high formal oxidation state (highly electrophilic, easily reducible).
• The ligands have relatively high energy lone pairs and low electronegativity (readily oxidizable).

Key Exam Examples:

• Permanganate ($MnO_4^-$): Manganese is $Mn(VII)$ with a $d^0$ electron configuration. Because there are no $d$-electrons, $d-d$ transitions are impossible. The intense purple color ($\epsilon \approx 2,500\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$) is caused by an LMCT transition from oxygen $2p_\pi$ non-bonding lone pairs into vacant manganese $3d$ orbitals.
• Chromate ($CrO_4^{2-}$) and Dichromate ($Cr_2O_7^{2-}$): Chromium is $Cr(VI)$ ($d^0$); bright yellow and orange colors arise from $O^{2-} \rightarrow Cr(VI)$ LMCT.
• Iron(III) Thiocyanate ($[Fe(H_2O)_5(SCN)]^{2+}$): Deep blood-red color used in qualitative analysis arises from $SCN^- \rightarrow Fe^{3+}$ LMCT.

2. Metal-to-Ligand Charge Transfer (MLCT)

An electron is promoted from a metal-dominated $d$-orbital into a low-lying, unoccupied $\pi^*$ antibonding orbital of the ligand. MLCT is favored when:

• The central metal is in a low formal oxidation state (easily oxidizable).
• The ligands possess accessible, low-energy $\pi^*$ acceptor systems (conjugated aromatic amines like 2,2'-bipyridine and 1,10-phenanthroline, or $CO$ and $CN^-$).

Key Exam Examples:

• Tris(1,10-phenanthroline)iron(II) ($[Fe(phen)_3]^{2+}$, Ferroin): Intense blood-red redox indicator ($\epsilon \approx 11,000\,\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$) arising from $Fe(II) \rightarrow phen\,\pi^*$ MLCT.
• Tris(bipyridine)ruthenium(II) ($[Ru(bpy)_3]^{2+}$): Vibrant orange complex widely utilized in photoredox catalysis and solar energy conversion.

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Russell-Saunders Term Symbols and Free-Ion Ground States

In multi-electron ions, electron-electron repulsions and spin-orbit couplings split electron configurations into discrete energy states designated by Russell-Saunders (L-S coupling) term symbols:

$^{2S+1}L_J$

• $S$: Total spin angular momentum quantum number ($S = \sum m_s$). The term $2S+1$ is the spin multiplicity (1 = singlet, 2 = doublet, 3 = triplet, 4 = quartet, 5 = quintet, 6 = sextet).
• $L$: Total orbital angular momentum quantum number ($L = \sum m_l$). Letter codes denote $L$ values:
  • $L = 0 \rightarrow S$
  • $L = 1 \rightarrow P$
  • $L = 2 \rightarrow D$
  • $L = 3 \rightarrow F$
  • $L = 4 \rightarrow G$
  • $L = 5 \rightarrow H$
• $J$: Total angular momentum quantum number, taking values from $|L - S|$ to $L + S$.

Hund's Rules for Ground State Terms

1. Highest Multiplicity: The state with the maximum value of $S$ (highest $2S+1$) lies lowest in energy.
2. Highest $L$: For states of identical spin multiplicity, the state with the maximum value of $L$ lies lowest.
3. $J$ Assignment:
   • If the subshell is less than half-full, the state with the lowest $J$ lies lowest ($J = |L - S|$).
   • If the subshell is more than half-full, the state with the highest $J$ lies lowest ($J = L + S$).
   • If the subshell is exactly half-full, $L = 0$ and $J = S$.

Free-Ion Ground Terms for $d^n$ Configurations

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Tanabe-Sugano Diagrams: Structure and Interpretation

Tanabe-Sugano diagrams provide complete energy-level maps for transition metal complexes across all crystal field strengths. They differ fundamentally from simple Orgel diagrams because:

1. The ground electronic state is always plotted as the horizontal zero-energy baseline on the x-axis ($E/B = 0$).
2. Both the energy of the electronic states ($E$) and the crystal field splitting ($\Delta_o$ or $10Dq$) are divided by the Racah parameter $B$ (plotting $E/B$ versus $\Delta_o/B$) to account for interelectronic repulsion in a single, dimensionless graph.
3. They cover both weak-field (high-spin) and strong-field (low-spin) configurations.

Key Structural Features of Tanabe-Sugano Diagrams

• Spin Crossover Boundary: For configurations that exhibit high-spin/low-spin choices ($d^4, d^5, d^6, d^7$), a vertical dashed line marks the critical field strength ($\Delta_o/B$) where the ground state changes. For example, in the $d^6$ diagram, the ground state transitions from $^5T_{2g}$ (high-spin) to $^1A_{1g}$ (low-spin), causing an abrupt change in the slopes of all energy levels.
• Predicting Absorption Spectra: Spin-allowed transitions are vertical promotions from the ground state to excited states possessing the exact same spin multiplicity (identical superscript numbers).
  • Example ($d^3$, $[Cr(H_2O)_6]^{3+}$): The ground state is $^4A_{2g}$. Looking up the diagram, three spin-allowed quartet states exist: $^4T_{2g}$, $^4T_{1g}(F)$, and $^4T_{1g}(P)$. Thus, $Cr^{3+}$ displays three distinct absorption bands:
    1. $\nu_1: \; ^4A_{2g} \rightarrow ^4T_{2g} = \Delta_o$
    2. $\nu_2: \; ^4A_{2g} \rightarrow ^4T_{1g}(F)$
    3. $\nu_3: \; ^4A_{2g} \rightarrow ^4T_{1g}(P)$
• Band Widths and Slopes:
  • States represented by horizontal lines have electron configurations identical to the ground state; their energy is independent of $\Delta_o$. Transitions to these states yield sharp, narrow peaks (e.g., the spin-forbidden $^4A_{2g} \leftrightarrow {}^2E_g$ lines of $Cr^{3+}$; ruby's 694 nm emission is $^2E_g \rightarrow {}^4A_{2g}$).
  • States with steeply sloping lines involve promoting an electron between $t_{2g}$ and $e_g$. Molecular vibrations modulate the metal-ligand distance and hence $\Delta_o$, broadening the absorption band.

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Magnetochemistry: Paramagnetism, Diamagnetism, and Magnetic Moments

When placed into a magnetic field ($H$):

• Diamagnetic substances possess only paired electrons. They create an opposing internal field, are weakly repelled by magnetic field gradients, and have a negative magnetic susceptibility ($\chi < 0$).
• Paramagnetic substances possess one or more unpaired electrons. Their permanent magnetic dipoles align parallel to the external field, drawing the substance into the magnetic field ($\chi > 0$).

Experimental Susceptibility Measurements

1. Gouy Balance: A cylindrical sample is suspended between the poles of a powerful electromagnet. When the magnetic field is energized, a paramagnetic sample is pulled downward toward the high-field region, manifesting as an apparent increase in sample mass ($\Delta m$). The mass change directly yields the molar magnetic susceptibility ($\chi_M$).
2. Evans NMR Method: Paramagnetic metal ions in solution induce a chemical shift displacement ($\Delta \nu$) in an inert solvent reference signal (e.g., $t$-butanol or TMS). By measuring the frequency difference between pure solvent in an inner coaxial tube and the paramagnetic solution in the outer tube, $\chi_M$ is calculated:

$\chi_M = \frac{3\,\Delta \nu}{4\pi\,\nu_0\,m} + \chi_0$

Temperature Dependence: Curie and Curie-Weiss Laws

Paramagnetic susceptibility decreases with increasing thermal agitation:

$\chi_M = \frac{C}{T} \quad (\text{Curie's Law}) \qquad \chi_M = \frac{C}{T - \theta} \quad (\text{Curie-Weiss Law})$

where $C$ is the Curie constant and $\theta$ is the Weiss constant (positive $\theta$ indicates ferromagnetic exchange coupling; negative $\theta$ indicates antiferromagnetic interactions).

Spin-Only Magnetic Moment Formula

For first-row ($3d$) transition metal complexes, the effective magnetic moment ($\mu_{eff}$) is primarily governed by the total electron spin ($S$) and number of unpaired electrons ($n$):

$\mu_s = \sqrt{n(n+2)}\,\mu_B = 2\sqrt{S(S+1)}\,\mu_B$

where $\mu_B$ is the Bohr Magneton ($\mu_B = \frac{e\hbar}{2m_e} = 9.274 \times 10^{-24}\,\text{J}\cdot\text{T}^{-1}$).

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Orbital Contribution to Magnetic Moments and Quenching

When both spin ($S$) and orbital ($L$) angular momenta contribute, the theoretical magnetic moment is given by:

$\mu_{S+L} = \sqrt{4S(S+1) + L(L+1)}\,\mu_B$

In most transition metal complexes, however, the observed magnetic moment is very close to the spin-only value $\mu_s$. This phenomenon is known as orbital quenching.

Mechanism of Orbital Quenching

In a free gaseous atom, electrons can circulate freely between degenerate $d$-orbitals, generating an orbital magnetic dipole. In a coordination complex, the surrounding ligand field destroys spherical symmetry and splits the $d$-orbitals. Because the orbitals are locked in specific spatial orientations along or between the ligand axes, the orbital angular momentum is largely "quenched" by the electrostatic field.

Criteria for Unquenched Orbital Angular Momentum

For an orbital angular momentum contribution to persist in a coordination complex, an electron must be able to circulate among degenerate orbitals that can be converted into one another by a rotation about an axis without altering its electron spin. This requires that:

1. The $d$-orbitals involved must be orbitally degenerate (i.e., the ground term must be an $E$ or $T$ state).
2. The orbitals must be unevenly occupied.

In an octahedral geometry, the $e_g$ orbitals ($d_{z^2}$ and $d_{x^2-y^2}$) cannot be interconverted by any symmetry rotation of the octahedron; hence, an unevenly populated $e_g$ manifold ($E_g$ term) cannot generate orbital angular momentum.

In contrast, the three $t_{2g}$ orbitals ($d_{xy}, d_{xz}, d_{yz}$) are interconvertible by $90^\circ$ rotations about Cartesian axes. Therefore, an unquenched orbital angular momentum contribution occurs whenever the $t_{2g}$ subshell is asymmetrically occupied, generating a $T_{1g}$ or $T_{2g}$ ground term:

• Orbital contribution possible ($\mu_{eff}$ deviates from $\mu_s$, usually upward):
  • $t_{2g}^1$ ($^2T_{2g}$, e.g., $Ti^{3+}$, typically $\mu_{eff} \approx 1.7 - 1.8\,\mu_B$ because spin-orbit coupling partly offsets the orbital term)
  • $t_{2g}^2$ ($^3T_{1g}$, e.g., $V^{3+}$, $\mu_{eff} \approx 2.8 - 3.1\,\mu_B$)
  • Low-spin $d^4$ ($t_{2g}^4$, $^3T_{1g}$)
  • Low-spin $d^5$ ($t_{2g}^5$, $^2T_{2g}$, e.g., low-spin $Fe^{3+}$, $\mu_{eff} \approx 2.0 - 2.5\,\mu_B > 1.73$)
  • High-spin $d^7$ ($t_{2g}^5 e_g^2$, $^4T_{1g}$, e.g., high-spin $Co^{2+}$, $\mu_{eff} \approx 4.8 - 5.2\,\mu_B \gg 3.87$)
• Orbital contribution quenched ($\mu_{eff} \approx \mu_s$):
  • $t_{2g}^3$ ($^4A_{2g}$, half-filled $t_{2g}$, e.g., $Cr^{3+}$, $\mu_{eff} \approx 3.87\,\mu_B$)
  • High-spin $d^5$ ($t_{2g}^3 e_g^2$, $^6A_{1g}$, e.g., high-spin $Fe^{3+}$ and $Mn^{2+}$, $\mu_{eff} \approx 5.92\,\mu_B$)
  • $t_{2g}^6 e_g^2$ ($^3A_{2g}$, e.g., $Ni^{2+}$, $\mu_{eff} \approx 2.9 - 3.3\,\mu_B$, slight deviation due to second-order spin-orbit coupling)