4.3 Electron Transfer Reactions

Inner-Sphere vs Outer-Sphere Electron Transfer

Electron transfer between metal centers is one of the most fundamental reaction types in coordination chemistry. Every redox process involving two metal complexes must proceed by one of two pathways: inner-sphere or outer-sphere. Distinguishing between them, and predicting which one operates in a given system, is a central skill in this unit.

Inner-Sphere Mechanism and Bridging Ligands

In an inner-sphere electron transfer, the two metal centers share a bridging ligand during the electron transfer step. The electron moves through that bridging ligand rather than jumping across empty space.

The mechanism proceeds in three stages:

1. Substitution — One metal center (typically the more labile one) loses a ligand and bonds to a ligand already coordinated to the other metal, forming a bridged binuclear intermediate.
2. Electron transfer — The electron passes from donor to acceptor through the bridging ligand's orbitals.
3. Dissociation — The binuclear intermediate breaks apart, and the bridging ligand may end up on either metal center.

The classic demonstration is Taube's experiment: reduction of $[\text{Co(NH}_3)_5\text{Cl}]^{2+}$ by $[\text{Cr(H}_2\text{O)}_6]^{2+}$. After the reaction, the chloride ends up on chromium, proving it served as a bridge during electron transfer.

Common bridging ligands include $\text{Cl}^-$, $\text{CN}^-$, $\text{NCS}^-$, and pyrazine ($\text{C}_4\text{H}_4\text{N}_2$). Good bridging ligands generally have lone pairs or $\pi$-systems that can mediate electronic coupling between the two metals.

Outer-Sphere Mechanism and Encounter Complexes

In an outer-sphere electron transfer, both metal centers retain their full coordination spheres. No bridging ligand forms. Instead, the two complexes approach each other to form a loosely associated encounter complex, and the electron tunnels through space (or through the intervening solvent/ligand shell).

The mechanism has three stages:

1. Formation of the encounter complex — The two complexes diffuse together and sit at close range, held by electrostatic and van der Waals interactions.
2. Electron transfer — The electron tunnels from donor to acceptor without any covalent bridge.
3. Dissociation — The products separate.

A well-known example is the reduction of $[\text{Fe(CN)}_6]^{3-}$ by $[\text{Ru(NH}_3)_6]^{2+}$. Both complexes are substitutionally inert, so bridge formation is not feasible, and the reaction proceeds outer-sphere.

Factors Influencing the Choice of Mechanism

Several factors determine which pathway a given reaction follows:

• Ligand lability — At least one metal center must be labile enough to open a coordination site for bridge formation. If both complexes are inert (slow ligand exchange), the inner-sphere pathway is blocked, and the reaction goes outer-sphere.
• Availability of a suitable bridging ligand — The ligand must have the right geometry and electronic structure to bridge two metals. Monodentate ligands with lone pairs on more than one atom (e.g., $\text{Cl}^-$, $\text{CN}^-$, $\text{SCN}^-$) are good candidates.
• Steric effects — Bulky ligands around either metal center can physically prevent the close approach needed for bridge formation, pushing the reaction toward outer-sphere.
• Solvent — High-dielectric solvents stabilize the charged encounter complex in outer-sphere reactions, making that pathway more competitive.

Factors Influencing Electron Transfer Rates

Driving Force and Reorganization Energy

The driving force ($-\Delta G^\circ$) for an electron transfer reaction comes from the difference in reduction potentials of the two couples. A larger potential difference means a more thermodynamically favorable reaction and, up to a point, a faster rate.

The reorganization energy ($\lambda$) is the energy cost of distorting the reactants and surrounding solvent into the geometry they will have in the products, before the electron actually transfers. It has two components:

• Inner-sphere reorganization ($\lambda_{\text{in}}$) — changes in metal-ligand bond lengths and angles. For example, when $\text{Fe}^{2+}$ is oxidized to $\text{Fe}^{3+}$, the metal-ligand bonds shorten significantly, and this structural adjustment costs energy.
• Outer-sphere reorganization ($\lambda_{\text{out}}$) — reorientation of solvent dipoles around the changing charge distribution.

Higher reorganization energies mean a larger activation barrier and slower rates. Complexes that change geometry very little upon oxidation or reduction (like $[\text{Ru(bpy)}_3]^{2+/3+}$) have small $\lambda$ values and undergo fast electron transfer.

Distance and Electronic Coupling

The rate of electron transfer decreases exponentially with the distance between donor and acceptor:

$k_{\text{ET}} \propto e^{-\beta d}$

where $d$ is the donor-acceptor distance and $\beta$ is a decay parameter that depends on the intervening medium. Through vacuum, $\beta$ is large (roughly 3–4 Å$^{-1}$). Through a conjugated bridging ligand, $\beta$ can be much smaller (0.2–0.7 Å$^{-1}$), meaning the electronic coupling decays more slowly with distance.

This is why the nature of the bridge matters so much in inner-sphere reactions. A conjugated, $\pi$-electron-rich bridge like pyrazine provides much stronger electronic coupling than a simple chloride bridge, enabling faster electron transfer over longer distances.

Solvent Effects and Spin States

Solvent effects operate primarily through the outer-sphere reorganization energy. Polar solvents with high dielectric constants (like water) stabilize charge-separated states and can lower the barrier for reactions that involve significant charge redistribution. However, highly polar solvents also increase $\lambda_{\text{out}}$, which can raise the barrier. The net effect depends on the specific system.

Spin-state considerations add another layer. Electron transfer is fastest when it is spin-allowed. Transferring an electron between two metal centers that share the same spin state (e.g., both high-spin, or both low-spin) avoids the need for a spin flip, which would make the process spin-forbidden and much slower.

For example, self-exchange between high-spin $\text{Fe}^{2+}$ ($d^6$) and high-spin $\text{Fe}^{3+}$ ($d^5$) in aqua complexes is relatively straightforward. But if one center is low-spin and the other high-spin, the electron transfer requires a simultaneous spin-state change, adding a significant barrier.

Thermodynamics and Kinetics of Electron Transfer

Marcus Theory

Marcus theory is the central quantitative framework for electron transfer kinetics. It connects the activation free energy ($\Delta G^\ddagger$) to two quantities you already know: the driving force ($\Delta G^\circ$) and the reorganization energy ($\lambda$).

The Marcus equation for the activation barrier is:

$\Delta G^\ddagger = \frac{(\lambda + \Delta G^\circ)^2}{4\lambda}$

This parabolic relationship has three important regimes:

• Normal region ($|\Delta G^\circ| < \lambda$): Increasing the driving force lowers the barrier and speeds up the reaction. This is the intuitive case.
• Optimal point ($-\Delta G^\circ = \lambda$): The barrier drops to zero, and the rate reaches its maximum.
• Inverted region ($|\Delta G^\circ| > \lambda$): Increasing the driving force raises the barrier and actually slows the reaction down. This counterintuitive prediction was one of Marcus's most important contributions and was experimentally confirmed decades later.

Gibbs Free Energy and the Nernst Equation

The thermodynamic driving force is calculated from electrochemical data:

$\Delta G^\circ = -nFE^\circ_{\text{cell}}$

where $n$ is the number of electrons transferred, $F$ is Faraday's constant (96,485 C/mol), and $E^\circ_{\text{cell}}$ is the standard cell potential (difference in standard reduction potentials of the two couples).

For non-standard conditions, the full Nernst equation applies:

$E = E^\circ - \frac{RT}{nF}\ln Q$

where $Q$ is the reaction quotient.

Rate Constants and the Marcus Cross Relation

The overall rate constant for electron transfer can be expressed using the Eyring or Arrhenius framework:

$k = A \, e^{-\Delta G^\ddagger / RT}$

where $A$ is the pre-exponential factor (which contains the electronic coupling and nuclear frequency terms) and $\Delta G^\ddagger$ comes from the Marcus equation above.

The Marcus cross relation is a powerful predictive tool. It lets you estimate the rate constant for a cross reaction (between two different metal couples) using only the self-exchange rate constants and the equilibrium constant:

$k_{12} = (k_{11} \, k_{22} \, K_{12} \, f)^{1/2}$

• $k_{11}$ and $k_{22}$ are the self-exchange rate constants for each metal couple (e.g., $[\text{Fe(CN)}_6]^{3-/4-}$ exchanging an electron with itself).
• $K_{12}$ is the equilibrium constant for the cross reaction, calculated from the difference in reduction potentials.
• $f$ is a correction factor, usually close to 1, defined as:

$\ln f = \frac{(\ln K_{12})^2}{4 \ln(k_{11} k_{22} / Z^2)}$

where $Z$ is the collision frequency (often approximated as $10^{11}$ M$^{-1}$s$^{-1}$ in solution).

The cross relation works remarkably well for outer-sphere reactions and provides a quick check: if the experimentally measured $k_{12}$ is much faster than the cross relation predicts, an inner-sphere mechanism may be operating.