Structure and Properties of the Precursor/Successor Complex and Transition State of the CrCl²⁺/Cr²⁺ Electron Self-Exchange Reaction via the Inner-Sphere Pathway

The electron self-exchange reaction CrCl­(OH₂)₅²⁺ + Cr­(OH₂)₆²⁺ → Cr(OH₂)₆²⁺ + CrCl­(OH₂)₅²⁺, proceeding via the inner-sphere pathway, was investigated with quantum-chemical methods. Geometry and vibrational frequencies of the precursor/successor (P/S) complex, (H₂O)₅Cr^IIIClCr^II(OH₂)₅⁴⁺/(H₂O)₅Cr^IIClCr^III(OH₂)₅⁴⁺, and the transition state (TS), (H₂O)₅CrClCr­(OH₂)₅^4+ ⧧, were computed with density functional theory (DFT) and conductor polarizable continuum model hydration. Consistent data were obtained solely with long-range-corrected functionals, whereby in this study, LC-BOP was used. Bent and linear structures were computed for the TS and P/S. The electronic coupling matrix element (H_ab) and the reorganizational energy (λ) were calculated with multistate extended general multiconfiguration quasi-degenerate second-order perturbation theory. The nuclear tunneling factor (Γ_n), the nuclear frequency factor (ν_n), the electronic frequency factor (ν_el), the electron transmission coefficient (κ_el), and the first-order rate constant (k_et) for the electron-transfer step (the conversion of the precursor complex into the successor complex) were calculated based on the imaginary frequency (ν^⧧) of the TS, the Gibbs activation energy (ΔG^⧧), H_ab, and λ. The formation of the precursor complex via water substitution at Cr­(OH₂)₆²⁺ was also investigated with DFT and found to be very fast. Thus, the electron-transfer step is rate-determining. For the substitution reaction, only a bent TS structure could be obtained. The overall rate constant (k) was estimated as the product K_Ak_et, whereby K_A is the equilibrium constant for the formation of the ion aggregate of the reactants Cr­(OH₂)₆²⁺ and CrCl­(OH₂)₅²⁺, Cr­(H₂O)₆·CrCl­(OH₂)₅⁴⁺ (IAR). k calculated for the bent and linear isomers agrees with the experimental value.