The self-exchange of a nonbonding electron via the outer-sphere pathway: reorganizational energy and electronic coupling matrix element for the V(OH2)62+/3+, Ru(OH2)62+/3+, V(OH2)63+/4+, and Ru(OH2)63+/4+ couples

Abstract

The electron self-exchange reaction via the outer-sphere pathway of V(OH₂)₆^2+/3+, Ru(OH₂)₆^2+/3+, V(OH₂)₆^3+/4+, and Ru(OH₂)₆^3+/4+ was investigated with quantum chemical methods. The reorganizational energy (λ) and the nuclear frequency factor (ν_n) were computed on the basis of M(OH₂)₆·(OH₂)₁₂ⁿ⁺ (M = V, Ru; n = 2, 3, 4) model compounds, in which the 12 water molecules represent the second coordination sphere. The constant for the association of the reactants (K_A) was calculated via the Fuoss equation. The electronic coupling matrix element (H_ab) was computed on the basis of [M(OH₂)₆]₂ⁿ⁺ (M = V, Ru; n = 5, 7) dimers, and the electronic frequency factor (ν_el) was determined from H_ab and λ. Because, for the present redox couples, ν_n is much greater than ν_el, the second-order rate constant (k) is equal to K_Aν_ele^−ΔE‡/RT (whereby ΔE^‡ = λ/4). The experimental rate constant for the V(OH₂)₆^2+/3+ self-exchange reaction is too low because side reactions involving ClO₄⁻, the anion of the supporting electrolyte, have not been considered. The present computations suggest a rate constant in the range of 0.14–0.19 M⁻¹ s⁻¹ (25 °C, I = 2.0 M). The calculated rate constant for the Ru(OH₂)₆^2+/3+ self-exchange reaction agrees with experiment. For the V(OH₂)₆^3+/4+ self-exchange reaction, an estimated rate constant of (1–6) × 10⁻⁶ M⁻¹ s⁻¹ (25 °C, I = 2.0 M) is predicted. The second-order rate constant for the Ru(OH₂)₆^3+/4+ self-exchange process is estimated as 0.07 or 18 M⁻¹ s⁻¹ (25 °C, I = 5.0 M), respectively, for a δ or a σ donor–acceptor interaction, whereby the preferred pathway is yet unknown.