Abstract:

n the 1950s of the 20th century, the non-equilibrium solvation theory for ultra-fast processes such as electron transfer and light absorption/emission was paid particular attention to. A number of scientists made efforts to study this area and various models, which give reasonable qualitative descriptions for solvent reorganization energy in electron transfer and spectral shift in solution, were developed within the framework of continuous medium theory. However, in a series of publications by the authors (see for example J. Comput. Chem. 2004, 25: 500; J. Comput. Chem. 2004, 25: 835; J. Comput. Chem. 2005, 26: 399; Chinese Sci. Bull. 2003, 48: 965; J. Mole. Struct.Theochem 2005, 715: 157), it was clarified that the expression of the nonequilibrium electrostatic free energy, which is at the dominant position of nonequilibrium solvation and serves as the basis of various models, was incorrectly formulated. In this work, the authors argue that reversible charging work integration δW=∫_VΦδρdV  was inappropriately applied to an irreversible path linking the equilibrium state and the non-equilibrium one in the past. Because the step from the equilibrium state to the non-equilibrium one is factually thermodynamically irreversible, the conventional expression for nonequilibrium free energy, G^non₂(M)=(1/2)∫_V(ρ₂Φ^non₂+ρ₂Φ^eq₁-ρ₁Φ^non₂)dV  that was deduced in different ways, is unreasonable. Here the authors derive the non-equilibrium free energy to a quite different form of  G^non₂=(1/2)∫_V ρ₂Φ^non₂dV according to Jackson integral formula,dG= (1/2)∫_V(Φδρ+ρδΦ)dV. Such a difference throws doubts to the models including the famous Marcus two-sphere model for solvent reorganization energy of electron transfer and the Lippert-Mataga equation for spectral shift. By introducing the concept of “spring energy” arising from medium polarizations, the energy constitution of the non-equilibrium state is highlighted. For a solutesolvent system, the authors separate the total electrostatic energy into different components: the selfenergies of solute charge and polarized charge, the interaction energy between them and the“spring energy” of the solvent polarization. With detailed reasoning and derivation, our formula for non-equilibrium free energy can be reached throughdifferent ways. Based on the new expression of nonequilibrium free energy, the generalized form for solvent reorganization energy,λ_av=(1/4)∫_VΔρ(Δφ_op-Δφ_s)dV, has been attained. A new twosphere model for solvent reorganization energy is proved to have the form of  λ_av=(1/2)Δq²(1/ε_op-1/ε_s)(1/(2r_D)+1/(2r_A)-1/d). Compared with Marcus′ expression, this new formula estimates the solvent reorganization energy only one half of the latter. This difference provides a pretty explanation for why Marcus′ theory often overestimated the solvent reorganization energy by a factor about two in the past. With the single-sphere model and point dipole approximation, the authors argue that the total spectral shift should look like Δhν_total=(1/2)R_slow(μ₁-μ_m)², and this is also one half of the Lippert-Mataga result. The novel expressions for the spectral shifts for individual absorption and emission have also been given. Finally, a numerical algorism for the solution of Poisson equation is presented and the total nonequilibrium solvation energy is deduced to a quite different and much more compact form as ΔF^non₂=〈Ψ^non₂|H⁰+(1/2H''|Ψ^non₂〉-〈Ψ^gas₂|H⁰|Ψ^gas₂〉+(1/2)∑_i∑_j［q₂,_fast(i)+q_1,slow(i)］Z_j/(|r_i-R_j|) when compared with the most recently developed expression by other authors. As an application, the numerical algorism incorporated with COSMO was applied to a model system, and the solvent reorganization energy is found in excellent agreement with the experimental fitting,while the conventional theories always estimate twice this quantity.

Key words: Nonequilibrium solvation; Electron transfer; Spectral shift; Numerical algorism; Two sphere model