Theoretical calculation of reorganization energy for electron self-exchange reaction by constrained density functional theory and constrained equilibrium thermodynamics

Within the framework of constrained density functional theory (CDFT), the diabatic or charge localized states of electron transfer (ET) have been constructed. Based on the diabatic states, inner reorganization energy λ_in has been directly calculated. For solvent reorganization energy λ_s, a novel and reasonable nonequilibrium solvation model is established by introducing a constrained equilibrium manipulation, and a new expression of λ_s has been formulated. It is found that λ_s is actually the cost of maintaining the residual polarization, which equilibrates with the extra electric field. On the basis of diabatic states constructed by CDFT, a numerical algorithm using the new formulations with the dielectric polarizable continuum model (D-PCM) has been implemented. As typical test cases, self-exchange ET reactions between tetracyanoethylene (TCNE) and tetrathiafulvalene (TTF) and their corresponding ionic radicals in acetonitrile are investigated. The calculated reorganization energies λ are 7293 cm^-1 for TCNE/TCNE^- and 5939 cm^-1 for TTF/TTF⁺ reactions, agreeing well with available experimental results of 7250 cm^-1 and 5810 cm^-1, respectively.