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 λin has been directly calculated. For solvent reorganization energy λs, a novel and reasonable nonequilibrium solvation model is established by introducing a constrained equilibrium manipulation, and a new expression of λs has been formulated. It is found that λs 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.

A modified two-sphere model for solvent reorganization energy in electron transfer.

In this work, the solvent reorganization energy is formulated within the framework of classical thermodynamics, by adding some external charges to construct a constrained equilibrium state. The derivation clearly shows that the reorganization energy is exactly the polarization cost for the inertial part of the polarization. We perform our derivation just within the framework of the first law of thermodynamics, and the final form of the reorganization energy is completely the same as that we gave in our recent work by defining a nonequilibrium solvation free energy. With the two-sphere model approximation, our solvent reorganization energy is derived as λ(0) = Δq(2)/2[1/r(D) + 1/r(A) - 2/d][(ε(-1)(op) - ε(-1)(s))/(1 - ε(-1)(s))]. This amends Marcus' model by a factor of (ε(-1)(op) - ε(-1)(s))/(1 - ε(-1)(s)), which is coupled with the solvent polarity. Making use of the modified expression of solvent reorganization energy, two recently reported electron transfer processes are investigated in representative solvents. The results show that our formula can well reproduce the experimental observations.

Spectral shifts of the n → π* and π → π* transitions of uracil based on a modified form of solvent reorganization energy.

According to our recent studies on the nonequilibrium solvation, the solvent reorganization energy is found to be the cost of maintaining the residual polarization P', which equilibrates with the extra electric field E(ex). On the basis of this solvent reorganization energy and the well-established equilibrium solvation energy, a novel and reasonable expression for the spectral shift of the electronic absorption spectra is proposed in this work. Furthermore, the two lowest transitions of uracil in aqueous solution are investigated as test cases with the TDDFT/6-311++G** method. The obtained spectral shift is 0.48 eV for n → π* transition and -0.14 eV for π → π* transition, agreeing well with available experimental results. The contributions to the shift are discussed and the electrostatic plus polarization components are found to be crucial for the electronic absorption spectra of uracil in aqueous solution.