To the best of my knowledge, Marcus theory is the benchmark for electron transfer kinetics.

If you think of the electron transfer process as occurring by the system switching from one potential energy surface (PES) to another, where one PES corresponds to the "reactant" state and the other to the "product" state then you need to search for the intersection between the two PESs to find the barrier. Numerically locating such a point with electronic structure calculations can be very challenging.

The Marcus expression is based on the assumption that the states may be modeled by two intersecting parabolas. Given this simple functional form, the intersection point is determined by the distance between the minima and the curvatures, so it can be determined with just a couple of energy calculations at the equilibrium geometries, from which the reorganization energy may be determined. Page 16 of this document shows how the crossing point is determined: http://www.acmm.nl/molsim/han/2011/ensing_lectures1.pdf

I suppose that one could assume a more flexible functional form for each of the two states, but it would then be necessary to determine more parameters to fully specify the functions, hence more calculations would be needed. Basically the idea would be to more accurately locate the intersection by more accurately specifying the shapes of the intersecting functions. II don't know if this would be worth the effort.

Thank you very much for answers

Marcus expression for the estimation of energy barrier, ΔG#, has two important assumes for PESs which are two-dimensional PESs with similar curvatures. Therefore, I feel that geometry determination of intersection point according to expression reported in page 16 of above document (in Karl answer) is simple only for either system with similar geometries of donor and acceptor (ΔG0 ~ 0) or systems with a few vibrational frequencies involved within electron transfer process. This means that it is very difficult to calculate geometry of intersection point for complex systems with multi-dimensional PESs.

Regarding exercise (Gert says), when reorganization energy is going to zero value, geometries of before and after electron transfer are approaching together, parametrically. This can infer based on the definition of reorganization energy. This is reason for rate of zero or infinite energy barrier.

There is significant point about transition state geometry in electron transfer reactions. In some systems, electron transfer accompanies by severe change of some structural parameters (angles or bond lengths). For example, electron transfer from donor species to CO2 molecules can produce the [CO2]−, [C2O4]− and [C2O4]2− subunits which results to conversion of linear geometry of neutral CO2 molecule to bent geometries in the [CO2]−, [C2O4]− and [C2O4]2−. In these systems, we may able to achieve transition state geometry, directly. It should be very interesting to compare calculated energy barrier by this transition state geometry with that of Marcus equation in these systems.

The constrained density functional theory (CDFT) is the useful method to obtain optimized geometries along with arbitrary constraints. For example, equilibrium geometry of before electron transfer (D1e-A1) and after electron transfer (D2-A2e) can obtain through electron constraint on donor geometry (D) and acceptor geometry (A), respectively. For calculation of reorganization energy (λ) as shown in the attached file, we need non-equilibrium geometry of D2e-A2. Note that non-equilibrium geometry of D2e-A2 has structural properties of D2-A2e and electronic properties of (D1e-A1), simultaneously. Application of electron constraint on D2-A2 geometry can result to non-equilibrium energy of D2e-A2 which is necessary for the calculation of λ factor. In spite of significant role of CDFT method for the calculation of λ factor through estimation of non-equilibrium energy of

D2e-A2, this method has not used as an interesting strategy to estimate the energy of interesting point of two PESs for the calculation of energy barrier. Application of CDFT method can be good solution to obtain the geometry of transition state. But, I feel that it is not easy.

Dear Gert

I don’t know any method for obtaining of transition state geometry in electron transfer reaction which be reported in the literature. I feel that the estimation of transition state geometry in electron transfer reaction is an open field in literature. The CDFT method is just my suggestion for the solution of this problem (estimation of a height of energy barrier). I am planning to use the CDFT method for obtaining of transition state geometry in well-known simple systems. It should be interesting to compare the predicted energy barrier by CDFT method (through estimation of transition state geometry) with that of Marcus equation (without transition state geometry).

Dear Gert

As I know, there are three conventional methods for the calculation of reorganization energy, computationally. The first method is Nelson's four-point method (H. Sahu, A.N. Panda, Phys.Chem.Chem.Phys. 16 (2014) 8563-8574) that is just suitable for systems with weak interactions between donor and acceptor subunits. The second method is based on vibrational frequencies that requests the calculation of all vibrational frequencies (J. Phys. Chem. B 2006, 110, 18904-18911). The third method is direct method that requests both geometries of before and after electron transfer (the requested geometry can obtain by CDFT (K.M. Rosso, D. M. A. Smith, M. Dupuis, J. Chem. Phys. 118 (2003) 6455)). I have used all these methods for different systems and have noted that the direct method (by CDFT) is simpler than other methods.