Electron transfer reactions (ET) are fundamental processes involved in most biological phenomena such as respiration, oxidation [1], photosynthesis [2] or DNA processes [3]. From a mechanistic point of view, ET begins when diabatic states from reactants (R) and products (P) cross each other (Fig.1) according to the Frank-Condon principle.

Figure 1. Electron transfer process diagram in Marcus Theory.


The rate constant of the process is given by the Marcus equation:

where h is the Planck's constant, kB is the Boltzmann's constant, T is the temperature, ∆G is the reaction Gibbs energy, λ is the reorganization energy and VDA is the electronic coupling between donor and acceptor. Out of these parameters, VDA is the most geometrically dependent element in the kinetic constant because its value depends on the donor-acceptor distance and geometry of the system (orbitals orientation). Besides, VDA is sensible to modifications in the system as solvent, temperature , distance [4] , etc. , being a good estimator of the optimum reaction's conditions. Moreover, it affects quadratically the rate and it is commonly used as an overall indication of the ET propensity. Its exact calculation through quantum chemistry methods is a difficult task, involving the localization and overlap of diabatic states, which are quite difficult to characterize. For all these reasons simpler approximations have been developed in last years to compute VDA in complex biological systems. This server allows you to compute VDA using two of these approximations: GMH and FCD.


DONOR-ACCEPTOR TWO STATE MODEL

For symmetric and simple systems, the effective VDA could be defined as the half of the splitting energy at the seam between two adiabatic states (Fig.2)

Figure 2. Detail of Adiabatic states (a) and diabatic states (d) at the crossing.


However, in biological systems (~weak electronic coupling regime), donor and acceptor are far from each other and the crossover is difficult to obtain. Hence, several approximations have been developed to calculate the matrix by orthogonal transformation of adiabatic states to diabatic states including the Generalization of Mulliken Hush (GMH [5]) and the Fragment Charge Difference (FCD [6]) approximations.

In these methods an extra operator is applied to modify the Hamiltonian of the system, which should have the same value for donor and acceptor in the crossover.

If we consider a D-A (donor-acceptor) model described with 2 diabatic states (initial and final states), the diabatic hamiltonian of the system can be defined as a matrix where energies from diabatic states are the diagonal elements and the off-elements are the electronic couplings.

In the adibatic representation, however, the hamiltonian matrix is a diagonal matrix whit the energies from the adiabatic states ψ1 and ψ2 from donor and acceptor, respectively.

Both diabatic and adiabatic Hamiltonians are connected: From the diabatic hamiltonian , the energy matrix E is obtained by application of the orthogonal matrix CT, resulting in a diagonal matrix.

From this matrix is possible to compute the electronic coupling from the diabatic hamiltonian, by applying the transformation matrix T which is equal to the transposed matrix C (T = CT )

As a conclusion, the diabatic Hamiltonian Hd is expressed through the diagonal matrix E of the adiabatic energy and therefore, the diabatic parameters can be computed from the adiabatic energies (readily obtained in quantum chemistry calculations) if we are able to compute the transformation matrix T .
There are several approximations to compute the transformation matrix (T). For example (table 1) in the GMH method the operator is the dipole and in the FCD is the charge.

The operator employed in the GMH method (Cave & Newton (1996)) is the adiabatic dipole moment matrix. Under this approximation (and in the weak coupling regime), the coupling is calculated as:

where ΔE is the orbital energy difference and Δμ12 is the difference of the diabatic state dipole moments.

In FCD method (Voityuk and Rösch (2002) ) the matrix T is obtained by diagonalization of the charge difference matrix ∆Q.

where ∆Q12 is the transition charge difference, ∆Q1, ∆Q2 are the charge difference in the adiabatic states, and ΔE is the orbital energy difference. This method requires to define acceptor and donor sites involved in the process.

Table 1. Summary of GMH and FCD methods in the two state model .


Method

Operator

Diabatic electronic coupling matrix


GMH


Δμ1-2



FCD


ΔQ12



MULTISTATE E-COUPLING

In the two state model we have considered one electronic state for each donor and acceptor. However, if there are energy degenerate (or quasidegenerate - close in energy) orbitals, we cannot rule out them in the calculation. The effective coupling (V2 effDA) takes into account the multi state situation:


where N1 and N2 are the number of degenerate states from the Donor and from the Acceptor and rmsVDA is the root mean square of the electronic coupling between donor and acceptor for each D-A combination (all taken into account):




BRIDGE MEDIATOR E-COUPLING

Electon/hole transfer from donor to acceptor is sometimes mediated by a bridge. This phenomena called, superexchange is specially important in some physical and biological processes as long-range charge migration in DNA, long range ET in proteins or in some wires or semiconductors and thus, its effect can not be neglected.

The effective bridge assisted electron coupling can be defined as:



where VDA is the electron coupling between donor and acceptor applied to the whole donor-bridge-acceptor system (two state or multistate model) and VSE is the superexchange term.

VSE depends on the coupling between donor-bridge and bridge-acceptor and the energy barriers between orbitals , following the formula:


where :




1 DN Beratan, JN Onuchic, Winkler JR, and HB Gray(1992)Science 258 (5089), 1740-1741.

2 H.B.Gray and J.R.Winkler, (2009)Chem.Phys.Lett. 483,1.

3 Y.A.Berlin, I.V.Kurnikov,D.Beratan,M.A.Ratner and A.L.Burin(2004)Chem.Phys.Lett. 267, 234-243.

4 Kaniyankandy, S., Rawalekar, S., Sen, A., Ganguly, B., & Ghosh, H. N. (2011). Does Bridging Geometry Influence Interfacial Electron Transfer Dynamics? Case of the Enediol-TiO2 System. The Journal of Physical Chemistry C, 116(1), 98-103.

5 Cave, R. J., & Newton, M. D. (1996). Generalization of the Mulliken-Hush treatment for the calculation of electron transfer matrix elements. Chemical physics letters, 249(1), 15-19.

6 A.AVoityuk, N. Rösch. Fragment Charge Difference Method for Estimating Donor-Acceptor Electronic Coupling: Application to DNAπ-Stacks. J. Chem. Phys.2002,117, 5607–5616