Modeling Electron Transfer Reactions

Along with hydrogen transfers (in the form of protons, hydrogens, or hydrides), electron transfer is one of the most fundamental chemical processes in biological systems. Electron transfers are found in the respiratory chain, in photosynthesis, and in uncountable metabolic pathways. Enzymes such as the family of oxidoreductases have evolved to catalyze chemical reactions such as the functiona-lization of C–H bonds. Inner-sphere or outer-sphere (Long Range) electron

Table3FivepopulationapproachescommonlyusedincDFTcomputationstodefinetheweightmatrixelements PADefinitionofthechargeinthevolumeOWeightmatrixelementscDFTweightwi MullikenQMulliken A¼ZA
P m2AðPSÞmm Wmn¼Smnifmandn2A Wmn¼1 2Smnifmorn2A Wmn¼0ifmandn=2A

– Lo¨wdinQL€owdin A¼ZA
P m2AðS12=PS12=ÞmmWmn¼P lS12= mlS12= ln– BeckeQBecke A¼ZA
Ð BeckecellofArðrÞdrWmn¼P iaiwimðriÞnðriÞ wbecke i¼1:0insidethecell wbecke i¼0:0outsidethecell +smoothingfunction Hirshfeld QHirshfeld A¼ZA
Ð rðrÞra AðrÞ X allatomsraðrÞdrWmn¼P iaiwimðriÞnðriÞ wi¼

X atomsinAraðriÞ X allatomsraðriÞ VDD QVDD A¼Ð VoronoicellofArðrÞ
P allatomsraðrÞ drWmn¼P iaiwimðriÞnðriÞ wVDD i¼1:0insidethecell wVDD i¼0:0outsidethecell

transfers are usually at play within the catalytic cycles of these enzymes [239].

However, the precise molecular strategies enabling efficient catalysis are often elusive. Computational chemistry, and in particular DFT, provides a valuable means to complement biochemical studies to reach a detailed understanding of these biological processes. From a theoretical point of view, the most popular conceptual framework for dealing with the kinetics of ET is the Marcus Theory (MT) [213]. The theory assumes the definition of two phenomenological electronic states corresponding to situations where the electron to be transferred is localized on the reductant (initial redox state) or on the oxidant (final redox state). These diabatic electronic states correspond to the empirical description familiar to the chemist. In MT two limiting regimes are considered. First when the quantum coupling between the two diabatic states is strong as in the case of inner-sphere ET, an adiabatic rate constant can be employed:

k^A_ET¼ v exp
ðDG^þ lÞ² 4lkBT

!

; (28)

where DG^is the free energy of the reaction,l is the reorganization energy, and n is the effective frequency along the reaction coordinate. Actually the diabatic energy gap is often a good choice of reaction coordinate [240]. The other terms have their usual meaning. When the coupling between the two diabatic states is weak, as in the case of a long-range ET or of symmetry-forbidden ET, a nonadiabatic expression applies

k^NA_ET¼2p

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4plkBT

p jHDAj²exp
ðDG^þ lÞ² 4lkBT

!

: (29)

whereHDAis the electronic coupling factor between the initial and final redox states (respectivelyj i and ’’D j i):A

HDA¼ ’h jHD elj i;’A (30) Helbeing the electronic Hamiltonian.

The cDFT formalism of Dederich et al. and Wu et al. has been employed by various groups to model the two diabatic states and to compute the terms entering the adiabatic and nonadiabatic rate constant expressions [241–244]. In particular, cDFT is one of the few DFT-based approaches that enable the evaluation of the electronic coupling terms between the diabatic states with no need for the adiabatic representation [242, 245]. In addition, the possibility of performing BOMD simulations on cDFT potential energy surfaces makes possible the evaluation of free energy terms. In this sense, cDFT holds great promise for the modeling of ET reactions at the DFT level, especially thanks to the computational advantages of the cADFT framework. We already mentioned the cDFT method of Wesolowski et al.

that also has found numerous applications in the field [246,247].

In [226] we investigated the decay of the electronic coupling as a function of the donor to acceptor distance for the electron transfer between two lithiums separated by a polyglycine peptide of increasing length (Fig.19). The simulations were run in the gas and aqueous phases thanks to the QM/MM scheme described in Sect.5.

Note that the BOMD simulations were biased so as to sample exclusively the molecular configurations corresponding to the degeneracy of the two diabatic states. Note also that the lithium and its first coordination shell were fixed during the simulations to control theD to A distance. In this work we also reported the first estimates of the characteristic decay times of the Franck–Condon factors by means of cDFT/MM MD simulations. This term reflects the time the molecular system remains in the Franck–Condon region that is active for the ET. It can be computed by the following expression:

whereFxnandan,respectively, denote the forces felt by thenth degree of freedom in the electronic state 1 or 2 and the typical width of the wave packet associated with this degree of freedom. Actually Eq. (31) provides a means to evaluate the decoherence time in the context described in Sect. 4.3, but under a short time approximation and within the high temperature limit [139]. Note that this approxi-mate equation enables an estimation of the decoherence time without resorting to computationally expensive diverging trajectories (see Sect.4.3).

The graphs shown in Fig. 20 are in line with common knowledge of the electronic coupling decaying exponentially with theD to A distance. Characteristic decays of 0.52 and 0.58 A˚
¹ are found for the gas phase and aqueous phase, respectively, in good agreement with the experimental values reported in the literature [248] for LRET rates through b-strands. The characteristic decoherence times are found to be around 4.5 and 1.5 fs in the gas and aqueous phases, but quite interestingly independent of the bridge length. This constant trend can be under-stood by looking at Eq. (31). The atoms that contribute the most to decoherence are those which feel the different forces in the two redox states. Therefore, the molecular fragments that are far from the donor and from the acceptor do not appreciably contribute to decoherence. And, similarly, when the bridge length is increased the new intervening atoms contribute less and less to decoherence.

Recall, however, that in these simulations the most contributing atoms were fixed n

Fig. 19 Li₂^+/0redox pairs separated by polypeptide chains of increasing lengths investigated in [226]. Reproduced with permission from [226]

in space for practical reasons so that the actual characteristic decoherence time for this reaction is probably much below 1 fs. In fact the study reported in [226] is more to be seen as a proof of principles of the use of cDFT/MM MD approaches for the estimation of decoherence times in large molecular systems. Investigations of decoherence times in real enzymatic systems are currently being performed in our laboratories using this methodology and will be reported in due course.