Electron Transfer Rates in Solution: Toward a Predictive First Principle Approach

Abstract

Using a very recently proposed theoretical model, electron transfer rates in solution are calculated from first principles for different donor-acceptor pairs in tetrahydrofuran. We show that this approach, which integrates tunneling effects into a classical treatment of solvent motion, is able to provide reliable rate constants and their temperature dependence, even in the case of highly exergonic reactions, where Marcus’ theory usually fails.

1. Introduction

Electron transfer (ET) reactions are at foundation of modern chemistry, being exploited in numerous applications (e.g., photovoltaics, biosensing, molecular electronics), and are still the subject of active and vibrant research. The seminal work of Marcus paved the way to the development of this vast field [1,2,3]. Marcus’ theory relates the kinetics of ET process to two well-defined physical quantities: (i) the $\mathsf{\Delta }{G}^0$ of the whole ET reaction and (ii) the reorganization energy associated with the motions of both solvent ${\lambda }_{\mathrm{solv}}$ and reactants ${\lambda }_{\mathrm{int}}$. While this elegant theory is still successfully used to address ET phenomena, as for instance the rational design of organic molecules to be used in technological devices, its classical treatment of nuclear quantum motion does not allow to explore regimes in which tunneling is crucial [4,5,6], and drammatically fails in reproducing ET rates of strongly exothermic reactions [7].

Alternative theories have been formulated [8], and several papers have focused on implementing quantum effects arising from high frequency vibrational modes into Marcus’ theory [9,10,11,12,13,14,15,16], but only recently the task of including the whole heat bath provided by intramolecular coordinates of the redox pair—which represented the real breakthrough for understanding the unusual temperature dependence of an early ET step occurring in bacterial photosynthetic reaction center [17,18,19]—has been undertaken [20]. That objective has been achieved by introducing a multistep kinetic model of ET reactions, described in refs. [21,22], in which the motion of the solvent is separated from that of the redox pair [23,24]. That separation of motion makes it possible the employment of an effective treatment of tunnelling effects, which incorporates the whole set of nuclear coordinates of the redox pair and takes into account changes of both the equilibrium nuclear positions and of vibrational frequencies upon electronic transition, allowing to include effects due to normal mode mixing too [17,25,26]. The approach has been tested on different D-A pairs in both polar and non-polar solvents: it proved to provide ET rates in excellent agreement with measurements [27,28] and to reproduce the expected temperature dependence [20]. The achievement of such a satisfactory result was however subdued to the use of solvent reorganization energy as extrapolated from available experimental data. Nevertheless, to make the model predictive, an ab initio determination of such a quantity would be highly desirable. The evaluation of the solvent reorganization energy based on molecular dynamics with an explicit treatment of solvent molecules [29,30,31,32] is certainly a possible choice, but it might be unaffordable for high-throughput screening procedures. Therefore, the use of implicit solvent methods would be desirable, provided that their accuracy is properly tested. Herein, we show that the simple non-equilibrium Marcus’ approach for the evaluation of solvent reorganization energy is extremely effective and leads to calculated ET rates which are in excellent agreement with the observed ones, yielding also the correct temperature dependence.

2. Theory

2.1. The Kinetic Model

For a given donor-acceptor system, the multistep mechanism reads as Scheme 1:

Where ${\mathrm{D}}^{-}\mathrm{A}$ and $\mathrm{D}{\mathrm{A}}^{-}$ represent the initial and the final states, respectively, and $\left\{{\mathrm{D}}^{-}\mathrm{A}\right\}*$ and $\left\{\mathrm{D}{\mathrm{A}}^{-}\right\}*$ indicate the ensembles of transient structures in vibronic resonance with each other.

First, we consider the activation step bringing the donor and the acceptor species into electronic degeneracy, which is mandatory when initial and final states are not in vibronic resonance at frozen solvent coordinates. This may typically occur in polar solvents for which ${\lambda }_{\mathrm{solv}}>\mathsf{\Delta }{G}^0$. The rate constant of this step is hence estimated assuming the sole involvement of the solvent coordinate and a typical Arrhenius dependence on temperature:

$$k_{act}=k_{0}exp\left(-\frac{\mathsf{\Delta }{G}^{\#}}{k_{B}T}\right)$$

(1)

where $k_0$ is a transmission coefficient and $\mathsf{\Delta }{G}^{\#}$ is the standard activation free energy, i.e. the free energy difference of $\left\{{\mathrm{D}}^{-}\mathrm{A}\right\}*$ with respect to ${\mathrm{D}}^{-}\mathrm{A}$. Applying Marcus’ reasoning to frozen intramolecular coordinates [3], $\mathsf{\Delta }{G}^{\#}$ is defined as [1,3]:

$$\mathsf{\Delta }{G}^{\#}=\frac{{(\mathsf{\Delta }{G}_{fi}^0+\lambda )}^2}{4{\lambda }_{\mathrm{solv}}}.$$

(2)

In Equation (2), $\mathsf{\Delta }{G}_{fi}^0$ is the free energy difference between initial and final states, $\lambda$ is the total reorganization energy, composed by the sum of the solvent and reactants contributions. These in turn may be splitted in the individual contributions pertaining to the donor and the acceptor, e.g., for the solvent ${\lambda }_{\mathrm{solv}}={\lambda }_{\mathrm{solv}}\left(\mathrm{D}\right)+{\lambda }_{\mathrm{solv}}\left(\mathrm{A}\right)$. Following Ref. [20], $k_0$ appearing in Equation (1), is defined as:

$$k_0=k_{dact}\frac{g_{act}}{g_{eq}},$$

(3)

where $g_{act}$ and $g_{eq}$ are the total degeneracies of the activated and the equilibrium states and $k_{dact}$ is the rate constant for deactivation, available from experiments, in consistence with the principle of microscopic reversibility [33].

The second step is the elementary ET process in resonant conditions, for which the rate can be calculated within the framework of the Fermi Golden Rule, considering a non-radiative transition between two electronic states $|i\rangle$ and $|f\rangle$:

$$k_{i\to f}=\frac{2\pi }{\hslash }F(\mathsf{\Delta }{E}_{fi},T),$$

(4)

where $F(\mathsf{\Delta }{E}_{fi},T)$ is:

$$\begin{array}{cc}\hfill F(\mathsf{\Delta }{E}_{fi},T)& =|\langle f|{\mathcal{H}}^{\prime }{|i\rangle |}^2\sum _{{\mathbf{v}}^{\prime }}\sum _{\mathbf{v}″}{w}_{{\mathbf{v}}^{\prime }}\left(T\right){|\langle \mathbf{v}″\left|{\mathbf{v}}^{\prime }\right.〉|}^2\delta (E_{i{\mathbf{v}}^{\prime }}-E_{f\mathbf{v}″}-\mathsf{\Delta }{E}_{fi})\hfill \\ \hfill & =|V_{fi}{|}^2\rho (\mathsf{\Delta }{E}_{fi},T)\hfill \end{array}$$

(5)

where ${\mathcal{H}}^{\prime }$ is the electronic coupling element for ET reaction, assumed to be independent of the vibrational coordinates, ${\mathbf{v}}^{\prime }$ and ${\mathbf{v}}^{″}$ indicate the vectors of the vibrational quantum states of the $|i\rangle$ and $|f\rangle$, respectively, $E_{i{\mathbf{v}}^{\prime }}$ and $E_{f{\mathbf{v}}^{″}}$ are the vibronic energies of $|i{\mathbf{v}}^{\prime }\rangle$ and $|f\mathbf{v}″\rangle$, respectively, and $w_{{\mathbf{v}}^{\prime }}$ is the equilibrium (Boltzmann) population of $|i{\mathbf{v}}^{\prime }\rangle$. Overall, $F(\mathsf{\Delta }{E}_{fi},T)$ consists of an electronic coupling term $V_{fi}$ and of $\rho (\mathsf{\Delta }{E}_{fi},T)$, the Franck-Condon weighted density of states of the elementary $|i\rangle$→$|f\rangle$ transition at the $\mathsf{\Delta }E$ of the reaction, averaged over a thermal equilibrium distribution of initial vibrational states. For weakly interacting molecules, the vibrational motions of each molecular site upon electron transfer can be assumed to be independent from each other. It follows that:

$$\rho (\mathsf{\Delta }E,T)={\int }_{-\infty }^{\infty }dE\mathcal{D}\left(E\right)\mathcal{A}(\mathsf{\Delta }E-E).$$

(6)

where $\mathcal{D}\left(E\right)$ and $\mathcal{A}\left(E\right)$ are the spectral distributions of the donor photoelectronic and of the acceptor electron attachment spectra [34,35], which can be measured [36,37,38] and reliably estimated from ab initio calculations [17,35,39,40,41,42,43,44,45].

From the description of the kinetic scheme, it is clear that the determination of the solvent reorganization energy is apical in the model. In fact, while ET reactions are exothermic when both solvent (Q) and molecular ($\overrightarrow{q}$) nuclear coordinates are in their equilibrium conditions, we here assume the elementary ET step to occur at fixed Q. In particular, we define:

$$\mathsf{\Delta }{G}_{eff}^0=\mathsf{\Delta }{G}_{fi}^0+{\lambda }_{\mathrm{solv}},$$

(7)

and consider two cases: (i) $\mathsf{\Delta }{G}_{eff}^0<0$, for which ET occurs at $Q=Q_{0i}$, the equilibrium solvent coordinate of the initial state, without thus requiring any solvent activation; (ii) $\mathsf{\Delta }{G}_{eff}^0>0$, for which ET occurs at $Q=Q_c$, the point at which the potential energy surfaces of the initial and final states cross each other when intramolecular coordinates are kept fixed at their initial equilibrium value. From Equation (7), it follows that ${\lambda }_{\mathrm{solv}}$ determines the occurrence and the rate of the activation step. Furthermore, even in cases when ET is exothermic and activation is not required, i.e. the kinetic model is not applied, ${\lambda }_{\mathrm{solv}}$ still influences the ET rate, inasmuch as the Franck-Condon weighted density of states is evaluated from Equation (4) at $\mathsf{\Delta }{E}_{fi}=\mathsf{\Delta }{G}_{eff}^0$, which in turns depend on ${\lambda }_{\mathrm{solv}}$, Equation (7).

2.2. Solvent Reorganization Energy

In a previous study [20], ${\lambda }_{\mathrm{solv}}$ in tetrahydrofuran (THF) has been directly estimated from the experimental rate constants of the extremely exothermic BIP-BQO or BIP-NQO pairs, see Figure 1, and from the electronic coupling elements reported in ref. [29]. However, this approach provides values of ${\lambda }_{\mathrm{solv}}$ independent of the chemical nature of the acceptor, an approximation which works well for the A/D pairs of Figure 1 [29], but likely impracticable in other cases, and, more importantly, restricts the feasibility of the model to the study of cases in which ${\lambda }_{\mathrm{solv}}$ can be extrapolated from measurements, thus limiting its predictive power.

Here, we calculate ${\lambda }_{\mathrm{solv}}$ in THF using Marcus’ non-equilibrium approach, according to which [1,46]:

$${\lambda }_{\mathrm{solv}}={\left(ne\right)}^2\left(\frac{1}{2r_{\mathrm{A}}}+\frac{1}{2r_{\mathrm{D}}}-\frac{1}{R}\right)\left(\frac{1}{{ϵ}_{\infty }}-\frac{1}{{ϵ}_0}\right),$$

(8)

where n is the number of exchanged electrons, e is the electron charge, $r_{\mathrm{A}}$ and $r_{\mathrm{D}}$ are the radii of cavity accommodating the donor and acceptor, respectively, R the intermolecular distance, ${ϵ}_{\infty }$ and ${ϵ}_0$ the high-frequency and static dielectric constants of the solvent, respectively.

Since solvent reorganization energies are sensitive to the chosen radii of the dielectric cavities, only for comparison purposes, we have also employed a nonlocal response function theory, formulated in the inverted-space representation of the electrostatic fields, in which the cavity is determined from the atomic radii of A/D pair and solvent [47]. Within this theory, the solvent reorganization is given by two contributions [47]:

$${\lambda }_{\mathrm{solv}}={\lambda }_p+{\lambda }_d.$$

(9)

where ${\lambda }_p$ is the usual orientational reorganization energy, associated with the energetic strength of anisotropic fluctuations of the solvent polarization, and ${\lambda }_d$ the density reorganization energy, which describes how the translational motion of the solvent alters its response by locally changing solvent density around the solute. We will not consider here the latter effect, which is not included in Marcus’ theory.

3. Computational Details

Equilibrium geometries, normal coordinates and vibrational frequencies of the neutral and anionic species are calculated at the density functional theory (DFT) using Gaussian 16 [48]. We employ the B3LYP (Becke [49], 3-parameter [50], Lee–Yang–Parr) [51] functional with the 6-31+G(d,p) basis set. In all the calculations, solvent effect was included by using the continuum polarizable medium (PCM) approach [52,53]. To determine ${\lambda }_{\mathrm{solv}}$ in Equation (8), the radii of cavities have been obtained from the computed PCM cavity volumes, assuming spherical shapes, whereas R has been evaluated as the sum of $r_{\mathrm{A}}$, $r_{\mathrm{D}}$ and the androstane spacer length, the latter taken from geometry optimization of the whole molecules.

Solvent reorganization energies from nonlocal response theory are evaluated employing the freely available SOLVMOL package [54]; for each redox state, ESP charges evaluated by Gaussian 16 at the same level of the theory cited above have been employed [55].

Franck-Condon weighted density of states (FCWDs) are calculated using a development version of the MolFC package, available on request. The internal (curvilinear) representation of normal coordinates have been adopted in all the cases [56]. The electronic coupling elements reported in ref. [29] have been employed throughout. When activation step is not necessary, ET rates are calculated by Equation (4) with the FCWD evaluated at $\mathsf{\Delta }{G}_{eff}^0$. In the other case, the kinetic scheme is used and the set of coupled ordinary differential equations (ODEs) is solved by using the Dormand-Prince method of order 4, a member of the Runge-Kutta family of ODE solvers, as implemented in MATLAB package [57] with $k_{act}$ from Equation (1) and $k_{ET}$ from Equation (4) but with the FCWD evaluated at $\mathsf{\Delta }G=0$.

4. Results and Discussion

Calculated values of $\mathsf{\Delta }{G}_{fi}^0$ s and ${\lambda }_{\mathrm{solv}}$ s for each A/D pair in THF are listed in

Table 1. Calculated equilibrium energy changes ($\mathsf{\Delta }{G}_{fi}^0$), reorganization energies (${\lambda }_{\mathrm{solv}}$), activation free energy ($\mathsf{\Delta }{G}^{\#}$), and rate constants (k) for ET of each BIP-bridge-A pair. All energies are in eV. Experimental rate constants from ref. [27].

		${\mathit{\lambda }}_{\mathbf{solv}}$			k(s${}^{-1}$)
Acceptor	$\mathsf{\Delta }{\mathit{G}}_{\mathit{f}\mathit{i}}^0$	Equation (8)	SolvMol	$\mathsf{\Delta }{\mathit{G}}^{\#}$	Theo	Exp
BQO	−2.4	0.78	0.71	-	1.2 × 10${}^8$	(2.5 ± 0.3) × 10${}^8$
NAP	−0.03	0.74	0.71	0.2	4.4 × 10${}^6$	(1.5 ± 0.5) × 10${}^6$
NQO	−2.1	0.73	0.64	-	6.7 × 10${}^8$	(3.8 ± 1) × 10${}^8$
PHN	−0.1	0.71	0.68	0.2	1.3 × 10${}^7$	(1.2 ± 0.2) × 10${}^7$
PYE	−0.5	0.66	0.68	0.01	4.0 × 10${}^9$	(1.5 ± 0.5) × 10${}^9$

, along with values of $\mathsf{\Delta }{G}^{\#}$ s, the standard activation free energy (c.f. Equation (2)), and the calculated and the observed rate constants. Solvent reorganization energies calculated by Marcus’ solvent treatment and by nonlocal response theory function are very similar each other, when only solvent polarization effects are considered; we have thus considered only the former ones for computing ET rates. Herein, assuming that the entropy of reaction due to intramolecular vibrations is negligible, we consider the calculated energy differences as free energies. The extremely exergonic ET reactions involving BIP-BQO and BIP-NQO pairs do not require an activation step because they retain exergonicity also at frozen solvent coordinates (see Table 1), so that their rate constants can be directly evaluated by Equations (4)–(6), using $\rho (\mathsf{\Delta }{G}_{eff},T)$. Vice versa, all the other A/D pairs require a solvent activation step, so that their rates have been obtained by numerically solving the systems of ordinary differential equations associated to the multistep kinetic ET mechanism, c.f. Scheme 1. The rate of the solvent response to a nonequilibrium charge distribution of the solute ($k_{dact}$) has been taken from time dependent spectroscopic measurements (Stokes shifts) [58], and the transmission coefficient $k_0$ has been set to 5$\times {10}^{12}$ s${}^{-1}$ for all species, as in previous work [20]. Table 1 shows that for all systems considered here the calculated rates differ from the experimental ones by a factor 2–3, the order of magnitude being well reproduced in all the cases.

In Figure 2, the temperature dependence of ET for two acceptors, BQO and NAP, the only two for which experimental results are available [59], is reported. ET from BIP to NAP needs activation by solvent motion and therefore the T dependence of ET rate constant is dominated by the Arrenhius exponential factor, and thus ET rates change by over 4 orders of magnitude. Conversely, ET from BIP to BQO occurs by tunneling and it is almost temperature independent. In this case, we have considered that the dielectric constant of THF exhibits a significant temperature dependence, which, in the temperature range $-78$ to 30 °C, can be expressed by the following relation: $ϵ\left(T\right)=-1.50+2650/T$ [60]. It is interesting to remark as $ϵ\left(T\right)$ increases with T, so does solvent reorganization energy. However, since ET rates occurs by tunneling, the larger values of ${\lambda }_{\mathrm{solv}}$ do not produce an exponentially decrease of the rate. In fact, we observed an increase of the Franck-Condon weighted density of states with a mild effect on ET rates, c.f. Figure 2.

5. Conclusions

In our approach, we have assumed that elementary ET reactions always occurs by tunneling and that solvent motion modulates the energy condition under which this phenomenon occurs. Herein, at variance with a previous study, we calculate solvent reorganization energy using Marcus’ original expression, showing that it yields satisfying results for the systems considered here. That is a significant step toward the development of a predictive first-principles approach for the calculation of ET rates, which ultimately tends to avoid the long and laborious calculations of molecular dynamics, routinely used for determining ET rates in solution. However, the approach still makes use of an adjustable parameter to be set from experimental data, i.e. the transmission coefficient $k_0$, and this parameter has to be related to solvent properties; work is in progress along this line.