We present a straightforward and low-cost computational protocol to estimate the variation of the charge transfer rate constant, kCT, in a molecular donor–acceptor caused by an external electric field. The proposed protocol also allows for determining the strength and direction of the field that maximize the kCT. The application of this external electric field results in up to a >4000-fold increase in the kCT for one of the systems studied. Our method allows the identification of field-induced charge-transfer processes that would not occur without the perturbation caused by an external electric field. In addition, the proposed protocol can be used to predict the effect on the kCT due to the presence of charged functional groups, which may allow for the rational design of more efficient donor–acceptor dyads.

Charge transfer (CT), also known as electron transfer (ET) reactions, is ubiquitous in biology and chemistry. In biology,1 ET reactions can be found, for instance, in redox enzymes2,3 or in the activation of sensory proteins,4 but also in fundamental processes such as nerve impulse transmission,5 photosynthesis,6 cellular respiration,7 or DNA UV-damage repair,8 among others. In chemistry, there are many reactions that can be classified as ET reactions. The most obvious example is constituted by the group of oxidation-reduction reactions.9 Other examples include the proton-coupled electron transfer (PCET) reactions4 as well as all reactions carried out under photocatalytic conditions.10,11

Charge transfer reactions are also in the core of organic solar cells (OSCs). Solar cells are needed to harvest solar energy and convert it into electricity, reducing the use of coal and oil. Such a transition toward competitive low carbon-fingerprint energy harvesting technologies is the seventh goal of the United Nations Sustainable Development Agenda for 2030 and UNESCO’s World Heritage.12 OSCs represent a promising alternative to building photovoltaic devices owing to their easier manufacturing, lower weight, flexibility, and associated cost.13–15 When designing OSCs constructed using the molecular heterojunctions (MHJs)16 approach, the electron acceptor and electron donor are covalently linked, forming a donor–acceptor (D–A) dyad. Compared to OSCs based on bulk heterojunction structures, MHJs allow better structural control and charge mobility tuning, which are notable advantages for the difficult task of optimizing the performance of charge separation processes. Although these MHJ organic cells are not implemented in real organic cells, the studied dyads can be used as model systems to understand the photoinduced electron transfer processes that occur in OSCs.

Computational modeling can help improve the design of D–A dyads with high charge-transfer rate constants.17–20 In this work, we present a computationally inexpensive protocol to determine the effect of the oriented external electric fields (OEEFs) on the rate of the charge transfer process, kCT. Our methodology can be used to speed up any charge transfer process by applying an OEEF, although, in this work, we decided to focus our study on the changes in kCT induced by an OEEF in the case of four fullerene-based dyads (Fig. 1): trans-2 C60-ZnTPP (ZnTPP),21 C60-triphenylamine (TPA),22,23 C60-3,6-ditBu-Azulene (Az),24 and C59N-phthalocyanine (PC).25 

FIG. 1.

The donor–acceptor dyads studied in this work and their orientation in the Cartesian space.

FIG. 1.

The donor–acceptor dyads studied in this work and their orientation in the Cartesian space.

Close modal
Within Marcus theory,26–28 the charge transfer rate depends exponentially on the negative of the square of the sum of the reorganization energy, λ, and the Gibbs energy change in the electron transfer process, ΔG,
kCT=2πVDA214πλkBTexpλ+ΔG24λkbT,
(1)
where refers to the reduced Planck constant and kB to the Boltzmann constant. In most D–A dyads, the excitation is delocalized over D and A and oscillates back and forth between them to finally populate the most stable local excited state [LES1, (D–A)*],29 which can be located on D or A or delocalized among D and A. The charge transfer preferably occurs in the transition from LES1 to the most stable charge transfer state (CTS1, D+–A).30–32 Therefore, the ΔG to be estimated in Eq. (1) is ΔG = ΔGLES1→CTS1 = ΔGCTS1 − ΔGLES1, where ΔGi is the Gibbs energy difference between the ith excited state and the ground state. A good D–A dyad must show fast charge separation (i.e., high kCT) and slow charge recombination. Here, we investigate how OEEF can be used to increase the kCT in D–A dyads. When the D–A dyads present a random orientation in space, an OEEF cannot be directly applied to improve the efficiency of their charge transfer process. However, once the direction and strength of the most suitable electric field for optimizing the charge-transfer process for a given dyad are determined, it is possible to redesign the dyad by placing charged or polar functional groups at certain positions to locally generate the required electric field and, therefore, increase the efficiency of the charge transfer process.

According to Eq. (1), the maximum value of the rate constant is observed when ΔG = −λ. This constraint can be imposed by modifying ΔG, λ, or both with an external perturbation, such as an OEEF. Despite the fact that the effect of internal or local EF has been acknowledged,33 measurements of charge transfer rates in D–A dyads under OEEFs are very scarce.34–36 In order to identify the optimal OEEF to speed up a charge transfer process, it would be ideal to develop an inexpensive computational tool to predict the impact of the OEEF on kCT.

The study of the effect of OEEF on reactivity is a hot topic nowadays.37–44 In a previous study, some of us reported a method to predict the effect of an OEEF on the rate and selectivity of a chemical reaction based on the Taylor expansion of the field-dependent energy of the reactants and transition states in terms of their field-free dipole moments and electrical (hyper)polarizabilities.45 Here, we propose to use an equivalent approach to evaluate the changes induced by an OEEF in the relative energy of the excited states of a molecule. The field-dependent relative Gibbs energy of the ith excited state, ΔGi(F), is given by
ΔGi(F)=ΔGi(0)ΔμiFΔαiF2+O(F3),
(2)
where ΔGi(0), Δμi, and Δαi correspond, respectively, to the field-free relative Gibbs energy of the ith excited state and the differences between the electronic dipole moment and electronic polarizability of the ith excited state and the ground state. Equation (2) allows the estimation of the change in the relative Gibbs energy of the excited states due to the presence of an arbitrary OEEF only from data obtained in field-free calculations and, therefore, without the need to perform calculations including the OEEF explicitly.

For practical use, Eq. (2) must be truncated at a point that balances the desired accuracy and the computational cost. In a previous study, some of us showed that the truncation of the Taylor expansion of the field-dependent energy at the quadratic (i.e., polarizability) term accurately predicts the changes in the rate and the selectivity of a chemical reaction at a very low computational cost.45 Here, we have used Eq. (2) to predict the field-dependent Gibbs energies of the excited states of the photoactive systems ZnTTP, TPA, Az, and PC and, subsequently, determine the optimal OEEF that maximizes their kCT by imposing the constraint ΔG = −λ. For ZnTTP, we have also compared the kCT obtained from Eq. (2) with corrections up to the second order with the one obtained by means of explicit OEEF calculations as a benchmark of the accuracy of the presented method.

As for many D–A dyads, the first and second excited states of ZnTTP correspond to the most stable charge transfer state (CTS1) and the most stable local excited state (LES1), respectively. Specifically, the electron-transfer LES1→CTS1 can be conceptualized as the movement of one electron from the porphyrinic ring (the donor unit) to the C60 (the acceptor unit). Interestingly, a judiciously applied OEEF enhances the kCT value (vide infra), while the charge recombination process is expected to be slowed down as the OEEF pulls the electron density away from the generated hole. Once the methodology was validated for ZnTTP, it was also applied to the other systems under study, namely TPA, Az, and PC. We finalize our work by studying the kCT of ZnTTP with the point-charges located in space to generate an OEEF similar to the optimal OEEF determined with our new methodology. We show that in this point-charge model, the kCT is enhanced, thus paving the path toward the design of more efficient dyads.

We have obtained the ground-state equilibrium geometry at the B3LYP-D3(BJ)/6-311G(d,p) level of theory46–50 for each structure from somewhere else.51 The donor and acceptor geometries needed to estimate the internal reorganization energy were obtained at the same level of theory as the ground state equilibrium geometries.

Twenty to one hundred lowest-lying singlet excited states for each D–A pair were computed using the time-dependent density functional theory (TDDFT) formalism52–58 at the CAM-B3LYP level of theory expanding the orbitals with the double-ζ with polarization Def2SVP59 basis set at the ground-state equilibrium geometry. As several works60–63 prove, except for a few cases,64 changes in kCT are minor if one considers the effect of the geometrical relaxation in the LES1. Indeed, in the case of fullerenes, the change in the equilibrium geometry in the transition from the ground state to the LES1 is minor, as can be seen from the root mean square deviation, RMSD, values given in Table S5 of the supplementary material. CAM-B3LYP65 has been reported to be one of the best density functional approximations for the evaluation of CTS.51 Both the ground and excited state calculations were performed with the Gaussian16 package.66 

A quantitative analysis of exciton delocalization and charge separation is carried out in terms of the transition density matrix T0i of the ith excited state (Φi*). This analysis is performed on the more convenient Löwdin orthogonal basis. The matrix λC of the molecular orbital (MO) coefficients expanded in a basis of orthogonalized atomic orbitals is obtained from the coefficients C in the original atomic basis λC = S1/2C, where S is the atomic orbital overlap matrix. The transition density matrix T0i for an excited state Φi* is constructed as a superposition of singly excited configurations67,68 where an occupied MO ψj in the ground state is replaced by a virtual MO ψa and is computed as
Tαβ0i=jaAjaiλCαjλCβa,
(3)
where Aij→a are the expansion coefficients corresponding to the ith excited state and alpha and beta are atomic orbitals.
The excitation weight Ωi(D, A) is determined by
Ωi(D,A)=1/2αD,βATαβ0i2.
(4)
The weights of local excitations on D and A are Ωi(D, D) and Ωi(A, A), respectively. The weight of electron transfer configurations D → A and A → D is represented by Ωi(D, A) and Ωi(A, D). Therefore, the quantity CSi = Ωi(D, A) − Ωi(A, D) describes charge separation between D and A, and the CTi = Ωi(D, A) + Ωi(A, D) is the total weight of CT configurations in the excited state Φi*. With this methodology, CT states (CTS) and local excited states (LES) can be easily identified. In LES, the excitation is mostly localized on a single fragment (CS < 0.1 e), whereas in CTS, the electron density is transferred between D and A (CS > 0.9 e).
We used the Fragment Charge Difference (FCD) method to derive the coupling of LES and CTS calculated with TDDFT.69 Within the two-state model, the D–A coupling is given by
VDA=EiEjΔqijΔqiΔqj2+4Δqij2,
(5)
where Δqi and Δqj are the difference in the donor and acceptor charges in the adiabatic states Φi and Φj, respectively, and Δqij is the charge difference computed from the Φi → Φj transition density matrix. Several years ago, the Fragment Charge Difference (FCD) method was extended to calculate the electronic couplings and diabatic energies for photoinduced reactions.69 FCD was shown to provide consistent values of the ET parameters for two- and multi-state model systems. It was suggested how to identify situations where the two-state scheme can be applied and where it will fail to provide satisfactory results. In our present work, we used these criteria69 to thoroughly check whether the two-state model can be applied to derive electronic couplings.
The total reorganization energy can be decomposed into the internal and external contributions (λint and λext). λint is the average of the energy required to distort the nuclear configuration from the D+–A or (D–A)* equilibrium geometry to the equilibrium geometry of the (D–A)* or D+–A state without transferring an electron. λext is the corresponding energy required to change the slow (reorientational) part of the solvent reorganization between both equilibrium geometries. In this study, λint was computed considering isolated donor and acceptor fragments, which contribute separately to the internal reorganization energy,
λint=λD+λA,
(6)
where λD and λA are the reorganization energies of the donor and acceptor, respectively. In turn, λD was estimated as
λD=12λD+λD,
(7a)
λD=En(D)En(D),
(7b)
λD=Eion(D)Eion(D),
(7c)
where En(D) and Eion(D) are the electronic energies of the neutral and ionic states of the donor computed at their ground-state equilibrium geometry, and En′(D) is the energy of the neutral state computed at the equilibrium geometry of the ionic state D+. Eion′(D) is the energy of D+, estimated at the equilibrium geometry of neutral D. Similarly, we calculated λA using equilibrium geometries of A and A.

The λext(F) is difficult to estimate using polarizable continuum models of the solvent. In this work, and as a first approximation, we have not considered the dependence on the OEEF of the λext. The reason lies in the fact that in the presence of an OEEF (especially if it is intense), the solvent molecules will be oriented in the direction of the field instead of following the electron density of the solute. In this case, nothing or little will change after a charge-transfer, and the solvent molecules will continue to be oriented in the direction of the field, i.e., a zero (or low) value for λext(F). We have calculated λext only for the field-free calculation.

The CT rates were computed within the nonadiabatic electron transfer theory, where the CT process DA → D+A can be described by the Marcus equation [Eq. (1)]. Taking into account the special treatment that λ deserves, the field-dependent rate-constants have been computed as the combination of the field-free rate constant, kCT(0), where the λext has been taken into account,51 and a correction for the presence of the field by the addition of the field-dependent quantity kCT(F) − kCT(0), computed setting λext = 0 for both rates, the kCT(F) and kCT(0). The final derived kCT(F) is given by kCT(F) = kCT(0, λ = λint + λext) + kCT(F, λ = λint) − kCT(0, λ = λint).

Owing to the zwitterionic nature of the CTSs, such states are expected to have considerably larger dipole moments than the ground state or the LES. In other words, the change of the dipole moment in the CT reaction is large, and it dominates the Taylor expansion (2). Consequently, for the CT reaction, Eq. (2) can be safely truncated down to the following equation:
ΔGCT(F)=ΔGCT(0)ΔμCTF.
(8)
Yet, for ZnTTP, we have tested the accuracy of Eq. (8) to compute the difference between the field-dependent energies of CTS1 and LES1 with respect to the field-dependent energies obtained with Eq. (2), including the second term correction, i.e., the polarizability changes. To our delight, the maximum difference including or not including the second-order corrections was less than 0.02 eV for the range of electric fields studied (for details, see Table S1 in the supplementary material). Therefore, we concluded that Eq. (8) can be safely used as it retains a proper description of the field-dependent difference in energy between the states of interest.
To impose the desired constraint [i.e., ΔG(F) = −λ(F)], one should determine the strength and orientation of F for which the value of ΔG(F) given by Eq. (8) is equal to the value of −λ(F) provided by an homologous Taylor expansion for −λ [i.e., −λ(F) = −λ0 − (∂λ/∂F) · F…). Nevertheless, it turns out that the change of −λ due to the OEEF is negligible compared to the change of the Gibbs energy in the LES1→CTS1 transition in the presence of an OEEF, and therefore the approximation −λ(F) = −λ0 − (∂λ/∂F) · F ≈ −λ0 can be safely used (see Fig. 2). Therefore, the final working equation to obtain the optimal field-strength (OFS) that maximizes kCT is (see derivation in the supplementary material),
OFS=ΔG(0)+λΔμ.
(9)
The optimal direction in the space of the OFS is given by the difference between the dipole moments of the CT excited state and the LES1 state, Δµ. In this work, we have aligned the Z axis with the Δµ vector (see Fig. 1), which generally coincides with the axis generated by the region of the space where the electrons and holes are created during the CT process.
FIG. 2.

Field-dependent LES1 to CTS1 ΔG (red), −λ(0) (blue), −λ(F) (blue crosses), and determination of OFS (crossing point, black) for ZnTTP. LES1 (green) and CTS1 (black) ΔG energies are given with respect to the ground-state Gibbs energy.

FIG. 2.

Field-dependent LES1 to CTS1 ΔG (red), −λ(0) (blue), −λ(F) (blue crosses), and determination of OFS (crossing point, black) for ZnTTP. LES1 (green) and CTS1 (black) ΔG energies are given with respect to the ground-state Gibbs energy.

Close modal

The difference between the OFS predicted with Eq. (9) and the corresponding equation, including the dependence of λ with the OEEF strength for ZnTTP, is lower than the resolution of possible experimental setups (i.e., <1 mV/Å). The same assumption holds for the other systems studied in this work, with a maximum difference of 1.2 mV/Å (see Table S2 in the supplementary material for further details). We will refer to the new methodology to compute OFS and determine kCT’s as Field-Dependent-Barrier for Charge-Transfer (FDB-CT) reactions.

Figure 2 summarizes the field-induced changes on Gibbs excitation energies for CTS1 and LES1 [ΔGCTS1(F), ΔGLES1(F)]; for the transition between these two states, ΔGCT(F); the negative of field-dependent reorganization energy, −λLES1→CTS1(F); and the OFS. For convenience, hereafter, ΔGCT(F) and −λLES1→CTS1(F) are referred to as ΔG and −λ, respectively. The distance between the blue and red lines of Fig. 2 determines the value of the term (ΔG − λ) in Eq. (1) as a function of the F strength.

To test the validity of the FDB-CT approach, we have computed for ZnTTP the variation of ΔG, λ, (λ + ΔG)2, and kCT in the presence of explicit OEEFs judiciously oriented such that the positive and negative poles lie in the direction where the electron and the hole of the CTS are located at F = 0, respectively. For this test, the ZnTTP optimal geometry in the presence of the EF was used. The strength of such OEEFs was selected to be F1 = 0, and F2 = 1.45 × 10−3 a.u. (74.7 mV/Å); being F2 the OFS of ZnTTP computed with Eq. (9). The largest kCT predicted by the FDB-CT method is the one given by the OFS (3.34 × 1011 s−1). kCT(OFS) is almost twice as large as the field free kCT(0) (1.8 × 1011 s−1) and corresponds to the theoretical limit of the kCT for this system. Calculations using explicit electric fields (i.e., without any approximation) at F2 field-strength give a smaller value for (λ + ΔG)2 than the one at F = 0, which then leads to a larger kCT (3.31 × 1011 s−1). To our delight, the rate constants computed considering explicit external electric fields are very similar to the predictions obtained with the FDB-CT method. The origin of the small differences is (i) the first-order truncation used in the application of Eq. (2); (ii) the field-dependence of λ and the coupling term, which in the FDB-CT are considered constant and equal to the field-free value; and (iii) the minor changes in equilibrium geometry induced by the external electric field.

Deviations of FDB-CT may also be expected when a very intense field is applied (i.e., F ≥ 10−2 a.u.; see Table S4 in the supplementary material), as it can trigger perturbations and transformations of the excited states that are beyond the scope of prediction of our computational model. Despite the FDB-CT approximations, considering that FDB-CT has a far lower computational cost than the calculations performed with explicit OEEF, it becomes a low-cost and efficient tool to explore the dependence of kCT with respect to different OEFFs and to determine the OFS for a given D–A dyad.

Once the performance of our computational model was validated for ZnTTP, we used the FDB-CT method to determine the OFS for three more dyads, namely TPA, Az, and PC. For TPA, the predicted field-free rate constant is kCT = 2.46 × 1010 s−1. FDB-CT predicts for this system an OFS equal to −1.54 × 10−3 a.u. (−79.2 mV/Å), and the rate constant under the presence of such an OEEF is predicted to increase 4-fold to kCT = 1.03 × 1011 s−1 (see Fig. S2 in the supplementary material).

Regarding the Az system, without an electric field, the rate constant kCT is 3.29 × 109 s−1.51 However, by applying the OFS (1.58 × 10−3 a.u.), the kCT is boosted to 1.45 × 1014 s−1 (see Fig. S3 in the supplementary material). Such a sharp enlargement of the rate constant corresponds to a 4400-fold increase and is the highest reported in this manuscript. Although the Marcus equation is less accurate for the prediction of kCT values in very fast CT processes, the OFS predicted by our method will be quite similar to the electric field that maximizes the actual value of kCT.

The enhancement in the CT rate for Az can be explained by the change in the thermodynamics of the process induced by the electric field. Specifically, Az is the only system that presents an unfavorable CT (ΔG = 0.20 eV, see Table I), while λ and Δµ are similar to the rest of the systems. The OFS transforms the process from endergonic to exergonic (ΔG = −0.24 eV) by stabilizing the charge-transfer state, thus largely facilitating the CT and then increasing its CT rate constant.

TABLE I.

Predicted and experimental charge-transfer rate parameters for ZnTTP, TPA, Az, and PC computed through the FDB-CT method or under explicit judiciously selected OEEFs.a Experimental values are always measured at F = 0.b

OEEF (a.u.)ΔG (eV)λint (eV)(λ + ΔG)2 (eV2)kCT (s−1)
ZnTTP 
Experimental21     2.9 × 1010 
0.00 −0.125 0.281 0.057 1.11 × 1010 
1.45 × 10−3 (OFS) −0.281 0.281 0.0 1.87 × 1011 
1.45 × 10−3 (OFS) −0.260c 0.264c 1.6 × 10−5c 1.83 × 1011c 
Point charge (1.29 × 10−3−0.281 0.204 5.9 × 10−3 1.51 × 1011 
TPA 
Experimental23     6 × 1010 
0.00 −0.96 0.210 0.563 2.46 × 1010 
−1.54 × 10−3 (OFS) −0.210 0.210 0.000 1.03 × 1011 
Az 
Experimental24     2 × 1010 
0.00 0.20 0.240 0.194 3.29 × 109 
1.58 × 10−3 (OFS) −0.24 0.240 0.000 1.45 × 1014 
PC 
Experimental25     1.25 × 1012 
0.00 −0.178 0.134 0.078 3.75 × 1013 
−3.47 × 10−4 (OFS1−0.134 0.134 0.000 5.79 × 1013 
−8.9 × 10−3 (OFS2−0.366 0.366 0.000 2.98 × 1011 
OEEF (a.u.)ΔG (eV)λint (eV)(λ + ΔG)2 (eV2)kCT (s−1)
ZnTTP 
Experimental21     2.9 × 1010 
0.00 −0.125 0.281 0.057 1.11 × 1010 
1.45 × 10−3 (OFS) −0.281 0.281 0.0 1.87 × 1011 
1.45 × 10−3 (OFS) −0.260c 0.264c 1.6 × 10−5c 1.83 × 1011c 
Point charge (1.29 × 10−3−0.281 0.204 5.9 × 10−3 1.51 × 1011 
TPA 
Experimental23     6 × 1010 
0.00 −0.96 0.210 0.563 2.46 × 1010 
−1.54 × 10−3 (OFS) −0.210 0.210 0.000 1.03 × 1011 
Az 
Experimental24     2 × 1010 
0.00 0.20 0.240 0.194 3.29 × 109 
1.58 × 10−3 (OFS) −0.24 0.240 0.000 1.45 × 1014 
PC 
Experimental25     1.25 × 1012 
0.00 −0.178 0.134 0.078 3.75 × 1013 
−3.47 × 10−4 (OFS1−0.134 0.134 0.000 5.79 × 1013 
−8.9 × 10−3 (OFS2−0.366 0.366 0.000 2.98 × 1011 
a

Electronic couplings for ZnTPP, TPA, Az, and PC are 0.0031, 0.0015, 0.0735, and 0.0525 eV, respectively.

b

Root-mean square errors in vertical excitation energies calculated with popular density functionals vary from ∼0.3 to 0.7 eV, with range corrected functionals such as CAM-B3LYP being among the most accurate.70 These errors translate into similar errors for ΔG and, consequently, differences between experimental and computed kCT of one order of magnitude or even larger are commonly found.51 

c

Values computed using explicit OEEF.

The high boost reported for Az clearly demonstrates that although using OEEFs to enhance the rate constant is always a valid strategy, for some particular D–A dyads, it has a greater impact due to their intrinsic chemical nature. Specifically, the room for improvement in terms of kCT enhancement for a particular D–A dyad is directly proportional to its (ΔG − λtot)2 value at F = 0, and whether such improvement is reached at reasonable fields depends on the ΔμCT. FDB-CT can be used to find the D–A dyads for which kCT has a stronger dependence on the OEEF.

However, it should be mentioned that in particular scenarios where there are several low-lying LESs below the CTS, the chances of state recombinations or other situations that are not accounted for by the FDB-CT method increase, and therefore it may lead to a divergence between the kCT predicted by FDB-CT and the experimental observations or the use of explicit OEEFs.

In the case of the PC system, we have determined that there exist several excited states that correspond to an electron transfer from the phthalocyanine group (Pht) to the C59N, as for instance, CTS1 [charge separation (CS) = 0.870 e], CTS2 (CS = 0.853 e), and CTS3 (CS = 0.926 e); while some other excited states are better described as the electron transfer from the C59N to the Pht unit in PC, as it turns out to be the case for CTS4 (CS = 0.738 e) or CTS5 (CS = 0.822 e) (see the supplementary material for further details). Since the sign of the electric field represents its orientation in space, this distinct behavior is very easily recognized by looking at the sign of the slopes that such states present in Fig. 3. It is worth remarking that the slopes are given by the change of the dipole moment in the charge transfer process from LES1 to CTSn and, therefore, are directly proportional to the charge-separation parameter, CS [CSi = Ωi(D, A) − Ωi(A, D)].

FIG. 3.

Field-dependent excitation energy for the relevant excited states in PC: LES1, CTS1, CTS2, CTS3, CTS4, and CTS5.

FIG. 3.

Field-dependent excitation energy for the relevant excited states in PC: LES1, CTS1, CTS2, CTS3, CTS4, and CTS5.

Close modal

A given state can be stabilized at will by fine-tuning the direction and strength of the OEEF. In the particular case of PC (Fig. 3), by placing an OEEF with the positive pole on the C59N side (we have adopted an F > 0 convention) and the negative pole on the Pht side, one can obtain the most stable excited state, the CTS1 when F < 8 × 10−3 a.u. and the CTS3 when F > 8 × 10−3 a.u.

On the other hand, by switching the direction of the OEEF (F < 0), one obtains the most stable excited state CTS5 for |F| > 6.7 × 10−3 a.u., while for 6.7 × 10−3 > |F| > 1.5 × 10−3 a.u., the most stable excited state becomes LES1 (thus, in this range of fields, the CT reaction is endergonic). Finally, for |F| < 1.5 × 10−3 a.u., CTS1 is the most stable state. Note that at F = 0, CTS1 is the first excited state.

Having this in mind, we have determined two different OFSs, namely OFS1 and OFS2, which will correspond to the OFS for the Pht-to-C59N (LES1 → CTS1) and the C59N-to-Pht (LES1 → CTS2) electron transfers, respectively (Fig. 4). The field-free kCT is 3.75 × 1013 s−1 for the Pht-to-C59N charge transfer. The predicted OFS1 is located at −3.47 × 10−4 a.u. (17.8 mV/Å), and its corresponding predicted kCT is 5.79 × 1013 s−1, which represents a slight increase with respect to the F = 0 scenario. For the reverse charge transfer, the OFS2 is −8.9 × 10−3 a.u. (−457.7 mV/Å), and the charge-transfer rate is 2.2 × 1011 s−1. However, the charge transfer occurring at OFS2 is purely a field-induced process since it was non-existent at field-free, as CTS4 is more than 1 eV higher in energy than the other lower-lying CTSs. Taking into account the large magnitude of OFS2, the reliability of the FDB-CT prediction for this particular OEEF is lower than for the other values presented in this paper.

FIG. 4.

Energy dependence of ΔG in PC: solid for CTS1 and dashed for CTS5; −λ (blue), and determination of OFS1 (black dot) and OFS2 (black triangle). The LES1, CTS1, and CTS5 energies are given with respect to the ground-state energy.

FIG. 4.

Energy dependence of ΔG in PC: solid for CTS1 and dashed for CTS5; −λ (blue), and determination of OFS1 (black dot) and OFS2 (black triangle). The LES1, CTS1, and CTS5 energies are given with respect to the ground-state energy.

Close modal

To finalize, we have examined the effect on the charge-transfer rate constant of two opposite point-charge models, located at ±20.8 Å from the D–A junction of ZnTTP, which generate a non-homogeneous electric field of analogous strength and direction to the OFS predicted by Eq. (8). The calculated ΔG = −0.281 eV matches the values predicted using the FDB-CT approach for the OFS (with the number of figures reported), and it is in very good agreement with the value of −0.260 eV predicted by explicit OEEFs; analogously, the obtained rate constant of kCT = 2.98 × 1011 s−1 is very close to the 3.31 × 1011 s−1 (3.34 × 1011 s−1) value obtained with an explicit homogeneous electric field (FDB-CT method). Such results open the door for using FDB-CT to simulate the effect of charged functional groups on the kCT, which could be one of the keys to improving the rational design of highly efficient DSSCs based on D–A dyads.

We have proposed a simple analytical approach to compute in a straightforward manner the field-dependent excitation energies for any D–A dyad (it can also be applied to D–A–D triads, etc.). With a very simple linear formula, our computational approach determines the optimal direction and strength of the OEEF needed to maximize the charge-transfer rate in the framework of Marcus theory. FDB-CT predicts the field-induced changes of the energy of any excited state, and therefore all possible combinations of LES to CTS transitions can be studied separately. To validate the FDB-CT method, we have performed charge-transfer rate calculations in the presence of explicit electric fields, obtaining OFS results matching those obtained with the FDB-CT approach. We have analyzed several fullerene-based molecular dyads and, for each of them, determined the optimal electric field that maximizes its charge-transfer rate constant, normally by stabilizing the charge-transfer states. For all the systems studied, OFS has yielded a larger kCT than the one obtained at F = 0. The potential enhancement of kCT depends on the value of (ΔG + λtot)2 for each particular system calculated at F = 0, while ΔμCT controls whether such enhancement is feasible to be obtained at reasonable small fields. The highest kCT enhancement predicted in this work is about 4400-fold with respect to the field-free value. Furthermore, visual inspection of the figures representing the results obtained from Eq. (8) gives insight into (i) the predicted changes induced by the OEEF in the energy of the relevant states; (ii) the order and nature (LES/CTS) of excited states as a function of the electric field applied; and (iii) the possibility of controlling the movement of the electron and hole participating in the charge transfer process.

There are two clear limitations in our approach: (i) the calculation of kCT with the Marcus approach in the case of very fast charge transfer processes is unreliable, and (ii) the reliability of the FDB-CT predictions decreases when the strength of the electric field is very high because of the truncation error in Eq. (2) due to the first-order approximation used. Despite these limitations, we think that our FDB-CT approach can assist in the rational design of D–A dyads with large kCT induced by an OEEF or by a local field generated by charged or polar functional groups.

Full derivation and justification of Eq. (9), ΔG(F) and Δλ(F) for all the systems, analysis of the C59N excited states, graphical representation of Eq. (8) for TPA and Az systems, and Cartesian coordinates.

This work was supported with funds from the Spanish Ministerio de Ciencia e Innovación (Project Nos. PID2020-13711GB-I00 and PGC2018-098212-B-C22) and the Generalitat de Catalunya (Project No. 2021SGR623). We acknowledge the Spanish government for the predoctoral grant to P.B.-S. (Grant No. FPU17/02058). We are also grateful for the computational time financed by the Consorci de Serveis Universitaris de Catalunya (CSUC).

The authors have no conflicts to disclose.

P.B.-S. and A.A.V. performed all the calculations. P.B.-S. analyzed the results and wrote the first draft of the manuscript. A.A.V., J.M.L., and M.S. devised and supervised the project. The manuscript was written with the contributions of all authors. All authors have given approval to the final version of the manuscript.

Pau Besalú-Sala: Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (equal); Writing – review & editing (equal). Alexander A. Voityuk: Formal analysis (equal); Investigation (supporting); Methodology (lead); Writing – review & editing (equal). Josep M. Luis: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Miquel Solà: Conceptualization (equal); Funding acquisition (lead); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

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