J. Phys. Chem. C 2008, 112, 3429-3433
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Quantum-Chemical Approach to Electronic Coupling: Application to Charge Separation and Charge Recombination Pathways in a Model Molecular Donor-Acceptor System for Organic Solar Cells Tsutomu Kawatsu,† Veaceslav Coropceanu,* Aijun Ye,‡ and Jean-Luc Bre´ das* School of Chemistry and Biochemistry & Center for Organic Photonics and Electronics, Georgia Institute of Technology, Atlanta, Georgia 30332-0400 ReceiVed: NoVember 26, 2007
We have developed a new computational method to evaluate in a general way the electronic coupling between electronic states of any spin multiplicity. This method is especially relevant to the description of chargeseparation and charge-recombination processes in molecular donor-acceptor complexes; in such instances, the electronic states can correspond, for example, to a locally excited initial state and a charge-separated final state. Here, we describe our approach in a simple way by deriving the excited states of the donor and acceptor moieties with Zerner’s intermediate neglect of differential overlap method coupled to a configuration interaction scheme involving single and double electron excitations. In order to increase the efficiency of the calculations, the same Hartree-Fock molecular orbital basis set is employed to compute the electronic states of both neutral and charged configurations. We illustrate the application of the present approach by computing the rates of charge generation and charge recombination in a model phthalocyanine/perylene-tetracarboxylicdiimide donor/acceptor molecular system. The role of triplet channels is underlined.
1. Introduction There are currently major efforts devoted to the development of renewable energy sources. Conventional silicon-based solar cells, which have been optimized over the past decades, have now reached a high level of efficiency and reliability;1 however, the cost of their generated power remains high when compared to the cost of power from the electrical grid. As a result, there is a significant amount of research focusing on potentially less expensive organics-based solar cells.2-4 The applications of organic photovoltaics could additionally benefit from other advantages offered by plastic materials such as structural tunability, light weight, and mechanical flexibility. Typically, an organic solar cell is a two-component system, based on an electron-donating (D) material and an electronaccepting (A) material combined in the form of either a multilayered structure or a blend. In the case of silicon-based solar cells, light absorption and dissociation of the electronhole pairs into free charges occur at the same location. In contrast, in the case of organic solar cells, the conversion of light into current (that is, into separated charges collected at different electrodes) takes place in several steps and at different locations. Initially, light is absorbed by one (or both) of the donor and acceptor components to form singlet excitons. These excitons then have to diffuse to the donor-acceptor interface where they are expected to dissociate, via an electron- or holetransfer process, into a charge-separated state. The generated * Corresponding authors. E-mail: veaceslav.coropceanu@ chemistry.gatech.edu (V.C.) and
[email protected] (J.L.B.). † Present address: Fukui Institute for Fundamental Chemistry, Kyoto University, Takano-Nishihiraki-cho 34-4, Sakyo-ku, Kyoto 606-8103, Japan. ‡ Present address: Department of Chemistry, State University of New York at Buffalo, New York 14260-3000.
charges may then either escape their mutual Coulomb attraction to produce free charge carriers or recombine. It is remarkable to realize that, at this point in time, our understanding of the charge separation and recombination processes is still rather limited. Therefore, in order to eventually develop valuable strategies to increase the efficiency of organic solar cells, it is crucial to obtain a much better, fundamental understanding of the charge separation (CS) and charge recombination (CR) pathways. In general, several pathways can be followed for both the CS and the CR processes. First, multiple donor and/or acceptor excited states could be populated by the incident light, including triplet states formed via intersystem crossing. Next, while only the lowest D+ 0 + A0 charge-transfer state is often considered in the literature, the CS process could actually involve other, more - 5 efficient excited charge-transfer states D+ k +Al . The charge recombination process could also take place via several pathways; for instance, CR via triplet states could in some cases be more efficient than the direct transition back to the singlet ground state. The electron-transfer (ET) rate of any CS or CR process depends on several factors such as electron-vibration coupling, electronic coupling between reactants and products, and relative free energies of reactants and products.4,6 In the weak electronic coupling limit, a number of the relevant microscopic parameters can be estimated from calculations performed on the isolated molecules.3,4,6 However, the calculations of the electronic couplings themselves are more challenging, since their values are determined by the overlap between the relevant donor and acceptor wavefunctions. This issue has for a long time received significant attention in many areas of biology, chemistry, and physics.6-14 Most rigorously, the magnitude of the electronic coupling is defined by the matrix element (transfer integral) Vab ) 〈Ψa|H|Ψb〉, where H is the electronic Hamiltonian of the
10.1021/jp711186j CCC: $40.75 © 2008 American Chemical Society Published on Web 02/09/2008
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system and Ψa and Ψb are the wavefunctions of two localized states (diabatic states). In the case of optically induced ET processes, the estimation of the transfer integrals is currently usually based on the two-state generalized Mulliken-Hush model,9,12,15 where Vab is obtained as:
Vab )
Mab∆Eab
x(∆µab)2 + 4(Mab)2
(1)
Here, ∆µab and Mab are the change in permanent dipole moment and the change in transition dipole moment, respectively, between the adiabatic states associated with the considered diabatic states; ∆Eab is the free energy difference between these adiabatic states. The dipole moments and transition energy can be derived from excited-state calculations on the whole donor-acceptor complex. It must be stressed that the application of eq 1 is based on the assumption that the two diabatic states involved in the ET process exclusively interact with one another, so that they can be represented as a linear combination of only two adiabatic states. In many cases, however, a given initial diabatic state could be electronically coupled with more than one final state; in addition, a particular adiabatic state could be formed by the superposition of charge-transfer and locally excited states. Therefore, the application of eq 1 can be rather limited. In the present work, we move beyond this significant limitation and develop an original approach that allows for a direct calculation of the electronic coupling between any two diabatic states of a donor-acceptor complex. In contrast to other approaches reported in the literature, the present method allows one to evaluate the electronic couplings between electronic states of any spin multiplicity. As a result, for instance, ET rates involving singlet and triplet states can be computed; both types of CS and CR channels are expected to be operative in organic solar cells. To illustrate the new methodology, we discuss its application to the evaluation of the electronic couplings for different CS and CR pathways in a molecular donor/acceptor complex composed of a phthalocyanine (Pc) donor molecule and a perylene-tetracarboxylic-diimide (PTCDI) acceptor molecule (see the chemical structures in Figure 1); we specifically chose this model complex in order to compare our results to those previously obtained on the same system via the generalized Mulliken-Hush method.3 2. Methodology We assume that the electronic coupling between the donor and the acceptor molecules is weak. In this case, the diabatic states can be constructed as antisymmeterized products of the isolated donor (ψDi ) and acceptor (ψAi ) wavefunctions. Spinadapted diabatic states can be then obtained as linear combinations of these products by means of Clebsch-Gordan coefficients:16
Ψij (SM) )
C SSMM ;S M |ψXi (SiMi)ψYj (SjMj)| ∑ MM i
i
i
j
j
(2)
j
Here, the S and M terms denote the total spin and respective spin projections for the donor-acceptor complex; the SiMi terms denote the same quantum numbers for the donor or acceptor; the CSSM terms represent the Clebsch-Gordan coefficients; iMi;SjMj X and Y stand for D and A or D+ and A-, respectively. Equation 2 provides a general way to construct diabatic states that involve donor and acceptor states with neutral or charged configurations of any spin multiplicity. The electronic coupling between a
Figure 1. (a) Chemical structure of the phthalocyanine (Pc) molecule (donor). (b) Chemical structure of the perylene-tetracarboxylic-diimide (PTCDI) molecule (acceptor). (c) Illustration of the cofacial geometry of the molecular Pc/PTCDI complex considered in the ET calculations; the molecular planes are separated by 4 Å.
locally excited initial state and a charge-transfer final state is then given by:
Vij;km )
C SSMM ;S M C SSMM ;S M ∑ MMM M i
i
j
k
i
j
j
k
k
m
m
×
m
+
-
〈|ψDi (SiMi)ψAj (SjMj)||H||ψDk (SkMk)ψ Am (SmMm)|〉 (3) Equation 3 takes into account that the electronic coupling is diagonal in the representation of quantum numbers S and M for initial and final states of the complex. In order to make use of eq 3, the ground and excited states of the donor (ψ Di ) and acceptor (ψ Ai ) moieties, as well as their respective charged + states (ψ Di and ψ Ai ) should be derived first. For the sake of simplicity, we use here Zerner’s intermediate neglect of differential overlap (ZINDO) Hamiltonian17 coupled to a configuration interaction scheme involving single and double electron excitations (CISD). We stress that the methodology represented by eq 3 is completely general and any other quantum-chemical approach could be used as well. Any of the relevant wavefunctions can be written as a linear combination of Slater determinants:
ψZi )
∑l c ilΦZl
(4)
where Z ) D, A, D+, or A-, and ΦZl is a Slater determinant given in terms of the corresponding molecular orbitals (MOs), φZj . By substituting eq 4 into eq 3, the electronic coupling can be expressed in terms of transfer and overlap integrals between molecular orbitals.
Quantum-Chemical Approach to Electronic Coupling
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TABLE 1: Transfer Integrals (in meV) between Selected Phthalocyanine (Pc) and Perylene-tetracarboxylic-diimide (PTCDI) Molecular Orbitals Pc\PTCDI
HOMO
LUMO
LUMO+1
LUMO+2
LUMO+3
HOMO LUMO LUMO+1 LUMO+2 LUMO+3
78.3 0.0 0.0 15.8 0.0
0.0 48.4 0.1 0.0 0.0
3.5 0.0 0.0 20.5 0.0
0.0 0.0 0.0 0.0 7.4
0.0 0.0 0.0 0.0 16.8
The problem can be significantly simplified by using the same Hartree-Fock (HF) molecular orbital (MO) basis set to compute the electronic states of both neutral and charged configurations; we used here the basis sets obtained from self-consistent-field (SCF) calculations on the neutral molecules. In this approximation, because of the fact that only the one-electron part of the INDO Hamiltonian contributes to the ET integrals, the matrix elements that remain between two Slater determinants are those that differ by at most one electron occupation. The electronic coupling between the initial a≡ij diabatic state and the final b≡km diabatic state can be finally represented as:
Vab )
DA γab ∑ rs t rs rs
(5)
Here, the γab rs coefficients account for contributions of a particular pair of determinants in eq 3 and is given by a combination of products formed from configuration-interaction coefficients cim (see eq 4) and Clebsch-Gordan coefficients. D A The matrix element t DA rs ≡ 〈φr |H|φs 〉 represents the transfer integral between donor and acceptor molecular orbitals; in the framework of the INDO method, its value is defined by the overlap between the involved atomic orbitals.18 Thus, the calculation of the electronic coupling between any initial a state and final b state is reduced to the computations of the related DA γab rs and trs coefficients. Our methodology has been applied to a Pc-PTCDI complex similar to the one considered earlier in ref 3; the two molecular planes are set to be cofacial and are separated by 4 Å; see Figure 1. The geometry of each molecule was optimized at the HF semiempirical AM1 level by means of the AMPAC program.19 The total energies of the initial and final states are given by eqs 6 and 7, respectively (see ref 3): D A DA,coul E (i) ij ) E i + E j + E ij +
-
+ -,coul
D A D A E (f) km ) E k + E m + E km
(6) (7)
EXn denotes the energy of state n in molecule X. The EXY,coul np term represents the Coulomb energy between molecules X and Y in states n and p, respectively:
) E XY,coul np
∑g∈ X,h∈Y(q ng q ph/rgh)
(8)
where q mh is the partial charge on atom h in the mth excited state; is the dielectric constant of the medium; and rgh is the distance between atoms g and h. For the sake of comparison with earlier work,3 we estimate the ET rate using the Marcus semi-classical model:
x
kab ) Vab2
π exp[-(∆G + λ)2/4λkBT] λkBT p2
(9)
Here, λ is the reorganization energy that accounts for the contributions from both the molecular geometry modifications
Figure 2. Electron-transfer pathways and rate constants in Pc-PTCDI complex for (a) charge-separation processes and (b) charge-recombination processes.
and the modifications in the surrounding medium that occur when an electron is added to or removed from a molecule; ∆G is the Gibbs free energy change of reaction (driving force) and is approximated here as the energy difference between the initial (i) and final states, ∆G ) E (f) km - E ij ; kB denotes the Boltzmann constant, and T is the temperature. 3. Results and Discussion We assume that both molecules are initially in their singlet ground states (S0). The ZINDO-CISD calculations predict two strong dipole-allowed optical peaks for phthalocyanine correpc pc pc sponding to the S pc 0 f S 1 and S 0 f S 2 transitions. The pc pc transition from S 0 to S 1 is dominated by a single HOMO to LUMO+1 (here, HOMO and LUMO stand for highest occupied molecular orbital and lowest unoccupied molecular orbital, pc respectively) excitation, while the Spc 0 f S 2 transition is dominated by a single HOMO to LUMO excitation. The computed transition energies [oscillator strengths] for these excitations are 2.2 eV [0.64] and 2.4 eV [0.74], respectively; these energy values are about 0.4 eV larger than the experimental results.20,21 ZINDO-CISD calculations on PTCDI give a strong absorption band at 3.4 eV [1.02 oscillator strength], corresponding to the S ptcdi f S ptcdi excitation dominated by a 0 1 single HOMO to LUMO transition. In the case of Pc, there are two triplet states (T1 and T2) located below the first singlet excited-state by 1.0 and 0.9 eV, respectively; in PTCDI, only T1 has a lower energy (by 1.2 eV) than S1. The calculated transfer integrals between the donor and acceptor MOs are shown in Table 1. The estimated electronic coupling between several intramolecular states and chargetransfer states are collected in Tables 2 and 3; only states belonging to the spin singlet, doublet, and triplet manifolds of the system have been considered. At this stage, it is useful to recall that a simplistic scheme involving just the HOMO and LUMO levels of the donor and acceptor molecules is often considered. In that instance, (i) photoinduced electron transfer is depicted as an electron jumping from the photoexcited donor LUMO to the acceptor LUMO; (ii) photoinduced hole transfer is depicted as a hole jumping from the photoexcited acceptor HOMO to the donor HOMO; and (iii) charge recombination is depicted as an electron jumping from the acceptor LUMO to the donor HOMO. Our calculations confirm that the transfer integral between the Pc HOMO and the PTCDI LUMO is nearly zero;3 in contrast, the transfer integrals for the Pc LUMO-PTCDI LUMO and Pc HOMO-PTCDI HOMO interactions are large. As a consequence, as seen from Table 2, the electronic coupling
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TABLE 2: Transfer Integrals (in meV) between Selected Intramolecular and Charge-Transfer Singlet States of the Pc-PTCDI Complex +
ptcdi D pc 0 X D0 ptcdi S pc 0 X S0 pc S 1 X S ptcdi 0 ptcdi S pc 2 X S0 ptcdi S pc 3 X S0 ptcdi S pc X S 4 0 pc S 0 X S ptcdi 1 ptcdi { T pc }S)0 1 X T1
-
+
ptcdi D pc 0 X D0
0.0 0.2 39.5 0.0 0.0 71.3 0.0
-
+
ptcdi D pc 0 X D0
8.0 0.0 0.0 0.0 14.5 0.0 24.7
+
-
ptcdi D pc 0 X D0
0.0 0.0 0.0 33.6 0.3 0.0 0.0
-
4.6 0.0 0.0 0.2 15.2 0.0 2.0
TABLE 3: Transfer Integrals (in meV) between Selected Intramolecular and Charge-Transfer Triplet States of the Pc-PTCDI Complex +
ptcdi D pc 0 X D0 ptcdi T pc 1 X S0 pc T 2 X S ptcdi 0 ptcdi S pc 0 X T1 pc S 1 X T ptcdi 1 ptcdi S pc 2 X T1
43.6 0.2 67.3 0.0 0.0
-
+
ptcdi D pc 1 X D0
0.0 0.0 0.0 0.0 45.2
ptcdi between the ground state S pc and the lowest charge0 X S0 pc+ ptcdiis nearly zero, which makes the transfer state D 0 X D 0 direct transition between these two states negligible. There are, ptcdi however, weak electronic couplings between S pc and 0 X S0 charge-transfer states involving the first excited state of either + ptcdithe charged Pc or the charged PTCDI molecule (D pc 1 X D0 pc+ ptcdiand D 0 X D 1 ). The calculations also indicate that the + ptcdi X D ptcdi state is strongly coupled with the S pc D pc 0 0 2 X S0 pc ptcdi and S 0 X S 1 states: the transfer integrals are equal to 39.5 and 71.3 meV, respectively. By using a dielectric constant of ) 2, our calculations yield a driving force of 0.02 and 1.0 eV + + ptcdi ptcdiptcdi for the S pc f D pc and S pc f D pc 2 X S0 0 X D0 0 X S1 0 X ptcdireactions, respectively. By taking a typical value of λ ) D0 0.5 eV for the overall reorganization energy, eq 9 provides comparable ET rates for these reactions: kET ) 4.1 × 1011 s-1 and kET ) 1.0 × 1012 s-1, respectively (see also Figure 2a). It is useful to note that, because of the significant mixing of + ptcdi pc ptcdi X D ptcdi , S pc closely lying D pc 0 0 1 X S 0 , and S 2 X S 0 states, direct application of eq 1 would yield erroneous electronic couplings.3 As a result, in order to apply eq 1, the authors of ref 3 were forced to perform their calculations with an active space from which all electronic configurations responsible for electron delocalization had to be removed. We also note that, in contrast to our model, the electronic couplings derived within the approach worked out in ref 3 strongly depend on the value of the dielectric constant. The electronic couplings calculated in ref 3 using static dielectric constants of 3 and 5 were obtained, + ptcdi respectively, as 58 and 96 meV for the S pc f D pc X 2 X S0 0 ptcdi pc ptcdi pc+ pathway and 46 and 67 meV for the S 0 X S 1 f D 0 D0 X D ptcdi pathway.3 These values compare well with our 0 results of 39.5 and 71.3 meV for these two pathways, although some discrepancy between the two sets of results is also evident. We turn now to the discussion of the CR pathways. The CR + ptcdi transitions from D pc X D ptcdi to the S pc and S pc 0 0 0 X S0 1 X ptcdi S 0 states are negligible because of the very weak electronic couplings (see Table 2). Thus, the CR pathway involving singlet molecular states can take place only via back ET to the S pc 2 X S ptcdi state. The rate of this transition clearly critically depends 0 on the energy separation between the states. Another possibility for a singlet CR pathway could be a transition to a singlet state formed from a product of triplet states; however, the lowest ptcdi state from this manifold, {T pc 1 X T 1 }S)0, is calculated to lie
-
+
ptcdi D pc 2 X D0
+
-
ptcdi D pc 0 X D0
0.0 0.0 0.0 35.9 0.0
-
0.0 0.0 0.0 0.0 2.9 +
-
ptcdi }S)0 state. Thus, in the about 1.05 eV above the {D pc 0 X D0 present system, this manifold of states does not contribute to CR. +
-
Importantly, the singlet {D pc X D ptcdi }S)0 state can also 0 0 pc+ }S)1 state, which decay to a lower-lying triplet { D 0 X D ptcdi 0 thereby provides additional CR pathways. Indeed, in the Pc/ PTCDI complex, there exist three intramolecular triplet states, ptcdi pc ptcdi pc ptcdi S pc which are located 0 X T 1 , T 1 X S 0 , and T 2 X S 0 pc+ ptcdistate by 0.18, 1.15, and 1.06 eV, below the D 0 X D 0 respectively. The first two of these states, see Table 3, exhibits + a significant electronic coupling with the {D pc X D ptcdi }S)1 0 0 state. The ET rates for all CR pathways (calculated by assuming the same reorganization energy of λ ) 0.5 eV and negligible + + energetic splitting between the {D pc X D ptcdi }S)1 and {D pc 0 0 0 X D ptcdi }S)0 states) are shown in Figure 2b. As seen from this 0 + ptcdiptcdi f S pc pathway is figure, the rate for the D pc 0 X D0 0 X T1 calculated to be extremely fast, in the subpicosecond regime. Finally, we were interested in getting some understanding of the role of the driving force on the CR processes. To do so, we took a simple approach and computed the ET rates as a function ptcdi state of the energy difference between the ground S pc 0 X S0 pc+ ptcdiand the lowest charge-transfer D 0 X D 0 state while keeping the intramolecular state energies fixed. The results of + ptcdi pc+ the calculations for the D pc X D ptcdi f S pc X 0 0 2 X S0 , D0 + ptcdi ptcdi ptcdi D ptcdi f S pc , and D pc f T pc CR 0 0 X T1 0 X D0 1 X S0 + transitions are shown in Figure 3. We find that the D pc X 0 ptcdipc ptcdi D0 f S 2 X S 0 singlet channel significantly contributes to the CR process only for a narrow range of energy difference + ptcdi between the D pc X D ptcdi and S pc states. When the 0 0 2 X S0 + pc ptcdienergy of the D 0 X D 0 state gets increasingly stabilized, as can occur in more polar media, the singlet pathway provides for very slow recombination,3 while the triplet pathways are strongly picking up. These results point to an important added complexity in the description of the CR/CS processes. They are consistent with the significant enhancements in intramolecular triplet generation that have been experimentally observed for a number of materials upon blend formation. This effect was explained in terms of possible transitions from CS states to lower-lying intramolecular triplets,22-24 as shown for the model Pc-PTCDI system here.
Quantum-Chemical Approach to Electronic Coupling
J. Phys. Chem. C, Vol. 112, No. 9, 2008 3433 systems of current interest and especially for those involving large molecules such as C60. Work along these lines is currently in progress. Acknowledgment. This work has been partly supported by the Office of Naval Research, the National Science Foundation (through the STC Program under Award No. DMR-0120967 and the CRIF Program under Award No. CHE-0443564), the Georgia Tech “Center on Organic Photonics and Electronics (COPE)”, and Solvay. References and Notes
+
-
ptcdi Figure 3. Rate constants for the D pc X D ptcdi f S pc 0 0 2 X S0 pc+ ptcdi(S2-S0; located 2.4 eV above the ground state), D 0 X D 0 f S pc 0 + ptcdi X T 1 (S0-T1; located 2.2 eV above the ground state) and D pc X 0 pc ptcdi D ptcdi (T1-S0; located 1.4 eV above the 0 - f T1 X S0
ground state) transitions in the Pc-PTCDI complex as a function of the energy difference between the lowest charge-transfer state + ptcdi (D pc X D ptcdi ) and the ground state (S pc ). 0 0 0 X S0 4. Synopsis In this work, we have developed a new computational method to evaluate the electronic coupling between intramolecular and charge-transfer states of a donor-acceptor complex. The ground and excited states of the donor and acceptor moieties, as well as the states corresponding to their respective charged configurations, were derived at the ZINDO-CISD level of theory. We employed the same HF orbital basis set to compute the electronic states of both neutral and charged configurations; this approach simplifies the working expressions and significantly increases the speed of the calculations. The versatility of the method was demonstrated by calculating the electronic couplings for different CS and CR pathways in a model phthalocyanine/perylene-tetracarboxylic-diimide complex. It was shown in particular that the electronic couplings between charge-transfer and intramolecular triplet states can be similar to those arising from associated intramolecular singlet states. This result is consistent with recent findings that suggest that, in a number of materials, transitions to intramolecular triplet states could prove to be the main channel for deactivation of the charge-transfer states. We expect that the method developed in this work will be useful for the calculations of the electronic couplings in many
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