Electronic Couplings for Photoinduced Electron Transfer and

Article pubs.acs.org/JPCA

Electronic Couplings for Photoinduced Electron Transfer and Excitation Energy Transfer Computed Using Excited States of Noninteracting Molecules Alexander A. Voityuk* Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain Institut de Química Computacional i Catàlisi (IQCC), Universitat de Girona, 17071 Girona, Spain ABSTRACT: A new computational scheme to calculate electronic coupling for photoinduced electron transfer and excitation energy transfer is described. The transfer integral between predefined quasi-diabatic states is expressed through adiabatic excitation energies of the system and expansion coefficients derived by decomposition of the transition density matrix of the reference states. To demonstrate the feasibility of the developed approach, electronic couplings for charge separation and exciton transfer in a heterojunction composed of quaterthiophene and C60 fullerene are computed at the DFT/ω-B97XD level.

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Each state ψi is represented by a linear combination of the reference states defined above. The extent, in which two reference states ϕα and ϕβ mix, depends on the ratio of their coupling Vαβ and the energy gap |εα − εβ|. Multiple transitions can occur in the system (ABC)*: 6 different EET X*Y → XY*, 12 CS processes X*Y → X+Y− and X*Y → Y+X−, 6 charge recombination reactions X+Y− → XY, and hole and electron shifts X+Y → XY+ and X+Y → XY+. To describe the ET and EET reactions, the Hamiltonian in the quasi-diabatic representation (εα and Vαβ) is required. Because quantummechanical (QM) calculations provide adiabatic states (wave functions ψi and energies Ei), the required parameters εα and Vαβ should be expressed through adiabatic properties. One can use, for example, transition dipole moments16 or fragment charges17,24 to transform the adiabatic Hamiltonian to its quasidiabatic representation. In most studies, one assumes that two reference states of interest (e.g., X*Y and X+Y−) are well represented by a combination of only two adiabatic states. In this case, the two-state model is applied to derive the diabatic parameters. In general, however, N adiabatic states (N > 2) should be treated simultaneously within a multistate model. Because the density of states increases rapidly with the size of the system, a large number of adiabatic states have to be included in the model. The multistate treatment is not uniquely determined and has some restrictions preventing the derivation of reliable coupling values.25−27 An alternative strategy to calculate ET and EET couplings is based on the “direct” construction of diabatic states.28 Because

INTRODUCTION Computational modeling of light-induced electron transfer (ET) and excitation energy transfer (EET) is widely employed to understand these elementary reactions in biological molecules and organic materials.1−7 In many cases, Marcus equation and its extensions provide satisfactory phenomenological description of ET.1,2,8−10 More sophisticated simulations rely on the nonadiabatic molecular dynamics method. Efficient computational algorithms11,12 based on the tight-binding Hamiltonian make it possible to apply this approach to extended systems.13−15 The key parameter that controls the ET and EET is electronic coupling of excited states involved in the reactions. Coupling matrix elements could be obtained by orthogonal transformation of the adiabatic states to quasi-diabatic states.16−23 This approach works well for small and medium-size systems but becomes quite cumbersome when applied to extended molecular complexes. The main problem here is a proper selection of adiabatic states involved in the transformation. To explain this point, let us consider a simple system consisting of three molecules A, B, and C. Each subsystem has only one HOMO (HA, HB, and HC) and one LUMO (LA, LB, and LC). An electron transition HX → LX leads to a locally excited (LE) state X* (X = A, B, or C). Excited-state configurations HX → LY and HY → LX correspond to charge-transfer (CT) states X+Y− and Y+X−. Thus nine reference (three LE and six CT X+Y−) states have to be considered to describe one-electron excitations in the simple system. The electronic Hamiltonian of this model is defined by 45 matrix elements, 9 diabatic energies εα (diagonal matrix elements), and 36 electronic couplings Vαβ (off-diagonal matrix elements). Diagonalization of this matrix gives nine adiabatic states ψi with the energy Ei. © 2017 American Chemical Society

Received: April 26, 2017 Revised: July 5, 2017 Published: July 5, 2017 5414

DOI: 10.1021/acs.jpca.7b03924 J. Phys. Chem. A 2017, 121, 5414−5419

Article

The Journal of Physical Chemistry A

the transition density matrix T(i). In AO basis, T(i) is represented by a nonsymmetric square matrix of size M × M (M is the number of AOs in the system). Matrix elements of T(i) are expressed through amplitudes of each determinant in ψi and MO coefficients. In the CIS (configuration interaction of singles) scheme, which in DFT is known as the Tamm− Dankov approximation (TDA), T(i) is defined by

the diabatic states depend weakly on the molecular geometry, they provide a more suitable basis for the QM treatment of electronic transitions. A well known scheme to construct diabatic states employs block-diagonalization (BD) of the Hamiltonian matrix.29,30 Several “direct” approaches are employed to estimate electronic coupling for ET reactions.31−34 The Förster formula for the EET coupling of donor (D) and acceptor (A) VDA =

μD⃗ μA⃗ 3 RDA

−

vac occ

Tμν(i) =

3(μD⃗ R⃗DA )(μA⃗ R⃗DA ) 5 RDA

a

(1)

(3)

where Dka is the amplitude of in ψi and cμk and cνa are AO coefficients in MO k and a. The adiabatic states ψi of the system can be represented by linear combinations of the reference states ϕα ψi ≈

∑ uαiϕα i

(4)

The following procedure can be applied to obtain the quasidiabatic states ϕα.29,30 A reference structure with the noninteracting donor and acceptor sites is constructed using the actual geometry of the system as the starting point. The excited states computed for the reference structure serve as quasidiabatic states for the system of interest. For instance, the quasidiabatic LE and CT states for a system AB can be represented by excited states of the reference system [A···B], where the molecules A and B are separated from each other so that the LE and CT reference states do not mix (their couplings are negligibly weak). Note that the LE reference states A* or B* can also be obtained from QM calculations of individual molecules A and B. Transformation of eq 4 in the opposite direction gives

1

HIF − 2 SIF(HII + HFF) 2 1 − SIF

k

Φak

is an example of how to apply the direct method. Within this approach, excited states of individual molecules D and A are considered as reference states. The Coulomb interaction of the transition dipole moments μD and μA of the reference states within the complex DA gives the coupling matrix element. In general, to estimate the coupling of an initial state (I) and a final state (F) for some electronic transition, one constructs the reference states ϕI and ϕF that properly describe the essential nature of I and F. Then, one computes Hamiltonian matrix elements HII, HFF, and HIF and the overlap integral SIF in the system using the reference states and estimates the coupling VIF VIF =

∑ ∑ Dka(i)cμkcνa

(2)

Note that VIF = HIF if ϕI and ϕF are orthogonal. To make feasible such calculations for large systems, the states ϕI and ϕF are often approximated by localized molecular orbitals. For instance, the first excited state of fragment X in the system XY can be represented by a one-electron transition between HOMO and LUMO localized on X, whereas charge-separated state X+Y− is described by HOMO of X and LUMO of Y. Hole transfer between the sites is treated in terms of the HOMOs of the fragments, whereas excess electron transfer is described by their LUMOs. Despite the fact that the orbital approach has been widely employed,8,13,15,35−37 the accuracy of the oneelectron approximation remains unclear. A new method to estimate ET and EET couplings is developed. Within this scheme, the quasi-diabatic states that describe the initial and final states of the electronic transitions are represented by multiconfigurational wave functions computed for the reference structures. The key feature is that the reference states are expanded in terms of adiabatic states by decomposition of the transition density matrix. This permits us to express electronic couplings via excitation energies of the system. It is explained in detail how to exploit this method in combination with DFT calculations. Electronic coupling values are derived for different ET and EET processes in a solar-cell model consisting of a thiophene oligomer and fullerene C60.

ϕα ≈

∑ uiαψi i

(5)

Our aim now is to determine the expansion coefficients uiα in eq 5. This can be done by taking advantage of the condition38 T +(i)ST (j) = δij

(6)

where S is the overlap matrix in AO basis. In the symmetrically orthogonalized AO basis, T(i) matrixes are orthonormal (in terms of the Frobenius norm)

∑ Tμν(i)Tμν(j) = δij (7)

μ,ν

Then, the coefficients uiα can be expressed using the transition density matrix Θα for the reference state ϕα uiα =

∑ Θμν(α)Tμν(i) μ,ν

■

(8)

If the matrixes Θ(α) and T(i) are of the same size, then the indexes μ and ν in eq 8 run over all AOs. If Θ(α) is built for a part of the system (e.g., for fragment A in AB), then the sum is taken over the orbitals of A. The use of the orthogonalized AO basis makes the computations much more efficient. The accuracy of the expansion 5 is determined by the quantity Uα

METHODS QM calculations of a molecular system provide electronic properties for its ground and excited states. The ground state ψo is usually described by a single determinant Φ0, whereas the excited states ψi (i = 1, ..., n) are represented by a linear combination of singly excited determinants Φak constructed from Φ0 by replacing an occupied MO k by a vacant MO a and may also include doubly excited determinants Φab kl and higher excited configurations. For each vertical excitation ψo → ψi, we will consider only two quantities: the excitation energy Ei and

Uα =

∑ ui2α i

(9)

In the ideal case, Uα = 1 for all states ϕα in hand. If necessary, the number of the adiabatic states ψi in expansion 5 can be 5415


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(1) One constructs reference structures and performs QM calculations of their excited states to obtain quasi-diabatic states for the system. The excited states of individual molecules A, B, and C can serve as quasi-diabatic LE states of the system. In a similar manner, one uses excited states of complexes [A···B], [A···C], and [B···C] to define quasi-diabatic CT states ϕα. The transition density matrix Θ(α) is calculated for each reference state ϕα. Analysis of Θ(α), eq 14, helps to identify the CT states of interest and determine their quality (the corresponding XAB values should equal 1). Note that the orientation of fragments in the reference structures and in the system has to be the same; that is, the reference structures are generated from the actual geometry by shifting of the fragments. If for some reason the orientation is changed, then the transition matrixes Θ(α) should be transformed by the corresponding rotation of AOs. (2) Expansion coefficients uiα and uiβ(i = 1, ..., n) are determined using eq 8. The overlap integral Sαβ and the matrix elements Hαα, Hββ, and Hαβ are computed with eqs 10 and 11. Finally, the coupling Vαβ for the electronic transition between the reference states α and β is estimated using eq 12. The suggested scheme differs essentially from the BD approach,29,30 which operates with canonical or natural molecular orbitals for the reference and actual geometries: (1) MOs are determined for a reference geometry, (2) the diabatic MOs for the structure of interest are obtained by maximizing their overlap with the reference orbitals, and (3) CI coefficients are transformed to the new basis on which the block diagonalization of the CI matrix is performed. In our scheme, only transition density matrixes are considered.

increased. The overlap and Hamiltonian matrix elements for the reference states α and β are expressed via the amplitudes uiα Sαβ ≈

1 UαUβ

∑ uiαuiβ

1 UαUβ

Hαβ ≈

(10)

i

∑ uiαEiuiβ (11)

i

In the last expression, Ei is the excitation energy of the adiabatic state ψi. By definition, the energy of the ground state ψ0 is zero, E0 = 0, and this state does not contribute to the matrix element Hαβ. The quantities Uα and Uβ defined by eq 9 are used to normalize the expansion coefficients. The diabatic energy εα is defined by Hαα. The coupling Vαβ of the diabatic states is defined by 1

Vαβ =

Hαβ − 2 Sαβ(Hαα + Hββ) 2 1 − Sαβ

(12)

If the overlap of the diabatic states is vanishingly small, Sαβ = 0, then the matrix element Vαβ is equal to Hαβ. The described approach can be employed to estimate the electronic coupling of any two reference states. If the state α is localized on molecule A and the state β on B, then the quantity Vαβ determines the coupling for excitation energy transfer between A and B. Charge separation AB* → B+A− is described by coupling of the LE state B* and the CT state B+A−. The initial and final states for photoinduced hole shift B+ → C+ in ABC with an excess electron on A are represented by reference states B+A− and C+A−. The following procedure can be employed to determine ET and EET electronic couplings for the system ABC. (1) QM calculations of n excited states of the system are carried out. For each state ψi, the transition density matrix T(i) is constructed using the symmetrically orthogonalized AO basis. (2) The obtained excited states are analyzed by means of T(i).38−40 Although this step is not compulsory, it provides helpful information on the nature of ψi. (3) Let us briefly consider how to perform this analysis. The weight of local excitations A* in ψi is determined by XAA (i) =

∑

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COMPUTATIONAL DETAILS In our case study, we consider a model donor−acceptor heterojunction [QT C60] composed of quaterthiophene (QT) and C60 fullerene. The model is shown in Figure 1. Cartesian

2 Tμν (i )

μ∈A

(13)

ν∈A

The sum is taken over AOs of the domain A. Similarly, one gets the contribution of LE states of B and C. The weight of CT configurations A+B− in ψi is given by XAB(i) =

∑

2 Tμν (i )

Figure 1. Molecular complex of quaterthiophene (QT) and C60 fullerene.

μ∈A

(14)

ν∈B

with μ and ν running over AOs of A and B, respectively. The 2 (i). weight of CT states B+A− is defined by XBA (i) = ∑ μ ∈ B Tμν

coordinates of the complex were taken from the supporting materials of ref 41. To obtain quasi-diabatic states, we also treated a reference structure where the orbital interaction of the donor and acceptor sites is negligibly weak. The geometry of the reference system was constructed from the optimized structure by shifting the thiophene tetramer along the Z axis by 10 Å. 100 vertical excitations were calculated for both systems with TDA DFT using the long-range corrected ω-B97XD

ν∈A

Note that the quantities XAA and XAB describe the character of adiabatic states but do not provide any link to quasi-diabatic states. In particular, the quantity XAA(i) gives the extent of exciton localization of ψi on fragment , but does not suggest whether only one or several excitations of A contribute to this state. 5416


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Figure 2. Character of excited states in the reference system with noninteracting QT and C60 (A) and in the heterojunction model (B). The oscillator strength, f, the contributions of local excitations X(C60) and X(QT), and the contribution of CT configurations are presented in the upper, middle, and lower panels, respectively.

functional42−44 and the standard 6-31G* basis set. The corresponding transition density matrixes were calculated by eq 3. The DFT calculations were performed by Gaussian 09.45

acceptor sites decreases. Some CT states in the complex become weakly absorbing (f ≈ 0.02) due to their mixing with QT*. In distinction to the reference system, many excitations in the complex are delocalized over both molecules and contain substantial contributions of CT states. Because of that, it is quite difficult to unambiguously select adiabatic states in the complex for estimating ET and EET couplings within the multistate model. In contrast, the use of the predefined reference states allows one to compute systematically the matrix elements of interest. Electronic Couplings for Charge Separation. The squared coupling of the highly absorbing QT* state with the CT states determines the probability of the corresponding charge-separation processes. Decomposition of the reference QT* state over adiabatic states, eq 8, suggests that two states at ∼3.7 eV contribute to the QT* state. This can also be seen from the top panel of Figure 2B. A more complicated situation is found for the CT states. In this case, linear combinations of up to 10 adiabatic states have to be taken to represent the quasi-diabatic states. Obviously, the two-state model, which describes both the initial and final ET states using only two adiabatic states, cannot be applied. Also, the multistate treatment is difficult to perform because there is no clear procedure to select N adiabatic states that can be linearly transformed into N reference states of interest. The suggested scheme, eqs 10−12, provides, however, the direct way to the transfer integrals. Table 1 shows the diabatic energies and electronic couplings for charge separation and excitation energy transfer in the system. The CT and LE states that exhibit negligibly weak interaction with QT*, |V| < 10−6 eV, are not listed in the Table. The reference CT state at 3.39 eV appears to be most important. It has lower energy that QT* and is strongly coupled to this state, V = 0.016 eV. Although two other CT states at ∼4.40 eV have even larger couplings (0.028 and 0.034 eV), they can hardly be involved in the ET process because they lie higher in energy than QT* (εQT*= 3.687 eV).

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RESULTS AND DISCUSSION The decay of the first LE state QT* of quaterthiophene in the complex [QT C60] may occur due to ET from QT to the fullerene acceptor resulting in several CS states [QT+ C60−]. Also, excitation energy transfer between the molecules can be responsible for the decay of QT*. Because excited states of C60 are lower in energy than QT*, EET from QT to C60 in the heterojunction is energetically feasible. Let us first consider excited states in the reference system with vanishing orbital interaction between the molecules. The character of these states described in terms of the quantities XAA and XAB, eqs 13 and 14, is shown in Figure 2A. The excited states of the reference structure can be well described either as excitations completely localized on the subunits or as pure CS states with an electron fully transferred from QT to C60 (Figure 2A). Fifteen lowest excitations (2.67 to 3.10 eV) are localized on C60 and have negligibly weak oscillator strengths. The lowest excited state of QT, QT*, is found at 3.56 eV. This highly absorbing state ( f = 1.043) plays the key role in photophysics of the heterojunction. Three almost degenerate CT states [QT+ C60−] lie at 4.33 eV. In these states, an electron is completely transferred from the QT moiety to the fullerene molecule. Absorption at 4.47 eV corresponds to three LE states (T1u) of C60. These quasidiabatic states will be used to calculate the coupling matrix elements in the model with the interacting donor and acceptor sites. The character of excitations changes considerably when passing from the reference system (Figure 2A) to the optimized structure (Figure 2B). Although the overall picture is still retained, the electronic interaction of the subunits leads to splitting and mixing of the excited states. The CT states are stabilized by ∼1 eV as the separation between the donor and 5417


The Journal of Physical Chemistry A Table 1. Diabatic Energies ε and Electronic Couplings V for CS and EET in the Complex [QT C60] Calculated Using TDA DFT with the ω-B97xd Functional charge separation |V| (eV)

ε (eV)

3.39 4.40 4.43

1.613 × 10−2 2.281 × 10−2 3.351 × 10−2

2.72 2.74 2.77 2.78 2.80 2.81 3.17 3.19

|V| (eV) 4.238 3.983 2.458 3.130 5.528 2.572 2.051 1.929

× × × × × × × ×

10−3 10−4 10−3 10−3 10−5 10−3 10−2 10−2

ACKNOWLEDGMENTS

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REFERENCES

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Electronic Couplings for Excitation Energy Transfer. Let us consider the interaction of QT* with the lower excitations localized on C60. The coupling squared determines the probability of EET between the subunits. As seen from Figure 2, the low-lying LE states of C60 cannot be populated by light absorption because their oscillator strength is vanishingly weak, f = 0. Because the transition dipole moment of the LE states of C60 is zero, the EET between QT* and the LE states is forbidden within the Förster theory (in line with eq 1, the EET coupling is zero). Dipole allowed transitions 11Ag →1T1u in C60 lie ∼1 eV higher than QT*, and the EET to these states is energetically unfavorable. However, in molecular complexes where donor and acceptor are in contact, the short-range electronic interaction that includes both the orbital and exchange terms may allow EET between states with the zeroed transition dipole moment. Table 1 lists excitonic couplings of QT* with LE states of C60. Relatively strong EET coupling, V = 0.02 eV, is found for the LE states at 3.2 eV. Because the EET couplings are comparable to the CS coupling, both CS and EET processes are likely to occur in heterojunctions composed of thiophene oligomers and C60 fullerene.

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CONCLUSIONS A new computational scheme to calculate electronic couplings and diabatic energies for photoinduced electron transfer and excitation energy transfer has been developed. It implies that excited electronic states computed for the reference geometries serve as the diabatic states for the system at the actual geometry. The one-particle transition density matrixes for ψo → ψi excitations are computed for the system of interest and the corresponding reference structures. The transfer integrals are expressed through the excitation energies of the system and the expansion coefficients derived by decomposition of the reference transition density matrix, eqs 8−12. The scheme can be employed in combination with TDA DFT calculations of excited electronic states for large systems. ET and EET couplings in the heterojunction of quaterthiophene and C60 fullerene have been estimated at the DFT/ω-B97XD level.

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I thank Dr. Anton Stasyuk for his kind help. Financial support from the Spanish Ministerio de Economiá y Competitividad (MINECO) (project CTQ2015-69363-P), the Catalan DIUE ́ (Project 2014SGR931), Xarxa de Referència en Quimica Teòrica i Computacional), and the FEDER fund (UNGI104E-801) is gratefully acknoledged.

excitation energy transfer

ε (eV)

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Alexander A. Voityuk: 0000-0001-6620-4362 Notes

The author declares no competing financial interest. 5418


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Electronic Couplings for Photoinduced Electron Transfer and

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