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Insight into the Excited State Electronic and Structural Properties of the Organic Photovoltaic Donor Polymer Poly(thieno[3,4‑b]thiophene benzodithiophene) by Means of ab Initio and Density Functional Theory Itamar Borges, Jr.,*,†,‡,§ Elmar Uhl,§ Lucas Modesto-Costa,‡ Adélia J. A. Aquino,† and Hans Lischka*,†,∥ †

Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States Departamento de Química, Instituto Militar de Engenharia, Praça General Tibúrcio, 80, 22290-270 Rio de Janeiro, Brazil § Programa de Pós-Graduaçaõ em Engenharia de Defesa, Instituto Militar de Engenharia, Praça General Tibúrcio, 80, 22290-270 Rio de Janeiro, Brazil ∥ Institute for Theoretical Chemistry, University of Vienna, 1090 Vienna, Austria ‡

S Supporting Information *

ABSTRACT: Structural and electronic properties of the ground and lowest excited states of the electron-donor conjugated copolymer poly(thieno[3,4-b]thiophene benzodithiophene) (PTB1) are reported based on high-level theoretical investigations. PTB1 combined with phenyl-C61butyric acid methyl ester (PCBM) results in a bulk heterojunction blend with promising properties for use in organic solar cells. The ab initio algebraic diagrammatic construction method to second order, ADC(2), was used to obtain benchmark data for excited state energies, oscillator strengths, and bond length alternation (BLA) analysis. Time dependent density functional theory (TDDFT) calculations using the exchange-correlation functionals PBE, B3LYP, BHandHLYP, CAM-B3LYP, and LC-wPBE were also performed and compared with ADC(2) calculations. It was shown that a minimum of 20% Hartree−Fock exchange in the functional is necessary to reproduce the major features of the ADC(2) results. Analysis of the BLA results indicates the possibility of exciton trapping by geometry relaxation occurring in the middle of the polymer chain. The corresponding exciton binding energy is about 0.4 eV. Charge distributions in the ground and lowest excited singlet state were analyzed as well. The natural population analysis (NPA) confirms the electronegative character of the benozodithiophene group and the corresponding positive one of the thienothiophene moiety leading to an alternant chain polarization of positive and negative charges. Electron density differences between the S0 and S1 states show a transfer of electron density from double bond regions to areas of single bonds, a feature which parallels nicely the lengthening of double bonds and shortening of single bonds in the S1 state.

1. INTRODUCTION The continuous increase of global demand for energy along with the well-known environmental problems associated with fossil fuels have stimulated intensive research for alternative energy sources. The use of organic semiconductors for generation of electricity from sunlight through the photovoltaic effect has emerged as a promising low-cost alternative.1 Photoconducting polymers have several other important applications such as thin film transistors, sensor technology, and organic light emitting diodes (OLEDs).2 Although semiconducting organic polymers have a similar functionality as compared to inorganic semiconductors, there are significant differences that strongly complicate the processes in the former case and require, therefore, a more detailed knowledge of the different photophysical processes with special emphasis on an understanding at a molecular level and the capability of quantitative predictions. © 2016 American Chemical Society

The initial photophysical process in a photovoltaic cell after light absorption is the formation of excitons (closely coupled electron−hole pairs).3 In inorganic solar cells, dielectric constants are usually large; hence exciton dissociation, the fundamental step for production of the electric current, is readily observed.4,5 However, an organic solar cell exciton is strongly bound at room temperature due to the low dielectric constants of the polymer and a strong electron−phonon coupling.5 Furthermore, given the large number of internal degrees of freedom and the multitude of electronic states of strongly varying character, the dynamical properties of the excitons in organic conducting polymers are much more involved in contrast to inorganic materials. Thus, a Received: July 30, 2016 Revised: August 30, 2016 Published: August 31, 2016 21818

DOI: 10.1021/acs.jpcc.6b07689 J. Phys. Chem. C 2016, 120, 21818−21826

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The Journal of Physical Chemistry C Scheme 1

capabilities of the CC2 method can be found in ref 26. The capabilities of ab initio methods in comparison to the more frequently used TDDFT method for charge transfer states in stacked oligothiophenes were systematically studied, and good agreement between ADC(2) and range-separated TDDFT/ CAM-B3LYP was observed.16 Recently, RI-ADC(2) calculations were performed to investigate vertical excitations, excitonic self-trapping, and torsional potentials of poly(p-phenylenevinylene) (PPV)20 which is of paradigmatic importance in studies of photovoltaic processes in conjugated systems. The vibrational broadening of UV absorption and fluorescence spectra of PPV were studied as well.21,27,28 RI-ADC(2) calculations on a large PTB1/C60 model (over 200 atoms) were performed. Analysis of the manifold of electronic states in this model system showed that a series of charge transfer states were located well below the bright π → π* state, a fact which should facilitate the first step of the charge separation process via internal conversion.11 Even though PTB1 has been used with great success in several experimental investigations to improve the efficiency of organic photovoltaic materials, a detailed ab initio description of its electronic structure and an investigation of the hypotheses on which the functionality is built have not been given yet. For that purpose, in this work, the electronic spectra of the PTB1 oligomers and electronic and structural changes on electronic excitations were investigated in detail applying the ADC(2) method to calculations on extended PTB1 oligomers. Knowledge of these features is important for judging the capabilities of π-conjugated semiconductors in terms of intrinsic molecular properties. Our ADC(2) results were also compared with those of TDDFT calculations using different exchange-correlation functionals. It has been demonstrated previously for PPV29 that the amount of Hartree−Fock (HF) exchange admixture in the functionals crucially affects bond length alternation (BLA) and excitonic trapping in the S1 state. This is a question of high relevance since the proper selection of a suitable density functional model should produce not only correctly excitation energies but also equally important properties such as the geometrical relaxation in the excited states and the correct character of the electronic excitation. The comparison between benchmark ADC(2) data and those obtained from different types of functionals used in the TDDFT calculations will aid the selection of appropriate functionals for the accurate description of excitonic states and considerably facilitate reliable predictions of the excited state properties of complex organophotovoltaic materials.

comprehensive modeling of the different fundamental photophysical mechanisms is still a great challenge6 and provides strong support for the development of efficient photovoltaic materials. In contrast to the inorganic solar cell, the components of the organic counterpart are bound by weak intermolecular interactions, thereby forming discrete electronic states rather than regular band structures. 7,8 Moreover, the exciton dissociation occurs in organic solar cells mostly at the interface between two distinct materials, an electron-donor material and an electron acceptor. The electronic properties of these components must be carefully selected and tuned to produce effective photocurrents. One important approach to construct these devices is the bulk heterojunction (BHJ),4−7,9−11 a bicontinuous composite of donor and acceptor phases prepared in such a way that maximizes the interfacial area between them. Promising new BHJ materials with increased conversion efficiency have been synthesized.5,7,9,12,13 Among them, the composite polymer PTB1/PCBM (poly(thieno[3,4-b]thiophene benzodithiophene)/[6,6]phenyl-C61-butyric acid methyl ester) was developed according to the concept that a thienothiophene−benzodithiophene acting as a basic unit in the donor polymer PTB1 can support a quinoidal structure that produces a narrow polymer band gap. Furthermore, the unique zigzag structure of PTB1 favors a “face-on” backbone orientation leading to a stacking of the polymer chains that preserves the extended π system and facilitates charge transport in the PTB1/PCBM blend.13,14 Each PTB1 monomer has two moieties, the electron rich benzodithiophene (BDT) and the electron deficient thienothiophene (TT).15 Due to the large size of the oligomers to be considered in realistic simulations, the time dependent density functional theory (TDDFT) would be the most efficient method for computing the electronic spectra of such large π-conjugated polymers. However, TDDFT has well-known problems in describing excited charge transfer (CT) states.16 Rangeseparated functionals especially with the inclusion of tuning procedures may improve on these artifacts.17−19 Therefore, it is important to employ also other methods that are well-suited to deal with the CT states and are computationally efficient. The algebraic diagrammatic construction method to second order, ADC(2), an ab initio wave-function-based polarization propagator method, combined with the resolution of identity (RI) method, provides such a good compromise for dealing with these systems.20,21 The closely related approximate coupled cluster method to second order (CC2)22 gives similarly good results23,24 for excited state computations and was also successfully applied in calculating the electronic spectrum of methylene-bridges oligofluorenes22 and oligo-pphenylenes.25 A systematic benchmark investigation on the

2. COMPUTATIONAL METHODS Ground state geometries up to the (PTB1)5 pentamer were optimized using DFT with the generalized gradient approx21819


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The indices of the bond distances Rij are defined in Scheme 2.

imation Perdew−Burke−Ernzerhof (GGA-PBE) exchangecorrelation energy functional.30 The motivation for choosing this nonhybrid functional lies in the computational efficiency in the joint use with the RI method.31 In the optimized structures, the ester side chain of the PTB1 was removed since the purpose was to focus on the role of the π conjugation involving the basic BDT and TT subunits. DFT geometries were used for the computation of the vertical excitations. Ground state geometry of the model (PTB1)4 (see Scheme 1) was optimized using Møller−Plesset perturbation theory to second order (MP2)32 for the BLA analysis. The RI method for computing electron repulsion integrals was employed in both MP2 and DFT optimizations.20,21 The geometries of the PTB oligomers in the first (S1) excited singlet state were optimized using ADC(2) and TDDFT with different exchange-correlation functionals described below. Unless stated otherwise, in all geometry optimizations the core orbitals were kept frozen (denoted as standard freezing scheme). All geometries of the model PTB1 without the ester side chain were optimized in Cs symmetry without any further constraints. Because of the computational demands of the larger oligomers, for the entire oligomer series from n = 1 to n = 5 the polarized split valence SVP basis set for sulfur33 and the SV Gaussian basis set for the remaining atoms, thereby denoted SV-SVP, were employed for all geometry optimizations at MP2 and DFT/PBE levels. Comparative test optimizations performed for the trimer using the SV(P) basis set on all atoms showed that the smaller SV-SVP basis set reproduced bond distances within deviations of ∼0.02 Å in the maximum. To compute electronic transitions, ADC(2) and, for comparison, TDDFT were used. In the latter case, the effect of a series of exchange-correlation potentials described below was investigated. Additionally, two larger basis sets, TZVP and TZVP-def2-S′34 were used for optimizations and spectral calculations up to the PTB1 trimer. TZVP-def2-S′ denotes a TZVP basis set for the H and C atoms which is extended by the def2-TZVP basis set for sulfur,34 which includes an extended 2d/1f set of polarization functions. The complete def2-TZVP basis set for all atoms was used for a series of high-level ADC(2) and TDDFT calculations on (PTB1)3.

Scheme 2. Atomic Numbering of the PTB1 Model

From the above definition of BLA values, four types of structural deformation of the PTB1 oligomers are obtained. BLA1 monitors thienothiophene and BLA3 the benzodithiophene subunits. Both represent the difference between C−C and CC bond distances in these moieties, thereby highlighting intra-ring effects. On the other hand, BLA2 and BLA4 describe inter(sub)unit distances; namely, BLA2 values characterize intersubunit effects through the bonds connecting the thionethiophene and benzodithiophene moieties in each unit (i.e., monomer) whereas BLA4 represents the bond between the units. For TDDFT calculations, as previously suggested,29 the following hierarchy of exchange-correlation functionals ranging from pure GGA to long-range hybrids, were employed: PBE30 (a = 0), B3LYP36 (a = 20%), BHandHLYP (a = 50%)̧36 CAMB3LYP (a = 20/65%),37 and LC-wPBE (a = 0/100%),38 where a is the percentage of HF exchange EHF X in the functional according to the expression E XC = aE XHF + (1 − a)E XGGA + ECGGA

This set of exchange-correlation functionals varies from pure generalized gradient correction (GGA) to long-range corrected hybrids. The long-range corrected functionals behave at shortrange typically as a hybrid or GGA functional. However, they possess an increasing HF exchange component at long distances up to a maximum value of 65% for CAM-B3LYP and 100% for LC-wPBE. To alleviate the still significant computational effort of the RI-ADC(2) approach for the large systems examined in this work, several freezing schemes with regard to occupied and virtual orbitals were tested and adopted. For oligomers up to n = 4, ADC(2) spectra calculations included only the freezing of the core orbitals (standard freezing). For n = 5, 60% of the lowest energy occupied orbitals (250 frozen out of 416) and 50% of the highest energy virtual orbitals (364 out of 728) were frozen. Comparative calculations performed on (PTB1)1 for 18 different freezing schemes are collected in the Supporting Information (see discussion before Figure S1). Excited states were characterized according to a recently developed analysis of the transition density matrix39 based on previous work.40,41 In this scheme, the transition matrix D0α,[AO] between the ground state 0 and the excited state α is divided into contributions to fragments A, B, and so on of the oligomer chain. Afterward, charge transfer numbers (diagonal and nondiagonal values) from fragment A to B of the transition is computed according to the expression 1 ΩαAB = ∑ (D0α ,[AO]S[AO])ab (S[AO]D0α ,[AO])ab 2 a∈A

3. THEORETICAL BACKGROUND AND DEFINITIONS The geometrical relaxation of the polymer geometry upon electronic excitation29 will be quantified by bond length alternation values35 computed from bond length differences. BLA values provide a compact characterization of geometry changes that reflect localization of the exciton.29,35 The BLA analysis was performed for the (PTB1)4 oligomer using the MP2/SV-SVP geometry of the ground (S0) state and the ADC(2)/SV-SVP and TDDFT/SV-SVP structures of the excited S1 state. Four different types of BLA’s descriptors were defined according to the following expressions: BLA1 = 1/2(R 2,3 + R 2,5) − 1/3(R1,2 + R3,4 + R 5,6)

(1)

BLA 2 = R 4,7

(2)

BLA3 = 1/5(R 8,9 + R 9,10 + R12,13 + R11,14 + R14,15) − 1/5(R 7,8 + R 9,12 + R10,11 + R13,14 + R15,16)

(6)

b∈B

(3)

BLA4 = R16,1

(5)

ΩαAB

The descriptor is computed by summing over basis functions a and b located on the respective fragments A, B, and

(4) 21820


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The Journal of Physical Chemistry C so on; the overlap matrix S[AO] is used to account for nonorthogonality of the AOs. The charge transfer numbers therefore describe the excited state as an electron−hole pair with the probability of finding the hole on fragment A and the electron on fragment B. Two-dimensional plots of ΩαAB for each excitation allowed the visualization of the spatial exciton distribution of each transition. The S1 excitation energy was extrapolated to infinite chain length according to the expression: ⎛ π ⎞ ⎟ E(n) = E0 1 − 2α cos⎜ ⎝ n + 1⎠

(7)

Figure 1. ADC(2)/SV-SVP (a) transition energies ΔE (eV) and (b) optical oscillator strengths ( f) as function of 1/n, n being the number of units of the oligomer. Dashed lines are linear extrapolated values of the transition energies. The continuous line is the Kuhn extrapolation of the PTB1 S1 transition energy.

suggested by Kuhn by considering a chain of coupled oscillators;42,43 n is the number of oligomers varying from 1 to 5. Excitation energies of the S2−S4 states and also of S1 were extrapolated linearly against 1/n. Natural population analysis (NPA)44 charges for (PTB1)5 in the S0 and S1 states were computed. The Turbomole program version 7.01 was used for the ADC(2), MP2 and PBE calculations including geometry optimizations. TDDFT spectra calculations using the hierarchy of exchange-correlation functionals were performed with the Gaussian 09 suite of programs.45

HOMO−LUMO π → π* excitation. For all excitation energies, there are ranges of n values that display a linear behavior with 1/n; in particular, the bright π → π* transition (S1) transition energies mostly vary linearly with 1/n for the whole range of the computed oligomers. The S1 transition energies were extrapolated to n → ∞ (infinite chain limit) using the nonlinear Kuhn formula (eq 5) whereas the S2 − S4 states were linearly extrapolated. The Kuhn extrapolation value of the S1 transition energy is 2.330 eV. For the states S2−S4, the linear extrapolated values are as follows: S2, 2.057 eV; S3, 2.132 eV; S4, 2.665 eV. The linearly fitted transition energies of the three lowest excited singlet states S1−S3 converge asymptotically to the same value of ∼2.1 eV (Figure 1a) implying that they belong to the same exciton band; an identical situation appears in PPV.27 Furthermore, similarly to PPV, the S4 state of PTB1 asymptotically converges to a higher energy value as compared to the states S1−S3 and, hence, corresponds to the next energy band (Figure 1). The measured bright π → π* transition of PTB1 is a broad and flat peak in the range of 1.80−1.98 eV.12 The extrapolated infinite chain length value of 2.330 eV for the S1 state was obtained with the SV-SVP basis set. The discussion in the beginning of this section on the basis set effect for (PTB1)n=1−3 showed that the S1 transition energy is further reduced by ∼0.3 eV based on SV-SVP, SVP, TZVP, and TZVP-def2-S/ ADC(2) calculations. Reducing the extrapolated value by this amount leads to good agreement with the experimental absorption maximum. In Figure 1b, the oscillator strengths are plotted as a function of the inverse number n of units of the oligomers. The first transition (S1) carries almost all the oscillator strength and its value steeply increases with the length of the chain. On the other hand, the S3 (41A′) state starts to acquire a nonnegligible, though small, oscillator strength as compared to the S1 value when the chain length increases. The other two transitions remain dark. As mentioned in the Introduction, the proper selection of the exchange-correlation functional is crucial for the successful application of TDDFT to the description of excited properties of π-conjugated polymers. To evaluate the effect of the exchange-correlation functional on the vertical transition energies, TDDFT excitation energies from the ground to S1− S4 states using several functionals are compared in Figure 2 with

4. RESULTS AND DISCUSSION 4.1. Vertical Excitations. The major cost factor in the computational demand of the present calculations is the basis set size. To cope with this problem, the effect of the basis set quality on the excitation energies of (PTB1)n was investigated in the oligomer range n = 1−3 using ADC(2). The standard freezing scheme, i.e., freezing only the core orbitals, was employed. The SV-SVP, SVP, TZVP, and TZVP-def2-S′ basis sets were chosen for that purpose. Results are collected in Figure S2 and Tables S1−S3 of the Supporting Information. Increasing the basis set size reduces the excitation energies for the examined oligomers by about 0.3 eV between the smallest SV-SVP and the larger TZVP and TZVP-def2-S′ basis sets. It is important to note that transition energies and the character of the states using the SV-SVP basis set as compared to the larger ones are well-reproduced and exhibit regular behavior (Tables S1−S3). This regular behavior of the ab initio ADC(2) calculations in combination with the SV-SVP basis set is especially important because the use of frozen occupied and virtual orbitals in a controlled way allowed the investigation of excited states of quite large systems (oligomers up to n = 5), as was done before for the PTB1/PCBM system.11 The dependence of the transition energy on the amount of frozen occupied and virtual orbitals, as investigated employing 18 different freezing and the ADC(2)/SV-SVP method for (PTB1)1, is less than 0.4 eV in the whole range of tested freezing schemes (see Supporting Information and Figure S1). The difference between TZVP and TZVP-def2-S′ excitation energies is much smaller, both values being essentially identical (Figure S2 and Tables S1−S3). The experience with the different basis sets and freezing schemes described in the previous paragraph was used for extrapolating the lowest four excitation energies in the (PTB1)n series using the ADC(2) method with standard freezing and the SV-SVP basis. In Figure 1a, excitation energies are plotted as a function of 1/n where n is the number of oligomer units. For all PTB1 oligomers, the first (S1) transition refers to the bright 21821


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BLA1 (Figure 3a) represents the difference between carbon− carbon single and double bond lengths in the electron deficient thienothiophene subunit. The MP2 and ADC(2) results are discussed first. At the MP2 level, the BLA1 plot shows that in the ground state the CC single bonds in thienothiophene in the four subunits are larger by ∼0.05 Å as compared to the average of the double bond distances; for subunit 1, where thienothiophene is the leftmost moiety, this difference is a bit larger. Upon excitation to S1, a significant tip appears in BLA1 for the inner subunits 2 and 3 and the four BLA1 values decrease significantly. The behavior of BLA3, which measures the difference between the average single and double carbon−carbon bond distances in the benzodithiophene subunit, is depicted in Figure 3b (the vertical scale differs from Figure 3a). The overall behavior of BLA3 is similar to BLA1: the S0 curve is flattened, and now it is the rightmost moiety, a benzodithiophene, which has a somewhat larger value (i.e., a border effect) and the S1 curve displays a tip in the inner subunits. The magnitude of this tip in the S1 state is about 0.03 Å for BLA1 and 0.02 Å for BLA3. By examining the S0 and S1 BLA1 and BLA3 values (see also Figures S4 and S5), it is seen that upon excitation in both thionethiophene and benzodithiophene moieties in average CC bonds stretch as compared to C−C bonds especially in the inner subunits. For subunit 2 (M2) of benzodithiopheone BLA3 is slightly negative (∼−0.005 Å); hence its CC bonds increase even more. This behavior agrees well with the picture that excitation occurs mostly from occupied molecular orbitals (MOs) located at double CC bonds to MOs with major contributions at single bonds, as discussed in the next section. BLA2 values describe the effects on the bond between the thienothiophene and benzodithiophene moieties in each unit (Figure 3c) whereas BLA4 values represent carbon−carbon bond distances between different units (Figure 3d). In the ground state, both BLA values do not depend significantly on the position of the segment. On the other hand, both the S1

Figure 2. (PTB1)3 TDDFT and ADC(2) vertical transition energies using the def2-TZVP basis set. The PBE/SV-SVP optimized geometry was used in all cases.

ADC(2) results for (PTB1)3 using the def2-TZVP basis. BHandHLYP and CAM-B3LYP give the best agreement with the ADC(2) excitation energies. This is expected especially because these functionals contain a balanced description and amount of HF exchange, a crucial feature to accurately compute the S1−S4 states that have some charge transfer character (see the next section). Not surprisingly, B3LYP having even less amount of HF exchange, and especially PBE having none, give too low excitation energies as compared to ADC(2). 4.2. Geometry Effects in the Ground and Lowest Excited State. The BLA values defined in eqs 1−4 were used to describe the geometry effect due to electronic excitation to the S1 state. In this section, the results were obtained for the (PTB1)4 oligomer unless stated otherwise. MP2, ADC(2), and TDDFT geometries were optimized employing five distinct exchange-correlation functionals. As the ground-state counterpart for ADC(2) S1 calculations, the MP2 method was used. Results are collected in Figure 3. Detailed results for individual bond distances are displayed in Figures S3−S5.

Figure 3. Bond length alternation (BLA) analysis of (PTB)4 S0 (solid lines) and S1 (dashed lines) states using MP2, ADC(2), and DFT methods with different exchange-correlation functionals. MP2 S0 values are represented by solid black lines and ADC(2) S1 by the dashed black lines. (a) BLA1, (b) BLA3, (c) BLA2, and (b) BLA4. Labels M1−M4 denote the monomer number. 21822


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PBE provide less than half of the relaxation energies as compared to ADC(2). 4.3. Character of the Excited States. To characterize the excited states of the PTB1 polymer, ΩαAB plots (eq 6) were computed for (PTB1)5 using ADC(2) and DFT methods including five exchange-correlation functionals in the latter case. (PTB1)5 charge transfer q(CT) in the four excited states and natural population analysis (NPA)44 charges of the S0 and S1 states were computed as well. (PTB1)5 has five repeat units. For each unit, two adjacent subunits are defined: the first one represents the benzodithiophene moiety and the second one thionethiophene. Each of the squares in the plot indicates the probability of simultaneously finding the electron in a subunit numbered in the horizontal axis and the hole in a subunit numbered in the vertical axis. The extension of the exciton along the chain relates to the diagonal in the plot whereas the electron−hole separation (i.e., the charge transfer character) corresponds to the off-diagonal width. Hence, the ΩαAB plots in Figure 4 provide a detailed picture of the exciton wave function of each transition in (PTB1)5.

BLA2 and BLA4 values decrease respectively by 0.03 and 0.02 Å, a similar magnitude of the BLA1 and BLA3 tips. As expected, BLA2 and BLA4 values do not show edge effects. Turning to the DFT results in Figure 3, one observes that the five ground state BLA values have overall the same flattened curves as is the case of the MP2 result though their values differ. Whereas for BLA1 and BLA3, related to intrasubunits effects, PBE results are the closest to MP2; for BLA2 and BLA4 the closest curve to MP2 is the LC-wPBE. For the S1 state, excepting the PBE functional, the other functionals display the expected tip. The above BLA analysis clearly indicates that excitons can be trapped at the center of the (PTB)4 oligomer due to geometry relaxation upon excitation to the S1 state both in intra- and interoligomer bonds. Moreover, excepting PBE, the other four exchange-correlation functionals that have at least 20% of HF exchange localize (i.e., traps) the exciton. A similar conclusion using the same hierarchy of functionals was found before for PPV oligomers.29 Furthermore, exciton trapping or defect localization in the S1 excited state was also found in PPV using the ADC(2)/SV(P)27 and PPP/force field approaches.35 Geometry relaxation for oligomers n = 1 to n = 4 in the S1 state leads to an energetic stabilization by 0.4−0.5 eV with respect to the vertical excitation computed at the ADC(2) level (Table 1). This stabilization in the S1 state arises from the Table 1. ADC(2)/SV-SVP Lowest Excited State (S1) Adiabatic and Vertical Transition Energies (eV) of the (PTB1)1 to (PTB1)4 Oligomers state

ΔEadiabatic

ΔEvertical

difference

(PTB1)1 (PTB1)2 (PTB1)3 (PTB1)4

3.308 2.501 2.348 2.279

3.779 3.057 2.810 2.702

0.471 0.556 0.462 0.423

Figure 4. (PTB1)5 ΩαAB plots of the first four electronic transitions (S1 to S4) employing ADC(2), LC-wPBE, and the SV-SVP basis set. The vertical axis indicates the position of a hole and the horizontal axis the position of the electron. Each square of the plot represents either a benzodithiophene or a thienothiophene subunit. The square in the lowest leftmost corner corresponds to the electron rich benzodithiophene moiety and in the highest rightmost corner to the electron deficient thienothiophene. The shades represent the probability values according to the scale on the right of each panel.

exciton localization at the center of the chain in this state as was just discussed. Starting with n = 2, the energetic stabilization of the S1 transition energy decreases but still remains at a value larger than 0.4 eV. To examine the effect of the type of exchange-correlation functional on the relaxation energy, Table 2 collects adiabatic Table 2. (PTB1)4 Lowest Excited State (S1) Adiabatic and Vertical Transition Energies (eV) Using the Five Exchange− Correlation Functionals in Order of Increasing Percentage of HF Exchange and the SV-SVP Basis Set method

ΔEvertical

ΔEadiabatic

difference

PBE B3LYP BHandHLYP CAM-B3LYP LC-wPBE

1.322 1.850 2.457 2.522 2.992

1.224 1.637 2.103 2.181 2.578

0.098 0.213 0.354 0.341 0.414

Charge transfer of (PTB1)5 in the four states occurs primarily between two or three neighboring subunitsi.e., between adjacent thioneothiophene and benzodithiophene moieties (squares in the plots), as seen from the small offdiagonal widths; in particular, it occurs from thienothiophene to benzodithiophene subunits, as expected and confirmed by the NPA charges (see below). Hence, these excitons are rather tightly bound. This behavior reflects the dipolar effect (see below). Charge transfer concentrates in the center of the oligomer (S1 state), on the edges (S2 and in a less balanced way in S3), and toward one of the edges (S4). The S3, and especially the S4, state displays edge effects: they show charge transfer mostly toward the electron rich edge, benzodithiophene (lowest leftmost square). In the S1 state the exciton is quite localized whereas for the S2−S4 states the exciton stretches out practically over the whole chain that is 58 Å long in (PTB1)5 (Figure 4). Charge transfer numbers, obtained from the ΩαAB matrix elements, show that the largest CT values for the four states are about 0.040 e in the most concentrated regions

and vertical S1 transition energies of (PTB1)4 for the five functionals investigated in this work. The ADC(2) reference value can be found in Table 1 for comparison. ADC(2) shows the largest stabilization energy (0.423 eV), followed closely by LC-wPBE (0.414 eV) and farther by BHandHLYP (0.345 eV) and CAM-B3LYP (0.341 eV); B3LYP (0.213 eV) and PBE (0.098 eV) give even smaller stabilizations. Hence, B3LYP and 21823


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function analysis which also shows a concentration in the center of the chain (Figure 4). The latter effect also parallels the reduction of the different BLA values upon excitation to the S1 state as seen in the discussion of Figure 3. It is believed that local charges and local dipole values occurring in different repeat subunits play an important role in photovoltaic properties of donor polymers.15 The dipolar effect discussed in that work relies on the fact that the exciton right before it splits is polarized with opposed partial charges on thionethiophene and benzodithiophene subunits. To examine this effect and the charge character of the subunits, computed NPA S0 and S1 net charges of the moieties (subunits) are collected in Figure 6.

(Figure 4). The other CT numbers between adjacent moieties are typically in the 0.02−0.03 e range whereas edge CT values (toward the edge benzodithiophene moiety) are about 0.010 e (S1), 0.020 e (S2), and 0.030 e (S3 and S4). In comparison to the S1 state, the S2 ΩαAB plot (Figure 4) has a region of low density in the center corresponding to a nodal plane of the excitonic wave function. The S3 plot likewise displays two nodal planes. However, the S4 ΩαAB plot does not follow the behavior of the first three states because the dominant contributions in S4 are not quite symmetric at the edges. According to this analysis, the ΩαAB plots of the four states can be interpreted according to a particle-in-a-box model46 for the nodal progression of the exciton wave function in polymeric molecules. The S1 state wave function has no nodal plane whereas S2 has one and S3 has two, depicted accordingly in the corresponding ΩαAB plots. Thus, the first three excited states of (PTB1)5 belong to the same excitation band arising at infinite chain length according to the extrapolation of the S1−S3 transition energies which leads to essentially the same value (see discussion above). Conversely, since the S4 transition energy does not converge to the S1−S3 values (a 0.5 eV difference; see above), this state does not belong to the first band although its exciton wave function has a particle-in-a-box shape of higher energy. This overall picture of (PTB1)5 resembles the planar PPV8 unit oligomer: its S1−S3 states have a similar excitonic wave function progression according to ΩαAB plots, and the three energies also converge to the same value in the infinite chain length limit.27 The ΩαAB plots obtained from TDDFT calculations depend significantly on the type of exchange-correlation functional. Taking the ADC(2) as standard, the different TDDFT calculations overall behave as expected: the BHandHLYP (not shown), CAM-B3LYP (not shown), and LC-wPBE (Figure 4) functionals that accurately describe charge transfer properties have ΩαAB plots similar to ADC(2). On the other hand, B3LYP (Figure S6) and PBE plots clearly differ from the ADC(2) results. Not only the B3LYP charge transfer in (PTB1)5 is much more spread out as compared to ADC(2) because it has a larger off-diagonal width, but the overall shapes of the ΩαAB plots also differ considerably. Charge density differences between the bright S1 π−π* state and the S0 ground state computed at the ADC(2)/SV-SVP level for (PTB1)5 are shown in Figure 5. The charge depletion and charge accumulation regions are depicted separately. The electron density is mostly transferred from double bond (blue) to single bond (red) carbon−carbon regions and accumulates in the center of the chain. This agrees with the S1 exciton wave

Figure 6. NPA charges (e) for each benzodithiophene and thienothiophene subunit in (PTB1)5 computed at MP2 (S0 state) and ADC(2) (S1 state), respectively, using the SV-SVP basis set.

NPA charges of the S0 and S1 states in both types of subunits display the expected behavior of electron rich benzodithiophene and electron deficient thienothiophene. They show a polarization of the chain with small negative charges (−0.03 e) on the benozodithiophene group and corresponding positive ones on the thienothiophene part. Electron excitation to S1 increases the polarity slightly. Hence, the dipolar effect is nicely seen in both ground and excited states: for alternating adjacent subunits, opposite charges are found and their largest magnitudes are found in the inner moieties.

5. CONCLUSIONS The evolution of the electronic spectrum of the poly(thieno[3,4-b]thiophene benzodithiophene) polymer (PTB1) with chain length was investigated using the ab initio ADC(2) method, and extrapolation to infinite chain length was performed. After taking into account basis set effects, the computed transition energy agrees well with the experimentally observed band maximum of PTB1. The analysis of the Ω matrices based on the transition density shows that the excitons in the S1−S4 states are tightly bound. Both the nodal shape found in these matrices as well as extrapolated transition energies show that the S1−S3 states correspond to the same band system. Comparison of the bond length alternation (BLA) parameters in ground and first excited states show a significant structural relaxation of the PTB1 polymer in the inner part of the chain demonstrating the possibility of exciton trapping. The corresponding exciton binding amounts to about 0.4 eV based on the values for the PTB1 tetramer. The results of TDDFT calculations obtained with five different exchangecorrelation functionals with varying amounts of HF exchange were explored in comparison with ADC(2) reference data. Taking into account excitation energies, BLA and exciton binding energies, this comparison shows that a sufficient amount of Hartree−Fock exchange (>20%) is necessary for a proper description of excitation energies and exciton binding.

Figure 5. ADC(2)/SV-SVP charge difference diagram of the (PTB1)5 S0 → S1 transition. Red: electron accumulation (+0.0003e). Blue: electron depletion (−0.0003e). 21824


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Sandia National Laboratories (Contract DE-AC0494AL85000).

In terms of excitation energies, BHandHLYP and CAM-B3LYP are found to give the best agreement with the ADC(2) results. In addition to the energetic and structural characterization of the excited state properties of the PTB1 polymer, the charge distributions and their changes on excitation in the benzodithiophene and thienothiophene subunits were investigated. NPA analysis based on the MP2 density shows a negative charge of −0.03 e on the benzodithiophene moiety and a corresponding +0.03 e on the thienothiophene group in the center of the (PTB1) pentamer chain in line with the classification of electron rich and electron deficient character of these subgroups. Electron excitation to S1 enhances the polarity. This polarization agrees well with the dipolar effect proposed by Lu and Yu15 for the splitting of the exciton. The electron density difference diagram between the first excited and the ground state shows that electron density is transferred by the electronic excitation from the double bond regions to single bonds, which illustrates nicely the structural changes of increasing the double bond lengths and reducing those of the single bonds found from the analysis of the BLA. In summary, a comprehensive characterization of a number of important structural and electronic factors characterizing the properties of PTB1 has been given and support for selected classes of density functionals has been found.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b07689. Testing of freezing schemes, plots of bond distance differences between S0 and S1 geometries of (PTB1)n=4, basis set effects on the transition energies, B3LYP Omega plots and character of the states are reported (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*(I.B.) E-mail: [email protected]. *(H.L.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I.B. thanks the Brazilian Agencies CNPq, Faperj and Capes for support of this work. H.L. and I.B. acknowledge support from Capes in the framework of the Science without Borders Brazilian Program. I.B. thanks the Fulbright Commission for support of a stay at Texas Tech in 2012 where part of this work was done. E.U. thanks Capes and Programa de Pós-Graduaçaõ em Engenharia de Defesa and L.M.-C. the Programa de PósGraduaçaõ em Quı ́mica for postdoc fellowships. This material is based upon work supported by the National Science Foundation under Project No. CHE-1213263 and by the Austrian Science Fund within the framework of the Special Research Program F41. Computer time at the Vienna Scientific Cluster (Project Nos. 70019 and 70376) is gratefully acknowledged. Support was also provided by the Robert A. Welch Foundation under Grant No. D-0005 and by the Center for Integrated Nanotechnologies (Project No. C2013A0070), an Office of Science User Facility operated for the U.S. Department of Energy Office of Science by Los Alamos National Laboratory (Contract DE-AC52-06NA25396) and 21825


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