Quantitative Estimation of Exciton Binding Energy of Polythiophene

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Quantitative Estimation of Exciton Binding Energy of Polythiophene-Derived Polymers Using Polarizable Continuum Model (PCM)-Tuned Range-Separated Density Functional Haitao Sun, Zhubin Hu, Cheng Zhong, Shian Zhang, and Zhen-Rong Sun J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b01975 • Publication Date (Web): 24 Mar 2016 Downloaded from http://pubs.acs.org on March 29, 2016

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The Journal of Physical Chemistry

Quantitative Estimation of Exciton Binding Energy of Polythiophene-Derived Polymers Using Polarizable Continuum Model (PCM)-Tuned RangeSeparated Density Functional

Haitao Sun†,*, Zhubin Hu†, Cheng Zhong‡, Shian Zhang†, and Zhenrong Sun†,*

†

State Key Laboratory of Precision Spectroscopy and Department of Physics East China Normal University (ECNU) Shanghai 200062, P. R. China ‡

Department of Chemistry Wuhan University

Hubei 430072, P. R. China

*Corresponding authors: [email protected]; [email protected]

ACS Paragon Plus Environment


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ABSTRACT. Exciton binding energies of six polythiophene-derived polymers were studied through using a non-empirical, optimally-tuned range-separated (RS) functional combining with the polarizable continuum model (PCM). We demonstrate that this approach predicts ionization energies (IE), electron affinities (EA), transport gaps, optical gaps and exciton binding energies of six different polymer chains in both vacuum and solid (dielectric medium) with accuracy comparable to many-body perturbation theory within the GW approximation and Bethe-Salpeter equation (BSE). Furthermore, the behavior of exciton binding energy versus dielectric constant was also reasonably described by the PCM-tuned RS functional while the conventional functionals such as PBE, B3LYP, M062X, and non-tuned LC-ωPBE completely fail. We believe our method provides for a reliable and computationally efficient tool for future investigation of efficiency-enhancing mechanism and molecular design in organic solar cells.

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1. Introduction π-Conjugated polymers currently have attracted considerable attention due to their wide applications in novel organic-based electronic or opto-electronic devices, such as organic solar cells (OSCs) and organic lighting-emitting diodes (OLEDs).1-11 It is well-known that the solution-processed bulk heterojunction (BHJ) blends comprising polythiophene-derived polymer donor and fullerene derivative acceptor are among the highest-efficiency OSCs.12, 13 In general, after light-irradiation, the electron donor layer would absorb photons in the range of its absorption band to generate Frenkel excitons (or coulombically bound electron-hole pairs). The excitons that subsequently diffuse to the BHJ donor/acceptor interface separate into mobile free electrons and holes.14 However, it is not guaranteed to produce free charge carriers directly but the electron-hole pair located on the donor layer is expected to firstly overcome the exciton binding energy (Eb).15 Recently, reducing the Eb of the donor layer has been proved to be a useful approach to simultaneously achieve efficient exciton separation efficiency, increase the open-circuit voltage, and finally realize high-efficiency organic photovoltaics.16,

17

It is also

worth pointing out that the magnitude of Eb in organic conducting polymers such as poly(phenylenevinylene) (PPV) has been intensively investigated in the past and still remains controversial.18 Therefore, a quantitative prediction of Eb is critical to understand the efficiencyenhancing mechanisms and to design novel materials before actual synthesis. Definition of the Eb is typically given as the difference between the transport gap (Eg) and the optical gap (Eo): Eb = Eg – Eo. The former Eg, also known as the fundamental gap, is defined as the minimum energy of formation of a pair of separated free electron and hole. The Eg is calculated as the difference between the ionization energy (IE) and electron affinity (EA), which

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are experimentally determined via ultraviolet photoelectron spectroscopy (UPS) and inverse photoemission spectroscopy (IPES), respectively.19 The later Eo, defined as the vertical excitation energy from the ground state to the first dipole-allowed excited state, corresponds to the formation of a Frenkel exciton with the hole and electron bound each other in organic systems.20 However, the direct measurements of Eb are not straightforward except in very pure crystals. The transport gap does not consistently follow the optical measurements due to the differences in samples and experimental techniques. Furthermore, the determinations using peak onset or peak maximum values, polarization effect, molecular relaxation, electronic correlation and electron-photo coupling bring forward the questions on the comparability for those measurements.21, 22 From a theoretical view, the investigations of Eb of conjugated polymers have been of course important but also challenging. A reliable description of Eb requires quantitative predictions of Eg and Eo simultaneously. Various approaches from semi-empirical model23, 24 to the “state-of-the-art” many-body perturbation theory within the GW approximation and BetheSalpeter equation (BSE)25-33 have been employed to study the behavior of Eb. However, the semi-empirical methods usually give only qualitative understanding although they can be easily explored to large-size systems. The GW-BSE approach from first-principles has been shown to provide quantitative predictions for the quasi-particle and optical spectrum of organic and inorganic systems. However, the issues of expensive computational demanding and significant starting-point dependence greatly limit its application as a routine task.34-40 Density functional theory (DFT) and time-dependent density functional theory (TDDFT) are widely employed to determine the ground- and excited-state electronic structures of molecules and solids due to the well balance between the computational cost and the accuracy of 4


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results. However, recent studies have showed that the most frequently applied functionals such as conventional (semi-)local exchange-correlation (XC) functionals (i.e. PBE41 and B3LYP42, 43) fail in predicting the band gaps and especially the charge-transfer (CT) excitation energy in organic donor-acceptor systems.44-48 Furthermore, the “fortuitous” agreement between the results obtained by these conventional functionals and the experimental data has to be reexamined. For example, Schwenn et al. and Phillips et al. found that the B3LYP-calcualted energies of highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of a series of gas-phase organic semiconductor molecules agree quite well with the solid-state IE and EA experimental measurements of corresponding molecular crystals or thin films.49, 50 Zade and Bendikov calculated the HOMO-LUMO energy gap of organic conjugated polymers using B3LYP functional and found excellent agreement with the optical gap measured from UV-vis spectrum.51, 52 Here, we may attribute the fortuitous accuracy of the orbital eigenvalues to the cancellation of the errors because the former completely neglects the polarizable solid state environment and the later ignores the existence of Eb. Furthermore, the failure of conventional XC functionals may be attributed to the lack of derivative discontinuity (DD),53-55 large self-interaction error (SIE)44, 56, 57 or delocalization error (DE)45,58 and incorrect asymptotic behavior,59-62 which make the HOMO and LUMO do not correspond to the IE and EA, respectively. In solid state such as molecular crystal and thin film, the polarization effect obviously exists and can be regarded as a phenomenon of nonlocal correlation, but such nonlocal correlation cannot be effectively captured by semilocal XC functionals.63 Very recently, our group developed a new method64 which combines the concept of “optimal tuning”47, 48 based on the range-separated (RS) functionals59-62 and the polarizable continuum model (PCM).65 (see the Computational Methods for details) The optimal tuning

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method has been demonstrated to provide reliable descriptions of both ground-state properties44, 66-69

and excited-state properties51,

70-72

of organic molecular systems. More importantly, the

novelty of this PCM-tuned method comes from the fact that it connects the optimal tuning concept with the description of solid-state screening effects via applying a magnitude-equivalent dielectric constant (ε) that can potentially benefit the calculation of general solid systems. In this work, we investigate six thiophene-based polymers, namely polythiophene (PT), poly(dithiophene ethyne) (PDTE), poly(dithiophene vinylene) (PDTV), poly(dithiophene dicyanovinylene)

(PDTDCNV),

poly(dithiophene

difluorovinylene)

(PDTDFV),

and

poly(dithiophene dichlorovinylene) (PDTDCV) (see Figure 1). We performed DFT and TDDFT calculations on the transport gap, optical gap and the resulting Eb of gas-phase isolated polymer chain (see Figure 2) using the previously-used optimally-tuned RS functional and those of solid thin-films in dielectric medium using our recently-developed PCM-tuned approach. The frontier orbital energies (HOMO and LUMO) computed from such tuning methods can be adjusted to represent IE and EA, respectively, in a direct manner through applying an asymptotically correct XC potential. The calculated results are further compared to the G0W0@PBE(-BSE) benchmark data from the work of Samsonidze et al.33 In addition, computations using a series of conventional functionals are also performed for comparison. The behavior of Eb as a function of dielectric constant ε was also quantitatively studied herein.

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Figure 1. (a) Molecular structures of polythiophene (PT), poly(dithiophene ethyne) (PDTE), poly(dithiophene

vinylene)

(PDTV),

poly(dithiophene

dicyanovinylene)

(PDTDCNV),

poly(dithiophene difluorovinylene) (PDTDFV), poly(dithiophene dichlorovinylene) (PDTDCV). (b) Polymer models used in this work with a long chain limit, i.e. PT20, PDTE10, PDTV8, PDTDCNV8, PDTDFV8, and PDTDCV8. These numbers indicate the repeat units. The grey, white, blue, cyan, green, and yellow atoms represent carbon, hydrogen, nitrogen, fluorine, chlorine, and sulfur atoms, respectively.

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Figure 2. Schematic energy diagram of quasi-hole (electron) from gas isolated chain to solid thin film; IE: ionization energy; EA: electron affinity; -εH: negative HOMO energy; -εL: negative LUMO energy; Eg: transport gap; Eo: optical excitation gap; P+/P- : polarization energy for hole/electron; and Eb: exciton binding energy.

2. Computational Methods The optimal tuning method is typically performed based on the RS functionals. In these functionals, the interelectronic repulsion is separated into short- and long-range domain, which is given for the interelectronic distance r12 by the standard error function erf(x) shown as: erfc ω erf ω , where erfc(x) = 1 – erf(x). The exchange is further split into long-range HF-exact-exchange component and short-range DFT component of the exchange. The range-separation parameter ω is the inverse of a distance at which the functional switches from DFT-like to HF-like. Because the range-separation parameter ω is considered to be a functional of the density,54, 73 it has been shown to be strongly system-dependent.51, 74 Therefore, in this work the ω value for the molecule in the gas-phase is not necessarily the same as the one

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in the solid state since the electron density will be influenced by polarization effects. Baer, Kronik, and co-workers proposed a non-empirical, optimal tuning procedure which is based on the IP (ionization potential)-theorem, which enforces the negative HOMO energy (−εH) in exact Kohn-Sham DFT equals to the vertical IE.48 Then the range-separation parameter ω is determined non-empirically through minimizing | | for the neutral system. Later, a refined target functional as shown below was proposed particularly for “better” description of HOMO-LUMO gap or transport gap47:

[ ] (1) The above equation simultaneously applies the IP-criterion for both neutral (N) and anion (N + 1) systems. However, although the optimal tuning approach has been shown to provide reliable prediction for finite-sized molecules, the approach as given here is inapplicable to solids or periodic systems such as thin film and molecular crystals. The reason is that to add or remove an electron explicitly is not possible due to periodic boundary conditions.54, 64 Hence, to circumvent such an issue and to benefit from the optimal tuning concept for the description of polymer thin film in this work, we introduce the PCM-tuned approach that performs the above tuning procedure by adding a magnitude-equivalent dielectric constant (ε) of the thin film within the polarizable continuum model.65 To allow for a direct comparison with the G0W0@PBE(-BSE) calculations from work of Samsonidze et al,33 we employed the consistent static dielectric constant ε = 1.7 used in the PCM to simulate the solid thin-film environment. It has been proved that non-self-consistent G0W0 calculations based on a (semi-)local DFT starting point are sufficiently reliable to reproduce the experimental solid-state determinations, although the calculations on transport gap and optical

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gap of gas-phase molecules are typically underestimated to different extent.29, 32, 33, 75, 76 The static Coulomb-hole-plus-screened-exchange (COHSEX) approximation was included to account for the contributions to the GW electron self-energies from dielectric medium.33 All the geometries of polymers were optimized using the B3LYP functional and 6-31G(d) basis set with C2h symmetry as shown in Figure 1. The optimization of the range-separation parameter ω, as described in our previous work,71, 77 was performed based on the LC-ωPBE61, 78 functional and cc-pVDZ basis set. The default PCM model using the integral equation formalism variant (IEFPCM) was employed throughout this work to simulate the environmental solid polarization effects by adding the “scrf(pcm, read)” keyword and defining the magnitude of the dielectric constant. The IEs, EAs, Eg and Eo values were then calculated in both gas-phase and solid using the optimally tuned LC-ωPBE functional (referred as LC-ωPBE*) and cc-pVDZ basis set. We also examine the performance of four conventional density functionals with increasing amount of HF%: PBE(0%HF)41, B3LYP(20%HF)42,

43

, M062X(56%HF)79, and

default LC-ωPBE(0~100%). The default settings for PCM parameters were employed except where explicitly stated otherwise. All calculations were performed using the Gaussian 09 software.80 Table 1. Calculated IE(–εH), EA(–εL), Eg, Eo and Eb of isolated polymer chains in the gas phase at the optimally-tuned LC-ωPBE*/cc-pVDZ level. The GW(-BSE) data are listed for comparison. Gas Isolated chain PT20 PDTE10 PDTV8 PDTDCNV8 PDTDFV8 PDTDCV8 MAE

ε 1.0 1.0 1.0 1.0 1.0 1.0

PCM-tuned LC-ωPBE*/cc-pVDZ ω IE/EA Eg Eo Eb 0.123 5.61/1.80 3.81 1.88 1.93 0.126 5.76/1.98 3.77 1.89 1.88 0.111 5.51/2.22 3.29 1.63 1.66 0.100 6.17/3.33 2.84 1.32 1.52 0.119 5.63/2.07 3.57 1.76 1.81 0.117 5.80/2.23 3.57 1.78 1.79 0.23/0.43 0.66 0.31 0.35

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G0W0(@PBE)+BSE IE/EA Eg Eo 5.35/2.25 3.10 1.48 5.52/2.39 3.13 1.79 5.14/2.52 2.62 1.23 6.04/3.86 2.18 1.00 5.48/2.54 2.94 1.45 5.58/2.64 2.94 1.46

Eb 1.62 1.34 1.39 1.18 1.49 1.47

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Over/Undera +/– + + + a The symbols “+” and “–” mean the calculated values are overestimated and underestimated, respectively, compared to the G0W0(@PBE)-BSE data. Table 2. Calculated IE(–εH), EA(–εL), Eg, Eo and Eb of solid thin films of polymers at the optimally-tuned LC-ωPBE*/cc-pVDZ level. The GW(-BSE) data are listed for comparison. Solid Thin film ε PT20 1.7 PDTE10 1.7 PDTV8 1.7 PDTDCNV8 1.7 PDTDFV8 1.7 PDTDCV8 1.7 MAE Over/Underb a The contributions to

PCM-tuned LC-ωPBE*/cc-pVDZ G0W0(@PBE+COHSEXa)+BSE ω IE/EA Eg Eo Eb IE/EA Eg Eo Eb 0.050 4.99/2.60 2.39 1.46 0.97 4.93/2.72 2.21 1.39 0.82 0.047 5.00/2.67 2.33 1.43 0.90 5.12/2.85 2.27 1.59 0.68 0.047 4.78/2.63 2.15 1.25 0.90 4.76/2.96 1.80 1.11 0.69 0.043 5.56/3.77 1.79 0.99 0.80 5.69/4.26 1.43 0.84 0.59 0.049 4.96/2.67 2.29 1.36 0.94 5.08/2.99 2.09 1.33 0.76 0.048 5.13/2.83 2.29 1.38 0.92 5.18/3.09 2.08 1.34 0.74 0.08/0.28 0.23 0.10 0.19 +/– + + + the electron self-energy from dielectric medium were computed in the

static Coulomb-hole-plus-screened-exchange (COHSEX) approximation. bThe symbols “+” and “–” mean the calculated values are overestimated and underestimated, respectively, compared to the G0W0(@PBE)-BSE data.

Results and Discussion This section is organized as follows: we first test the rationality of polymer models studied in this work. Optimal tuning based on the gas-phase isolated polymer chain and also the chain in the dielectric medium is performed next, followed by calculations of IE, EA, transport gap Eg, optical gap Eo and exciton binding energy Eb. The results of optimally-tuned RS functional and various conventional functionals are further analyzed by comparison to the GW-(BSE) benchmark data. The behavior of Eb as a function of dielectric constant ε is finally demonstrated.

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Typically, the term band gap is used to represent a property of the finite system that converges to the infinite band-structure limit as the oligomer size increases.51 Conwell suggested that the exciton binding energy Eb for a conducting polymer can only be defined unambiguously in the long chain limit.18 To obtain a reasonable polymer model of long chain limit, we tested the behavior of convergence of HOMO-LUMO gaps of various polymers as shown in Figure S1. The limiting HOMO-LUMO gap is found to be converged and independent of the conjugation length of polymer chain for PT20, PDTE10, PDTV8, PDTDCNV8, PDTDFV8 and PDTDCV8, which were then used to represent the polymer models. In addition, although the converged behavior of a much longer oligomer model or a periodic polymer chain can be guaranteed, this will not benefit the SCF convergence during tuning process because the tuning approach requires a large number of single-point calculations of neutral, cation and anion systems and a periodic boundary condition (PBC) calculation is impossible for charged systems. Table 1 and Table 2 collects the optimally-tuned ω values derived for the LC-ωPBE functional for the 6 long polymer chains in both gas phase and solid. The simulation of the polymer thin-film employs the PCM model with the respective dielectric constant of ε = 1.7 during the tuning process. Compared to the default ω = 0.400 Bohr-1 for LC-ωPBE, the optimal ω values significantly reduce to roughly 0.10-0.13 Bohr-1 for gas-phase isolated chain. For the case of solid thin film, the ω values further decrease to roughly 0.04-0.05 Bohr-1. Körzdörfer et al. found that there is an inverse relationship between the tuned ω value and the spatial extension of the conjugation or delocalization.74 The smaller ω values indicate that the electron density in the solid environment is more delocalized than that for the isolated polymer chain in the gas phase. As we introduced previously, the range-separation parameter corresponds to an inverse distance where the exchange will switch from DFT-like to HF-like. Hence, a decreased ω value in LC-

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ωPBE functional indicates that the short-range DFT-PBE exchange will be replaced by longrange HF exchange at larger distance. We then take PT20 as an example. The tuned ω value for gas-phase isolated PT20 is 0.123 bohr-1 that can represent the saturation on the extrapolation to infinite chain51 and the character of functional switches from DFT-PBE exchange to HF exchange at the distance r12 = (1/ω) = 5.4 atomic unit. For the thin film of PT20, the tuned ω value is 0.050 bohr-1 and the resulting switching distance is much larger at roughly r12 = 20.0 atomic unit. In other words, there is less HF exchange and more DFT-PBE exchange in the short range exchange interactions, indicating that the reasonable description of solid thin film requires the functionals to include less “localized” HF exchange and more “delocalized” DFT-PBE exchange. After establishing the tuning procedure, we calculated the IEs, EAs, and Eg values from gasphase isolated chain to solid environment as shown in Table 1 and Table 2. The results are compared to those calculated from G0W0@PBE and BSE calculations of the work of Samsonidze et al.33 The numbers computed by a series of conventional functionals were also presented in Table S1. The mean absolute errors (MAEs) for various functionals are also graphically represented in Figure 3. For isolated chain in gas-phase, the LC-ωPBE* functional overestimate the ionization energies with the MAE of 0.23 eV and underestimate electron affinities with the MAE of 0.43 eV compared to the GW IEs and EAs. The resulting MAE of the transport gap Eg is collected to be 0.66 eV. It is interesting to recall the fact that Sharifzadeh et al. found the nonself-consistent G0W0 calculations based on a PBE starting point underestimate the transport gap Eg of pentacene in the gas phase by as much as 0.7 eV compared to the experimental data.29 Our results are in line with their finding and confirm the reliable calculations of Eg from the LCωPBE* functional. The calculated optical gap Eo predicted by the LC-ωPBE* functional agree

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reasonably well with the G0W0@PBE-BSE results, with MAEs of 0.31 eV. For solid thin films, all the MAEs are significantly reduced as expected. For example, the LC-ωPBE* functional yields ionization energies that are in excellent agreement with the GW IEs (MAD= 0.08 eV). The MAEs for EAs, Eg, and Eo are calculated to be 0.28 eV, 0.23 eV, and 0.10 eV, respectively. For comparison, the IEs, EAs, Eg, and Eo are also calculated by using PBE, B3LYP, M062X and default LC-ωPBE functionals. All the numbers are listed in Table S1 and the corresponding MAEs are also graphically shown in Figure S2-S3. Not surprisingly, the above results are as expected that the optimally-tuned LC-ωPBE* functional overall produces the smallest MAEs for IEs, EAs, Eg and Eo, which agree well with our previous findings.46, 51, 64, 68, 71, 77

Figure 3. Calculated MAEs of exciton binding energy Eb for both gas isolated chain and solid thin film using various functionals compared to G0W0@PBE-BSE benchmark data. Negative values mean the numbers waer underestimated by the functional. Here, we focus on the prediction of Eb and Figure 3 displays the MAEs of calculated Eb using various functionals in this work compared to the GW(-BSE) benchmark data. It can be

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seen that the PBE (0%HF) and B3LYP (20%HF) functionals possessing low amount of HF% tend to underestimate the Eb for both gas isolated chain and solid thin film. Conversely, the M062X (56%HF) and default LC-ωPBE functionals with relatively larger HF% overestimate the corresponding Eb. The small MAE of 0.08 eV for the gas isolated chain by M062X functional may be attributed to an error cancellation resulting from the simultaneous overestimation of transport gap and optical gap (as seen in Figure S2). Overall, the above results indicate that the (PCM-)tuned LC-ωPBE* approach yields reliable IEs, EAs, transport gaps, optical gap and also exciton binding energy for isolate polymer chain in both gas and solid. By the assistance of PCM-tuned method, we first discuss the relationship of the calculated Eb as a function of dielectric constant ε as shown in Figure 4. We now evaluate the Eb as a function of ε from 1.5 to 15.5. The calculated Eb decrease from 1.93 eV in gas-phase to 0.51 eV when ε = 3.0. van der Horst et al. performed the GW@LDA-BSE calculation and found the exciton binding energy is drastically reduced from 1.86 eV for a polythiophene chain in vacuum to 0.76 eV for a chain in the bulk by including interchain contributions into the screened Coulomb interaction in polythiophene.75,

76

Our finding is in excellent agreement with these

GW(-BSE) data and also very close to the experimental Eb of 0.55 eV determined by Sakurai et al.81 Furthermore, the decreased exciton binding energies indicate that the solid environment possessing a larger polarizability can result an easier separation of hole-electron pairs. It is worth pointing out that none of the conventional functionals including PBE, B3LYP, M062X, and LCωPBE studied in this work can correctly describe the behavior of the decrease of Eb from gasphase to solid in dielectric medium, which was quantitatively predicted by GW(-BSE) method. Furthermore, compared to the experimental Eb of 0.55 eV for the polythiophene thin film, the PBE functional completely fails to predict due to the calculated negative Eb values as shown in

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Figure 4. The non-tuned LC-ωPBE functionals produce significantly large Eb values. The hybrid M062X functional with 56%HF produces reasonable Eb value but only for the isolated chain approaching a vacuum case. Interestingly, the hybrid B3LYP with 20%HF yields the Eb close to the solid experimental value, however, the solid polarization effect causing the phenomenon of decreasing Eb is still not captured.

Figure 4. Calculated exciton binding energy Eb (eV) of PT20 thin film, as a function of ε using various functionals. The G0W0@LDA(-BSE) numbers from the work of van der Horst et al.75,76 (circles in green) and experimental data from the work of Sakurai et al.81 (short dash line in grey) are also represented in the figure.

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Figure 5. Calculated exciton binding energy Eb (eV) of PT20 thin film as a function of 1/ε. Straight dash lines are a linear fit. To further explore the physics behind the Eb and test if our PCM-tuned approach indeed captures the nature of the behavior of Eb versus ε, we consider a classic model of exciton binding energy associated with the Coulombic potential.16, 17, 82 In this model, the Eb is given by

/4# $ , where e is the electron charge, ε0 is the permittivity of vacuum, εr is the permittivity

of the materials and r is the distance between the hole and electron. For the same type of polymer in different dielectric medium, the hole-electron distance r is fixed and the only variable is εr. If the Eb is reasonably described, the calculated Eb should be linear in 1/εr for any εr. Figure 5 shows the Eb values of PT20 computed as a function of 1/εr. It can be seen that the relation is indeed linear, indicating our approach successfully captures behavior of Eb versus ε and further the quantitative description of Eb is not a coincidence.

3. Conclusion and Outlook

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In conclusion, we have demonstrated the efficiency of a nonempirically tuned rangeseparated exchange functional in reproducing the exciton binding energies Eb of 6 polythiophene-based polymers in both vacuum and thin films with an accuracy comparable to GW(-BSE) calculations. To simulate the solid environment, we combine the optimal tuning concept for RS functionals with a magnitude-equivalent dielectric constant ε in the polarizable continuum model. With respect to the G0W0@PBE(-BSE) benchmarks, the optimally tuned LCωPBE* functional leads to very good theoretical estimates with MAEs of 0.35 eV for the Eb in vacuum and of 0.19 eV in solid thin films. For comparison, the MAE values of Eb were also collected using other conventional functionals. Generally, the functionals with low amount of HF% such as PBE and B3LYP underestimate the Eb values, and those with relatively high HF% inversely overestimate the Eb. Therefore, a suitable amount of HF% included in the functional is important to achieve reasonable description for the exciton binding energy. We attribute the success of our tuning method to a favorable balance between localization and delocalization effects included in the functionals. The Eb for polythiophene significantly decreased from 1.91 eV (in the gas isolated chain) to 0.51 eV (in the thin film of polythiophene with the ε of 3.0), indicating that the solid polarization effect in bulk systems are critically important and should not be ignored. In addition, through combining the merits of optimal tuning and PCM, we reproduce the correct behavior of exciton binding energy as a function of dielectric constant, which cannot be captured by those conventional functionals. Considering its comparable accuracy to GW, high computational efficiency and simple application, the PCM-tuned RS approach are expected to further study the efficiency-improving mechanism in organic solar cells and used as an efficient tool for the molecular design of novel organic solid materials.

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ASSOCIATED CONTENT Supporting Information. Behavior of convergence of HOMO-LUMO gap versus number of unit (N) for various polymer chains calculated at the B3LYP/6-31G(d) level. Calculated MAEs for the ionization energy, electron affinity, transport gap and optical gap for both gas isolated chain and solid thin film using various functionals compared to G0W0@PBE(-BSE) data. Detailed information for the calculations of IE(–εH), EA(–εL), Eg, Eo and Eb of various polymers in both gas- and solid- states using PBE, B3LYP, M062X and default LC-ωPBE functionals. This information is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Authors* E-mails: [email protected]; [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS This work has been partly supported by National Natural Science Fund (No. 11474096, 11004060,

11027403

and

51203121)

and

China

Postdoctoral

Science

Foundation

(2014M561435). H.S. sincerely appreciates Professor Jean-Luc Brédas for the stimulating discussions.

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Quantitative Estimation of Exciton Binding Energy of Polythiophene

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