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Article Cite This: ACS Omega 2019, 4, 5758−5767
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Comparative Computational Study of Electronic Excitations of Neutral and Charged Small Oligothiophenes and Their Extrapolations Based on Simple Models Marta Kowalczyk,†,§ Ning Chen,†,‡ and Seogjoo J. Jang*,†,‡ †
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Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367, United States ‡ Ph.D. Programs in Chemistry and Physics, and Initiative for the Theoretical Sciences, Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, United States S Supporting Information *
ABSTRACT: This work reports electronic excitation energies of neutral and charged oligothiophenes (OTn) with repeat unit n = 2−6 computed by routinely used semiempirical and time-dependent density functional theory (TD-DFT) methods. More specifically, for OTn, OT+n , and OT−n , we calculated vertical transition energies for electronic absorption spectroscopy employing the Zerner’s version of intermediate neglect differential overlap method for structures optimized by the PM6 semiempirical method and the TD-DFT method with three different functionals, B3LYP, BVP86, and M06-2X, for structures optimized by the groundstate DFT method employing the same functionals. We also calculated vertical transition energies for the emission spectroscopy from the lowest singlet excited states by employing the TD-DFT method for the structures optimized for the lowest singlet excited states. In addition to computational results in vacuum, solution phase data calculated at the level of polarizable continuum model are reported and compared with available experimental data. Most of the data are fitted reasonably well by two simple model functions, one based on a Frenkel exciton theory and the other based on the model of independent electrons in a box with sinusoidal modulation of potential. Despite similar levels of fitting performance, the two models produce distinctively different asymptotic values of excitation energies. Comparison of these with available experimental and computational data suggests that the values based on the exciton model, while seemingly overestimating, are closer to true values than those based on the other model. This assessment is confirmed by additional calculations for a larger oligomer. The fitting parameters offer new means to understand the relationship between electronic excitations of OTs and their sizes and suggest the feasibility of constructing simple coarse-grained exciton-bath models applicable for aggregates of OTs. tested extensively.15 This is due to their relatively good charge hole mobility and ease of chemical modification/material processing.16 Another reason is that high polarizability of sulfur in thiophene rings stabilizes conjugation of π-bonds and promotes charge transport capability significantly. Although the efficiencies of OPV devices utilizing PTs are not among the highest ones at present, they remain highly robust systems available to date. In addition, an OPV device utilizing a related molecule, benzodithiophene polymer (PTB7), was shown to exhibit about 7.4% energy conversion efficiency.17 Recently synthesized polymers constituting organic photovoltaic devices,18,19 which can attain about 10% efficiency, are partly based on thiophene units as well. Thus, thiophene-based oligomers/polymers are expected to still serve as important active components for potentially more efficient OPV devices that may be available in the future. PTs or related CPs, which are typically employed as OPV materials, have at least hundreds of repeat units. However,
I. INTRODUCTION It is widely recognized that organic conjugated polymers (CPs) and oligomers (COs) serve as unique optoelectronic materials thanks to their rich and highly tunable optical/electronic properties.1−5 Remarkable progress has been made in the application of such properties, e.g., for organic light-emitting diodes6−9 and field-effect transistors.2,10−12 As yet, there remains ample possibility for further advances in their utilization, through more satisfactory characterization and control of the electronic properties. For example, while significant progress has been made in the utilization of CPs for organic photovoltaic (OPV) devices employing bulk heterojunction morphology,13 their efficiencies are not sufficient for large-scale commercialization yet. Major factors contributing to the electronic excitation and charge mobility of CPs/COs need to be characterized better, which are essential for developing new advanced morphologies promoting more efficient migration of excitons and charge carriers in OPV devices.14 Accurate theoretical understanding of the spectroscopy of CPs/COs is an important step in achieving this goal. Polythiophene (PT) and its derivatives are among the earliest components used for OPV devices and have been © 2019 American Chemical Society
Received: January 10, 2019 Accepted: February 25, 2019 Published: March 25, 2019 5758
DOI: 10.1021/acsomega.8b02972 ACS Omega 2019, 4, 5758−5767
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Table 1. (a) Vertical Absorption Transition Energies for OTn in eVa n 2 3 4 5 6
v s v s v s v s v s
ZINDO
B3LYP
BVP86
M06-2X
3.99(0.60) 3.35(0.65) 3.11(0.87) 3.12(0.81) 2.86(1.17) 3.01(1.01) 2.72(1.44) 2.76(1.46) 2.62(1.69) 2.54(1.49)
3.97(0.40) 3.84(0.50) 3.26(0.76) 3.10(0.90) 2.85(1.11) 2.68(1.29) 2.58(1.49) 2.43(1.67) 2.39(1.84) 2.26(2.03)
3.71(0.39) 3.59(0.50) 2.93(0.74) 2.77(0.89) 2.48(1.08) 2.31(1.27) 2.18(1.39) 2.03(1.59) 1.97(1.62) 1.82(2.03)
4.22(0.40) 4.12(0.50) 3.77(0.73) 3.64(0.86) 3.32(1.08) 3.22(1.23) 3.23(1.46) 3.10(1.62) 3.10(1.85) 2.94(2.03)
exp.61 4.11 3.49 3.18 2.98 2.87
a
Both computational data in the vacuum (v) and solution (s) phases and experimental data in the solution phase61 are shown. Oscillator strengths are shown in parentheses. The solvent of the solution phase is CHCl3, which is represented by the method of PCM in the calculation. The ZINDO results are for structures optimized by the PM6 method. TD-DFT calculations with three different functionals were conducted for structures optimized by the DFT method with the same functional. The basis of 6-31+G(d,p) was used for all of the DFT and TD-DFT calculations.
against experimental ones, and fitting of these data based on two simple models. The paper concludes with a summary and discussion of the implications of our results in Section III.
direct ab initio calculation of such large molecules with reasonable accuracy is still difficult, let alone their aggregates that ultimately need to be investigated for accurate characterization of electronic properties of OPV materials.11,15 In practice, computational studies have been conducted for oligothiophenes (OTs), from which properties of PTs can be extrapolated.20−25 The computational methods that have been employed for studying OTs range from semiempirical21,24,26−31 to ab initio methods at various levels of approximation.20,24,32−39 Earlier computational studies of excitation energies in large molecules mostly used Zerner’s version of intermediate neglect differential overlap (ZINDO) method.40,41 More recently, the time-dependent density functional theory (TD-DFT) method42 has become a popular tool due to its modest computational cost and reasonable accuracy. TD-DFT tends to overestimate the electronic correlation for long oligomers.43 However, errors can be reduced through judicious choice of functionals and methods to correct self-interaction error,24,37,43−47 although identifying a reliable scheme applicable to all major spectroscopic data remains challenging.47 More recently, many-body perturbation theory approaches such as Green’s function with GW approximation and Bethe− Salpeter equation (BSE) have emerged as new promising computational tools.38,48,49 Even at the level of simple computational methods and small OTs, a comprehensive set of data comparing their accuracy against well-established experimental results are not easy to find. Such data set can help assess the performance of computational methods and establish simple correction schemes applicable to more complex and larger systems. Motivated by this need, in this work, we report results based on a computational study of OTn and its charged species, OT+n and OT−n , with the repeat unit of n = 2−6. For these, relatively well-established experimental data are available. We here consider simple computational methods only, which are the ZINDO method40,41 and the TD-DFT method42 based on three functionals with distinctively different Hartree−Fock contributions,50 B3LYP,51,52 BVP86,53−55 and M06-2X.56 Although the scope of computational methods considered here is limited, the analysis presented is general and can be extended for more comprehensive computational studies in the future. The paper is organized as follows. Section II provides calculation results, comparative analysis of the computed data
II. COMPUTATIONAL RESULTS II.I. Absorption Spectra. II.I.I. Neutral Oligothiophenes. For each OTn (n = 2−6), we determined optimized structures of the ground electronic states based on the semiempirical PM6 method57 and the density functional theory (DFT) method.58 For DFT calculations, three functionals, B3LYP,51,52 BVP86,53−55 and M06-2X,53−55 were used. Both vacuum state and solution phase calculations were performed. For the latter, the polarizable continuum model (PCM)59 was employed to account for the solvation effect. We tried a range of different basis sets and determined that 6-31+G(d,p) serves as a reasonable choice with acceptable accuracy for all of the DFT calculations. Thus, only the data for this basis are reported here. Excitation energies were then calculated using the semiempirical ZINDO method for structures optimized by the PM6 method57 and using the TD-DFT method for structures optimized by the DFT method based on the same functionals and the basis set. All calculations were conducted by employing the Gaussian 09 package.60 For a thorough investigation of the effects of structure and methods of calculation, it is ideal to compare all different methods of excitation energy calculation for each structure determined. However, considering the resulting size of the data set expected, we here limit the scope of our study only to comparing fully semiempirical ones and with fully DFT-based methods. Cross-examination of the effects of structures will be the topic of future study when necessary. In other words, the objective of this work is not to offer comprehensive benchmarking but to compare and assess outcomes of select approaches that are typically viable for large molecules. In addition, considering the fact that the structures obtained from the PM6 method are not particularly worse than those based on DFT calculations, we believe the outcomes of calculations being presented here offer reasonable assessment of the ZINDO calculation methods, which in turn can help reassess old conclusions drawn from similar semiempirical calculations. In addition, considering the efficiency of the semiempirical methods, it is worth examining how the combination of the two best available approaches performs against the DFT/TDDFT combination. 5759
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Table 2. Parameters of Equations 1 and 2 Fitting the Data in Table 1 ZINDO v
s
E∞,1 J E∞,2 A α E∞,1 J E∞,2 A α
2.23 3.39 2.24 3.79 0.00 2.52 1.81 0.00 32.22 3.38
± ± ± ± ± ± ± ± ± ±
0.07 0.26 0.18 0.87 2.8 0.12 0.43 2.8 49.68 3.04
B3LYP 2.06 3.89 1.32 8.47 0.77 1.91 3.91 1.33 7.14 0.60
± ± ± ± ± ± ± ± ± ±
BVP86
0.05 0.18 0.01 0.06 0.01 0.03 0.12 0.03 0.23 0.04
1y n+1 i En = E∞ ,2jjj1 − zzz + A n ( n + α)2 k {
± ± ± ± ± ± ± ± ± ±
0.06 0.20 0.01 0.10 0.01 0.04 0.16 0.03 0.21 0.03
n
Detailed structural data are provided in the Supporting Information (SI). The resulting excitation energies are listed in Table 1 and are compared with solution phase experimental data. Different methods produce distinctively different trends in the dependence of excitation energies on the number of repeat units n, as has been observed before.62 The result of the TD-DFT method with the B3LYP functional is fairly close to the experimental value for bithiophene (n = 2), which is consistent with its good performance for calculating the structure in the ground electronic state.63,64 However, its accuracy degrades quickly as n increases. The BVP86 functional underestimates excitation energies even more. These data for B3LYP and BVP86 reflect well-known selfinteraction errors,65 which appear to make them even worse than those based on the PM6/ZINDO data. However, the M06-2X functional does not exhibit such tendency and demonstrates the best performance for all of the cases considered here. These results also suggest that the performance of M06-2X is comparable to that of range-separated functionals37,47 and the GW/BSE approach,38 although structures employed in these latter cases are somewhat different. Inclusion of solvation effect causes shifts of excitation energies for all of the DFT data, whereas the ZINDO calculation results do not exhibit a consistent trend. Of the three functionals tested for the DFT method, the BVP86 functional produces results with the largest red shift due to solvation and thus leads to further underestimation of excitation energies. The smallest change due to solvation can be seen for the M06-2X functional. The resulting values of excitation energies, as can be seen from Table 1, are in relatively good agreement with experimental data estimated from the position of absorption peak maxima. Various models and fitting functions are available for the extrapolation of computational data.20−25 In this work, we employ the following two model functions i i π yzyzz zzzz En = E∞ ,1 + J jjjj1 − cosjjj k n + 1 {{ k
1.61 4.28 0.77 8.87 0.83 1.44 4.36 0.73 7.74 0.68
Ĥ ex =
exp.61
M06-2X 2.84 2.84 2.35 7.06 0.63 2.69 2.94 2.11 7.57 0.72
± ± ± ± ± ± ± ± ± ±
0.06 0.22 0.40 2.74 0.44 0.06 0.20 0.31 2.20 0.34
2.57 3.09 2.26 5.74 0.40
± ± ± ± ±
0.01 0.04 0.03 0.21 0.04
n−1
∑ (E∞ ,1 + J )|k⟩⟨k|− J ∑ (|k⟩⟨k + 1| k=1
+ |k + 1⟩⟨k|)
2 k=1
(3)
where |k⟩ represents the state where the excitation is localized at each monomer unit. Equation 2 is the excitation energy68 for a system of independent electrons confined in a onedimensional box of length (n + α)lm, where lm is the length of the monomer unit, with sinusoidal potential of amplitude E∞,2. This function is based on a well-defined single-particle model,68 and the case with α = 0 was previously used for the modeling of experimental data.69 Table 2 provides all of the parameters of eqs 1 and 2 for the data of Table 1. Figure 1 compares the two functions with
Figure 1. Calculated and experimental absorption transition energies of OTn versus 1/n in solvent CHCl3. The solid lines are fitting curves based on eq 1, and the dot-dashed lines are those based on eq 2. The values of parameters are listed in Table 2.
parameters determined to fit the solution phase data. For eq 1, the quality of fitting is reasonable except for the ZINDO data. Overall, the fitting by eq 2 appears to be more satisfactory, including the case for ZINDO data, for which restriction of A and α to positive values was necessary. On the other hand, the values of E∞,2 are significantly lower than those of E∞,1. The experimental data20,70,71 for maximum absorption frequencies of PTs are in the range of 2.5−2.6 eV. In the low temperature limit,72 the vertical excitation energy has also been estimated to be 2.54 eV. This is closer to E∞,1 than E∞,2. Thus, the former, which is based on the exciton model, appears to serve as a better estimate for the true asymptotic value than the latter. We believe that it is a general trend that E∞,1 is higher and E∞,2 is lower than the true asymptotic value. The physical implication of this is that the exciton model of eq 3 underestimates the delocalization, whereas the free particle
(1)
(2)
Equation 1 represents the lowest exciton state (for J > 0) of the simple Frenkel exciton model66,67 with the following Hamiltonian 5760
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Table 3. Theoretical and Experimental (in Solution of CH2Cl2)73 Values of Vertical Absorption Transition Energies for OT+n in eVa n 2
v s
3
v s
4
v s
5
v s
6
v
s
ZINDO
B3LYP
BVP86
M06-2X
2.82(0.27) 2.02(0.24) 1.99 (0.30) 2.79 (0.20) 1.71(0.40) 2.36(0.19) 1.77(0.41) 2.53(0.14) 1.75(0.57) 2.25(0.18) 1.93(0.60) 2.49(0.10) 1.63(0.62) 2.13(0.14) 1.82(0.63) 3.33(0.11) 1.23(0.50) 1.43(0.40) 2.33(0.30) 1.52(0.30) 1.06(0.10)
3.31(0.49) 2.26(0.04) 3.14(0.58) 2.23(0.06) 2.60(0.88) 1.63(0.07) 2.44(0.94) 1.62(0.12) 2.21(0.94) 1.30(0.16) 2.03(1.32) 1.29(0.27) 1.93(1.57) 1.08(0.28) 1.80(1.58) 1.06(0.46) 1.75(1.78) 0.93(0.44)
3.29(0.47) 2.18(0.02) 3.13(0.56) 2.17(0.04) 2.56(0.86) 1.52(0.04) 2.39(0.97) 1.51(0.07) 2.17(0.97) 1.21(0.10) 1.97(1.05) 2.07(0.26) 1.85(1.38) 1.01(0.17) 1.71(1.57) 0.99(0.31) 1.65(1.66) 0.87(0.27)
3.48(0.45) 2.36(0.09) 3.33(0.51) 2.32(0.15) 2.81(0.76) 1.77(0.19) 2.69(0.75) 1.73(0.31) 2.43(1.06) 1.42(0.37) 2.33(1.08) 1.40(0.52) 2.23(1.37) 1.17(0.58) 2.15(1.27) 1.14(0.79) 2.09(1.50) 0.99(0.81)
1.65(1.75) 0.91(0.68)
1.53(1.76) 0.84(0.75)
2.05(1.41) 0.99(1.01)
exp.73
2.92
2.28
1.91
1.70
1.57
a Oscillator strengths are shown in parentheses. The results for solution phase are for PCM models of CH2Cl2. The semiempirical calculation by the ZINDO method was made for structures optimized by the PM6 method. TD-DFT calculations with three different functionals were conducted for structures optimized by the DFT method with the same functional. The basis of 6-31+G(d,p) was used for all of the DFT and TD-DFT calculations.
Table 4. Parameters of Equations 1 and 2 Fitting the Data in Table 3 ZINDO v
s
E∞,1 J E∞,2 A α E∞,1 J E∞,2 A α
0.98 3.47 0.90 3.06 0.00 1.62 0.76 0.00 22.83 3.92
± ± ± ± ± ± ± ± ± ±
0.19 0.69 0.84 3.99 1.16 0.14 0.48 15.65 315.98 29.1
B3LYP 1.43 3.84 0.70 7.73 0.80 1.31 3.71 0.79 6.07 0.58
± ± ± ± ± ± ± ± ± ±
BVP86
0.05 0.18 0.05 0.40 0.06 0.03 0.10 0.07 0.48 0.09
1.33 4.01 0.41 9.11 0.98 1.18 3.95 0.54 6.92 0.69
± ± ± ± ± ± ± ± ± ±
0.07 0.23 0.14 1.12 0.15 0.04 0.14 0.02 0.12 0.02
M06-2X 1.77 3.46 1.36 5.69 0.47 1.73 3.22 1.45 4.74 0.33
± ± ± ± ± ± ± ± ± ±
0.02 0.07 0.06 0.39 0.07 0.01 0.05 0.09 0.55 0.12
exp.73
1.26 3.37 0.81 5.42 0.54
± ± ± ± ±
0.02 0.09 0.07 0.44 0.09
data, which are listed in Table 3, and will be used for the comparison of computational data and fitting functions. We calculated excitation energies of OT+n with n = 2−6 employing the ZINDO/PM6 and TD-DFT/DFT methods in the same manner as was done for neutral molecules. The effect of solvation was also calculated employing the method of PCM for dichloromethane. The results are shown in Table 3 along with the experimental data obtained from absorption spectroscopy in dichloromethane.73 All of these data were fitted by eqs 1 and 2 as in the case of neutral molecules. The resulting parameters are listed in Table 4. Figure 2 compares these fitting curves with the solution phase data in Table 3. For the experimental data, E∞,1 is about 0.16 eV higher and E∞,2 is about 0.3 eV lower than the estimate by Salzner,35 which is 1.1 eV for the corresponding E2 states. This is consistent with our view that E∞,1 might serve as an upper bound for the true asymptotic value and is closer to this than E∞,2. The ZINDO/PM6 method produces excitation energies reasonably close to the experimental value for each molecule. However, the relative positions of other excitation energies and
model of eq 2 exaggerates its effect. Further assessment of these trends requires more extensive computational study. As can be seen from Table 2, the TD-DFT method with the M06-2X functional produces fitting parameters that are in best agreement with experimental ones. The values of E∞,1 and J differ from those of the experimental data only by about 5%. The values of E∞,2 between the two also differ by about 7%. Based on the fact that E∞,1 for the case of experimental data serves as a better estimate for the asymptotic value than E∞,1, we believe that 2.69 eV, the value of E∞,1 for the M06-2X functional, is closest to the theoretical PT limit of the lowest singlet excitation energy in the solution phase. II.I.II. Cations. There have been absorption spectra for OT+ns created through pulse radiolysis73,74 and laser flash photolysis.75 OT+n s are often the products of chemical oxidation (doping process) and are known to lead to higher electronic conductivity. It is known that35 OTn+s have two major absorption bands for n < 8, with the third one appearing for n ≥ 8. Of the two transitions for n < 8, the major bands lying in the higher-energy side have more well-established experimental 5761
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Tables 3 and 4 suggest that this is due to the cancellation of two opposing errors. For OT+2 , both functionals overestimate the major excitation energies by about 0.2 eV, most likely due to underestimation of the electronic correlation effect. On the other hand, the larger values of J for these two methods relative to the experimental one, as can be seen from Table 4, show that there is significant self-interaction error, which causes steeper decrease of the excitation energy with the length of OT than the experimental one. On the other hand, M06-2X overestimates the transition energies by about 0.4−0.5 eV for all of the cases. However, this is a systematic error relatively independent of the length of the oligomer. Indeed, the value of J fitted from the data of M06-2X is quite close to that from the experimental data even in this case, differing less than 1.5%. II.I.III. Anions. Similar calculations were performed for OT anions, and the results are provided in Table 5. Published experimental results for anions are difficult to find, and we here provide unpublished experimental data offered by the John Miller group for comparison. The overall trends of the data based on different calculation methods are similar to those of cations. These results are consistent with the interpretation based on a simple molecular orbital picture. The excitation energies for OT+n can be understood as mainly due to transitions from the singly occupied molecular orbital (SOMO) to the lowest unoccupied molecular orbital (LUMO). On the other hand, the excitation energies of OT−n can be understood as being mainly due to transitions from SOMO − 1 to SOMO. The similarity of the excitation energies for cations and anions can be understood from similar magnitudes of these gaps, as expected from the fact that SOMO and LUMO for OT+n become SOMO − 1 and SOMO for OT−n , respectively. The trends of parameters of eqs 1 and 2 fitting the data for anions are also similar to those of cations (see Figure 3 and Table 6). Both B3LYP- and BVP86-based calculation data
Figure 2. Calculated and experimental absorption transition energies of OT+n versus 1/n in solvent CHCl3. The solid lines are fitting curves based on eq 1, and the dot-dashed lines are fitting curves based on eq 2. The values of fitting parameters are shown in Table 4.
the magnitudes of the oscillator strengths in general are not consistent with the experimental data. The effects of solvation also vary significantly depending on the length of the molecule, causing a red shift for n = 2 but blue shift for larger molecules. On the other hand, the qualitative features of all of the TDDFT/DFT data are consistent with experimental results in that the major transitions have higher energies of excitation than the minor ones. In contrast to neutral molecules, the results for B3LYP and BVP86 agree fairly well with the experimental data. For further examination of the accuracy, we made additional calculations employing the extended basis set for the case of the BVP86 functional. Even when the second diffuse basis functions as well as multiple polarization basis functions were added, we did not notice significant change in the values of excitation energies. This confirmed that the 6-31+G(d,p) basis was sufficient for this case. The B3LYP and BVP86 functionals exhibit relatively good performance for these cations. However, the trends shown in
Table 5. Theoretical and Experimental Values of Transition Energies for OT−n in eVa n 2
v s
3
v s
4
v s
5
v s
6
v s
ZINDO
B3LYP
BVP86
M0-62X
2.09(0.38) 2.56(0.04) 1.87(0.31) 2.71(0.06) 1.77(0.46) 2.95(0.12) 1.87(0.37) 2.58(0.03) 1.77(0.59) 2.34(0.01) 1.87(0.51) 2.59(0.01) 1.67(0.60) 2.75(0.04) 1.87(0.53) 2.58(0.01) 1.71(0.67) 2.30(0.01) 1.85(0.50) 2.58(0.08)
2.82(0.17) 1.83(0.07) 2.72(0.40) 1.88(0.09) 2.30(0.67) 1.39(0.12) 2.21(0.75) 1.40(0.18) 2.01(1.00) 1.13(0.20) 1.91(1.06) 1.13(0.30) 1.81(1.30) 0.96(0.31) 1.72(1.31) 0.95(0.45) 1.66(1.53) 0.83(0.44) 1.59(1.51) 0.83(0.63)
2.80(0.17) 1.70(0.06) 2.67(0.34) 1.75(0.08) 2.25(0.64) 1.29(0.09) 2.16(0.75) 1.30(0.14) 1.95(0.99) 1.05(0.14) 1.84(1.09) 1.05(0.22) 1.73(1.27) 0.89(0.20) 1.63(1.39) 0.88(0.32) 1.56(1.42) 0.78(0.29) 1.47(1.57) 0.76(0.45)
3.01(0.24) 1.96(0.11) 3.00(0.36) 1.99(0.15) 2.57(0.58) 1.49(0.22) 2.52(0.63) 1.49(0.30) 2.31(0.84) 1.21(0.36) 2.25(0.85) 1.20(0.47) 2.14(1.07) 1.01(0.53) 2.10(1.05) 1.02(0.65) 2.03(1.26) 0.87(0.71) 2.01(1.21) 0.90(0.82)
exp.
2.58
2.06
1.82 1.81
1.59 1.44
a Oscillator strengths are shown in parentheses. The results for the solution phase are for PCM models of CH2Cl2. TD-DFT calculations with three different functionals were conducted for structures optimized by the DFT method with the same functional. The basis of 6-31+G(d,p) was used for all of the DFT and TD-DFT calculations.
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Table 7. Theoretical and Experimental Values of Emission Transition Energies from S1 to S0 for OTn in eVa n 2
v sa sb
3
4
5
Figure 3. Calculated and experimental absorption transition energies of OT−n versus 1/n in solvent CH2Cl2. The solid lines are fitting curves based on eq 1, and the dot-dashed lines are fitting curves based on eq 2. The values of fitting parameters are shown in Table 6.
6
reproduce the experimental results fairly well. However, as in the case of cations, this is mostly likely due to the cancellation of two opposing errors, one that leads to the overestimation of excitation energy as apparent for OT−2 and the other due to self-interaction error. As in the case of cations, the M06-2X functional results in overestimation of E∞ by 0.42 eV, but the value of J for these data is virtually the same as that for experimental results. Thus, the discrepancies of M06-2X appear to be mostly due to systematic errors independent of the size of oligomers. II.II. Emission Spectra. Calculation of emission spectra requires optimization in the excited state, for which we here consider the TD-DFT method only. For the same three functionals and the basis set, we optimized OTs for the first excited states and calculated their vertical emission energies. As expected, all of the OTs have quinoidal and planar structures in the excited electronic states. Calculation based on the BVP86 functional produces carbon bond lengths that are slightly longer than those based on the B3LYP functional (see SI), whereas lengths of bonds containing sulfur remain almost the same. Table 7 lists the values of vertical electronic emission transition energies calculated as described above in both vacuum and solution phases. Experimental data for two different solvents are also shown for comparison. As in the case of absorption spectra, both B3LYP and BVP86 functionals tend to underestimate the transition energies substantially, with the errors worsening as n becomes larger. On the other hand, the performance of the M06-2X functional is fairly satisfactory. The solution phase results for the M06-2X functional are smaller than the experimental data by about
sa sb v sa sb v sa sb v sa sb
B3LYP
BVP86
M06-2X
3.31(0.40) 3.14(0.51) 3.16(0.52) 2.72(0.80) 2.54(0.93) 2.55(0.94) 2.39(1.22) 2.11(1.38) 2.21(1.39) 2.16(1.63) 1.97(1.82) 1.99(2.23) 2.00(2.04) 1.81(2.24) 1.83(2.23)
3.23(0.36) 3.09(0.49) 3.10(0.52) 2.63(0.78) 2.46(0.92) 2.46(0.93) 2.26(1.16) 2.08(1.35) 2.09(1.36) 2.00(1.50) 1.83(1.76) 1.84(1.76) 1.80(1.79) 1.65(2.13) 1.84(2.23)
3.47(0.41) 3.31(0.49) 3.33(0.50) 2.94(0.76) 2.75(0.88) 2.77(0.89) 2.64(1.15) 2.44(1.30) 2.47(1.30) 2.46(1.54) 2.26(1.69) 2.29(1.70) 2.34(1.93) 2.16(2.07) 2.19(2.08)
exp. 3.44 3.43 2.94 2.91 2.61 2.59 2.59 2.41
2.31
a
For theoretical values, the TD-DFT method was used employing the basis of 6-31+G(d,p) for all of the calculations. Oscillator strengths are shown in parentheses. sa is for acetonitrile from ref 76 and sb is for dioxane.
0.1−0.13 eV, but their dependence on the length of OT is very close to that for experimental data as can be seen from Table 8 and Figure 4. The value of E∞,1 fitting the M06-2X data is smaller than that based on experimental results by about 0.13 eV. On the other hand, the value of J fitting the M06-2X data is different from that for the experimental results by less than 1.5%.
III. DISCUSSION For neutral OTs, we find that the performance of ZINDO/ PM6 in vacuum is reasonable and can be viewed to be even better than that of the TD-DFT method with B3LYP and BVP86 functionals. However, its solution phase results with PCM are somewhat erratic in the sense that the solvation effect causes a blue shift of the excitation energy for n = 3−5. The overall ZINDO results for OT cations and anions are also unreliable. This is not unexpected considering that parameters of ZINDO and PM6 were determined for neutral molecules and may also account for some solvation effect already through fitting to solution phase experimental data. The TD-DFT/DFT calculations employing B3LYP and BVP86 underestimate the excitation energies for neutral OTs, but their performance for OT cations and anions appears to be reasonable at least up to n = 6. However, even in these cases, a comparison of fitting parameters clearly indicates that these
Table 6. Parameters of Equations 1 and 2 Fitting the Data in Table 5 ZINDO v
s
E∞,1 J E∞,2 A α E∞,1 J E∞,2 A α
1.56 0.98 1.62 1.68 0.00 1.86 0.03 1.82 2.11 0.57
± ± ± ± ± ± ± ± ± ±
0.05 0.19 0.19 0.9 0.48 0.01 0.03 0.04 0.26 0.14
B3LYP 1.43 2.84 0.85 6.34 0.82 1.35 2.79 0.84 5.77 0.74
± ± ± ± ± ± ± ± ± ±
BVP86
0.04 0.14 0.05 0.34 0.07 0.03 0.12 0.02 0.12 0.02
1.32 3.02 0.65 7.04 0.92 1.24 2.94 0.51 7.36 1.02 5763
± ± ± ± ± ± ± ± ± ±
0.05 0.17 0.09 0.70 0.12 0.05 0.18 0.02 0.16 0.03
M06-2X 1.82 2.42 1.37 5.69 0.71 1.77 2.47 1.49 4.51 0.45
± ± ± ± ± ± ± ± ± ±
0.03 0.11 0.02 0.15 0.03 0.01 0.05 0.05 0.30 0.07
exp.73
1.35 2.46 1.19 3.35 0.25
± ± ± ± ±
0.01 0.02 0.01 0.04 0.01
DOI: 10.1021/acsomega.8b02972 ACS Omega 2019, 4, 5758−5767
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Table 8. Parameters of Equations 1 and 2 Fitting the Data in Table 7 B3LYP v
s
E∞,1 J E∞,2 A α E∞,1 J E∞,2 A α
1.73 3.21 1.10 7.13 0.79 1.55 3.27 0.98 6.54 0.71
± ± ± ± ± ± ± ± ± ±
BVP86
0.04 0.16 0.04 0.26 0.04 0.04 0.13 0.02 0.15 0.03
1.54 3.49 0.59 9.50 1.12 1.47 3.28 1.21 4.41 0.30
± ± ± ± ± ± ± ± ± ±
M06-2X
0.07 0.24 0.05 0.40 0.05 0.04 0.16 0.28 1.57 0.36
2.09 2.80 1.68 5.77 0.56 1.92 2.85 1.61 4.98 0.43
± ± ± ± ± ± ± ± ± ±
0.02 0.09 0.02 0.16 0.03 0.01 0.05 0.04 0.27 0.06
exp.73
2.05 2.81 1.65 5.67 0.55
± ± ± ± ±
0.03 0.09 0.09 0.60 0.12
We also determined new fitting parameters including the data for OT10. The resulting values are listed in Table 9. The new parameters for eq 1 changed slightly from those in Tables 2 and 8 and became even closer to experimental values. On the other hand, the new values for eq 2 are quite different from those in Tables 2 and 8 and do not appear to have converged well enough. The good performance of eq 1 in fitting both experimental and computational data also confirms the Jaggregate-like character of intrachain couplings within a single oligothiophene.5 Utilization of this model for aggregates so as to explain H-aggregate-like behavior for intermolecular couplings5 is an interesting possibility, for which further test of the exciton model for the length dependence of oscillator strengths is needed.
Figure 4. Calculated and experimental emission transition energies of OTn versus 1/n in dioxane. The solid lines are fitting curves based on eq 1, and the dot-dashed lines are fitting curves based on eq 2. The values of fitting parameters are shown in Table 6.
IV. CONCLUSIONS In this paper, we presented vertical transition energies for electronic absorption spectroscopy of OTn, OT+n , and OT−n and for electronic emission spectroscopy of OTn with n = 2−6. The methods of calculation employed here were the semiempirical ZINDO/PM6 method and the TD-DFT/DFT method with three different functionals, B3LYP, BVP86, and M06-2X. Comparison of these with experimental results demonstrates the best performance of the TD-DFT method with the M062X functional for neutral molecules. Although this was expected, the level of agreement with experimental results as confirmed here is certainly notable. For cations and anions, the results of the TD-DFT/DFT method with B3LYP and BVP86 appear to be closer to experimental results. However, the distance dependences of results for the M06-2X functional are still in best agreement with experimental trends. Although they are based on simple assumptions, the fitting results of both the computational and experimental data by the two model functions, eqs 1 and 2, have significant implications. First, they show that such fitting can serve as an effective means for comparing the performance of different computational methods. Second, as discussed in the last section, the analysis of the fitting parameters shows that the exciton model is more reliable and offers better estimates for the asymptotic values of the transition energies. This suggests the feasibility of
overestimate the lowering of excitation energies with respect to the length of OT. Overall, the performance of M06-2X is most satisfactory, producing correct trends with the increase of oligomer length, although excitation energies for OT+n and OT−n are overestimated systematically by about 0.4−0.5 eV. However, whether this is purely due to computational error or the effect of counter charge effect needs to be understood better. What is certain at this point is that, in all of the cases, the M06-2X functional serves as a reliable method for evaluating the coupling constant J, defined within the exciton model of eq 1. This is consistent with other results where the M06-2X functional is shown to provide reasonable estimate of electronic couplings while overestimating excitation energies.77 To test further the reliability of the model parameters, we conducted additional calculation of absorption and emission energies based on the M06-2X functional for OTn with n = 10. The resulting value of absorption energy in CHCl3 was found to be 2.75 eV and that of emission energy in dioxane was 2.05 eV. These values are fairly close to those based on eq 1, with parameters in Tables 2 and 8, which are 2.81 and 2.05 eV. On the other hand, the values based on eq 2 are 2.62 and 2.01 eV. This confirms that the latter model is less reliable than the former.
Table 9. Parameters of Equations 1 and 2 for the Absorption and Emission Data for the Neutral OTn Including Those for OT10 Calculated by the DFT/TD-DFT Method with the M06-2X Functional
absorption emission
E∞,1
J
E∞,2
A
α
2.67 ± 0.04 1.92 ± 0.01
3.01 ± 0.16 2.83 ± 0.04
2.36 ± 0.13 1.77 ± 0.06
6.02 ± 0.94 4.10 ± 0.36
0.47 ± 0.18 0.24 ± 0.09
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constructing coarse-grained exciton-bath models that can be useful for the modeling of aggregates of OTs and PTs. There are two important issues to be addressed in the future as the next step of this work. First, for the development of complete exciton-bath models and also for a more accurate comparison of computational and experimental data, it is important to account for Huang−Rhys factors of all major vibrational modes, possible Duschinsky effects, differences of vibrational zero point energies between the excited and ground electronic states, and thermal fluctuations. Some of these data are already available,5,15 but more extensive calculations and simulations are needed to obtain comprehensive enough data for all of neutral and charged OTs and PTs. Second, for OT+n and OT−n , the effects of counterions need to be understood better although their effects were assessed to be relatively minor for small oligomers.35,36 For the case of polyaniline, strong interactions between ions and the backbone of molecules result in significant modification of UV−vis spectra. Thus, inclusion of the counterion effect may still turn out to be necessary for more accurate calculations of spectroscopic data for OT+n and OT−n as well. Future effort will be devoted to addressing these issues.
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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/acsomega.8b02972. Detailed structural data for some of the computational results including schematics of OT6 and labeling convention, dihedral angles for the optimized structures in the ground electronic state calculated by PM6 and DFT calculations with the 6-31+G(d,p) basis set, and bond distances between rings for the S1 state of OTn (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Seogjoo J. Jang: 0000-0002-9975-5504 Present Address §
Natural Sciences Department, LaGuardia Community College, City University of New York, 31-10 Thomson Avenue, Long Island City, New York 11101, United States (M.K.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was mainly supported by the Office of Basic Energy Sciences, Department of Energy (DE-SC0001393), and in part by the National Science Foundation (CHE-1362926) during the early stage. S.J.J. thanks John Miller for providing experimental data for excitation energies of anions. The authors acknowledge computational support by the CUNY High Performance Computing Center, Queens College Center for Computational Infrastructure for the Sciences, and the Center for Functional Nanomaterials at Brookhaven National Laboratory. 5765
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DOI: 10.1021/acsomega.8b02972 ACS Omega 2019, 4, 5758−5767
