Conjugated Polymers: Evaluating DFT Methods for More Accurate

May 14, 2013 - Materials Research Express 2018 5 (1), 015314 .... Tuning range-separated DFT functionals for accurate orbital energy modeling of conju...
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Conjugated Polymers: Evaluating DFT Methods for More Accurate Orbital Energy Modeling Theresa M. McCormick, Colin R. Bridges, Elisa I. Carrera, Paul M. DiCarmine, Gregory L. Gibson, Jon Hollinger, Lisa M. Kozycz, and Dwight S. Seferos* Lash Miller Chemical Laboratories, Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S 3H6 S Supporting Information *

ABSTRACT: Density functional theory (DFT) calculations are useful to model orbital energies of conjugated polymers, yet discrepancy between theory and experiment exist. Here we evaluate a series of relatively straightforward calculation methods using the standard Gaussian 09 software package. Five calculations were performed on 22 different conjugated polymer model compounds at the B3LYP and CAM-B3LYP levels of theory and results compared with experiment. Chain length saturation occurs at approximately 6 and 4 repeat units for homo- and donor−acceptor type conjugated polymers, respectively. The frontier orbital energies are better approximated using B3LYP than CAM-B3LYP, and the HOMO energy can be reasonably correlated with experiment [mean signed error (MSE) = 0.22 eV]. The LUMO energies, however are poorly correlated (MSE = 0.59 eV), and we show that the molecular orbital energy of the triplet state gives a much better estimate of the experimentally determined LUMO level (MSE = −0.13 eV).

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INTRODUCTION The frontier energy levels of conjugated polymers are critical to their performance in organic electronic devices such as organic solar cells and electrochemical supercapacitors.1−3 In organic solar cells the magnitude of the HOMO−LUMO energy gap directly affects the short circuit current, the HOMO energy of the electron donor relative to the LUMO of the electron acceptor is proportional to the open-circuit voltage, and the offset between the LUMO of the donor and acceptor enables (or prohibits) charge separation at the donor/acceptor interface leading to a photocurrent.4−6 In polymer-based electrochemical supercapacitors the HOMO and LUMO levels play a significant role in device voltage as well.7,8 Recently, density functional theory (DFT) calculations have become useful for predicting molecular geometries, energy levels, and absorption spectra of conjugated organic molecules.9−14 Performing DFT calculations on conjugated polymers is challenging due to their size (large number of atoms); however, they can be approximated using oligomeric model compounds.15−17 Although HOMO energies from B3LYP calculations have been accepted as good predictors for oxidation potentials, the calculated LUMO energy, or first virtual orbital, consistently gives values that are less negative (higher lying) than those experimentally determined, in some cases by ∼1.0 eV.18−21 The lack of an electron in this orbital is thought to bring about the discrepancy between calculated and experimental LUMO energy values.22 The most common way to account for this is to apply a correction factor to the predicted energies, however one must determine correction factors experimentally to ensure that they are consistent across many classes of compounds.9,10 More accurate methods to predict the frontier © XXXX American Chemical Society

orbital energies of conjugated polymers are therefore of great value. Herein, we present a series of computational methods to estimate the HOMO, LUMO, and HOMO−LUMO gap energies for 22 different conjugated polymer model compounds. Five computations were conducted on each model compound. The orbital energies of the HOMO and LUMO were predicted from the geometry optimization (computation A, Figure 1a). The ionization energy was predicted from the

Figure 1. Representation of electronic distribution in (a) computation A, the neutral ground state, (b) computation B, singly oxidized doublet, (c) computation C, singly reduced doublet, (d) computation D, neutral triplet state, and (e) computation E, excitation energy added to HOMO energy. Received: March 8, 2013 Revised: April 24, 2013

A

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R groups shown in Scheme 1. We would not expect significant changes in the electrochemical oxidation and reduction potentials as a function of R group, as previously reported.23 The experimental procedure for determining these values can be found in the references provided. We used six and four repeat unit oligomeric models for homopolymers and donor−acceptor copolymers, respectively. Only small changes in the HOMO and LUMO energy occur by adding more repeat units, thus we conclude that these models accurately represent the saturation length of the polymers (Figure S1, Supporting Information). This result is also consistent with several experimental studies in which oligomers of known composition were prepared to determine the point at which optical and electrochemical properties saturate as a function of chain-length.15,24 The total length of the oligomers vary significantly depending on the number of rings in the repeat unit, however to keep the computational method consistent we do not correct for this, and have not found evidence of systematic error that is attributable to chain length (see Table 4). Some systems are correlated well with smaller oligomers and dimers with individually optimized condition.25,26 The use of larger oligomers more accurately models polymers, and can be computed within a time period we that we feel is acceptable. We chose not to use the periodic boundary condition (PBC) calculations because the TD-DFT calculations cannot be conducted using this method (for more detailed discussion the reader is referred to the literature).26 Basis Set and Level of Theory. The B3LYP level of theory is often used for calculating the geometry and orbital energy levels of monomeric or oligomeric model compounds to approximate conjugated polymers.11 Recent reports have suggested using the long-range corrected Coulomb attenuated method of B3LYP (CAM-B3LYP) to better model charge transfer states.16,27−32 In this paper, we compare both the B3LYP33,34 and CAM-B3LYP35 levels of theory with 6311G(d) basis set,36 a relatively large yet computationally viable basis set. Computation. Alkyl side chains on the polymers were replaced with methyl groups to reduce computational time and geometries were optimized to a local minimum. If several conformations were possible (such as structures with different rotational angles between connected units) the optimized geometries of all possible conformations were calculated and the lowest energy geometry was used. Five separate computations were performed on each model compound for each level of theory (for representative input sections see Supporting Information). In computation A, the geometry was optimized at the neutral singlet ground state and all remaining computations were performed using this geometry. Three subsequent single point energy calculations were performed on the oxidized (+1) doublet (computation B), the reduced (−1) doublet (computation C), and the neutral triplet state (computation D). Finally, a TD-DFT computation was performed to determine the three lowest singlet excitation energies from the neutral singlet state (computation E).37 Two different methods were evaluated for the HOMO: (1) the energy of the HOMO from the optimized geometry (computation A); and (2) the difference between the total energy of computation A (optimization) and the total energy of computation B (oxidized) (ionization energy). Six different methods were evaluated for the LUMO: (1) the energy of the first virtual orbital from computation A; (2) the

difference between total energy of the singly oxidized doublet (computation B, Figure 1b) and the total energy from computation A. The electron affinity was predicted from the energy difference between singly reduced (−1) and neutral compound (computation C, Figure 1c). The energy of the αHOMO (details of this orbital are provided below) was also determined from computation C. The energy of the triplet state was determined (computation D, Figure 1d). Finally, the lowest excitation energy was calculated from time-dependent DFT (TD-DFT) (computation E, Figure 1e). Each of these computations is then compared with the experimental values to determine which best predicts frontier orbital energies.

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RESULTS AND DISCUSSION Compounds of Interest. We chose to study both homopolymers as well as alternating donor−acceptor copolymers where the HOMO (Table 1) energy was determined by

Table 1. Experimental HOMO (Cyclic Voltammetry, As Indicated in the References), Calculated HOMO Energy (Neutral Singlet), and Calculated Ionization Energy for All 22 Studied Compoundsa compound 141 241 311 442 543 641 744 845 946 1041 1147 1248 b 49 13 b 50 14 159 1623 b 51 17 b 52 18 b 53 19 2054 219 2254 MSEc MAEd

HOMO experimental (eV) −5.16 −5.05 −5.61 −5.07 −4.55 −5.10 −4.90 −4.76 −5.50 −4.88 −4.62 −5.43 −5.60 −5.60 −5.45 −5.50 −5.17 −5.14 −5.43 −5.25 −5.53 −5.16

HOMO neutral singlet B3LYP (CAM-B3LYP) (eV) −5.19 −5.14 −4.81 −5.04 −4.12 −5.14 −5.06 −4.78 −5.33 −5.18 −5.07 −5.46 −5.03 −4.88 −5.04 −5.08 −4.74 −4.68 −4.77 −5.00 −5.14 −4.98 0.22 0.32

(−6.44) (−6.42) (−6.02) (−6.35) (−5.35) (−6.42) (−6.29) (−6.08) (−6.79) (−6.39) (−6.36) (−6.66) (−6.20) (−6.16) (−6.32) (−6.36) (−5.95) (−5.93) (−6.70) (−6.21) (−6.42) (−6.18) (−1.07) (1.07)

ionization energy B3LYP (CAMB3LYP) (eV) −5.83 −5.80 −5.38 −5.81 −5.06 −5.76 −5.68 −5.43 −6.28 −5.79 −6.35 −6.09 −5.93 −5.69 −5.53 −5.57 −5.38 −5.31 −5.94 −5.54 −5.63 −5.51 −0.49 0.51

(−6.62) (−6.69) (−6.19) (−6.54) (−5.50) (−6.59) (−6.44) (−6.22) (−6.97) (−6.54) (−6.66) (−6.84) (−6.37) (−6.33) (−6.48) (−6.53) (−6.05) (−6.02) (−6.91) (−6.36) (−6.59) (−6.31) (−1.24) (1.24)

a

Calculations were conducted using B3LYP and CAM-B3LYP with the 6-311G(d) basis set. bDetermined by film electrochemistry. cThe mean signed error determined by average difference of the calculated energies from experiment. dThe mean absolute error determined by average absolute value of the difference of the calculated energies from experiment.

cyclic voltammetry (CV) and the LUMO (Table 2) energy was either determined by CV or estimated from the HOMO energy plus the optical band gap (Table 3), as determined by the onset of absorption (Scheme 1). Some variation in experimental oxidation potentials and optical HOMO−LUMO gaps are expected. The experimental values we report are, when possible, from the seminal report of these polymers, with the B

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Table 2. Experimental (Cyclic Voltammetry, or Optical Band Bap Added to HOMO) and Calculated LUMO Energies (B3LYP and CAM-B3LYP) with 6-311G(d) Basis Set for All 22 Studied Compounds compound 141 241 311 442 543 641 744 845 946 1041 1147 1248 a 49 13 a 50 14 159 1623 a 51 17 a 52 18 a 53 19 2054 219 2254 MSEb MAEc

LUMO expt (eV) −2.67 −2.69 −2.83 −2.90 −2.95 −3.19 −3.20 −3.26 −3.30 −3.33 −3.34 −3.40 −3.50 −3.60 −3.60 −3.60 −3.68 −3.71 −3.76 −3.77 −3.80 −3.84

LUMO neutral B3YLP (CAM-B3LYP) (eV) −2.58 −2.57 −1.92 −2.39 −1.64 −3.17 −2.91 −3.11 −2.52 −3.26 −2.26 −3.10 −2.41 −2.67 −2.86 −2.92 −3.12 −3.16 −2.26 −3.45 −3.12 −3.52 0.59 0.59

(−1.48) (−2.03) (−0.82) (−1.23) (−0.55) (−2.03) (−1.79) (−1.88) (−1.25) (−2.21) (−1.18) (−2.03) (−1.32) (−1.50) (−1.74) (−1.79) (−2.05) (−2.07) (−0.74) (−2.36) (2.02) (−2.46) (1.70) (1.70)

EA B3YLP (CAMB3LYP) (eV) −1.93 −1.88 −1.36 −1.62 −0.71 −2.53 −2.28 −2.47 −1.57 −2.62 −0.89 −2.45 −1.53 −1.87 −2.36 −2.43 −2.48 −2.53 −1.28 −2.90 −2.61 −2.98 1.30 1.30

(−1.32) (−1.84) (−0.65) (−1.04) (−0.38) (−1.84) (−1.65) (−1.74) (−1.06) (−2.03) (−0.76) (−1.88) (−1.12) (−1.30) (−1.57) (−1.62) (−1.90) (−1.91) (−0.50) (−2.20) (−1.85) (−2.32) (1.88) (1.88)

ΔE triplet B3YLP (CAM-B3LYP) (eV) −1.98 −1.92 −2.28 −1.92 −1.62 −1.42 −1.59 −1.16 −1.92 −1.38 −1.83 −1.75 −1.82 −1.49 −1.51 −1.50 −1.10 −1.00 −1.77 −0.99 −1.46 −0.99 1.77 1.77

(−2.59) (−2.00) (−2.77) (−2.57) (−2.12) (−2.00) (−2.44) (−1.89) (−2.68) (−1.78) (−1.63) (−2.31) (−2.32) (−1.99) (−1.92) (−1.90) (−1.63) (−1.55) (−2.95) (−1.30) (−1.86) (−1.22) (1.30) (1.30)

αHOMO triplet B3YLP (CAM-B3LYP) (eV) −3.31 −3.31 −2.66 −3.19 −2.52 −3.79 −3.53 −3.69 −3.44 −3.87 −3.22 −3.83 −3.25 −3.43 −3.62 −3.66 −3.67 −3.70 −3.21 −4.17 −3.76 −4.03 −0.13 0.29

(−4.23) (−4.91) (−3.78) (−4.16) (−3.35) (−4.91) (−4.12) (−4.56) (−4.38) (−5.04) (−4.69) (−4.58) (−4.04) (−4.34) (−4.76) (−4.79) (−4.63) (−4.71) (−3.91) (−5.14) (−4.94) (−5.19) (−1.15) (1.15)

HOMO + excitation B3YLP (CAM-B3LYP) (eV) −2.89 −2.90 −2.28 −2.70 −1.86 −3.48 −3.21 −3.33 −2.87 −3.56 −2.67 −3.43 −2.69 −2.93 −3.18 −3.24 −3.37 −3.41 −2.38 −3.70 −3.33 −3.76 0.31 0.40

(−3.36) (−3.81) (−2.79) (−3.25) (−2.53) (−3.81) (−3.64) (−3.70) (−3.49) (−3.99) (−3.31) (−3.92) (−3.26) (−3.44) (−3.66) (−3.71) (−3.82) (−3.85) (−3.11) (−4.10) (−3.86) (−4.17) (−0.21) (0.35)

a The sum of the optical HOMO−LUMO and oxidation potential determined from electrochemistry, as reported in the references. bThe mean signed error determined by average difference of the calculated energies from experiment. cThe mean absolute error determined by average absolute value of the difference of the calculated energies from experiment.

the oxidation energies are again better approximated using B3LYP than CAM-B3LYP (the MSE is −0.49 and −1.24 eV, respectively). Consistent with this, the ionization energy determined by UPS (ultraviolet photoelectron spectroscopy) are lower than the oxidation potential determined by CV.38 No data points were removed from the analysis. The difference between the calculated and experimental HOMO levels (computation A) for compounds 3 and 14 lie outside two standard deviations (0.34 eV) of the MSE; however, these outliers are structurally different. We compared the MSE for structural subgroups of polymers (donor−acceptor polymers, homopolymers, polymers containing Se, polymers containing benzothiadiazole and polymers containing benzodithiophene) to the MSE for the entire set of polymers. Some of the methods worked slightly better for some types of polymers, but not consistently or significantly (Table 4). We were surprised to find that the CAM-B3LYP does not give more accurate results for the donor−acceptor polymers (i.e., those that possess internal charge-transfer). The calculations can be applied to a large variety of polymer types and give similar results. The LUMO Energy. The first unoccupied or virtual orbital from computation A (ground state optimization) consistently gives values that are less negative (more higher lying) than experiment (Table 2). Similar to the HOMO energy calculations, B3LYP gives better estimations than CAMB3LYP (MSE = 0.59 and 1.70 eV, respectively), however the correlation is still poor (Figure 3). To achieve more accurate calculated values we populate this orbital through a one electron reduction, giving the compounds an overall negative charge (computation C). Since the

difference between the total energy from computation A and computation C (reduced) (the electron affinity); (3) the energy of the αHOMO (see the discussion below) from computation C; (4) the difference in the total energy of computation A and computation D (triplet); (5) the energy of the αHOMO (see the discussion below) from computation D; and (6) the energy of the HOMO from computation A plus the energy of the lowest electronic transition with significant oscillator strength (greater than 0.1; the orbitals involved in this transition are HOMO and LUMO) from computation E. Two different methods were evaluated for the HOMO− LUMO energy gap: (1) the difference between the HOMO and LUMO energies from the optimized neutral singlet calculation; and (2) the energy of the lowest singlet excited state transition from the TD-DFT calculation. All values are reported in eV relative to the vacuum, and compared to experimental data (Tables 1, 2, 3). The HOMO Energy. We found that the HOMO energies from computation A for B3LYP and CAM-B3LYP are significantly different (Figure 2). B3LYP consistently gives better correlation to experimental energies than CAM-B3LYP [the mean signed error (MSE) is 0.22 and −1.07 eV, respectively; Table 1]. HOMO energies calculated for oligomeric model compounds (4−6 repeat units, ca. 20−40 Å in length) with the B3LYP level of theory and 6-311G(d) basis set most closely match polymer experimental oxidation potentials. The calculated ionization energy (the difference between the total energy of computation A and computation B) does not correlate well with experiment oxidation potentials. However, C

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D

0.20 (−1.07) −0.47 (−1.23) 0.44 (2.86) 0.18 (0.98) 0.54 (1.68) 1.21 (1.85) -0.16 (−1.15) 0.27 (−0.19) 0.23 (−1.06) −0.48 (−1.21) 0.49 (2.92) 0.21 (1.02) 0.58 (1.73) 1.30 (1.91) -0.13 (−1.11) 0.31 (−0.18) 0.23 (−1.06) −0.46 (−1.23) 0.47 (2.85) 0.19 (0.94) 0.55 (1.64) 1.22 (1.82) -0.17 (−1.16) 0.27 (−0.26) 0.22 (−1.07) −0.49 (−1.23) 0.50 (2.90) 0.21 (0.98) 0.59 (1.70) 1.30 (1.88) -0.13 (−1.15) 0.31 (−0.21) LUMO

GAP

neutral singlet ionization energy HOMO−LUMO TD-DFT neutral singlet EA αHOMO triplet HOMO + excitation HOMO

all compounds B3LYP (CAM-B3LYP) eV MSE

Table 4. MSE of Polymer Types, with the Best Methods in Bold

negatively charged compound has one unpaired electron the calculation must be unrestricted. In an unrestricted calculation the spin up and spin down orbitals are denoted α and β, respectively. When there are unequal numbers of electrons in the α and β orbitals these levels are no longer degenerate and the highest occupied orbital is the αHOMO which in the reduced molecule, is the same MO as the LUMO in computation A. The αHOMO energies from computation C are poorly correlated with the LUMO energies (MSE = 1.99 eV for B3LYP and 2.06 eV for CAM-B3LYP; Table S1, Supporting Information). We can also compare the change in energy between computation C and computation A, which is equivalent to the electron affinity. For both B3LYP and CAM-B3LYP, the calculated electron affinity poorly predicts the LUMO energy (MSE = 1.30 eV for B3LYP and 1.88 eV for CAM-B3LYP) (Table 2). For negatively charged computations, diffuse functions are usually added to the basis set, however we chose not to employ them to keep the calculations feasible and consistent across all methods. To avoid having a charged species, we populate what would be the neutral singlet LUMO with a triplet state (computation D). Again, we consider the change in total energy of the system (computation D versus computation A) and the energy of the αHOMO. For both B3LYP and CAM-B3LYP, the change in total energy of the system results in poorer correlation with experiment than computation A. The energy of the αHOMO of

donor−acceptorsa

homob

a

a Compounds: 6−8, 10−12, 14−18, 20−22. bCompounds: 1, 5, 9, 13, 19. cCompounds: 10, 14, 18, 19, 22. dCompounds: 6, 15, 16, 17, 20, 21, 22. eCompounds: 1, 2, 6, 7, 10, 12. fCompounds: 5, 9, 13, 19. gCompounds: 15, 16, 20, 21, 22.

Determined by film electrochemistry. bMeasured by the onset of absorption. cThe mean signed error determined by average difference of the calculated energies from experiment. dThe mean absolute error determined by average absolute value of the difference of the calculated energies from experiment.

0.21 (−1.03) −0.58 (−1.20) 0.51 (2.78) 0.21 (0.89) 0.66 (1.69) 1.47 (1.89) -0.06 (−1.10) 0.36 (−0.20)

(3.08) (2.61) (3.23) (3.10) (2.81) (2.61) (2.65) (2.39) (3.30) (2.40) (3.05) (2.74) (2.94) (2.72) (2.66) (2.65) (2.13) (2.08) (3.59) (2.11) (2.56) (2.01) (0.98) (0.98)

0.30 (−0.91) −0.30 (−1.07) 0.45 (2.76) 0.17 (0.90) 0.53 (1.62) 1.15 (1.78) -0.17 (−1.16) 0.25 (−0.24)

2.30 2.24 2.53 2.34 2.26 1.66 1.86 1.45 2.46 1.61 2.40 2.04 2.34 1.95 1.86 1.84 1.37 1.27 2.39 1.29 1.80 1.22 0.21 0.26

0.24 (−1.13) −0.41 (−1.30) 0.43 (2.98) 0.17 (1.03) 0.66 (1.84) 1.28 (2.01) -0.06 (−1.05) 0.40 (−0.11)

(4.96) (4.39) (5.20) (5.12) (4.79) (4.39) (4.50) (4.20) (5.54) (4.19) (5.24) (4.63) (4.88) (4.66) (4.57) (4.57) (3.90) (3.86) (5.96) (3.85) (4.40) (3.72) (2.90) (2.90)

small (1 ring)f

2.61 2.57 2.89 2.65 2.48 1.96 2.15 1.68 2.81 1.92 2.82 2.36 2.63 2.21 2.18 2.16 1.63 1.52 2.51 1.55 2.02 1.47 0.50 0.50

benzodithiophenee

2.13 2.06 2.30 2.14 1.60 1.70 1.62 1.30 1.80 1.52 1.20 1.82 2.10 1.98 1.88 1.90 1.49 1.43 1.67 1.20 1.75 1.04

TD-DFT B3YLP (CAM-B3LYP) (eV)

thia-diazoled

141 241 311 442 a 43 5 641 744 845 946 1041 1147 1248 b 49 13 b 50 14 159 1623 b 51 17 b 52 18 b 53 19 2054 219 2254 MSEb MAEc

HOMO−LUMO gap B3YLP (CAM-B3LYP) (eV)

Sec

compound

experimental optical gap (eV)

large (5 rings)g

Table 3. Experimental (Electrochemical or Optical As Indicated) and Calculated (B3LYP and CAM-B3LYP with 6311G(d) Basis Set) HOMO−LUMO Energy Gaps for all 22 Studied Compounds

0.37 (−0.98) -0.32 (−1.14) 0.54 (3.06) 0.28 (1.12) 0.67 (1.84) 1.32 (2.01) -0.06 (−1.00) 0.41 (−0.10)

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Scheme 1. Structure of Polymers and Modeled Oligomers

Figure 2. HOMO energy from the optimized geometry of the neutral singlet model compounds; comparing B3LYP to CAM-B3LYP level of theory and experiment.

The final method is carried out by adding the lowest singlet excitation energy from the TD-DFT calculations (computation E) to the energy of the HOMO from computation A. In this case, values obtained from the CAM-B3LYP calculations better match experiment (MSE = 0.21 eV, Figure 4) than those from B3LYP (MSE = 0.31 eV), and both methods give more accurate values than those obtained from computation A (Figure 4). Caution should be taken when considering this method; although CAM-B3LYP appears to give slightly better results than B3LYP, this is caused by a combination of

the triplet state from B3LYP (computation D) gives the best correlation with experiment (MSE = −0.13 eV) (Figure 4). The magnitude of this MSE is comparable to the magnitude of MSE obtained for the best HOMO energy approximations, discussed above, and once again the CAM-B3LYP level of theory gives less accurate results (MSE = −1.15 eV). No correlation between the type of polymer and the error is observed. The αHOMO of the triplet state from B3LYP is well-correlated with experiment for a wide variety of polymer types (Table 4). E

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Figure 3. LUMO energy from the optimized geometry of the neutral singlet model compounds; comparing B3LYP to CAM-B3LYP level of theory and experiment.

Figure 4. LUMO energy level calculated from αHOMO of the triplet state from B3LYP, the LUMO energy level calculated by adding the first excitation energy to the HOMO energy of the optimized neutral singlet at the CAM-B3LYP level of theory and LUMO energies from experiment.

Figure 5. HOMO−LUMO energy gap (the first excitation energy) from the TD-DFT calculation; comparing B3LYP to CAM-B3LYP level of theory and experiment.

underestimating the HOMO energy and overestimating the

with the calculated HOMO−LUMO energy gap from computation A, and then with the first excitation energy from computation E. The HOMO−LUMO energy gap from computation A overestimates the optical gap due to the

excitation energy thus canceling out more significant error. The HOMO−LUMO Energy Gap. We further compare the experimentally reported optical HOMO−LUMO energy gap F

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ACKNOWLEDGMENTS This work was supported by the University of Toronto, NSERC, the CFI, and the Ontario Research Fund. D.S.S. is grateful to the Connaught Foundation (Innovation Award), MaRS Innovation (Proof-of-Principle Grant), and DuPont (Young Professor Grant).

predicted high lying LUMO level. This error is larger in the CAM-B3LYP calculations (Figure 5). The energy gap is better modeled using the lowest excitation energy from computation E and B3LYP (MSE = 0.21 eV) (Table 3).

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CONCLUSIONS The frontier energy levels of conjugated polymers are important parameters for optimizing materials for organic electronic applications. Computational methods are becoming more common for understanding and predicting properties of conjugated polymers. Before carrying out a multistep synthesis, it is beneficial to have an estimate of the optoelectronic properties of the target as a preliminary screening measure. We have learned that DFT predicted chain length effects on the frontier orbital energies saturate relatively quickly, at approximately 6 and 4 repeat units for homo- and donor−acceptor type conjugated polymers, respectively. Thus, very large models (on the order of the polymer chain length) are not needed at the levels of theory reported here. B3LYP is more accurate than CAM-B3LYP, and accurately predicts the HOMO energy; however the LUMO energy is predicted much more high-lying than the measured value. The LUMO energy is most accurately approximated from the αHOMO energy of the triplet state. The HOMO−LUMO energy gap is best predicted by the first excitation energy from TD-DFT calculations. Good approximation of the LUMO energy of conjugated homopolymers and donor−acceptor copolymers is achieved by performing one additional single point calculation after the geometry optimization. The HOMO−LUMO gap can be predicted by performing the TD-DFT calculation.

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REFERENCES

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The geometries of all the model compounds were optimized with the nonlocal hybrid Becke three-parameter Lee−Yang−Parr (B3LYP) functional33 and Handy and co-workers’ long-range corrected version of B3LYP using the Coulomb-attenuating method as implemented in Gaussian 09 (CAM-B3LYP)35 with the 6-311g(d) basis set. Calculations were performed with Gaussian 0939 using Gaussview40 to generate structures. Single point calculations from this geometry at both levels of theory were conducted for the oxidized doublet, reduced doublet, and neutral singlet with the same basis set. The first three singlet excited-states were calculated with TD-DFT37 using the optimized geometries using both B3LYP and CAM-B3LYP. The energies of the orbitals were taken from the Gaussian output file (“Population analysis using the SCF density”, HOMO is the last “Alpha occ. eigenvalues”, LUMO is the first “Alpha virt. Eigenvalues”). The total energies were taken from energy given from the last “SCF Done”: line in the output file. All values were converted to eV from the reported Hartree energies.

S Supporting Information *

Saturation length plot, energy of the αHOMO level from the reduced doublet calculation, representative syntax, optimized coordinates, and complete citation for reference 21. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

* E-mail: (D.S.S.) [email protected]. Notes

The authors declare no competing financial interest. G

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