Solar Energy Materials & Solar Cells 97 (2012) 139–149

Contents lists available at SciVerse ScienceDirect

Solar Energy Materials & Solar Cells

0927-02

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/solmat

Comparison between theoretical and experimental electronic propertiesof some popular donor polymers for bulk-heterojunction solar cells

Luca Longo, Chiara Carbonera, Andrea Pellegrino, Nicola Perin, Giuliana Schimperna,Alessandra Tacca, Riccardo Po n

Research Center for Non-Conventional Energies Istituto Eni Donegani, ENI S.p.A, Via Fauser 4, 28100 Novara, Italy

a r t i c l e i n f o

Available online 4 October 2011

Keywords:

Polymer solar cell

Electron donor polymer

Properties prediction

Frontier orbital energy

Energy gap

Density functional theory

48/$ - see front matter & 2011 Elsevier B.V. A

016/j.solmat.2011.09.035

esponding author. Tel.: þ39 0321 447541; fa

ail address: riccardo.po@eni.com (R. Po).

a b s t r a c t

The theoretical estimation of energy levels and energy gaps of conjugated polymers for polymer solar

cells represents in principle an useful tool for an a priori screening of new donor systems. Fourteen

polymers, whose energy gaps vary in the range 1.2–3.1 eV have been selected and their HOMO, LUMO

and gap energies have been calculated by applying Density Functional Theory methods. In spite of the

variety of the examined molecular structures, nice correlations between theoretical and experimental

electronic parameters were found. In particular, optical gaps and, to a lesser extent, electrochemical gap

very well correlate with theoretical gaps, while for the other parameters (oxidation and reduction

potentials) the general trend is reproduced. It is shown that, in general, the theoretical energies of the

base repeating units have values close to the experimental energies, but the linear fittings are better

when the theoretical data of much longer chains are considered: infinitely long chains must be used to

predict the optical gaps, while long oligomers are more appropriate to estimate the electrochemical

properties. Criteria and relationships for the prediction of energy data from theoretical ones are

provided.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Bulk heterojunction (BHJ) polymer solar cells based on con-jugated polymer donors and functionalized fullerene acceptorsare a new and expanding research field. Many materials havebeen studied during the last ten years to reach the best perfor-mances [1–6]. To date, the most efficient polymer solar cellsexhibit a power conversion efficiency of 8.3% [7].

Energy level (HOMO, Highest Occupied Molecular Orbital, andLUMO, Lowest Unoccupied Molecular Orbital) engineering is apromising strategy to improve the performance of these devicesthrough the design of tailored structures able to optimize the lightharvesting and the photoinduced electron transfer process from thedonor to the acceptor. There are several approaches and design rulesthat can drive the chemist in the synthesis of effective donorsystems [8–14]. Most reliable and applied criterions are (i) to pushthe polymer HOMO level toward lower energy in order to maximizethe difference with the LUMO level of the acceptor, and (ii) tomaintain the energy gap of the donor in the range 1.6–1.9 eV. Inturn, several empirical and semi-empirical rules exist to translatethe above mentioned criterions into molecular structures. Indeed,

ll rights reserved.

x: þ39 0321 447241.

there are other important factors affecting the device overallefficiency and performance, i.e. the charge carrier mobility, thecharge extraction at the electrodes [15], the photochemical stability[16], but it is by far more difficult to define rules of thumb for theseaspects. A couple of recently published papers [2,6] made an effortto establish some empirical correlations between several experi-mental parameters (HOMO, LUMO, energy gap, open circuit voltage,charge carrier mobility, and power conversion efficiency) of themost efficient and popular donor polymers used in bulk-hetero-junction polymer solar cells.

Going back to HOMO and LUMO energies and energy gaps inconjugated polymers, these values can be estimated by quantummechanical calculations based on Density Functional Theory(DFT) [17–20]. Many papers on novel photoactive materials fororganic solar cells report the use of DFT calculations to validateand rationalize the experimental observation [21–28]. However,these studies are generally focussed on families including only avery small number of related structures, and there are nopublished studies reporting a comprehensive comparison oftheoretical and experimental data. On the other hand a fewpapers have been published [29,30] on quantum mechanicalcalculations of the energy levels of the frontier orbitals of full-erenes derivatives, in particular the LUMO [29].

In this work we have selected both from the literature and amongthe products synthesized in our labs a number of representative

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149140

conjugated polymer structures for which we collected/measured theelectronic properties, derived from cyclic voltammetry and UV-visiblespectroscopy data. In parallel, we calculated the frontier orbitals andgap energies for the corresponding model molecules. Finally, thetheoretical/experimental data sets (HOMO, LUMO, optical gaps, andelectrochemical gaps) have been correlated to assess the reliability ofquantum mechanical calculations for the prediction of energy levelsin real systems.

2. Experimental part

Poly[2,7-(9,9-bis(2-ethylhexyl)fluorene)-alt-4,7-(benzo-2,1-3-thia-diazole)] (P1), poly[2,7-(9,9-bis(2-ethylhexyl)fluorene)-alt-4,40-tri-phenylamine)] (P2), poly[2,7-(9,9-bis(2-ethylhexyl)fluorene)-alt-5,5-(40,70-di-2-thienyl-20-(2-ethylhexyl)benzo-10,20,30-triazole)] (P5), andpoly[2,7-(9,9-bis(2-ethylhexyl)fluorene)-alt-5,5-(40,70-di-2-thienyl-benzo-20,10,30-thiadiazole)] (P7) have been synthesized by Suzukipolycondensation following a previously reported procedure [31]starting from boronic acid and dibromide monomers. Poly(3-hex-ylthiophene) (P3) was purchased from Plextronics Inc. (grade Plex-core OS2100).

UV–vis absorption spectra were recorded at room temperaturewith a Lambda 950 spectrophotometer (Perkin-Elmer Inc. Waltham,MA, USA). Thin films (�100 nm) of pristine polymers were preparedby spin-coating from chlorobenzene solution on quartz Suprasilsubstrates.

The optical energy gaps were evaluated by the edge corre-sponding to the intersection between the negative tangent line inthe inflection point of lowest energy absorption band and thetangent line to linear portion of the absorption tail.

Electrochemical measurements were performed with an AutolabPGSTAT30 potentiostat/galvanostat (EcoChemie, The Netherlands)run by a PC with GPES software, in a one-compartment three-electrode cell in argon-purged acetonitrile with 0.1 M Bu4NBF4 as

P1 2P

P4 P7

P5 P8

P6 P9P6 P9

P10

Scheme 1. Structures of the repeating units of the studied poly

supporting electrolyte. A CHCl3 solution 1 mg/ml of the compoundwas coated on the Glassy Carbon working electrode (Amel Electro-chemistry, Milano, Italy) having a surface of 0.071 cm2. A Platinumcounterelectrode and an aqueous saturated calomel (SCE) referenceelectrode were used. The film formed on the electrode was analyzedat a scan rate of 200 mV/s. The data have been referred to the Fcþ/Fcredox couple (ferricenium/ferrocene), according to IUPAC [32]. Themolecular models were obtained by Density Functional Theoryoptimization [33] and the chosen software was Amsterdam DensityFunctional (ADF) from SCM Inc. [34,35]. The basic functions adoptedfor all those models are TZP Slater Type Orbitals (Triple Zeta – moreaccurately a Double Zeta for the core and Triple Zeta the upperelectronic shells – with the addition of a further additional polariza-tion function) [36–39]. In order to reduce the computationalcomplexity of the SCF calculations, the core functions ‘‘Small Core’’were implemented (the atomic internal levels below the valenceshell are kept ‘‘frozen’’ while they are independently orthogonalizedwith respect to the above levels). The DFT calculations wereperformed with the Vosko–Wilk–Nusair Local Density Approxima-tion (LDA) [40] combined with two Generalized Gradient Approx-imations (GGA): the method in which the BP-86 exchangefunctional proposed by Becke [41,42] is combined with the correla-tion functional developed by Perdew [43] and the hybrid methodB3-LYP [44]. The latter method is an extension of the threeparameters method by Becke using the correlation functionaldeveloped by Lee et al. [44] and ensures a better accuracy with DFT.

3. Results and discussion

The structure of polymers P1–P14 selected for this study[45–63] are reported in Scheme 1. Most of them represent someof the most effective donors currently used in polymer solar cells[2,6,64]; a few materials, e.g. P2, have been chosen to expand therange of examined energies, although they are not really useful in

3P

P11

P12

P13P13

P14

mers. For each of them, R groups are specified in Table 1.

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149 141

affording efficient devices. Thus, on the whole, materials with gapenergies ranging from 1.2 to 3.1 eV have been considered(Table 1).

A few remarks are necessary to place this study in the ‘‘right’’context. As we are conscious of the fact that in general great careshould be taken in comparing measurements coming fromdifferent laboratories [17,65], we established some criteria onwhich we based our choice for the selection of: (a) the polymerstructures to be analyzed among the multiple variants reported inthe literature and (b) the proper values of experimental data to beused in the fittings. By the way, in previous papers in which thecorrelation between device performance and HOMO, LUMO, gapsand charge carrier mobilities are studied [2,6,17], the respectiveauthors faced the same dilemma.

As a general statement, it should be noted that HOMO andLUMO are not physical entities that can be experimentallymeasured, but are only theoretical constructs. Nevertheless,HOMO and LUMO energies values can be derived from physicalentities. From an electrochemical point of view, the most com-monly accepted and used relationships [66] extrapolate HOMOand LUMO (and consequently electrochemical gaps) from cyclicvoltammetry measurements and are based on the absolute valuefor the normal hydrogen electrode (NHE) [67]:

EHOMO ¼ ½�eðEoxidation onsetðvs:SCEÞ2Eoxidation onset ðFc=Fcþ vs: SCEÞÞ��4:8eV

ð1aÞ

ELUMO ¼ ½�eðEreduction onset ðvs:SCEÞ2Ereduction onset ðFc=Fcþ vs: SCEÞÞ��4:8eV

ð1bÞ

However this is a widely debated point: (a) other equationsare proposed in literature [65], also in order to estimate optical

Table 1Experimental HOMO and LUMO energies, energy gaps and onsets of oxidation (Eox) a

conjugated polymers P1-P14 calculated from electrochemical (CV) and optical (UV–vis

Polymer Alkyl chains Reference HOMO (e

P1 R¼2-ethylhexyl This work �5.90

P2 R¼2-ethylhexyl This work �5.30

P3 R¼hexyl [45] �5.2

R¼hexyl [46] �5.2

R¼hexyl [47] �4.8

R¼hexyl [48] �4.9

R¼hexyl [49] �5.02

R¼hexyl This work �4.89

P4 R¼octyl [50] �5.3

P5 R¼2-ethylhexyl, R1¼2-ethylhexyl This work �5.38

P6 R¼octyl, R1¼2-hexyldecyl [51] �5.48

R¼octyl, R1¼2-butyloctyl [51] �5.47

R¼octyl, R1¼2-ethylhexyl [51] �5.43

P7 R¼2-ethylhexyl This work �5.38

R¼2-ethylhexyl [52] �5.5

R¼octyl [53] �5.3

P8 R¼2-ethylhexyl [54,55] �5.39

P9 R¼1-octylnonyl [56] �5.5

P10 R¼2-ethylhexyl, R1¼hexyl [57] �5.23

P11 R¼2-ethylhexyl [58] �5.3

P12 R¼2-ethylhexyl [59] �5.05

P13 R¼dodecyl [60] �4.65

R¼1-octylnonyl [61] �4.89

R¼1-hexylheptyl [61] �4.86

R¼1-pentylhexyl [61] �4.81

P14 R¼octyl, R1¼dodecyl [62,63] �4.90

R¼octyl, R1¼2-butyloctyl [63] �5.01

R¼2-ethylhexyl, R1¼octyl [63] �5.01

HOMO and LUMO energies [68]; (b) the evaluation criteria of thepotential can also be different: as a matter of fact the onset, peakmaximum or half-wave criteria are more widespread than formalpotentials E10 [69]; (c) it is worthwhile noticing that the maindiscrepancies between the data sets arise from the experimentaldifferences, i.e. reference electrode, solvent and electrolyte, ana-lysis in solution or on film. All these aspects have been discussedin detail in the paper of Cardona et al. [65] Accordingly, the datacollected from the literature were checked to assess the homo-geneity of the conditions adopted for the measurements. Theonsets of oxidation (Eox) and reduction (Ered) potentials have beengathered, either from the raw electrochemical data reported inthe papers (P6, P9, P11, P12, P13) or calculated backwards fromthe frontier orbital energies (P4, P8, P10, P14), paying attention tonormalize the values to a common reference electrode, Fc/Fcþ

(P9, P11). These values are reported in Table 1.Optical gaps, values which are usually lower than the electro-

chemical gaps (a difference commonly found in the literature [70]and related to the formation of charge carriers in voltammetricanalysis, which requires higher energy than optical absorption)have been taken from the literature paying attention if they havebeen measured in solution or in solid films and discarding thefirst alternative when both were reported. Worth noticing, bothelectrochemical and optical gaps are sensitive to packing effects[19] and to the lateral chains of the polymer when measured onthin films [69].

As for the multiple variants of the same base structure, fromthe literature it appears that either the same polymers have beenprepared and characterized by different research groups or thesame macromolecular skeletons bear different alkyl chains.Hence, Table 1 reports different sets of data for P3, P6, P7 andP13. For poly(3-hexylthiophene), P3, the spread of values might

nd reduction (Ered) potentials (normalized to a common reference electrode) of

spectroscopy) measurements.

V) LUMO (eV) Energy gap (eV) Eox (V) Ered (V)

CV optical

�3.00 2.90 2.43 1.10 �1.80

�2.20 3.10 2.97 0.50 �2.60

�3.0 2.2 1.9 – –

�3.53 1.67 1.92 – –

n.a. n.a. n.a. – –

�2.7 2.2 n.a. – –

�2.83 2.19 n.a. – –

�2.60 2.29 1.91 0.09 �2.2

�4.0 1.3 1.2 0.5 �0.8

�2.67 2.71 2.22 0.58 �2.13

�3.69 1.79 1.74 – –

�3.66 1.81 1.76 – –

�3.67 1.76 1.77 0.19 �1.57

�3.21 2.17 1.94 0.58 �1.59

n.a. n.a. 1.9 – –

�3.3 2.0 2.23 – –

�3.6 1.79 1.82 0.6 �1.2

�3.6 1.9 1.88 0.38 �1.49

n.a. n.a. 1.85 0.43 n.a.

�3.57 1.73 1.4 �0.09 �1.82

�3.27 1.78 1.45 0.25 �1.52

�3.13 1.52 1.20 – –

�3.08 1.81 1.41 – –

�3.07 1.79 1.42 – –

�3.08 1.73 1.43 0.01 �1.72

�3.20 1.70 1.58 – –

�3.17 1.84 1.61 – –

�3.24 1.77 1.62 0.21 �1.56

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149142

in part arise from the regioregularity. As in the DFT simulationsperfectly regioregular segments were considered, the used Eox,Ered and gap energies were those measured in our laboratories ona highly regioregular polymer; noteworthy, the optical gaps of aseries of poly(3-alkylthiophene)s (alkyl¼butyl, hexyl, octyl anddodecyl) have been measured, and in all cases the same value of1.91 eV was found. In some cases the differences of electroche-mical data appear to be consequence of the structure of the alkylchains [14,71], which have been approximated by methyl groupsin the theoretical calculations. In the case of P6 (polymers bearingdifferent chains prepared by the same authors) the energydifferences are lower than 0.05 eV, but in the case of P13(different polymers by different authors) they are as high as0.24 eV. When multiple values were available, the data obtainedin our laboratories and those of the structures bearing the short-est alkyl chains have been selected.

DFT calculations have been performed on finite model systemscorresponding to the general structures H�(Px)n�H, where Px

(x¼1–14) represents the repeating units sketched in Schemes 1and n¼1, 2, 4, 8. In all cases, the alkyl groups on the Px fragmentshave been approximated with methyl groups, to shorten thecomputational times. However, as already noted, a few studies havebeen reported recently, where the effect of the length and thestructure of the lateral alkyl chains of conjugated polymers on theirelectronic properties and photovoltaic behaviour is demonstrated[14,71]. The determination of the frontier orbital energies wascarried on in several steps. First of all, the more stable conformationin vacuum for each model structure was determined.

Geometry optimization in vacuo of the model systems wasperformed by a combined BP-86/B3-LYP functional using the TripleZetaþPolarization Slater type orbitals basis set. The basis set andfunctional chosen for this work had been selected as an acceptablecompromise between computational complexity and the need touse extended basis sets and strong exchange/correlation functionalsin order to correctly estimate fine electronic properties. However,before the beginning of the systematic study of all the molecularsystems discussed in this paper, a set of parallel computations on asubset of them had been performed applying different combinationsof basis sets (DZP, TZP, TZ2P, etc.) and functionals (BP86, XLYP) inorder to identify the best computational method according to thelimits described above. We had also noted that in practically allcases, even if the results are quantitatively different for each one ofthe described approaches, they all remain in qualitative agreementbetween them.

During the energies optimization process, the s bonds con-necting the aromatic units tend to assume a planar conformation.In this way the hyperconjugation of the p-electron system overthe whole molecular backbone is maximized and the overallstability of the molecule is increased. This simple planar approx-imation has a significant influence on the final optical propertiesand leads to underestimation of energy gaps [19,72]. Thus, theplanar conformers were the starting point for the structuredetermination: for each single bond all the s-cis/s-trans con-formation have been systematically explored and the conformerexhibiting the lowest enthalpy of formation has been selected.Then, this conformer has been allowed to relax its conformationwithout any symmetry constrains to account for steric hindrancesand to further reduce the formation enthalpy. In other words, theoptimized conformation is the result of two opposite effects: asexplained above, electronic effects would favour a planar, hyper-conjugated system; on the other hand, steric effects would berelieved when the aromatic rings are almost perpendicular one toeach other. For example, the dihedral angles of an optimizedthiophene–fluorene–thiophene fragment (present in P7) areabout 141, while the corresponding angles for a thiophene–carbazole–thiophene fragment (present in P9) are about 181.

This can be explained by recognizing that the higher dipolemoment of carbazole compared to fluorene disturbs the hyper-conjugation, reducing the importance of the electronic effectcompared to steric effects in determining the more stable con-former [69]. The molecular system resulting from geometryoptimization has been used for the calculation of the frontierorbital’s energy and of their electron density distribution. Fig. 1shows how the LUMO are always localized on the electron-poorregion of the structures, in particular when the benzothiadiazoleunit is present. On the other hand, the HOMO are in most casesdelocalized over a large part of the structure.

The values of the calculated HOMO and LUMO energies for themodel oligomers are reported in Table 2. For each oligomer thenumber of electron pairs (EP) is also reported; this was calculatedby adding to the double bonds of the basic unit the number ofelectron pairs contributing to the conjugation and to the aroma-ticity. EP has been suggested as a better parameter than n for theproper comparison of structures having different number ofdouble bonds in the basic repeating unit [17]. The calculatedHOMO and LUMO energies have been plotted against the reci-procal ‘‘electronic length’’, 1/EP. The calculated HOMO energiesfor P3, poly(3-hexylthiophene, and P9, poly[2,7-(N-(1-octylnonyl)-carbazole)-alt-5,5-(40,70-di-2-thienylbenzo-20,10,30-thiadiazole)], vs.1/EP are reported as examples in Fig. 2, where the interpolatinglines that allow for the calculation of HOMON, i.e. the HOMO energyfor a polymer of infinite length are shown as well. The plot of Fig. 2clearly shows also how – for instance – the oligomer of P3 with n¼6should be compared to the oligomer of P9 with n¼1, having thesame number of electron pairs. According to the literature a secondorder polynomial is the more appropriate functional form for theinterpolation of experimental HOMO–LUMO gaps vs. 1/n [73]; onthe other hand, ionization potentials linearly correlate with 1/na

(a¼empirical parameter). In our case, we both tested a linearand second order polynomial model. The fitting is slightly betterwith the latter, nevertheless the coefficients of determination aregenerally very high (40.94, and in most cases 40.98) also in thecase of the linear model. The extrapolated HOMON values differ inthe worst case of less than 1% for the two models; therefore, at therequired level of approximation, the linear model can be consideredsatisfactory. A similar procedure was applied to estimate theLUMON values (Table 2), finding an even better general agreementwith the linear model. Only for two polymers, P6 and P10, R2 iso0.97. Fig. 3 depicts graphically the evolutions of the calculatedHOMO energies of the studied systems with n. For most of them, thetheoretical HOMO energies decrease rapidly, approaching the linecorresponding to n¼N, with increasing the length of the modelchain, a rapidity that reflects the values of the slopes of thecorrelation lines reported in Table 2.

From the plot is also evident that, on the opposite, P6 and –especially – P10 undergo energetic saturation very rapidly. P10shows the flattest correlation line (the slope is only �0.33 eV)and the fastest energetic saturation: the behavior is attributed tothe alkyl groups on the thiophene rings that cause a strongdistortion of the dihedral angles thiophene–benzothiadiazole(451), which in turn decreases the conjugation.

The energy gaps for infinite polymer chains have beenobtained in two different ways, i.e. (a) from the difference ofthe extrapolated HOMO and LUMO energies, (LUMON–HOMON)and (b) from the linear extrapolation of the HOMO–LUMOdifferences, (LUMO–HOMO)N. Both methods afforded the sameresult; the values of LUMON–HOMON are reported in Table 2.The parameters of Table 2 can be also used to estimate the HOMOand LUMO energies for any given EP value.

As it can be seen from Fig. 3, the theoretical HOMO energieshave values that are closer to the experimental data when theoligomers with n¼1 are taken into account, but apparently the

Fig. 1. Graphic representation of HOMO and LUMO wavefunction distributions of P1–P14.

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149 143

Fig. 1. (continued)

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149144

correlation theory-experiment is poor; for LUMO (figure notreported) the theoretical values are close to the oligomers withn¼1 and n¼2. Therefore, the fitting was applied to the energies of

all the series of oligomers to find which one exhibits the bestcorrelation. Both the reciprocal length n and the reciprocalelectronic length EP have been considered for these calculations,

Table 2DFT Calculated HOMO/LUMO energies and energy gaps for H�(Px)n�H model oligomers, and fitting parameters for the linear correlations.

Polymer n EPa�HOMOn (eV) HOMO¼k/EPþHOMON �LUMOn (eV) LUMO¼h/EPþLUMON LUMON–HOMON (eV)

k (eV) �HOMON (eV) R2 h (eV) �LUMON (eV) R2

P1 1 11 5.30 �4.58 4.88 0.9981 3.31 2.35 3.52 0.9889 1.36

2 22 5.08 3.40

4 44 4.99 3.47

8 88 4.94 3.50

P2 1 16 4.53 �2.40 4.37 0.9839 2.09 6.1 2.48 0.9876 1.89

2 32 4.44 2.31

4 64 4.41 2.38

8 128 4.40 2.42

P3 1 3 5.60 �4.77 4.04 0.9945 1.32 5.20 3.00 0.9843 1.04

2 6 4.90 2.02

4 12 4.42 2.56

8 24 4.21 2.86

P4 1 25 4.71 �7.16 4.43 0.9799 3.91 3.74 4.05 0.9857 0.38

2 50 4.58 3.97

4 100 4.48 4.02

8 200 4.48 4.04

P5 1 17 4.79 �5.71 4.45 0.9914 2.83 6.59 3.23 0.9739 1.22

2 34 4.60 3.07

4 68 4.54 3.14

8 136 4.50 3.16

P6 1 18 4.63 �1.13 4.57 0.9457 3.38 6.14 3.72 0.9978 0.85

2 36 4.59 3.54

4 72 4.58 3.64

8 144 4.58 3.68

P7 1 17 4.86 �4.20 4.61 0.9900 3.51 3.02 3.68 0.9978 0.93

2 34 4.72 3.59

4 68 4.67 3.64

8 136 4.65 3.66

P8 1 17 4.88 �4.23 4.63 0.9966 3.51 3.12 3.69 0.9994 0.94

2 34 4.75 3.60

4 68 4.69 3.65

8 136 4.67 3.67

P9 1 18 4.84 �4.52 4.59 0.9906 3.48 2.88 3.64 0.9936 0.95

2 36 4.70 3.55

4 72 4.66 3.60

8 144 4.62 3.62

P10 1 18 4.76 �0.33 4.74 0.9684 3.43 3.60 3.63 0.9934 1.11

2 36 4.75 3.53

4 72 4.74 3.57

8 144 4.74 3.61

P11 1 11 4.83 �5.87 4.30 0.9997 3.35 5.62 3.86 0.9958 0.44

2 22 4.56 3.58

4 44 4.43 3.73

8 88 4.37 3.80

P12 1 11 4.92 �5.57 4.41 0.9978 3.37 4.70 3.80 0.9969 0.61

2 22 4.65 3.60

4 44 4.53 3.70

8 88 4.48 3.74

P13 1 12 4.80 �5.65 4.32 0.9876 3.33 5.75 3.80 0.9974 0.52

2 24 4.53 3.55

4 48 4.43 3.68

8 96 4.40 3.75

P14 1 12 4.69 �2.06 4.52 0.9978 3.04 8.96 3.79 0.9998 0.73

2 24 4.61 3.41

4 48 4.56 3.60

8 96 4.54 3.70

a electron pairs¼total double bondsþaromatic lone pairs.

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149 145

and the coefficients of determination have been used to measurethe goodness of linear fits. To do this, the renormalized values ofEox and Ered reported in Table 1 have been considered instead ofthe ‘‘experimental’’ HOMO and LUMO energies. Indeed, accordingto Koopmans theorem, the electron affinity of the neutral groundstate of a molecular system is given by the negative of the LUMOsingle particle eigenvalue, and the ionization potential is given bythe negative of the HOMO energy [74]. If required, Eox and Ered

may be easily converted in corrected (comparable) HOMO/LUMOenergies using Eqs. (1a) and (1b).

The R2 coefficients for the experimental vs. theoretical valuesas a function of 1/n and 1/EP are reported in Fig. 4a and b,respectively. In general, three elements clearly emerge from thediagrams. First of all, the simulated oligomers should be longenough (n45–8 or EP425) in order to observe the best fittings;secondly, the electronic length EP instead of the physical length n

must be taken into account to have the highest values of R2,suggesting that any comparison between theory and experimentsshould be done on model structures having the same number ofelectron pairs, and not simply on models with the same number

4

5

6

01 / EP

- HO

MO

n (e

V)

P3

P9

n=2

n=1n=8

n=4

n=8

n=4 n=2 n=1

HOMO

0.1 0.2 0.3 0.4

Fig. 2. Calculated HOMO energies vs. 1/EP of P3 and P9.

exptl.

n = 1

n = 2n = 4

n = ∞

P10

P1

n = 8

P2P12P4P5P14

P6P9P7P8

P11P13

P3

6.0

5.5

5.0

-HO

MO

n (e

V)

4.5

4.04.0 4.5 5.0

-HOMO∞ (eV)

Fig. 3. Evolution of the calculated HOMO energies of P1–P14, ordered along x-axis

according to their HOMON energy, with respect to the number of repeating units.

Asterisks represent the uncorrected experimental values reported in the literature.

The lines only serve as visual connections for the values in the same series.

0.0

0.2

0.4

0.6

0.8

1.0

0

1/n

R2

0.2 0.4 0.6 0.8 1

Fig. 4. Coefficients of determination R2 for the linear fittings of the energetic paramete

and optical gaps (full diamonds) vs. (a) the reciprocal length and (b) the reciprocal ele

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149146

of repeating units. Regrettably, in the literature the second type ofcriterion is the most often adopted.

Finally, while the optical gaps very well correlate when themodel chains are very long (above EP¼50, and until an infinitelylong chain, R2 is greater than 0.97 and almost constant), theelectrochemical properties are found to exhibit better correlationsif the models with 1/EP�0.04 (EP�25) are considered. This isattributed to the intrinsic different nature of the phenomenabehind the two physical processes, and in a naıve picture it can beseen as a consequence of the fact that electrochemical gaps arehigher in energy than optical gaps [75]. Noteworthy, in somecases the choice of EP¼25 could mean to consider models that areactually shorter than the basic unit.

Thus, the Eox/HOMO(1/EP¼0.04), Ered/LUMO(1/EP¼0.04) andelectrochemical energy gaps/[LUMO(1/EP¼0.04)–HOMO(1/EP¼0.04)]correlation plots are reported in Figs. 5 and 6, respectively, whileoptical energy gaps vs. LUMON–HOMON are reported in Fig. 7. Inall cases, the gap energies are appreciably underestimated com-pared to the experimental values, because the estimated HOMOlevels are shallower than the experimental ones while the LUMOlevels are in most cases deeper. This is not surprising, because thetheoretical values refer to isolated model molecules in vacuum at0 K, a very different situation compared to the real systems.Nevertheless a linear trend can be clearly caught for all the datasets. While for Eox and Ered the general trend is traced but thecoefficients of determination are only about 0.59 and 0.79,respectively, the gap energies – especially the optical gaps(Fig. 7) – are pretty well fitted. The linear regression parametersare summarized in Table 3. Noteworthy, if the HOMO/LUMOenergy values (or, equivalently, Eox/Ered) reported in the originalpapers [45–63] were plotted against the theoretical HOMON/LUMON, the coefficients of determination would be lower.

Finally, the correlation between experimental Voc of solar cells,when available, and theoretical HOMON energies has beenstudied. As Voc is related to the difference between the ionizationpotential of the donor and the electron affinity of the acceptor[76], and similar acceptors (PC61BM or PC71BM, that havecomparable LUMO levels) were used for fabricating the devicesreported in the selected literature [51,52,55,56,58,59,61,63,77], inprinciple the HOMO energy of the donor should exhibit a linear

0.0

0.2

0.4

0.6

0.8

1.0

0

1/EP

R2

0.02 0.04 0.06 0.08 0.1

rs Eox (empty squares), Ered (full squares), electrochemical gaps (empty diamonds)

ctronic length of the model molecule.

0

1

2

3

210LUMO(1/EP=0.04) -HOMO(1/EP=0.04) (eV)

ener

gy g

ap, C

V (e

V)

Fig. 6. Experimental electrochemical gaps of polymers P1–P9 and P11–P14 vs.

theoretical gaps (LUMO–HOMO) interpolated at EP¼25. ‘‘P’’ omitted for clarity in

samples labels.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

4

-HOMO (eV)

Voc

(V

)

89

12

13

1114

6

3

7

4

4.2 4.4 4.6 4.8 5

Fig. 8. Experimental open circuit voltages of photovoltaic devices of polymers P3–

P4, P6–P9 and P11–P14 vs. theoretical HOMO energy extrapolated at infinite

polymer chain length. ‘‘P’’ omitted for clarity in samples labels.

-3

-2

-1

0

1

2-LUMO(1/EP=0.04) (eV)

Eox

(V)

-3

-2

-1

0

12

-HOMO(1/EP=0.04) (eV)

Ere

d (V

)

3 4 5

3 4 5

Fig. 5. Onsets of oxidation (Eox) and reduction (Ered) potentials vs. theoretical

HOMO/LUMO energies of polymers P1–P14 interpolated at EP¼25. ‘‘P’’ omitted

for clarity in samples labels.

0

1

2

3

210LUMO∞ -HOMO∞ (eV)

ener

gy g

ap, o

pt. (

eV)

1

8

79

1213

5

4

2

3

1114

106

Fig. 7. Experimental optical gaps of polymers P1–P14 vs. theoretical gaps

(LUMON–HOMON) extrapolated at infinite polymer chain length. ‘‘P’’ omitted

for clarity in samples labels.

Table 3Linear regression parameters for oxidation and reduction potentials, energy gaps

and open circuit voltages.

a b R2

Eox¼�a �HOMO(1/EP¼0.04)þb 1.22 V/eV �5.32 V 0.59

Ered¼�a � LUMO(1/EP¼0.04)þb 0.91 V/eV �4.75 V 0.79

Energy gap (optical)

¼a � (LUMON–HOMON)þb

1.13 0.80 eV 0.97

Energy gap (electrochemical)

¼a � (LUMO(1/EP¼0.04) –HOMO(1/EP¼0.04))þb

1.30 0.40 eV 0.89

Voc¼�a �HOMONþb 0.69 V/eV �2.31 V 0.65

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149 147

correlation with the open circuit voltages. Polymer P10 wasexcluded from the fitting because the Voc of the correspondingdevices are very sensitive to annealing (0.84 V vs. 0.58 V for the

not-annealed one [57]), so the reported value might be unopti-mized. The correlation plot between theoretical and experimentalparameters is reported in Fig. 8. The theoretical HOMO energiesextrapolated at infinite chain lengths give the best, although notso good, correlation (R2

¼0.65). The HOMO energies of modelshaving finite lengths give lower coefficients of determination.

4. Conclusions

In spite of the variety of the examined molecular structure andof the inherent uncertainties of characterization methods, it waspossible to find a correlation between some theoretical andexperimental electronic characteristics of conjugated polymersrelevant for photovoltaic applications. In particular:

�

the optical energy gap (Egopt) well correlates with theoretical

values from HOMON and LUMON values extrapolated atinfinitely long chains. The equation that allows the estimationof optical gap is:

EgoptðVÞ ¼ 1:13� ½LUMO1�HOMO1�þ0:80;

�

the general trends of Eox and Ered (and, correspondingly, HOMOand LUMO energies) are modeled when oligomers with anelectronic length (EP) equal to 25 are considered. However theinterpolation with the fitting parameters gives only approx-imate values of redox potentials.

To sum up, although care must be taken in considering theabsolute calculated values, DFT methods, if properly applied, are a

L. Longo et al. / Solar Energy Materials & Solar Cells 97 (2012) 139–149148

powerful tool to predict the energy gaps of new conjugatedpolymers, and can give to the synthetic chemist useful hints inthe design of novel materials. Hence, as a general criterium for theuse of theoretical DFT calculations for the estimation of electronicproperties of conjugated polymers, the following practical proce-dure is proposed:

(1)

Electrochemical and spectroscopic measurements for thedetermination of redox potentials and energy gaps of a rangeof polymers should be performed in the same laboratoryand under the same experimental conditions [65], to build adatabase of experimental data for the fittings.

(2)

HOMO and LUMO energies for each model structure should becalculated for at least four different chain lengths and plottedagainst the reciprocal electronic length (1/EP). The equationsof the interpolating lines are derived from the plots.

(3)

The experimental electronic properties are then plotted againstthe theoretical energies interpolated from the previous plotsfor a fixed EP, for which the coefficient of determination ismaximum (for the data used in the present paper, EP¼25 forelectrochemical data and EP¼N for spectroscopic data).

(4)

The resulting regression lines can be used to extrapolatethe properties of unknown, not yet synthesized, polymerswhose theoretical parameters have been previously calcu-lated according to step (2).

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