ElectronâPhonon Coupling in Phenyleneethynylene Oligomers: A

5770

J. Phys. Chem. C 2007, 111, 5770-5782

Electron-Phonon Coupling in Phenyleneethynylene Oligomers: A Nonlinear One-Dimensional Configuration-Coordinate Model Lu Tian Liu,† David Yaron,*,† and Mark A. Berg‡ Department of Chemistry, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213, and Department of Chemistry and Biochemistry, UniVersity of South Carolina, Columbia, South Carolina 29208 ReceiVed: September 18, 2006; In Final Form: February 8, 2007

In many conjugated polymers, coupling of the torsional motions to the electronic excitation plays a major role in the spectroscopy and photophysics. In a recent paper [J. Phys. Chem. B 2006, 110, 18844], we developed a multidimensional exciton model of electron-torsion coupling that accounts for a number of unusual features in the spectroscopy of poly(phenyleneethynylene). This paper reduces that multidimensional exciton model to a one-dimensional configuration-coordinate (CC) model to facilitate interpretation of the spectroscopy and to allow extension to more complex systems. Because the CC model is derived from a more exact multidimensional model, the approximations inherent in the CC model can be examined quantitatively. The CC model accurately accounts for the line shapes in absorption and emission and the evolution of those shapes with oligomer length. Both fully anharmonic torsional potentials and nonlinear coupling are needed to predict these properties. In addition, the entropic contributions in the CC model give rise to strong effects that are not evident in the potential energy of a single torsion. When the number of torsions is small, the shape of the entropic portion of the potential is far from the harmonic shape it acquires in the limit of large dimensionality. Strongly non-Gaussian line shapes result. The CC model accurately predicts the nonlinearity of the transition energy versus 1/N without including broken conjugation. Such plots may therefore not be an unambiguous method for measuring the effective conjugation length in polymers.

I. Introduction The coupling of nuclear motion to electronic transitions is a familiar problem in both gas-phase and condensed-phase spectroscopy. However, the torsional motions of conjugated oligomers and polymers present new challenges that require going beyond standard approaches in either field. First, the number of torsional coordinates varies from only a few in oligomers to essentially infinite in polymers. Neither full quantum calculations for every degree of freedom nor limiting statistical assumptions are appropriate across this entire range. Second, the torsional barriers are often low in the groundelectronic state, leading to a large degree of torsional disorder. Harmonic approximations to the torsional potential are insufficient across this broad range of torsional angles. Finally, the torsional barriers often increase markedly in the excited state. As a result, the torsional motions couple strongly to both the spectroscopy and photophysics of these materials. Strong electron-torsion coupling makes perturbative approximations to the coupling unsuitable. In a previous paper, we developed an exciton model for torsion-electron coupling in conjugated polymers and showed that it correctly accounts for anomalous features in the spectroscopy of oligomers of poly(phenyleneethynelene) (PPE).1 Previous work has also considered the role of torsions in the electronic spectroscopy of conjugated polymers and particularly their role in destroying mirror-image symmetry between absorption and emission spectra.2-6 The exciton model went beyond previous models by including both the strongly anharmonic * Address correspondence to this author. E-mail: [email protected]. † Carnegie Mellon University. ‡ University of South Carolina.

shape of the torsional potentials and a full sampling of torsional conformations, even in long oligomers. With these features, the exciton model is not only able to account for the lack of mirrorimage symmetry of the spectra, but also explains the anomalous evolution of the absorption line shape with oligomer length. Configuration-coordinate (CC) models reduce a complex set of multidimensional nuclear motions to a single coordinate. They were originally used to describe the coupling of impurity electronic states to phonons of a crystal in either semiclassical or quantized versions.7,8 Starting with Marcus,9 they have become standard in the theory of electron transfer in condensed phases. Configuration-coordinate descriptions of solvent motion are also standard in the theory of electronic-state solvation.10 More recently, they have been extended to describe nonlinear spectroscopies.11 In the current paper, we reduce the multidimensional exciton model to a one-dimensional configuration-coordinate (CC) model. In this model, the full torsional configuration is represented by a single, average torsional angle. Electronic-state and coupling potentials as a function of average torsion are developed. The multidimensionality of the exciton model is reduced to an effective entropy at each average torsional angle. Despite the approximations involved, the unusual spectroscopic effects seen in PPE are reproduced by this model. These effects arise from several features that are retained in the CC model, but are often neglected in simpler systems. The potential energies are strongly anharmonic functions of the average torsion, creating non-Gaussian populations. The coupling potential is also fully nonlinear, causing the line shapes to be strongly distorted from the population distributions. The competition between energy and entropy effects can lead to line shapes that are bimodal and that evolve with the length of the

10.1021/jp066104z CCC: $37.00 © 2007 American Chemical Society Published on Web 03/27/2007

Phenyleneethynylene Oligomers oligomer. For low dimensions, the entropic contribution to the free energy potentials is also strongly anharmonic, further complicating the line shapes. These effects were previously accounted for by the exciton model. Relative to the exciton model, there is a small reduction in the quantitative accuracy of the CC model. However, the simplicity of the CC model makes the physical origin of the torsional effects on the photophysics much clearer. Because this CC model is derived from a well-defined multidimensional model, this model also provides an opportunity to test the underlying assumptions of CC models in general. These effects of torsional motion are common to many conjugated polymers and oligomers,2-6,12-16 but are especially pronounced in PPE, where the ground-state torsional barrier is near or below kT.17 Bunz and co-workers have proposed that the properties of the “aggregates” seen in solid-phase PPE are primarily due to torsional conformations induced by packing, rather than due to intermolecular interactions, as is commonly assumed.18-20 Kim and Swager have shown that by manipulating the conformation of PPE, its photophysics can be controlled.21 For this reason, PPE provides a demanding test case for treating torsional effects, and it will be the focus of this paper. In a following paper, poly(phenylenevinylene) (PPV), a polymer with a larger ground-state torsional barrier, will be treated with the same model. This paper is focused on torsional effects. There is also substantial coupling to high-frequency vibrations in real polymers. The torsional line shapes calculated by the CC model must be convolved with the high-frequency vibronic structure before comparing to experimental data. Although the vibronic contributions to the line shapes are important, the torsional contributions are more variable, and ultimately are of primary importance in understanding changes in the spectra. II. Existing Approaches for Coupling Electronic States to Nuclear Motion Broadly speaking, there are two approaches to coupling electronic and nuclear degrees of freedom: using a complete set of exact modes or using a single, collective configuration coordinate to describe the nuclear motion. A set of normal modes is standard for treating high-frequency, intramolecular vibrations in all molecules. In small, isolated molecules, such as conjugated dimers and trimers, a similar set of modes can be used to describe the low-frequency torsions as well.2-6 Highquality electronic structure calculations can be performed to account for anharmonicity, and all the modes can be treated explicitly. As the number of torsional coordinates grows, this approach becomes increasingly difficult, both in terms of computation and in terms of interpreting the spectrum from the coupled, multidimensional potential surfaces. As we shall see, the multidimensionality of the problem leads to important entropic effects that are not apparent in the single-mode potentials of dimers. In condensed-phase spectroscopy, electronic transitions are coupled to the nuclear motions of a bath. In solutions, these motions are highly damped and do not even have the vibrational character needed for a normal-mode approach. In solids, the phonon bath is vibrational, but the number of modes is essentially infinite, and it is not possible to treat them all individually. In these circumstances, a configuration-coordinate approach is often used.7-9,11 A single, spectroscopically active coordinate is projected out of the large number of bath modes. Population distributions and transition probabilities are derived from the resulting one-dimensional potentials, much as they are

J. Phys. Chem. C, Vol. 111, No. 15, 2007 5771

Figure 1. Linear-coupling model. Left: The ground-state (Gg) and excited-state (Ge) potentials (heavy curves) are harmonic, leading to Gaussian equilibrium population distributions in both the ground and excited states. The potentials are displaced, but have the same curvature, leading to linear coupling ∆E (heavy line). Right: The absorption (upper) and emission (lower) spectra result from projecting the groundand excited-state populations, respectively, onto the coupling.

for a single-mode potential. However, the CC potentials are freeenergy potentials; they include entropic effects from the spectroscopically inactive coordinates that have been projected out of the description. The representation of a complex set of nuclear motions by a single configuration coordinate is often introduced as an uncontrolled approximation. In this paper, the validity of the CC model will be explicitly tested against the multidimensional exciton model for its ability to describe equilibrium and spectroscopy. In general, the question of the correct equation of motion needed to describe dynamics on the effective potentials requires another set of assumptions. That question is not addressed in this paper. In either the normal-mode or CC approaches, it is common to approximate the ground- (Gg) and excited-state potentials (Ge) as harmonic functions of the nuclear coordinate q,

1 Gg(q) ) kgq2 2 1 Ge(q) ) Eo + ke(q - d)2 2

(1)

where Eo is the adiabatic or relaxed excitation energy. It is also common to approximate the difference between these potentials, the electron-nuclear coupling, as linear in the nuclear coordinate

1 ∆EL(q) ) Ge - Gg ) Eo + kd2 - kdq 2

(

)

(2)

Linear coupling is equivalent to assuming the ground- and excited-state potentials have equal curvature, k ) ke ) kg, and differ only by a displacement of the minima, d. The linear-coupling model is illustrated in Figure 1. The harmonic potentials lead to Gaussian population distributions at equilibrium in either the ground or excited state. For torsions, the frequencies are low enough that a semiclassical treatment is adequate near room temperature. To predict the spectral line shapes, the population distributions are projected vertically onto the difference potential. For a linear ∆EL, both absorption and emission spectra have a Gaussian line shape. The absorption and emission line widths are identical, but there is a displacement between the spectral maxima, the Stokes shift. The size of the Stokes shift is directly linked to the line widths by their common dependence on the slope of the coupling, kd. Although the linear-coupling model is widely applicable in many areas, it clearly fails for many conjugated polymers.2,15,16

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Figure 4. Structures of the PPE oligomers used in the experiments. Lower: The dominant ground-state configuration, which allows relatively free rotation about the torsional angles θi. Upper: An example of a cumulenic structure. This and similar structures contribute more in the excited state, resulting in higher torsional barriers. Figure 2. Absorption (a) and emission (b) spectra of PPE oligomers shifted to align the 0 f 0 peaks in the emission spectra: polymer, N ≈ 100 (black), N ) 9 (red), N ) 4 (blue), N ) 3 (orange), N ) 2 (green).

Figure 3. Quadratic-coupling model. Left: The ground-state (Gg) and excited-state (Ge) potentials are harmonic, leading to Gaussian equilibrium population distributions in both the ground and excited states. The potentials have different curvatures, but are not displaced, leading to a quadratic coupling ∆E. Right: The absorption and emission spectra result from projecting the ground- and excited-state populations, respectively, onto the coupling. Both spectra have a sharp cutoff at low frequency, but they have different high-frequency tails (upper, absorption; lower, emission).

Absorption spectra are much broader than emission spectra. Despite the large absorption line widths, the Stokes shifts (measured between onset frequencies) are small. Examples of these phenomena in PPE are shown in Figure 2. It has been proposed that these effects are due to energy migration between effectively conjugated segments of a polymer.22 However, the same phenomena occur in isolated oligomers where energy migration is not possible.4-6,12-14,17 In the case of PPE, one of us has shown that these phenomena are linked to the viscosity of the solvent, strongly implicating largeamplitude torsional motions rather than small-amplitude vibrations.17 Moreover, the time-resolved Stokes shift in PPE is also inconsistent with the linear-coupling model.17 Several papers have tried to account for these unusual spectral features by invoking quadratic coupling, either to high-frequency vibrations3 or to torsions.5,6,17 The quadratic-coupling model6,8,22 is illustrated in Figure 3. In this case, the ground- and excitedstate potentials are still harmonic, but with no displacement of the minima, d ) 0. Instead, the curvatures of the potentials are unequal, ke * kg, leading to quadratic coupling

1 ∆EQ(q) ) Eo + (ke - kg)q2 2

(3)

In this model, the equilibrium population distributions in both the ground and excited states remain Gaussian, but when these distributions are projected onto the coupling potential ∆EQ, the spectra become strongly non-Gaussian. The shapes are asymmetric with a sharp cusp at low frequency and an exponential tail to high frequency. Both absorption and emission spectra peak at the same wavelength. In other words, there is no Stokes shift. At the same time, the absorption line width is broad, and the emission spectrum is narrow, in general agreement with the experimental observations. The reason that strong quadratic coupling to torsions is an attractive model for conjugated polymers is illustrated for the case of PPE in Figure 4. In the ground electronic state, the repeat unit contains single bonds with a low barrier to torsion (Figure 4, bottom). In the excited state, cumulenic configurations (Figure 4, top) play a much greater role, resulting in a torsional potential that favors planarity much more strongly. Although the quadratic-coupling model is conceptually attractive, we showed that it is unable to describe the evolution of the absorption line shape in PPE with oligomer length, even at a qualitative level.1 The observed absorption spectrum shown in Figure 2 has sharp features for N ) 2, shows apparent additional peaks or shoulders for N ) 3 and 4, and then partially narrows again for N ) 9 and the polymer. This behavior cannot be explained by either the linear- or quadratic-coupling models. We developed an exciton model that accurately and efficiently calculates the excitation frequency even in long oligomers and with large ensembles of torsional configurations.1 Each torsional coordinate was treated individually, as in a normal-mode approach. However, the electronic-state and coupling potentials were not limited to linear or quadratic approximations, nor was the problem limited to independent modes of nuclear motion. With all these features, the absorption line shapes as a function of oligomer length were quantitatively accounted for. Although the exciton model is successful in predicting experimental results, it lacks the ease of interpretation and the physical transparency of the one-dimensional models, such as those shown in Figures 1 and 3. For this reason, the next section develops a one-dimensional CC model based on the multidimensional exciton model. The CC model retains the important physical features of the exciton model that are needed to describe the experimental results, but has the simplicity of the linearand quadratic-coupling models.

Phenyleneethynylene Oligomers


Figure 5. Torsional line shapes without vibronic structure: Multidimensional exciton (dashed) and one-dimensional CC (solid) models for PPE oligomers with different numbers of phenyl rings N. Top: Absorption. Bottom: Equilibrated emission. The ground-state barrier is V ) 1 kT for all N. The exciton results are smoothed with 100 cm-1 of additional broadening.

III. Development of a Nonlinear, One-Dimensional Configuration-Coordinate Model A. Exciton Model. The exciton model was developed fully in a previous paper,1 and only its essential elements are reviewed here. In the exciton model, the electronic-state energies are a function of the set of torsional angles {θj} in the oligomer. Each θj is a localized torsion, i.e., the angle between neighboring phenyl groups in PPE (Figure 4). The multidimensional groundstate energy has the molecular-mechanics form

Eg({θj}) )

VN-1

∑ 1 - cos(2θj)

2 j)1

(4)

where V is the torsional barrier height and N is the number of phenyl rings in an oligomer. Equation 4 assumes that the energy to rotate one torsion is independent of the other torsional coordinates. This assumption has been verified by AM1 calculations on a PPE trimer.1 In these calculations, the barrier to torsion in one of the angles varies by less than 1% as the other torsion varies from 0° to 90°. The excited-state energy Ee is derived by adding the multidimensional excitation energy ∆Em to the ground-state energy

Ee({θj}) ) Eg({θj}) + ∆Em({θj})

(5)

The excitation energy is simulated as an exciton that hops between sites on the oligomer. The excitation energy is the lowest eigenvalue of a Hu¨ckel-like Hamiltonian

[

H) R1 βo cos (θ1) 0 l 0

βo cos(θ1) R2 βo cos(θ2) 0 ‚‚‚

0 βo cos(θ2) l l 0

‚‚‚ 0 l Rn βo cos(θn)

]

0 l 0 βo cos(θn) RN (6)

The energy of an excitation on a single site is R. A small improvement in fits to data is obtained by allowing R of the end sites to differ from that in the middle. However, all the sites are assumed to have the same R here. The amplitude for coherent hopping between adjacent sites is βo cos(θj). This amplitude has a cosine dependence on torsional angle, as expected for the overlap of p-orbitals. This cosine dependence has also been verified in PPE by INDO calculations.1 The values of the model parameters, R and βo, were determined by fitting to the results of INDO calculations on about 2000 torsional configurations with N ) 2 to 5. There is a small dependence of the parameters on N, but this variation has little effect on the predicted spectra, and it is ignored here. In addition, fits to the experimental spectra lead to a 60% increase in βo relative to that derived from the INDO calculation. The larger value is used here: R ) 34560 cm-1 and βo ) 6086 cm-1.1 Absorption (emission) spectra are calculated by sampling a Boltzmann distribution in the ground-state (excited-state) potential [eqs 4 and 5] and assuming a δ-function absorption at the excitation frequency derived from eq 6. Figure 5 shows the torsional line shapes from the exciton model for oligomers of various lengths with a fixed torsional barrier. The absorption line shapes are strongly non-Gaussian for small oligomers and evolve rapidly with the length of the oligomer. These effects are similar to the ones seen in experiment. The torsional effects predicted by the exciton model are compared to experimental absorption spectra in Figure 6. The torsional line shape is convolved with the vibronic structure from the N ) 3 emission spectrum. Quantitative fits to the PPE absorption spectra are obtained by letting the barrier height vary from V(2) ) 5 kT for N ) 2 to V(20) ) 1 kT for N ) 20. The agreement is good. In particular, both the change in line shape and the red-shift with increasing oligomer length are correctly described by the exciton model. However, the physical origin of these effects is not readily discernible from these calculations. B. Potential Energies in the Configuration-Coordinate Model. In a configuration-coordinate model, the true multidi-


Liu et al.

Figure 7. The dependence of excitation energy obtained from the mutidimensional exciton (dots) and 1-D CC (black lines) models on Θ0, for oligomers with 2 (maroon), 3 (green), 4 (red), and 9 (blue) phenyl rings. Figure 6. Total absorption spectra (torsional and vibronic): Multidimensional exciton (dashed) and one-dimensional CC (solid) models compared to experimental spectra (thick solid, from Figure 2A). Both models use the vibronic structure from the N ) 3 emission spectrum. The ground-state torsional barriers are as follows: V ) 2 kT for N ) 3; V ) 1.5 kT for N ) 4; V ) 1.5 kT for N ) 9; and V ) 1 kT for N ) 20.

mensional potential energy surfaces are represented by onedimensional curves along a collective coordinate. This simplification assumes that there is some coordinate transformation that will yield one spectroscopically active coordinate θ0 and N - 2 inactive coordinates. The other N - 2 coordinates are integrated to yield an entropy at each value of θ0. This entropy is added to the energy as a function of θ0 to give onedimensional free-energy potentials from which spectra can be predicted. Often the underlying multidimensional potential energy surfaces and the transformation to collective coordinates are not specified. The existence of a collective coordinate is assumed and the free-energy surfaces are developed to fit experimental data. Here, the collective coordinate is explicitly derived from the exciton model, and the accuracy of the reduction to one coordinate can be tested against that more accurate model. We make Eg a function of one variable through the following transformation from local torsional angles {θj} to the collective coordinates {Θk}

cos 2Θk )

N-1

1

(

k

∑ cos N - 1π(j - 1) N - 1 j)1

)

cos 2θj

(7)

The ground-state energy [eq 4] is only a function of Θ0,

Eg(Θ0) )

(N - 1)V (1 - cos 2Θ0) 2

1

(10)

The validity of approximating ∆E as a single-valued function of Θ0 is tested for PPE in Figure 7. A uniform sampling of torsional configurations was generated, and the excitation energy for each configuration was calculated from the exciton model. For N ) 2 and 3, the CC approximation is exact. For larger N, the exact excitation energy shows some dispersion. The degree of dispersion represents an error in the CC model. To some extent, this error could be compensated by adding an extra phenomenological broadening to the line width. This would be in addition to broadening from other factors, such as the additional 100 cm-1 broadening included in the spectral simulations reported here. Both of these additional broadenings can be absorbed into a single additional broadening, but this minor correction has not been needed here. For the CC model, we need to choose a formula for ∆E that approximates the average of this distribution. One choice would be to take the mean or mode of the distribution at each value of Θ0. We make a different choice, an analytical formula that is the exact result for N ) 2 and 3, but that can be extended to all values of N as an approximation

∆E(Θ0) ) R - b(N)βo cos(Θ0)

N-1

cos2 θj ) 〈cos2 θj〉 ∑ N - 1 j)1

Ee(Θ0) ) Eg(Θ0) + ∆E(Θ0)

(8)

The average torsional angle Θ0 is the configuration coordinate for our model and is defined by

cos2 Θ0 )

coordinate Θ0. It is always possible to force one potential surface to be a function of one coordinate by the choice of coordinate transformation.11 In the current model, we force the groundstate energy to be a single valued function of Θ0. The key to the success of the CC model is whether the second of the three potentialssin the current case, the coupling ∆Eswill be singlevalued. Once two of the three potentials are single-valued functions of Θ0, the third (here Ee) will also be single-valued by the relationship

(9)

Note that the appropriate ground-state potential in the CC model [eq 8] has a barrier that increases linearly with the length of the oligomer. The essential approximation of the CC model is that the three potentialssthe ground-state, excited-state, and coupling potentialssare all single-valued functions of the transformed

(11)

When all the torsional angles are 90° (Θ0 ) (π/2)), the excitation energy is the single-site energy R; when the oligomer is completely planar (Θ0 ) 0°), conjugation reduces the excitation energy by an amount b(N)βo. Specific results from the exciton model include the following: b(2) ) 1, b(3) ) x2, b(4) ) 1.62, b(9) ) 1.90, b(20) ) 1.98, and b(∞) ) 2. The potential energies in the CC model are now completely defined by eqs 8, 10, and 11. C. Entropy and Free-Energy Potentials in the Configuration-Coordinate Model. To convert the potential energies of the CC model to free energies, the entropy at each value of



Θ0 must be calculated. We begin from the distribution of individual torsional angles, which is governed by a simple Boltzmann distribution

Fθ({θj} ) )

(

)

E({θj}) 1 exp Q kT

(12)

If the polymer is in a frozen matrix, it may be reasonable to assume the above Boltzmann distribution with an effective temperature, such as the temperature at which the torsions became immobile. In the collective coordinates, the probability density also contains the Jacobean of the coordinate transformation J

( ) (

)

E({Θi)} 1 {Θi} exp FΘ({Θi} ) ) J Q {θj} kT

(13)

Integrating over Θ1 ... ΘN-2 yields the one-dimensional probability distribution in Θ0

( (

) )

Ee/g(Θ0) kT Fe/g Θ (Θ0) ) Ω(Θ0) Ee/g(Θ0) π/2 exp dΘ0 0 kT exp -

∫

(14)

where e or g represent the excited or ground state, respectively. This expression contains the density of torsional configurations, Ω(Θ0), at each value of Θ0

Ω(Θ0) )

( ) ∫ ( ) ∫J J

{Θm} {θn}

{Θm} {θn}

dΘ1 ... dΘN-2

0

lim S(Θ0) ) k(N - 2) ln Θ0

Θ0f0

S(Θ0) ) k ln Ω(Θ0)

(16)

The probabilities in the CC model are governed by a Boltzmann distribution in the free energy

Ge/g(Θ0) ) Ee/g(Θ0) - TS(Θ0)

σox2π 1 N-1 π2 Θ + k ln SCLT(Θ0) ) k 0 2 σ2 4 xN - 1

)

(18)

However, for oligomers with small values of N, the degeneracy and entropy must be calculated explicitly. For N ) 3, eq 15 can be solved analytically; for N > 3, it must be solved numerically (see Appendix A). The results of these calculations are shown in Figure 8. The left panel of the figure shows the degeneracy Ω, which is equivalent to the distribution of average torsions in the absence of any torsional barrier. For large N (N ) 9 or 20), the

π 2

)

(19)

Near 45°, the central-limit expression [eq 18] is valid for large N, and the entropy is parabolic. Again, the potentials do not approach this limiting behavior smoothly. There are cusps and sharp breaks at intermediate values of N. D. Transforming Populations to Spectra. The absorption cross section σ(ω) for a single molecule is

σ(ω) ∝

(17)

Note that the entropy is independent of the electronic state. As a result, entropy effects cancel in the difference potential. For N ) 2, the entropy is simply a constant. For large N, it is common to invoke the central-limit theorem (CLT), for which the degeneracy is Gaussian and the entropy is harmonic in Θ0

(

lim S(Θ0) ) k(N - 2) ln Θ0 -

Θ0fπ/2

(15)

This degeneracy creates an angle dependent entropy

(

distribution is nearly Gaussian and peaks at Θ0 ) 45°, as expected from the central-limit theorem. However, for small N, the distributions are strongly non-Gaussian. For N ) 3, there is a divergence at Θ0 ) 45°. For N ) 4, the distribution has a flattened top. For N ) 5 and 6 (not shown), the width of the flattened region narrows, and the shape becomes close to Gaussian for N g 7. The corresponding entropy per angle (-(S/[(N - 1)k]) is shown on the right of Figure 8. For angles near Θ0 ) 0° or 90°, the entropy diverges logarithmically

dΘ0 ... dΘN-2

∫Θ )const dθ0 ... dθN-2 ) ∫ dθ0 ... dθN-2

o

Figure 8. Left panel: The density of torsional configurations Ω as a function of the average planarity Θ0 for various length oligomers: N ) 2 (red), N ) 3 (blue), N ) 4 (green), N ) 9 (maroon), and N ) 20 (purple). Right panel: The corresponding entropy per torsional angle, -(S/(N - 1)k).

ωifµif2|〈f|i〉|2Fi ∑ i,f

(20)

where 〈f|i〉 is the Franck-Condon overlap between initial and final nuclear wave functions, Fi is the population of the initial state, and µif is the electronic transition moment between the initial and final states. Because torsional motions are low frequency, we use a semiclassical approximation7 in which the summation over states is replaced with an integration over nuclear coordinates {θj}, and the transitions are strictly vertical in these coordinates, |〈f|i〉|2 f δ(θfj - θij):

χ(ω) ) ∝

nc σ(ω) 4πω

∫ µ2({θj})δ(∆E({θj}) - pω)F({θj}) d{θj}

(21)

where χ(ω) is the molecular susceptibility, and n is the indexof-refraction. In the Condon approximation, the transition moments are independent of the torsional configuration. (This assumption was verified within the full multdimensional exciton


Liu et al.

model where, for oligomers with N up to 9, the use of a constant transition moment was found to give line shapes that agreed well with those obtained by using the configuration-dependent transition moments. On longer oligomers, localization of the excitation may cause this approximation to break down, as discussed in Section V.) In the configuration-coordinate model, the integrals are reduced to a single coordinate. Neglecting constant factors,

χ(ω) ) µ2

∫0π/2 δ(∆E(Θ0) - pω)Fe/g Θ (Θ0) dΘ0

(22)

The spectrum is calculated as a susceptibility χ, which does not contain any of the frequency factors that occur in the experimental absorbance or emission intensity. As a result, both ground-state absorption and excited-state emission spectra can be calculated from eq 22 by using the appropriate populations. The experimental data presented in this paper have also been corrected to susceptibilities.23 Constant factors are neglected in eq 22. To integrate over the δ function, the coordinate must be transformed from Θ0 to ∆E

χ(ω) ) )

dΘ

0 d∆E ∫0∞ δ(∆E - pω)Fe/g Θ (Θ(∆E)) d∆E

( ) d∆E dΘ0

-1

Fe/g Θ (Θ0(pω))

(23)

∆E)pω

The dependence of Θ0 on ∆E is determined by inverting eq 11

Θ0(∆E) ) cos-1

(

)

R - ∆E b(N)βo

(24)

The line shape is thereby distorted from the population distribution by the inverse of the derivative of the coupling potential. For the linear-coupling model, this term has no effect. For the quadratic-coupling model, this term leads to a factor of sin(Θ0)-1 in the line shape, a factor that has sometimes been neglected. (The occurrence of divergences in the spectrum is an unimportant consequence of the CC approximation. Higher order corrections to the theory will result in convolution with additional line broadening factors that remove the divergence.) Combining eqs 14 and 23 leads to the final formula for absorption and emission spectra

(

χe/g(ω) ) csc(Θ0) Ω(Θ0) exp -

)

Ee/g(Θ0) kT

(25)

where Θ0 is a function of ω ) (∆E/p) from eq 24. IV. Application of the Configuration-Coordinate Model to Line Shapes in PPE Oligomers Figure 9 shows results from the CC model for parameters appropriate to PPE oligomers. Figure 9a shows the free-energy potentials and populations versus the average torsional angle; Figure 9b shows the corresponding torsional contribution to line broadening in the absorption and emission spectra. The behavior of the excited state is relatively simple. For all oligomer lengths, the effective potential is deep enough to confine the population near the minimum. Near these minima, the potentials are close to harmonic, and the populations are close to Gaussian. The only complication to the line shape is the nonlinear transformation between population and spectrum. In the N ) 2 case, the population is peaked close to Θ0 ) 0°, where the coupling potential has a minimum. The result is very

Figure 9. (a) Effective one-dimensional free-energy potentials (dotted), Ge/g(Θ0), versus average torsional angle in the ground (GS) and excited (ES) states for oligomers of varing lengths. Thermal populations at T ) 300 K are shown for the equilibrated ground (solid) and excited (dashed) states. (b) The torsional line shapes, χ(ω), for absorption (solid) and emission (dashed) from the populations in part a. V ) 1 kT is used for the ground-state barrier and βo ) 29.25 kT is used in the CC model.

similar to the quadratic-coupling model (Figure 3); the spectrum is highly asymmetric with a cusp on the low-frequency edge and an exponential tail on the high-frequency side. As the oligomer length increases, the population shifts to larger angles and less highly curved regions of the coupling. The spectrum evolves away from the quadratic-coupling model and toward the linear-coupling model (Figure 1). For N ) 20, the emission spectrum is nearly Gaussian. For all oligomer lengths, the torsional contribution to the line shape is dominated by a sharp, narrow peak. The result is consistent with the well-resolved vibronic structure in the experimental emission spectra (Figure 2). The behavior in the ground state is more complicated because of the low-energy barrier to torsion. For small oligomers, a wide range of torsions are populated, and the anharmonicity of the potential becomes evident. In addition, the entropy competes effectively against the energy, and the cusps and breaks in the entropy for small N play an important role. For the N ) 2 ground state, the results are similar to the quadratic-coupling case (Figure 3). The cusp on the lowfrequency side of the spectrum allows vibronic structure to be seen in the experimental absorption spectrum. The large distribution of angles in the ground state leads to substantial intensity in a broad tail to the high-frequency side of the line shape and a large unresolved component in the absorption spectrum (Figure 2). For long oligomers, competition between energetic and entropic effects confines the population to a relatively narrow range of angles away from Θ0 ) 0°. Over this range, the free energy is approximately harmonic, and the coupling is approximately linear. The spectrum approaches the linear-coupling model (Figure 1)snearly Gaussian with modest broadening. At intermediate lengths (N ) 3 and 4), the spectra are more complex. Population starts to build up at large angles in response to the entropy, but some population remains near Θ0 ) 0°, where it samples the minimum in the nonlinear coupling. As a result, the spectrum becomes bimodal: one peak at the population maximum, one cusp at the minimum of the coupling. The accuracy of these predictions can be judged by comparison to the torsional line shapes from the full exciton model



Figure 10. An illustration of the various nonstandard effects in the CC model that contribute to the final torsional line shape of absorption. Top row: Anharmonic ground-state torsional barrier. Middle row: Anharmonic ground-state torsional barrier with entropy effect. Bottom row: Anharmonic ground-state torsional barrier with both entropy and nonlinear coupling effects. Bottom x-axis is the excitation energy. Top x-axis is the average torsional angle, Θ0.

(Figure 5). The CC model has the same unusual features predicted by the exciton model. Thus the discussion of the line shapes in terms of the CC model just given is a qualitatively correct explanation of the origin of the line shapes calculated from the exciton model. To compare to the experimental spectra, the torsional line shapes predicted by the CC or exciton model must be convolved with the vibronic structure coming from high-frequency vibrations of the oligomer. This process was discussed at length in our previous paper.1 A comparison of both the exciton and CC models to the experimental spectra is shown in Figure 6. The agreement is quite good on the low-frequency edge and across the peak of the spectrum, where the torsional broadening is most important. The disagreement between the CC and exciton models for N ) 20 can be attributed to the assumption, in the CC model, that the exciton is delocalized uniformly over the entire oligomer. This assumption breaks down for very long oligomers such as N ) 20 (see Appendix B). The basic features of the ground-state torsional line shape are discernible in the experimental absorption spectra (Figure 2), especially on the low-frequency edges. For the dimer, there are sharp vibronic peaks on top of a broad background, reflecting the cusp and high-frequency tail of the torsional line shape. In the trimer, there is a poorly resolved peak at low frequencies preceding the main peak, reflecting the two peaks in the torsional line shape. For N ) 4, the low-frequency edge is unusually broad, reflecting the broad shoulder in the torsional line shape at frequencies below the peak. For high values of N, the torsional line shape narrows and becomes essentially Gaussian. The absorption spectra also narrow and show somewhat better defined vibronic structure for long oligomers and the polymer. For all lengths, the torsional broadening of the emission spectrum is small, leading to well-resolved vibronic structure. Having established that the CC model reproduces the essential features of the experimental line shapes, we can discuss the origin of these features in more detail. Equation 25 for the torsional line shape is written as a product of three terms, each of which isolates one nonstandard effect in the CC model. The

effect of adding each term is shown in one row of Figure 10; the effect on a given oligomer length is shown in each column. The top row of Figure 10 shows the line shape using only the right-most, exponential term in eq 25. These line shapes include the effect of the anharmonicity of the torsional potential. The lines are strongly asymmetric and deviate strongly from the predictions of a linear-coupling model. However, the lines are qualitatively similar to the results from a quadratic-coupling model. The middle row of Figure 10 shows the line shape using only the two right-most terms in eq 25. These line shapes include the effects of both entropy and energy. For long oligomers, the major effect of the entropy is to shift the spectrum to higher frequencies, an effect that will be discussed in more detail in the next section. For small oligomers, the entropy creates structure in the spectrum. The bottom row of Figure 10 shows the line shape using all the terms in eq (25). Compared to the middle row, the effect of the nonuniform density of torsion angles with respect to transition frequency has been included. Figure 11 illustrates the origin of this effect in more detail for the N ) 3 absorption spectrum. At large values of Θ0, ∆E is approximately linear, and the spectrum at high frequencies reflects the shape of the population distribution. At small values of Θ0, ∆E is strongly curved causing spectral intensity to pile up near a low-frequency cutoff. As a result, a new, low-frequency peak appears in short oligomers. Figure 12 shows the same series of calculations for the excited-state emission spectrum. The same effects are present as in the ground state, but they cause less dramatic changes in the line shape. The emission line remains relatively narrow under all conditions. It is easy to regard the ground state of PPE as a free rotor and the excited state as fully planarized. Both ideas are oversimplifications. The ground-state populations (Figure 9a) are significantly affected by the torsional barrier, even though it is only 1 kT. Without the effect of the barrier, the population distributions would be symmetrical about 45° and identical with


Liu et al.

Figure 11. An example of the effect of nonlinear coupling. The ground-state population distribution (N ) 3 shown on bottom left) is projected onto the coupling potential ∆E to generate the absorption spectrum (right). Absorption intensity piles up in regions where the slope of ∆E is small and is spread out where the slope is large.

Figure 12. Same as Figure 10 except for the excited state.

the shape of Ω(Θ0) (Figure 8). Conversely, the excited-state populations (Figure 9a) are significantly affected by entropy. Despite the high torsional barrier, the mean torsional angle shifts away from planarity by a substantial amount for large oligomers. The consequences of this shift are considered in more detail in the next section. V. Red-Shifts versus Number of Torsions and the Effective Conjugation Length It is common to assume that in a fully planar and conjugated polymer, the transition energy will approach a limiting value,

∆E(N) ) ∆E(∞) + c/N, as N f ∞. It is also common to assume that in a real polymer, the conjugation is “broken” occassionally along the backbone, leading to multiple segments that behave spectroscopically as shorter oligomers. Experimental plots of ∆E versus 1/N deviate from linearity for long oligomers and polymers, which is taken as verification of the broken conjugation, and the length where deviations become prominent is taken as a measure of the effective length of the conjugated segments.24,25 These arguments are contingent on the idea that the polymer is normally planar, or at least that the torsional distribution is


Figure 13. Frequency of the emission maximum versus length of the PPE oligomer N: experiment, solid circles; exciton model, open diamonds; configuration-coordinate model, open circles; and planar oligomer (exciton model), pluses. The deviation from linearity in the experiment is explained without introducing an “effective conjugation length”.

centered on a planar conformation. However, the CC model shows that this assumption is not true in PPE, even in the excited state, where the torsional barrier is high (see Figure 9a). The effect on the red-shift with oligomer length is illustrated in Figure 13. The peak frequency of the experimental PPE emission spectra (solid line, from Figure 2) is linear in 1/N for small oligomers, but deviates from linearity for long oligomers and the polymer. This behavior is typical of conjugated polymers. The exciton model has been used to calculate the emission frequency of completely planar oligomers. As expected, these values are nearly linear when plotted against 1/N (pluses). However, when the exciton model is applied to the full ensemble of torsional conformations, the mean emission energy versus 1/N deviates from linearity and is in excellent agreement with the experimental data (diamonds). (Note that the parameters of this model were adjusted to match the shape of the absorption spectra, but not to match the emission frequencies.) The results from both the experiments and the exciton model are consistent with the idea that the conjugation of the polymer is “broken” by torsions. This interpretation is acceptable whether one believes that the important breaks are single, large twists or that they are due to the cumulative effects of disorder in the coupling along the chain (Anderson localization). In either case, the exciton model contains the full torsional disorder seen in the real system, and would be expected to show effects attributed to broken conjugation. The unexpected result is that the CC model also reproduces the nonlinearity of the experimental data (dotted line with circle, Figure 13). This result is surprising because the CC approximation eliminates all of the disorder that is invoked as a cause of broken conjugation. The approximation to the transition energy and transition moment used in the CC model [eq 11] is the exact solution for an oligomer in which every individual angle is equal to the average angle. Thus the CC approximation can be viewed as taking the real ensemble of torsional configurations and replacing it with an ensemble in which each member of the ensemble is uniformly twisted. As a result, the CC model does not include any mechanism for breaking the conjugation. Nonetheless, this model faithfully reproduces the saturation of the red-shift with oligomer length. The interpretation implied by this result differs from the standard one in subtle, but important ways. Long oligomers have

J. Phys. Chem. C, Vol. 111, No. 15, 2007 5779 a greater average twist per torsion angle due to entropy effects. As a result, the coupling is reduced on average. Accounting for this reduction in coupling, even in an averaged fashion, is sufficient to reproduce the experimental spectral shifts, even without introducing disorder along the oligomer chain. This interpretation does not deny that the conjugation is broken in conjugated polymers. It only implies that measuring spectral shifts versus length is not an accurate method for measuring the conjugation length. The conjugation length could be substantially longer than the length implied by such measurements. Although the absorption frequency is not sensitive to torsional disorder, the line shape is. In the limit of increasing oligomer length, the CC model predicts that the torsional line width will decrease slowly to zero. This result is not realistic. For example, the experimental absorption spectrum of the polymer (N ) 100) is only marginally narrower than the spectrum of the N ) 9 oligomer. At some length, the torsional disorder in the real system causes localization of the excitation to various regions. A long chain is then expected to have a number of relevant excited states with different energies and transition moments. This causes deviations from the CC model, which assumes a single state that is delocalized over the entire oligomer. VI. Discussion and Conclusions This paper has developed a one-dimensional configurationcoordinate model for the effect of low-frequency torsions on the absorption and emission spectra of conjugated oligomers. The resulting model extends standard CC models, by calculating realistically anharmonic potentials and a nonlinear coupling. In an earlier paper, one of us argued that a quadratic-coupling model was appropriate for PPE oligomers.17 The argument was based on the idea that the symmetry of the torsional energy barrier about zero degrees prevented a linear coupling term from appearing, such that the line shape would have a maximum at a point corresponding to a planar system. The current CC model shows that this argument is flawed. Entropy effects cause the minimum in the free energy to differ considerably from zero degrees, an effect that is not evident in the potential of a single torsion. In fact, the CC model predicts a transition from quadraticcoupling-like spectra for small oligomers to linear-couplinglike spectra for long oligomers. This transition can be identified in the experimental spectra by the characteristically nonGaussian line shapes that occur for intermediate oligomer lengths. The CC model is derived from a more complete multidimensional exciton model. Despite the relative simplicity of the CC model, it replicates the torsional line shapes from the exciton model. Both models quantitatively account for the anomalous features seen in experimental spectra, in particular, the strong asymmetry in widths of the absorption and emission spectra and the complex evolution of the absorption line shape with oligomer length. The success of these two models in explaining the experiments clearly indicates that torsions are a large source of spectral line broadening in PPE and the largest source of variation in the spectra. The primary difference between the configuration-coordinate and exciton models relates to the effects of torsional disorder. The CC model is equivalent to a model in which every individual chain is uniformly twisted with a torsional angle Θ0 for each angle on the chain. The only disorder is between chains, each of which has a different uniform twist angle Θ0. The CC model does not include any localization of the excitation due

5780 J. Phys. Chem. C, Vol. 111, No. 15, 2007 to disorder in the torsional angles. The CC model also does not include “broken conjugation” due to a large twist at a single site. The simplicity of the configuration-coordinate model permits a relatively clear interpretation of the band shapes seen in experimental spectra. They are due to three major effects: the nonlinearity of the torsional potential, the increasing effect of entropy as the number of torsions increases, and the nonlinearity of the electron-torsion coupling potential. None of these effects are included in standard configuration-coordinate models. Of these effects, the nonlinearity of the torsional potential produces the mildest effects. The competition between energy and entropy produces strong effects on the absorption line shape. A surprising effect of the entropy is a strong violation of the central-limit theorem, which leads to strongly non-Gaussian populations for small oligomers. In combination with the nonlinear coupling, multiple peaks or distinct shoulders on the absorption spectrum can result. These complex line shapes are, however, confined to short oligomers, and near-Gaussian line shapes return for long oligomers. In the excited state, the higher torsional barrier prevents these complex line shapes from emerging. However, the CC model does predict a more subtle evolution with oligomer length from a strongly asymmetric line, typical of quadratic coupling, to a symmetric line, typical of linear coupling. In PPE, the torsional line widths in emission are too small for this effect to be seen experimentally. The CC model makes a significant experimental prediction regarding the emission frequency versus oligomer length. This model correctly predicts the observed deviation from linearity on a plot of emission frequency versus inverse oligomer length. This type of deviation has typically been attributed to an effective conjugation length that is less than the full length of the oligomer. However, the CC model does not include any mechanism for limiting the conjugation length. In this model, the deviation from linearity is caused by entropy forcing the mean torsional angle significantly away from zero, and consequently, a weakening of the average level of conjugation. Assignments of conjugation lengths based on oligomer frequency shifts should be reconsidered in the light of this result. The ground state of PPE provides a particularly good test case for torsional models, because of its very low torsional barrier. However, torsional effects are not limited to this case. Despite the much higher torsional barrier in PPE’s excited state, mean torsions as large as 25° are predicted for long oligomers. This finding is consistent with the proposal of Bunz and coworkers that PPE “aggregates” in the solid state are chains that are more planar than they are in solution.18-20 This method has also been applied to poly(phenylvinylene) (PPV), which has a higher barrier than PPE, and similar effects are still observable (to be published). We note that the a priori quantitative prediction of spectral line shapes is still difficult. In part, this difficulty arises from the need to include the vibronic structure from high-frequency vibrations in addition to the broadening from low-frequency torsions. This vibronic structure varies with oligomer length. Moreover, it may vary with torsional angle. More planar structures favor a more cumulenic electronic structure (Figure 4), which in turn alters the intensity of the vibronic coupling for high-frequency vibrations. Although these higher order effects are not included in the configuration-coordinate model, it does provide a good semiquantitative explanation and guide to the important role of torsion in the spectra and photophysics of conjugated oligomers and polymers.

Liu et al. Appendix A: Calculation of Ω(Θ0) and S(Θ0) This Appendix derives the density of torsional configurations, eq 15, as a function of the number of torsional angles, n. Since the approach builds up the density of states from that of shorter oligomers, we need to change the notation somewhat from the main text. Θ h (n) will be used to denote the average planarity, Θ0 of the main text, for an oligomer with n dihedral angles. The density of torsional configurations of an oligomer with n dihedral angles will be denoted as ΩΘh (n)(Θ h (n)). The approach begins with (1) (1) ΩΘh (Θ h ) and builds up higher ΩΘh (n)(Θ h (n)). For n ) 1, h (1)) is a uniform distribution in the range of [0, (π/2)] ΩΘh (1)(Θ

ΩΘh (1)(Θ h (1)) )

2 π for 0 e Θ e π 2

(A1)

It is convenient to introduce the following two variables,

h (n)) Y h (n) ) cos(2Θ

(A2a)

n

h (n) ) Y(n) ) nY

cos(2θj) ∑ j)1

(A2b)

and to transform the probability distribution of eq A1 from a function of Θ h (n) to a function of Y(n),

dY h (n) h (n)) ) ΩYh (n)(Y h (n)) (n) ΩΘh (n)(Θ dΘ h ) 2 sin(2Θ h (n)) ΩYh (n)(Y h (n)) ) 2x1 - (Y h (n))2 ΩYh (n)(Y h (n))

(A3a)

with

h (n)) ) ΩY(n)(Y(n)) ΩYh (n)(Y

dY(n) dY h (n)

) n ΩY(n)(Y(n))

(A3b)

Since Y(n) in eq A2b is a summation of indentical independent variables, ΩY(n)(Y(n)) can be obtained through statistical convolution. To accomplish this, we first obtain the form of ΩY(1)(Y(1)) for n ) 1 using eqs A1, A3a, and A3b,

h (1)) ΩY(1)(Y(1)) ) ΩYh (1)(Y ) )

ΩΘh (1)(Θ h (1)) 2 sin(2Θ h (1)) 1 π x1 - (Y(1))2

(A4)

where Y(1) ∈ [-1, + 1]. We next note that, from eq A2b, Y(n) can be viewed as a sum of two independent variables, Y(n-1) and Y(1),

Y(n) ) Y(n-1) + Y(1)

(A5)

The constraints that -1 e Y(1) e +1 and -(n - 1) e Y(n-1) e + (n - 1) lead to the following allowed range for Y(1) for a given Y(n)



max(-1, Y(n) - (n - 1)) e Y(1) e min(+1, Y(n) + (n - 1)) (A6) The distribution function for Y(n), ΩY(n)(Y(n)), is then given by the integral of ΩY(n-1)(Yn-1)ΩY(1)(Y(1)) over all values of Y(n-1) and Y(1) that sum to Y(n). This corresponds to the following convolution function,

ΩY(n)(Y(n)) )

min(+1,Y +n-1) I(Y(1)) dY(1) ∫max(-1,Y -n+1) (n)

(A7)

(n)

with I(Y(1)) ) ΩY(1)(Y(1))ΩY(n-1)(Y(n) - Y(1)). This allows ΩY(2)(Y(2)) to be obtained from ΩY(1)(Y(1)) of eq A4 as

ΩY(2)(Y(2)) )

( (x

))

(2 - Y )(2 + Y ) 2

1 π F , π2 2

(2)

(2)

h for i ∈ [1, ..., m] 2

(A8)

(A9)

where m is the number of bins in the interval of [-1, +1], h ) (2/m) is the step size, and i is the array index. These grid points avoid the end points of -1 and +1 since, according to eq 31, ΩY(1)(Y(1)) is singular at these points. This singularity is, however, integratable and so we choose the values of ΩY(1) (1) (Y(1) i ) at the numerical grid points such that ΩY(1)(Yi ) are obtained from a midpoint integration. This leads to

ΩY(1)(Y(1) i )=

1 h

1 ) π

+h/2 ΩYh (1)(Y h (1)) dY h (1) ∫YhYh -h/2 (1) i (1) i

(

arccos Y h (1) i +

h h - arccos Y h (1) i 2 2 (A10) h

)

(

2 sin(2Θ h (n)) ΩYh (n)(Y h (n)) b(n)βo sin(Θ h (n))

where F(φ,k) is the elliptic integral of the first kind.26 For n g 3, we could not obtain an analytical form for ΩY(n)(Y(n)) and instead used the following numerical integration approach. The integration was done by using the following grid for Y(1)

Y(1) i ) -1 + ih -

generate ΩY(n)(Y(n)) for any value of n. The configurational h (n)), can then be obtained through eqs A3a and entropy, ΩΘh (n)(Θ A3b. In Section IV, the spectral density, χ(ω), as a function of frequency is given by eq 25. Since the sin (Θ h (n)) term in the (n) denominator of eq 25 goes to zero at Θ h ) 0, the behavior of the density of states depends on the behavior of the numerator, ΩΘh (n)(Θ h (n)) at Θ h (n) ) 0. For n ) 1, ΩΘh (n)(Θ h (n)) is a constant at Θ h (n) ) 0 and this leads to a cusp in the density of states. For n g 2, substitution of eqs A3a and A3b into the first two terms of eq 25 leads to the form,

)

A discrete version of eq A7 can now be constructed as

which reduces to

h (n))) 4n cos(Θ h (n))ΩY(n)(n cos(2Θ b(n)βo

) ΩY ∑ΩY(n-1)(Y(n-1) j

(Y(1) i )

(1)

{i,j}

(A11)

Appendix B: Configuration-Coordinate Model in the Infinite Polymer Limit Although this paper focuses on predicting the properties of oligomers, it is useful to know if the CC model behaves well in the limit that N goes to infinity. In this case, the centrallimit theorem is valid, and the leading term in the entropy is linear in N - 1 [eq 18]. The energy also is linear in N [eq 8]. Thus, as N increases, the competition between energy and entropy will stabilize, and the position of the free-energy minimum will reach a limiting position between 0° and 45°. This behavior can be seen in Figures 9, 10, and 12. It is commonly assumed that the transition energy of a fully conjugated and planar polymer will approach a fixed value as N increases. In terms of our model [eq 11]

Y(n) k ) -n + kh +

(n - 2)h 2

for

k ∈ [1, ..., mn - n + 1]

(A12)

Given this definition for the grid points Y(n) k , the restricted sum of eq A11 corresponds to a summation over all pairs of {i, j} that satisfies k ) i + j - 1. This procedure may be used to

(

lim b(N) ) b(∞) 1 -

nf∞

where the summation is over all pairs of {i, j} that satisfies Y(n) k ) Y(n-1) + Y(1) j i . The numerical grid for Y(1) in eq A9 spans the range [-1 + (h/2), 1 - (h/2)] in a step size of h. Equation A11 generates Y(2) on a grid with a range of [-2 + h, +2 - h] and a step size of h. In general, the grid for Y(n) will span the range [-n + (nh/2), + n - (nh/2)] with a step size of h. The grid points for Y(n) are then

(A14)

h (2))) is given by the elliptical integral For n ) 2, ΩY(2)(2 cos(2Θ of eq A8, which has a constant value at Θ h (2) ) 0°. Therefore, for n ) 2, χ(ω) has a finite value at Θ h (2) ) 0°. For n g 3, ΩY(n)(n cos(2Θ h (n))) is equal to zero at Θ h (n) ) 0° and so χ(ω) (n) goes to zero at Θ h ) 0°.

′

ΩY(n)(Y(n) k ))h

(A13)

c N

)

(B1)

where b(∞) and c are constants. Thus the coupling potential reaches a fixed form as N becomes large. Combined with the fact that the free-energy minima and therefore the peaks of the populations reach stable values at large N, the CC model predicts that the peaks of the absorption and emission spectra will reach fixed frequencies in the polymer limit. However, that limit does not correspond to the planar form of the polymer, even in the more strongly planarized excited state. Although the CC model makes well-behaved predictions of the peak frequency for large N, the predictions of the line widths must fail for sufficiently large N. As argued at the beginning of this section, the free-energy potentials scale linearly with N for large lengths. Thus the curvature at the minimum of the free-energy potentials will increase without bound and the predicted line widths will go to zero. The reason for this artifact can be seen in Figure 7, which shows the distribution of excitation energies for different mean torsional angles. The total range of excitation energies is due

5782 J. Phys. Chem. C, Vol. 111, No. 15, 2007 to two factors: the thermally excited distribution in Θ0 and the range of other collective angles {Θ1 ... ΘN-2}. The former is included in the CC model; the latter is not. The CC model is reasonable, so long as the range attributable to Θ0 is larger than the range due to {Θ1 ... ΘN-2}. As the length of the oligomer increases, the thermal distribution of Θ0 becomes narrower and narrower. When the range of excitation energies due to Θ0 becomes smaller than the range due to {Θ1 ... ΘN-2}, the CC results for line widths are no longer valid. Alternatively, this artifact can be viewed as arising from breakdown in the CC model’s assumption that the excitation is delocalized uniformly over the entire oligomer. Such an assumption is not made in the exciton model, which explicitly computes the wave function and so includes localization induced by the torsional disorder. The agreement between the CC and exciton models for N up to 9 (Figure 6) implies that localization is not an important contributor to the line shape for these length oligomers. However, for N ) 20, the disagreement between the exciton and CC models seen in Figure 6 can be attributed to the average delocalization of the wave function, as measured by participation ratios, being considerably less than the full length of the N ) 20 oligomer. Acknowledgment. We thank Adelheid Godt for providing the oligomer samples and Uwe H. F. Bunz for providing the polymer sample. This work was supported by the National Science Foundation (CHE-0220986 and CHE-0316759). References and Notes (1) Liu, L. T.; Yaron, D.; Sluch, M.; Berg, M. A. J. Phys. Chem. B 2006, 110 (38), 18844-52. (2) Heimel, G.; Daghofer, M.; Gierschner, J.; List, E.; Grimsdale, A.; Mu¨llen, K.; Beljonne, D.; Bre´das, J.; Zojer, E. J. Chem. Phys. 2005, 122, 054501.

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