Theory of optical transitions in conjugated polymers. I. Ideal systems

Abstract

We describe a theory of linear optical transitions in conjugated polymers. The theory is based on three assumptions. The first is that the low-lying excited states of conjugated polymers are Frenkel excitons coupled to local normal modes, described by the Frenkel-Holstein model. Second, we assume that the relevant parameter regime is ℏω ≪ J, i.e., the adiabatic regime, and thus the Born-Oppenheimer factorization of the electronic and nuclear degrees of freedom is generally applicable. Finally, we assume that the Condon approximation is valid, i.e., the exciton-polaron wavefunction is essentially independent of the normal modes. Using these assumptions we derive an expression for an effective Huang-Rhys parameter for a chain (or chromophore) of N monomers, given by S(N) = S(1)/IPR, where S(1) is the Huang-Rhys parameter for an isolated monomer. IPR is the inverse participation ratio, defined by IPR = (∑(n)|Ψ(n)|(4))(-1), where Ψ(n) is the exciton center-of-mass wavefunction. Since the IPR is proportional to the spread of the exciton center-of-mass wavefunction, this is a key result, as it shows that S(N) decreases with chain length. As in molecules, in a polymer S(N) has two interpretations. First, ℏωS(N) is the relaxation energy of an excited state caused by its coupling to the normal modes. Second, S(N) appears in the definition of an effective Franck-Condon factor, F(0v)(N) = S(N)(v)exp ( - S(N))/v! for the vth vibronic manifold. We show that the 0 - 0 and 0 - 1 optical intensities are proportional to F00(N) and F01(N), respectively, and thus the ratio of the 0 - 1 to 0 - 0 absorption and emission intensities are proportional to S(N). These analytical results are checked by extensive DMRG calculations and found to be generally valid, particularly for emission. However, for large chain lengths higher-lying quasimomentum exciton states become degenerate with the lowest vibrational excitation of the lowest exciton state. When this happens there is mixing of the electronic and nuclear states and a partial breakdown of the Born-Oppenheimer approximation, meaning that the ratio of the 0 - 0 to 0 - 1 absorption intensities no longer increases as fast as the IPR. When ℏω/J = 0.1, a value applicable to phenyl-based polymers, the critical value of N is ~20 monomers.

Ultra-fast relaxation, decoherence, and localization of photoexcited states in π-conjugated polymers

“…One reason for this lack of theoretical insight is the inherent challenge in simulating the exciton relaxation dynamics numerically, due to the strong exciton-phonon interactions synonymous with polymer systems. One common way to treat exciton-phonon interactions within these systems is to use the one dimensional Frenkel-Holstein model, 10,11 which consists of a linear array of sites through which the exciton can propagate (where each site corresponds to a moiety in the polymer chain), and where the nuclear degrees of freedom are represented by a single harmonic oscillator per site, which couples locally to the exciton. The size of the Hilbert space associated with this model grows exponentially with the number of sites in the chain, which means that computationally solving the time-dependent Schrödinger equation using standard differential equation solvers becomes unfeasible for even modest chain lengths.…”

Theory of optical transitions in curved chromophores

“…In particular, these authors showed that the emission intensity ratio satisfies I 01 /I 00 = S(1)/N, where S(1) is the Huang-Rhys parameter for a single monomer which forms the aggregate or polymer and N is the exciton coherence length. In related work, Barford and co-workers 3,4 showed that within the Born-Oppenheimer approximation both for emission and absorption the intensity ratios satisfy I 01 /I 00 ≃ A 01 /A 00 = S(1)/IPR, where IPR (the exciton inverse participation ratio) can be regarded as the average chromophore size in conformationally disordered π-conjugated polymers. From an experimental perspective, these results are useful as they allow the chromophore sizes to be determined, and in principle polymer conformations to be found.…”

Intrachain exciton dynamics in conjugated polymer chains in solution

“…arises from the ratio between the root-mean-square spread of a LEGS and the length of chain it occupies, as shown by Makhov and Barford 1. The exciton localization length is a physically important value, as it can be directly related to the relative peak heights in the vibronic progression in both absorption and emission spectra of conjugated polymers26,27 as well as potentially playing an important role in an explanation of other effects, such as ultrafast fluorescence depolarization20,28,29 and exciton dynamics (as discussed The exciton localization length at various finite temperatures for the 'free' (i.e., 'untrapped') exciton (red circles) and exciton-polaron (i.e., 'self-trapped') (black squares). Although increasing the temperature causes a slight increase in the size of the exciton-polaron, this is only a very minor effect.…”