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