Theoretical and computational studies of excitons in conjugated polymers

Abstract

We present a theoretical and computational analysis of excitons in conjugated polymers. We use a tight-binding model of $\ensuremath{\pi}$-conjugated electrons, with $1/r$ interactions for large r. In both the weak-coupling limit (defined by $W\ensuremath{\gg}U)$ and the strong-coupling limit (defined by $W\ensuremath{\ll}U),$ where W is the bandwidth and U is the on-site Coulomb interaction, we derive and analyze effective-particle models. We compare these to density matrix renormalization group (DMRG) calculations, and find good agreement in the extreme limits. We use these analytical results to interpret the DMRG calculations in the intermediate-coupling regime (defined by $W\ensuremath{\sim}U),$ most applicable to conjugated polymers. We make the following conclusions. (1) In the weak-coupling limit the bound states are Mott-Wannier excitons, i.e., conduction-band electrons bound to valence-band holes. Singlet and triplet excitons whose relative wave functions are odd under a reflection of the relative coordinate are degenerate. Thus, the $2{}^{1}{A}_{g}^{+}$ and $1{}^{3}{A}_{g}^{\ensuremath{-}}$ states are degenerate in this limit. (2) In the strong-coupling limit the bound states are Mott-Hubbard excitons, i.e., particles in the upper Hubbard band bound to holes in the lower Hubbard band. These bound states occur in doublets of even and odd parity excitons. Triplet excitons are magnons bound to the singlet excitons, and hence are degenerate with their singlet counterparts. (3) In the intermediate-coupling regime Mott-Wannier excitons are the more appropriate description for large dimerization, while for the undimerized chain Mott-Hubbard excitons are the correct description. For dimerizations relevant to polyacetylene and polydiacetylene both Mott-Hubbard and Mott-Wannier excitons are present. (4) For all coupling strengths an infinite number of bound states exist for $1/r$ interactions for an infinite polymer. As a result of the discreteness of the lattice and the restrictions on the exciton wave functions in one dimension, the progression of states does not follow the Rydberg series. In practice, excitons whose particle-hole separation exceeds the length of the polymer can be considered unbound. (5) The DMRG calculated exciton excitation energies scale as the inverse of the chain length for short chains and the inverse of the square of the chain length for long chains. This fits the effective-particle-in-a-box model.