Diagrammatic exciton-basis valence-bond theory of linear polyenes

Abstract

Understanding the photophysics of $\ensuremath{\pi}$-conjugated polymers requires a physical understanding of the excited states involved in the photophysics. Detailed physical understanding is difficult because of the extensive configuration interaction that occurs within realistic theoretical models for these systems. We develop a diagrammatic exciton-basis valence-bond representation that is particularly suitable for the intermediate magnitude of the Coulomb interactions in these systems. We present detailed comparisons of our exact exciton-basis treatment and previous approximate approaches, focusing on the specific many-body and single-particle interactions that have been ignored in the past, and the consequences thereof. Following this, we present the results of exact numerical calculations for the noninteracting band limit, the limit of isolated dimers interacting through Coulomb interactions, and for the Pariser-Parr-Pople Ohno Coulomb interactions with two different bond-alternation parameters for the ten-carbon linear polyene. Simple pictorial descriptions of the eigenstates relevant in photophysics are obtained in each case, and taken together, these results provide a systematic characterization of both low- and high-energy excited states in linear chain $\ensuremath{\pi}$-conjugated systems for realistic parameters. Two different quantities, the number of effective excitations within the exciton basis, and the particle-hole correlation length for the one-excitation eigenstates are defined and calculated for further quantitative comparisons between the eigenstates. A pictorial description of optical nonlinearity is obtained thereby. For both small and large bond alternation, it is found that the two-photon state that dominates third order optical nonlinearity in the low-energy region is the lowest even parity one-excitation state with a larger particle-hole correlation length than the ${1B}_{u}$ exciton. The reason for the dominance by this ${\mathrm{mA}}_{g}$ state can be understood within the exciton basis from the nature of the current operator. It is shown that the relationship between the correlated ${\mathrm{mA}}_{g}$ and the correlated ${1B}_{u}$ is identical to that between the uncorrelated ${2A}_{g}$ and the uncorrelated ${1B}_{u}.$ In the high-energy region of the spectrum evidence for stable biexcitons is found from the nature of the singlet-singlet two-excitation wave functions.