Quantum-confinement effects on the ordering of the lowest-lying excited states in conjugated chains

The symmetrized density-matrix renormalization-group approach is applied within the extended Hubbard-Peierls model (with parameters $U/t,$ $V/t,$ and bond alternation δ) to study the ordering of the lowest one-photon ${(1\mathrm{}}^1{B}_u^{-})$ and two-photon ${(2\mathrm{}}^1{A}_g^{+})$ states in one-dimensional conjugated systems with chain lengths $N$ up to $N=80\mathrm{}$ sites. Three different types of crossovers are studied, as a function of $U/t,$ δ, and $N.\mathrm{}$ The “$U$ crossover” emphasizes the larger ionic character of the ${2A}_g$ state compared to the lowest triplet excitation. The “δ crossover” shows strong dependence on both $N$ and $U/t.\mathrm{}$ the “$N$ crossover” illustrates the more localized nature of the ${2A}_g$ excitation relative to the ${1B}_u$ excitation at intermediate correlation strengths.