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 (1¹Bᵤ^-) and two-photon (2¹Ag⁺) states in one-dimensional conjugated systems with chain lengths N up to $N=80$ sites. Three different types of crossovers are studied, as a function of $U/t,$ {δ}, and $N.$ The ``U crossover'' emphasizes the larger ionic character of the 2Ag state compared to the lowest triplet excitation. The ``{δ} crossover'' shows strong dependence on both N and $U/t.$ the ``N crossover'' illustrates the more localized nature of the 2Ag excitation relative to the 1Bᵤ excitation at intermediate correlation strengths.