Correlation-Driven Singlet--Triplet Inversion and Optical Spectra of N-Substituted Phenalenyls: Insights from PPP-FCI and TDDFT

Abstract

Several researchers have reported an atypical inversion of the lowest singlet ($S_{1}$) and triplet ($T_{1}$) excited states in nitrogen-substituted planar hydrocarbons. Contrary to Hund's rule of maximum multiplicity, poly-aza derivatives of phenalenyl, such as cyclazine and heptazine, have long been known to possess a negative singlet-triplet gap, $ΔE_{ST}=E(S_{1})-E(T_{1})$. Using the well-known Pariser-Parr-Pople (PPP) model, a semi-empirical, wave-function-based Hamiltonian for correlated $π$-electrons, we investigate the singlet-triplet gap in ten N-substituted phenalenyl motifs. Two schemes of long-range Coulomb interactions are considered, and key parameters, including the on-site orbital energy, Hubbard U, and hopping integrals, are systematically tuned to benchmark the model for reliably estimating negative singlet-triplet gaps. Full configuration-interaction (FCI) method is employed to incorporate electron-correlation effects. Ground-state and $T_{1}$ absorption spectra are subsequently calculated using the optimized PPP parameters that best reproduce the available experimental data. For validation, first-principles TDDFT calculations using various exchange--correlation functionals are also performed. Conventional adiabatic TDDFT fails to reproduce the inverted ($S_{1}-T_{1}$) ordering, indicating that the inclusion of the correlation driven multiconfigurational effects are essential for describing it. We find that the singlet TDDFT ground-state absorption spectra reproduce several qualitative features of the PPP-FCI spectra, although the TDDFT excitation energies are systematically higher. Detailed analysis of the ground-state and $T_{1}$ absorption spectra reveals significant configuration mixing, providing a comprehensive characterization of the optical response of the N-substituted phenalenyl derivatives studied.