A first principles examination of phosphorescence

Abstract

This paper explores phosphorescence from a first principles standpoint, and examines the intricacies involved in calculating the spin-forbidden T ₁ → S ₀ transition dipole moment, to highlight that the mechanism is not as complicated to compute as it seems. Using gas phase acridine as a case study, we break down the formalism required to compute the phosphorescent spectra within both the Franck-Condon and Herzberg-Teller regimes by coupling the first triplet excited state up to the S ₄ and T ₄ states. Despite the first singlet excited state appearing as an L _b state and not of nπ* character, the second order corrected rate constant was found to be 0.402 s^-1, comparing well with experimental phosphorescent lifetimes of acridine derivatives. In showing only certain states are required to accurately describe the matrix elements as well as how to find these states, our calculations suggest that the nπ* state only weakly couples to the T ₁ state. This suggest its importance hinges on its ability to quench fluorescence and exalt non-radiative mechanisms rather than its contribution to the transition dipole moment. A followup investigation into the T ₁ → S ₀ transition dipole moment's growth as a function of its coupling to other electronic states highlights that terms dominating the matrix element arise entirely from the inclusion of states with strong spin-orbit coupling terms. This means that while the expansion of the transition dipole moment can extend to include an infinite number of electronic states, only certain states need to be included.

Fig. 1. Schematic representation of the acridine chromophore.

Fig. 2. Molecular orbitals for acridine in the gas phase.

Fig. 3. Second order corrected phosphorescence emission spectra predicted from first principles for acridine in the gas phase.

Fig. 4. Fine-structure of the phosphorescence spectra. Second-order corrected spectra as a function of (A) temperature, and (B) Gaussian damping, in units of K and cm⁻¹ respectively. Gaussian damping spectra have been offset to more clearly show the minute differences in the fine-structure. (C) Infra-red spectra of the T₁ state, with a Gaussian broadening of 4.5 meV. (D) Phosphorescence due to only Franck–Condon terms are shown to illustrate possible convolution of modes resulting in an almost Gaussian lineshape. Spectra are not normalised.

Fig. 5. Scaling of M as a function of inter-state couplings. For an arbitrary state, M evolves by increasing either transition dipole matrix elements, spin–orbit coupling matrix element, or the energy separation between states.