Density operator description of excitons in molecular aggregates: Optical absorption and motion. I. The dimer problem

Abstract

A compact formalism is proposed for a rigorous calculation of the optical absorption and of the subsequent exciton motion and thermalization in molecular aggregates. The formalism is based on superoperators technique which is used to derive a macroscopic evolution operator from Liouville's equation written in Lie's algebra, the basis of which is formed with eigenoperators of the constants of motion. The compactness advantages of such an algebra are displayed in actual calculations of observables, where the evolution equation is decoupled in a number of independent equations of easier handling, each one corresponding to a specific physical process. Comparison is made with the technique used by Haken et al. to derive the same density-operator equation. The formalism allows simple derivation of optical line shapes, for different rates of coherent to incoherent couplings ($\frac{V}{{\gamma }_0}$). Actual calculations are presented for a simple case of a statistical ensemble of pairs of anthracene molecules, in their crystal-cell mutual orientation. In that case, the line-shape formula reduces to that obtained by Haken and Reineker. The formalism is also used to derive a time-dependent exciton density operator $\mathcal{E}$, which describes rigorously the time-dependent nonstationary state of an ensemble of molecular aggregates excited optically. The initial state, the motion of the excitation, and its thermalization are discussed in terms of time-dependent first and second moments and coherence of the excitation, calculated with the kernel of $\mathcal{E}$ and using the stochastic exciton Hamiltonian proposed by the precited authors. The results are paralleled with those provided by models of motion such as coherent and incoherent hopping excitons; cases where models might provide artificial motions are discussed. The two moments, taken as position and volume, lead to a consistent and pictorial description of the exciton statistical motion towards its thermalized state. Recipe formulas and tables are provided to use the superoperators technique, which is shown to be very flexible to allow exact solutions for "one-dimension exciton" crystals, for clusters or larger aggregates observed as traps in optical and ESR experiments in mixed crystals.