Density-operator description of excitons in molecular aggregates. III. Excitonic motion in one-dimensional finite chains

Abstract

The exciton density-operator and the superoperator technique introduced in parts I and II of this series are now used as a theoretical frame to derive a rigorous description by spatial moments of the exciton motion in finite linear chains. Expressions for exciton spatial moments are deduced from this operator by conventional procedures. The exciton motion is discussed in terms of the dispersion tensor of the wave packet and of the displacement of its center of gravity which actually moves in finite systems and exhibits important edge effects. The packet itself is considered as a coherent superposition of physically accessible collective excitations of the chain. The motion is discussed in detail in short and long chains, and the problem of the recurrence time is examined. This work may be a prerequisite for a microscopic description of coherentmotion-dependent exciton-exciton interactions, for which it is shown that the billiardball site-to-site motion is not valid. Conditions of validity---a narrow exciton band and strongly localized-site wave functions---are provided for models using site populations. Under these conditions, a moment-generating function is derived which allows standard derivation of all moments. Diffusion constants and exciton velocities are calculated for finite systems in initial times, their correlation with formulae obtained by other authors in infinite systems is shown. The failure of the dispersion tensor to provide a full description of the delocalization of the wave packet is pointed out and illustrated. An entropylike function is introduced, which contains information from all the moments of the probability distribution, and which is shown to describe correctly the delocalization of the exciton in time. Actual calculations of exciton motion in 3- to 50-site chains are presented in the cases of purely coherent, purely incoherent, and general motion. The moment expressions are analytical except for the latter situation where a numerical procedure is used. The analysis of the motion shows that it cannot be separated into coherent and incoherent components since the relaxation operator $\stackrel{^}{\ensuremath{\Gamma}}$ strongly couples the eigenstates of the energy operator ${\stackrel{^}{H}}_{0}$. Finally, we examine the equilibrium density operator when represented on two basis sets and discuss the main limitation of the stochastic model, namely that it leads to equal populations at equilibrium, whatever the temperature.