Diffusion of Frenkel excitons

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

In the diffusive regime, the quantity of interest is the exciton concentration c(r,t) as a function of position r and time t. It obeys the diffusion equation 

A useful measure for exciton migration is the diffusion length L = (Dto)1/2. Experimental data show that for Frenkel excitons in molecular crystals at room temperature the diffusion coefficient D 10 3 cm2/s and the lifetime of singlet excitons to 10 8s. This gives a typical diffusion length L 10 6cm (for anthracene crystals L 5 10 6cm). 

In the remainder of this chapter, we will focus on various ways to calculate the diffusion constant from microscopic principles. We will start by considering 

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