Excitonic States, Fundamental Equations

We assume that the crystal consists of N unit cells. We will limit ourselves here to crystals with no more than two molecules in the unit cell. The Hamiltonian H of the pure molecular crystal is given by 

Hma is the Hamiltonian of the molecule a in the m-th unit cell of the crystal, describes the pairwise interaction between the molecule a in unit cell m with molecule in unit cell n (cf. the fundamental work of Silbey et al. [22]). With two molecules in the unit cell, the indices a and p mn over these two sites, and m and n ran over aU the N unit cells. 

For the ground state of the crystal, in analogy to Eq. (5.1) we obtain with the ground-state functions (j ma of the individual molecules the function 

A is here an antisymmetrisation operator. For the excited state in which one molecule na is excited and aU the others remain in their ground states, the wavefunction 

The localised single-exdton wavefunctions are not eigenfunctions of the crystal Hamiltonian. A wavefunction appropriate to the crystal symmetry and the periodic potential can be found using the Bloch-wave ansatz. From the localised basis functions 1 , one obtains the delocalised wavefunction 

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