Small corrections to Heitler-London approximation

To demonstrate the idea of the calculations we consider below the case of a molecular crystal with one two-level molecule in the unit cell (Subsection 3.4.1). For this model the second-quantized Hamiltonian of the crystal has the form 

Here Eq is the energy of the molecular excitation where also the gas-condensed matter shift is taken into account, and Pn, Pn are the exciton creation and annihilation Pauli operators on the molecule n, where n labels the unit cells of the 

To estimate the importance of such corrections it is reasonable to compare them with the width of the exciton states. For the lowest energy exciton in anthracene this correction is of the order of a few cm-1, while the width of the exciton level even at low temperatures is of the order of several tens of cm-1. Thus, for the lowest energy in anthracene the correction may be neglected. However, for example, for the lowest energy exciton in a-quaterthienyl (T4) this correction is already a few hundred cm-1, and for the second (intense) transitions in anthracene and in many other molecular solids this correction can be of the order of a thousand cm-1. And it is clear that the corrections to the HLA can be particularly important in the theory which takes into account the mixing of molecular configurations with the distance between mixed excited states smaller or of the order of the intermolecular interaction. 

The contribution to the ground state energy (equal to zero in the HLA) in the boson approximation, when all terms quadratic in Bose amplitudes are taken into accout, to the lowest order is equal to 

To calculate the corrections to the HLA we may take advantage of the smallness of V/E0 explicitly, using perturbation theory with respect to the terms that do not conserve the number of particles. All the first-order corrections to the energies vanish since the perturbation has no diagonal matrix elements in the HLA eigenstate basis. The correction to the energy Es of the state s) due to the perturbation V is given by 

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