Derivation of the effective-particle Schrodinger equation

The Schodinger equation for the effective-particle model of excitons was introduced in Chapter 6. In this appendix we derive that equation. 

First we need to derive an exciton Hamiltonian in the weak-coupling limit. To do this it is necessary to recast the Pariser-Parr-Pople model (see Section 2.8.3), 

As a simplification, we assume that the Wannier molecular orbitals for a linear, dimerized chain are localized on a particular dimer, that is, 

Substituting eqns (D.4) and (D.5) into the Pariser-Parr-Pople model, eqn (D.l), we obtain the molecular orbital exciton Hamiltonian,  

The first two terms are the Coulomb interactions between electrons on the same dimer, while the third term is the Coulomb interaction between electrons on dimers t units apart. 

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