Comparisons to the numerical calculations

It is instructive to compare the approximate weak-coupling theory to essential exact, numerical (density matrix renormalization group) calculations on the same model (namely the Pariser-Parr-Pople model). The numerical calculations are performed on polymer chains with the polyacetylene geometry. Since these chains posses inversion symmetry the many-body eigenstates are either even (Ag) or odd By). As discussed previously, the singlet exciton wave function has either even or odd parity when the particle-hole eigenvalue is odd or even. Conversely, the triplet exciton wavefunction has either even or odd parity when the particle-hole eigenvalue is even or odd. As a consequence, we can express a B state as 

Generally, the sums will be dominated by one component (except at anticrossings, as discussed shortly). The other contributions to the state vectors include components not described by the exciton basis, for example, covalent and holon-doublon terms. These are expected to be neghgible in the weak-coupling limit. 

Particle-hole separation, m (in units of the repeat distance) 

For large N the energies scale as 1/AT. A detailed analysis (Barford et al. 

2 Particle-hole correlation function The exciton component of the numerical many-body wavefunction can be obtained using the operator defined in eqn (6.8). Using eqns (6.7) and (6.12) it follows that projecting S )J GS) onto the exciton state gives the exciton wavefunction nj(u R) - 

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