The Pariser-Parr-Pople-Peierls model

the Pariser-Parr-Pople-Peierls model is defined as 

The variables and parameters in eqns (7.2) - (7.5) are defined in Chapter 4. Vmn (defined in eqn (2.55)) is the Ohno potential for the undistorted structure. For generality, we also include an extrinsic dimerization, represented by the term 

The second term on the right-hand side of eqn (7.1) is the change in the Coulomb interactions arising from changes in bond length, where 

7 = (C/14.397) and ro is the undistorted average bond length. It is instructive to rewrite this term as 

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We start this investigation by treating the electronic degrees of freedom within the Born-Oppenheimer approximation, where the nuclear degrees of freedom are static, classical variables. The 7r-electron model that describes both electron-electron and electron-phonon interactions in the Born-Oppenheimer approximation is known as the Pariser-Parr-Pople-Peierls model. This is described and its predictions are analyzed in the following sections. Chapter 10 will deal with quantum phonons in an interacting electron model, specifically for trans-polyacetylene. 

First, we examine the relaxed and vertical energies of the Pariser-Parr-Pople-Peierls model as a function of the interaction strength. These transition energies are illustrated in Fig. 7.4. We first note the crossover in the vertical energies of the and states as a function of C (as already discussed in Chapter... 

Table 7.1 The vertical and relaxation energies (in eV) of a site linear polyene calculated from the Pariser-Parr-Pople-Peierls model with extrinsic dimerization (t = 2.5 eV, U = 10.0 eV, A = 0.1, and <5e = O.i ...

The predictions of the Pariser-Parr-Pople-Peierls model (defined by eqn (7.1)) as a function of the Coulomb interaction strength for the linear polyene structure were described in Chapter 7. In this section we discuss the predictions taken from Barford et al. (2001) of the model for the particular parameter set relevant for trans-polyacetylene. 

Fig. 10.4. The staggered, normalized bond dimerization, S , (defined in eqn (4.27)) as a function of bond index from the centre of the chain of various states of trans-polyacetylene calculated from the Pariser-Parr-Pople-Peierls model. 1 A+ (crosses), (squares), 1 5 (triangles), 2 (diamonds) andpolaron (circles).

The adiabatic approximation is widely accepted as being applicable to the electronic states of conjugated polymers. As described above, solutions of an adiabatic Hamiltonian (namely, the Pariser-Parr-Pople-Peierls model) agree remarkably well with experimental observations for short polyenes. A linear extrapolation in inverse chain length of the experimental observations coincide with the experimental observations of the energies of the and states in thin... 

The Pariser-Parr-Pople-Peierls model is the adiabatic limit of this model, taken by setting M oo and treating the nuclear displacements classically. However, now we intend to quantize the nuclear degrees of freedom. To do this... 

To investigate the combined effects of electron-lattice and electron-electron interactions we again employ the Pariser-Parr-Pople-Peierls model introduced in Section 7.2. Notice that the calculations described here do not describe free rotations of phenyl rings relative to one another. Thus, their applicability are to ladder poly(para-phenylene), where the stereochemistry causes the rings to have a planar geometry, or polymers in the solid state, where ring rotations are more restricted. 

Fig. 11.19. The fractional change in transfer integrals of eight-ring para-phenylene oligomers from the uniform value, t, in the interacting limit. The parameters used in the Pariser-Parr-Pople-Peierls model (eqn (7.1)) are U = 10.06 eV, t = 2.514 eV, and A = 0.12. The labels refer to the bonds shown in Fig. 11.16. Only the upper rung of bonds are shown.

In the next section we describe how the semiempirical Pariser-Parr-Pople-Peierls model - a correlated 7r-electron model with electron-lattice coupling -quantitatively predicts the excitation spectrum of polyene oligomers, while it qualitatively predicts the spectrum for frons-polyacetylene thin films. 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

Pariser-Parr-Pople-Peierls Static-nuclear hmit of the P-P-P-SSH model 7... 

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