The Pariser-Parr-Pople Model

Interacting electrons with fixed nuclei satisfy the Pariser-Parr-Pople model (Pariser and Parr 1953a, b Pople 1953, 1954), defined as 

Propagator or Green s function methods are employed in this chapter to analyze the many-electron problem in planar unsaturated molecules as treated within the Pariser-Parr-Pople (PPP) model. A derivation of the model in many-electron theory serves to demonstrate the nature of the approximations involved. Applications are presented for the case of weakly interacting atoms. A decoupling procedure for Green s functions proposed by the authors is shown capable of yielding a correct description of this case. 

Propagators have the advantage of giving direct information about transition energies and amplitudes from a reference state, but like density matrices, they suffer from a lack of simple ways generally to ensure so-called A-representability or correspondence to proper many-electron state vectors. Nevertheless, the propagator approach to semi-empirical many-electron theory appears to have certain advantages over other methods. Such treatment has led to useful relations between matrix elements in the PPP-model P 

The molecular orbital method is a very flexible and often successful tool for analyzing electronic structure-dependent properties. Its deficiencies are intimately connected with the treatment of superpositions of configurations. In particular, the molecular orbital model is not satisfactory when the overlap between relevant valence orbitals on adjacent atoms is smaller than 1/2. This result was particularly well illustrated by Coulson and Fischer in their well-known study of the hydrogen molecule, and it is relevant for the molecular orbital treatment of TT-electron systems, where the typical overlap is in the range 1/3-1/4. Evidence has also been presented for the insufficiency of the PPP-model when 

An alternative approach to extend the molecular orbital method is offered by the work of Hubbard for the study of narrow energy bands in solids with the aim to study magnetism. The main idea of this work is to analyze the many-electron problem for the case of separated atoms, which means the limit of zero bandwidth. 

In order to derive the PPP-hamiltonian, it is assumed that the nuclear framework of the molecular system is invariant under the point group Cg, which contains the operations of identity and a reflection. This is not always true for all molecules to which the model is applied, and then the magnitude of the perturbation caused by the noninvariant part of the nuclear potential must be examined. The electron field operators t) are expressed as the sum of two components, each transforming according to an irreducible representation of C,  

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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-... 

If the basis set is restricted to one pn basis function on each sp2 carbon, if the two-electron integrals ignore all three-center or four-center ones, and if we exclude exchange components, one has the Pariser-Parr-Pople model. If, further, all two-electron integrals are set to zero except for the repulsion between opposite spins on the same site and the one-electron tunneling terms are restricted to nearest neighbors, the result is the Hubbard Hamiltonian... 

Table 3. One and two-photon thresholds (in eV) of trans-polyenes, CnHn+2, in alkane matrices[58] and in the Pariser-Parr-Pople model[21]. The solid-state shift of 0.40eV of the ionic 1 1B state yields gas-phase values up to n = 12 covalent 2M+ shifts of 0.05eV are neglected.

J. l.inderberg. Chon. Phys. Lett., 1,39 (I 967). Consistency Requirement in the Pariser-Parr-Pople Model. J. Linderberg and L. Seamans, Int. J. Quantum (.hem., 8, 925 (1974). Matrix Elements in All Valance Electron Models. 

From the high extinction values and the Pariser-Parr-Pople model... 

Figure 25 Evolution of E lBu) — E 2Ag), as a function of polyene chain length, in the Pariser-Parr-Pople model.

The orbital expansions Eq. (11.10) and Eq. (11.12) are inserted in the hamil-tonian in Eq. (11-7), resulting in the Pariser-Parr-Pople model hamiltonian... 

Huckel theory for the even alternant hydrocarbons leads to the Coulson-Rushbrooke theorem and some other characteristic results shown by McLach-lan to be valid also in the Pariser-Parr-Pople model. These are the well-known pairing relations between electronic states of alternant hydrocarbon cat-and anions. This particle-hole symmetry is analogous to the situation discussed in Chapter 4 for electrons and holes in atomic subshells. 

The Pariser-Parr-Pople model is known as the extended Hnbbard model in solid state physics. 

Fig. 5.4. The phase diagram of the Pariser-Parr-Pople model at half-filling, t = 2.5 eV.

Having discussed the weak and strong coupling limits of the Pariser-Parr-Pople model, we can now qualitatively explain the behaviour of the excitation energies shown in Fig. 5.2. 

Table 5.1 The molecular orbital eigenstates of the dimer (namely the noninteructing limit of the Pariser-Parr-Pople model) expressed within the valence bond basis...

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... 

Fig. 6.3. The transition energies of the lowest momentum (j = 1) n = 1 singlet (solid curve), n = 1 triplet (dotted curve), n = 2 singlet and triplet (short-dashed curve) excitons, and the charge gap (long-dashed curve) in the weak-coupling limit, t = 2.5 eV and 6 = 0.2. The symbols are the DMRG calculations of the Pariser-Parr-Pople model on 102 site chains filled circles, (n = l,j = 1 singlet) full circles,...

Theoretical work also suggests the important role of electronic interactions in linear polyenes. By performing a double-configuration-interaction calculation on the Pariser-Parr-Pople model, Schulten and Karplus (1972) demonstrated that the 2 A+ state has a strong triplet-triplet contribution, and has a lower energy than the state. The triplet-triplet and correlated nature of the... 

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