Noninteracting limit

In this section we discuss the geometrical structures and relaxed energies of the phenyl-based systems. We take the noninteracting and interacting electron limits in turn. 

As discussed in Chapter 4, the noninteracting limit in the adiabatic approximation is described by the Peierls model (defined in Section 4.2). The ground and excited state structures are easily obtained via the Hellmann-Feynman procedure, described in Section 4.4. 

To see this effective bond alternation we define the summed bond distortions 

Then we define the normalized, staggered and summed bond alternation, as 

Next consider the lBi excited state structure, shown in Fig. 11.15. This is the quinoid structure, illustrated in Fig. 11.16 (b). In contrast to the ground state, there is now a significant variation in the bond lengths in the phenyl-ring bonds labelled 1 shorten, while bonds labelled 2 lengthen. The bridging bond also shortens. 

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In other words, we get the same result by considering (A+B) as a supersystem as when handling A and B subsystems separately. This is especially important for extended systems, involving n subsystems, in which case the limiting process n —> oo will only make sense if the energy is strictly linear in n in the noninteracting limit. 

As far as explicit approximations for the polarisation functions Tltidq) are concerned only very little is known, even in the static limit. The complete frequency dependence is available for the noninteracting limit ( ). i e. the relativistic generalisation of the Lindhard function [95, 114]. In addition to its vacuum part (A.26) one has... 

Novel effects for partially filed shells. Elementary arguments (18) are sufficient to demonstrate that, in the noninteracting limit, versus n should exhibit kinks as the added electrons complete closed shells this forms the basis for an elementary discussion of the stability of aromatic molecules. What Fig. 2 shows is that similar kinks are found even when a shell (namely the lowest unoccupied level) is only partially filled. This is entirely an electronic correlation effect and signifies a novel mechanism for the stability of certain partially filled shells. 

Beyond the noninteracting limit only the vacuum part of the 2-loop contribution to the polarization function has been evaluated [2()4,205]. Moreover, the screening length n rf(0,0) is related to the energy density via the compressibility sum rule [206],... 

The induced current (308) is automatically UV-finite if the expansion is based on renormalized response functions, i.e. A(r) just sums up the terms required for the transition from the Xr l to their renormalized counterparts. It is in-structive to analyze the corresponding counterterms for the noninteracting limit of (308) given graphically by... 

Electron-phonon coupling plays a crucial role in one-dimensional systems. For any value of the electron-phonon coupling an infinite, undistorted polymer chain is unstable with respect to a lower symmetry, distorted structure. This is a consequence of the well-known Peierls theorem (Frohlich 1954 Peierls 1955), which states that a one-dimensional metal is unstable with respect to a lattice distortion that opens a band gap at the Fermi surface. A proof of bond-alternation in conjugated polymers in the noninteracting limit was first presented independently by Ooshika (1957, 1959), and Longuet-Higgins and Salem (1959). 

In the noninteracting limit, the band gap is directly proportional to the dimerization gap. In fact, this prediction is violated in traris-polyacteylene, indicating the importance of electron-electron interactions - as we describe in Chapter 7. 

Fig. 4.11. Probability distribution functions of the soliton defects in the noninteracting limit on a 102-site chain for the 1B state. Left defect, or soUton (filled symbols), right defect, or antisoliton (open symbols) extrinsic dimerization, = 0 (circles), Je = 0.1 (squares) and A = 0.1.

Figure 7.10(a) and (b) shows the geometric structures of the 1 B and l B states, respectively. As the extrinsic dimerization causes a confinement of the soliton-antisoliton pair, the geometrical structures are polaronic in the noninteracting limit for both cases. For the l B state, as before, increased Coulomb interactions bind the particle-hole pair into an exciton, resulting in very little change to the geometrical structure. For the state, however, electronic... 

We now turn to a discussion of the low-energy spectrum of biphenyl, again starting from the noninteracting limit. It is convenient to regard biphenyl as two benzene molecules (stripped of one hydrogen atom each) bonded together, as illustrated in Fig. 11.5. Since biphenyl possesses D h symmetry it is convenient to use the D2h symmetry-adapted molecular orbitals of benzene (shown in Fig. 11.4) to construct its molecular orbitals. 

Fig. 11.15. The fractional change in transfer integrals of poly(para-phenylene) from the uniform valne, t, in the noninteracting limit. The electron-phonon parameter used in the Peierls model (eqn (4.1) is A = 0.12. The labels refer to the bonds shown in Fig. 11.16. Only the upper rung of bonds are shown. Notice that the change in transfer integrals is opposite to the change in bond lengths.

In Section 11.2.1 the electronic spectrum of benzene was discussed from the molecular orbital (or noninteracting) limit. However, as experiments and the exact solution of the Pariser-Parr-Pole model indicate, the molecular orbital approach fails to qualitatively predict the low-lying singlet spectrum. The covalent j = 3 transition, namely the state, lies energetically well below the ionic... 

Stevens and Kremer [82-84] calculated the osmotic pressure of a system of up to 16 chains of 64 beads. Their simulation showed two scaling regimes. In the high density regime they found a scaling exponent of 9/4, characteristic of semi-dilute neutral polymer solutions. For densities below the overlap concentration, the data are consistent with a 9/8 value as predicted by Odijk s scaling theory. The simulation extended to lower densities where the noninteracting limit n = kTCp(l -I- 1/N) is reached. 

Like the experimental data (Fig. 3.16) the simulation data (Fig. 3.20) show two scaling regimes for the osmotic pressure. The simulations were able to reach the dilute noninteraction limit, where II = cksT + 1/N). Thus, at the lowest concentrations there is a deviation from the experimentally observed c / dependence. Also, at these very dilute concentrations the osmotic pressure does not exhibit a chain-length dependence. At the intermediate concentrations, the c / behavior fits the simulation data very well in agreement with experiment, and there appears some chain-length dependence. 

Fig. 7.31 Data such as shown in Fig. 7.30(b) replotted in scaled form on a log-log plot vs. Ne/k T. Three values of N are included, Y= 16 (squares), N=32 (triangles) and N=M (circles). Upper part refers to the mean-square end-to-end distance and lower part to the mean-square gyration radius of the minority chain, in both cases normalized by the value of the noninteracting limit, e/kaT- O. Straight lines with slope -1/3 indicate the collapsed behavior. (From Sariban and Binder, " in replotted form.)...

For asymmetric composition already in the disordered phase the peak position of the collective structure factor depends on / (cf. also Figs 7.20[a], [b]) and this problem has already been addressed by Leibler s RPA theory. Figure 7.39 shows that in the noninteracting limit the theory predicts this position (f correctly, but with increasing strength of the interaction eN the peak position q T) decreases, and in the transition region X = has only about 80% of its value in the noninteracting limit. 

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