E-ph coupling

The ground states of equations (27), (29) and (31) were found by minimalization of the expressions against the involved VPs. The respective ground state energies /iopl, a=0 E°n=o< anc T -o.A-o.opi will be compared mutually and with the exact value from numerical simulation in order to find out importance of the VPs 17, A in different regions of model parameters y= (/3/a) (reflection) and fL = (a2/fl2) (effective e-ph coupling). The place of E(f)e JT model will clearly come out as an important special case. 

Fig. 7 According to bi-polaron model of [115], the isotopic shift shows the largest signal only for intermediate values of hopping, t, which were carefully chosen to mimic the microstrain. It is for e-ph couplings. A, above A = 12 where the isotopic shift takes negative values, and this is possibly connected to the relevant properties of HTS (Upper panel). Conversely, the isotopic change of the polaron correlation shows a maximum for a given value of A (Lower panel), [116]...

In the case of the inelastic/incoherent transport regime, the phonon and the electron-phonon (e-ph) coupling term Hph and He.ph, respectively) should be added to the molecular Hamiltonian ... 

The potential V R) introduces electron-electron (e-e) interactions, and Taylor expansion of t(R) or V(R) about equilibrium generates electron-phonon (e-ph) coupling. Conjugated polymers abundantly illustrate [12,13] e-ph and e-e contributions whose joint analysis is difficult mathematically. But a joint analysis will undoubtedly emerge, and this review is a step in that direction. We seek sufficiently powerful e-ph descriptions for detailed fits of vibrational spectra and sufficiently accurate correlated states to understand excitations, including a host of recent nonlinear optical (NLO) spectra. Both e-ph and e-e interactions appear naturally in models, and both lead to characteristic susceptibilities. A related issue for vibrational spectra is the precise identification of IT-electronic contributions. We will emphasize the advantages of models for microscopic descriptions of conjugated polymers. To develop these themes. 

Fig. 6.1 Idealized linear chain with alternating transfer integrals r(l 5) and spring constants /f(l A) for double and single bonds, with 8 = A = 0.2. (a) Valence and conduction bands, Eq. (2), of the Hiickel chain Ho 8) with unit cell 2a, a = 1. (b) Optical and acoustical phonon frequencies, Eq. (3), of the chain the anomalous dispersion (dashed line) is for about 50% stronger e-ph coupling than in polyacetylene.

The Peierls instability of PA modeled by SSH is based on 7/0(8), a harmonic lattice with A = 0 for the a electrons, and linear e-ph coupling constant... 

Even at the 7r-electron level, conjugated polymers have additional vibrational modes, quadratic e-ph coupling, and Coulomb interactions V(/ ) that generate other e-ph coupling constants. More seriously, V(7 ) spoils the convenient single-particle solutions in Eq. (2) or Fig. 6.1a and alters the spectrum drastically. 

The structure is an input for 77-eIectron models. Once V(R) and t(R) have been specified, excitations and e-ph coupling can be treated comprehensively. Any structural change has both 77 and o-components. Conversely, e-ph analysis is not restricted to a particular V(/ ). Although both reflect 77 electrons, vibrational and electronic models focus on different issues and have been developed separately. Debates about the relative importance of e-ph and e-e interactions miss their common origin, their close connection to delocalization, and the many guises of tt electrons. Both e-ph and e-e interactions are in fact crucial. 

Our goal is to model quantitatively 7r-electronic contributions to both vibrational and electronic spectra. The general e-ph analysis introduced in Section II combines the microscopic AM formalism [18,19] with the spectroscopic ECC model [22]. The reference force field F for PA provides an experimental identiHcation of delocalization effects. Transferable e-ph coupling constants are presented in Section III for polyenes and isotopes of trans- and a s-PA. The polymer force field in internal coordinates directly shows greater delocalization in t-PA, while coupling to C—C—C bends illustrates V(/ ) participation and different coupling constants a(/ a) and a(Jis) in Eq. (3) support an exponential r(/ ). NLO spectra of PDA crystals and films are presented in Section IV, with multiphoton resonances related to excited states of PPP models and vibronic contributions included in the Condon approximation. Linear and electroabsorption (EA) spectra of PDA crystals provide an experimental separation of vibrational and electronic contributions, and the full tt-tt spectrum is needed to model EA. We turn in Section V to correlated descriptions of electronic excitations, with particular attention to theoretical and experimental evidence for one- and two-photon thresholds of centrosymmetric backbones. The final section comments on parameters for conjugated polymers, extensions, and open questions. 

Once the reference force field F is properly derived, diagonalization of the GF° matrix [see Eq. (8)] yields the reference frequencies normal coordinates Q, . We use capital Q s to distinguish normal coordinates from the generic nuclear coordinates q in Eq. (9). This is a partial solution, since we still have to account for linear e-ph coupling. The expansion in Eq. (9) is conveniently carried out on the basis of the reference normal coordinates. The simplest case arises when, at least for a given symmetry subspace, only one electronic operator, 0, is coupled to the phonons. 

Here g, is the e-ph coupling constant for mode QP. In spite of its simplicity, //e-ph is rather general. It accounts for the coupling of g molecular vibrations to CT excitations in organic ion-radical crystals [29-32] and is implic-ity assumed in both AM [18,19] and ECC [22] models. 

The electronic susceptibility in Eq. (14) can now be explicitly evaluated for Hiickel, PPP, or other alternating chains. Estimates of the bare e-ph coupling constants are thus possible. The g, represent the modulation of t(R) in the reference normal coordinates and can be compared with theoretical estimates [56,57]. Moreover, the dependence of g, on isotopic substitution is completely given by the reference normal coordinates and can be treated by standard spectroscopic methods. 

In the previous. section we discussed the reference force field of /-PA (see Table 6.2) derived from the force field of butadiene. In the Og symmetry block, the high frequency C—H stretch is decoupled from the other modes and thus from tt electrons. We are left with three relevant Ug modes their reference and experimental frequencies are reported in Table 6.3 and, as discussed in Section II, fix the matrix and the x cd) curves in Fig. 6.4. The A matrix is written on the basis of the reference normal coordinates Q . It consequently depends on both the G and F matrices and varies with molecular or polymeric structure. The e-ph coupling constants g, thus vary even with isotopic substitution. To define coupling constants independent of mass, we use the symmetry coordinates to solve the GF problem for the reference state. In fact, diagonalization of GF gives both eigenvector matrix L in the S basis. The L matrix is used to transform Jin Eq. (12) back to the S basis ... 

We stress that five parameters (three e-ph coupling constants and two relative x values) fixed by experimental frequencies allow us to reproduce quantitatively 24 experimental frequencies. The same parameters also give semiquantitative estimates of the relative intensities of the coupled modes in the Raman spectrum of pristine samples and in the IR spectra of doped and photoexcited... 

Since the symmetry coordinates for Og modes are the same as in /-PA, we can test the transferability of e-ph coupling constants. We account for e-ph coupling in the flg block through F = F - F, with the same J F as in /-PA, and plot the frequencies as functions of rj in Fig. 6.7. The best fit to the experimental Raman frequencies [70,71] of m-PA occurs at t = 0.56. Within Huckel theory, t corresponds to cis)lx(trans). The predicted intensities are again in qualitative agreement with experiment [70,71]. The calculated Raman intensities of pz and... 

Huckel theory clearly provides a useful analytical model for e-ph coupling in polyenes and PA, but some limita-... 

What is more interesting is the coupling of these modes to tt electrons. We have already seen that Hiickel theory accounts approximately for e-ph coupling in /-PA. By adopting a Hiickel model we have found the electronic susceptibilities relevant to all normal modes in the 102- and 101-site chains, as shown in Fig. 6.9. Small deviations of the relevant to the soliton with respect to the perfect chain are observed around q = n and q = ttH. However, only one mode, corresponding to the infrared-active out-of-phase combination of C—C stretches shows sizable deviations. This mode is sketched in Fig. 6.9 and is a damped ECC coordinate. 

The nature of the excited states jF) is left open in formal SOS expressions for NLO coefficients [86-89] that include all states. The same e-ph coupling governs the vibrational properties of ground and excited states, but exhaustive treatment of e-ph coupling in the latter requires more vibrational information than is presently available. Indeed, accurate excited states for conjugated polymers are difficult even at the 7r-electron level. Since large responses are due to dipole-allowed virtual excitations, vibronic analysis can be carried out with crude... 

The strength of e-ph coupling is usually measured by the dimensionless constant [17]... 

Hiickel analysis of /-PA with 5 = 0.18 matches E]b 1.8 eV and suffices for most aspects of e-ph coupling in Section III, where analytical results for the infinite chain are particularly important. Smaller 5 - 0.07 occurs in PPP models of polyenes or PA, since V R) also contributes to fie, and yields comparable but less precise X- for vibrations on extrapolating to the polymer (Fig. 6.8). Different 77-electron models are therefore suitable for different applications. Wider applications place more restrictions on the form and parameters of He - The inherently approximate nature of models, however, limits quantitative comparisons even if parameters are readjusted for each polymer, as is often done in solid-state discussions, and leaves open the range of optical, vibrational, NLO, or other spectra described by More detailed analyses and novel applications are certain in view of the generality of alternating chains and the flexibility of models. 

---