Quantum phonons

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 fully quantized model is described by the following Hamiltonian, 

The electron Hamiltonian is the usual Pariser-Parr-Pople model, 

Finally, assuming that electron-phonon coupling arises from hnear deviations in bond lengths, the electron-phonon coupling Hamiltonian is the same as that introduced in Chapter 7, namely, 

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 

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Let us note that the quantum phonon assistance of the electron tunneling (/3-term in equations (1) and (2)) constitutes the difference of the model from the related dimer and exciton quantum models where instead of /3 Y.n (h2n + 2n)° n °f equation (1) there stands Acrxn, where A is the distance between the levels [4,5],... 

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. 

The predictions presented in this chapter are all within the Born-Oppenheimer approximation. Quantum phonons will reduce the amplitude of the bond alternation in the ground state (Fradkin and Hirsch 1983 McKenzie and Wilkin 1992) - for realistic models of fraras-polyactylene by about 20% (Barford et al. 2002b), as described in Chapter 10. Quantum phonons also prevent self-trapping. This latter has a rather significant affect on the relaxed energies of the 1 B+ and 2 A+ states in linear polyenes, as explained in Chapter 10. 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Cerullo G, De Silverstri S and Banin U 1999 Size-dependent dynamics of coherent acoustic phonons in nanocrystal quantum dots Phys. Rev. B 60 1928... 

Finally, it can be shown from dre quantum dreoiy of vibrational energy in dre solid state drat, at temperatures above dre Debye temperature 0d, dre density of phonons, p, is inversely related to 6 according to dre equation... 

At low temperatures, when the bath is quantum (icoj P 1), the rate expression, expanded in series over the coupling strength, breaks up into the contributions from the various processes involving the bath phonons... 

As argued in section 2.3, when the asymmetry e far exceeds A, phonons should easily destroy coherence, and relaxation should persist even in the tunneling regime. Such an incoherent tunneling, characterized by a rate constant, requires a change in the quantum numbers of the vibrations coupled to the reaction coordinate. In section 2.3 we derived the expression for the intradoublet relaxation rate with the assumption that only the one-phonon processes are relevant. 

DSP crystal, a detailed picture of the lattice motion and related displacements was constructed and related to the topochemical postulate and the mechanism of phonon assistance. Holm and Zienty (1972) have measured the quantum yield for the overall polymerization process of a,a -bis(4-acetoxy-3-methoxybenzylidene)-p-benzenediacetonitrile (AMBBA) crystals in slurries and reported it to be 0.7 on the basis of the disappearance of two double bonds ( = 1.4 if assigned on the basis of the number of double bonds consumed). 

With the knowledge of g, we can estimate the inverse mean free path of a phonon with frequency co. As done originally within the TLS model, the quantum dynamics of the two lowest energies of each tunneling center are described by the Hamiltonian //tls = gcTz/2 + Aa /2. This expression, together with Eqs. (15) and (17), is a complete (approximate) Hamiltonian of... 

Section V, other quantum effects are indeed present in the theory and we will discuss how these contribute both to the deviation of the conductivity from the law and to the way the heat capacity differs from the strict linear dependence, both contributions being in the direction observed in experiment. Finally, when there is significant time dependence of cy, the kinematics of the thermal conductivity experiments are more complex and in need of attention. When the time-dependent effects are included, both phonons and two-level systems should ideally be treated by coupled kinetic equations. Such kinetic analysis, in the context of the time-dependent heat capacity, has been conducted before by other workers [102]. 

According to the quantum transition state theory [108], and ignoring damping, at a temperature T h(S) /Inks — a/ i )To/2n, the wall motion will typically be classically activated. This temperature lies within the plateau in thermal conductivity [19]. This estimate will be lowered if damping, which becomes considerable also at these temperatures, is included in the treatment. Indeed, as shown later in this section, interaction with phonons results in the usual phenomena of frequency shift and level broadening in an internal resonance. Also, activated motion necessarily implies that the system is multilevel. While a complete characterization of all the states does not seem realistic at present, we can extract at least the spectrum of their important subset, namely, those that correspond to the vibrational excitations of the mosaic, whose spectraFspatial density will turn out to be sufficiently high to account for the existence of the boson peak. 

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