Polarons electronic wave function

As discussed by Travers [102], this can be due to either (1) an increase in the intrachain diffusion rate or (2) a decrease in the electron-proton coupling constants. In case 1, hydration has an effect on the polaron mobility, i.e., the latter is enhanced in case 2, hydration modifies the polaron electronic wave function. In addition, T p shows that the low frequency contribution to the proton relaxation is not affected, which is not consistent with a change in the coupling constant. It can be concluded that the increase in the macroscopic conductivity observed on hydration is related to an increase of the on-chain polaron mobility. A possible explanation can be proposed in terms of a solvation effect of the counterions resulting in a depinning of the polarons. 

An additional complication arises from the fact that the probability of an electron (or hole) being self-trapped due to the electron - phonon interaction increases strongly as the electronic wave function shrinks in size to the order of atomic dimensions (Emin, 1982). A consequence of this is that electrons in disorder-induced localized states are believed to be more susceptible to small polaron formation and self-trapping than are ordinary extended-state electrons (Emin, 1984 Cohen et al, 1983). Thus, not only does the disordered structure of amorphous semiconductors introduce new physical phenomena, namely, the mobility edge, but also the effect of known phenomena, such as the electron - phonon interaction, can be qualitatively different. 

The polaron model is properly an extension of the primitive cavity model. In both, the electron is considered to be solvated by a number of ammonia molecules. However, in the cavity model one considers the localization or solvation of the electron as described by a cavity of some shape whose boundaries act as limiting points for the potential or the electronic wave function. In the polaron model, on the other hand, the electron is considered to polarize the surrounding ammonia molecules in such a way as to provide a trapping potential for itself. The potential is derived from the laws of electrostatics adapted to the quantum mechanical description of the electron density in terms of the electronic wave function. In the final development of the theory one would of course, require self-consistency between the wave function of the electron and the potential in which it moves. It is possible that the end result may indicate that the electronic wave function is in fact almost localised within a definite volume of certain shape. However, no such assumption is made a priori as in the cavity model. 

For the cavity model, a careful analysis of the two-electron polaron is necessary to test the correctness of the conclusion that it is unstable. Also, for the one-electron polaron, the orthogonality of the unpaired electron wave function to the wave functions of the electrons on neighboring ammonia molecules has to be considered carefully for possible contributions to the observed and proton Knight shifts in N.M.R. measurements. 

In discussing low temperature-dependent mobility, we should mention charge transport by polarons, an intermolecular phonon-assisted hopping process 24>25>. Polarons (charge carriers trapped in their polarization field) arise from a strong electron-phonon interaction where there is a weak overlap of wave functions of... 

For the case of typical ionic crystals aP 1-10, and the weak coupling limit is applicable. The most important conclusion from this treatment is that the weak coupling limit leads to a perturbed Bloch type wave function characterized by equal probability for finding the electron at any point of the medium. Thus, in the case of the ionic crystals, the current description of the polaron is that of a mobile electron followed by lattice polarization. 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

For our formal treatment of the fermionic 2-level system we assume that we may describe the behaviour of electrons in the one-electron approximation. Then each electron is represented by a wave function that is independent of the wave functions of other electrons, and the individual wave functions may be linearly superimposed. This picture often proves useful in the context of inorganic semiconductors [2,3]. However, it may be highly questionable in organic and molecular matter, where excitonic [4] and polaronic effects are often predominant [5]. 

A small proton polaron is different in some aspects from the electron polaron that is, the hydrogen atom is able to participate in the lattice vibrations in principle (in any case it is allowable for excited states see Section II.F), but the electron cannot. This means that one more mechanism of phonon influence on the proton polaron is quite feasible. That is, phonon fluctuations would directly influence wave functions of the protons and thereby contribute to the overlapping of their wave functions. In other words, phonons can directly increase the overlap integral in concept. Such an approach allows one to describe the proton transfer without using the concept of transfer from site to site through an intermediate state. 

To obtain the binding energy of the electron in the monomer, Becker, Lindquist, and Alder performed an approximate calculation. An effective dielectric constant D oi 7 was assumed in the region over which the monomer electron moves. This value of D gave the maximum of the Is wave function of the monomer at a distance 3.5 A, less than the radius 4.53 A of the solvated ion. The ionization energy is then obtained as 0.28 ev as compared with the value of 1.415 ev found for fFu by Jortner using the polaron model. To determine... 

This effect can be seen e.g. for a nitrobenzene solution of poly(3-butylthiophene) [28]. In a chloroform solution of PHT where the PHT chains assume a coiled conformation [42, 61], the polaron wave function is expected to be localized on account of the disorder and the electron-lattice interaction [12] both of which act cooperatively. In fact, Nowak et al. [61] inferred from analysis of the ESR line with hyperfine splitting that the polaron wave function is localized primarily on a single thiophene ring with only a small spin density on the neighboring rings on either side. They... 

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