Excitons, Frenkel theory

Hamiltonians formally similar to Eq. (2.4) are encountered not only in the central problems of lattice dynamics and electron propagation, but also in a large variety of other problems. Among them we mention the Frenkel theory of excitons, the coupled electron-lattice impurities in the entire range of coupling, the Jahn-Teller (or pseudo-Jahn-Teller) systems, interacting spins, and so on. 

The first section recalls the Frenkel-Davydov model in terms of a set of electromagnetically coupled point dipoles. A compact version of Tyablikov s quantum-mechanical solution is displayed and found equivalent to the usual semiclassical theory. The general solution is then applied to a 3D lattice. Ewald summation and nonanalyticity at the zone center are discussed.14 Separating short and long-range terms in the equations allows us to introduce Coulomb (dipolar) excitons and polaritons.15,16 Lastly, the finite extent of actual molecules is considered, and consequent modifications of the above theory qualitatively discussed.14-22... 

The spectra may also be described in the language of solid state theory. The atomic excited states are the same as the excitons that were described, for semiconductors, at the close of Chapter 6. They are electrons in the conduction band that are bound to the valence-band hole thus they form an excitation that cannot carry current. The difference between atomic excited states and excitons is merely that of different extremes the weakly bound exciton found in the semiconductor is frequently called a Mott-Wannier exciton-, the tightly bound cxciton found in the inert-gas solid is called a Frenkel exciton. The important point is that thecxcitonic absorption that is so prominent in the spectra for inert-gas solids does not produce free carriers and therefore it docs not give a measure of the band gap but of a smaller energy. Values for the exciton energy are given in Table 12-4. 

The foundation of excitons theory was formulated by Frenkel, Peierls and Wannier (l)-(5) more than 70 years ago. After that time the theory has been enriched by many new aspects. The theory has also been exposed to continuous experimental verification, which has confirmed the role of excitons in such processes as absorption of light, luminescence and energy transfer, photochemical processes, etc. Before we examine the experiments, which illustrate the presence and the role of excitons in crystals, we will briefly describe the basic models of excitons, which are mostly used in the interpretation of experimental results. 

The theory of Frenkel excitons will be given in Chapt. 2 and Chapt. 3. 

The physics of exciton ST has many features in common with electron (hole) ST. Therefore, the methods and results of the theory of electron ST have been widely used in the development of the theory of exciton ST. The effects of strong exciton-phonon coupling in organic crystals and the possibility of exciton ST were discussed in the papers by Peierls (35), Frenkel (36), and Davydov (37). 

Numerous theoretical and experimental studies have been carried out in this field so that a whole branch of molecular optics - the optics of molecular crystals and molecular liquids - has been established. Even before Frenkel put forward his exciton concept, workers in this branch of optics had developed a variety of exact and approximate methods for the theoretical description of optical phenomena many of these methods were also substantiated in experimental studies. However, after the discovery of excitons the use of these methods became increasingly rare and many of the results obtained with them have not been sufficiently understood in the framework of exciton theory. Therefore, further development and generalization of these methods were impeded. On the other hand, since the results of pre-excitonic molecular optics were underestimated, the optical properties of crystals were treated in terms of only exciton theory even in those cases when this could be done much more easily by using the earlier, simpler... 

This phenomenon (Davydov splitting) has been studied in many systems. Although it was found from the application of Frenkel s exciton theory, actually... 

To go further we recollect how the gas-condensed matter shift can be calculated. It is known from the theory of molecular (Frenkel) excitons (5) that this shift appears due to the difference between the energies of interaction of the excited molecule (molecular state /) and the unexcited molecule (molecular ground state 0) with all other molecules of the same crystal in the ground state ... 

In the same way as in the theory of Frenkel exciton mixing (Ch. 3), we have to diagonalize the total Hamiltonian H to consider the mixing of Frenkel and CT states and to find the new mixed states. We use the linear transformation to new operators ( and where... 

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