Frenkel, Coulomb, and mechanical excitons

Thus in crystals where unit cells contain a molecules, to any single nondegenerate excited state of a free molecule in the crystal corresponds not one, but a bands of excited states and correspondingly several absorbtion lines. Such a splitting was first discussed by Davydov ((9)—(11)) and is usually called the Davydov splitting,8 to distinguish it from the Bethe splitting (14). 

The degeneracy of a molecular term is usually related to a sufficiently high symmetry of molecules. In crystals, however, the symmetry of fields in the place where the molecule a is located depends on the intermolecular interactions and can be lower than the symmetry of isolated molecules. In this case the degeneracy can be removed and a splitting can appear, which is just the Bethe splitting. 

To demonstrate the mechanism of Bethe splitting in molecular crystals we assume that a unit cell contains only one molecule but the term / under consideration is ro-fold degenerate. In the case being considered, quite analogous to that of the equation system (2.19), we obtain a system of ro equations for coefficients u r The solution of the secular equation shows that in crystals with one molecule in the unit cell ro excitonic bands appear which correspond to an ro-fold degenerate molecular term. 

If an impurity molecule is characterized by a degenerate term, the Bethe splitting removes partially or totally the degeneracy and instead of one term a multiplicity of terms appears. In crystals, due to their translation symmetry single degenerate terms expand to several excitonic bands. 

Let us note that both Bethe and Davydov splittings are not consequences of specific quantum-mechanical properties of molecules. They appear also in the crystals where the molecules are modeled, for example, by classical harmonic oscillators. 

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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... 

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