Exciton/excitonic degeneracy

The vibronic exciton approximation restricts H to a subspace corresponding to a given vibronic molecular state. In this subspace the degeneracy of the localized vibronic states is lifted by the interactions JnmB Bm. Using the translational invariance, the eigenstates of the crystal are seen to be the vibronic excitons, or vibrons ... 

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. 

Application of group theory can also explain the important problem of degeneracy of excitonic states. This degeneracy can be conditioned by invariance of the crystal Hamiltonian upon elements of its space group (in this case it is sometimes called compulsory degeneracy, see (21), 40) or, as shown by Herring (23), it can result from the crystal invariance upon the time-reversal operation. 

Let us first consider the case of compulsory degeneracy. To this end we assume that an exciton state (u(k) is degenerated for k = ko, which means there are p excitonic states associated with the wavevector ko, t = 0,1,..., p,... 

As an example we consider the case of the naphthalene crystal, G ft being its space group. The point group C-2h (see Table 2.1) has only one-dimensional representations. Since any subgroup of the group C h can also have only onedimensional representations, it is clear that in crystals of naphthalene type the compulsory degeneracy for excitonic states inside the first Brillouin zone is not possible. 

As a second example consider the case of a quartz crystal, with the space group Ds. One of the representations of the point group >3 has dimension two. Thus, if the vector ko is parallel to the three-fold symmetry axis when the point group Gk0 coincides with the point group >3, a double degeneracy of excitonic terms is, in general, possible. 

There is another reason for degeneracy of excitonic states, being a consequence of the structure of the Schrodinger equation. Indeed, since the Hamiltonian is a self-conjugated operator, wavefunctions l>ko (( = 1,2,..., p), where the star means complex conjugate, as well as wavefunctions kotJ ( = 1,2,..., p),... 

When the long-range part of the Coulomb interaction is taken into account, in the second case the degeneracy is lifted, while in the first case we observe only a frequency shift. It is appropriate to note here that when the Coulomb interaction is fully accounted for, the exciton energy, in accordance with the results of the microscopic theory (see Chs. 2 and 3), depends on s when k —> 0, i.e. reveals a nonanalytic function of k for k —> 0. 

---