Theory of the Vibrational Structure of Molecular Excitons. Soluble One-“Phonon” Models

Solutions of several models of vibronic coupling in molecular crystals are presented using the techniques of second quantization. In the first, the VEP model, vibronic exciton $V$ transmutes into another vibronic exciton $E$ by emitting a vibrational exciton $P$ (the phonon). Stable $V$ excitons are regarded as bound states of $E$ and $P$ . If Pauli commutators are replaced by Bose ones, the VEP Hamiltonian admits two constants of motion, and the Hilbert space breaks into sectors belonging to distinct eigenvalues of the constants. Weak and strong coupling correspond to stable and unstable $V$ excitons, respectively. For the latter case, decay of $V$ into $\mathrm{E–P}$ scattering states is briefly considered. Explicit formulas are given for the $\mathrm{E–P}$ scattering states for stable and unstable $V$ . In the second model, interactions neglected in the first are crudely approximated by a separable pairing interaction. A new bound state of $E$ and $P$ results, illustrating the influence of the previously ignored terms in creating bound states. The third model is based on the Hamiltonian of Rashba in which electronic and vibrational excitation are treated separately. In this model, exciton and phonon numbers are constants and a solution exists if terms comparable to those neglected in the VEP model are ignored. When simplified the secular equation is found to have the same form as an earlier solution of the VEP model. Other avenues of investigation opened by the present work are discussed.