R. E. Merrifield

Ionized States in a One‐Dimensional Molecular Crystal

Generalization of the theory of excitons in molecular crystals to include ionized states is carried out for an idealized one‐dimensional model for which the exact wave functions and energy levels can be derived. The ionized states are found to form a progression of bands corresponding to bound states of the electron and hole. The bound levels converge to a band of free states in which the electron and hole move independently and can carry an electric current. The lowest‐lying ionized states may interact significantly with the un‐ionized molecular exciton states and give rise to weak charge‐transfer absorption bands.

Theory of the Vibrational Structure of Molecular Exciton States

The interaction of excitons with molecular vibrations is formulated in the second quantization formalism. The Hamiltonian which describes interacting excitons and molecular phonons is derived. The exciton creation and destruction operators can be eliminated resulting in a reduced Hamiltonian which contains phonon operators only. A natural variational function for the lowest state of the reduced Hamiltonian is suggested by the forms of the eigenstates for the two exactly soluble limiting cases of vanishing exciton—phonon coupling and vanishing exciton bandwidth. Numerical calculations of the properties of the exciton in the lowest state are carried out for a simple one‐dimensional nearest‐neighbor interaction model.

Propagation of Electronic Excitation in Insulating Crystals

A general method for the description of nonstationary excited states of an insulating crystal is outlined. For the general case, the time‐dependent wave function is given implicitly by means of a simple generating function. In the nearest‐neighbor approximation, the solution can be given explicitly in terms of Bessel functions. The effect of an idealized crystal surface on exciton propagation is examined. The exciton behaves as though repelled by the surface.

Interaction of Excitation Waves in a One‐Dimensional Crystal

The problem of exciton‐exciton interaction in molecular crystals is discussed in terms of the stationary two‐quantum excited states of an idealized one‐dimensional crystal. If only a single molecular excited state is considered, the wave functions and energy levels of the two‐quantum states may be obtained directly from those of the corresponding one‐quantum problem with the proviso that the two excitons have different wave numbers. If in addition, a second molecular excited state is included, there can exist states in which two excitons remain bound to each other as they propagate through the crystal.

Moment Theorems for Vibronic Exciton Spectra of Molecular Crystals

The mean energy and second moment of an absorption band can be calculated exactly without a knowledge of the energies and eigenstates of the upper levels of the transition. Application of this theorem to exciton spectra of molecular crystals having two molecules per unit cell shows that the purely electronic part of the factor‐group splitting is given by the difference of the mean energies of absorption in the two relevant polarizations and that the second moment of the absorption band is independent of the intermolecular coupling strength. These results are independent of the vibronic coupling strength and of whether or not the vibrations are harmonic. The finite‐temperature form of the mean energy theorem is derived and is employed to discuss the broadening of exciton spectra resulting from interaction with lattice vibrations.

Exciton Analog of the Peierls Instability of One‐Dimensional Metals

A one‐dimensional metal in which the conduction electrons interact with excitons in an array of polarizable side groups is shown to possess an inherent instability analogous to the phonon‐induced Peierls instability. The instability manifests itself by the simultaneous appearance of an energy gap at the center of the conduction band and a ferroelectric polarization of the side groups. An approximate quantitative treatment of this effect is presented. The principal conclusions are: (1) the system is unstable for all values of the electron—exciton coupling constant, (2) the resulting semiconducting gap is at least as large as the BCS superconducting gap for the same system, and (3) while it cannot be said with certainty that this precludes superconductivity in such a system it suggests that if it does not become superconducting it will become ferroelectric.