Conditions for zeros in the generalized oscillator strength: One-electron atom and diatomic molecule examples

Requirements for the existence of isolated zeros in the generalized oscillator strength (GOS) for one-electron atoms and molecules are considered. It is shown that in certain limits the atomic GOS cannot be zero for any value of the momentum-transfer magnitude ℎK≠0 unless it is zero for all values. A relationship between the existence of a zero and the angular momentum of the target's states is pointed out for the atomic case and a numerical example is provided. The conditions for the existence of an isolated zero for a molecular GOS are derived and, using the atom case as a model, they indicate that one is unlikely for 0<K<∞ and the internuclear separation R restricted to 0<R<∞. Minima, or possibly zeros, in the molecular GOS occur in both experiment and theory. It is postulated here that these structures are minima and not zeros. They appear to be due to zeros in a matrix element related to the leading term of the small-K GOS expansion while higher terms remain finite. Several numerical examples are provided and the speculation is supported by the correlation of the GOS minimum as a function of K and R to a zero in the dipole oscillator strength. Attention is brought to the existence of zeros in the molecular dipole oscillator strength when a nodeless function appears in this matrix element, contrary to the atomic case, and an explanation for this difference is given