  Perturbative contributions to single-beam two-photon transition rates may be
divided into two types. The first, involving low-energy intermediate states,
require a high-order perturbation treatment, or an exact diagonalization. The
other, involving high energy intermediate states, only require a low-order
perturbation treatment. We show how to partition the effective transition
operator into two terms, corresponding to these two types, in such a way that a
many-body perturbation expansion may be generated that obeys the linked cluster
theorem and has a simple diagrammatic representation.
