  Coupled cluster theory provides hierarchical many-particle models and is
presently considered as the ultimate benchmark in quantum chemistry. Despite is
practical significance, a rigorous mathematical analysis of its properties is
still in its infancy. The present work focuses on nonlinear models within the
random phase approximation (RPA). Solutions of these models are commonly
represented by series of a particular class of Goldstone diagrams so-called RPA
diagrams. We present a detailed asymptotic analysis of these RPA diagrams using
techniques from singular analysis and discuss their computational complexity
within adaptive approximation schemes. In particular, we provide a connection
between RPA diagrams and classical pseudo-differential operators which enables
an efficient treatment of the linear and nonlinear interactions in these
models. Finally, we discuss a best $N$-term approximation scheme for
RPA-diagrams and provide the corresponding convergence rates.
