In the collision of electrons with molecules and molecular ions, excitation and dissociation are dominated by resonant processes, where the electron becomes temporarily trapped, changing the forces felt by the nuclei. In these processes, not only must the static and exchange forces be treated correctly, but correlation and polarization effects become critical. Cross sections and branching ratios are sensitive to the crossing points of the resonant and target potential energy curves and the shape of the curves, primarily near the Franck-Condon region but also asymptotically. In order to study such processes, it is clearly necessary to treat the polarization and correlation, as well as the balance between the resonant and target calculations. A proper treatment of correlation/polarization becomes critical when describing the collision of a positron and a molecule. The simple picture - just change the sign of the potential and drop the exchange interaction - hides the dramatic changes this causes to the scattering. As the low-energy positron approaches the molecule, there is a large distortion of electronic wave function of the target. This long-range polarization effect and short-range correlation can not be described with simple static approximations, but must use sophisticated methods that can correctly describe this interaction. We propose to use these methods study a number of systems where correlation plays a significant role in the dynamics of the interaction. The theory will use modern ab initio techniques to explore how correlation affects the flow of energy in the system and to elucidate the mechanisms that underlie this interaction. The systems range from studies of dissociative recombination, which are critical in understanding the formation of molecules in interstellar clouds, dissociative attachment of systems of biological relevance, to the collision of positrons with molecules, a new unfolding area of applications, where the physics of the collision process is dominated by correlation. In addition, we will elucidate the foundation for the basic equations (the local or boomerang equations) that have been employed by most researchers to study resonance problems, carrying out a rigorous study for the convergence and application of such approximations.
