The focus of this thesis research is the vibrational Feshbach resonance (VFR) mechanism for positron annihilation on molecules below the threshold for positronium formation. This process results in resonances in the positron-molecule annihilation rate when the incident positron has energy [epsilon]=[omega]v - [epsilon]b, where [omega]v is the energy of a molecular vibration and [epsilon]b is the positron-molecule binding energy. To understand this process, annihilation rates are measured as a function of positron energy for a variety of molecular species. These experiments provide new insight into the VFR process. In small molecules, the annihilation spectrum can be described well by a recent theory, which was brought to fruition with the assistance of data presented here. It is shown that the magnitudes of the VFR resonances in these molecules depend only on a simple scaling factor g = [sqroot][epsilon]b/[epsilon]. This theory fails in larger molecules, where the magnitudes of annihilation resonances rise rapidly with molecular size. However in hydrocarbons, when the scaling factor g is normalized out, the resonance due to the C-H stretch mode follows a universal scaling with the number of vibrational degrees of freedom. This is interpreted as evidence that the VFR are being enhanced by intramolecular vibrational relaxation (IVR). To date, only fluoroalkane molecules deviate from this trend, exhibiting a suppression of annihilation above a certain energy threshold. It is demonstrated that a resonant inelastic process involving the C-F stretch mode is responsible for this behavior. Data are presented for a number of deeply bound species and compared to molecules of similar size. The relationship between binding energy and various physical parameters is explored. A number of other phenomena are discussed, including the observation of combination-mode VFR, providing added insight into the annihilation process
