Projection-operator calculations for shape resonances: A new method based on the many-body optical-potential approach

The projection-operator formalism of Feshbach defines a separation of the T matrix into a smooth background term and a resonant T matrix which may vary rapidly with energy. The resonance is characterized by an unperturbed energy d, a width function (E), and a level-shift function (E). Such a separation of the fixed-nuclei electron-molecule scattering T matrix is of considerable practical relevance for the treatment of nuclear dynamics in resonant electron-molecule scattering. We present an explicit realization of the projection-operator formalism for electron-molecule scattering within the framework of the many-body optical-potential approach. In contrast to the approach of Hazi [J. Phys. B 11, L259 (1978)] which is based on the use of Stieltjes moment techniques to compute (E), we obtain explicitly the background T matrix as well as the information on the angular distribution of the resonant scattering. The performance of the method is illustrated for the well-known 2.3-eV shape resonance in electron scattering from the nitrogen molecule. The two-particle-hole Tamm-Dancoff approximation (2ph-TDA) is adopted for the optical potential and the Schwinger variational principle is used to solve the background scattering problem. The resulting resonance parameters d, (E), (E), and the resonant eigenphase sum are in excellent agreement with results obtained previously by Hazi using different computational methods.