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 ${\epsilon }_d$, a width function $\Gamma \mathrm{}\left(E\right)\mathrm{}$, and a level-shift function $\Delta \mathrm{}\left(E\right)\mathrm{}$. 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 $\Gamma \mathrm{}\left(E\right)\mathrm{}$, 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 ${\epsilon }_d$, $\Gamma \mathrm{}\left(E\right)\mathrm{}$, $\Delta \mathrm{}\left(E\right)\mathrm{}$, and the resonant eigenphase sum are in excellent agreement with results obtained previously by Hazi using different computational methods.