Shape resonances in electron–molecule scattering. A proposed model in terms of S matrix poles and associated generalized quantum numbers

Abstract

AbstractAs an extension of the recent atomic work of Krylstedt et al. [1, 2], we propose the synthesis of the static exchange plus polarization model within the multiple scattering Xα formalism and the exterior complex rotation method for the study of shape resonances occurring in molecular collisional systems. The bound state multiple scattering Xα method is reviewed with special attention to the form of the various wave functions for the different molecular regions. In connection with the inclusion of continuum electron exchange into the scattering model, we analyze a possible solution to the problem of asymptotical behavior caused by the local density‐based free electron gas approximation used in this case. We also propose a method to fix the cutoff parameter, inherent in the polarization potential, for symmetry‐related molecular participants in the scattering process, thereby obtaining a possible predictive power for the one‐parameter scattering model.Titchmarsh–Weyl's theory is used to formally connect the above formulation to scattering theory. The theoretical difficulties in obtaining a unique meromorphic continuation of the S matrix are investigated in connection with the requirements on the actual potential to be exterior dilation analytic. Furthermore, the occurrence of asymptotic quantum numbers is noted and discussed in conjunction with muffin tin‐like approximations and related exterior complex rotated schemes.It is found that the electron exchange part of the nonstatic one‐body potential exhibits a functional form that is not exterior dilation analytic, albeit the general electron and nuclear many‐body problem involving Coulomb forces are dilation analytic. Although the immediate consequences indicate a rotation angle (θ) dependence on the S matrix continuation, a uniquely defined assignment of the cutoff parameter r0 = r0(θ) makes the present nonstatic model “exterior dilation analytic” in the sense that it mimics the dilation analytic structure of the full many‐body problem at the same time obeying “asymptotic” spherical symmetry via the associated constant of evolution. However, in the static exchange approximation the above difficulties are shown to be circumvented via a certain reformulation, leading to a regular analytic asymptotic form for the interaction potential.