Reconstruction of intermolecular potentials at fixed energy: Functional sensitivity analysis approach

The iterative, perturbative inversion procedure of Mackintosh et al. is modified for the inversion of (noise free) elastic molecular scattering data at fixed energy, in conjunction with singular system analysis and the Tihkonov-Miller-type regularization method. The singular system consisting of the first order functional relation between infinitesimal changes in the phase shifts and the intermolecular potential provides a natural basis for solving the corresponding linear Fredholm integral equation of the first kind. The regularization procedure stabilizes the ill-conditioned inverse problem arising from the above linear integral equation. Furthermore, in regions where the wavelength is small compared to the length scale of the potential, the first order functional derivatives or sensitivity densities of the phase shifts with respect to the potential δη_l/δV(R) are approximated by their mean values to facilitate the regularization procedure and, therefore, the convergence of the problem. This semiclassical modification and the adoption of the singular system analysis are necessary for intermolecular potentials possessing a rapidly varying region that cannot be appropriately represented by a small number of spline functions. For illustration, a model of the He-Ne system has been adopted and the applicability of the proposed method at diffrent energies has been examined in detail.