Global, nonlinear algorithm for inverting quantum-mechanical observations

Inverting laboratory measurements of quantum-mechanical observables to recover the underlying molecular potential typically produces nonunique solutions. Without quantifying the full family of potentials consistent with the measurements, it is impossible to fully determine how experimental error and limited data affect the inversion, or to assess the quality of the recovered potential. Here, we present a global, nonlinear algorithm for extracting molecular potentials from measurements of quantum-mechanical observables. The method utilizes a mapping technique to learn the relationship between a broad domain of potentials and their resulting observables to facilitate the inversion. Once constructed, the maps reduce the arduous task of repeatedly solving the Schrödinger equation for each trial potential tested during the inversion and permit the use of normally expensive, global optimization procedures to thoroughly explore the distribution of potentials consistent with the data. As a demonstration, the new algorithm is applied to quantum collision cross sections to illustrate the effect of experimental error and finite resolution of the scattering observables on the recovered potential. A series of simulated inversions were performed to examine these issues along with the inversion of laboratory differential cross-section data for He+Ne scattering. These illustrations show that laboratory errors can have a nonlinear effect on the family of extracted potentials. Furthermore, the examples provide a benchmark for the capabilities of the proposed algorithm to stably reveal the full distribution of potentials consistent with the data. The algorithm may be applied to other observables and molecular systems with more spatial coordinates.