Potential surfaces from the inversion of time dependent probability density data

This article presents an algorithm to assess the feasibility of inverting probability density data to extract potential surfaces. Such data admit the generation of a noniterative quantum inversion algorithm that does not require the solution of the Schrödinger equation. Tikhonov regularization is employed to manage the singular nature of the problem. The inversion in regular regions has excellent accuracy, and an error analysis also indicates that the potential in the regular regions is stable under perturbations from noisy data. The regular regions of the potential are identified by the algorithm. The algorithm does not require knowledge of the excitation process initiating the evolution of the system. Analysis indicates that the most detailed potential surface information will result from broadband excitation leaving the molecule with significant population in as many quantum states as possible. The inversion algorithm is tested in a simulation for the O-H potential, which shows that the algorithm is very fast and reliable.