Regularized inversion of diatomic vibration-rotation spectral data: A functional sensitivity analysis approach

We present a stable and accurate inversion method for extracting potentials from spectroscopic data of diatomic molecules. The method, which was developed previously for inverting scattering data, is based on first-order functional sensitivity analysis in conjunction with the Tikhonov regularization, singular system analysis, and an exact transformation technique. Besides being numerically stable, it requires neither explicit functional forms nor special basis function expansions for the potential corrections when solving the corresponding linearized integral equation. Instead, we solve the linear equation directly in terms of the probability densities of the unperturbed vibrotation eigenstates. For illustration, we consider the ground electronic state of the H₂ molecule. Inversions have been carried out for simulated data free of noise and for those with noise of magnitude comparable to realistic experimental errors. It is found that in both cases, a relatively large deviation of the starting reference potential from the truth may be tolerated to still accurately recover the intended one. The propagation of the spectral errors is analyzed in detail based on the linearization assumption. The variance of the recovered potential reveals the reliability of various regions of the recovered potential.