From PECs to Spectrum and From Spectrum to PECs: A Morse Protocol for Diatomic X-ray Absorption

A central pursuit in molecular science is to bridge molecular potential energy surfaces (PESs) with spectroscopic observables, both predicting spectra from potentials and inferring potentials from spectra. Here, we present a unified framework for diatomic systems by establishing and validating a bidirectional bridge between Morse potential parameters and vibrationally resolved X-ray absorption spectra. Using quantum wavepacket simulations, we first decode the forward relationship by linking specific potential parameters to distinct spectral features: the equilibrium bond displacement (ΔR) dictates Franck-Condon progressions, the anharmonicity parameter (α) modulates intensity ratios, and the well depth (D_e) governs the absolute energy scaling. These systematic parameter-sensitivity scans vividly illustrate Franck-Condon progressions and produce a trusted library to assess the physical plausibility of fitted parameters in the reverse procedure. Further quantitative analysis of spectral descriptors transforms these intuitive trends into explicit scaling laws within the scanned ranges. Conversely, we then validate this correspondence in reverse by inverting high-resolution experimental spectra of prototypical CO and CO⁺ systems, directly extracting both ground and core-excited state potential energy curves (PECs) with quantitative accuracy. Furthermore, two complementary diagnostic methods─spectral root-mean-square error (RMSE) and parameter-space deviations─are employed to comprehensively analyze the performance of different electronic-structure methods. The resulting physics-informed framework decodes explicit structure-spectrum relationships, providing both a powerful tool for spectral analysis and a valuable diagnostic for assessing novel electronic structure methods across broad regions of the PES.