Franck-Condon simulation of vibrationally resolved x-ray spectra for diatomic systems: Validation of the harmonic approximation and density functional theory

Under the Franck-Condon approximation, we systematically validated the performance of density functional theory (DFT) and the effects of anharmonicity in simulating C, N, and O $K$-edge vibrationally resolved x-ray spectra of common diatomic molecules and cations. To get ``transparent'' validations, vibronic fine structures of only the lowest $1s$ excited or ionized state in x-ray absorption spectroscopy (XAS) or x-ray photoelectron spectroscopy (XPS) were investigated. All six systems $({\mathrm{N}}_{2}, {\mathrm{N}}_{2}^{+}, \mathrm{NO}, {\mathrm{NO}}^{+}, \mathrm{CO}, \text{and} {\mathrm{CO}}^{+})$ were studied within the harmonic oscillator (HO) approximation using DFT with four functionals (BLYP, BP86, B3LYP, and M06-2X) for ten XAS and four XPS spectra, and excellent agreement between theoretical and experimental spectra was found in most systems, except O $1s$ XAS of NO, CO, and ${\mathrm{NO}}^{+}$. We analyzed and established a connection between their complex vibronic structures (many weak oscillating features within a broad peak) and the significant geometrical changes induced by the O $1s$ hole. The three spectra were well reproduced with anharmonic (AH) calculations by using quantum wave-packet dynamics based on potential energy curves (PECs) generated by DFT methods or multiconfigurational levels, highlighting sensitivity to the anharmonic effect and the PEC quality. In other examples of XAS $({\mathrm{CO}}^{+}, \mathrm{C} 1s \text{and} \mathrm{O} 1s; \mathrm{NO}, \mathrm{N} 1s)$ corresponding to smaller structural changes, HO and AH approaches lead to similar fine structures, which are dominated by 0-0 and 0-1 transitions. This study highlights the use of DFT with selected functionals for such diatomic calculations due to its easy execution and generally reliable accuracy. Functional dependence in diatomic systems is generally more pronounced than in polyatomic ones. We found that the BLYP, BP86, and B3LYP functionals consistently exhibited high accuracy in predicting spectral profiles, bond lengths, and vibrational frequencies, which slightly outperformed M06-2X.