Mapping Hydrogen Positions along the Proton Transfer Pathway in an Organic Crystal by Computational X-ray Spectra

Understanding the proton transfer dynamics through hydrogen bonds is a fundamental issue in chemistry, especially in condensed phases. While time-resolved X-ray spectroscopy offers a unique probe localized within the hydrogen bonds, accurate interpretation remains a challenge and relies on high-quality theoretical spectral references that map the proton motions. Here, with hybrid quantum mechanical and molecular mechanical (QM/MM) simulations, we computed a two-dimensional (2D) map of the N1s X-ray photoelectron/absorption spectra (XPS/XAS) for an organic crystal, composed of protonated 4,4'-bipyridine (BpyH$^+$, acceptor) and 5-sulfosalicylic acid (Sulfo$^-$, donor), with respect to varying hydrogen positions at nitrogens N1 and N2 of BpyH$^+$. We obtained a continuous picture of each spectrum and the chemical shift, mapping the proton transfer processes from O--H$\cdots$N to O$^{-}\cdots$H$^{+}$--N at N1 and from O$^{-}\cdots$H$^{+}$--N to O--H$\cdots$N at N2. We demonstrate that N1s transient XPS/XAS spectra are sensitive probes for hydrogen positions and proton transfer processes. We observe that reducing the O--H distance at the N1 site by only 0.2 {\AA}, compared to the crystal structure determined from X-ray diffraction (XRD), provides an excellent match between the QM/MM and experimental spectra, consistently in both XPS and XAS. Our calculations also demonstrate that geometry optimizations with periodic boundary conditions are difficult to refine the proton positions in experimental crystal structures, whereas our scaled snapshot protocol offers a more effective way. Our study reveals a nonlinear behavior of the absorption energy for the $\pi_1^*/\pi_2^*$ peak in XAS with increasing N--H distance, exhibiting a distinct barrier at around 0.95 \AA. This study provides a clear mapping of the two correlated proton transfer dynamics into X-ray spectra within a complex crystal, offering insights for future transient experimental studies.