Applying Bayesian Inference and deterministic anisotropy to retrieve the molecular structure $|\Psi(\boldsymbol{R})|^2$ distribution from gas-phase diffraction experiments

Currently, our general approach to retrieving molecular structures from ultrafast gas-phase diffraction heavily relies on complex ab initio electronic or vibrational excited state simulations to make conclusive interpretations. Without such simulations, inverting this measurement for the structural probability distribution is typically intractable. This creates a so-called inverse problem. In this work, we develop a broadly applicable method that addresses this inverse problem by approximating the molecular frame structure $|\Psi(\boldsymbol{R}, t)|^2$ distribution independent of these complex simulations. We retrieve the vibronic ground state $|\Psi(\boldsymbol{R})|^2$ for both simulated stretched NO$_2$ and measured N$_2$O. From measured N$_2$O, we observe 40 mAngstroms coordinate-space resolution from 3.75 inverse Angstroms reciprocal space range and poor signal-to-noise, a 50X improvement over traditional Fourier transform methods. In simulated NO$_2$, typical to high signal-to-noise levels predict 100--1000X resolution improvements, down to 0.1 mAngstroms. By directly measuring the width of $|\Psi(\boldsymbol{R})|^2$, we open ultrafast gas-phase diffraction capabilities to measurements beyond current analysis approaches. This method has the potential to effectively turn gas-phase ultrafast diffraction into a discovery-oriented technique to probe systems that are prohibitively difficult to simulate.