Imaging the Shape of Electron Orbitals with Twisted X-rays

Electron orbitals are quantum-mechanical wave functions that describe the spatial distribution of an electron, with the square of their magnitude representing the electron density. The concept of orbital forms foundation for visualizing and understanding the microscopic structure of atoms and molecules. We theoretically demonstrate how twisted x-ray diffraction can be employed to characterize the spatial profile of electron orbitals unambiguously. Using the hydrogen atom as a model system, we derive exact analytical expressions for its ground and excited states and show that twisted x-rays enable a clearer distinction between these states compared to standard x-ray diffraction. In particular, we highlight how the characteristic spatial profile of twisted beams, arising from their orbital angular momentum, overcomes technical limitations of standard x-ray diffraction, providing precise and unambiguous sensitivity to orbital spatial structure. We further demonstrate the applicability of this framework to realistic molecular systems by presenting ab initio x-ray diffraction signatures of Rydberg-excited oxazole, showing that large orbital angular momentum of the incident twisted x-rays enhances the distinction between the π − 3 s and π − 3 p Rydberg states. Our results lay the groundwork for the application of twisted x-ray diffraction to more complex systems with potential extensions to valence-bonded molecular orbitals.