Wave functions consistent with experimental x-ray diffraction data: A hircocervus becomes reality

Since the early days of quantum physics, the possibility of obtaining wave functions consistent with experimental x-ray diffraction data has been envisioned. The idea is firmly grounded in the postulates of quantum mechanics and finds full support in the Hohenberg and Kohn theorem and Levy-Lieb search formulation of density functional theory. Within this framework, a rich history of research has unfolded over the years, introducing various strategies to obtain plausible one-electron reduced density matrices or wave functions that are compatible with x-ray structure factors. Approximately twenty-five years ago, all of this culminated with the development of the x-ray restrained wave function (XRW) approach. This method aims to determine wave functions that minimize the electronic energy of the examined systems while maximizing the statistical agreement between experimental and calculated x-ray diffraction data. Presently, the XRW technique stands as a well-established strategy, manifesting in various forms, and addressing numerous problems and challenges across chemistry, physics, and materials science. Moreover, there remains large room for improvement and extensions in the coming years. This paper will comprehensively review the current state of the x-ray restrained wave function approach, discussing its underlying foundations, historical background, theoretical details and extensions, practical applications, and forthcoming perspectives.