Inversion of the X-ray restrained wavefunction equations: a first step towards the development of exchange–correlation functionals based on X-ray data

The X-ray restrained wavefunction (XRW) method is a quantum crystallographic technique that enables the determination of wavefunctions compatible with experimental X-ray diffraction data. Extensive research has demonstrated that this strategy inherently captures electron correlation and polarization effects on the electron density, while also providing consistent electron distributions. These findings suggest that the approach could be valuable in the development of new exchange–correlation (xc) functionals for density functional theory (DFT) calculations. This is particularly relevant in light of recent observations and recommendations by Medvedev et al. [Science (2017), 355, 49–52], who stressed the importance that xc functionals give both accurate energy values and exact electron densities, in line with the original spirit of DFT. Motivated by this perspective, this paper presents a preliminary investigation that aims at extracting and visualizing for the first time the perturbation potentials arising from the use of X-ray diffraction data as restraints in XRW calculations. In the present work, these potentials are simply obtained as orbital-averaged potentials through straightforward inversions of the XRW equations, where theoretical or high-quality experimental X-ray structure factors are employed in XRW computations at the restricted Hartree–Fock level for atoms (neon, argon and krypton) and simple molecules (dilithium and urea). Features and limitations of the resulting preliminary potentials are illustrated, while future perspectives on the use of the XRW method for the development of xc functionals are also discussed.