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I know it is possible to measure electron densities from certain types of scattering experiments.
Have these densities ever been fed into density functionals to calculate energies and other properties? If so, is this useful for testing the veracity of functionals in any way? Can functionals be designed such that the converged density is consistent with these experimental densities?
If not, what are the difficulties associated with doing these calculations?
The combination of the diffraction technique with functionals of the density functional theory has been used. It allows one to describe both the static charge distribution and picture the electron motion in terms of the local energy and related functions. It gives scope to calculate energies and other properties. Refer to paper.
There have been instances where researchers have tried to find the discrepancy between energy density (ED) functions calculated from the Hartree-Fock electron density using approximation and functions calculated using the model ED derived from X-ray diffraction experiments. The following reasons for discrepancy were mentioned:
First, the rapid variation of the electron density in the vicinity of the nuclei and its slow variation in the valence electron shells prevents the existence of the density functional approximation for g(r), which provides a good description everywhere in the position space.
Second,the leading term in the kinetic energy density expansion (5) comes from the statistical Thomas-Fermi theory, which is valid for high-density regions. Quantum corrections improve the local behavior of this function; however, discrepancy with the Hartree-Fock kinetic energy density still remains.
For deeper understanding, refer to paper.
Although multiple rigorous methods of density functional theory and correction terms have been mentioned in the book 'Electron Density and Bonding in Crystals:Principles, Theory and X-ray Diffraction Experiments in Solid State Physics and Chemistry' which provide less relative error, however I am sceptical about the part on converged density being consistent with the experimental densities as there are difficulties associated with construction of general rigorous DFT with electron density as basic variable.As written in the book aforementioned:
The main difficulty of the DFT lies in the construction of approximation to functional G[ρ] which contains both kinetic T[ρ] and exchange-correlation Fxc[ρ] energies in the total energy functional E[ρ].
