The Total Energy from X-ray Electron Density?

Context: This approach to quantum crystallography ensures the satisfaction of three quantum conditions, namely, idempotency, hermiticity, and normalization of the one-body density matrix, simultaneously as the X-ray diffraction problem is solved to reproduce the observed structure factors. The variational theorem used to optimize energy will only hold true if the density matrices used for such purpose are N-representable. The point of N-representability is to ensure the mapping of a density matrix to an N-body antisymmetric wavefunction. The antisymmetry is consistent with the experimental indistinguishability of fermions. This article develops a procedure for the fast and accurate application of quantum crystallography to large systems while guaranteeing that the results are N-representable.

Methods: For large molecules, it is advantageous to have the one-body density matrix assembled from sub-matrices of fragments within a scheme known as the kernel energy method (KEM). KEM accounts for up to two-body interactions between all fragments and ignores higher order interactions, an approximation that proved accurate through extensive past numerical testing. Since this approach to quantum crystallography rests on a single-determinant description, an explicit form of the corresponding N-representable two-body density matrix, which is determined by its one-body counterpart, is also given along with its use to calculate the total energy. This approach can be applied to extract conceptual density functional theory properties, quantum theory of atoms in molecules (QTAIM) properties, etc. from experimental structure factors.