Orbital-free quantum crystallography: view on forces in crystals

Abstract

Quantum theory of atoms in molecules and the orbital-free density functional theory (DFT) are combined in this work to study the spatial distribution of electrostatic and quantum electronic forces acting in stable crystals. The electron distribution is determined by electrostatic electron mutual repulsion corrected for exchange and correlation, their attraction to nuclei and by electron kinetic energy. The latter defines the spread of permissible variations in the electron momentum resulting from the de Broglie relationship and uncertainty principle, as far as the limitations of Pauli principle and the presence of atomic nuclei and other electrons allow. All forces are expressed via kinetic and DFT potentials and then defined in terms of the experimental electron density and its derivatives; hence, this approach may be considered as orbital-free quantum crystallography. The net force acting on an electron in a crystal at equilibrium is zero everywhere, presenting a balance of the kinetic F kin( r ) and potential forces F ( r ). The critical points of both potentials are analyzed and they are recognized as the points at which forces F kin( r ) and F ( r ) individually are zero (the Lagrange points). The positions of these points in a crystal are described according to Wyckoff notations, while their types depend on the considered scalar field. It was found that F ( r ) force pushes electrons to the atomic nuclei, while the kinetic force F kin( r ) draws electrons from nuclei. This favors formation of electron concentration bridges between some of the nearest atoms. However, in a crystal at equilibrium, only kinetic potential v kin( r ) and corresponding force exhibit the electronic shells and atomic-like zero-flux basins around the nuclear attractors. The force-field approach and quantum topological theory of atoms in molecules are compared and their distinctions are clarified.

The explicit role of electron exchange in the hydrogen bonded molecular complexes

“…They locally compensate the F kin at each position point for systems in the equilibrium state. 68 The component F xc changes its direction in the HÁÁÁO w region similar to the F P (Figure 4 Figure 6), which characterize bonding in molecular systems. 60,61,86 However, neither v PAEM nor…”

“…It is zero everywhere in position space of the equilibrium system. 68 The consideration of the F KS eff r ð Þ, arising from the effective Kohn-Sham potential, can be reduced to the analysis of the force acting on one electron within a molecule, F FAEM (r). 57 Indeed, the v KS eff can be expressed as…”

“…where ρ(r 1 ) is an electron density, Γ xc (r 1 , r 2 ) is an exchange-correlation two-electron density, 89 It also corresponds to the effective potential in the Euler equation for 68,90 Then, the kinetic Kohn-Sham force F KS kin in Equation (4) will be replaced by…”

“…It is a measure of electron pair localization. 93 The Pauli potential as it follows from the Euler equation 68,90 and Equation (9), can be calculated as…”

“…The Weizsäcker term evaluates spin-independent steric effects in the molecular systems, [63][64][65] including investigation of their role in the H-bonded dimer formation, 66 while Pauli term describes the exchange correlation contribution to the electron kinetic energy. 67,68 To our best knowledge, there is still no study of H-bonds from the perspective of the non-local bonding descriptors (SF, DI) and energy decomposition analysis coupled with consideration of oneelectron potentials and corresponding forces. Here we fill this gap and focus on the following issues: (i) the interplay between different physical effects and atomic influence on the electron density in the Hbond region, and (ii) the reasons of the H-bond energy changes due to variations in chemical composition and structure of interacting molecules.…”