An intrinsic criterion of defining ionic or covalent character of AB-type crystals based on the turning boundary radii calculated by an ab initio method

Electronic Force Density Fields: Insights into Partial Bonds, Transition States, and Chemical Structure Evolution

This paper presents the quantum-topological binding approach, in which the electrostatic and total static force density fields, Fes(r) and F ( r ) , together with the electron density gradient field ∇ρ(r), are simultaneously analyzed to elucidate the chemical structure of transition states and the nature of interatomic interactions for semibroken semiformed partial chemical bonds. The approach attributes the discrepancies between the force fields F ( r ) and Fes(r) to the nonclassical electron-electron interaction effects. The internuclear gap between the zero-flux boundaries of Fes(r) and ∇ρ(r) indicates the interatomic charge transfer phenomenon (ICT) that occurs upon the formation of a system from free atoms. Concomitantly, the mismatch of the zero-flux surfaces defined in - F ( r ) and ∇ρ(r) can be interpreted as a phenomenon of the electron-transfer-induced quantum chemical response (QCR), which originates from the electron exchange correlation. Our study permits the assertion of parallels between partial bonds and noncovalent interactions, as both typically exhibit incomplete QCRs, indicating the partial electron sharing of the transferred density. The changes in atomic and pseudoatomic charges are employed to describe the evolution of the chemical structure upon the substitution reaction. It is observed that the acquired difference in the actual atomic electronegativity causes polarization upon the heterolytic breaking of virtually nonpolar bonds. It is further proposed that the proximity of closely related stationary states along the reaction path on a potential energy hypersurface implies their similarity in the manifestation of the ICT and sympathetic QCR. Furthermore, the involvement of an electron pair in a partial bond facilitates its delocalization through the attraction by the static forces F ( r ) and Fes(r) to a neighboring nucleus and through the smearing by the Pauli kinetic force FP(r).

Real-Space Interpretation of Interatomic Charge Transfer and Electron Exchange Effects by Combining Static and Kinetic Potentials and Associated Vector Fields

Intricate behaviour of one-electron potentials from the Euler equation for electron density and corresponding gradient force fields in crystals was studied. Channels of locally enhanced kinetic potential and corresponding saddle Lagrange points were found between chemically bonded atoms. Superposition of electrostatic ϕ e s r and kinetic ϕ k r potentials and electron density ρ r allowed partitioning any molecules and crystals into atomic ρ - and potential-based ϕ -basins; ϕ k -basins explicitly account for the electron exchange effect, which is missed for ϕ e s -ones. Phenomena of interatomic charge transfer and related electron exchange were explained in terms of space gaps between zero-flux surfaces of ρ - and ϕ -basins. The gap between ϕ e s - and ρ -basins represents the charge transfer, while the gap between ϕ k - and ρ -basins is a real-space manifestation of sharing the transferred electrons caused by the static exchange and kinetic effects as a response against the electron transfer. The regularity describing relative positions of ρ -, ϕ e s -, and ϕ k - basin boundaries between interacting atoms was proposed. The position of ϕ k -boundary between ϕ e s - and ρ -ones within an electron occupier atom determines the extent of transferred electron sharing. The stronger an H⋅⋅⋅O hydrogen bond is, the deeper hydrogen atom's ϕ k -basin penetrates oxygen atom's ρ -basin, while for covalent bonds a ϕ k -boundary closely approaches a ϕ e s -one indicating almost complete sharing of the transferred electrons. In the case of ionic bonds, the same region corresponds to electron pairing within the ρ -basin of an electron occupier atom.

Orbital-free quantum crystallography: view on forces in crystals

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.

Partitioning a Molecule into the Atomic Basins and the Resultant Atomic Charges from Quantum Chemical Topology Analysis of the Kohn-Sham Potential

Quantum chemical topology (QCT) solidifies the chemical basic concepts demonstrating how a molecular system is intrinsically partitioned into its components and what the interaction lines between them are. Here, QCT analysis using a Kohn-Sham one-electron potential (KSpot) in KS equation as a scalar function is initiated and explored, showing KSpot and its resultant electron force lines have novel spatial features which reveal that an atom in a molecule is a spatial basin governed by its nucleus as a 3D-attractor that terminates all the electron force lines defined by the negative gradient of KSpot and that a chemical bond line is just a minimum path of KSpot for the electron motion. Particularly, the atomic charges from this KSpot QCT analysis are moderate and good, having much lower dependence on basis sets chosen for computation. This may provide a platform for the study of molecular structures and properties, intra- and intermolecular electrostatic interaction, energy decomposition, and construction of force field.

What do we learn from the classical turning surface of the Kohn-Sham potential as electron number is varied continuously?

The classical Kohn-Sham turning radius R_t of an atom can be defined as the radius where the Kohn-Sham potential is equal to the negative ionization potential of the atom, i.e., where v_s(R_t) = ϵ_h. It was recently shown [E. Ospadov et al., Proc. Natl. Acad. Sci. U. S. A. 115, E11578-E11585 (2018)] to yield chemically relevant bonding distances, in line with known empirical values. In this work, we show that extension of the concept to non-integer electron number yields additional information about atomic systems and can be used to detect the difficulty of adding or subtracting electrons. Notably, it reflects the ease of bonding in open p-shells and its greater difficulty in open s-shells. The latter manifests in significant discontinuities in the turning radius as the electron number changes the principal quantum number of the outermost electronic shell (e.g., going from Na to Na²⁺). We then show that a non-integer picture is required to correctly interpret bonding and dissociation in H₂ ⁺. Results are consistent when properties are calculated exactly or via an appropriate approximation. They can be interpreted in the context of conceptual density functional theory.

Visualizing atomic sizes and molecular shapes with the classical turning surface of the Kohn-Sham potential

The Kohn-Sham potential [Formula: see text] is the effective multiplicative operator in a noninteracting Schrödinger equation that reproduces the ground-state density of a real (interacting) system. The sizes and shapes of atoms, molecules, and solids can be defined in terms of Kohn-Sham potentials in a nonarbitrary way that accords with chemical intuition and can be implemented efficiently, permitting a natural pictorial representation for chemistry and condensed-matter physics. Let [Formula: see text] be the maximum occupied orbital energy of the noninteracting electrons. Then the equation [Formula: see text] defines the surface at which classical electrons with energy [Formula: see text] would be turned back and thus determines the surface of any electronic object. Atomic and ionic radii defined in this manner agree well with empirical estimates, show regular chemical trends, and allow one to identify the type of chemical bonding between two given atoms by comparing the actual internuclear distance to the sum of atomic radii. The molecular surfaces can be fused (for a covalent bond), seamed (ionic bond), necked (hydrogen bond), or divided (van der Waals bond). This contribution extends the pioneering work of Z.-Z. Yang et al. [Yang ZZ, Davidson ER (1997) Int J Quantum Chem 62:47-53; Zhao DX, et al. (2018) Mol Phys 116:969-977] by our consideration of the Kohn-Sham potential, protomolecules, doubly negative atomic ions, a bond-type parameter, seamed and necked molecular surfaces, and a more extensive table of atomic and ionic radii that are fully consistent with expected periodic trends.