Zero-flux Surfaces Research Articles

Quantum Topological Atomic Properties of 44K molecules

We present a data set of quantum topological properties of atoms of 44K randomly selected molecules from the GDB-9 data set. These atomic properties were obtained as defined by the quantum theory of atoms in molecules (QTAIM) within an atomic basin, a region of real space bounded by zero-flux surfaces in the electron density gradient vector field. The wave function files were generated through DFT static calculations (B3LYP/6-31G), and the atomic properties were calculated using QTAIM. The calculated atomic properties include the energy of the atomic basin, the electronic population, the magnitude of the total dipole moment, and the magnitude of the total quadrupole moment. The atomic properties allow one to understand the chemical structure, reactivity, and molecular recognition. They can be incorporated into force fields for molecular dynamics or for predicting reactive sites. We believe that this data set could facilitate new studies in chemical informatics, machine learning applied to chemistry, and computational molecular design.

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 , 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 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 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 and Fes(r) to a neighboring nucleus and through the smearing by the Pauli kinetic force FP(r).

Toward the Chemical Structure of Diborane: Electronic Force Density Fields, Effective Electronegativity, and Internuclear Turning Surface Properties.

The chemical structure of diborane was elucidated through the superposition of the vector fields of the electron density gradient ∇ρ(r), the electrostatic force Fes(r), and the kinetic force Fk(r), together with the analysis of the cumulative charges of the atoms and pseudoatoms delimited in the aforementioned fields. It was proposed that the Fk-pseudoatomic charge could be employed as a metric for quantifying the ionic component of a related atomic charge. The electron permeability across an internuclear turning surface─specifically, the zero-flux surface in Fk(r)─was characterized by probing it through mapping the total static potential φem(r). The conceptualization of post hoc electronegativity was presented for consideration. Our analysis revealed that the ordinary B-H and bent B-μ-H bonds in diborane demonstrate the polar covalent character with minor contributions of the ionic component. The former bond exhibits a greater electron permeability through the internuclear turning surface, indicating a stronger tendency for electron sharing between the corresponding nucleus-dominated regions. The electron density accumulation along the bent B-μ-H bond path diverges from the minimum action trajectories of the forces Fes(r) and Fk(r). This phenomenon can be associated with the structural strain within the angled, three-center two-electron bonding B-μ-H-B. The internuclear B···B paths were identified in Fes(r) and Fk(r), in contrast to ∇ρ(r). This fact, in conjunction with a pronounced electron permeability through the mutual turning surface between the two boron nuclei, implies a certain degree of electron exchange between the boron-dominated pseudoatomic regions. Furthermore, the anomalous [B-]H···μ-H intermolecular polar interactions were described between the strongly negatively charged hydrogen atoms in the monoclinic crystalline β-phase of diborane. In fact, the ionic contributions to the charges of the hydrogen atoms are shown to be relatively small.

Applicability of transferable multipole pseudo-atoms for restoring inner-crystal electronic force density fields. Chemical bonding and binding features in the crystal and dimer of 1,3-bis(2-hydroxyethyl)-6-methyluracil.

We considered it timely to test the applicability of transferable multipole pseudo-atoms for restoring inner-crystal electronic force density fields. The procedure was carried out on the crystal of 1,3-bis(2-hydroxyethyl)-6-methyluracil, and some derived properties of the scalar potential and vector force fields were compared with those obtained from the experimental multipole model and from the aspherical pseudo-atom model with parameters fitted to the calculated structure factors. The procedure was shown to accurately replicate the general vector-field behavior, the peculiarities of the quantum potentials and the characteristics of the force-field pseudoatoms, such as charge, shape and volume, as well as to reproduce the relative arrangement of atomic and pseudoatomic zero-flux surfaces along internuclear regions. It was found that, in addition to the quantum-topological atoms, the force-field pseudoatoms are spatially reproduced within a single structural fragment and similar environment. In addition, the classical and nonclassical hydrogen bonds in the uracil derivative crystal, as well as the H...O, N...O and N...C interactions in the free π-stacked dimer of the uracil derivative molecules, were studied using the potential and force fields within the concepts of interatomic charge transfer and electron lone pair donation-acceptance. Remarkably, the nitrogen atoms in the N...O and N...C interactions behave rather like a Lewis base and an electron contributor. At the same time, the hydrogen atom in the H...O interaction, being a Lewis acid, also participates in the interatomic electron transfer by acting as a contributor. Thus, it has been argued that, when describing polar interatomic interactions within orbital-free considerations, it makes more physical sense to identify electronegative (electron occupier) and electropositive (electron contributor) atoms or subatomic fragments rather than nucleophilic and electrophilic sites.

Electronic and Crystal Packing Effects in Terms of Static and Kinetic Force Field Features: Picolinic Acid N-Oxide and Methimazole

Herein, we experimentally obtained and studied the inner-crystal scalar potential fields and the associated vector force fields of static and kinetic nature in picolinic acid N-oxide (PANO) and methimazole. The “through-bond” and “through-space” electronic effects were defined and distinguished via the vector fields and concurred with the penetration of the electron contributor’s electrostatic and kinetic force field pseudoatoms (φes- and φk-basins) into the occupier’s chemical atom (ρ-basin). A special focus was given to the dipolar N+–O– bond as well as to the thioamide N–C═S and carboxylic O═C–OH fragments. An unusual way of interatomic charge transfer was revealed between two nonbonded hydrogen atoms [COO−]H···H[−C] in the PANO crystal. The experimental electric and kinetic force fields in the molecular crystals were compared to the theoretical ones for the free molecules and hydrogen-bonded associates, which helped figure out the natural consequences of the crystal packing effect. As expected, the appearance of neighboring attractors in a dense crystal packing requires zero-flux surfaces (ZFSs) emerging in the vector fields and the compression of the outer force field pseudoatoms of a molecule. We proposed to consider a ZFS in the kinetic force field to be a turning surface for electrons in the sense that an electron passed through the boundary is immediately affected by the redirected force attributed to another attractor. The strengthening and noticeable increase observed in the covalency of the intramolecular hydrogen bond O–H···O in the PANO crystal is a direct consequence of such compression of the hydrogen atom and pseudoatoms by the inner-crystal environment. The Pauli and fermionic scalar and vector fields were applied to locate electron lone pairs and describe their involvement in noncovalent interactions within the donor–acceptor mechanism.

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 and kinetic potentials and electron density allowed partitioning any molecules and crystals into atomic - and potential-based -basins; -basins explicitly account for the electron exchange effect, which is missed for -ones. Phenomena of interatomic charge transfer and related electron exchange were explained in terms of space gaps between zero-flux surfaces of - and -basins. The gap between - and -basins represents the charge transfer, while the gap between - and -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 -, -, and - basin boundaries between interacting atoms was proposed. The position of -boundary between - and -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 -basin penetrates oxygen atom's -basin, while for covalent bonds a -boundary closely approaches a -one indicating almost complete sharing of the transferred electrons. In the case of ionic bonds, the same region corresponds to electron pairing within the -basin of an electron occupier atom.

Study of Condensate Absorption Capacity in Exposed Soil when Water Recedes at the Bottom of Hoh Xil Lake, Qinghai

In order to investigate the characteristics of the condensate absorption capacity in an exposed sandy bottom when water recedes, the characteristics of condensate variation, condensate formation time, condensate volume, and its absorption capacity were investigated in July 2021, using a micro-osmometer. The research area was the artificial water-retention layer and bare ground of the exposed sandy bottom, formed under the influence of the warming-wetting trend that occurs when water recedes in the salt-lake area of the Qinghai-Tibet Plateau, as well as two conditions: underpass and under sealed. According to the results, the time of condensation generation during the observation period in the salt-lake area of Hoh Xil begins at about 0:00 and ends at about 10:00 The artificial water-retention layer had little influence on the condensation generation time, and the trend of the condensation rate is the same. The unidirectional condensation of water in near-surface air is significantly better under artificial water-retention layer conditions than under bare ground conditions, with condensation occurring three times more frequently than under bare ground conditions. The amount of water condensation in the lower part of the soil under artificial aquifer conditions is 2.588 times greater than that in the near-ground air, while the amount of water condensation in the lower part of the soil under bare ground conditions is 1.783 times greater than that in the near-ground air. The total amount of bi-directional condensation under artificial water-retention layer conditions is slightly less than that in bare ground conditions, while the total amount of unidirectional condensation under artificial water-retention layer conditions is significantly higher, indicating that the artificial water-retention layer contributes to the absorption of water from near-surface air. Due to the presence of permafrost in the Qinghai-Tibet Plateau region, the zero-flux surface transport of evanescent heat in the salt-lake area of Hoh Xil lies approximately within 30 cm from the surface to the ground. The analogous humidity coefficient characterizes the condensate absorption capacity coefficient as the storage capacity of condensate in the surface layer of the soil in a certain area, providing strong evidence that the condensate absorption capacity is higher under the artificial water-retention layer conditions than in bare ground, regardless of whether the condensation is bi-directional or unidirectional. The results of this study can provide a theoretical basis for condensate absorption capacity and vegetation restoration in the bare ecologically degraded areas of the lake bottom.

Electron Density Geometry and the Quantum Theory of Atoms in Molecules.

A novel form of charge density analysis, that of isosurface curvature redistribution, is formulated and applied to the toy problem of carbonyl oxygen activation in formaldehyde. The isosurface representation of the electron charge density allows us to incorporate the rigorous geometric constraints of closed surfaces toward the analysis and chemical interpretation of the charge density response to perturbations. Visual inspection of 2D isosurface motion resulting from applied external electric fields reveals how the isosurface curvature flows within and between atoms and that a molecule can be uniquely and completely partitioned into chemically significant regions of positive and negative curvatures. These concepts reveal that carbonyl oxygen activation proceeds primarily through curvature and charge redistribution within rather than between Bader atoms. Using gradient bundle analysis─the partitioning of formaldehyde into infinitesimal volume elements bounded by QTAIM zero-flux surfaces─the observations from visual isosurface inspection are verified. The results of the formaldehyde carbonyl analysis are then shown to be transferable to the substrate carbonyl in the ketosteroid isomerase enzyme, laying the groundwork for extending this approach to the problems of enzymatic catalysis.

Bonding features in Appel's salt from the orbital-free quantum crystallographic perspective

Bonding properties in the crystal of 4,5-dichloro-l,2,3-dithiazolium chloride (Appel's salt) were studied using a combination of single-crystal high-resolution X-ray diffraction data and the orbital-free quantum crystallography approach. A QTAIM-based topological model shows the proximity of S—C and S—N bonds to the sesquialteral type and establishes the low S—S bond order in the l,2,3-dithiazolium heterocycle. It is found that the electrostatic potential carries the traces of a common positive area on the junction of interatomic zero-flux surfaces of S1 and S2 atomic basins; meanwhile the exchange energy density per particle shows perfectly here two separate minima through which the two bond paths run. Thus, the pair intermolecular interactions Cl−...S1 and Cl−...S2 formed by the common chloride anion placed near the center of the S—S bond are categorized as chalcogen bonds.

Manipulation of Diatomic Molecules with Oriented External Electric Fields: Linear Correlations in Atomic Properties Lead to Nonlinear Molecular Responses.

Oriented external electric fields (OEEFs) have been shown to have great potential in being able to provide unprecedented control of chemical reactions, catalysis, and selectivity with applications ranging from H2 storage to molecular machines. We report a theoretical study of the atomic origins of molecular changes because of OEEFs since understanding the characteristics of OEEF-induced couplings between atomic and molecular properties is an important step toward comprehensive understanding of the effects of strong external fields on the molecular structure, stability, and reactivity. We focus on the atomic and molecular (bond) properties of a set of homo- (H2, N2, O2, F2, and Cl2) and heterodiatomic (HF, HCl, CO, and NO) molecules under intense external electric fields in the context of quantum theory of atoms in molecules (QTAIM). It is shown that the atomic properties (atomic charges, energies, and localization indices) correlate linearly with the field strengths, but molecular properties (bond length, electron density at the bond critical point, and electron delocalization index) exhibit nonlinear responses to the imposed fields. In particular, the changes in the electron density distribution alter the shapes and locations of the zero-flux surfaces, atomic volumes, atomic electron population, and localization/delocalization indices. The topography and topology of the molecular electrostatic potential undergo dramatic changes. External fields also perturb the covalent-polar-ionic characteristic of the studied chemical bonds, hallmarking the impact of electric fields on the stability and reactivity of chemical compounds. The findings are well-rationalized within the framework of the QTAIM and form a coherent conceptual understanding of these effects in prototypical diatomic molecules.

Magnetically Induced Current Density Spatial Domains.

The pseudostagnation graphs of the current density induced by a magnetic field perpendicular to the plane of the carbon atoms of benzene and cyclopropane, chosen as archetypal of π- and σ-aromatic molecules, have been worked out. Saddle nodes forming the saddle stagnation lines of the pseudostagnation graph have been connected by means of trajectories of the current density, which delimit the border of four different sectors of flow around each saddle line. The merging of all such boundary trajectories provides local zero-flux surfaces in the pseudocurrent density, i.e., separatrices, which split the induced current density field in a number of three-dimensional domains of flow. As a result, a new partition scheme, corresponding to the physical flow that one could observe, has been obtained. For benzene and cyclopropane, by integration of the current density over the various spatial domains, the parallel component of the nuclear magnetic shielding and magnetizability tensors has been decomposed into contributions that are hard to reconcile with any atomic models of partition.

The general setting for the zero-flux condition: The lagrangian and zero-flux conditions that give the heisenberg equation of motion.

Generalizing our recent work on relativistic generalizations of the quantum theory of atoms in molecules, we present the general setting under which the principle of stationary action for a region leads to open quantum subsystems. The approach presented here is general and works for any Hamiltonian, and when a reasonable Lagrangian is selected, it often leads to the integral of the Laplacian of the electron density on the region vanishing as a necessary condition for the zero-flux surface. Alternatively, with this method, one can design a Lagrangian that leads to a surface of interest (though this Lagrangian may not be, and indeed probably will not be, "reasonable"). For any reasonable Lagrangian for the electronic wave function and any two-component method (related by integration by parts to the Hamiltonian) considered, the Bader definition of an atom is recaptured. © 2018 Wiley Periodicals, Inc.

Toward a Rigorous Definition of a Strength of Any Interaction Between Bader's Atomic Basins.

Strength of interaction between Bader's atomic basins, enclosed by zero-flux surfaces of electron distribution, was proposed to be a measure of elastic deformation of an interaction. The set containing 53 atomic aggregate and covering all range of interaction strength (from van der Waals interactions to triple covalent bonds) was calculated by DFT and perturbation theory methods. Further analysis was performed to seek correlations between various local quantities based on electron density and effective force constants of stretching diatomic vibrations. The linear trend between effective force constants and the potential energy density at the (3, -1) critical point of electron distribution was found. This correlation was improved by the integration of the potential energy density over an interbasin zero-flux surface of electron density. Simple mechanical explanation of established trends is presented. The correlations can be further used to at least semiquantitatively compare any pair of interactions between Bader's atomic basins.

Exploring the Gradient Paths and Zero Flux Surfaces of Molecular Electrostatic Potential.

The gradient vector field of molecular electrostatic potential, ∇V(r), has remained relatively unexplored in molecular quantum mechanics. The present article explores the conceptual as well as practical aspects of this vector field. A three-dimensional atomic partition of molecular space has been achieved on the basis of zero flux surfaces (ZFSs) of ∇V(r). Such ZFSs may completely enclose some of the atoms in the molecule, unlike what is observed in density-based atomic partitioning. The demonstration of this phenomenon is elucidated through typical examples, e.g., N2, CO, H2O, H2CO, OF(•), :CH2, and NH3BF3, where the electronegative atoms or group of atoms (group electronegativity) exhibits a closed ZFS of ∇V(r) around them. The present article determines an explicit reason for this phenomenon and also provides a necessary and sufficient condition for such a closed ZFS of ∇V(r) to exist. It also describes how the potential-based picture of atoms in molecules differs from its electron density-based analogue. This work further illustrates the manifestation of anisotropy in the gradient paths of MESP of some molecular systems, with respect to CO, (•)OH, H2O, and H2CO, and points to its potential in understanding the reactivity patterns of the interacting molecules.

Molecular Electrostatic Potential-Based Atoms in Molecules: Shielding Effects and Reactivity Patterns

The Atoms in Molecules (AIM) concept based on the zero-flux surface (ZFS) of the gradient of molecular electrostatic potential (MESP) has been recently proposed by the present authors. The nature of MESP-based atomic basins brings out the asymmetric electronic distribution in a molecule. An electron-rich atom among the two bonded atoms is seen to possess a completely closed MESP-based atomic basin. The present article illustrates the nature of atomic basins for a variety of molecules such as BF, BH3, AlCl3, B2H6, and Al2Cl6, and a Lewis acid–base pair, viz. NH3BH3 wherein the electronic distribution is not merely guided by difference in the electronegativity of the atoms. The study also explores some transition metal complexes, viz. Ni(CO)4, Fe(CO)5, Cr(CO)6, Mn2(CO)10, Co2(CO)8, Fe(η5-C5H5)2, Co(η3-C3H5), and Co(η3-C3H5)(CO)3, which show a similar phenomenon of intricate charge transfer among the ligands and the metal centre. The present article employs MESP-based AIM for a qualitative explanation of the shielding or deshielding effects revealed by NMR data as well as susceptibility of an atomic region towards an electrophilic or nucleophilic attack. Because the topographical features of MESP and thus the nature of atomic basins are not very sensitive to the level of theory and basis set, the present article demonstrates the capability of MESP as a consistent and simple tool for the portrayal of asymmetry in molecular charge distribution.

The influence of zero-flux surface motion on chemical reactivity.

Visualizing and predicting the response of the electron density, ρ(r), to an external perturbation provides a portion of the insight necessary to understand chemical reactivity. One strategy used to portray electron response is the electron pushing formalism commonly utilized in organic chemistry, where electrons are pictured as flowing between atoms and bonds. Electron pushing is a powerful tool, but does not give a complete picture of electron response. We propose using the motion of zero-flux surfaces (ZFSs) in the gradient of the charge density, ∇ρ(r), as an adjunct to electron pushing. Here we derive an equation rooted in conceptual density functional theory showing that the movement of ZFSs contributes to energetic changes in a molecule undergoing a chemical reaction. Using a substituted acetylene, 1-iodo-2-fluoroethyne, as an example, we show the importance of both the boundary motion and the change in electron counts within the atomic basins of the quantum theory of atoms in molecules for chemical reactivity. This method can be extended to study the ZFS motion between smaller gradient bundles in ρ(r) in addition to larger atomic basins. Finally, we show that the behavior of ∇ρ(r) within atomic basins contains information about electron response and can be used to predict chemical reactivity.

Ad hoc methods for accurate determination of Bader's atomic boundary

In addition to the recently published triangulation method [P. M. Polestshuk, J. Comput. Chem. 34, 206 (2013)], two new highly accurate approaches, ZFSX and SINTY, for the integration over an atomic region covered by a zero-flux surface (zfs) were developed and efficiently interfaced into the TWOE program. ZFSX method was realized as three independent modules (ZFSX-1, ZFSX-3, and ZFSX-5) handling interatomic surfaces of a different complexity. Details of algorithmic implementation of ZFSX and SINTY are discussed. A special attention to an extended analysis of errors in calculations of atomic properties is paid. It was shown that uncertainties in zfs determination caused by ZFSX and SINTY approaches contribute negligibly (less than 10(-6) a.u.) to the total atomic integration errors. Moreover, the new methods are able to evaluate atomic integrals with a reasonable time and can be universally applied for the systems of any complexity. It is suggested, therefore, that ZFSX and SINTY can be regarded as benchmark methods for the computation of any Quantum Theory of Atoms in Molecules atomic property.

Accurate integration over atomic regions bounded by zero‐flux surfaces

The approach for the integration over a region covered by zero-flux surface is described. This approach based on the surface triangulation technique is efficiently realized in a newly developed program TWOE. The elaborated method is tested on several atomic properties including the source function. TWOE results are compared with those produced by using well-known existing programs. Absolute errors in computed atomic properties are shown to range usually from 10(-6) to 10(-5) au. The demonstrative examples prove that present realization has perfect convergence of atomic properties with increasing size of angular grid and allows to obtain highly accurate data even in the most difficult cases. It is believed that the developed program can be bridgehead that allows to implement atomic partitioning of any desired molecular property with high accuracy.

Quantum Theory of Atoms in Molecules: Results for the SR-ZORA Hamiltonian

The quantum theory of atoms in molecules (QTAIM) is generalized to include relativistic effects using the popular scalar-relativistic zeroth-order regular approximation (SR-ZORA). It is usually assumed that the definition of the atom as a volume bounded by a zero-flux surface of the electron density is closely linked to the form of the kinetic energy, so it is somewhat surprising that the atoms corresponding to the relativistic kinetic-energy operator in the SR-ZORA Hamiltonian are also bounded by zero-flux surfaces. The SR-ZORA Hamiltonian should be sufficient for qualitative descriptions of molecular electronic structure across the periodic table, which suggests that QTAIM-based analysis can be useful for molecules and solids containing heavy atoms.

The experimental and theoretical QTAIMC study of the atomic and molecular interactions in dinitrogen tetroxide

The atomic and molecular interactions in a crystal of dinitrogen tetraoxide, alpha-N2O4, have been studied in terms of the quantum topological theory of molecular structure using high-resolution, low-temperature X-ray diffraction data. The experimental electron density and electrostatic potential have been reconstructed with the Hansen-Coppens multipole model. In addition, the three-dimensional periodic electron density of crystalline alpha-N2O4 has been calculated at the B3LYP/cc-pVDZ level of theory with and without the geometry optimization. The application of the quantum theory of atoms in molecules and crystals (QTAIMC) recovered the two types of intermolecular bond paths between O atoms in crystalline alpha-N2O4, one measuring 3.094, the other 3.116 A. The three-dimensional distribution of the Laplacian of the electron density around the O atoms showed that the lumps in the negative Laplacian fit the holes on the O atoms in the adjacent molecules, both atoms being linked by the intermolecular bond paths. This shows that the Lewis-type molecular complementarity contributes significantly to intermolecular bonding in crystalline N2O4. Partial overlap of atomic-like basins created by zero-flux surfaces in both the electron density and the electrostatic potential show that attractive electrostatic interaction exists between O atoms even though they carry the same net formal charge. The exchange and correlation contributions to the potential energy density were also computed by means of the model functionals, which use the experimental electron density and its derivatives. It was found that the intermolecular interactions in alpha-N2O4 are accompanied by the correlation energy-density ;bridges' lowering the local potential energy along the intermolecular O...O bond paths in the electron density, while the exchange energy density governs the shape of bounded molecules.