Zero-flux Surfaces Research Articles

On the Volume Chemistry of Solid Compounds: the Legacy of Wilhelm Biltz

AbstractFor the calculation of the atomic or ionic volumes the Quantum Theory of Atoms In Molecules method was applied. The regions (basins) around the nuclei confined by the zero‐flux surfaces in the electron density gradient are called QTAIM atoms. They are non‐overlapping and completely fill the space. The volume of the basins gives volumes of atoms or ions. The integration of the electron density within the volumina yields effective charges, defining neutral or ionic character of the given QTAIM species. Present investigations refer to metal hydrides, metal nitrides and to intermetallic compounds of the system Al‐Pt. A linear relation between the ionic volumina of hydrogen or nitrogen established according to QTAIM and after Biltz has been found with (homodesmic) binary metal hydrides and binary metal nitrides, but has been observed merely as a trend with stronger deviations for heterodesmic compounds, such as ternary hydrido‐ and nitridometallates Aa[MmXx] (A – alkali or alkaline earth metal, M – transition metal and X – H or N). The deviation from linearity for heterodesmic compounds is caused by the different kinds of chemical bonds being present within the [MmXx] anions on the one hand and between the anions and the cations on the other hand reflected by the calculated volumes and the QTAIM charges of M and X components. Concerning the intermetallic compounds of the system Al‐Pt, the quantum chemical calculations reveal negative charges for the platinum atoms and positive ones for the aluminium atoms in accordance with their electronegativities. Introducing the variation of the atomic volume with the composition extends the Vegard's approach and gives a non‐linear slope for the concentration dependence of mean atomic volume which explains qualitatively the experimental results.

Zero-Flux Surfaces of the Electrostatic Potential: The Border of Influence Zones of Nucleophilic and Electrophilic Sites in Crystalline Environment

The topology of the electrostatic potential varphi(r) has been studied for single molecules using geometries and electron distributions rho(r) determined from high-resolution single-crystal X-ray diffraction data. The electrostatic potential gradient nablavarphi(r), which is the negative of the electric field E = -nablavarphi, has been represented, revealing the position of zero-flux surfaces and critical points. Local maxima and minima of the electrostatic potential are interpreted in terms of electrophilic and nucleophilic sites, which present influence zones delimited by zero-flux surfaces containing saddle points. The influence zones of the nucleophilic and electrophilic sites define two alternative partitions of the space in disjoint volumes, the completeness of these partitions depending on either the neutral or ionic character of the molecule. The results obtained by using this methodology are useful for the interpretation of the saddle points of the electrostatic potential, which are related to the limits of the influence zones and reveal the path for preferred attack on reactive sites with finite influence zones.

Polarised neutron diffraction and molecular magnetic materials

The topology of the electrostatic potential has been studied for single molecules using geometries and electron distributions determined from high-resolution singlecrystal X-ray diffraction experiments. The electrostatic potential gradient, which is the negative of the electric field, has been represented, revealing the position of zeroflux surfaces and critical points [1]. Through the representation of the electric field lines, the relationship between the topology of the electrostatic potential and the reactivity of the molecule is explored. Local maxima, corresponding to the nuclei, are associated to electrophilic sites, while local minima, which are related to local accumulations of electron density (i.e. in lone pairs) are identified as nucleophilic sites. Both kind of reactive sites present influence zones delimited by zeroflux surfaces. Space can be partitioned in disjoint volumes (primary bundles) [2], each of them corresponding to the intersection of the influence zones of at most one nucleophilic and one electrophilic sites, which are both situated on the surface of the primary bundle. As a result of the interaction with the whole molecule, a probe charge inside one primary bundle is directed to, or away from, the reactive sites on the surface of this volume. The bundles can be added to generate two separate partitions in nucleophilic and electrophilic influence zones, the completeness of these last partitions depending on either the neutral or ionic character of the molecule. Analogous to the bond and ring critical points in the electron density topology, electrostatic potential saddle critical points appear on the zero-flux surfaces. They are related to the limits of the electrophilic and nucleophilic influence zones, which can extend to infinite or present a finite volume. In this last case, a saddle point reveals the limit of the influence zone and the path for preferred attack on the reactive site.

Electron Density Analyses of Opioids: A Comparative Study

The electron densities of five morphine related molecules (codeine, diprenorphine, naltrexone in the neutral and protonated states, and dextromethorphan) were determined from high-resolution X-ray diffraction experiments (Mo Kalpha and synchrotron primary radiation) at low temperature and CCD area detection techniques. Bond topological analyses were applied, and a partitioning of the molecules into atomic regions making use of Bader's zero flux surfaces yielded atomic volumes and charges. The obtained atom and bonding properties were compared to the results of a previous experimental study of morphine and to theoretical calculations. Experimental and theoretical properties for all chemically equivalent bonds agree within an uncertainty range as is otherwise seen for different theoretical calculations. Hence, the transferability of chemically equivalent submolecular properties, being a key issue of the atoms in molecules (AIM) theory, has been verified experimentally in this class of chemically related molecules. On the other hand, topological differences could clearly be verified in regions with different chemical environments. Electron density differences between the two forms of naltrexone were examined and made visible in an extended region around the nitrogen atom which is once in a neutral state and once in a positively charged state.

The relationship between the molecular structure and the chemical properties: what about the topological analysis of the electron density distribution? An application to hydrogen bonds

Molecular structure is determined by nuclei positions. From the Hellmann-Feynman theorem we know that the exact ground state electron density distribution ρ(r) depends on the nuclei positions only. Furthermore, the Hohenberg-Kohn theorem states that the total energy of a system can be written in terms of its ρ(r) distribution. In this context, the relationship between the structure of a molecular system and its physical and chemical properties should be reflected by the electron distribution. As far as the molecular structure is the straightforward consequence of interatomic interactions, a correspondence between these interactions and the observed ρ(r) applies. Accordingly, the interatomic interactions described in terms of ρ(r) can be considered as a fundamental subject of study to get insight on the structure-properties relationship, as the former is a conceptual bridge between the latter. The topological analysis of ρ(r) developed by Bader and co-workers [1] is a useful tool for characterizing atomic interactions in internuclear regions. It permits to obtain the molecular space partition into atomic basins that are separated by zero-flux surfaces S(r )o f the electron distribution and behave as quantum objects. Along the bond path directions ρ(r) is minimum at the surfaces S(r), where topological bond critical points rBCP appear and ρ(r) exhibits saddle distributions. Analysis of topological and energetic magnitudes of ρ(r )a trBCP permits a deep characterization of interactions. Following this description we have analyzed experimental and theoretically calculated energetic properties of hydrogen bonded systems in terms of the ρ(r) distribution at their rBCP’s [2-7].

Chemical bonding in pentaerythritol at very low temperature or at high pressure: an experimental and theoretical study

Chemical bonding in the pentaerythritol crystal based on the experimental electron density at 15 (1) K, and theoretical calculations at the experimental molecular geometries obtained at room and low (15 K) temperatures have been analyzed and compared in terms of the topological analysis. Topological electron-density features corresponding to the high-pressure (1.15 GPa) geometry are also reported. In addition to the bond critical points (CPs) within the molecular layers, CPs between the atoms of different molecular layers have been located and the bonding character of these relatively weak interactions discussed. Atomic charges and energies have been integrated over the atomic basins delimited by the zero-flux surfaces, and the intermolecular interaction energies have been calculated. The interaction between molecular layers in the crystal becomes stronger both at very low temperature and high pressure, as demonstrated by the more negative intermolecular interaction energies, higher electron density and energy density values at the CPs, and sharper electronic-energy density profiles.

Dipole and quadrupole moments of molecules in crystals: A novel approach based on integration over Hirshfeld surfaces

Elegant expressions are derived for the computation of dipole and quadrupole moments of molecules using the electrostatic potential and electric field evaluated on an oriented molecular surface. These expressions are implemented for Hirshfeld surfaces, applied to various molecular crystals, and compared with the results from the quantum theory of atoms in molecules. The effect of intermolecular interactions is also explored by examining the differences between electrostatic moments derived from a periodic Hartree-Fock electron density and an electron density resulting from a superposition of noninteracting molecules. The enhancement of the dipole moment for hydrogen bonded molecular crystals is typically 30%-40% and shown to be largely independent of the partitioning scheme. Dipole moments calculated from Hirshfeld surfaces systematically underestimate those from zero-flux surfaces, a result attributed to the translation of the Hirshfeld surface relative to the zero-flux surfaces for these molecules. For acetylene and benzene, the differences between a crystal calculation and the sum of noninteracting molecules are small, and both partitioning schemes yield quadrupole and second moment results in close agreement.

Why Are Carborane Acids so Acidic? An Electrostatic Interpretation of Brønsted Acid Strengths

Acidity of Brønsted acids is explained in terms of the electrostatic potentials of the corresponding conjugate bases. The electrostatic potential distribution on the zero-flux surface of the strongest isolable carborane anions is seen to provide a good measure of their acidities. Increasing value of the lowest minimum in the electrostatic potential is observed to be a signature of increasing acidity.

Submolecular partitioning of morphine hydrate based on its experimental charge density at 25 K

The electron density distribution of morphine hydrate has been determined from high-resolution single-crystal X-ray diffraction measurements at 25 K. A topological analysis was applied and, in order to analyze the submolecular transferability based on an experimental electron density, a partitioning of the molecule into atomic regions was carried out, making use of Bader's zero-flux surfaces to yield atomic volumes and charges. The properties obtained were compared with the theoretical calculations of smaller fragment molecules, from which the complete morphine molecule can be reconstructed, and with theoretical studies of another opiate, Oripavine PEO, reported in the literature.

Transferability of Atomic Volumes and Charges in the Peptide Bond Region in the Solid State

Bader's partitioning scheme that makes use of the zero flux surfaces (ZFS) in the electron density gradient vector field has been applied to the peptide bond regions of three oligopeptides (two dipeptides, one hexapeptide), yielding eight peptide bond regions to compare. From integration over the atomic volume, very reproducible atoms in molecules (AIM) charges were calculated which agree within the given atom types by 0.04−0.08 e. The polarization of the peptide bond atoms is high compared to the charges normally used in force field parametrization. The positive charges of the Cα, C‘, and H atoms sum up to ≈ +1.6 e, while the negative charges of N and O amount to ≈ −1.85 e, so that for each peptide bond region an excess of −0.25 e has to be compensated by the Cα hydrogen and the side chains.

Experimental charge density of octafluoro-1,2-dimethylenecyclobutane: atomic volumes and charges in a perfluorinated hydrocarbon.

Octafluoro-1,2-dimethylenecyclobutane, mp. 238 K, was crystallized in situ on a SMART 1000-CCD diffractometer, and high order X-ray diffraction data were collected at 100 K for a charge density determination. A topological analysis was applied and a partitioning of the molecule into atomic regions making use of Bader's zero flux surfaces yielded atomic volumes and charges. Corresponding atomic properties were also derived theoretically from B3LYP/6-311++G(3df,3pd) wavefunctions. While for carbon the volumes and charges are largely dependent on their bonding environment, fluorine has an almost constant atomic volume around 16-17 A3 and a charge between -0.6 and -0.7e, not only in the title compound, but also in two further perfluorinated hydrocarbons, of which the charge densities were determined earlier.

Electron density and energy density view on the atomic interactions in SrTiO3.

The results of topological analysis of the electron density in an SrTiO3 crystal based on the experimental (at 145 K) and theoretical data are presented and discussed. The features of the electron density lead to the conclusion that the Ti-O interaction is of the partly polar covalent (or intermediate) type. Complicated atomic shapes defined by the zero-flux surfaces in the electron density are revealed. It is found that, in general, they are far from spherical and have very slight asphericity in the close-packed layers. The topological coordination numbers of Sr and Ti are the same as the geometrical numbers, whereas the topological coordination for the O atom (6) differs from the geometrical value (12). The latter results from the specific shape of the Ti-atom basin, which prevents bond-path formation between the O atoms. The analysis of the kinetic and potential energy densities derived from the electron density using the density functional theory formulae revealed the stabilizing crystal-forming role of the O atoms in SrTiO3. Structural homeomorphism between the experimental electron density and the potential and kinetic energy densities is observed.

Atomic properties of N(2)O(4) based on its experimental charge density.

Nitrogen dioxide, being known to exist as a dimer N2O4 in the crystal with a very long N-N bond length of 1.76 A, was crystallized at low-temperature conditions on a diffractometer. High-resolution X-ray data (sin(theta/lambda) = 1.249 A-1) were recorded with a CCD area detector to allow the generation of an experimental charge density distribution. By making use of Bader's AIM theory, zero-flux surfaces were calculated, and we examined atomic volumes and atomic charges obtained from this experiment and various theoretical calculations. Four commonly used methods of computing atomic charges (Mulliken, AIM, NPA, and CHELP) were considered. The AIM charges are rather independent from the used basis set. Interestingly, the evaluated atomic volumes are very similar between experiment and theory, although in theory isolated molecules are considered. For the long N-N bond a bond order n of approximately 0.5 was derived from a comparison with appropriate model compounds.

The zero-flux surface and the topological and quantum definitions of an atom in a molecule

The partitioning of a charge distribution by surfaces exhibiting a local zero flux in the gradient vector field of the electron density leads to an exhaustive and disjoint division of the system into a set of mono-nuclear regions or atoms, provided the only local attractors present in the system are isolated nuclear attractors and the electronic energy is less than that required to produce the plasma state. The existence of non-isolated attractors, whose limited occurrence is confined primarily to excited state charge distributions of one-electron systems, is shown to be readily encompassed within the topological theory of molecular structure, a theory whose purpose is to relate a system's properties to the observed topology of its density distribution. The zero-flux surface serves as the necessary boundary condition for the application of Schwinger's principle of stationary action to define the physics of an atom in a molecule as an open system. Schwinger's principle requires the use of a special class of trial functions: those whose variation is to be equated to the action of smooth, continuous changes in the coordinates of the physical system caused by the action of generators of infinitesimal transformations, the very requirement needed to ensure the applicability of the zero-flux surface condition as the defining constraint of an open system.

THE ZERO FLUX SURFACE OF THE ELECTRONIC WAVEFUNCTION'S GRADIENT

In a previous publication we derived a criterion for defining interatomic surfaces in condensed systems based on the zero flux surface (ZFS) of the electronic wavefunction's gradient which holds the condition of flux zero at every point. Here this criterion is examined and physical as well as mathematical properties are illustrated and discussed; in the light of this discussion we conclude that the definition illustrated in this work represents a natural extension of Bader's definition of atoms in molecules.

Electron density of KNiF3: analysis of the atomic interactions

The topological analysis of the electron density in the perovskite KNiF3, potassium nickel trifluoride, based on the accurate X-ray diffraction data, has been performed. The topological picture of the atomic interactions differs from that resulting from the classic crystal chemistry consideration. The shapes of atoms in KNiF3 defined by zero-flux surfaces in the electron density are, in general, far from spherical. At the same time, their asphericity in the close-packed layer is very small. The topological coordination numbers of K and Ni are the same as the geometrical ones, whereas topological coordination for the F atom (6) differs from the geometrical value. The latter results from a specific shape of the Ni-atom basin preventing the bond-path formation between F atoms in the same atomic close-packed layer, in spite of the fact that the closest F-F distance is the same as K-F. Judging by the electron density value and curvature at the bond critical points, the K-F interaction in KNiF3 can be considered ionic, while the Ni-F bond belongs to the polar covalent type. No correlation of the topological ionic radii with crystal or ionic radii was found in KNiF3. Critical points in the electrostatic potential have also been studied.

Atomic force microscope as an open system and the Ehrenfest force

Two systems in contact, such as the tip of an atomic force microscope (AFM) and a sample, share a common surface. Each exerts an equal and opposite force on the other determined by the pressure it exerts on every element of the surface of separation, as required by the physics of an open system. In a quantum system, the force exerted on the tip is the Ehrenfest force, a force that is balanced by the pressure exerted on every element of its surface, as determined by the quantum stress tensor. The surface separating the tip from the sample is one of local zero flux in the gradient vector field of the electron density, the surface that separates two neighboring atoms. A zero-flux surface also defines a proper open system, one whose observables are governed by the equations of motion, the equation for the electronic momentum yielding the Ehrenfest force theorem. Thus the force measured in the AFM is exerted on a surface determined by the boundaries separating the atoms in the tip from those in the sample, and its response is a consequence of the atomic form of matter. This approach to the determination of the force measured in the AFM is contrasted with results reported in the literature that equate it to the Hellmann-Feyman forces exerted on the nuclei of the atoms in the tip.

Elastic sheet method for identifying atoms in molecules

We have developed a new method for finding and representing dividing surfaces which can, for example, be used to identify “atoms” in molecules or condensed phases based on Bader’s definition. Given the total electron density of the system, the dividing surface is taken to be the zero-flux surface, i.e., the surface on which the normal component of the gradient vanishes. Our method for finding this surface involves creating an “elastic sheet” represented by a swarm of fictitious particles which interact with each other so as to give a nearly uniform distribution of points on the sheet. Two kinds of forces act on the particles: (1) the component of the gradient of the density normal to the elastic sheet, and (2) an interparticle force which only acts in the local tangent plane of the sheet. Starting with a spherical surface and applying an optimization algorithm that minimizes the forces leads to convergence of the particles to the zero-flux surface. The elastic sheet tends to round off regions where the zero-flux surface has sharp cusps or points, but this appears not to be a serious problem in cases we have studied. The elastic sheet method is robust and can converge in situations where currently used methods fail. We demonstrate the method with a study of water clusters and a Si interstitial in a Si crystal.

Analytical derivatives of atomic zero-flux surfaces and properties of atoms in molecules with respect to external perturbations

A complete theory of the response of the atomic zero-flux surfaces to external perturbations is presented. The resulting expressions for the analytical derivatives of the properties of atoms in molecules (AIMs) are implemented in an efficient computer code. Several test calculations demonstrate the clear advantages of the new approach over methods based upon numerical differentiation. The present development opens an avenue to routine calculations of the second-order response properties of AIMs.

Determination of zero-flux surfaces and integrated properties from experimentally determined charge densities in crystals

Determination of zero-flux surfaces and integrated properties from experimentally determined charge densities in crystals