Covalent bond orders and atomic anisotropies from iterated stockholder atoms

Iterated stockholder atoms are produced by dividing molecular electron densities into sums of overlapping, near-spherical atomic densities. It is shown that there exists a good correlation between the overlap of the densities of two atoms and the order of the covalent bond between the atoms (as given by simple valence rules). Furthermore, iterated stockholder atoms minimise a functional of the charge density, and this functional can be expressed as a sum of atomic contributions, which are related to the deviation of the atomic densities from spherical symmetry. Since iterated stockholder atoms can be obtained uniquely from the electron density, this work gives an orbital-free method for predicting bond orders and atomic anisotropies from experimental or theoretical charge density data.     

Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.     

Remarks on the concept of an atom in a molecule and on charge transfer between atoms on molecule formation

Density functional theory provides a natural and rigorous definition of an atom in a molecule in its ground state: The molecular electron density is the sum of atomic densities, the atoms have the same chemical potential as does the molecule, and the atoms are minimally promoted from their ground states. These atoms in general are not spherical, and in general they bear nonintegral charges. Charge transfer on molecule formation is thereby uniquely defined. Calculations by Palke and by Guse are reviewed, in which the hydrogen atom is identified in the hydrogen molecule.     

Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory

This work presents a method to move beyond the recently introduced atomic fragment approximation. Like the bare atomic fragment approach, the new method is an ab initio, parameter-free, orbital-free implementation of density functional theory based on the bifunctional formalism that treats the potential and the electron density as two separate variables, and provides access to the Kohn–Sham Pauli kinetic energy for an appropriately chosen Pauli potential. In the present ansatz, the molecular Pauli potential is approximated by the sum of the bare atomic fragment approach, and a so-called deformation potential that takes the interaction between the atoms into account. It is shown that this model can reproduce the bond-length contraction due to multiple bonding within the list of second-row homonuclear dimers. The present model only relies on the electron densities of the participating atoms, which themselves are represented by a simple monopole expansion. Thus, the bond-length contraction can be rationalized without referring to the angular quantum numbers of the participating atoms.     

Concerning atomic radii

The identification of the atomic radius with the distance from the nucleus to the position of the minimum in the internuclear electronic density is studied. It is shown that a set of statistically significant radii may be defined for the atoms of a given column of the periodic table bound to any of the atoms of another column. Sets of radii calculated by modified Anderson-Parr relations are presented. The values obtained are consistent with radii obtained using a minimum in electronic density criterion with electron densities calculated from molecular orbital wave functions or approximated by a sum of atomic densities.     

Chemistry and the near-sighted nature of the one-electron density matrix

Two different macrospopic pieces of copper have different external potentials and, because of the unique functional relationship between the electron density and the external potential as demanded by density functional theory, should possess different electron density distributions. Experimentally, however, an atom in the bulk exhibits the same electron density in both samples and they possess identical sets of intensive properties. Density functional theory does not account for the fundamental observation underlying the theory of atoms in molecules: that what are apparently identical distributions of charge can be observed for an atom or a grouping of atoms in systems with different external potentials and that these atoms contribute essentially identical amounts to the energies and all other properties of the systems in which they occur. It is shown that, unlike the external potential, the kinetic energy density and the potential energy density, defined by the virial of the Ehrenfest force acting on electron density, are short-range functions. As recorded in the first article on atoms in molecules, they exhibit a local dependence on the electron density that causes them to faithfully mimic the transferability of the atomic charge distributions from one system to another. The electron, the kinetic energy, and the virial densities are all determined directly by the one-electron density matrix, a function termed near-sighted by Professor Kohn. It is this near-sighted property of the one-matrix that underlies the working hypothesis of chemistry—that of a functional group exhibiting a characteristic set of properties. The observations obtained from the theory of atoms in molecules and the atomic theorems it determines demonstrate the existence of a local relationship between the electron density and all properties of a system. © 1995 John Wiley & Sons, Inc.     

The topology of the Ehrenfest force density revisited. A different perspective based on Slater‐type orbitals

The topology of the Ehrenfest force density was studied with Slater‐type orbitals (STO). At larger distances from the nuclei, STOs generate similar artefacts as noticed before with Gaussian‐type orbitals. The topology of the Ehrenfest force density was found to be mainly homeomorphic with the topology of the electron density. For the first time, reliable integrations of several properties over force density atomic basins were performed successfully. Integration of the electron density of a number of hydrides, fluorides, and chlorides of first row elements over force density basins indicate substantial differences between the partial charges of the atoms as compared with those obtained from electron density basins. Calculations on saturated hydrocarbons confirm that the electronegativity of carbon atoms increases with increasing geometrical strain. Atomic interaction lines are observed to exist in the Ehrenfest force density between the hydrogen atoms of several so‐called “congested” molecules, and also in some inclusion complexes of alkanes with helium. However, interaction lines are lacking in several other controversial cases. © 2015 Wiley Periodicals, Inc.     

A fast and accurate algorithm for QTAIM integration in solids

A new algorithm is presented for the calculation of atomic properties, in the sense of the quantum theory of atoms in molecules. This new method, named QTREE , applies to solid‐state densities and allows the computation of the atomic properties of all the atoms in the crystal in seconds to minutes. The basis of the method is the recursive subdivision of a symmetry‐reduced wedge of the Wigner‐Seitz cell, which in turn is expressed as a union of tetrahedra, plus the use of β‐spheres to improve the performance. A considerable speedup is thus achieved compared with traditional quadrature‐based schemes, justified by the poor performance of the latter because of the particular features of atomic basins in solids. QTREE can use both analytical or interpolated densities, calculates all the atomic properties available, and converges to the correct values in the limit of infinite precision. Several gradient path tracing and integration techniques are tested. Basin volumes and charges for a selected set of 11 crystals are determined as a test of the new method. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Hartree-Fock energy partitioning in terms of Hirshfeld atoms.

A Hirshfeld decomposition scheme of the Hartree-Fock total molecular energy into atomic energies is presented. The calculations are performed by direct numerical integration and the results are compared for a set of 28 molecules containing different kinds of atoms. The calculated atomic energies show a strong dependency on changes of atomic electron population and hybridization. Linear correlations are found between the energy and the population for H, these being related to the electronegativity of this atom and to the external potential created by the remaining atoms. The proposed energy partitioning scheme appears to be useful for studies such as proton acidity, the anomeric effect and group transferability, and allows atomic virial ratios to be obtained. Finally, the atomic potential energies are found to mimic trends based on exact expressions as well as trends displayed by molecular quantities, thus lending credibility to the partitioning scheme used.     

Generalized electron number distribution functions: real space versus orbital space descriptions

The algebraic expressions previously derived to compute the electron number distribution functions (EDF) for exhaustive partitions of the physical space into sharp-boundary atoms are generalized to allow for the use of fuzzy atoms and orbital-based partitions. In some of the latter, the atomic overlap matrix required to obtain the EDF is analytical. This makes them attractive alternatives to other definitions, as the one based on the atomic basins of the quantum theory of atoms in molecules (QTAIM), which are more physically sound but also much more demanding computationally. We will compute the EDF for a series of test molecules using different fuzzy and orbital-based partitions and compare them to QTAIM EDF. The effects of electron correlation and the use of the core approximation on the EDF will also be explored.     

On the relationship between densities of Shannon entropy and Fisher information for atoms and molecules

An analytical relationship between the densities of the Shannon entropy and Fisher information for atomic and molecular systems has been established in this work. Two equivalent forms of the Fisher information density are introduced as well. It is found that for electron densities of atoms and molecules the Shannon entropy density is intrinsically related to the electron density and the two forms of the Fisher information density. The formulas have been confirmed by the numerical results for the first two-row atoms.     

Measuring electron sharing between atoms in first-principle simulations

Calculations of large scale electronic structure within periodic boundary conditions, mostly based on solid state physics, allow the modeling of atomic forces and molecular dynamics for atomic assemblies of 100–1000 atoms, thus providing complementary information in material and macromolecular sciences. Nevertheless, these methods lack connections with the chemistry of simple molecules as isolated entities. In order to contribute to establish a conceptual connection between solid state physics and chemistry, the calculation of the extent of electron sharing between atoms, also known as delocalization index, is performed on simple molecules and on complexes with transition metal atoms, using density functional calculations where the Kohn–Sham molecular orbitals are represented in terms of plane waves and in periodic boundary conditions. These applications show that the useful measure of electron sharing between atomic pairs can be recovered from density functional calculations using the same set-up applied to large atomic assemblies in condensed phases, with no projections of molecular orbitals onto atomic orbitals.     

Information distance analysis of molecular electron densities

The information‐theoretic basis of the Hirshfeld partitioning of the molecular electronic density into the densities of the “stockholder” atoms‐in‐molecules (AIM) is summarized. It is argued that these AIM densities minimize both the directed divergence (Kullback–Leibler) and divergence (Kullback) measures of the entropy deficiency between the AIM and their free atom analogs of the promolecule. The local equalization of the information distance densities of the Hirshfeld components, at the local value of the corresponding global entropy deficiency density, is outlined and several approximate relations are established between the alternative local measures of the missing information and the familiar function of a difference between the molecular and promolecule densities. Various global (of the system as a whole) and atomic measures of the entropy deficiency or the displacements relative to the isoelectronic promolecule, defined for densities or probabilities (shape functions) in both the local resolution and the Hirshfeld AIM discretization, are introduced and tested. This analysis is performed also for the valence electron (frozen‐core) approximation. Illustrative results for representative linear molecules, including diatomics, triatomics, and tetraatomics, are reported. They are interpreted as complementary characteristics of changes in the net AIM charge distribution and of the displacements in the information content of the electron distributions of bonded atoms. These numerical results confirm the overall similarity of the stockholder AIM to their free atom analogs and reflect the information displacements due to the AIM polarization and charge transfer in molecules. They also demonstrate the semiquantitative nature of the approximate relations established between the entropy deficiency densities and the related functions involving the density difference function. This development extends the range of interpretations based on the density difference diagrams into probing the associated information displacements in a molecule accompanying the formation of the chemical bonds. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002     

Na-beam cooling by a mode-locked laser

Controlled mode-locked laser cooling of sodium atoms has been demonstrated for the first time. The broadband laser spectrum covers most of the Doppler-broadened atomic resonance spectrum. Atoms of a broad velocity range are thus decelerated simultaneously. In order to stop this process at a defined atomic velocity a two-mode laser beam tuned to resonance with both sodium groundstate levels is copropagating with the atomic beam and counterpropagating to the mode-locked laser beam. In this optical molasses sodium atoms ofv=20 m/s have been piled up to densities of about 2·10⁶ cm^?3. The density dependence on variable laser parameters has been investigated systematically.     

Exploring the lower limit in hydrogen bonds: analysis of weak C-H...O and C-H...pi interactions in substituted coumarins from charge density analysis

Experimental electron densities in coumarin, 1-thiocoumarin, and 3-acetylcoumarin have been analyzed based on the X-ray diffraction data at 90 K. These compounds pack in the crystal lattice with weak C-H...O and C-H...pi interactions, and variations in charge density properties and derived local energy densities have been investigated in the regions of intermolecular interactions. Theoretical charge density calculations on crystals using the B3LYP/6-31G* method show remarkable agreement with the derived properties and energy densities from the experiment. The intermolecular interactions follow an exponential dependence of electron density and energy densities at the bond critical points. The Laplacian follows a "Morse-like" dependence on the length of the interaction line. Based on the set of criteria defined using the theory of "atoms in molecules", it has become possible to distinguish between a hydrogen bond (C-H...O) and a van der Waals interaction (C-H...pi). This has resulted in the identification of a "region of overlap" in terms of electron densities, energy densities, and mutual penetration of the hydrogen and acceptor atoms with respect to the interaction length. This approach suggests a possible tool to distinguish between the two types of interactions.     

Nonempirical statistical theory for atomic evaporation from nonrigid clusters: applications to the absolute rate constant and kinetic energy release

A high energy atomic cluster undergoing frequent structural isomerization behaves like a liquid droplet, from which atoms or molecules can be emitted. Even after evaporation, the daughter cluster may still keep changing its structure. We study the dynamics of such an evaporation process of atomic evaporation. To do so, we develop a statistical rate theory for dissociation of highly nonrigid molecules and propose a simple method to calculate the absolute value of classical phase-space volume for a potential function that has many locally stable basins. The statistical prediction of the final distribution of the released kinetic energy is also developed. A direct application of the Rice-Ramsperger-Kassed-Marcus (RRKM) theory to this kind of multichannel chemical reaction is prohibitively difficult, unless further modeling and/or assumptions are made. We carry out a completely nonempirical statistical calculation for these dynamical quantities, in that nothing empirical is introduced like remodeling (or reparametrization) of artificial potential energy functions or recalibration of the phase-space volume referring to other "empirical" values such as those estimated with the molecular dynamics method. The so-called dividing surface is determined variationally, at which the flux is calculated in a consistent manner with the estimate of the phase-space volume in the initial state. Also, for the correct treatment of a highly nonrigid cluster, the phase-space volume and flux are estimated without the separation of vibrational and rotational motions. Both the microcanonical reaction rate and the final kinetic energy distribution thus obtained have quite accurately reproduced the corresponding quantities given by molecular dynamics calculations. This establishes the validity of the statistical arguments, which in turn brings about the deeper physical insight about the evaporation dynamics.     

Atomic radii scales and electron properties deduced from the charge density

A procedure to represent Hartree-Fock electron densities in atoms [L. Fernandez Pacios, J. Comp. Chem., 14 , 410 (1993)] defines ρ(r) as a reduced expansion of exponential functions. These analytically modeled densities (AMDs) are used in this article to develop a simple computational procedure for analyzing different atomic radii scales implemented in the commercial software system MATHEMATICA. The analysis is focused on the physical information associated to a given atomic radius as deduced from calculations depending on ρ(r). The amount of electron charge contained in the sphere of the given radius as well as the distinct contributions to the potential energy integrated up to that radius are obtained within the AMD formulation for main-group atoms H—Kr. The ASCII file needed to run the procedure within MATHEMATICA is also presented. © 1995 by John Wiley & Sons, Inc.     

Mapping the intramolecular signal propagation pathways in protein using Bayesian change point analysis of atomic motions

We propose to use change points of atomic positions in the molecular dynamics trajectory as indicators of the propagating signals in protein. We designate these changes as signals because they can propagate within the molecule in the form of “perturbation wave”, transmit energy or information between different parts of protein, and serve as allosteric signals. We found that change points can distinguish between thermal fluctuations of atoms (noise) and signals in a protein despite the differences in the motility of amino acid residues. Clustering of the spatially close residues that were experiencing change points close in time, allowed us to map pathways of signal propagation in a protein at the atomic level of resolution. We propose a potential mechanism for the origin of the signal and its propagation that relies on the autonomic coherence resonance in atomic fluctuations. According to this mechanism, random synchronization of fluctuations of neighboring atoms results in a resonance, which increases amplitude of vibration of these atoms. This increase can be transmitted to the atoms colliding with the resonant atoms, leading to the propagating signal. The wavelet-based coherence analysis of the inter-atomic distances between carbon-alpha atoms and surrounding atoms for the residue pairs that belong to the same communication pathway allowed us to find time periods with temporarily locked phases, confirming the occurrence of conditions for resonance. Analysis of the mapped pathways demonstrated that they form a network that connects different regions of the protein.