Electron localization function from density components

This work addresses the decomposition of the Electron Localization Function (ELF) into partial density contributions using an appealing split of kinetic energy densities. Regarding the degree of the electron localization, the relationship between ELF and its usual spin‐polarized formula is discussed. A new polarized ELF formula, built from any subsystems of the density, and a localization function, quantifying the measure of electron localization for only a subpart of the total system are introduced. The methodology appears tailored to describe the electron localization in bonding patterns of subsystems, such as the local nucleophilic character. Beyond these striking examples, this work opens up opportunities to describe any electronic properties that depend only on subparts of the density in atoms, molecules, or solids. © 2016 Wiley Periodicals, Inc.     

Electron localization functions and local measures of the covariance

The electron localization measure proposed by Becke and Edgecombe is shown to be related to the covariance of the electron pair distribution. Just as with the electron localization function, the local covariance does not seem to be, in and of itself, a useful quantity for elucidating shell structure. A function of the local covariance, however, is useful for this purpose. A different function, based on the hyperbolic tangent, is proposed to elucidate the shell structure encapsulated by the local covariance; this function also seems to work better for the electron localization measure of Becke and Edgecombe. In addition, we propose a different measure for the electron localization that incorporates both the electron localization measure of Becke and Edgecombe and the Laplacian of the electron density; preliminary indications are that this measure is especially good at elucidating the shell structure in valence regions. Methods for evaluating electron localization functions directly from the electron density, without recourse to the Kohn-Sham orbitals, are discussed.     

Electron Density Analysis of Hyperconjugation

Hyperconjugation is analyzed through the electron density of orbitals responsible for hyperconjugative interactions, which cannot be detected by means of conventional electron‐density‐based calculations. This interaction is detected through the π electron density topology, by excluding σ electron density from the total. As the presence of the hyperconjugation phenomenon in carbocation systems is well understood, several carbocations are benchmarked, and the results show that the positive carbon atom establishes a hyperconjugative critical point with the adjacent methyl group(s). Also, π localization and delocalization indices are employed to support the conclusions made by the topological analysis.     

Thermal dissociation of 1,2-dioxethane. III. Localized molecular-orbital study

The localized MO 's (LMO 's) of 1,2-dioxethane in its ground state and along the dissociation reaction path (to formaldehyde products) are generated using Boys' criteria for localization. The total charge density in each LMO is partitioned into atomic and overlap densities and the binding or antibinding character of each LMO is discussed in terms of the forces exerted on the nuclei by these densities. The driving force for the dissociation reaction is shown to arise essentially from the atomic dipole forces exerted on the oxygen nuclei by their lone-pair LMO 's. The characterization of a saddle point on the potential energy surface has been discussed in terms of the electrostatic equilibrium between forces exerted by the electron clouds “incomplete following” and “preceding.” The differences between the LMO 's obtained from the two Hartree–Fock solutions to which the SCF procedure converges have been discussed.     

A quantitative measure of bond polarity from the electron localization function and the theory of atoms in molecules

 A quantitative measure of the polarity of a bond can be obtained through combining the two complementary topological partitionings of the electron density obtained from the atoms in molecules theory, on the one hand, and the electron localization function, on the other. This requires an integration of the electron density in the atomic subbasins of a common bond electron localization basin. We present the first numerical application of the resulting topological definition of bond polarity to a set of small linear systems consisting of the FCN, HF, HCl, HBr, and NaCl molecules and the NeAr van der Waals dimer. It is shown that the findings are essentially in line with common expectation for these simple molecules, thus confirming the potential value of the novel bond polarity index for the analysis of controversial bonding situations. Additional insight is provided through the detailed investigation of fluctuations in the basin populations. Received: 10 January 2001 / Accepted 12 February 2001 / Published online: 22 May 2001     

A computer program for calculating Ge(Li) detector counting efficiencies for Marinelli beakers

A novel computing method has been developed to calculate the absolute photopeak efficiency of a Ge(Li) detector for Marinelli beakers of different heights and diameters and with variable density. For each point in the cylindrical sample the detection efficiency is calculated taking into account the distance from the detector and gamma-ray attenuation and the efficiency is integrated numerically over the volume of the sample. The detector is approximated as a point detector with an experimentally determined effective interaction depth. It is necessary to measure the absolute efficiency for a point source located on the detector axis and on a line beside the detector parallel to the axis. The computer program calculates the absolute counting efficiency for Marinelli beakers of any geometry and for any density. The measured and calculated values for three different densities give a good (–2.2%) overall agreement.     

Electron localization in a finite one-dimensional chain

The localization characteristics of the electronic wave functions in a finite one-dimensional chain with the diagonal or the off-diagonal disorder of the potentials have been studied. It has been shown that the eigenfuction at the frontier level is relatively “strong” against the temptation to localize caused by the existence of the random potentials. It has also been pointed out that the spatial behavior of the total density reflects that of the diagonal random potentials, but that under the off-diagonal random potentials the total density is spatially uniform (completely extended).     

Electron localization function as information measure

The conditional two-electron probability function, which defines the electron localization function (ELF) of Becke and Edgecombe in the Kohn-Sham theory, is interpreted as the nonadditive (interorbital) Fisher information contained in the electron distribution. The probability normalization considerations suggest a use of the related information measure defined in terms of the unity-normalized probability distributions (shape factors of the electron densities), as the key ingredient of the modified information-theoretic ELF. This modified Fisher information density is validated by a comparison with the original two-electron probability function. Illustrative applications to typical molecular systems demonstrate the adequacy of the modified information-theoretic ELF in extracting the key features of the electron distributions in molecules. The overall Fisher information itself and the associated information-distance quantities are also proposed as complementary localization functions.     

On the convergence of isolated neutral oxygen vacancy and divacancy properties in metal oxides using supercell models

The properties of isolated neutral oxygen vacancies and divacancies of metal oxides of increasing complexity (MgO, CaO, alpha-Al2O3, and ZnO) have been studied by means of density-functional theory within a supercell periodic approach. Vacancy formation energies, vacancy-vacancy interactions, and geometry rearrangements around these point defects have been investigated in detail. The characterization of the electronic structure of these point defects has been established by analysis of the density of states and of the topology of the electron density and of electron localization function. It is found that the chemical character of the oxide determines the properties of the oxygen vacancies. For the covalent ZnO oxide, a more complex scheme arises in which the relaxation around the oxygen vacancy is much larger leading to the formation of Zn4-like almost metallic particles in the crystal. The relationship of these structures with the crystal shear planes is discussed. The present study shows that supercells containing approximately 200-300 atoms provide converged values for the geometric and electronic structure of oxygen vacancies of these metal oxides in the point defect low concentration limit.     

Reliable measurement of near-critical adsorption by gravimetric method

A gravimetric apparatus is used to measure the excess adsorption at high pressure. The equipment consists of a Rubotherm magnetic suspension balance, which allows to measure also the density of the fluid. In order to obtain the excess adsorbed amount, the measured weight has to be corrected with a buoyancy term, for which the density of the adsorbing fluid has to be known at each experimental conditions. Therefore the homogeneity of density in the high-pressure cell plays a fundamental role in determining the accuracy of the measured excess adsorbed amounts. This paper is intended to show the impact of the actual approach to thermostating the unit on the density distribution of the adsorbing fluid inside the high-pressure cell. Namely, by changing the inlet position of the heating fluid, large differences in the measured excess adsorption are produced. The closer to the critical point of the fluid, the stronger are these differences. An optimum configuration for our measuring device has been found and it has been used to study the adsorption of carbon dioxide (CO₂) on Filtrasorb 400 activated carbon at supercritical and near-critical conditions.     

Information-theoretic approach to kinetic-energy functionals: the nearly uniform electron gas

We strengthen the connection between information theory and quantum mechanical systems using a recently developed dequantization procedure which results in a decomposition of the kinetic energy as the sum of a classical term and a purely quantum term. For the nearly uniform electron gas, we thereby approximate the noninteracting kinetic energy as the sum of the Thomas-Fermi term, which is exact for the uniform electron gas, and the Weizsäcker term, which is proportional to the Fisher information. Electron correlation is included via a nonlocal analytical expression which is a functional of the (N-1)-conditional probability density. This expression is evaluated via a statistically rigorous Monte-Carlo procedure to obtain the correlation energy as a functional of the electron density. We show that this functional is well aproximated by a term which is proportional to the Shannon entropy. Thus the kinetic energy is expressed as the standard Thomas-Fermi term plus terms which are proportional to two of the cornerstones of information theory: the Fisher information, which is a measure of localization, and the Shannon entropy, which is a measure of delocalization.     

On the Electride Nature of Na-hP4

Early quantum mechanical models suggested that pressure drives solids towards free-electron metal behavior where the ions are locked into simple close-packed structures. The prediction and subsequent discovery of high-pressure electrides (HPEs), compounds assuming open structures where the valence electrons are localized in interstitial voids, required a paradigm shift. Our quantum chemical calculations on the iconic insulating Na-hP4 HPE show that increasing density causes a 3s→3pd electronic transition due to Pauli repulsion between the 1s2s and 3s states, and orthogonality of the 3pd states to the core. The large lobes of the resulting Na-pd hybrid orbitals point towards the center of an 11-membered penta-capped trigonal prism and overlap constructively, forming multicentered bonds, which are responsible for the emergence of the interstitial charge localization in Na-hP4. These multicentered bonds facilitate the increased density of this phase, which is key for its stabilization under pressure.     

Reductively Caged,Photoactivatable DNA‐PAINT for High‐Throughput Super‐resolution Microscopy

In DNA points accumulation in nanoscale topography (DNA‐PAINT), capable of single‐molecule localization microscopy with sub‐10‐nm resolution, the high background stemming from the unbound fluorescent probes in solution limits the imaging speed and throughput. Herein, we reductively cage the fluorescent DNA probes conjugated with a cyanine dye to hydrocyanine, acting as a photoactivatable dark state. The additional dark state from caging lowered the fluorescent background while enabling optically selective activation by total internal reflection (TIR) illumination at 405 nm. These benefits from “reductive caging” helped to increase the localization density or the imaging speed while preserving the image quality. With the aid of high‐density analysis, we could further increase the imaging speed of conventional DNA‐PAINT by two orders of magnitude, making DNA‐PAINT capable of high‐throughput super‐resolution imaging.     

Mutual information and electron correlation in momentum space

Mutual information and information entropies in momentum space are proposed as measures of the nonlocal aspects of information. Singlet and triplet state members of the helium isoelectronic series are employed to examine Coulomb and Fermi correlations, and their manifestations, in both the position and momentum space mutual information measures. The triplet state measures exemplify that the magnitude of the spatial correlations relative to the momentum correlations depends on and may be controlled by the strength of the electronic correlation. The examination of one- and two-electron Shannon entropies in the triplet state series yields a crossover point, which is characterized by a localized momentum density. The mutual information density in momentum space illustrates that this localization is accompanied by strong correlation at small values of p.     

Approximate charge density localization of molecular orbitals

A NDO approximate procedure based on the indirect intrinsic ab initio localization method of von Niessen is developed. It is shown that only when the NDO approximations are introduced at the two electron level, expressions are obtained which are the charge density counterpart of those found in the approximate energy localization methods. The results of these two methods are quite similar both in the CNDO and INDO approximations. The indeterminacies observed in the CNDO localization for unsaturated systems and for molecules with two or three lone pairs on the same atom, are removed by localizing up to an INDO level. The approximate charge density localization is however computationally much easier than the approximate energy localization method and should be more appropriate in LMO studies of large organic molecules.     

Analysis of the delocalization in the topological theory of chemical bond

The topological analysis of the gradient field of the electron localization function provides a convenient theoretical framework for the partition of the molecular space into basins of attractors having a clear chemical meaning. The basin populations are evaluated by integrating the one-electron density over the basins. The variance of the basin population provides a measure of the delocalization. The behavior of the core C(X) and protonated valence basins V(X, H) populations were investigated. The analysis of the population variance in terms of cross-contributions is presented for aromatic and antiaromatic systems, hypervalent molecules and hydrogen-bonded complexes. For hypervalent molecules this analysis emphasizes the importance of the ionic resonance structures.     

Influence of core–valence separation of electron localization function

The electron localization function (ELF) shows too-high values when computed from valence densities only (instead of using the total density). This effect is mainly found when d electrons are present in the outermost shell of the core. Although no pronounced qualitative differences could be noticed in the examples studied up to now, it is found that the quantitative differences between the values of ELF obtained from the valence densities only or from the total densities can be large. We also show, for the first time, an example (the Be atom) where ELF is obtained directly from the density. This exemplifies the possibility of computing ELF from highly accurate calculations (or from experimental data). © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1431–1439, 1997     

Adsorption of fluids in slitlike pores containing a small amount of mobile ions

We apply density functional theory to investigate changes in the phase behavior of a fluid caused by the presence of mobile ions inside the pore. The approach has been based on the fundamental measure density functional theory and on the theory of nonuniform electrolytes developed recently by O. Pizio, A. Patrykiejew, S. Soko?owski [J. Chem. Phys. 121 (2005) 11,957]. We have evaluated capillary condensation phase diagrams for pores of different widths and for different concentrations of confined ions. The calculations have demonstrated that the presence of ions leads to lowering the critical temperature and to an increase of the value of the chemical potential at the capillary condensation point.     

An analysis of two local measures of the electronic localization: a comparison with the ELF and the exchange-correlation density results

In this work we have explored the performance of two functions, recently proposed by Ayers [J. Chem. Sci., 2005, 117, 441], with the purpose of quantifying local electron localization. The first function, ζ(h), measures the total fluctuation per electron in the number of electrons at a given position r(1), while the second one, ζ(R), is a local representation of the minimum fluctuation criterion for electron localization. The study is carried out through a set of diatomic molecules that covers a wide range of covalent/polar character. Additionally, we have also calculated the electron localization function and the exchange-correlation hole along the internuclear axis. We have found that, for all the studied molecules, the numerical integration involved in computing ζ(h) did not converge. We think that this is so because the hole correlation calculations are not able to yield its correct asymptotic decaying behavior for large absolute values of the internuclear distances. On the other hand, the calculation of ζ(R) has proved to be feasible, and the information obtained from it has been concluded to be compatible to that rendered by the electron localization function (ELF) and the exchange-correlation density. Moreover, it has been also found that the results for ζ(R) allow to quantify the relative degree of electron localization within different molecular regions.