On the local representation of the electronic momentum operator in atomic systems

The local quantum theory is applied to the study of the momentum operator in atomic systems. Consequently, a quantum-based local momentum expression in terms of the single-electron density is determined. The limiting values of this function correctly obey two fundamental theorems: Kato's cusp condition and the Hoffmann-Ostenhof and Hoffmann-Ostenhof exponential decay. The local momentum also depicts the electron shell structure in atoms as given by its local maxima and inflection points. The integration of the electron density in a shell gives electron populations that are in agreement with the ones expected from the Periodic Table of the elements. The shell structure obtained is in agreement with the higher level of theory computations, which include the Kohn-Sham kinetic energy density. The average of the local kinetic energy associated with the local momentum is the Weizsacker kinetic energy. In conclusion, the local representation of the momentum operator provides relevant information about the electronic properties of the atom at any distance from the nucleus.     

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.     

Kinetic energy density study of some representative semilocal kinetic energy functionals

There is a number of explicit kinetic energy density functionals for noninteracting electron systems that are obtained in terms of the electron density and its derivatives. These semilocal functionals have been widely used in the literature. In this work, we present a comparative study of the kinetic energy density of these semilocal functionals, stressing the importance of the local behavior to assess the quality of the functionals. We propose a quality factor that measures the local differences between the usual orbital-based kinetic energy density distributions and the approximated ones, allowing us to ensure if the good results obtained for the total kinetic energies with these semilocal functionals are due to their correct local performance or to error cancellations. We have also included contributions coming from the Laplacian of the electron density to work with an infinite set of kinetic energy densities. For all but one of the functionals, we have found that their success in the evaluation of the total kinetic energy is due to global error cancellations, whereas the local behavior of their kinetic energy density becomes worse than that corresponding to the Thomas-Fermi functional.     

Bond length and local energy density property connections for non-transition-metal oxide-bonded interactions

For a variety of molecules and earth materials, the theoretical local kinetic energy density, G(r(c)), increases and the local potential energy density, V(r(c)), decreases as the M-O bond lengths (M = first- and second-row metal atoms bonded to O) decrease and the electron density, rho(r(c)), accumulates at the bond critical points, r(c). Despite the claim that the local kinetic energy density per electronic charge, G(r(c))/rho(r(c)), classifies bonded interactions as shared interactions when less than unity and closed-shell when greater, the ratio was found to increase from 0.5 to 2.5 au as the local electronic energy density, H(r(c)) = G(r(c)) + V(r(c)), decreases and becomes progressively more negative. The ratio appears to be a measure of the character of a given M-O bonded interaction, the greater the ratio, the larger the value of rho(r(c)), the smaller the coordination number of the M atom and the more shared the bonded interaction. H(r(c))/rho(r(c)) versus G(r(c))/rho(r(c)) scatter diagrams categorize the M-O bonded interactions into domains with the local electronic energy density per electron charge, H(r(c))/rho(r(c)), tending to decrease as the electronegativity differences for the bonded pairs of atoms decrease. The values of G(r(c)) and V(r(c)), estimated with a gradient-corrected electron gas theory expression and the local virial theorem, are in good agreement with theoretical values, particularly for the bonded interactions involving second-row M atoms. The agreement is poorer for shared C-O and N-O bonded interactions.     

Si-O bonded interactions in silicate crystals and molecules: a comparison

Bond critical point, local kinetic energy density, G(rc), and local potential energy density, V(rc), properties of the electron density distributions, rho(r), calculated for silicates such as quartz and gas-phase molecules such as disiloxane are similar, indicating that the forces that govern the Si-O bonded interactions in silica are short-ranged and molecular-like. Using the G(rc)/rho(rc) ratio as a measure of bond character, the ratio increases as the Si-O bond length, the local electronic energy density, H(rc)= G(rc) + V(rc), and the coordination number of the Si atom decrease and as the accumulation of the electron density at the bond critical point, rho(rc), and the Laplacian, inverted Delta2 rho(rc), increase. The G(rc)/rho(rc) and H(rc)/rho(rc) ratios categorize the bonded interaction as observed for other second row atom M-O bonds into discrete categories with the covalent character of each of the M-O bonds increasing with the H(rc)/rho(rc) ratio. The character of the bond is examined in terms of the large net atomic charges conferred on the Si atoms comprising disiloxane, stishovite, quartz, and forsterite and the domains of localized electron density along the Si-O bond vectors and on the reflex side of the Si-O-Si angle together with the close similarity of the Si-O bonded interactions observed for a variety of hydroxyacid silicate molecules and a large number of silicate crystals. The bond critical point and local energy density properties of the electron density distribution indicate that the bond is an intermediate interaction between Al-O and P-O bonded interactions rather than being a closed-shell or a shared interaction.     

Quantum chemistry aspects of the solvent effects on the ene reaction of 1‐Phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene

A density functional theory study was used to investigate the quantum aspects of the solvent effects on the kinetic and mechanism of the ene reaction of 1‐phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene. Using the B3LYP/6–311++ G(d,p) level of the theory, reaction rates have been calculated in the various solvents and good agreement with the experimental data has been obtained. Natural bond orbital analysis has been applied to calculate the stabilization energy of N18? H19 bond during the reaction. Topological analysis of quantum theory of atom in molecule (QTAIM) studies for the electron charge density in the bond critical point (BCP) of N18? H19 bond of the transition states (TSs) in different solvents shows a linear correlation with the interaction energy. It is also seen form the QTAIM analysis that increase in the electron density in the BCP of N18? H19, raises the corresponding vibrational frequency. Average calculated ratio of 0.37 for kinetic energy density to local potential energy density at the BCPs as functions of N18? H19 bond length in different media confirmed covalent nature of this bond. Using the concepts of the global electrophilicity index, chemical hardness and electronic chemical potentials, some correlations with the rate constants and interaction energy have been established. Mechanism and kinetic studies on 1‐phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene ene reaction suggests that the reaction rate will boost with interaction energy enhancement. Interaction energy of the TS depends on the solvent nature and is directly related to electron density of the bonds involved in the reaction proceeding, global electrophilicity index and electronic chemical potential. However, the chemical hardness relationship is reversed. Finally, an interesting and direct correlation between the imaginary vibrational frequency of the N18? H19 critical bond and its electron density at the TS has been obtained. © 2014 Wiley Periodicals, Inc.     

The Relationship of the Energy of Interaction of Alkali Metal Cations with an Aprotic Solvent Molecule with Quantum Topological Electron Density Characteristics

The optimal geometry and wave functions of the complexes [M(Solv)]⁺ (M = Li, Na, K; Solv is an aprotic solvent molecule) were calculated and the topological characteristics of the electron density distribution at the (3,–1) critical points (CP) of ion–molecule bonds were analyzed by the density functional theory in the B3LYP/6-31+G(d, p) approximation. The parametric dependences for the energy of ion–molecule bonds in terms of the local kinetic and potential electron energy densities at the bond CTs were proposed.     

Evaluation eines Elektronendiffusionsmodelles zur Berechnung von nicht Gleichgewichts-Elektronenkonzentrationen in Induktiv gekoppelten Argonplasma für die spektrochemische Analyse

The electron density in argon ICP discharges has been found experimentally by other investigators to be higher than that calculated from the temperature distribution and Saha equation assuming local thermodynamic equilibrium (LTE). The results of the present study suggest that this non equilibrium concentration has mainly two causes: first, the kinetic energy of electrons owing to the power input of the rf field is higher than the kinetic energy of the gas particles. Second, as an effect of the extremely high gradients in the electron density and the temperature distribution, ambipolar diffusion of electrons results in a non LTE situation. With the help of the ambipolar diffusion constant and with recombination being taken into account, the electron concentration and the electron temperature in an ICP have been calculated. The so calculated electron density distributions are compared with literature values, found experimentally by other investigators. Finally a new model is proposed which explains the high ion concentration found experimentally for important analytical species.     

Alternative Representations of the Correlation Energy in Density‐Functional Theory: A Kinetic‐Energy Based Adiabatic Connection

The adiabatic‐connection framework has been widely used to explore the properties of the correlation energy in density‐functional theory. The integrand in this formula may be expressed in terms of the electron–electron interactions directly, involving intrinsically two‐particle expectation values. Alternatively, it may be expressed in terms of the kinetic energy, involving only one‐particle quantities. In this work, we explore this alternative representation for the correlation energy and highlight some of its potential for the construction of new density functional approximations. The kinetic‐energy based integrand is effective in concentrating static correlation effects to the low interaction strength regime and approaches zero asymptotically, offering interesting new possibilities for modeling the correlation energy in density‐functional theory     

Improved local density functional approach for atomic systems

Through a new local density approximation to the kinetic energy density functional introduced by us recently, a simple Thomas–Fermi-like scheme for the direct calculation of electron density in atoms is proposed. The calculated density is nonsingular at the nucleus and the energy values are in very good agreement with the corresponding Hartree–Fock results for atoms. © 1994 John Wiley & Sons, Inc.     

Generalized nonlocal kinetic energy density functionals based on the von Weizsäcker functional

We generalize the ideas behind the procedure for the construction of kinetic energy density functionals with a nonlocal term based on the structure of the von Weizs?cker functional, and present several types of nonlocal terms. In all cases, the functionals are constructed such that they reproduce the linear response function of the homogeneous electron gas. These functionals are designed by rewriting the von Weizs?cker functional with the help of a parameter β that determines the power of the electron density in the expression, a strategy we have previously used in the generalization of Thomas-Fermi nonlocal functionals. Benchmark calculations in localized systems have been performed with these functionals to test both their relative errors and the quality of their local behavior. We have obtained competitive results when compared to semilocal and previous nonlocal functionals, the generalized nonlocal von Weizs?cker functionals giving very good results for the total kinetic energies and improving the local behavior of the kinetic energy density. In addition, all the functionals discussed in this paper, when using an adequate reference density, can be evaluated as a single integral in momentum space, resulting in a quasilinear scaling for the computational cost.     

How electron delocalization influences the electron-withdrawing properties of isomeric benzobischalcogenadiazoles

The fermionic potential and delocalization indices for benzo-bis-1,2,5-chalcogenadiazoles reveal inhomogeneous electron delocalization in their benzene ring, which results in compactly localized lone electron pairs on the chalcogen atoms. These features of (de)localization are rooted in a local increase in the kinetic component of the electron correlation, which expresses the Fermi hole variability and the kinetic potential response to electron density variations in the benzene ring of benzobis-1,2,5-chalcogenadiazoles. This explains their better electron-withdrawing properties compared to benzobis-1,2,3-chalcogenadiazoles     

Strength and nature of hydrogen bonding interactions in mono- and di-hydrated formamide complexes

In this work, mono- and di-hydrated complexes of the formamide were studied. The calculations were performed at the MP2/6-311++G(d,p) level of approximation. The atoms in molecules theory (AIM), based on the topological properties of the electronic density distribution, was used to characterize the different types of bonds. The analysis of the hydrogen bonds (H-bonds) in the most stable mono- and di-hydrated formamide complexes shows a mutual reinforcement of the interactions, and some of these complexes can be considered as "bifunctional hydrogen bonding hydration complexes". In addition, we analyzed how the strength and the nature of the interactions, in mono-hydrated complexes, are modified by the presence of a second water molecule in di-hydrated formamide complexes. Structural changes, cooperativity, and electron density redistributions demonstrate that the H-bonds are stronger in the di-hydrated complexes than in the corresponding mono-hydrated complexes, wherein the σ- and π-electron delocalization were found. To explain the nature of such interactions, we carried out the atoms in molecules theory in conjunction with reduced variational space self-consistent field (RVS) decomposition analysis. On the basis of the local Virial theorem, the characteristics of the local electron energy density components at the bond critical points (BCPs) (the 1/4? (2)ρ(b) component of electron energy density and the kinetic energy density) were analyzed. These parameters were used in conjunction with the electron density and the Laplacian of the electron density to analyze the characteristics of the interactions. The analysis of the interaction energy components for the systems considered indicates that the strengthening of the hydrogen bonds is manifested by an increased contribution of the electrostatic energy component represented by the kinetic energy density at the BCP.     

Energy‐surfaces from the upper bound of the Pauli kinetic energy

Based on the Kohn–Sham Pauli potential and the Kohn–Sham electron density, the upper bound of the Pauli kinetic energy is tested as a suitable replacement for the exact Pauli kinetic energy for application in orbital‐free density functional calculations. It is found that bond lengths for strong and moderately bound systems can be qualitatively predicted, but with a systematic shift toward larger bond distances with a relative error of 6% up to 30%. Angular dependence of the energy‐surface cannot be modeled with the proposed functional. Therefore, the upper bound model is the first parameter‐free functional expression for the kinetic energy that is able to qualitatively reproduce binding curves with respect to bond distortions. © 2016 Wiley Periodicals, Inc.     

Application of AIM parameters at ring critical points for estimation of pi-electron delocalization in six-membered aromatic and quasi-aromatic rings

The AIM parameters at the ring critical point (the electron density and its Laplacian, the total electron energy density and both its components, potential and kinetic electron energy densities), have been intercorrelated with aromaticity indices: the geometry-based HOMA and the magnetism-based NICS, NICS(1), and NICS(1)(zz). A set of 33 phenylic rings having possibly a diversified aromatic character, and a set of 20 quasi-rings formed by intramolecular hydrogen and lithium bonds, have been taken into consideration. It has been found that the density of total electron energy, H, may serve as a new quantitative characteristic of pi-electron delocalization. The dependences between H values and aromaticity indices are correlated (cc(H/HOMA)=0.99, cc(H/NICS(1)zz)=0.95).     

Electron-molecule scattering calculations in a 3D finite element R-matrix approach

We have implemented a three-dimensional finite element approach, based on tricubic polynomials in spherical coordinates, which solves the Schrodinger equation for scattering of a low energy electron from a molecule, approximating the electron exchange as a local potential. The potential is treated as a sum of three terms: electrostatic, exchange, and polarization. The electrostatic term can be extracted directly from ab initio codes (GAUSSIAN 98 in the work described here), while the exchange term is approximated using different local density functionals. A local polarization potential approximately describes the long range attraction to the molecular target induced by the scattering electron.