Electron localization function and electron localizability indicator applied to study the bonding in the peroxynitrous acid HOONO

The ground‐state electronic structure of peroxynitrous acid (HOONO) and its singlet biradicaloid form (HO ··· ONO) have been studied using topological analysis of the electron localization function (ELF), together with the electron localizability indicator (ELI‐D), at the DFT (B3LYP, M05, M052X, and M06), CCSD, and CASSCF levels. Three isomers of HOONO (cis‐cis, cis‐perp, and trans‐perp) have been considered. The results show that from all functionals applied, only B3LYP yields the correct geometrical structure. The ELF and ELI‐D‐topology of the O? O and central N? O bonds strongly depends on the wave function used for analysis. Calculations carried out at CAS (14,12)/aug‐cc‐pVTZ//CCSD(T)/aug‐cc‐pVTZ level reveal two bonds of the charge‐shift type: a protocovalent N? O bond with a basin population of 0.82–1.08e, and a more electron depleted O? O bond with a population of 0.66–0.71e. The most favorable dissociation channel (HOONO → HO + ONO) corresponds to breaking of the most electron‐deficient bond (O? O). In the case of cis‐ and trans‐HO ··· ONO, the ELF, ELI‐D, and electron density fields results demonstrate a closed‐shell O ··· O interaction. The α‐spin electrons are found mainly (0.64e) in the lone pairs of oxygen V_{i = 1,2} (O) from the OH group. The β‐spin electrons are delocalized over the ONO group, with the largest concentration (0.34e) on the lone pair of nitrogen V(N). © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Electron localizability indicators ELI–D and ELIA for highly correlated wavefunctions of homonuclear dimers. I. Li2, Be2, B2, and C2

Electron localizability indicators based on the parallel‐spin electron pair density (ELI–D) and the antiparallel‐spin electron pair density (ELIA) are studied for the correlated ground‐state wavefunctions of Li₂, Be₂, B₂, and C₂ diatomic molecules. Different basis sets and reference spaces are used for the multireference configuration interaction method following the complete active space calculations to investigate the local effect of electron correlation on the extent of electron localizability in position space determined by the two functionals. The results are complemented by calculations of effective bond order, vibrational frequency, and Laplacian of the electron density at the bond midpoint. It turns out that for Li₂, B₂, and C₂ the reliable topology of ELI–D is obtained only at the correlated level of theory. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Charge decomposition analysis of the electron localizability indicator: a bridge between the orbital and direct space representation of the chemical bond

The novel functional electron localizability indicator is a useful tool for investigating chemical bonding in molecules and solids. In contrast to the traditional electron localization function (ELF), the electron localizability indicator is shown to be exactly decomposable into partial orbital contributions even though it displays at the single-determinantal level of theory the same topology as the ELF. This approach is generally valid for molecules and crystals at either the single-determinantal or the explicitly correlated level of theory. The advantages of the new approach are illustrated for the argon atom, homonuclear dimers N2 and F2, unsaturated hydrocarbons C2H4 and C6H6, and the transition-metal-containing molecules Sc(2)2+ and TiF4.     

Atomic shells from the electron localizability in momentum space

The electron localizability indicator in momentum space is proposed as a functional of the same‐spin momentum pair density. This functional yields a discrete distribution of values, which are proportional to the charge needed to form a fixed very small fraction of a same‐spin electron pair. It resolves all atomic shells for the examined atoms (Li–Kr) with reasonable occupation numbers, especially in the valence region. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Electron localizability indicator for correlated wavefunctions. I. Parallel-spin pairs

The electron localizability indicator (ELI) is based on a functional of the same-spin pair density. It reflects the correlation of the motion of same-spin electrons. In the Hartree–Fock approximation the ELI can be related to the electron localization function (ELF). For correlated wavefunctions the ELI formula differs from the one for the ELF.     

Electron localizability indicators ELI and ELIA: the case of highly correlated wavefunctions for the argon atom

Electron localizability indicators based on the same-spin electron pair density and the opposite-spin electron pair density are studied for correlated wavefunctions of the argon atom. Different basis sets and reference spaces are used for the multireference configuration interaction method following the complete active space calculations aiming at the understanding of the effect of local electron correlation when approaching the exact wavefunction. The populations of the three atomic shells of Ar atom in real space are calculated for each case.     

Electron localizability indicators ELI–D and ELIA for highly correlated wavefunctions of homonuclear dimers. II. N2, O2, F2, and Ne2

Electron localizability indicators based on the electron pair density ELI–D and ELIA Electron localizability indicators ELI‐D and ELIA based on the electron pair density are studied for the correlated ground‐state wavefunctions of N₂, O₂, F₂, and Ne₂ diatomics. Different basis sets and reference spaces are used for the multireference configuration interaction method following the complete active space calculations to investigate the local effect of electron correlation on the extent of electron localizability in position space determined by the two indicators. The results are complemented by calculations of effective bond order, vibrational frequency, and Laplacian of the electron density at the bond midpoint. It turns out that for O₂ and F₂, the reliable topology of ELI–D is obtained only at the correlated level of theory. © 2010 Wiley Periodicals, Inc. J Comput Chem 2010     

Topological analysis of the electron delocalization range

The electron delocalization range function EDR( ) (Janesko et al., J. Chem. Phys. 2014, 141, 144104) quantifies the extent to which an electron at point in a calculated wavefunction delocalizes over distance d. This work shows how topological analysis distills chemically useful information out of the EDR. Local maxima (attractors) in the EDR occur in regions such as atomic cores, covalent bonds, and lone pairs where the wavefunction is dominated by a single orbital lobe. The EDR characterizes each attractor in terms of a delocalization length D and a normalization , which are qualitatively consistent with the size of the orbital lobe and the number of lobes in the orbital. Attractors identify the progressively more delocalized atomic shells in heavy atoms, the interplay of delocalization and strong (nondynamical) correlation in stretched and dissociating covalent bonds, the locations of valence and weakly bound electrons in anionic water clusters, and the chemistry of different reactive sites on metal clusters. Application to ammonia dissociation over silicon illustrates how this density‐matrix‐based analysis can give insight into realistic systems. © 2016 Wiley Periodicals, Inc.     

Electron localizability indicators from spinor wavefunctions

For the fully relativistic 4‐component many‐electron wavefunction six flavors of electron localizability indicators (ELI) have been proposed. Their counterparts, suitable for the application to the 2‐component wavefunctions, have been also derived. Six proposed indicators have been tested on Ar and Rn atoms and one of them, the ELI‐D for spatially antisymmetrized electron pairs, has been found to reveal atomic shell structures at quantitative level. Shell structures of all the atoms of periods 4–7 of the periodic table have been obtained using this indicator and compared with these obtained from the nonrelativistic limit calculations as well as from scalar‐relativistic (zero‐order regular approximation) calculations. © 2014 Wiley Periodicals, Inc.     

Determination of the parameters of atomic pair correlations in nickel oxide films by the electron energy loss fine structure method

The electron energy loss extended fine structure (EELFS) spectra were obtained from the pure nickel surface (M _2,3 EELFS) of a stoichiometric NiO film (NiM _2,3 and OK EELFS spectra) and the “nonhomogeneous” oxide film on the surface of nickel Ni-O (NiM _2,3 and OK EELFS spectra). The amplitudes and intensities of electron transitions for the core levels of atoms were calculated with regard for the multiplicity of electron impact excitation of the corresponding core levels of atoms. The corresponding normalized oscillating terms were isolated using the results of calculations based on the experimental EELFS spectra. Agreement between the experimental and calculated (on Ni and NiO test objects) data showed that the theoretical approaches used and the calculated data for describing the EELFS spectra are good approximations. Using the results of calculations and the parameters of secondary electron elastic scattering (FEEF-8 data) we obtained the atomic pair correlation functions from the experimental normalized oscillating parts of the EELFS spectra by Tikhonov’s regularization method.     

Rules on pair electron correlation in MOH (M=H,Li, Na)

In this article, the transferable property of pair correlation energies of OH components is discussed for a series of OH containing compounds MOH (M=H, Li, Na). In this series of compounds, from OH free radicals through HOH, LiOH, NaOH to OH^?, both the intra‐ and interpair correlation energies and intra‐ and intershell correlation energies of the inner orbital electrons change little. The 1s$_{\mathrm{O}}^{2}$ is very much alike in all the above OH containing systems and such a pair correlation is transferable. But the interpair correlation and intrashell correlation energies of the valence electrons are large and change a lot in all systems. In MOH molecules, the OH correlation energy contribution increases with the increase of the ionic bond strength of the compound and this contribution is always between the correlation energy values of OH free radicals and OH^? atomic groups. For strong ionic compounds, we present a very simple method to estimate the correlation energy by adding the correlation energies of its component ions within the chemical accuracy (2 kcal/mol). © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 311–317, 2001     

Enhanced second‐order treatment of electron pair correlation

Ab initio calculations have been performed for F₂, HCCH, H₂O, HF, (HF)₂, and (H₂O)₂, comparing certain electron pair correlation methods, or methods for doubly substituted configurations. In these model systems, the reweighting of substituted configurations that occurs beyond a second‐order perturbative treatment of electron correlation can be partly built into the second‐order analysis in a computationally trivial step. Specific means for doing this are explored, and they offer improvement in certain cases or else very little change. A consistent improvement in the correlation energy when judged against treatment with double substitution coupled cluster theory for the test species is obtained through one of these schemes. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 226–236, 2000     

Pair density related to one‐electron information for the ground state of spin‐compensated two‐electron systems

The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr², with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Evaluation of the electron momentum density of crystalline systems from ab initio linear combination of atomic orbitals calculations

Alternative techniques are presented for the evaluation of the electron momentum density (EMD) of crystalline systems from ab initio linear combination of atomic‐orbitals calculations performed in the frame of one‐electron self‐consistent‐field Hamiltonians. Their respective merits and drawbacks are analyzed with reference to two periodic systems with very different electronic features: the fully covalent crystalline silicon and the ionic lithium fluoride. Beyond one‐electron Hamiltonians, a post‐Hartree–Fock correction to the EMD of crystalline materials is also illustrated in the case of lithium fluoride. © 2012 Wiley Periodicals, Inc.     

Role of electron–electron coalescence density in density functional theory

An expression for the evaluation of electron–electron coalescence density as a functional of the density for any electron system is proposed. The formula, clarifies previously advanced upper bounds for this quantity and provides a method to independently estimate the system‐averaged on‐top exchange–correlation hole. The relationship with the on‐top pair density shows that producing the true electron–electron coalescense should be considered as a leading physical requirement for trial wave functions in any energy minimization scheme. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

An orbital localization criterion based on the topological analysis of the electron localization function

This work proposes a new molecular orbital localization procedure. The approach is based on the decomposition of the overlap matrix in accordance with the partitioning of the three‐dimensional physical space into basins with clear chemical meaning arising from the topological analysis of the electron localization function. The procedure is computationally inexpensive, provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and basin occupancies, and preserves the σ/π‐separability in planar N‐electron systems. The localization algorithm is tested on selected molecular systems. © 2012 Wiley Periodicals, Inc.     

An orbital localization criterion based on the topological analysis of the electron localization function at correlated level

This work describes a procedure for localizing orbitals based on the topological analysis of the electron localization function at correlated level. The decomposition of the overlap matrix according to the partitioning of the three dimensional physical space into basins provided by that function allows us to define a localization index to be maximized using isopycnic orbital transformations. The localization algorithm has been computationally implemented and its efficiency tested on selected molecular systems at equilibrium, stretched, and twisted geometries. We report results which allow to analyze the influence of the correlated and uncorrelated treatments on the orbital localization.     

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