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 theory of "fuzzy" atoms

This work proposes a new procedure for localizing molecular and natural orbitals. The localization criterion presented here is based on the partitioning of the overlap matrix into atomic contributions within the theory of "fuzzy" atoms. Our approach has several advantages over other schemes: it is computationally inexpensive, preserves the sigma/pi-separability in planar systems and provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and atomic occupancies. The corresponding algorithm has been implemented and its efficiency tested on selected molecular systems.     

Determination of substitutional sites in heterocycles from the topological analysis of the electron localization function (ELF)

The topological analysis of the electron localization function ELF has been carried out on five‐membered (C₄H₄NH, C₄H₄PH, C₄H₄O, C₄H₄S) and six‐membered (C₅H₅N, C₅H₅P) heterocycles. The bonding in these molecules is discussed on the basis of the valence basin populations. It is shown that the values of the ELF function at the (3,−1) critical points between disynaptic basins related to a given center provide a criterion to determine substitutional sites. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 509–514, 2000     

Electron localization function at the correlated level

The electron localization function (ELF) has been proven so far a valuable tool to determine the location of electron pairs. Because of that, the ELF has been widely used to understand the nature of the chemical bonding and to discuss the mechanism of chemical reactions. Up to now, most applications of the ELF have been performed with monodeterminantal methods and only few attempts to calculate this function for correlated wave functions have been carried out. Here, a formulation of ELF valid for mono- and multiconfigurational wave functions is given and compared with previous recently reported approaches. The method described does not require the use of the homogeneous electron gas to define the ELF, at variance with the ELF definition given by Becke. The effect of the electron correlation in the ELF, introduced by means of configuration interaction with singles and doubles calculations, is discussed in the light of the results derived from a set of atomic and molecular systems.     

Electronegativity estimator built on QTAIM‐based domains of the bond electron density

The electron localization function, natural localized molecular orbitals, and the quantum theory of atoms in molecules have been used all together to analyze the bond electron density (BED) distribution of different hydrogen‐containing compounds through the definition of atomic contributions to the bonding regions. A function, g_AH, obtained from those contributions is analyzed along the second and third periods of the periodic table. It exhibits periodic trends typically assigned to the electronegativity (χ), and it is also sensitive to hybridization variations. This function also shows an interesting S shape with different χ‐scales, Allred–Rochow's being the one exhibiting the best monotonical increase with regard to the BED taken by each atom of the bond. Therefore, we think this χ can be actually related to the BED distribution. © 2014 Wiley Periodicals, Inc.     

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     

New insights on the bridge carbon-carbon bond in propellanes: a theoretical study based on the analysis of the electron localization function

The nature of the bonding between bridgehead carbon atoms (Ca, Ca') as well as the ring strain in a family of 10 propellanes formed by three-, four-, or five-member rings: [1.1.1] (I), [2.1.1] (II), [3.1.1] (III), [2.2.1] (IV), [3.2.1] (V), [2.2.2] (VI), [3.3.1] (VII), [3.2.2] (VIII), [3.3.2] (IX), and [3.3.3] (X) are studied by means of the electron localization function (ELF) at the DFT level (B3LYP/cc-pVTZ). The ELF analysis of smaller propellanes (I, II, and III) reveals the coexistence of two resonance forms: one with a nonbonding electron pair partially delocalized between Ca and Ca' atoms outside the cage (ionic) and the other with a bridge bond between the same atoms (covalent). The weights of each form are calculated according to the ELF-basin populations, yielding 94, 88, and 53% for the ionic structure of I, II, and III, respectively, while larger propellanes (IV-X) present only the covalent form. The question of the s-character of the bridge bond is addressed by dissecting the bridge-bond ELF basin into the molecular orbital contributions. Finally, sigma-aromaticity associated to surface electron delocalization has been analyzed by means of nucleus-independent chemical shift (NICS) calculations. The results point out that the stability of the fused ring structure of propellanes I, II, and III, can be assigned to the remarkable sigma-aromaticity of the involved three-member rings.     

Quantum chemical topology: The electronic structure of the alkaline nitrites MONO (M = Li,Na, K) studied by means of topological analysis of the electron localization function

In the search of the protocovalent bonding, previously recognized in the nitrous acid (HONO), a nature of the chemical bonds in the alkaline nitrites MONO (M = Li, Na, K) has been studied by means of the topological analysis of the Electron Localization Function (ELF) and Electron Localizability Indicator (ELI‐D). Calculations carried out with the B3LYP and MP2(full) methods, in conjunction with the aug‐cc‐pVTZ and 6‐311++G(3df,3pd) basis sets, revealed the cis (C_2v, more stable) and trans (C_s) isomers as minima on PES. Alkaline nitrites consist of the alkali metal cation M^δ+ interacting, mainly via electrostatic forces, with the nitrite anion [ONO]^δ− (δ ≈ 1e). The covalent N O bonds are characterized by disynaptic basins V(N,O) with the basin populations: 1.58÷1.62e for cis‐M^δ+[ONO]^δ− but 1.39÷1.49e for single N O bond and 1.81÷1.87e for formally double NO bond in trans M^δ+[O NO]^δ−. The protocovalent nitrogen–oxygen bond has not been observed. The N O bonds are slightly polarized towards the nitrogen atom with the polarity index p_NO ≤ 0.12. Two different sets of the hybrid (Lewis) structures are compared leading to different pictures of the bonding. According to NBO data there is a delocalization between the single N O and double NO type bonds, meanwhile results of the ELF analysis emphasize an electron delocalization between the single N O and ionic O⁻N⁺ hybrids. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Theoretical study of one‐electron bonds in a series of high‐spin lithium–beryllium–hydrogen clusters: “Valence shell single‐electron repulsion” rule and electron localization function analysis

A series of high‐spin clusters containing Li, H, and Be in which the valence shell molecular orbitals (MOs) are occupied by a single electron has been characterized using ab initio and density functional theory (DFT) calculations. A first type (⁵Li₂, ⁿ⁺¹LiH_n+ (n = 2–5), ⁸Li₂H) possesses only one electron pair in the lowest MO, with bond energies of ～3 kcal/mol. In a second type, all the MOs are singly occupied, which results in highly excited species that nevertheless constitute a marked minimum on their potential energy surface (PES). Thus, it is possible to design a larger panel of structures (⁸LiBe, ⁷Li₂, ⁸Li, ⁴LiH⁺, ⁶BeH, ⁿ⁺³LiH (n = 3, 4), ⁿ⁺²LiH (n = 4–6), ⁸Li₂H, ⁹Li₂H, ²²Li₃Be₃ and ²²Li₆H), single‐electron equivalent to doublet “classical” molecules ranging from CO to C₆H₆. The geometrical structure is studied in relation to the valence shell single‐electron repulsion (VSEPR) theory and the electron localization function (ELF) is analyzed, revealing a striking similarity with the corresponding structure having paired electrons. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Application of the topological analysis of the electronic localization function to archetypical [Pb(II)Ln]p complexes: The bonding of Pb2+ revisited

The coordination of neutral ligands (L = OC, HCN, NH₃, PH₃, SH₂, HNCO and H₂O) to Pb²⁺ is investigated and analyzed by means of the topological analysis of the Electronic Localization Function (ELF). It is shown that the mean charge density of the V(Pb) basin (〈ρ〉_V(Pb)) can reach a ligand‐independent limiting value from n = 6, a coordination number from which the [PbL_n]²⁺ complexes adopt holodirected structures. The investigations performed on anionic series (L = HS^?, OH^?, CN^?, F^?, Cl^?, and Br^?) lead to optimized stable structures in which the coordination number does not exceed n = 4, even in the presence of a model aqueous solvent. This different behavior with respect to the neutral ligand series is interpreted by means of natural populations and electrostatic repulsions. The main result of this contribution is that stable Pb(II) complexes could be those exhibiting reasonable values of 〈ρ〉_V(Pb), namely those not exceeding the saturation plateau evidenced in the present piece of work. © 2009 Wiley Periodicals, Inc. J Comput Chem 2010     

Application of the electron localization function to radical systems

An application of the topological electron localization function (ELF) analysis to free radical systems is presented. A separation of the ELF function into its α‐spin and β‐spin contributions has been performed. Methyl and phenyl radicals, ortho‐, meta‐, and para‐benzyne biradicals, and their corresponding radical anions have been chosen with the aim to validate the new ELF_α and ELF_β proposed functions. The results show that the ELF separation yields complementary information about the localization of the unpaired electron. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

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.     

Advancing beyond charge analysis using the electronic localization function: Chemically intuitive distribution of electrostatic moments

We propose here an evaluation of chemically intuitive distributed electrostatic moments using the topological analysis of the electron localization function (ELF). As this partition of the total charge density provides an accurate representation of the molecular dipole, the distributed electrostatic moments based on the ELF partition (DEMEP) allows computing of local moments located at non atomic centers such as lone pairs, sigma bonds and pi systems. As the local dipole contribution can be decomposed in polarization and charge transfer components, our results indicate that local dipolar polarization of the lone pairs and chemical reactivity are closely related whereas the charge transfer contribution is the key factor driving the local bond dipole. Results on relevant molecules show that local dipole contributions can be used to rationalize inductive polarization effects in alcohols derivatives and typical hydrogen bond interactions. Moreover, bond quadrupole polarization moments being related to a pi character enable to discuss bond multiplicities, and to sort families of molecules according to their bond order. That way, the nature of the C-O bond has been revisited for several typical systems by means of the DEMEP analysis which appears also helpful to discuss aromaticity. Special attention has been given to the carbon monoxide molecule, to the CuCO complex and to a weak intramolecular N|-CO interaction involved in several biological systems. In this latter case, it is confirmed that the bond formation is mainly linked to the CO bond polarization. Transferability tests show that the approach is suitable for the design of advanced force fields.     

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.     

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     

A joint study based on the electron localization function and catastrophe theory of the chameleonic and centauric models for the Cope rearrangement of 1,5-hexadiene and its cyano derivatives

A novel interpretation of the chameleonic and centauric models for the Cope rearrangements of 1,5-hexadiene (A) and different cyano derivatives (B: 2,5-dicyano, C: 1,3,4,6-tetracyano, and D: 1,3,5-tricyano) is presented by using the topological analysis of the electron localization function (ELF) and Thom's catastrophe theory (CT) on the reaction paths calculated at the B3LYP/6-31G(d,p) level. The progress of the reaction is monitorized by the changes of the ELF structural stability domains (SSD), each being change controlled by a turning point derived from CT. The reaction mechanism of the parent reaction A is characterized by nine ELF SSDs. All processes occur in the vicinity of the transition structure and corresponding to a concerted formation/breaking of C(1)-C(6) and C(3)-C(4) bonds, respectively, together with an accumulation of charge density onto C(2) and C(5) atoms. Reaction B presents the same number of ELF SSDs as A, but a different order appears; the presence of 2,5-dicyano substituents favors the formation of C(1)-C(6) bonds over the breaking of C(3)-C(4) bond process, changing the reaction mechanism from a concerted towards a stepwise, via a cyclohexane biradical intermediate. On the other side, reaction C presents the same type of turning points but two ELF SSD less than A or B; there is an enhancement of the C(3)-C(4) bond breaking process at an earlier stage of the reaction by delocalizing the electrons from the C(3)-C(4) bond among the cyano groups. In the case of competitive effects of cyano subsituents on each moiety, as it is for reaction D, seven different ELF SSDs have been identified separated by eight turning points (two of them occur simultaneously). Both processes, formation/breaking of C(1)-C(6) and C(3)-C(4) bonds, are slightly favored with respect to the parent reaction (A), and the TS presents mixed electronic features of both B and C. The employed methodology provides theoretical support for the centauric nature (half-allyl, half-radical) for the TS of D.     

Charge-shift bonding--a class of electron-pair bonds that emerges from valence bond theory and is supported by the electron localization function approach

This paper deals with a central paradigm of chemistry, the electron-pair bond. Valence bond (VB) theory and electron-localization function (ELF) calculations of 21 single bonds demonstrate that along the two classical bond families of covalent and ionic bonds, there exists a class of charge-shift bonds (CS bonds) in which the fluctuation of the electron pair density plays a dominant role. In VB theory, CS bonding manifests by way of a large covalent-ionic resonance energy, RE(CS), and in ELF by a depleted basin population with large variances (fluctuations). CS bonding is shown to be a fundamental mechanism that is necessary to satisfy the equilibrium condition, namely the virial ratio of the kinetic and potential energy contributions to the bond energy. The paper defines the atomic propensity and territory for CS bonding: Atoms (fragments) that are prone to CS bonding are compact electronegative and/or lone-pair-rich species. As such, the territory of CS bonding transcends considerations of static charge distribution, and involves: a) homopolar bonds of heteroatoms with zero static ionicity, b) heteropolar sigma and pi bonds of the electronegative and/or electron-pair-rich elements among themselves and to other atoms (e.g., the higher metalloids, Si, Ge, Sn, etc), c) all hypercoordinate molecules. Several experimental manifestations of charge-shift bonding are discussed, such as depleted bonding density, the rarity of ionic chemistry of silicon in condensed phases, and the high barriers of halogen-transfer reactions as compared to hydrogen-transfers.     

Fast topological analysis of 2D and 3D grids of data: application to the atoms in molecule (AIM) and the electron localization function (ELF)

The topological analysis of grids of data is used for determination of surfaces or volumes around maxima. The volumes are then related to chemical information such as atoms or bonds, and can be used for integration of local properties such as electronic population. The problem of global connectivity is reversed into the question of local connectivity yielding a linear scaling partition algorithm. Two packages are developed for a very fast analysis and partition of 2D or 3D grids of data, applications being made to C2H2, C2H4, C6H6, H2CO, and H2CS molecules using the Atoms in Molecule (AIM) or Electron Localization Function (ELF).