Electron pairing and chemical bonds: pair localization in ELF domains from the analysis of domain averaged Fermi holes

This article reports the application of a recently proposed formalism of domain averaged Fermi holes to the problem of the localization of electron pairs in electron localization function (ELF) domains and its possible implications for the electron pair model of chemical bond. The main focus was on the systems, such as H2O or N2, in which the "unphysical" population of ELF domains makes the parallel between these domains and chemical bond questionable. On the basis of the results of the Fermi-hole analysis, we propose that the above problems could be due to the fact that in some cases the boundaries of the ELF domains need not be determined precisely enough.     

电子定域化函数的含义与函数形式

电子定域化函数(ELF)是研究电子结构的重要工具.本文介绍了电子定域性的概念,从电子对密度和动能密度两个角度详细讨论了ELF的物理意义与其函数形式的联系,并将ELF推广到自旋极化形式.通过实例分析,指出了参考项在ELF 中起到了关键性作用.对两种自旋极化形式的ELF 的比较发现:CheckDen 和TopMoD程序中使用的形式并不合理,明显低估了单电子区域的定域性.最后指出了一些文献由于对ELF函数的错误理解而在引用时出现的错误.     

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.     

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 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 delocalization and bond formation under the ELF framework

A first approach to the relationship between the electron localization function (ELF) and electronic delocalization upon bond formation is provided. We show from first principles the ability of ELF at the bond critical points to act as an index of the electron reorganization involved in chemical bonding. Simultaneously, this index, that we shall call ELF delocalization index (EDI), constitutes a good measure of electron delocalization. We will show how the core of ELF is proportional to the Wiberg index under the valence bond approach. This relationship will be exploited for some representative examples where EDI is able to identify the stages of bond formation. Furthermore, a maximum in EDI along this process has been found to correlate with the molecular equilibrium configuration, allowing for a formulation of a ??maximal localization principle?? for the stable structure of covalent compounds in terms of ELF.     

Understanding reaction mechanisms in organic chemistry from catastrophe theory applied to the electron localization function topology

Thom's catastrophe theory applied to the evolution of the topology of the electron localization function (ELF) gradient field constitutes a way to rationalize the reorganization of electron pairing and a powerful tool for the unambiguous determination of the molecular mechanisms of a given chemical reaction. The identification of the turning points connecting the ELF structural stability domains along the reaction pathway allows a rigorous characterization of the sequence of electron pair rearrangements taking place during a chemical transformation, such as multiple bond forming/breaking processes, ring closure processes, creation/annihilation of lone pairs, transformations of C-C multiple bonds into single ones. The reaction mechanism of some relevant organic reactions: Diels-Alder, 1,3-dipolar cycloaddition and Cope rearrangement are reviewed to illustrate the potential of the present approach.     

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.     

Electron Domains and the VSEPR Model of Molecular Geometry

The valence shell electron pair repulsion (VSEPR) model—also known as the Gillespie–Nyholm rules—has for many years provided a useful basis for understanding and rationalizing molecular geometry, and because of its simplicity it has gained widespread acceptance as a pedagogical tool. In its original formulation the model was based on the concept that the valence shell electron pairs behave as if they repel each other and thus keep as far apart as possible. But in recent years more emphasis has been placed on the space occupied by a valence shell electron pair, called the domain of the electron pair, and on the relative sizes and shapes of these domains. This reformulated version of the model is simpler to apply, and it shows more clearly that the Pauli principle provides the physical basis of the model. Moreover, Bader and his co-workers' analysis of the electron density distribution of many covalent molecules have shown that the local concentrations of electron density (charge concentrations) in the valence shells of the atoms in a molecule have the same relative locations and sizes as have been assumed for the electron pair domains in the VSEPR model, thus providing further support for the model. This increased understanding of the model has inspired efforts to examine the electron density distribution in molecules that have long been regarded as exceptions to the VSEPR model to try to understand these exceptions better. This work has shown that it is often important to consider not only the relative locations and sizes, but also the shapes, of both bonding and lone pair domains in accounting for the details of molecular geometry. It has also been shown that a basic assumption of the VSEPR model, namely that the core of an atom underlying its valence shell is spherical and has no influence on the geometry of a molecule, is normally valid for the nonmetals but often not valid for the metals, including the transition metals. The cores of polarizable metal atoms may be nonspherical because they include nonbonding electrons or because they are distorted by the ligands, and these nonspherical cores may have an important influence on the geometry of a molecule.     

Comparison of the electron localization function and deformation electron density maps for selected earth materials

The electron localization function (ELF) and experimental and theoretical deformation electron density maps are compared for several earth materials and one representative molecule. The number and arrangement of the localized one-electron probability density domains generated in a mapping of the ELF correspond to the number and arrangement of the localized electron density domains generated in a mapping of the deformation electron density distribution, a correspondence that suggests that the two fields are homeomorphically related. As a homeomorphic relationship has been established previously between the Laplacian of the electron density distribution and the ELF, the relationship suggests that the deformation electron density distribution is also homeomorphically related to the Laplacian of the distribution.     

Electron localization function in full-potential representation for crystalline materials

The electron localization function (ELF) is implemented in the first-principles, all-electron, full-potential local orbital method. This full-potential implementation increases the accuracy with which the ELF can be computed for crystalline materials. Some representative results obtained are presented and compared with the results of other methods. Although for crystal structures with directed bonding only minor differences are found, in simple elemental metals, there are differences in the valence region, which give rise to different ELF topologies.     

Electron pairing and chemical bonds: Chemical bonds from the condition of minimum fluctuation of electron pair

The role of electron pairing in chemical bonding stressed by the Lewis electron-pair model of the chemical bond is analyzed and discussed from the point of view of the proposal that chemical bonds are the regions of space populated roughly by two electrons and which at the same time exhibit low fluctuation of an electron pair. Based on this assumption, we have been able to introduce a new localization procedure, the output of which are just the orbitals (chemical bonds) satisfying the criterion of minimum pair fluctuation. It has been shown that these orbitals remarkably well display the most important attributes of chemical bonds, namely, the localization in the regions where classical bonds are expected and there is very high transferability from one molecule to another. The applicability of this procedure as a new means of the analysis and the visualization of the molecular structure is also discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 193–200, 1998     

Electronic fluxes during diels‐alder reactions involving 1,2‐benzoquinones: mechanistic insights from the analysis of electron localization function and catastrophe theory

By means of the joint use of electron localization function (ELF) and Thom's catastrophe theory, a theoretical analysis of the energy profile for the hetero‐Diels‐Alder reaction of 4‐methoxy‐1,2‐benzoquinone 1 and methoxyethylene 2 has been carried out. The 12 different structural stability domains obtained by the bonding evolution theory have been identified as well as the bifurcation catastrophes (fold and cusp) responsible for the changes in the topology of the system. This analysis permits finding a relationship between the ELF topology and the evolution of the bond breaking/forming processes and electron pair rearrangements through the reaction progress in terms of the different ways of pairing up the electrons. The reaction mechanism corresponds to an asynchronous electronic flux; first, the O1? C5 bond is formed by the nucleophilic attack of the C5 carbon of the electron rich ethylene 2 on the most electrophilically activated carbonyl O1 oxygen of 1 , and once the σ bond has been completed, the formation process of the second O4? C6 bond takes place. In addition, the values of the local electrophilicity and local nucleophilcity indices in the framework of conceptual density functional theory accounts for the asychronicity of the process as well as for the observed regioselectivity. © 2012 Wiley Periodicals, Inc.     

Electron localization function comparative study of ground state,triplet state,radical anion,and cation in model carbonyl and imine compounds

The modifications of bonding in carbonyl and imine compounds upon excitation, electron attachment, and ionization were studied within the framework of the electron localization function (ELF). The topological analysis of ELF allows a partition of the molecular space into regions having a clear chemical meaning: the basins. The electronic populations of the basins associated with bonding and nonbonding character, as well as the basin spin densities, were calculated at the MP2 level of the quantum mechanical calculation. Good agreement was found with the classical view in terms of mesomeric structures corresponding to the dominant localization of electrons contained in frontier molecular orbitals (MOs). The variations of the basins population were compared to the predictions of MO theory. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 897–910, 1999     

通过价层电子密度分析展现分子电子结构

迄今已有众多实空间函数被提出用来揭示化学上感兴趣的分子电子结构特征,例如化学键、孤对电子和多中心电子共轭。在这些分析方法中,电子定域化函数(ELF)、电子密度的拉普拉斯(∇²ρ)和变形密度(ρ_def)被广泛用于实际研究。众所周知,分析分子的总电子密度无法像以上提及的方法那样展现出与分子电子结构有关的丰富的信息。但是,在本文中,通过数个实例以及通过与ELF、∇²ρ和ρ_def的对比,我们指出若只关注价层电子密度分布,分子电子结构特征也是可能被探究的。我们发现对大多数情况,对非常简单的价层电子密度的分析也可以给出与ELF、∇²ρ和ρ_def分析类似的信息,并且这种分析具有计算复杂度更低的额外优点。我们希望本文的工作可以使得化学家们关注长期被忽视的价层电子密度所具有的重要价值。也值得注意的是,价层电子密度分析并非完全没有缺点,当这种方法无法提供丰富信息的时候,研究者仍需借助于其它类型的分析手段。     

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 pair localization function: a practical tool to visualize electron localization in molecules from quantum Monte Carlo data

In this work we introduce an electron localization function describing the pairing of electrons in a molecular system. This function, called "electron pair localization function," is constructed to be particularly simple to evaluate within a quantum Monte Carlo framework. Two major advantages of this function are the following: (i) the simplicity and generality of its definition; and (ii) the possibility of calculating it with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, multiconfigurational wave functions, valence bond, density functional theory, variational Monte Carlo with explicitly correlated trial wave functions, fixed-node diffusion Monte Carlo, etc). A number of applications of the electron pair localization function to simple atomic and molecular systems are presented and systematic comparisons with the more standard electron localization function of Becke and Edgecombe are done. Results illustrate that the electron pair localization function is a simple and practical tool for visualizing electronic localization in molecular systems.     

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.