Electron locahzation function

Silvi and coworkers have explored the potential of the electron locahzation function (ELF) approach for the characterization of H bonds [71]. They discussed how ELF-derived quantities behave when a H bond exists. Fuster and Grabowski compared QTAIM and ELF as descriptors of H bonds. They found good correlations between ELF and QTAIM parameters, and demonstrated that many of their parameters, especially the core-valence bifurcation index and the total electron energy density at the HBCP, are particularly useful in describing the H-bond strength [72]. 

Shaik S, Danovich D, Silvi B, Lauvergnat DL, Hiberty PC (2005) Charge-shift bonding-A class of electron-pair bonds that emerges from valence bond theory and is supported by the electron locahzation function approach. Chem A Eur J 11(21) 6358-6371... 

It is also possible to prepare crystalline electrides in which a trapped electron acts in effect as the anion. The bnUc of the excess electron density in electrides resides in the X-ray empty cavities and in the intercoimecting chaimels. Stmctures of electri-dides [Li(2,l,l-crypt)]+ e [K(2,2,2-crypt)]+ e , [Rb(2,2,2-crypt)]+ e, [Cs(18-crown-6)2]+ e, [Cs(15-crown-5)2]" e and mixed-sandwich electride [Cs(18-crown-6)(15-crown-5)+e ]6 18-crown-6 are known. Silica-zeolites with pore diameters of vA have been used to prepare silica-based electrides. The potassium species contains weakly bound electron pairs which appear to be delocalized, whereas the cesium species have optical and magnetic properties indicative of electron locahzation in cavities with little interaction between the electrons or between them and the cation. The structural model of the stable cesium electride synthesized by intercalating cesium in zeohte ITQ-4 has been coirfirmed by the atomic pair distribution function (PDF) analysis. The synthetic methods, structures, spectroscopic properties, and magnetic behavior of some electrides have been reviewed. Theoretical study on structural and electronic properties of inorganic electrides has also been addressed recently. ... 

Markovian pair conditional probability density with the basic concepts of the catastrophe theory, he succeeded in introducing new Markovian ELF classes which generalize the previous Becke-Edgecombe definition. Going beyond the actual interpretation of ELF as the error in electron locahzation, this new approach provides a quantum step-function indicating where the electrons are trapped rather than where they have peaks of spatial density. 

The electronic wave function of a molecular system is a linear combination of the locahzed VB states, Tr ... 

With the advent of the density functional theory, i.e., with the growing recognition of the role that electronic density plays in describing quantum states of atoms and molecules, there also spears the possibility of visualizing bonds and electronic locahzation through procedures applied on electronic densities. 

It is prerequisite to define localized, diabatic state wave fimctions, representing specific Lewis resonance configurations, in a VB-like method. Although this can in principle be done using an orbital localization technique, the difficulty is that these localization methods not only include orthorgonalization tails, but also include delocalization tails, which make contribution to the electronic delocalization effect and are not appropriate to describe diabatic potential energy surfaces. We have proposed to construct the locahzed diabatic state, or Lewis resonance structure, using a strictly block-localized wave function (BLW) method, which was developed recently for the study of electronic delocalization within a molecule.(28-3 1)... 

In equation 17, N i2i)N i2j) is the number of the electron pairs classically expected from the basin population whereas N(Qi, f2 ) is the actual number of pairs obtained by integration of the pair-electron function over the basins and i2j. The variance is then a measure of the quantum mechanical uncertainty of the basin population which can be interpreted as a consequence of the electron delocalization, whereas the pair covariance indicates how much the population flucmations of two given basins are correlated. Within the AIM framework, the atomic locahzation and delocalization indices X A) and 5(A, B) have been introduced and defined by equations 18 and 19 ... 

Nevertheless this method has been successfully used by Van Vleck when establishing the nature of the effective Hamiltonian for a magnetic insulator [40]. In that case, the electrons involved in the magnetic properties are locahzed on each atom. In other words, an assumption of localization is made for each free-atom wave function its magnitude decreases exponentially when any electron is removed to a large distance from the center R m) of... 

The two fundamental building blocks of Hartree-Fock theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital carries two electrons, with opposite spin. The occupied orbitals themselves are only defined as an occupied one-electron subspace of the full space spanned by the eigenfunctions of the Fock operator. Transformations between them leave the total HF wave function invariant. Normally the orbitals are obtained in a delocalized form as the solutions to the HF equations. This formulation is the most relevant one in studies of spectroscopic properties of the molecule, that is, excitation and ionization. The invariance property, however, makes a transformation to locahzed orbitals possible. Such localized orbitals can be valuable for an analysis of the chemical bonds in the system. 

From a computational point of view, it should be stressed that the computational tool of Francisco et al. [35] results in obtaining the electron number probability distribution functions of an -electron molecule through an exhaustive partitioning of the real space into arbitrary regions. From the computed probabilities, several magnitudes relevant to chemical bonding theory are obtained, such as average electronic populations and locahzation/delocalization indices. 

Another example of electron delocalization can be seen in a comparison of the electrostatic potential maps of pyrrole and pyridine in Figure 18.16. The lone pair of pyrrole is delocaUzed due to its participation in estabhshing aromatic stabilization. As a result, pyrrole is an extremely weak base because protonation of pyrrole would result in the loss of aromaticity. In contrast, the lone pair in pyridine is locahzed and does not participate in estabhshing aromaticity. As a result, pyridine can function as a base without destroying its aromatic stabilization. For this reason, pyridine is a stronger base than pyrrole. In fact, inspection of the pA values in Table 23.1 reveals that pyridine is five orders of magnitude ( 100,000 times) more basic than pyrrole. 

The Siesta (Spanish Initiative for Electronic Simulations with Thousands of Atoms) method [383,384,400] achieves linear scaling by the exphcit use of locahzed Wannier-like functions and numerical pseudoatomic orbitals confined by a spherical infinite-potential wall [401]. As the restriction of the Siesta method we mention the difficulty of the all-electron calculations and use of only LDA/GGA exchange-correlation functionals. 

The PW basis set is universal, in the sense that it does not depend on the positions of the atoms in the unit cell, nor on their nature [458]. One does not have to construct a new basis set for every atom in the periodic table nor modify them in different materials, as is the case with locahzed atomic-hke functions and the basis can be made better (and more expensive) or worse (and cheaper) by varying a single parameter -the number of plane waves defined by the cutoff energy value. This characteristic is particularly valuable in the molecular-dynamics calculations, where nuclear positions are constantly changing. It is relatively easy to compute forces on atoms. Finally, plane-wave calculations do not suffer from the basis-set superposition error (BSSE) considered later. In practice, one must use a finite set of plane waves, and this in fact means that well-localized core electrons cannot be described in this manner. One must either augment the basis set with additional functions (as in linear combination of augmented plane waves scheme), or use pseudopotentials to describe the core states. Both AS and PW methods, developed in solid-state physics are used to solve Kohn-Sham equations. We refer the reader to recently published books for the detailed description of these methods [9-11]. 

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