Theory Electron Pair Localization

2 Theory Electron Pair Localization. - Bader and Heard found that the Laplacian of the conditional pair density, L(e, r) for same-spin electrons can be homeomorphically mapped onto L(r) = V p by placement of the reference electron e at positions of maximum localization of the Fermi hole for an (a, electron pair. Their mapping appeared faithfiil in that every maximiun in L(r) can be associated with a corresponding maximum in L(e, r). This mapping indicates that the charge concentrations of L(r) denote the spatial regions in which there is a partial condensation of the pair density towards individually localized electron pairs. The authors claim that, unlike L(r), which is the limiting form assumed by 

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We must now address a fundamental question. Are there C-H bonds in methane The answer from MO theory is clearly no. Population of the four bonding molecular orbitals with four pairs of electrons leads to a bonding interaction among the carbon atom and all of the hydrogen atoms (not just between carbon and the individual hydrogens). Thus, we should say that there is bonding in MO theory, but there are not distinct bonds formed by separate electron pairs localized between two atoms. 

Valence bond theory views bonding as arising from electron pairs localized between adjacent atoms. These pairs of electrons create bonds. Further, organic chemists commonly use the atomic orbitals involved in the three hybridization states of atoms sp, sp, and sp) to create the orbitals that hold these electrons because doing so allows the resulting orbitals to match the experimentally determined geometries around the atoms. Therefore, hybridization is also a VB theory concept. But how do we make the orbitals that contain the electrons and that reside between adjacent atoms This is where we return to MO theory. 

In molecular orbital theory, electrons occupy orbitals called molecular orbitals that spread throughout the entire molecule. In other words, whereas in the Lewis and valence-bond models of molecular structure the electrons are localized on atoms or between pairs of atoms, in molecular orbital theory all valence electrons are delocalized over the whole molecule, not confined to individual bonds. 

Redress can be obtained by the electron localization function (ELF). It decomposes the electron density spatially into regions that correspond to the notion of electron pairs, and its results are compatible with the valence shell electron-pair repulsion theory. An electron has a certain electron density p, (x, y, z) at a site x, y, z this can be calculated with quantum mechanics. Take a small, spherical volume element AV around this site. The product nY(x, y, z) = p, (x, y, z)AV corresponds to the number of electrons in this volume element. For a given number of electrons the size of the sphere AV adapts itself to the electron density. For this given number of electrons one can calculate the probability w(x, y, z) of finding a second electron with the same spin within this very volume element. According to the Pauli principle this electron must belong to another electron pair. The electron localization function is defined with the aid of this probability ... 

The theory as presented so far is clearly incomplete. The topology of the density, while recovering the concepts of atoms, bonds and structure, gives no indication of the localized bonded and non-bonded pairs of electrons of the Lewis model of structure and reactivity, a model secondary in importance only to the atomic model. The Lewis model is concerned with the pairing of electrons, information contained in the electron pair density and not in the density itself. Remarkably enough however, the essential information about the spatial pairing of electrons is contained in the Laplacian of the electron density, the sum of the three second derivatives of the density at each point in space, the quantity V2p(r) [44]. 

The fundamental principle of the Valence-Shell Electron-Pair Repnlsion theory is that the bonding pairs and lone, non-bonding pairs of electrons in the valence level of an atom repel one another. As you know, electron pairs of atoms are localized in orbitals, which are shapes that describe the space in which electrons are most likely to be found around a nucleus. 

One very important difference between VSEPR theory and MO theory should be noted. The MOs of the water molecule which participate in the bonding are three-centre orbitals. They are associated with all three atoms of the molecule. There are no localized electron pair bonds between pairs of atoms as used in the application of VSEPR theory. The existence of three-centre orbitals (and multi-centre orbitals in more complicated molecules) is not only more consistent with symmetry theory, it... 

The VSEPR assumption that there are four identical localized electron pair bonds in the four C-H regions, made up from sp3 hybrid carbon orbitals and the hydrogen Is orbitals, is not consistent with the experimentally observed photoelectron spectrum. The MO theory is consistent with the two ionizations shown in the photoelectron spectrum of CH4 and implies that the bonding consists of four electron pairs which occupy the la, and It, five-centre MOs. 

Ik this chapter we explore how symmetry considerations can be applied to one of the most pervasive concepts in all of chemistry bonding between atoms by the sharing of pairs of electrons. Though the idea of an electron-pair bond was first introduced in 1916 by G. N. Lewis, it was only after the advent of quantum mechanics that it could be given a proper theoretical basis. This came about through the development of two theories valence bond (VB) theory and localized MO theory both of which describe the electron pair in terms of orbitals of the component atoms of the bond. 

In fact, for tightly localized electron pairs, the dominant excitation level is the value of k nearest "vO.OlN (i.e., for about 200 electrons the double excitations in aggregate are more important than the SCF configuration and for 400 electrons quadruple excitations should dominate). Even for molecules with only 40 electrons quadruple and higher excitations must be considered in order to reproduce excitation energies (30) or potential surfaces to an accuracy of 0.1 eV. Thus, configuration interaction calculations for very large molecules are hopeless unless perturbation theory can be used to correct for unlinked cluster effects. 

Recently, Friesner et al.124 proposed a method referred to as J2 theory to predict accurate thermochemical data. This approach is based on the generalized valence bond-localized Moller-Plesset method (GVB-LMP2) and includes parameters that depend on the number of electron pairs and whether the pairs are a or 7t types. Thus, the parameterization in the J2 method is molecule dependent. The GVB-LMP2 method scales as n3 as opposed to n6 or n7 for the MP4, QCISD, or CCSD methods, so J2 is much faster than G2. The J2 method... 

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