Bonding Pauli repulsion

In the preceding section, we discussed the electron pair (2c-2e) bond and how it can be influenced by Pauli repulsion of the SOMOs with other electrons. In the three-electron (2c-3e) bond, Pauli repulsion plays an even more fundamental role, as we will see.72 The idea of the three-electron bond was introduced in the early 1930s by Pauling in the context of the valence bond (VB) model of the chemical bond.70 71 Since then, it has been further developed both in VB and in MO theory and has become a standard concept in chemistry.118-129 In VB theory,7°>71 118 123 the two-center, three-electron (2c-3e) bond between two fragments A and B is viewed as arising from a stabilizing resonance between two valence bond structures in which an electron pair is on fragment A and an unpaired electron on B (13a), or the other way around (13b) ... 

M. A. Buijse and E. J. Baerends,/. Chem. Phys., 93,4129 (1990). Analysis of Nondynamical Correlation in the Metal-Ligand Bond. Pauli Repulsion and Orbital Localization in MnOj. 

Multiple M=P bonding in (OC)5M=PR becomes evident with ADF s bond energy analysis in terms of electrostatic interactions, Pauli repulsion, and orbital interactions from which the a,Ti-separation is obtained using a symmetry decomposition scheme [21]. For singlet (OC)5Cr=PR, which has a BDEst of 40.5 kcal/mol, the a- and n-components are 62.4 and 40.9 kcal/mol, respectively. 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

The nonequivalence in the size and shape of bonding and nonbonding electron pair domains can alternatively be expressed in terms of the relative magnitude of their mutual Pauli repulsions, which decrease in the following order ... 

In a pericyclic reaction, the electron density is spread among the bonds involved in the rearrangement (the reason for aromatic TSs). On the other hand, pseudopericyclic reactions are characterized by electron accumulations and depletions on different atoms. Hence, the electron distributions in the TSs are not uniform for the bonds involved in the rearrangement. Recently some of us [121,122] showed that since the electron localization function (ELF), which measures the excess of kinetic energy density due to the Pauli repulsion, accounts for the electron distribution, we could expect connected (delocalized) pictures of bonds in pericyclic reactions, while pseudopericyclic reactions would give rise to disconnected (localized) pictures. Thus, ELF proves to be a valuable tool to differentiate between both reaction mechanisms. 

The nature of bonding of the adsorbed species to the model cluster of metal surfaces can be analyzed in terms of the so-called constrained space orbital variation (CSOV) method. For halogen anions adsorbed on various silver surfaces, it has been found that Pauli repulsion, metal polarization, and charge transfer to the metal surface mainly contribute to the binding energy of the ions [104, 301]. 

The bonding of ions to metals is dominated by Coulomb attraction since there is a significant difference in electron affinity between the metals and ions. The bonding also involves a redistribution of charge through intermolecular charge transfer (between adsorbed ions and the surface) and intramolecular polarization (in ions and on the surface), which reduces the Pauli repulsion. 

In the ethane case, however, the AIM analysis helps in understanding the overlap of the bonds and the location of the electrons as derived from the density picture, but it does not tell us anything about the origin of the rotational barrier. For that, we need methods that quantitatively give us energies that can be associated with the effects of donor-acceptor bonding (hyperconjugation) and electron-electron repulsion (Pauli repulsion) as noted above. 

Fig. 7.7 Anomeric effect in sugars (left) preferentially stabilising the axial position of the anomeric C—O bond, and isomer energy difference between 1,3- and 1,4-dioxane at B3LYP/6-31 G(d) as clear evidence against steric (Pauli) repulsion arguments for the anomeric effect.

The results of a valence bond treatment of the rotational barrier in ethane lie between the extremes of the NBO and EDA analyses and seem to reconcile this dispute by suggesting that both Pauli repulsion and hyperconjugation are important. This is probably closest to the truth (remember that Pauli repulsion dominates in the higher alkanes) but the VB approach is still imperfect and also is mostly a very powerful expert method [43]. VB methods construct the total wave function from linear combinations of covalent resonance and an array of ionic structures as the covalent structure is typically much lower in energy, the ionic contributions are included by using highly delocalised (and polarisable) so-called Coulson-Fischer orbitals. Needless to say, this is not error free and the brief description of this rather old but valuable approach indicates the expert nature of this type of analysis. 

We discuss below the physics of the classical electrostatic attraction AVe t and the steric or Pauli repulsion AEPauli. Thereafter, we turn to the stabilizing or bonding interactions, both electron-pair bond formation and donor-acceptor interactions, as well as stabilization coming from admixing of (higher) virtual orbitals on one fragment due to the potential field of the other fragment (polarization). Finally, in the last part of this section we discuss some aspects of the mutual influence between the various interactions. 

Transition metal complexes comprise a typical example. The lone pair orbitals of ligands like CO, H20, Cl-, and O2- experience significant repulsion from the upper core shells 3s and 3p (first transition series). The Pauli repulsion with these shells determines the repulsive wall in the metal-ligand -versus-R curve. Of course, there is also overlap with deeper core orbitals, and so their effect is less important. An illustration is provided elsewhere (Figures 2 and 3 of Ref. 35), where the behavior of the Pauli repulsion is demonstrated along the Mn—O bond distance in MnO. The fragments are Mn2+, which has 5 electrons with spin up in the 3d orbitals, and the 02 cage, which has 5 electrons... 

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