Valence natural hybrid orbitals

Covalent crystals are held together by strong, highly directional bonds usually described by the valence bond hybrid orbital method. Each atom is part of a large extended single molecule that is the crystal itself. Because of the nature of their bonds, covalent crystals have very high melting points and are hard and brittle. 

The extracted Natural Hybrid Orbitals (NHOs) are therefore not simply encoded forms of the molecular shape, as envisioned in valence shell electron pair repulsions (VSEPR)-type caricatures of hybridization theory. Instead, the NHOs represent optimal fits to the ESS-provided electronic occupancies (first-order density matrix elements cf. V B, p. 21ff) in terms of known angular properties of basis AOs. Thus, the NHOs predict preferred directional characteristics of bonding from angular patterns of electronic occupancy, and the deviations (if any) between NHO directions and the actual directions of bonded nuclei give important clues to bond strain or bending that are important descriptors of molecular stability and function. 

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

The HOMO, 4ai (only valence shell orbitals are numbered), represents a combination of halogen p AOs destabilized by an antibonding interaction with a central atom sp hybrid orbital. Thus this MO is antibonding E—Hal MO in nature and has quite a large contribution from a central atom valence s orbital. This explains the quite low IEs observed for ionizations from 4a 1 MO, incompatible with a lone-pair orbital mainly localized on the central atom. This MO determines the Lewis base properties of the central atom in EHal2. 

The analysis of results of delocalised molecular orbital calculations for more complex molecules is problematic because there is no longer a simple relationship with the Lewis-localised bond representations. The use of fragment analyses and overlap populations has been helpful, but requires some knowledge of perturbation theory. Several methods have been developed to attempt to bridge the gap between the molecular orbital calculations and Lewis structures - one that is widely used is natural bonding orbitals (NBOs). Each bonding NBO Oab (the donor) can be written in terms of two directed valence hybrids (NHOs) h, hs on atoms A and B, with corresponding polarisation coefficients Ca, Cb ... 

The cage system is treated quantum mechanically. In the original version of the model all valence electrons were included and to allow a natural definition of the cage, orthogonalized atomic hybrid orbitals were used as a basis set [215]. This allows to avoid problems with the saturation of dangling bonds since all hybrids on the same atom may belong to the cage with a wave function obtained by solution of a closed-shell secular equation. 

Atomic Structure The Nucleus Atomic Structure Orbitals 4 Atomic Structure Electron Configurations 6 Development of Chemical Bonding Theory 7 The Nature of Chemical Bonds Valence Bond Theory sp Hybrid Orbitals and the Structure of Methane 12 sp Hybrid Orbitals and the Structure of Ethane 13 sp2 Hybrid Orbitals and the Structure of Ethylene 14 sp Hybrid Orbitals and the Structure of Acetylene 17 Hybridization of Nitrogen, Oxygen, Phosphorus, and Sulfur 18 The Nature of Chemical Bonds Molecular Orbital Theory 20 Drawing Chemical Structures 21 Summary 24... 

When the atomic orbitals are located on the same atomic centre, it is often useful to consider the hybridization of some of them, i.e., to construct linear combinations of them. This may be done either by requiring that the energy of the linear combination in the molecule be a minimum, or that the bond-angles determine the nature of the hybridization". The latter is usually used for elementary discussions of (approximate) hybridization of orbitals in valence-bond stmctures, and is therefore appropriate for the valence bond treatments that we shall present in this book. For our purposes, the most relevant of the hybrid orbitals are the following, in which we have indicated the explicit forms of the linear combinations for only the first two, for the special cases of equivalent hybrids. 

However, a localized adaptation of the natural orbital algorithm allows one to similarly describe/civ-center molecular subregions in optimal fashion, corresponding to the localized lone pairs (one-center) and bonds (two-center) of the chemist s Lewis structure picture. The Natural Bond Orbitals (NBOs) that emerge from this algorithm are intrinsic to, uniquely determined by, and optimally adapted to localized description of, the system wavefunction. The compositional descriptors of NBOs map directly onto bond hybridization, polarization, and other freshman-level bonding concepts that underlie the modem electronic theory of valency and bonding. 

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