Schrodinger equation valence bond approach

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

Modem views of atomic structure are, as we have seen, based largely on the applications of wave mechanics to atomic systems. Modern views of molecular structure are based on applying wave mechanics to molecules such studies provide answers as to how and why atoms combine. The Schrodinger equation can be written to describe the behaviour of electrons in molecules, but it can be solved only approximately. Two such methods are the valence bond approach, developed by Heitler and Pauling, and the molecular orbital approach associated with Hund and MuUiken ... 

Following the work of Robin and Day, and Allen and Hush, the first rigorous solution to the vibronic Schrodinger equation for a mixed valence system was formulated by Piepho, Krausz, and Schatz. This model, based on the valence bond approach, is known as the PKS model. It is a model that has been modified to include orbital vibronic constants. A fully equivalent molecular orbital model developed by Piepho " is easier to conceptualize, however. The Piepho model also was the first to take advantage of the concept of orbital vibronic constants developed by Bersuker for Jahn-Teller systems (Figure 4). The Jahn-Teller theory is the basis for the PKS model. 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

The effective Hamiltonian and the metric (or overlap) operator J are Hermitian. The equation looks like those frequently used for solving the Schrodinger equation by means of a non-orthogonal basis set, as for instance in valence-bond (VB) theory. The operators Jf and y seem to have interesting transferability potentialities but up to now they do not seem to have been used for practical investigations. A more general approach has recently been given by Suzuki. 

The second broad approach to the description of molecular structure that is of importance in organic chemistry is molecular orbital theory. Molecular orbital (MO) theory discards the idea that bonding electron pairs are localized between specific atoms in a molecule and instead pictures electrons as being distributed among a set of molecular orbitals of discrete energies. In contrast to the orbitals described by valence bond theory, which are usually concentrated between two specific atoms, these orbitals can extend over the entire molecule. Molecular orbital theory is based on the Schrodinger equation,... 

Many computational methods exist, and, as yet, no one technique is ideally suited to all problems. For the sake of simplicity, these computational methods can be subdivided into molecular mechanical and quantum mechanical approaches, although each may encompass a wide range of techniques. The model used in computational chemistry refers to the type of approach—mechanics, semiempirical, or ab initio—that is adopted. The simplest model is that based on MM, in which atoms are considered to be solid spheres with ideal interatomic bond distances and angles the most complex models currently in use are those that attempt to solve the Schrodinger equations associated with molecular orbitals. In between are semiempirical models in which only valence electrons are considered in the quantum mechanical treatment, which in turn is greatly simplified... 

Another empirical method that has been extensively used by Warshel and co-workers is the empirical valence bond (EVB) theory. jn this approach, it is assumed that a reaction can be described by some VB resonance structures. The analytical form of these VB functions can be approximated by appropriate molecular mechanics potentials, and the parameters of these MM potentials are calibrated to reproduce experimental or ab initio MO data in the gas phase as well as in the condensed phase. The combined EVB/MM method and its unique calibration procedure have been recently reviewed. > 2 it should be noted that Kim and Hynes presented a similar method, yielding a nonlinear Schrodinger equation. However, the solvent was treated as a dielectric continuum in the Kim-Hynes theory. Nevertheless, an interesting feature in the latter method is a consideration of nonequilibrium coupling between the solute and solvent. ... 

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