Heitler-London approach generalization

The reader will recall that in Chapter 2 we gave examples of H2 calculations in which the orbitals were restricted to one or the other of the atomic centers and in Chapter 3 the examples used orbitals that range over more than one nuclear center. The genealogies of these two general sorts of wave functions can be traced back to the original Heitler-London approach and the Coulson-Fisher[15] approach, respectively. For the purposes of discussion in this chapter we will say the former approach uses local orbitals and the latter, nonlocal orbitals. One of the principal differences between these approaches revolves around the occurrence of the so-called ionic structures in the local orbital approach. We will describe the two methods in some detail and then return to the question of ionic stmctures in Chapter 8. 

Coulson described the first ten years of quantum chemists work on the electron valence bond (roughly 19281938) as work spent "escaping from the thought-forms of the physicist [my emphasis], so that the chemical notions of directional bonding and localization could be developed."45 Heisenberg earlier claimed that the Heitler-London treatment of the hydrogen molecule was not a characteristically physical approach, in contrast to Hund s more "general"... 

Fairly soon after the Heitler-London calculation, Slater, using his determi-nantal functions, gave a generalization to the n-electron VB problem[10]. This was a popular approach and several studies followed exploiting it. It was soon called the method of bond eigenfunctions. A little later Rumer[ll] showed how the use of these could be made more efficient by eliminating linear dependencies before matrix elements were calculated. 

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. 

As we have shown above, the exciton energy depends not only on the characteristics of a single molecular term /, as it would follow from an elementary approach based on the Heitler-London approximation, but, in general, depends on all excited states of the molecule. This property is reflected by the fact that, as we have shown, the energies of Coulomb excitons can be expressed in terms of the crystal dielectric tensor, which includes contributions of all resonances. 

Most of the approximate calculations are based on London s approach, which represents a generalization of the Heitler-London valence bond method for estimating the potential energy of the H2-mo-lecule. Thus, the wave functions cp and in (27 I) now describe the chemical bonds of the reacting H2-molecule (AB) and the product H2-molecule (BC), respectively. They are represented by the expressions... 

The third part deals with the theory of the chemical bond. It contains (of course) the seminal paper by Heitler zmd London, as well as a more general paper by London on the chemical bond and the quantum theory of homopolar valence numbers. The Heitler-London theory gave rise to the valence bond (VB) approach in quantum chemistry. There is also a paper with some exact calculations on H2 by Hylleraas, which present a computationally accurate view of the hydrogen bond. 

The first calculations on a two-electron bond was undertaken by Heitler and London for the H2 molecule and led to what is known as the valence bond approach. While the valence bond approach gained general acceptance in the chemical community, Robert S. Mulliken and others developed the molecular orbital approach for solving the electronic structure problem for molecules. The molecular orbital approach for molecules is the analogue of the atomic orbital approach for atoms. Each electron is subject to the electric field created by the nuclei plus that of the other electrons. Thus, one was led to a Hartree-Fock approach for molecules just as one had been for atoms. The molecular orbitals were written as linear combinations of atomic orbitals (i.e. hydrogen atom type atomic orbitals). The integrals that needed to be calculated presented great difficulty and the computations needed were... 

It is remarkable that the work by Heitler and London that outlined for the first time a physically correct description of the chemical bond did not replace the Lewis picture of electron-pair bonding that was based on intuition rather than on elementary physics. One reason is the dramatically different appeal of the two approaches for human imagination of the chemical bond. The Lewis picture is simple to use and it proved as extremely powerful ordering scheme for molecular structures and reactivities. Chemists are generally happy with such models. The quantum theoretical description of interatomic interactions introduced the wave function as the central term for chemical bonding, which is in contrast an elusive object for human imagination, as evidenced by the intensive discussions about the meaning and the interpretation of P mainly in the physics community. 

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