Heitler-London approach

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. 

Slater s bond eigenfunctions constitute one choice (out of an infinite number) of a particular sort of basis function to use in the evaluation of the Hamiltonian and overlap matrix elements. They have come to be called the Heitler-London-Slater-Pauling (HLSP) functions. Physically, they treat each chemical bond as a singlet-coupled pair of electrons. This is the natural extension of the original Heitler-London approach. In addition to Slater, Pauling[12] and Eyring and Kimbal[13] have contributed to the method. Our following description does not follow exactly the discussions of the early workers, but the final results are the same. 

Heitler-London approach to the chemical bond—and added some new ideas of his own. 

In the Heitler-London approach, the full Hamiltonian of equation (4) can be subdivided as follows... 

In constructing the spatial part of the electronic wave function the Valence-Bond (VB) method is useful. This method (similar to the Heitler-London approach) postulates that a linear combination of several product functions (called covalent structures or ionic structures ) is appropriate... 

The 3d electrons in a magnetic compound crystal are localized about single ions, so that they can be described fairly adequately by the Heitler-London approach. Usually, as explained in Section II, the orbital magnetic moment is quenched and only spin can contribute to the magnetic moment. When there are two magnetic ions with resultant spin and S, respectively at a reasonable distance from each other, we have the well-known magnetic dipolar interaction,... 

It will be recalled that the first quantum-mechanical explanation of the chemical bond is usually attributed to Heitler and London (1927). Their discussion of the hydrogen molecule, however, was not based on IPM concepts, with each electron assigned to a molecular orbital, but rather on an independent-atom approach in which the electrons were assigned to atomic orbitals. The Heitler-London approach was the basis for valence bond (VB) theory, which was important in the early days of quantum chemistry but later fell into disuse. There has been some revival of interest in VB theory now that more powerful computing facilities are available a full discussion of this method is deferred until Chapter 7, although a preliminary account will appear in Chapter 3. 

Slater developed an approach (the "determinantal method") that offers a way of choosing among linear combinations (essentially sums and differences) of the polar and nonpolar terms in the Hund-Mulliken equations to bring their method into better harmony with the nonpolar emphasis characteristic of the Heitler-London-Pauling approach in which polar terms do not figure in the wave equation. 72... 

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"... 

In the limit of infinite bond distance the width of the peak of the potential Vki approaches zero and the height of the peak approaches a value equal to the ionization energy of the molecule [43,78], The asymptotic expression for T, = T — T. can be obtained with T calculated with the Heitler-London... 

Calculations in which the orbitals used are restricted to being centered on only one atom of the molecule. They are legitimately called atomic orbitals . Treatments of this sort may have many configurations involving different orbitals. This approach may be considered a direct descendent of the original Heitler-London work, which is discussed in Chapter 2. 

The localized-electron model or the ligand-field approach is essentially the same as the Heitler-London theory for the hydrogen molecule. The model assumes that a crystal is composed of an assembly of independent ions fixed at their lattice sites and that overlap of atomic orbitals is small. When interatomic interactions are weak, intraatomic exchange (Hund s rule splitting) and electron-phonon interactions favour the localized behaviour of electrons. This increases the relaxation time of a charge carrier from about 10 s in an ordinary metal to 10 s, which is the order of time required for a lattice vibration in a polar crystal. 

An interesting approach to SN2 reactions has been recently advanced by Shaik which leads to verifiable predictions (Shaik, 1981 Pross and Shaik, 1981 Shaik, 1982 Shaik and Pross, 1982). The basic contention of this theory is that the origin for the barrier of SN2 reactions arises from an avoided curve-crossing between two curves containing the reactant and product Heitler-London VB forms. In this treatment, the reacting pair is analysed in terms of a nucleophile being the electron donor (D) and the substrate the electron acceptor (A). For the simple reactions of type (47), the... 

In point-charge simulation this electronic rearrangement is of no immediate consequence except for the assumption of a reduced interatomic distance, which is the parameter needed to calculate increased dissociation energies. However, in Heitler-London calculation it is necessary to compensate for the modified valence density, as was done for heteronuclear interactions. The closer approach between the nuclei, and the consequent increase in calculated dissociation energy, is assumed to result from screening of the nuclear repulsion by the excess valence density. Computationally this assumption is convenient and effective. 

The original Heitler-London calculation, being for two electrons, did not require any complicated spin and antisymmetrization considerations. It merely used the familiar rules that the spatial part of two-electron wave functions are symmetric in their coordinates for singlet states and antisymmetric for triplet states. Within a short time, however, Slater[10] had invented his determinantal method, and two approaches arose to deal with the twin problems of antisymmetrization and spin state generation. When one is constructing trial wave functions for variational calculations the question arises as to which of the two requirements is to be applied first, antisymmetrization or spin eigenfunction. 

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. 

The original Heitler-London treatment with its various extensions was a VB treatment that included several configurations, e.g., the total wave function is a sum of terms with spatial functions made up of different subsets of the orbitals. This is the essence of multiconfiguration methods. The most direct extension of this sort of approach is, of course, the inclusion of larger numbers of configurations and the application to larger molecules. The computational power allowed calculations of this sort. 

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. 

Historians would later credit Heitler, London, Slater, and Pauling with developing this approach to the chemical bond (in later years it would be called, in their honor, the HLSP theory). But most chemists would remember only Pauling, not only because he applied it to more molecules than the others, but because he alone among them was a chemist, able to communicate his results in a way that chemists understood. 

The Dlatomics-ln-Molecules Approach. The simple version of the DIM method that we employ is based on the Heitler-London approximation (28). In spirit, it is similar to the London-Eyring approach except that we use accurate diatomic potential curves (29a) rather than an approximate form for the diatomic triplet curve (e.g. the Sato parameter for LEPS surfaces) (29b). 

Initially, the MO and VB approaches were developed with different aims in view the primary task of MO theory was to explain the electronic spectra of molecules, while the VB method was concerned mainly with the problems of bonding and valency. This is directly reflected in the construction of the wavefunctions used in the most well-known examples of the two approaches, the Hartree-Fock (HF) method and classical (Heitler-London ) VB theory. 

---