Heitler-London problem

Figure 5.7 Definition of elec- In atomic units2 length is measured in units tron and nuclear coordinates in of ao = (47reo)/me2, energy in (double) ryd-the Heitler-London problem. berg units = 27.2 eV, and the Hamiltonian...

It is dear, therefore, that as the atoms are pulled apart the Hartree-Fock MO solution does not go over to that of two neutral-free atoms H°H°, but instead goes over to a mixed configuration, schematically represented by [2H°H° + H+H" + H H+]. Since the energy cost for the ionic configuration is / — A = (13.6 — 0.8) eV = 12.8 eV, the Hartree-Fock MO solution dissociates incorrectly to +6.4 eV rather than zero (We have assumed the Hartree-Fock treatment of H is exact.) The Heitler-London VB solution avoids this problem by working with only the covalent configurations in the first square bracket of eqn (3.37), so that it dissodates correctly. 

Shortly after quantum mechanics evolved Heitler and London[l] applied the then new ideas to the problem of molecule formation and chemical valence. Their treatment of the H2 molecule was qualitatively very successful, and this led to numerous studies by various workers applying the same ideas to other substances. Many of these involved refinements of the original Heitler-London procedure, and within three or four years, a group of ideas and procedures had become reasonably well codified in what was called the valence bond (VB) method for molecular structure. 

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. 

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. 

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. 

The problems for quantum chemists in the mid-forties were how to improve the methods of describing the electronic structure of molecules, valence theory, properties of the low excited states of small molecules, particularly aromatic hydrocarbons, and the theory of reactions. It seemed that the physics needed was by then all to hand. Quantum mechanics had been applied by Heitler, London, Slater and Pauling, and by Hund, Mulliken and Hiickei and others to the electronic structure of molecules, and there was a good basis in statistical mechanics. Although quantum electrodynamics had not yet been developed in a form convenient for treating the interaction of radiation with slow moving electrons in molecules, there were semi-classical methods that were adequate in many cases. 

Problems related to light absorption in the excitonic region of the spectrum will be discussed in the following chapters. At this point we only notice that the formulas (3.136), (3.137), and (3.145) allow one to establish the oscillator strength of an excitonic transition which are more exact than that by use of the Heitler-London approximation, and the differences can be substantial. 

We can obviously set up an heuristic primitive for the spatial part using the very idea which was used to generate Q/(. If each electron pair is a localised bond then the spatial part should be capable of being generated from a product of spatial orbitals similar to the primitive product used in the Heitler-London model of the electron-pair bond. Clearly, if these individual orbitals are formed from some set of basis functions, then the matrix which defines the orbitals in terms of the basis functions forms a familiar variational problem... 

In order to set up the hf we need the pair expansion coefficients Ck for each pair of electrons in exactly the same way as the PEMCSCF case. However, in this case the expansion lengths are much shorter but there are more of them. If we choose to work with the simplest, Heitler-London, case of two terms per pair there would be a 2 x 2 Cl problem to solve for the lowest eigenvalue of each pair the solutions here can be simply written down. Longer expansion lengths will require the lowest eigenvector of a small Cl problem for each pair. 

From this point the theory of valency develops in various directions. First, qualitative extensions of the Heitler-London result to atoms heavier than hydrogen are attempted. Secondly, efforts are made to improve the kind of approximation upon which the treatment of the hydrogen atom itself has been based. Thirdly, a number of more or less empirical rules, supported by but not strictly derivable from the principles of quantum mechanics, are introduced for the handling of special types of problem. 

The SIC-LSD still considers the electronic structure of the solid to be built from individual one-electron states, but offers an alternative description to the Bloch picture, namely in terms of periodic arrays of localized atom-centred states (i.e., the Heitler-London picture in terms of the exponentially decaying Warmier orbitals). Nevertheless, there still exist states that will never benefit from the SIC. These states retain their itinerant character of the Bloch form and move in the effective LSD potential. This is the case for the non-f conduction electron states in the lanthanides. In the SIC-LSD method, the eigenvalue problem, Eq. (23), is solved in the space of Bloch states, but a transformation to the Wannier representation is made at every step of the self-consistency process to calculate the localized orbitals and the corresponding charges that give rise to the SIC potentials of the states that are truly localized. TTiese repeated transformations between Bloch and Wannier representations constitute the major difference between the LSD and SIC-LSD methods. 

PROBLEMS of directed valence, like most problems of moiecular structure, can be attacked by either of two methods the method of locaiized electron pairs (Heitler-London) or the method of molecular orbitals (Hund-Mul-liken). While it is now realized that these methods are but different starting approximations to the same final solution, each has its advantages in obtaining qualitative results. Theories of directed valence based on the methods of focalized pairs have been developed by Slater and Pauling and extended by Hult-gren. The method of molecular orbitals has been developed principally by Hund and Mulliken. These methods have been compared extensively by Van Vleck and Sherman. ... 

In 19, four years after the breakthrough of the Heitler-London quantum treatment of the H + H problem, Eyring and Polanyi provided the first potential energy surface describing the motion of three atoms (Fig. 2) ... 

---