Heitler-London equation

If the overlap integral is neglected, the Heitler-London equation becomes... 

The energy corresponding to this state is given by the Heitler-London equation ... 

The generalization of the Heitler-London equation, Eq. (3.31), to three hydrogen atoms was also considered in the early days of quantum mechanics (around 1930). This description contains the essence of the energetics associated with bond breaking and bond making. 

Like Hund, Mulliken developed the basic Schrodinger equation in the direction of establishing the electron charge density resulting from a combination of the attractions of two or more nuclei and the averaged repulsions of other electrons in the system. This is a method that favors some particular region of space and disfavors others. In contrast to the Heitler-London method, it over-emphasizes, rather than underemphasizes, the ionic character of a molecule. For example, for the H2 molecule, Hund s wave function equation assumes that it is just as probable to have two electrons around the same nucleus as to have one electron around each nucleus. For a molecule made up of identical nuclei, this treatment is a considerable exaggeration of the ionic character of the molecule. 

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... 

Fig. 5. Contribution of ionic terms to the binding energy as a function of internuclear distance, (a) Contribution /i of ionic terms for H2 molecule (see equation 25) calculated with best screening constant of a Heitler-London wave function (after Coulson and Fischer (134)). (b) Variation of the binding energy...

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. 

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

Equations (1.105) and (1.106) are the Heitler-London counterpart of the corresponding quantities (Equations 3.4 and 3.5 on page 340 of Ruedenberg s paper (1962), which refers to a LCAO-MO wave function. Ruedenberg calls Equation (1.106) the modification of the quasi-classical density due to the interference effect , while we, more literally, speak of exchange[a(r)b(r)], [b(r)a(r)] and overlap[—Sa1(r)], [—Sfe2(r)] densities. Finally, it is worth noting that, while ... 

Equations (3.69), with regard to (3.60), yield for the Heitler-London approximation... 

Equations (3.70) and (3.71) coincide, as one might expect, with eqns (3.52) and (3.49) giving the energy and the wavefunctions in the Heitler-London approximation. For computing the excitonic energies within this approximation we notice that the quantities... 

As long as A-Y and X-Y are polar-covalent bonds, we can focus on the covalent parts of the wave function, denoted also as the Heitler-London wave function, called so after the names of the originators of this wave function [51]. Designating the active orbitals of the fragments X, A, and Y, respectively, as x, a and y, the wave functions of R is given in Eq. (7a), without the normalization constant. This equation describes X and the A-Y bond. The wave function for R is given in Eq. (7b) ... 

The Heitler-London VB wave function for ground-state Ha is [Eq. (13.101)] ls (l)lsj(2) ls (2)lsj(l) multiphed by a normalization constant and a spin function. The GVB ground-state Ha wave function replaces this spatial function by/(l)g(2) + /(2)g(l), where the functions/and g are found by minimization of the variational integral. To find / and g, one expands each of them in terms of a basis set of AOs and finds the expansion coefficients by iteratively solving one-electron equations that resemble the equations of the SCF MO method. 

In the third part of the paper Huckel derives energy expressions for the different electronic terms of benzene using method 1. The wavefunction is given as a Slater determinant, and the solution of the Schrodinger equation is expressed in analogy to the Heitler-London paper [1] as... 

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