Hydrogen Heitler-London model

The basic idea of the Heitler-London model for the hydrogen molecule can be extended to chemical bonds between any two atoms. The orbital function (10.8) must be associated with the singlet spin function cro,o(l > 2) in order that the overall wavefunction be antisymmetric [cf. Eq (8.14)]. This is a quantum-mechanical realization of the concept of an electron-pair bond, first proposed by G. N. Lewis in 1916. It is also now explained why the electron spins must be paired, i.e., antiparallel. It is also permissible to combine an antisymmetric orbital function with a triplet spin function, but this will, in most cases, give a repulsive state, such as the one shown in red in Fig. 10.2. 

Heitler-London model - An early quantum-mechanical model of the hydrogen atom which introduced the concept of the exchange interaction between electrons as the primary reason for stability of the chemical bond. 

Valence bond theory is usually introduced using the famous Heitler-London model of the hydrogen molecule [Heitler and London 1927] This model considers two non-interacting hydrogen atoms (a and b) in their ground states that are separated by a long distance. The wavefunction for this system is ... 

Only the first of these truly originates from him. In the first paper of the series, Pauling took up the idea of spatially directed bonds. By a generalization of the Heitler-London model for hydrogen, a normal chemical bond can be associated with the spin pairing of two electrons, one from each of the two atoms. While an s orbital is spherically symmetrical, other atomic... 

The exchange intCTaction is an electrostatic interaction between electrons and was first introduced in the theory of the Heitler-London model of a hydrogen molecule. The Hamiltonian for a molecule composed of two hydrogen nuclei (a and b) and two electrons (1 and 2) is given, on the basis of the Born-Oppenheimer approximation, by... 

Based on the Heitler-London model, when two hydrogen atoms located slightly away from each other (origin of energies) move closer, there are two possibilities ... 

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. 

When the ionization spheres of two neighbouring atoms interpenetrate, their valence electrons become delocalized over a common volume, from where they interact equally with both atomic cores. The covalent interaction in the hydrogen molecule was modelled on the same assumption in the pioneering Heitler-London simulation, with the use of free-atom wave functions. By the use of valence-state functions this H-L procedure can be extended to model the covalent bond between any pair of atoms. The calculated values of interatomic distance and dissociation energy agree with experimentally measured values. 

The origin of the VB model the Heitler-London treatment of the hydrogen molecule... 

The fundamental idea of the Heitler-London theory of valency bindiug is as follows. As a model of the hydrogen molecule we imagine two nuclei a and h on the c-axis at a distance It apart, and two electrons 1 and 2 revolving about the nuclei. To the state of two widely-separated neutral atoms there corresponds a large value of It and a motion of the electrons such that each one revolves round one of the two nuclei. Let the two atoms be in the ground state and have the... 

It is not the purpose of this chapter to produce and present a new force field. We rather want to provide a theoretical basis for MM and therefore also to be able to efficiently produce generic force-field parameters. As it stands, one parameter (ionization radius) is needed to initiate the derivation of all other parameters to model all bond orders of any covalent interaction. It is therefore reassuring to note that the uniform valence density within a characteristic atomic sphere has the same symmetry as the Is hydrogen electron. The first-order covalent interaction between any pair of atoms can therefore be modeled directly by the simple Heitler-London method for hydrogen to predict d, D and kr [44]. The results are in agreement with those of the simpler number-theory simulation [38], which is therefore preferred for general use. 

Originally one was satisfied to take over a model from macroscopic mechanics, namely that of completely hard spheres (b-correction of Van der Waals) or of almost hard spheres (Born repulsion, p. 36). It did indeed prove possible to obtain interesting results with this simple model (the ionic and Van der Waals radii). The work of Heitler and London on the hydrogen molec ule has laid the foundation for a more correct insight in this case also (p. 147). 

The calculation of Heitler and London (1927) of the energy of the hydrogen molecule must indeed be considered, together with the conception of the spatial model of the carbon atom by Van t Hoff and Le Bel, as the most important contribution to theoretical chemistry, since the advent of Dalton s atomic hypothesis. We shall, however, let the treatment of the hydrogen molecule itself be preceded by the discussion of the hydrogen molecule ion H2+, since this problem with only one electron is still simpler than that of the H2 molecule itself. 

The VB model given by Heitler and London was the first description of covalent bonding that followed the formulation of quantum mechanics. If < (1) and (j>b(2) are the one-electron wavefunctions or orbitals localized on neighboring atoms labeled a and b (e.g., the Is-orbitals on the two hydrogen atoms labeled a and b in molecular hydrogen) and each is occupied by one electron (the numbers 1 and 2 represent the coordinates of the two electrons and stand for x, z, X2, 2 ... 

A few months earlier, Heitler and London had published their calculation for the hydrogen molecule. This was too complicated for an exact solution, and their method also rested on a perturbation model, a combination of atomic wave functions in which the two electrons, with... 

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