Treatment of the Hydrogen Molecule

Condon s treatment is the prototype of the molecular-orbital method of discussing the electronic structure of molecules. In this method a wave function is formulated that involves the introduction of a pair of electrons in an electron orbital that extends about two or more atomic nuclei. 

Heitler and F, London, Z. Pkysik 44, 455 (1927). A mathematical improvement of this work was made by Y. Sugiura, ibid. 45, 484 (192 ). 

The second method of discussing the electronic structure of molecules, usually called the valence-bond method, involves the use of a wave function of such a nature that the two electrons of the electron-pair bond between, two atoms tend to remain on the two different atoms. The prototype of this method is the Heitler-London treatment of the hydrogen a olecule, which we shall now discuss. 

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One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

Coulson, C. A., and Fischer, I., Phil. Mag. 40, 386, Notes on the molecular orbital treatment of the hydrogen molecule."... 

The original VB wave function was introduced in the treatment of the hydrogen molecule by Heitler and London in 1932. This treatment considered only the one Is orbital on each hydrogen atom and assumed that the best wave function for a system of two electrons on two different atoms is a product of the two atomic Is orbitals i/ — XisXis- This wave function needs to be modified, however, to accommodate the antisymmetry of the wave function and to take into account the spin of the two electrons. 

The above perturbation treatment of the hydrogen molecule-ion has not before been published. 

A treatment of the hydrogen molecule by the Ritz method, applied to helium by Kellner (25), has been reported by S. C. Wang (Phys. Rev., 31, 579 (1928)). With this method the individual eigenfunctions p and

hydrogen-like eigenfunctions of an atom with atomic number Z differing from unity. The value found for Z is 1.166, and the... 

Two sets of papers in 1927 now provided successful treatments of the hydrogen molecule. One was written by Heitler and London, in residence at Schrodinger s institute in Zurich. The other was by Hund, Bom s assistant in... 

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"... 

The Heitler-London Treatment of the Hydrogen Molecule.—The hydrogen molecule consists of two nuclei, which may be designated A... 

Theory treatment of the hydrogen molecule. Their method has come to be known as the... 

Notes on the Molecular Orbital Treatment of the Hydrogen Molecule. 

Table 3.2.1 summarizes the results of various approximate wavefunctions for the hydrogen molecule. This list is by no means complete, but it does show that, as the level of sophistication of the trial function increases, the calculated dissociation energy and bond distance approach closer to the experimental values. In 1968, W. Kolos and L. Wolniewicz used a 100-term function to obtain results essentially identical to the experimental data. So the variational treatment of the hydrogen molecule is now a closed topic. 

It was Pauling and his collaborators who generalized the Heitler-London treatment of the hydrogen molecule to general polyelectronic systems and gave the rules for evaluating the Hamiltonian matrix elements [3,4], which could also... 

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. 

In the above treatment of the hydrogen molecule ion and the hydrogen molecule, the effect of electronic spin has been excluded, but the complete wave function of an electron must include not only the orbital motion, with which we have been concerned so far, but also a contribution for the spin. With single atoms it was possible to introduce a fourth quantum number s in addition to the three quantum numbers , I and m in order to account for the spin of the electron, and for polyelectronic molecules it is possible to proceed in an analogous manner. The complete wave function of an electron is considered to be the product of the orbital wave function, i,e. the wave function that we have been considering so far, and a wave function representing the orientation of the spin axis of the electron. 

The valence bond (VB) theory grew directly out of the ideas of electron pairing by Lewis and others. In 1927 W. Heiller and F. London proposed a quantum-mechanical treatment of the hydrogen molecule. Their method has come to be known as the valence bond approach and was developed extensively by men such as Linus Pauling... 

The hydrogen molecule H2 is the simplest molecule which forms an electron-pair bond. Many calculations have been made for this molecule, which is a prototype for many other chemical bonds. One of the two basic quantum-mechanical treatments of the hydrogen molecule involves constructing a molecular orbital for the bond from a linear combination of atomic orbitals (LCAO method). The other involves constructing the molecular orbital as the product of wave functions for each of the two electrons forming the bond. Both of these methods will be outlined. 

Sixty years ago, in 1949, Coulson and Fischer published a seminal paper [18] in the Philosophical Magazine, entitled Notes on the Molecular Orbital Treatment of the Hydrogen Molecule. In this note, they presented a wave function for the hydrogen molecule, which, whilst retaining a simple physical picture, combines the advantages of the two rival theories of molecular electronic structure, mo and VB theories. Let us briefly summarize the discussion given by Coulson and Fischer. 

The first quantum-mechanical treatment of the hydrogen molecule was by Heitler and London in 1927. Their ideas have been extended to give a general theory of chemical bonding, known as the valence-bond (VB) theory. The valence-bond method is more closely related to the chemist s idea of molecules as consisting of atoms held together by localized bonds than is the molecular-orbital method. The VB method views molecules as composed of atomic cores (nuclei plus inner-shell electrons) and bonding valence electrons. For H2, both electrons are valence electrons. 

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