1894-1954

The following text originates from:

G.G. Hall,

The Lennard-Jones paper of 1929 and the foundations of

molecular orbital theory.

Adv. Quant. Chem. 1991,

Molecular Orbital Theory

Shell Centre for Mathematical Education,

University of Nottingham,

Nottingham NG7 2RD

University of Nottingham,

Nottingham NG7 2RD

ABSTRACT

1. Introduction

In 1929 a Discussion of the Faraday Society was held in Bristol. lt was organized by W.E. Garner and J.E. Lennard-Jones, both local Professors. The subject was Molecular Spectra and Molecular Structure and it brought together many of those most active in these fields at that time including F. Hund, R.S. Mulliken, C.V. Raman, V. Henri and G. Herzberg. To this specialist audience Lennard-Jones presented his first paper on the Theory of Molecular Structure (1). To appreciate

The Discussion itself brought together both theorists and experimentalists and, following a trenchant paper by O.W. Richardson, much of the discussion centred on problems of notation and usage of symbols for various quantum numbers. Several of our presently accepted conventions were fixed during that meeting. As is typical of that time, papers on various kinds of spectra predominated and most of these concerned the interpretation of experimental spectra. In these circumstances the Lennard-Jones paper stood out as purely quantum mechanical and highly original.

2. Background

In 1926 Schrödinger published his epoch-making papers (2) on wavemechanics. Then Heisenberg applied perturbation theory (3) to the two-electron problem and showed how "resonance" arising from electron exchange could explain "exchange forces". Heitler and London (4) used this idea to explain the covalent bond as exemplified in the hydrogen molecule. From this beginning the electron-pair, or resonance, treatment of molecular structure developed rapidly. lt took over so many of the earlier ideas of the Lewis model of chemical bonding that it was easily accepted by many chemists and became the preferred mode of explanation.

At the same time Hund began to formulate, as an alternative, the Molecular Orbital (MO) theory (5). His objective was to determine the quantum numbers of all the electrons in a molecule and specify its state by vector coupling them, as had proved so successful for atoms. He postulated that each electron should have four quantum numbers which remained invariant with inter-nuclear distance R. Since the electronic structure of atoms could be assumed known, he considered the two limiting cases when R was infinite, giving separated atoms, or zero, giving the united atom. He drew correlation diagrams connecting the electron levels as a function of R and judged that levels going to unoccupied levels of the united atom would be unfavourable to bonding and those remaining occupied for all R would be favourable. With these ideas, and considerable experimental information from spectra of various kinds, he could assign the electron configurations of some diatomics. It emerged from his work that the conservation of the number of nodes in each orbital as R changed was very significant.

A very similar stand-point was taken by Mulliken (6) who adapted Hund's assignments and made more use of atomic data. His extensive spectroscopic experience enabled him to suggest electronic structures for many diatomics. In particular Mulliken emphasised the importance of the group theory classifications and his distinction between sigma, pi and delta bonds remains vital to us.

3. The Lennard-Jones paper

The starting-point of Lennard-Jones paper is his uncompromising insistence on the auf—bau principle. Just as Bohr built up atoms by adding electrons one by one, so molecules should be built up in the same way. (Mulliken had considered this possibility but thought it impractical.) He then considered what the Schrödinger equation for a typical electron in a diatomic would be. Since he was concerned only with homonuclear diatomics the differential equation can be written out in full but with unknown potentials expressing the effect of all the other electrons on the selected one. By treating these as perturbations he could start with the hydrogen molecular ion equation. Already an approximate solution for this was known using + and - combinations of the separated atomic orbitals. Thus his first approximation to the molecular orbital is a linear combination of H-like atomic orbitals suitably normalized using the overlap integral. The screening constants arc adjusted to allow for the other electrons. From bis evaluation of the integrals he then gave a correlation diagram having some quantitative significance and showing how different levels could both rise and fall with R.

For the diatomics with few electrons, this approach was quite successful and confirmed many of the assignments already made by Hund and Mulliken. For larger molecules Lennard-Jones introduced another idea. He accepted that atoms with the inert gas structure of completed outer shells could not show bonding. Except for the London force, they must repel one another. By starting from pairs of such atoms and removing electrons one by one he could arrive at the structure of these larger systems. Thus two Ne atoms repelled but the removal of two electrons and the reduction of the nuclear charges gave fluorine molecule with an "inverted H system" in the perturbation to the orbital equation. Thus the single bond of fluorine is due to the removal of two anti-bonding electrons. This gave a fast and easy explanation both for fluorine and oxygen molecules. His argument for the

In effect, this Lennard-Jones approach implied that the inner electrons in these molecules remained in atomic orbitals and only the valence electrons needed to be in MO involving both nuclei. This distinction was earlier made by Hund whereas Mulliken thought all electrons should be MO. Mulliken (7) eventually agreed that, while, strictly speaking, all should be in MO, it might be accurate enough to put the inner ones in AO.

As a final contribution, the paper also argued that the notation for the MO should begin with the principal quantum nurnber of the atomic orbital of the separated atom instead of using that of the united atom, as was the custom at that time.

4. The follow-up to the paper

Lennard-Jones himself did not rush to build on his beginning. He was involved in moving to Cambridge. By the time of the next Faraday Discussion on the subject, in 1933, the molecular orbital theory had become much more accepted as a valid and useful theory. In that discussion he presented a paper (8) on hydrocarbon free radicals, including CH

Perhaps this contact with Hückel prompted his next molecular paper (10) on the treatment of conjugated hydrocarbons. In this paper he introduced the concepts of compression energy of the sigma bonds and the variation of beta with distance, which opened up the whole subject of variable CC distances in conjugated hydrocarbons. This paper is still quoted as the formative paper in the treatment of polyacetylene since it allowed for the persistent alternation of double and single bond character despite the length of the molecule.

The first accurate calculation of a molecular orbital wavefunction was the Coulson (11) calculation on the hydrogen molecule. Although the method was then extended to the lithium molecule (12) it relied on the high symmetry of the homonuclear diatomic and did not generalize.

Although Lennard-Jones gave partial explanations much earlier, the final resolution of the problem of jusfifying the use of atomic orbitals for the inner electrons and diatomic orbitals only for the valence electrons was not achieved until twenty years later when he showed (13) that the determinantal wavefunction had unitary transformations which left it invariant but could be used to transform the molecular orbitals into localized equivalent orbitals without loss of accuracy. In these two papers he gave a rigorous derivation of the orbital equations from the Scbrödinger equation using a method he had introduced (14) for the atomic SCF equations. It was not until the third paper in this series (15) that the MOs were completely defined as eigenfunctions of the SCF Hamiltonian. At this point the MO theory became fully rigorous and consistent.

5. Conclusion

By any standards Lennard-Jones must be considered an outstanding scientist. He worked on many problems which continue to interest us and contributed his share of good ideas in eacb case. Perhaps bis work on intermolecular forces is best known today though his theories of liquids and of surface catalysis are still quoted and influence our thinking. His molecular structure work is represented in a relatively small number of papers but shows him trying to understand the fundamental aspects of the subject and injecting ideas which we still find useful.

Lennard-Jones has an important place in the history of theorefical studies in this country. N.F.Mott said (16) of him "If in his Bristol appointment he was the first man in this country to hold a Chair of Theoretical Pbysics within a Physics Department, certainly he was the first man to hold a chair of Theoretical Chemistry anywhere in the world." He set the style of the subject here and tried very hard, with some success, to explain it to experimental chemists in a very simple way.

2. Schrödinger, E. (1926) Ann.d.Phys

3. Heisenberg, W. (1926) Z.Phys.

4. Heitler, W. and London, F. (1927) Z.Phys.

5. Hund, F. (1926) Z.Phys.

6. Mulliken, R.S. (1927) Phys.Rev.

7. Mulliken, R.S. and Rieke, C.A. (1941) Rep.Progress Phys.

8. Lennard-Jones, J.E. (1934) Trans.Faraday Soc.

9. Hückel, E. (1934) Trans.Faraday Soc.

10. Lennard-Jones, J.E. (1937) Proc.Roy.Soc.

11. Coulson, C.A. (1938) Proc.Camb.Phil.Soc.

12. Coulson, C.A. and Duncanson, W.E. (1943)Proc.Roy.Soc.

13. Lennard-Jones, Sir John (1949) Proc.Roy.Soc.

14. Lennard-Jones, J.E. (1931) Proc.Camb.Phil.Soc.

15. Hall, G.G. and Lennard-Jones, Sir John (1950)Proc.Roy.Soc.

16. Mott, N.F. (1955) Biograph.Mem.Roy.Soc.

http://www.quantum-chemistry-history.com

Feb. 28, 2002 by U. Anders, Ph.D.

e-mail Udo Anders : udo39@t-online.de

Feb. 28, 2002 by U. Anders, Ph.D.

e-mail Udo Anders : udo39@t-online.de

Last updated : June 20, 2003 - 19:19 MEST