# General Considerations for Many Electron Systems

## Abstract

Thus far our considerations have in general been limited to systems containing only a single electron. While we have seen that many important principles and techniques can be developed using those cases, we shall now see that a major new concept is needed for systems containing more than one electron. That concept (the Pauli Exclusion Principle) will be developed in the sections to follow, along with a number of analyses and techniques that are of substantial importance in contemporary applications of quantum mechanics to chemistry. Before doing that, however, it is useful to introduce several conceptual approaches to many electron systems that were developed early, as well as the basic concepts of group theory. These will help us to understand and motivate the discussions to follow, as well as to provide useful tools for incorporating and understanding symmetry properties of molecules and wavefunctions.

## Keywords

Irreducible Representation General Consideration Symmetry Operation Virial Theorem Character Table## Preview

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## References

- 1.
- 4.
- 5.
- 6.
- 7.R. S. Mulliken,
*Phys. Rev*., 32, 186, 388, 761(1928);Google Scholar - 9.For additional discussion see, for example, M. Tinkham, Group Theory and Quantum
*Mechanics*, McGraw-Hill, New York, 1964; L. H. Hall,*Group Theory and Symmetry in Chemistry*, McGraw-Hill, New York, 1969; and G. G. Hall,*Applied Group Theory*, Longmans, Green and Co., London, 1967.Google Scholar - 20.The symbols used here to designate group names are known as the
*Schoenflies notation*. Other systems of nomenclature are also available, e. g., the Hermann-Mauguin nomenclature used for crystallographic point groups. For a detailed procedure to determine the point group for any molecule see, for example, L. H. Hall,*Group Theory and Symmetry in Chemistry*, McGraw Hill, New York, 1969, Chapter 4; see also F. A. Cotton,*Chemical Applications of Group Theory*, John Wiley, New York, 1963, Chapter 3, Sect. 10.Google Scholar - 21.R. E. Christoffersen, S. Hagstrom, and F. P. Prosser,
*J. Chem. Phys*., 40, 236 (1964).CrossRefGoogle Scholar - 25.These theorems will not be proved here, but the proofs are available in many group theory texts. See, for example, E. P. Wigner,
*Group Theory*, Academic Press, New York, 1959, Chapter 9.Google Scholar - 46.W. Heisenberg,
*Z. Phys*., 38, 411–426 (1926).CrossRefGoogle Scholar - 47.
- 58.P. O. Löwdin,
*Adv. Chem. Phys*., 2, 207–322 (1959).CrossRefGoogle Scholar - 60.P. O. Löwdin,
*Phys. Rev*., 97, 1474–1490 (1955).CrossRefGoogle Scholar - 66.J. O. Hirschfelder,
*J. Chem. Phys*., 33, 1462 (1960)CrossRefGoogle Scholar - 68.V. Fock,
*Z. Phys*., 63, 855 (1930)CrossRefGoogle Scholar - J. C. Slater,
*J. Chem. Phys*., 1, 687 (1933).CrossRefGoogle Scholar - 70.See J. I. Musher,
*Am. J. Phys*., 34, 267 (1966).CrossRefGoogle Scholar - 72.The analysis used in this section was given first by E. Hylleraas,
*Z. Phys*., 54, 347 (1929) and V. Fock,*Z. Phys*., 63, 855 (1930), and the form used here was outlined by P. O. Löwdin,*Adv. Chem. Phys*., 2, 207 (1959).Google Scholar - 78.
- R. F. W. Bader, P. M. Beddall, and J. J. Peslak,
*J. Chem. Phys*.,**58**, 557 (1973);CrossRefGoogle Scholar - R. F. W. Bader, A. J. Duke, and R. R. Messer,
*J. Amer. Chem. Soc*.,**95**, 7715 (1973);CrossRefGoogle Scholar - 83.Such an approach is frequently referred to as the “vector model” of the atom. For a detailed discussion see, for example, A. R. Edmonds,
*Angular Momentum in Quantum Mechanics*, Princeton University Press, Princeton, 1960.Google Scholar - 84.For a comprehensive discussion and listing of eigenfunctions of and SZ using Slater determinants, see J. C. Slater,
*Quantum Theory of Atomic Structure*, Volume 2, McGraw-Hill, New York, 1960, Chapter 21.Google Scholar