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Wave Functions of Many-Electron Atoms

Published online by Cambridge University Press:  24 October 2008

J. E. Lennard-Jones
J. E. Lennard-Jones
Affiliation:
University of Bristol
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Extract

1. The wave function of an atom containing many electrons has not yet been solved completely, even that of helium being as yet unknown. In the absence of a direct solution of the Schrödinger equation for the electrons in an atom, various attempts have been made to devise approximate methods of solution in particular cases. The particular case of helium, being the easiest, has received considerable attention and a number of approximate wave functions appropriate to the normal state have been constructed. These functions usually contain empirical constants which are adjusted to make the energy of the system a minimum. Zener has attempted the more ambitious programme of finding the wave functions of all the atoms in the first period of the Periodic Table (Lithium to Neon), and has made interesting discoveries as to the way in which the wave functions differ from atom to atom. This work also is based on the variation of parameters.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 3 , July 1931 , pp. 469 - 480
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Hylleraas, , Zeits. f. Physik, Vol. 54, p. 347 (1929)CrossRefGoogle Scholar; Slater, , Physical Review, Vol. 32, p. 349 (1928)CrossRefGoogle Scholar.

† Zener, , Physical Review, Vol. 36, p. 51 (1930)CrossRefGoogle Scholar; cf. also Slater, , Physical Review, Vol. 36, p. 57 (1930)CrossRefGoogle Scholar.

‡ Hartree, , Proc. Camb. Phil. Soc., Vol. 24, pp. 89, 111 (1928)CrossRefGoogle Scholar.

§ Fock, , Zeits. f. Physik, Vol. 61, p. 126 (1930).CrossRefGoogle Scholar

∥ Dirac, , Proc. Camb. Phil. Soc., Vol. 26, p. 376 (1930)CrossRefGoogle Scholar.

† Unit of length equals the radius of the hydrogen Bohr orbit, and unit of energy equals twice that of normal hydrogen.

* Dirac, , Proc. Roy. Soc., Vol. 112, p. 661 (1926)CrossRefGoogle Scholar.

† Slater, , Phys. Review, Vol. 34, p. 1293 (1929)CrossRefGoogle Scholar.

‡ Slater, loc. cit.

* Fock, loc. cit.,

* Fock, loc. cit., § 5.

† Slater, loc. cit.

‡ Hund, , Linienspektren u. periodisches System der Elemente, Springer (1927)CrossRefGoogle Scholar.

* In a paper which has just appeared (Proc. Camb. Phil. Soc. Vol. 27, p. 240 1931Google Scholar) Dirac gives a simple interpretation of the function ρ. The expression ρdτ 1 … dτ N is the probability that an electron shall be found in dτ 1, another in dτ 2, … and so on; in other words, it gives the probability of a specified configuration. The probability of finding N − 1 electrons in a prescribed configuration irrespective of the position of the Nth may therefore be obtained by integrating ρ over the variables of the Nth electron, and is therefore ρN−1. The probability of finding q of the electrons in prescribed elements of volume may be obtained by integrating over the variables dτ q+1 … dτ N in such a way that each configuration appears only once. Thus the configuration q + 1 = (q + 1)′, q + 2 = (q + 2)′, … N = N′, and any permutation of (q + 1)′, (q + 2)′, … N′ among the variables q + 1, q + 2, … N must be counted as the same. The result is just ρq.

† Fock, loc. cit., equations (83) and (84).