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On the expansion of a Coulomb potential in spherical harmonics

Published online by Cambridge University Press:  24 October 2008

B. C. Carlson  and
G. S. Rushbrooke
B. C. Carlson
Affiliation:
Clarendon LaboratoryOxford
G. S. Rushbrooke
Affiliation:
Clarendon LaboratoryOxford
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Extract

The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,

or, in an alternative notation,

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 4 , October 1950 , pp. 626 - 633
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

† Eyring, H., Walter, J. and Kimball, G. E., Quantum chemistry (New York, 1944), p. 371.Google Scholar

‡ Hobson, E. W., The theory of spherical and ellipsoidal harmonics (Cambridge, 1931), pp. 90, 99.Google Scholar

§ Our is the same for all m as defined by Condon, E. U. and Shortley, G. H., The theory of atomic spectra (Cambridge, 1935), p. 52.Google Scholar

† Terms up to an including R −6 are correctly given in Cartesian coordinates by Heller, R., J. Chem. Phys. 9 (1941), 156.CrossRefGoogle Scholar Elsewhere the quadrupole term has often been written with one or more errors of sign.

† Wigner, E., Gruppentheorie (Braunschweig, 1931), p. 206.Google Scholar Also Condon and Shortley, loc. cit. p. 75.

‡ Hobson, E. W., loc. cit. §§ 83, 85, 88, 101 and 103.

† Gatint, J. A., Philos. Trans. A, 228 (1929), 151.Google Scholar Gaunt's formula is written in notation very similar to ours by Condon and Shortley, loc. cit. p. 176. Our l 1, m 1, l 2, m 2, L are to be identified with their l, m, l′, m′, l″, and the integral on the left side of (16) is 1/√(2π) times the integral on the left side of their equation (11).