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In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance (), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the inter-electronic repulsion.

The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors and , then the Laplace expansion is

Here has the spherical polar coordinates and has with homogeneous polynomials of degree . Further r< is min(r, r′) and r> is max(r, r′). The function is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics,


The derivation of this expansion is simple. By the law of cosines,

We find here the generating function of the Legendre polynomials :
Use of the spherical harmonic addition theorem
gives the desired result.

Neumann expansion

A similar equation has been derived by Carl Gottfried Neumann[1] that allows expression of in prolate spheroidal coordinates as a series:

where and are associated Legendre functions of the first and second kind, respectively, defined such that they are real for . In analogy to the spherical coordinate case above, the relative sizes of the radial coordinates are important, as and .


  1. ^ Rüdenberg, Klaus (1951). "A Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. II. The Two‐Center Exchange Integrals". The Journal of Chemical Physics. 19 (12). AIP Publishing: 1459–1477. Bibcode:1951JChPh..19.1459R. doi:10.1063/1.1748101. ISSN 0021-9606.
  • Griffiths, David J (1981). Introduction to Electrodynamics. Englewood Cliffs, N.J.: Prentice-Hall.