Laplace expansion (potential)

In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.

The expansion

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

${\frac {1}{|\mathbf {r} -\mathbf {r} '|}}=\sum _{\ell =0}^{\infty }{\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }(-1)^{m}{\frac {r_{\scriptscriptstyle <}^{\ell }}{r_{\scriptscriptstyle >}^{\ell +1}}}Y_{\ell }^{-m}(\theta ,\varphi )Y_{\ell }^{m}(\theta ',\varphi ').$

Here r has the spherical polar coordinates (r, θ, φ) and r' has ( r', θ', φ'). Further r_< is min(r, r') and r_> is max(r, r'). The function $Y_{\ell }^{m}$ is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics,

${\frac {1}{|\mathbf {r} -\mathbf {r} '|}}=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {r} )R_{\ell }^{m}(\mathbf {r} ')\quad {\hbox{with}}\quad |\mathbf {r} |>|\mathbf {r} '|,$

where $R_{\ell }^{m}$ is a regular solid harmonic:

$R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),$

and $I_{\ell }^{m}$ is an irregular solid harmonic:

$I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}.$

Derivation

The derivation of this expansion is simple. One writes

${\frac {1}{|\mathbf {r} -\mathbf {r} '|}}={\frac {1}{\sqrt {r^{2}+(r')^{2}-2rr'\cos \gamma }}}={\frac {1}{r_{\scriptscriptstyle >}{\sqrt {1+h^{2}-2h\cos \gamma }}}}\quad {\hbox{with}}\quad h\equiv {\frac {r_{\scriptscriptstyle <}}{r_{\scriptscriptstyle >}}}.$

We find here the generating function of the Legendre polynomials $P_{\ell }(\cos \gamma )$ :

${\frac {1}{\sqrt {1+h^{2}-2h\cos \gamma }}}=\sum _{\ell =0}^{\infty }h^{\ell }P_{\ell }(\cos \gamma ).$

Use of the spherical harmonic addition theorem

$P_{\ell }(\cos \gamma )={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell }^{-m}(\theta ,\varphi )Y_{\ell }^{m}(\theta ',\varphi ')$

gives the desired result.