Spherical Multipole Moments - Interaction Energies of Spherical Multipoles

Interaction Energies of Spherical Multipoles

A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. Let the first charge distribution be centered on the origin and lie entirely within the second charge distribution . The interaction energy between any two static charge distributions is defined by

U \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r} \rho_{2}(\mathbf{r}) \Phi_{1}(\mathbf{r})

The potential of the first (central) charge distribution may be expanded in exterior multipoles

\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} \left( \frac{1}{r^{l+1}} \right) \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)

where represents the exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula

U = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} \int d\mathbf{r} \ \rho_{2}(\mathbf{r}) \left( \frac{1}{r^{l+1}} \right) \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)

Since the integral equals the complex conjugate of the interior multipole moments of the second (peripheral) charge distribution, the energy formula reduces to the simple form

U = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} Q_{1lm} I_{2lm}^{*}

For example, this formula may be used to determine the electrostatic interaction energies of the atomic nucleus with its surrounding electronic orbitals. Conversely, given the interaction energies and the interior multipole moments of the electronic orbitals, one may find the exterior multipole moments (and, hence, shape) of the atomic nucleus.

Read more about this topic:  Spherical Multipole Moments

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