Spherical multipole moments

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In physics, spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as ⁠${\displaystyle {\tfrac {1}{R}}.}$⁠ Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.

For clarity, we illustrate the expansion for a point charge,^[1] then generalize to an arbitrary charge density ${\displaystyle \rho (\mathbf {r} ').}$ Through this article, the primed coordinates such as ${\displaystyle \mathbf {r} '}$ refer to the position of charge(s), whereas the unprimed coordinates such as ${\displaystyle \mathbf {r} }$ refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector ${\displaystyle \mathbf {r} '}$ has coordinates ${\displaystyle (r',\theta ',\phi ')}$ where ${\displaystyle r'}$ is the radius, ${\displaystyle \theta '}$ is the colatitude and ${\displaystyle \phi '}$ is the azimuthal angle.

The electric potential due to a point charge located at ${\displaystyle \mathbf {r'} }$ is given by $${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}{\frac {1}{R}}={\frac {q}{4\pi \varepsilon }}{\frac {1}{\sqrt {r^{2}+r^{\prime 2}-2r'r\cos \gamma }}}.}$$ where ${\displaystyle R\ {\stackrel {\mathrm {def} }{=}}\ \left|\mathbf {r} -\mathbf {r'} \right|}$ is the distance between the charge position and the observation point and ${\displaystyle \gamma }$ is the angle between the vectors ${\displaystyle \mathbf {r} }$ and ${\displaystyle \mathbf {r'} }$. If the radius ${\displaystyle r}$ of the observation point is greater than the radius ${\displaystyle r'}$ of the charge, we may factor out 1/r and expand the square root in powers of ${\displaystyle (r'/r)<1}$ using Legendre polynomials $${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{\ell =0}^{\infty }\left({\frac {r'}{r}}\right)^{\ell }P_{\ell }(\cos \gamma )}$$ This is exactly analogous to the axial multipole expansion.

We may express ${\displaystyle \cos \gamma }$ in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2) $${\displaystyle \cos \gamma =\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\phi -\phi ')}$$

Substituting this equation for ${\displaystyle \cos \gamma }$ into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem $${\displaystyle P_{\ell }(\cos \gamma )={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$$ where the ${\displaystyle Y_{\ell m}}$ functions are the spherical harmonics. Substitution of this formula into the potential yields $${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{\ell =0}^{\infty }\left({\frac {r'}{r}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$$

which can be written as $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {Q_{\ell m}}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$$ where the multipole moments are defined $${\displaystyle Q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ q\left(r'\right)^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').}$$

As with axial multipole moments, we may also consider the case when the radius ${\displaystyle r}$ of the observation point is less than the radius ${\displaystyle r'}$ of the charge. In that case, we may write $${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r'}}\sum _{\ell =0}^{\infty }\left({\frac {r}{r'}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$$ which can be written as $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }I_{\ell m}r^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$$ where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics $${\displaystyle I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {q}{\left(r'\right)^{\ell +1}}}{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ')}$$

The two cases can be subsumed in a single expression if ${\displaystyle r_{<}}$ and ${\displaystyle r_{>}}$ are defined to be the lesser and greater, respectively, of the two radii ${\displaystyle r}$ and ${\displaystyle r'}$; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion $${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }{\frac {r_{<}^{\ell }}{r_{>}^{\ell +1}}}\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$$

It is straightforward to generalize these formulae by replacing the point charge ${\displaystyle q}$ with an infinitesimal charge element ${\displaystyle \rho (\mathbf {r} ')d\mathbf {r} '}$ and integrating. The functional form of the expansion is the same. In the exterior case, where ${\displaystyle r>r'}$, the result is: $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {Q_{\ell m}}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )\,,}$$ where the general multipole moments are defined $${\displaystyle Q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '\rho (\mathbf {r} ')\left(r'\right)^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').}$$

The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Y_ℓm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.

Similarly, the interior multipole expansion has the same functional form. In the interior case, where ${\displaystyle r'>r}$, the result is: $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }I_{\ell m}r^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi ),}$$ with the interior multipole moments defined as $${\displaystyle I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '{\frac {\rho (\mathbf {r} ')}{\left(r'\right)^{\ell +1}}}{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').}$$

A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. Let the first charge distribution ${\displaystyle \rho _{1}(\mathbf {r} ')}$ be centered on the origin and lie entirely within the second charge distribution ${\displaystyle \rho _{2}(\mathbf {r} ')}$. The interaction energy between any two static charge distributions is defined by $${\displaystyle U\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} \rho _{2}(\mathbf {r} )\Phi _{1}(\mathbf {r} ).}$$

The potential ${\displaystyle \Phi _{1}(\mathbf {r} )}$ of the first (central) charge distribution may be expanded in exterior multipoles $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}\left({\frac {1}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$$ where ${\displaystyle Q_{1\ell m}}$ represents the ${\displaystyle \ell m}$ exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula $${\displaystyle U={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}\int d\mathbf {r} \ \rho _{2}(\mathbf {r} )\left({\frac {1}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$$

Since the integral equals the complex conjugate of the interior multipole moments ${\displaystyle I_{2\ell m}}$ of the second (peripheral) charge distribution, the energy formula reduces to the simple form $${\displaystyle U={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}I_{2\ell m}^{*}}$$

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.

The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle ${\displaystyle \phi '}$). By carrying out the ${\displaystyle \phi '}$ integrations that define ${\displaystyle Q_{\ell m}}$ and ${\displaystyle I_{\ell m}}$, it can be shown the multipole moments are all zero except when ${\displaystyle m=0}$. Using the mathematical identity $${\displaystyle P_{\ell }(\cos \theta )\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell 0}(\theta ,\phi )}$$ the exterior multipole expansion becomes $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\left({\frac {Q_{\ell }}{r^{\ell +1}}}\right)P_{\ell }(\cos \theta )}$$ where the axially symmetric multipole moments are defined $${\displaystyle Q_{\ell }\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '\rho (\mathbf {r} ')\left(r'\right)^{\ell }P_{\ell }(\cos \theta ')}$$ In the limit that the charge is confined to the ${\displaystyle z}$-axis, we recover the exterior axial multipole moments.

Similarly the interior multipole expansion becomes $${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }I_{\ell }r^{\ell }P_{\ell }(\cos \theta )}$$ where the axially symmetric interior multipole moments are defined $${\displaystyle I_{\ell }\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '{\frac {\rho (\mathbf {r} ')}{\left(r'\right)^{\ell +1}}}P_{\ell }(\cos \theta ')}$$ In the limit that the charge is confined to the ${\displaystyle z}$-axis, we recover the interior axial multipole moments.

• Solid harmonics
• Laplace expansion
• Multipole expansion
• Legendre polynomials
• Axial multipole moments
• Cylindrical multipole moments