Multipole expansion (interaction)

In physics, two static, non-polarizable, electric charge distributions interact via Coulomb's law. When the charge distributions do not overlap the interaction can expanded in a convergent series of powers of 1/R. Here R is the distance between the charge distributions, for instance, the distance between the centers of charge of the two distributions. Because the factors of 1/R in this expansion are products of multipoles of the two charge distributions, this series is known as the multipole expansion of the interaction energy.

Mathematical formulation

A most important application of the multipole series is in the field of intermolecular forces, and because molecules are distributions consisting of discrete charges (electrons and nuclei), we consider the interaction between two discrete charge contributions. So, let us introduce two sets of point charges, one set {q _i } clustered around a point A and one set {q _j } clustered around a point B. By Coulomb's law the total electrostatic interaction energy U_AB between the two distributions is

$U_{AB}=\sum _{i\in A}\sum _{j\in B}{\frac {q_{i}q_{j}}{4\pi \varepsilon _{0}r_{ij}}},$

where ε₀ is the permittivity of the vacuum and r _ij is the distance between particle i and particle j. This energy can be expanded in a power series— the multipole expansion—in the inverse distance of A and B.

In order to derive this multipole expansion, we write r_XY ≡ r_Y − r_X, which is a vector pointing from X towards Y. Note that

$\mathbf {R} _{AB}+\mathbf {r} _{Bj}+\mathbf {r} _{ji}+\mathbf {r} _{iA}=0\quad \Longrightarrow \quad \mathbf {r} _{ij}=\mathbf {R} _{AB}-\mathbf {r} _{Ai}+\mathbf {r} _{Bj}.$

We assume that the two distributions do not overlap:

$|\mathbf {R} _{AB}|>|\mathbf {r} _{Bj}-\mathbf {r} _{Ai}|\quad {\hbox{for all}}\quad i,j.$

Under this condition we may apply the Laplace expansion in the following form

${\frac {1}{|\mathbf {r} _{j}-\mathbf {r} _{i}|}}={\frac {1}{|\mathbf {R} _{AB}-(\mathbf {r} _{Ai}-\mathbf {r} _{Bj})|}}=\sum _{L=0}^{\infty }\sum _{M=-L}^{L}\,(-1)^{M}I_{L}^{-M}(\mathbf {R} _{AB})\;R_{L}^{M}(\mathbf {r} _{Ai}-\mathbf {r} _{Bj}),$

where $I_{L}^{M}$ and $R_{L}^{M}$ are irregular and regular solid harmonics, respectively. The translation of the regular solid harmonic gives a finite expansion,

$R_{L}^{M}(\mathbf {r} _{Ai}-\mathbf {r} _{Bj})=\sum _{\ell _{A}=0}^{L}(-1)^{L-\ell _{A}}{\binom {2L}{2\ell _{A}}}^{1/2}$

$\times \sum _{m_{A}=-\ell _{A}}^{\ell _{A}}R_{\ell _{A}}^{m_{A}}(\mathbf {r} _{Ai})R_{L-\ell _{A}}^{M-m_{A}}(\mathbf {r} _{Bj})\;\langle \ell _{A},m_{A};L-\ell _{A},M-m_{A}|LM\rangle ,$

where the quantity between pointed brackets is a Clebsch-Gordan coefficient. Further we used

$R_{\ell }^{m}(-\mathbf {r} )=(-1)^{\ell }R_{\ell }^{m}(\mathbf {r} ).$

Use of the definition of spherical multipoles Q^m_l and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of L) gives finally

$U_{AB}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell _{A}=0}^{\infty }\sum _{\ell _{B}=0}^{\infty }(-1)^{\ell _{B}}{\binom {2\ell _{A}+2\ell _{B}}{2\ell _{A}}}^{1/2}\,$

$\times \sum _{m_{A}=-\ell _{A}}^{\ell _{A}}\sum _{m_{B}=-\ell _{B}}^{\ell _{B}}(-1)^{m_{A}+m_{B}}I_{\ell _{A}+\ell _{B}}^{-m_{A}-m_{B}}(\mathbf {R} _{AB})\;\left[\mathbf {Q} ^{\ell _{A}}\otimes \mathbf {Q} ^{\ell _{B}}\right]_{m_{A}+m_{B}}^{\ell _{A}+\ell _{B}}.$

This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance R_AB apart. Here the notation

$\left[\mathbf {Q} ^{\ell _{A}}\otimes \mathbf {Q} ^{\ell _{B}}\right]_{M}^{L}\equiv \sum _{m_{A}=-\ell _{A}}^{\ell _{A}}\sum _{m_{B}=-\ell _{B}}^{\ell _{B}}Q_{\ell _{A}}^{m_{A}}Q_{\ell _{B}}^{m_{B}}\;\langle \ell _{A},m_{A};\ell _{B},m_{B}|L,M\rangle ,$

is used for the Clebsch-Gordan series. Since

$I_{\ell _{A}+\ell _{B}}^{-(m_{A}+m_{B})}(\mathbf {R} _{AB})\equiv \left[{\frac {4\pi }{2\ell _{A}+2\ell _{B}+1}}\right]^{1/2}\;{\frac {Y_{\ell _{A}+\ell _{B}}^{-(m_{A}+m_{B})}({\widehat {\mathbf {R} }}_{AB})}{R_{AB}^{\ell _{A}+\ell _{B}+1}}}$

this expansion is manifestly in powers of 1/R_AB. The function Y^m_l is a normalized spherical harmonic, depending on the unit vector ${\widehat {\mathbf {R} }}_{AB}$ along the vector R_AB. Recall in this connection that a unit vector is in one-to-one correspondence with two spherical polar angles.

Finally, note that the existence of non-vanishing multipole moments is a matter of symmetry. For instance, a neutral, spherically symmetric charge distribution does not have any non-vanishing multipole moment. The interaction between two such non-overlapping systems is zero. An example of this is the interaction between two noble gas atoms that are so far apart that the overlap between their electronic wave functions is negligible.