Multipole Expansion - Interaction of Two Non-overlapping Charge Distributions

Interaction of Two Non-overlapping Charge Distributions

Consider two sets of point charges, one set {qi } clustered around a point A and one set {qj } clustered around a point B. Think for example of two molecules, and recall that a molecule by definition consists of electrons (negative point charges) and nuclei (positive point charges). The total electrostatic interaction energy UAB 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}}.

This energy can be expanded in a power series in the inverse distance of A and B. This expansion is known as the multipole expansion of UAB.

In order to derive this multipole expansion, we write rXY = rY-rX, 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\Leftrightarrow\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^M_{L}( \mathbf{r}_{Ai}-\mathbf{r}_{Bj}),

where and are irregular and regular solid harmonics, respectively. The translation of the regular solid harmonic gives a finite expansion,

R^M_L(\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^{m_A}_{\ell_A}(\mathbf{r}_{Ai}) R^{M-m_A}_{L-\ell_A}(\mathbf{r}_{Bj})\; \langle \ell_A, m_A; L-\ell_A, M-m_A| L M \rangle,

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

R^{m}_{\ell}(-\mathbf{r}) = (-1)^{\ell} R^{m}_{\ell}(\mathbf{r}) .

Use of the definition of spherical multipoles Qml 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})\; Q^{m_A}_{\ell_A} Q^{m_B}_{\ell_B}\; \langle \ell_A, m_A; \ell_B, m_B| \ell_A+\ell_B, m_A+m_B \rangle.

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

I_{\ell_A+\ell_B}^{-(m_A+m_B)}(\mathbf{R}_{AB}) \equiv \left^{1/2}\; \frac{Y^{-(m_A+m_B)}_{\ell_A+\ell_B}(\widehat{\mathbf{R}}_{AB})}{R^{\ell_A+\ell_B+1}_{AB}}

this expansion is manifestly in powers of 1/RAB. The function Yml is a normalized spherical harmonic.

Read more about this topic:  Multipole Expansion

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