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
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
We assume that the two distributions do not overlap:
Under this condition we may apply the Laplace expansion in the following form
where and are irregular and regular solid harmonics, respectively. The translation of the regular solid harmonic gives a finite expansion,
where the quantity between pointed brackets is a Clebsch-Gordan coefficient. Further we used
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
This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance RAB apart. Since
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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