by the Laplace expansion: where is an irregular solid harmonic (defined below as a spherical harmonic function divided by ) and is a regular solid harmonic (a spherical harmonic times ). We define the spherical multipole moment of the charge distribution as follows Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the N charges). A spherical harmonic depends on the unit vector . (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as so that the multipole expansion of the field at the point outside the charge distribution is given by This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the spherical multipole moments appear as coefficients in the expansion of the potential. It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks. Since the summand of the m summation is invariant under a unitary transformation of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a unitary transformation, we can simply substitute real irregular solid harmonics and real multipole moments. The term becomes This is in fact Coulomb's law again. For the term we introduce Then This term is identical to the one found in Cartesian form. In order to write the term, we have to introduce shorthand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation. Interaction of two non-overlapping charge distributions Consider two sets of point charges, one set clustered around a point and one set clustered around a point . 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 between the two distributions is This energy can be expanded in a power series in the inverse distance of and . This expansion is known as the multipole expansion of UAB. In order to derive this multipole expansion, we write , which is a vector pointing from towards . Note that We