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 and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of ) 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 . The function is a normalized spherical harmonic. Molecular moments All atoms and molecules (except S-state atoms) have one or more non-vanishing permanent multipole moments. Different definitions can be found in the literature, but the following definition in spherical form has the advantage that it is contained in one general equation. Because it is in complex form it has as the further advantage that it is easier to manipulate in calculations than its real counterpart. We consider a molecule consisting of N particles (electrons and nuclei) with charges eZi. (Electrons have a Z-value of −1, while for nuclei it is the atomic number). Particle i has spherical polar coordinates ri, θi, and φi and Cartesian coordinates xi, yi, and zi. The (complex) electrostatic multipole operator is where is a regular solid harmonic function in Racah's normalization (also known as Schmidt's semi-normalization). If the molecule has total normalized wave function Ψ (depending on the coordinates of electrons and nuclei), then the multipole moment of order of the molecule is given by the expectation (expected) value: If the molecule has certain point group symmetry, then this is reflected in the wave function: Ψ transforms according to a certain irreducible representation λ of the group ("Ψ has symmetry type λ"). This has the consequence that selection rules hold for the expectation value of the multipole operator, or in other words, that the expectation value may vanish because of symmetry. A well-known example of this is the fact that molecules with an inversion center do not carry a dipole (the expectation values of vanish for . For a molecule without symmetry, no selection rules are operative and such a molecule will have non-vanishing multipoles of any order (it will carry a dipole