and simultaneously a quadrupole, octupole, hexadecapole, etc.). The lowest explicit forms of the regular solid harmonics (with the Condon-Shortley phase) give: (the total charge of the molecule). The (complex) dipole components are: Note that by a simple linear combination one can transform the complex multipole operators to real ones. The real multipole operators are of cosine type or sine type . A few of the lowest ones are: Note on conventions The definition of the complex molecular multipole moment given above is the complex conjugate of the definition given in this article, which follows the definition of the standard textbook on classical electrodynamics by Jackson, except for the normalization. Moreover, in the classical definition of Jackson the equivalent of the N-particle quantum mechanical expectation value is an integral over a one-particle charge distribution. Remember that in the case of a one-particle quantum mechanical system the expectation value is nothing but an integral over the charge distribution (modulus of wavefunction squared), so that the definition of this article is a quantum mechanical N-particle generalization of Jackson's definition. The definition in this article agrees with, among others, the one of Fano and Racah and Brink and Satchler. Examples There are many types of multipole moments, since there are many types of potentials and many ways of approximating a potential by a series expansion, depending on the coordinates and the symmetry of the charge distribution. The most common expansions include: Axial multipole moments of a potential; Spherical multipole moments of a potential; and Cylindrical multipole moments of a potential Examples of potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. An example of a potential is the electric potential of an infinite line charge. General mathematical properties Multipole moments in mathematics and mathematical physics form an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of distribution theory, as directional derivatives. Multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of irreducible representations of the rotational symmetry group, which leads to spherical harmonics and