exterior case, where , the result is: where the general multipole moments are defined Note The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Yℓm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this. Interior spherical multipole moments Similarly, the interior multipole expansion has the same functional form. In the interior case, where , the result is: with the interior multipole moments defined as Interaction energies of spherical multipoles A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. Let the first charge distribution be centered on the origin and lie entirely within the second charge distribution . The interaction energy between any two static charge distributions is defined by The potential of the first (central) charge distribution may be expanded in exterior multipoles where represents the exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula Since the integral equals the complex conjugate of the interior multipole moments of the second (peripheral) charge distribution, the energy formula reduces to the simple form For example, this formula may be used to determine the electrostatic interaction energies of the atomic nucleus with its surrounding electronic orbitals. Conversely, given the interaction energies and the interior multipole moments of the electronic orbitals, one may find the exterior multipole moments (and, hence, shape) of the atomic nucleus. Special case of axial symmetry The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle ). By carrying out the integrations that define and , it can be shown the multipole moments are all zero except when . Using the mathematical identity the exterior multipole expansion becomes where the axially symmetric multipole moments are defined In the limit that the charge is confined to the -axis, we recover the exterior axial multipole moments. Similarly the interior multipole expansion becomes where the axially symmetric interior multipole moments are defined In the limit that the charge is confined to the -axis, we recover the interior axial multipole moments. See also Solid harmonics Laplace expansion Multipole expansion Legendre polynomials Axial multipole moments Cylindrical multipole moments Electromagnetism Potential theory Moment (physics)