Themes > Science > Physics > Electromagnetism > Electrostatics >  Electrostatic Potential > Multipole Expansion of the Electrostatic Potential For the calculation of the electrostatic potential U of a charge density one  can use the equation (in atomic units)   Inserting the Laplace expansion with the abbreviations , , and   and the complex spherical harmonics in the phase convention of Condon and Shortley  [126] as defined in [127], p. 3, Eq. (1.2-1), and interchanging the order of integration and  summation yields   This can be interpreted as orthogonal expansion in the complete orthonormal system of  spherical harmonics. If the charge density vanishes outside of a sphere of radius a with center at the origin and  if r>a holds then r>r'  is satisfied for all for which holds, and the relation follows. Introduction of the multipole moments allows to represent the potential by an infinite series of the form   that is usually called the multipole expansion. In particular, for a charge distribution that is invariant under rotations with axis , the multipole moments have the form   and the addition theorem   of the spherical harmonics yields an expansion in Legendre polynomials   This form of the electrostatic potential is also implied by the fact that the Laplace  equation  holds outside of a charge distribution. The only solutions of  this equation for in spherical coordinates are linear combinations of irregular  solid harmonics and thus, taking rotational symmetry  around the direction into account, only linear combinations of   occur in the multipole expansion of U where is the angle between the position vector and  the direction .  If the charge distribution does not have compact support in , the above multipole expansions  only hold approximately for large . In this case, the inequality r>r' does not hold for  all for which .  The difference between the exact potential U and the multipole expansion in Eq. (6) is given by Because of the difference can also be represented as If decays sufficiently rapidly for large arguments, the contributions with angular  momentum quantum number to the difference will go to zero for large r   (unless they vanish anyway by symmetry). In the case of the example   with , only the term with survives the  angular integration and one obtains  Hence, the difference vanishes for large r exponentially in this simple  model example. Note that the multipole expansion for is a solution of the Laplace equation .  But the exact electrostatic potential U satisfies the Poisson equation (atomic units) This implies that also the difference is a solution of this Poisson equation.  A further consequence  is that the multipole expansion can only be a good approximation  to U where the charge density is small, i.e., for large distances from the origin. An advantage of the multipole expansion is that the moments or , respectively, to a  given charge distribution  can be computed once and for all, and then, the multipole  approximation of can be computed very easily for very many arguments .  The exact expansion (3) of on the other hand is more demanding computationally.  It requires the calculation of the integrals  that depend also on the distance r to the expansions centre. If the charge density is rotationally  symmetric around the direction , then in analogy to Eq. (7) we have and hence, the expansion (3) simplifies to   This corresponds to the cylindrical symmetry of the problem since only the coordinates r   and with enter. Thus, one obtains an expansion in Legendre  polynomials that may or may not converge rapidly. Information provided by: http://www.chemie.uni-regensburg.de