text: Consider a nucleus with a non-zero quadrupole moment  and charge density , which is surrounded by a potential . This potential may be produced by the electrons as stated above, whose probability distribution might be non-isotropic in general. The potential energy in this system equals to the integral over the charge distribution  and the potential  within a domain :
  One can write the potential as a Taylor-expansion at the center of the considered nucleus. This method corresponds to the multipole expansion in cartesian coordinates (note that the equations below use the Einstein sum-convention):
  The first term involving  will not be relevant and can therefore be omitted. Since nuclei do not have an electric dipole moment , which would interact with the electric field , the first derivatives can also be neglected. One is therefore left with all nine combinations of second derivatives. However if one deals with a homogeneous oblate or prolate nucleus the matrix  will be diagonal and elements with  vanish. This leads to a simplification because the equation for the potential energy now contains only  the second derivatives in respect to the same variable:
  The remaining terms in the integral are related to the charge distribution and hence the quadrupole moment. The formula can be simplified even further by introducing the electric field gradient  , choosing the z-axis as the one with the maximal principal component  and using the Laplace equation to obtain the proportionality written above. For an  nucleus one obtains with the frequency-energy relation :