If two charges e and –e are separated by a distance d, they constitute an electric dipole of moment ed.  The units are Debye (10^-18 esu.cm), or in the MKS, Cm. (Coulomb meter).  More extended charge distributions such as two + and two – charges separated from one another lead to a quadrupole moment eQ which is defined as

eQ = ∫ρ(x,y,z) r² (3 cos²θ-1)dτ                                                                        (15.3)

Here, ρ(x,y,z) is the charge density at r = (x,y,z),   θ, the polar angle and dτ, the volume of integration.  Nuclei with spins I ≥ 1 have a quadrupole moment.  Nuceli with spins  ½ do possess a dipole moment, but their quadrupole moment is zero as the +ve charge density contribution is cancelled by the –ve contribution in  Eq. (15.3).

A point charge e interacts with the electrostatic potential V at its location to give an interaction energy eV.  A “point” dipole μ interacts with the electric field  E in which it is placed to give energy  - μ.E  The quadrupole interacts with the electric field gradient, whose components are given as the second derivative of the electrostatic potential

  eq_ij  =  ∂V / ∂x_i ∂x_j ;                    x_i = x,y,z,  x_j = x,y,z                                                       (15.4)

The quadrupole moment can be positive or negative, depending on the shape of the nucleus.  Two cases are shown in Fig 15.3.