Multi-pole expansion

That is, is due to the Coulomb and exchange interactions. Ignoring the exchange interaction and carrying out the multi-pole expansion for the Coulomb interaction yield... 

The l = 2 terms in the multi-pole expansion (8.32), which are linear in the electric field gradient (<9E/dz)r=0, are known as the quadrupole terms [89,90] and they produce a broadening of the shallow impurity lines known as the quadrupole broadening. At a difference with the quadratic Stark broadening, the quadrupole broadening results in a more symmetric profile of the perturbed line [90]. 

Note that the larger the dipole moment of the unit cell the worse the results. This is understandable, because the first non-vanishing contribution in the multi-pole expansion is the dipole-dipole term (cf. Appendix X). Note how considerably the unit cell dependence drops after this term is switched on... 

When the distance R is small when compared to the size of the interacting subsystems, multi-pole expansion gives bad results. To overcome this, the total charge distribution may be divided into atomic segments (see Appendix S available at booksite.elsevier.com/978-0 144-59436-5). Each atom would carry its charge and other multipoles, and the electrostatic energy would be the sum of the atom-atom contributions, any of which would represent a series similar to... 

---