Non-scalar Contribution of Potential in Crystals
  Ewald-parameter dependence of Coulomb interaction in ionic crystals was studied using a point-charge model. In the presence of the long-range interaction, the ion configuration breaks spherical symmetry of local potential and charge at each ion site, and gives non-scalar contributions to them. This non-scalar potential has similar effects to Heisenberg interaction, while is intrinsically distinct from conventional multipole expansions of the scalar potential. Symmetry and magnitude of the scalar and non-scalar potentials are similar for most materials despite the different definitions, but one exception can be seen in parent materials of hole-doped high-$T_{\mathrm{c}}$ cuprates. 
H. Uchiyama
cond-mat.supr-con cond-mat.str-el