The derivation of electric quadrupole for a periodic system proceeds in much the same way as for the magnetic dipole. The following will focus on the velocity form of the electric quadrupole operator, as this is the form used in both the modified velocity gauge (MVG) and origin-independent length gauge [LG(OI)]. We also use the non-traceless form of the operator for this derivation to simplify the equations. In the velocity gauge, the real space matrix elements of the electric quadrupole are written as