For a continuous charge distribution, the electric dipole moment is given by
where r points from positive to negative. For a discrete charge distribution
where r points from positive to negative.
As can be seen from the definition, the vector dipole moment can be dependent on the origin. To see when this is the case, define an arbitrary new origin offset from the original one by the vector and find when the two dipole moments are equal,
which will be true when
since is a constant. Therefore, p is independent of origin iff
The dipole field induced by an electric field E is
where is the atomic polarizibility tensor and
Atomic Polarizibility, Electric Dipole
