Electric dipole operator ($\mathbf{d}={\sum }_a{q}_a{\mathbf{r}}_a$)

The electric dipole operator is the charge-weighted position operator of a system. It is odd under parity and transforms as a rank-one spherical tensor operator. In the nonrelativistic electric-dipole approximation it acts on spatial coordinates without directly changing spin, explaining the parity, angular-momentum and approximate spin selection rules.

A parity eigenstate has zero expectation of an odd coordinate, hence no diagonal electric-dipole coupling to a uniform oscillatory field. Stationarity alone does not prove this: an oscillator displaced by $x_c$ has $⟨x⟩=x_c$. Degenerate subspaces or a parity-breaking static field require a separate symmetry analysis.