 Diagonal electric-dipole matrix elements and parity

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.