Diagonal electric-dipole matrix elements and parity
= 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 $\langle x\rangle=x_c$. Degenerate subspaces or a parity-breaking static field require a separate symmetry analysis.