Rotational Stark effect

An electric-field shift or splitting of molecular rotational levels. The observed line shift is the difference of upper- and lower-level shifts, subject to polarization-dependent selection rules; shifting only one endpoint gives an incorrect spectral splitting.

The dipole coupling $-pE\cos \theta$ connects only $J\pm 1$ at fixed $m$. For $J\ge 1$, second order gives $(pE{)}^2[J(J+1)-3m^2]/[2BJ(J+1)(2J-1)(2J+3)]$; the ground-state result is separately $-(pE{)}^2/(6B)$. Here $B$ has energy units and $\theta$ is the polar angle to the field.

For $J=0\to 1$, the $m=0$ and $|m|=1$ frequency shifts are $4(pE{)}^2/(15Bh)$ and $7(pE{)}^2/(60Bh)$. Their difference is $3(pE{)}^2/(20Bh)$, so $p=(h/E)\sqrt{(20/3)(B/h)\Delta \nu }$. The common lower-state shift cancels from the splitting but not from either absolute transition frequency.

Radiation polarized along the static field drives the zero-projection change, while transverse polarization supplies the two circular components with projection changes plus or minus one. The opposite signs of the excited magnetic quantum number remain degenerate in a pure static electric field; they need not be separate spectral lines.

Every state in a fixed-$J$ manifold has parity $(-1{)}^J$, while the dipole component is odd. Therefore its complete matrix within that degenerate manifold vanishes. One must check the whole degenerate block, not only one diagonal element, before concluding that first-order splitting is absent.