= Rotational Raman spectroscopy
{title2=$\Delta J=0,\pm2$}
<Rotational Raman spectroscopy> observes a <frequency> change of scattered light associated with <molecular rotation>. The coupling uses the induced <electric dipole moment>, $\mathbf d_{\rm ind}=\boldsymbol\alpha\mathbf E$, where $\boldsymbol\alpha$ is the <electric polarizability> tensor. For a <linear molecule>, the useful anisotropy is $\alpha_\parallel-\alpha_\perp$: the response differs for an <electric field> parallel or perpendicular to the molecular axis. A permanent <electric dipole moment> is not required.<Absorption lineshapes by Andrei Tokmakoff>[Andrei Tokmakoff, §6.2, printed pp. 6-7–6-8, equations (6.40)–(6.44): induced dipole, polarizability tensor and rotational Raman response]{ref}
For the spin-free <linear rigid rotor> in one internal <quantum state>, the anisotropic tensor depends quadratically on the molecular direction. Its angular part has even parity and degree two in <spherical harmonics>. The <triangle inequality> for coupled <angular momenta> permits $|\Delta J|\leq2$, and even parity removes odd $\Delta J$, leaving $\Delta J=0,\pm2$. The $\Delta J=0$ contribution is unshifted light; shifted rotational lines have $\Delta J=\pm2$. Consequently the rigid-model Stokes shifts, in ordinary <frequency> units, are
$$
\Delta\nu_J=\frac{E_{J+2}-E_J}{h}=B(4J+6),\qquad J=0,1,2,\ldots.
$$
Successive allowed shifts are separated by $4B$, rather than the $2B$ absorption spacing of <pure rotational electric-dipole selection rules>. In <spectroscopic wavenumber> units replace $B$ by $\widetilde B=B/c$ and $\Delta\nu$ by $\Delta\widetilde\nu=\Delta\nu/c$, using the <speed of light> in $\mathrm{cm\,s^{-1}}$ for $\mathrm{cm^{-1}}$.
A <homonuclear diatomic molecule> such as <molecular hydrogen> has no permanent <electric dipole moment> in its symmetric internal state, so it lacks the ordinary electric-dipole microwave ladder. Its anisotropic <electric polarizability> can nevertheless give <rotational Raman spectroscopy> lines. Nuclear-<spin> symmetry can change the populations of even and odd $J$ levels, so the spin-free population example above must not be reused unchanged for <molecular hydrogen>. The method of coupling, as well as the <energy eigenvalues>, determines what the experiment can see.
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