= Raman vibrational activity
{c}
{title2=$\left(\partial\alpha/\partial Q\right)_0\ne0$}
The simplest off-resonant explanation is a vibrating <polarizability>. Let $Q(t)=Q_0\cos(2\pi\nu_vt)$ be a small <normal mode> coordinate of <molecular vibration>, and use one scalar component of <polarizability>, $\alpha(Q)\simeq\alpha_0+\alpha'_0Q$. For incident <electric field> $E(t)=E_0\cos(2\pi\nu_0t)$, the induced <electric dipole moment> is
$$
\begin{aligned}
p(t)&=\alpha(Q(t))E(t)\\
&=\alpha_0E_0\cos(2\pi\nu_0t)\\
&\quad+\frac{\alpha'_0Q_0E_0}{2}
\left[\cos\bigl(2\pi(\nu_0-\nu_v)t\bigr)
+\cos\bigl(2\pi(\nu_0+\nu_v)t\bigr)\right].
\end{aligned}
$$
The <product-to-sum formula> produces the two displaced <frequencies>. A nonzero derivative $\alpha'_0$ gives vibrational <Raman scattering> in this linear model. Real <polarizability> is generally a <tensor>, and the selected components depend on incident and observed <wave polarization>. This condition differs from <vibrational infrared activity>, which requires a changing molecular <electric dipole moment> without an applied optical <electric field>. A vibration can therefore appear in one measurement and be absent in the other.
This classical modulation argument explains line locations but supplies equal sideband <wave amplitudes>; it does not explain their thermal asymmetry or the nonzero <Stokes Raman scattering> from a vibrational <ground state>. Those require the <quantum harmonic oscillator> population and transition strengths. The far-off-resonance drawing uses dashed virtual intermediate levels: they represent terms in a scattering <probability amplitude>, not a long-lived occupied electronic <energy eigenstate>. Near an electronic resonance, a more detailed treatment of the intermediate response is needed.
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