= Solution
{c}
<Infrared spectroscopy> measures absorption as the incident <photon> frequency is swept through molecular <normal mode> frequencies. In the electric-dipole approximation the interaction is $-\mathbf p\cdot\mathbf E$. Expanding the molecular <electric dipole moment> in a <normal coordinate> $Q$ gives
$$
\mathbf p(Q)=\mathbf p(0)+\left(\frac{\partial\mathbf p}{\partial Q}\right)_0Q+\cdots.
$$
The fundamental one-quantum vibrational <transition dipole moment> is proportional to the derivative multiplying $Q$. Thus
$$
\boxed{\left(\partial\mathbf p/\partial Q\right)_0\ne0}
$$
is the criterion for <vibrational infrared activity>; the derivative must also have a component along the incident <polarization of an electromagnetic wave>. A permanent molecular <electric dipole moment> is not necessary. Absorption occurs near $\hbar\omega=E_{v+1}-E_v$, with broadened lines reflecting finite lifetimes and the molecular environment.
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