= Vibrational infrared activity
{title2=$d'_e\ne0,\quad\Delta v=\pm1$}
In the ideal <electric-dipole interaction>, a vibrational transition needs a nonzero <matrix element> of the <electric dipole moment>. For a <diatomic bond stretch>, expand the body-axis dipole component as $d(q)=d_e+d'_eq+\cdots$. A changing <electric dipole moment>, rather than a permanent value $d_e$ alone, permits the harmonic fundamental. Keeping only the linear term and harmonic <wavefunctions> gives $\Delta v=\pm1$.<14: Harmonic Oscillators and IR Spectroscopy, UCD Chem 110A by Delmar S. Larsen>[Lecture 14, introductory recap and infrared-activity section: dipole change and adjacent-level selection]{ref}
The derivation is sharper than a picture of a shaking <molecule>. <Orthogonality> removes the constant term for $v'\ne v$, and <creation and annihilation operators> give
$$
\langle v'|q|v\rangle=\sqrt{\frac{\hbar}{2\mu\omega}}\left(\sqrt{v+1}\,\delta_{v',v+1}+\sqrt v\,\delta_{v',v-1}\right).
$$
Here $\delta$ is the <Kronecker delta>. Thus absorption raises $v$ by one when the dipole derivative and angular factor are nonzero, and its <photon> satisfies
$$
h\nu_\gamma=E_{v+1}-E_v=\hbar\omega,\qquad
\widetilde\nu_\gamma=\frac{E_{v+1}-E_v}{hc}=\widetilde\nu_{\rm harm}.
$$
A nonzero vibrational <matrix element> is only one factor: rotational <electric-dipole selection rules>, initial populations, and spectral resolution also constrain visibility.
For an isolated <diatomic molecule> with identical <atomic nuclei>, such as <molecular hydrogen> in its symmetric electronic <ground state>, stretching preserves zero <electric dipole moment>. By symmetry, this homonuclear stretch is infrared inactive in the ideal dipole model, although it still vibrates. Other interactions or environments can permit detection.<14: Harmonic Oscillators and IR Spectroscopy, UCD Chem 110A by Delmar S. Larsen>[Lecture 14, introductory recap: dipole-change criterion; the identical-nuclei case follows here by symmetry]{ref}
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