= Isotope scaling of diatomic vibration
{title2=$\nu'/\nu\simeq\sqrt{\mu/\mu'}$}
Near a stable equilibrium <bond length> $r_e$, a <Taylor expansion> of a <diatomic molecule>'s <potential energy> gives $U(r)\simeq U(r_e)+k(r-r_e)^2/2$, with $k=U''(r_e)>0$. Treat the internal stretch as a <harmonic oscillator> of <reduced mass>, initially omitting <molecular rotation>. If replacing an <isotope> leaves this local <potential energy> curvature unchanged, then
$$
\frac{\nu'}\nu\simeq\sqrt{\frac\mu{\mu'}},\qquad
\widetilde\nu_{\rm harm}=\frac1{2\pi c}\sqrt{\frac{k}{\mu}}.
$$
Here $c$ is the <speed of light> and $\widetilde\nu_{\rm harm}$ a <spectroscopic wavenumber>; use consistent length units for $c$ and $\widetilde\nu_{\rm harm}$. The <quantum harmonic oscillator> gives adjacent <energy eigenvalues> separated by $h\nu$, with $h$ the <Planck constant>. Consequently a vibrational <photon absorption> can probe the same <frequency> scale, subject to <electric-dipole selection rules>.
For an explicitly approximate <isotope> example, assign the <atomic masses> of $^{1}\mathrm H$, $^{2}\mathrm H$, $^{35}\mathrm{Cl}$, and $^{37}\mathrm{Cl}$ the mass-number values $1u,2u,35u,37u$, respectively, with $u$ the <unified atomic mass unit>. These are stipulated approximations to individual <atomic masses>, not measured <masses> of <molecules> or natural-abundance averages. They give
$$
\mu_{\mathrm H^{35}\mathrm{Cl}}=\frac{35}{36}u,\quad
\mu_{\mathrm D^{35}\mathrm{Cl}}=\frac{70}{37}u,\quad
\frac{\nu_{\mathrm D^{35}\mathrm{Cl}}}{\nu_{\mathrm H^{35}\mathrm{Cl}}}\simeq\sqrt{\frac{37}{72}}\approx0.716860,
\qquad
\frac{\nu_{\mathrm H^{37}\mathrm{Cl}}}{\nu_{\mathrm H^{35}\mathrm{Cl}}}\simeq\sqrt{\frac{665}{666}}\approx0.999249.
$$
The symbol $\mathrm D$ means <deuterium>, $^{2}\mathrm H$. Doubling the light partner's approximate <atomic mass> therefore produces a large predicted shift, whereas changing the already heavy <chlorine> partner by two mass units produces a much smaller one. A quantitative comparison with <infrared spectroscopy> requires actual <isotope>-specific <atomic masses>, fitted band parameters, and uncertainties, rather than identifying any arbitrary peak with $\widetilde\nu_{\rm harm}$.
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