= HCl and DCl harmonic stretch comparison
{c}
{title2=$\omega_{\rm DCl}/\omega_{\rm HCl}=\sqrt{37/72}$}
Keep the same <bond force constant> $k$ and approximate the nuclear <masses> of $^{1}\mathrm H$, $^{2}\mathrm H$, and $^{35}\mathrm{Cl}$ by $1u$, $2u$, and $35u$, respectively. Here $u$ is the <unified atomic mass unit>, and $\mathrm D$ denotes <deuterium>. These are chosen <mass number> approximations, not measured <atomic masses> or natural-abundance averages. With the same <chlorine> <isotope> in both <molecules>,
$$
\mu_{\rm HCl}=\frac{35}{36}u,\qquad
\mu_{\rm DCl}=\frac{70}{37}u,\qquad
\frac{\omega_{\rm DCl}}{\omega_{\rm HCl}}
=\frac{\widetilde\nu_{{\rm harm},\rm DCl}}{\widetilde\nu_{{\rm harm},\rm HCl}}
=\sqrt{\frac{\mu_{\rm HCl}}{\mu_{\rm DCl}}}
=\sqrt{\frac{37}{72}}\approx0.716860.
$$
This is the <isotope scaling of diatomic vibration> prediction. Doubling the light partner's <mass> nearly doubles the <reduced mass>, lowering the <frequency> by about $28.3\%$. The factor differs from $1/\sqrt2$ because the <chlorine> partner also moves.
\Image[/hcl-dcl-mass-and-stretch-response.png]
{title=<Isotope> substitution slows a stipulated harmonic <diatomic bond stretch>}
{id=image-hcl-dcl-mass-and-stretch-response}
{description=Original approximate <mass number> model: $1u,2u,35u$ and unchanged <bond force constant>. Left: calculated <reduced masses> and normalized <angular frequencies>. Right: free classical stretches released from the same small <displacement> and zero <velocity>, $q_{\rm HCl}/A=\cos\tau$ and $q_{\rm DCl}/A=\cos(\sqrt{37/72}\,\tau)$, with $\tau=\omega_{\rm HCl}t$. This is not a measured time trace or an <infrared> spectrum.}
For the same small initial stretch $A$ and zero <velocity>, both models start with the same internal <mechanical energy> $kA^2/2$, but the DCl <oscillation period> is longer by $\sqrt{72/37}\approx1.394972$. In the <centre of mass> frame, $u_{\rm H}=-35q/36$ and $u_{\rm Cl}=q/36$ for HCl; $u_{\rm D}=-35q/37$ and $u_{\rm Cl}=2q/37$ for DCl. These ratios explain which partner moves and are separate from the change in <frequency>.
The common-$k$ assumption belongs to the leading <Born-Oppenheimer approximation>: fixed nuclear positions and <electric charges> determine the electronic <potential energy>, while nuclear <masses> govern motion on that curve. Quantitative comparison requires actual <atomic masses>, departures from that approximation, and separation of <anharmonicity> and <molecular rotation> from the measured band. Even with common $k$, the <quantum harmonic oscillator> predicts a different <zero-point energy> for each <isotope>.<Rotation Vibration Spectrum of the HCl Molecule, University of Florida laboratory>[printed pp. 1 and 3, electronic motion; p. 12, Comprehension Question 2: common isotope potential and inferred spring constants]{ref}
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