Chemistry 352

Worked example with centrifugal distortion: HCl

Fit model for line positions (with centrifugal distortion):

\[ \tilde\nu_{J\to J+1} = 2B(J+1) - 4D(J+1)^3. \]

Fit results (least squares)

• Fitted rotational constant: \(B = 10.593257\ \text{cm}^{-1}\).
• Fitted centrifugal distortion constant: \(D = -8.09\times 10^{-7}\ \text{cm}^{-1}\) (magnitude ≈ \(8\times10^{-7}\ \text{cm}^{-1}\)).

Comment: the fitted value of \(D\) is essentially zero within the precision of these data (the sign is negative but magnitude is ≪ typical experimental distortion constants); the spectrum is extremely close to the rigid-rotor prediction over this range of \(J\).

Observed vs predicted line positions

Maximum absolute residual ≈ 0.0038 cm^-1, demonstrating an excellent fit.

Bond length from fitted \(B\)

Use the relation (with \(c\) in cm·s^-1 so that \(B\) is in cm^-1):

\[ B = \frac{h}{8\pi^2 c\,\mu r^2} \quad\Longrightarrow\quad r = \sqrt{\frac{h}{8\pi^2 c\,\mu B}}. \]

Using precise atomic masses for ^1H and ^{35}Cl gives a reduced mass \(\mu\approx 1.62665\times10^{-27}\ \text{kg}\). Substituting the fitted \(B=10.593257\ \text{cm}^{-1}\) yields

\[ r \approx 1.27456\times 10^{-10}\ \text{m} \;=\; 1.27456\ \mathrm{\AA}. \]

Interpretation: this bond length (~1.2746 Å) is consistent with literature H–Cl bond lengths.

Summary:

• The spectrum is extremely close to rigid-rotor behavior; the fitted distortion constant is negligibly small (|D|≈8×10⁻⁷ cm⁻¹).
• The fitted rotational constant B yields a bond length r ≈ 1.2746 Å.

Key points (one glance)

• The rigid rotor model assumes a fixed bond length, which is only an approximation for real rotating molecules.
• As rotational quantum number \(J\) increases, centrifugal forces stretch the bond slightly, increasing the moment of inertia.
• This bond stretching causes real rotational energy levels to lie slightly below the rigid-rotor predictions at high \(J\).
• Centrifugal distortion is accounted for by adding a correction term to the rotational term values: \(F_J = BJ(J+1) - D J^2(J+1)^2\).
• The centrifugal distortion constant \(D\) is typically much smaller than the rotational constant \(B\).
• Including centrifugal distortion leads to slight deviations from equal spacing between rotational spectral lines.
• High-resolution rotational spectra allow both \(B\) and \(D\) to be determined by fitting experimental data.
• The fitted rotational constant \(B\) still provides an accurate bond length, while \(D\) provides insight into bond flexibility.

Big picture: centrifugal distortion refines the rigid rotor model, revealing that rotational spectroscopy can probe not only bond lengths but also subtle bond stretching effects in rotating molecules.