The Rigid Rotor

Chem 3240 · Lecture 4.6

Davit Potoyan

The model

  • A mass \(\mu\) rotating at fixed radius \(r\)
  • No potential: pure rotational kinetic energy
  • Use spherical coordinates to exploit the symmetry
  • Hamiltonian becomes angular momentum: \[\hat{H}=\frac{\hat{L}^2}{2I}\]

Two angles, two quantum numbers

Eigenfunctions are the spherical harmonics \(Y(\theta,\phi)\): \[\hat{H}\,Y(\theta,\phi)=E_{J,m}\,Y(\theta,\phi)\]

  • \(J\) quantizes total angular momentum
  • \(M_J\) quantizes its projection
  • Quantization comes from the cyclic boundary condition \(\Phi(0)=\Phi(2\pi)\)

Rotational energy levels

\[E_J = \frac{\hbar^2}{2I}J(J+1) = B\,J(J+1)\]

Rotational constant (units of energy): \[B = \frac{h^2}{8\pi^2 I}\]

  • Energy depends only on \(J\)
  • No zero-point energy: \(J=0\) is allowed

Degeneracy and the moment of inertia

Each level is \((2J+1)\)-fold degenerate (the allowed \(M_J\))

For a diatomic: \[I = \mu r^2\]

  • Bigger \(I\) \(\Rightarrow\) smaller \(B\) \(\Rightarrow\) closer levels

Selection rules

The molecule must have a permanent dipole moment: \[\langle J' | \mu | J'' \rangle \neq 0\]

Only adjacent levels connect: \[\boxed{\Delta J = \pm 1}\]

  • Derived from the recursion relations of spherical harmonics

Evenly spaced lines

Line positions (\(\Delta J = +1\)): \[\tilde{\nu}_J = 2\tilde{B}(J+1)\]

Take one more difference: \[\tilde{\nu}_{J+1} - \tilde{\nu}_J = 2\tilde{B}\]

  • Successive lines at \(2\tilde{B},\,4\tilde{B},\,6\tilde{B},\dots\)
  • Equal spacing is the fingerprint of the rigid rotor

Reading the spacing

  • Measure line spacing \(\Rightarrow\) get \(\tilde{B}\)
  • \(\tilde{B}\Rightarrow I\Rightarrow\) bond length \(r\)
  • Different isotopes shift the lines
  • Intensities follow the Boltzmann populations \(g_J\,e^{-E_J/k_BT}\)

Rotation rides on vibration

Combine rotor and oscillator: \[\tilde{E}_{v,J} = \tilde{\omega}(v+\tfrac12) + \tilde{B}J(J+1)\]

  • R branch (\(\Delta J=+1\)): high side of \(\tilde{\omega}\)
  • P branch (\(\Delta J=-1\)): low side
  • Q branch (\(\Delta J=0\)) is forbidden

Where the model breaks

  • Rovibrational coupling: \(B\) shrinks with \(v\) \[B_v = B_e - \alpha_e(v+\tfrac12)\]
  • Centrifugal distortion: fast rotation stretches the bond \[\tilde{E}_r(J) = \tilde{B}J(J+1) - \tilde{D}J^2(J+1)^2\]
  • Both crowd the levels at large \(J\)

Takeaway

Rotational energies \(E_J = B\,J(J+1)\) are \((2J+1)\)-fold degenerate. With \(\Delta J=\pm 1\) this gives evenly spaced microwave lines separated by \(2\tilde{B}\), and the spacing reveals the bond length through \(I=\mu r^2\).