Molecular Vibrations

Chem 3240 · Lecture 4.3

Davit Potoyan

Bonds as Springs

A vibrating bond is a quantized harmonic oscillator.

  • Energy levels are evenly spaced: \[\tilde{E}_v = \tilde{\nu}\left(v + \frac{1}{2}\right), \qquad v = 0,1,2,\dots\]
  • A single transition energy, the fundamental: \[\Delta \tilde{E}_v = \tilde{E}_{v+1} - \tilde{E}_{v} = \tilde{\nu}\]

Vibrations are quantized: this underlies infrared and Raman spectroscopy.

Frequency, Mass, and Bond Strength

\[\tilde{\nu} = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}\]

  • Force constant \(k\): stiffness of the bond

  • Reduced mass \(\mu\) of the diatomic

  • Typical range 500 to 4000 cm\(^{-1}\)

  • Strong, stiff bonds vibrate at higher frequency

  • Heavy atoms vibrate at lower frequency

Harmonic Approximation

A parabola near the equilibrium bond length \(R_e\):

\[E(R) = \frac{1}{2}k(R - R_e)^2\]

  • It is the Taylor expansion of the true curve truncated at second order

  • Good near the bottom of the well, but it never dissociates

The Morse Potential

A realistic curve that dissociates:

\[V(R)=D_e\left(1-e^{-a(R-R_e)}\right)^2\]

  • \(D_e\): well depth from the minimum

  • \(D_0 = D_e - \tfrac{1}{2}h\nu\): from the ground level

  • Levels converge as \(v\) grows

Anharmonic Energy Levels

Add higher-order terms so levels are no longer evenly spaced:

\[\tilde{E}_v = \tilde{\nu}_e\left(v + \tfrac{1}{2}\right) - \tilde{\nu}_e x_e\left(v + \tfrac{1}{2}\right)^2 + \dots\]

  • \(\tilde{\nu}_e\): vibrational wavenumber, \(x_e\): anharmonicity constant
  • Transition energies shrink with \(v\): \[\tilde{\nu}_{v\rightarrow v+1} = \tilde{\nu}_e\left[1 - 2x_e(v+1)\right]\]

Normal Modes

A polyatomic molecule’s motion resolves into independent normal modes.

  • Linear molecules: \[N_{modes} = 3N - 5\]
  • Nonlinear molecules: \[N_{modes} = 3N - 6\]
  • Subtract translations and rotations from \(3N\)

IR Selection Rule

A transition is allowed only if the dipole changes with bond length:

\[\left( \frac{\partial \mu}{\partial x} \right)_{R_e} \neq 0\]

  • And the quantum number changes by one: \[\Delta v = \pm 1\]

The transition probability \(P_{v\rightarrow v'} \sim \langle v | x | v' \rangle\) vanishes unless both conditions hold.

IR-Active vs Inactive

  • Heteronuclear (HCl, CO): dipole changes, IR-active

  • Homonuclear (H\(_2\), O\(_2\), N\(_2\)): zero dipole, IR-inactive

  • IR-inactive stretches can still appear in the Raman spectrum

Overtones

  • Anharmonicity makes weak overtones appear: \(\Delta v = \pm 2, \pm 3, \dots\)

\[\tilde{\nu}_{0\rightarrow v} = \tilde{\nu}_e \cdot v - \tilde{\nu}_e x_e \cdot v(v+1)\]

  • Intensities fall off rapidly with each higher overtone

  • At room temperature almost all molecules sit in \(v=0\), so only the fundamental dominates the spectrum

Reading an IR Spectrum

  • Each bond type absorbs in a characteristic region

  • Stiff bonds and light atoms sit at high wavenumber

  • A spectrum is a fingerprint of functional groups

Takeaway

Bonds vibrate as quantized harmonic oscillators with \(\tilde{E}_v = \tilde{\nu}(v+\tfrac{1}{2})\); polyatomics split into \(3N-6\) (or \(3N-5\)) normal modes; a mode is IR-active only if the dipole changes, with \(\Delta v = \pm 1\); and the Morse potential captures the anharmonicity that lets molecules dissociate.