Morse Potential

Morse Potential

The Morse potential provides a more realistic description of molecular vibrations by accounting for bond dissociation. It is expressed as:

\[ V(r) = D_e \left( 1 - e^{-\alpha (r - r_0)} \right)^2 \]

Where:

  • \( V(r) \): Potential energy as a function of bond displacement (\(x\)).
  • \( D_e \): Dissociation energy (the energy required to break the bond completely).
  • \( \alpha \): A constant related to the stiffness of the bond and width of the potential well.
  • \( r_0 \): Equilibrium bond length.
  • \( r \): Bond length.

Graph Description

  • Shape: The Morse potential starts at \(x_0\) with zero energy and asymptotically approaches \(D_e\) as \(r \to \infty\), representing bond dissociation.
  • Asymmetry: The curve is asymmetric, with a steep rise in energy for compression (\(r < r_e\)) and a gradual leveling for bond stretching (\(r > r_e\)).
  • Realism: Unlike the harmonic oscillator, the Morse potential accurately represents molecular vibrations near bond dissociation and captures anharmonic behavior.

Comparison to the Harmonic Oscillator

  • The harmonic oscillator assumes infinite energy for bond stretching and has a symmetric potential around \(r_e\).
  • The Morse potential accounts for the finite dissociation energy \(D_e\) and the anharmonicity of real molecular bonds.