Morse Potential: Dissociation Energy and Anharmonicity

Vibrational spectroscopy is high-scoring in CSIR-NET, GATE, and IIT-JAM. While harmonic oscillator models treat chemical bonds as ideal springs, real bonds dissociate when stretched. The Morse potential accurately models dissociation and vibrational anharmonicity. Mastering Morse potential relations, D_e versus D₀, and Birge–Sponer extrapolation allows students to solve exam numericals quickly and reliably.

Why the Harmonic Model Fails for Real Bonds

The harmonic oscillator approximates potential energy as a parabola: V(r) = (1/2)k(r - r_e)². Near equilibrium bond distance r_e it works, but fails for real bonds. Nuclear repulsion at short distances (r < r_e) rises steeper than a parabola, while at large distances (r > r_e), bonds weaken and dissociate into neutral atoms. Real potential wells widen as energy rises, causing vibrational levels to converge. This deviation is called anharmonicity.

The Morse Potential Function

Philip M. Morse proposed an empirical potential reproducing real diatomic curves:

At r = r_e, V(r_e) = 0. When r → ∞, V(r) → D_e. If zero energy is set at dissociation, V(r) = D_e [1 - e^{-a(r - r_e)}]² - D_e.

Vibrational Term Values and Dissociation Energy

Solving the Schrödinger equation for a Morse oscillator gives quantized vibrational term values G(v) in cm^-1:

To convert cm^-1 to kJ/mol, multiply by N_Ahc = 0.01196266 kJ/(mol·cm^-1).

Transitions and Birge–Sponer Extrapolation

Anharmonicity relaxes harmonic selection rules, allowing overtones:

• Fundamental (0 → 1): ν̃₁ = ω_e - 2ω_ex_e
• First overtone (0 → 2): ν̃₂ = 2ω_e - 6ω_ex_e

Level spacing is ΔG_v+1/2 = ω_e - 2ω_ex_e(v + 1). Plotting ΔG against (v + 1) gives the linear Birge–Sponer plot. At dissociation (ΔG = 0), v_max + 1 = ω_e / (2ω_ex_e). The plot area gives D_e = ω_e² / (4ω_ex_e).

Worked Numerical Examples