Morse potential

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The Morse potential (blue) and harmonic oscillator potential (green). Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ħω, the Morse potential level spacing decreases as the energy approaches the dissociation energy. The dissociation energy D_e is larger than the true energy required for dissociation D₀ due to the zero point energy of the lowest (v = 0) vibrational level.

The Morse potential, named after physicist Philip M. Morse, is a convenient model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands.

[edit] Potential Energy Function

The Morse potential energy function is of the form

.

Here r is the distance between the atoms, r_e is the equilibrium bond distance, D_e is the well depth (defined relative the dissociated atoms), and a controls the 'width' of the potential. The dissociation energy of the bond can be calculated by subtracting the zero point energy E(0) from the depth of the well. The force constant of the bond can be found by taking the second derivative of the potential energy function, from which it can be shown that the parameter, a, is

,

where k_e is the force constant at the minimum of the well.

[edit] Vibrational Energy

Stationary states on the Morse potential have eigenvalues

E(v) = hν₀ * (v + 1 / 2) − (hν₀)² / 4D_e * (v + 1 / 2)²

where v is the vibrational quantum number, and ν₀ has units of energy, and is mathematically related to the particle mass, m, and the Morse constants via

.

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator is constant at hv₀, the energy between adjacent levels decreases with increasing n in the Morse oscillator. Mathematically, the spacing of Morse levels is

E(v + 1) − E(v) = hν₀ − (v + 1) * (hν₀)² / 2D_e.

This trend matches the anharmonicity found in real molecules. However, this equation fails above some value of v where E(v + 1) − E(v) is calculated to be zero or negative. This failure is due to the finite number of bound levels in the Morse potential, and some maximum v, v_m that remains bound. For energies above v_m, the all possible energy levels are allowed and the equation for E(v) is no longer valid.

Below v_m, E(v) is a good approximation for the true vibrational structure in non-rotating diatomic molecules. In fact, the real molecular spectra are generally fit to the form¹

E_v / hc = ω_e(v + 1 / 2) − ω_eχ_e(v + 1 / 2)²

in which the constants ω_e and ω_eχ_e can be directly related to the parameters for the Morse potential.

[edit] Solving Schroedinger's equation for the Morse oscillator

Like the quantum harmonic oscillator, the energies and eigenstates of the Morse potential can be found using operator methods. One approach involves generalized factorization of the Hamiltonian, of which a specific parameterization gives rise to the Morse potential oscillator functions.

[edit] See also

• Lennard-Jones potential

[edit] References

• ¹ CRC Handbook of chemistry and physics, Ed David R. Lide, 87th ed, Section 9, SPECTROSCOPIC CONSTANTS OF DIATOMIC MOLECULES p. 9-82
• P. M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 1929, 34, 57.
• I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry, Wiley, 2003, p207.