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 fine structure of the molecule because it implicitly includes the effects of bond breaking. A feature of the Morse potential which is absent for the harmonic oscillator model is that it accounts for unbounded states. It works better than the quantum harmonic oscillator model, because it accounts for anharmonicity, overtone frequencies, and combination frequencies. An overtone frequency occurs when a transition is n +/- 2 or greater and a combination frequency results from the addition or subtraction of two or more modes. 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 'depth' of the potential energy function (the limit of the function as the bond distance approaches infinity), T_o is the zero point energy, and a controls the 'width' of the potential and is equal to the square root of half the force constant divided by the morse potential, . The exponential functions are used as a power series cannot describe the spacing of experimentally observed energy levels. 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.

Vibrational Energy

n is the vibrational quantum number

E(n) = (n + 1 / 2)hv₀ − (n + 1 / 2)² * (hv₀)² / 4D_e.

E(n + 1) − E(n) = hv₀ − (n + 1) * (hv₀)² / 2D_e.

With the quantum harmonic oscillator, the energy between adjacent levels is constant, hv₀. With the Morse potential, the energy between adjacent levels decreases with increasing n as is seen in nature. It fails at the value of n where E(n + 1) − E(n) is calculated to be zero or negative. The Morse potential is a good approximation for the vibrational fine structure at n values below this limit.

[edit] Quantization in the Morse potential

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

• 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.