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Demos.MorseOscillator.Main

Burkhard Schmidt

WavePacket demo example: Morse oscillator

The Morse oscillator is the most prominent example of an anharmonic oscillator. Its main feature is decreasing energy level spacing with increasing energy. In molecular physics this model is frequently used to describe the anharmonic vibrational dynamics of a diatomic molecule accounting for the dissociation limit. as is illustrated in the examples given herein.

Model

As a standard example the OH bond in a water molecule is considered. The vibrational motion is modelled by a Morse oscillator

V(R) = De ( 1 − exp ( -α(R−Re) ) )2

with dissociation energy De=0.1994, equilibrium bond length Re=1.821, range parameter α=1.189, and reduced mass μ=1728.539 (all in atomic units: Hartree energy, Bohr radius, electron masses). Note that 1728.257 should be better for 16O-1H

Using the values for the harmonic frequency and anharmonicity,

ω = α (2De/μ)½ = 0.0180600870
χ = ω / (4De) = 0.022643038

the bound state energies can be expressed as

E_n=ω (n +½) − χ ω (n+½)2

with n ranging from 0 to INT((1/χ−1)/2)=21

Note that further results for the Morse oscillator model of OH can be found on our Wiki pages on molecular vibrations where we investigate the quantum dynamics induced by external fields, aiming at efficient and state-selective population transfer.

Bound state calculations

Although the bound state energies and wavefunctions are known from Morse's original work, we use the example of the Morse oscillator to illustrate the performance of the Fourier grid Hamiltonian algorithm as implemented in WavePacket. Learn more ...

Wavepacket dynamics

If a Gaussian wavepacket is chosen as an initial wavepacket, the wavefunction immediately changes its shape and undergoes complex dynamics. While the position/momentum densities appear completely unstructured, some structures are still visible in the phase space (Wigner) representation. At later times, (full or partial) revivals of the initial wavepacket occur. Learn more ...


Related

Wiki: Demos.Adi
Wiki: Demos.MolVibration.Main
Wiki: Demos.MolVibration.OH
Wiki: Demos.MorseOscillator.Bound1D
Wiki: Demos.MorseOscillator.Gaussian1D
Wiki: Demos.OptMorseOscillator.Main
Wiki: Numerics.AtomicUnits

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