The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the QHO (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. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. However, its mathematical form inspired the MLR (Morse/Long-range) potential, which is the most popular potential energy function used for fitting spectroscopic data.
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Potential energy function
The Morse potential energy function is of the form
Here is the distance between the atoms, is the equilibrium bond distance, is the well depth (defined relative to the dissociated atoms), and controls the 'width' of the potential (the smaller is, the larger the well). The dissociation energy of the bond can be calculated by subtracting the zero point energy from the depth of the well. The force constant (stiffness) of the bond can be found by Taylor expansion of around to the second derivative of the potential energy function, from which it can be shown that the parameter, , is
where is the force constant at the minimum of the well.
Since the zero of potential energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. When it is used to model the atom-surface interaction, the energy zero can be redefined so that the Morse potential becomes
which is usually written as
where is now the coordinate perpendicular to the surface. This form approaches zero at infinite and equals at its minimum, i.e. . It clearly shows that the Morse potential is the combination of a short-range repulsion term (the former) and a long-range attractive term (the latter), analogous to the Lennard-Jones potential.
Vibrational states and energies
Like the quantum harmonic oscillator, the energies and eigenstates of the Morse potential can be found using operator methods. One approach involves applying the factorization method to the Hamiltonian.
it is convenient to introduce the new variables:
Then, the Schrödinger equation takes the simple form:
with [x] denoting the largest integer smaller than x.
where and is a generalized Laguerre polynomial:
There also exists the following important analytical expression for matrix elements of the coordinate operator (here it is assumed that and )
The eigenenergies in the initial variables have form:
where is the vibrational quantum number, and has units of frequency, and is mathematically related to the particle mass, , and the Morse constants via
Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator is constant at , the energy between adjacent levels decreases with increasing in the Morse oscillator. Mathematically, the spacing of Morse levels is
This trend matches the anharmonicity found in real molecules. However, this equation fails above some value of where is calculated to be zero or negative. Specifically,
- integer part.
This failure is due to the finite number of bound levels in the Morse potential, and some maximum that remains bound. For energies above , all the possible energy levels are allowed and the equation for is no longer valid.
Below , 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
in which the constants and can be directly related to the parameters for the Morse potential.
An important extension of the Morse potential that made the Morse form very useful for modern spectroscopy is the MLR (Morse/Long-range) potential. The MLR potential is used as a standard for representing spectroscopic and/or virial data of diatomic molecules by a potential energy curve. It has been used on N2, Ca2, KLi, MgH, several electronic states of Li2, Cs2, Sr2, ArXe, LiCa, LiNa, Br2, Mg2, HF, HCl, HBr, HI, MgD, Be2, BeH, and NaH. More sophisticated versions are used for polyatomic molecules.
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