Morse potential
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 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.
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.
The Morse potential energy function is of the form Here
is the well depth (defined relative to the dissociated atoms), and
The force constant (stiffness) of the bond can be found by Taylor expansion of
to the second derivative of the potential energy function, from which it can be shown that the parameter,
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
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.
Like the quantum harmonic oscillator, the energies and eigenstates of the Morse potential can be found using operator methods.
[1] One approach involves applying the factorization method to the Hamiltonian.
To write the stationary states on the Morse potential, i.e. solutions
of the following Schrödinger equation: it is convenient to introduce the new variables: Then, the Schrödinger equation takes the simplified form: Its eigenvalues (reduced by
denoting the largest integer smaller than
is a generalized Laguerre polynomial: There also exists the following analytical expression for matrix elements of the coordinate operator:[3] which is valid for
The eigenenergies in the initial variables have the form: where
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
Mathematically, the spacing of Morse levels is This trend matches the inharmonicity found in real molecules.
Specifically, This failure is due to the finite number of bound levels in the Morse potential, and some maximum
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
can be directly related to the parameters for the Morse potential.
As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation in which
An extension of the Morse potential that made the Morse form useful for modern (high-resolution) spectroscopy is the MLR (Morse/Long-range) potential.
[4] 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,[5] Ca2,[6] KLi,[7] MgH,[8][9][10] several electronic states of Li2,[4][11][12][13][9] Cs2,[14][15] Sr2,[16] ArXe,[9][17] LiCa,[18] LiNa,[19] Br2,[20] Mg2,[21] HF,[22][23] HCl,[22][23] HBr,[22][23] HI,[22][23] MgD,[8] Be2,[24] BeH,[25] and NaH.
