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. The Morse potential can also be used to model other interactions, such as those 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, the most popular potential energy function for fitting spectroscopic data.

Potential energy function

The Morse potential energy function is of the form

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 to the second derivative, is

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 in 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

. 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.

To write the stationary states of the following Schrödinger equation:

it is convenient to introduce the new variables:

Then, the Schrödinger equation takes the simplified form:

Its eigenvalues) and eigenstates can be written as:

where

and

which satisfies the normalization condition

is a generalized Laguerre polynomial:

There also exists the following analytical expression for matrix elements of the coordinate operator:

The eigenenergies in the initial variables have the form:

is the vibrational quantum number and the Morse constants via

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator in the Morse oscillator. Mathematically, the spacing of Morse levels is

is calculated to be zero or negative. Specifically,

1. (integer part only).

This failure is due to the finite is no longer valid.

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 form

can be directly related to the parameters for the Morse potential. Specifically,

and

is in

As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation represents a wavenumber and not an angular frequency

Morse/Long-range potential

An extension of the Morse potential that made the Morse form useful for modern (high-resolution) 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.

See also

• Lennard-Jones potential

• Molecular mechanics

References

• 1Lide, David R. (ed.) CRC Handbook of Chemistry and Physics, 87th ed., Section 9, Spectroscopic Constants of Diatomic Molecules, pp.9–82.

• Khordad, R.; Edet, C. O.; and Ikot, A. N. (2022). "Application of Morse potential and improved deformed exponential-type potential (IDEP) model to predict thermodynamics properties of diatomic molecules". International Journal of Modern Physics C 33 (08): 2250106 https://www.worldscientific.com/doi/10.1142/S0129183122501066 doi:10.1142/S0129183122501066.

• Varshni, Yatendra Pal (1957). "Comparative Study of Potential Energy Functions for Diatomic Molecules". Rev. Mod. Phys. 29: 664 doi:10.1103/RevModPhys.29.664.

• Kaplan, I. G. (2003). Handbook of Molecular Physics and Quantum Chemistry, Wiley, p. 207.

• Haynes, W. M.; David, R.; Lide, T. J. B. (eds) (2017). CRC Handbook of Chemistry and Physics, Boca Raton, FL: CRC Press.