# Morse potential
The Morse potential is an empirical interatomic potential energy function,
\( V(r) = D_e \left(1 - e^{-a(r-r_e)}\right)^2 \), introduced by [Philip M. Morse](https://www.edgepedia.org/philip-m-morse) in 1929 to model the vibration of a diatomic molecule, in which
\( D_e \) is the bond dissociation energy,
\( r_e \) is the equilibrium bond length, and
\( a \) is a width parameter characteristic of the molecule[1](https://old.goldbook.iupac.org/html/M/MT07073.html) • [2](https://cds.cern.ch/record/436912?ln=en). It corrects the two main physical failures of the harmonic oscillator: the harmonic parabola rises without bound as the bond shortens and never approaches a dissociation limit as the bond stretches, so it predicts that bonds never break[3](https://qchem.qc-edu.org/HOandMorse.html).
| Key fact | Detail |
|---|---|
| Functional form | \( V(r) = D_e \left(1 - e^{-a(r-r_e)}\right)^2 \), with minimum \( V(r_e) = 0 \) and asymptote \( D_e \) at large \( r \)[1](https://old.goldbook.iupac.org/html/M/MT07073.html) • [3](https://qchem.qc-edu.org/HOandMorse.html) |
| Width parameter | \( a = \sqrt{\kappa_e / (2D_e)} \), where \( \kappa_e = V''(r_e) \) is the force constant at the minimum[3](https://qchem.qc-edu.org/HOandMorse.html) |
| Energy levels | \( E_k = \hbar\omega_0 (k+\tfrac{1}{2}) - [\hbar\omega_0 (k+\tfrac{1}{2})]^2 / (4D_e) \), with \( \omega_0 = \sqrt{\kappa_e/\mu} \)[3](https://qchem.qc-edu.org/HOandMorse.html) |
| Bound states | Finite in number: \( k_{\max} = \lfloor 2D_e/(\hbar\omega_0) - \tfrac{1}{2} \rfloor \), versus infinitely many for the harmonic oscillator[3](https://qchem.qc-edu.org/HOandMorse.html) |
| Origin | Philip M. Morse, Palmer Physical Laboratory, Princeton University; *Physical Review* **34**, 57 (1 July 1929)[2](https://cds.cern.ch/record/436912?ln=en) • [4](https://jupiter.chem.uoa.gr/thanost/papers/papers4/PR_34%281929%2957.pdf) |
| Known limitation | The exponential approach to the dissociation limit does not reproduce the correct inverse-power long-range behavior of real molecular potentials[5](https://onlinelibrary.wiley.com/doi/full/10.1002/qua.26808) |
| Practical use | Bond term in molecular force fields and a model for atom-surface interactions; most ambient-temperature force fields still use the harmonic form[6](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/molecular-dynamics/molecular_mechanics.html) • [7](https://www.sciencedirect.com/science/article/abs/pii/S0003491613001930) |
## Why the harmonic oscillator fails
Within the Born-Oppenheimer approximation, the nuclear potential \( V(R) \) of a bound diatomic molecule at low vibrational energy resembles a harmonic oscillator, a parabola \( V = \tfrac{1}{2}k(R-R_e)^2 \) with equally spaced energy levels[8](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/06%3A_Molecular_Spectroscopy_of_Iodine/6.03%3A_Potential_Energy_Curves). Real bonds are anharmonic: as a bond stretches it weakens and ultimately dissociates as \( R \to \infty \), while compression raises the energy faster than the harmonic form because of interatomic repulsion[8](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/06%3A_Molecular_Spectroscopy_of_Iodine/6.03%3A_Potential_Energy_Curves). The Morse potential captures both asymmetries with only two adjustable parameters beyond the minimum position, and it flattens to the dissociation limit \( D_e \) at large separation[3](https://qchem.qc-edu.org/HOandMorse.html).
**Anharmonic spectroscopy.** [Anharmonicity](https://www.edgepedia.org/anharmonicity) produces a finite number of vibrational levels whose spacing decreases with increasing energy; the difference between neighboring levels approaches zero as the energy approaches dissociation[8](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/06%3A_Molecular_Spectroscopy_of_Iodine/6.03%3A_Potential_Energy_Curves). The Morse potential reproduces the standard two-term expression
\( G_v = \omega_e (v+\tfrac{1}{2}) - \omega_e\chi_e (v+\tfrac{1}{2})^2 \) used to fit observed vibrational levels, and its parameters can be recovered from the spectroscopic constants \( \omega_e \) and \( \omega_e\chi_e \)[8](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/06%3A_Molecular_Spectroscopy_of_Iodine/6.03%3A_Potential_Energy_Curves).
## Exact solution
The [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation) for the Morse oscillator is exactly solvable, a rare property among nonlinear (anharmonic) potentials[3](https://qchem.qc-edu.org/HOandMorse.html) • [9](https://www.sciencedirect.com/topics/chemistry/morse-potential). The eigenfunctions are complicated functions of the associated [Laguerre polynomials](https://www.edgepedia.org/laguerre-polynomials), but the energies have the compact form[3](https://qchem.qc-edu.org/HOandMorse.html)
\[ E_k = \hbar\omega_0 \left(k+\tfrac{1}{2}\right) - \frac{\left[ \hbar\omega_0 \left(k+\tfrac{1}{2}\right) \right]^2}{4 D_e} \]
with \( \omega_0 = a\sqrt{2D_e/\mu} = \sqrt{\kappa_e/\mu} \), the same frequency formula as the harmonic oscillator with the same curvature at the minimum[3](https://qchem.qc-edu.org/HOandMorse.html). The quadratic term is the anharmonic correction that pulls the levels down and makes their spacing shrink with \( k \)[3](https://qchem.qc-edu.org/HOandMorse.html).
Unlike the harmonic oscillator, which binds infinitely many states, the Morse potential binds only a finite number, up to
\( k_{\max} = \lfloor 2D_e/(\hbar\omega_0) - \tfrac{1}{2} \rfloor \)[3](https://qchem.qc-edu.org/HOandMorse.html). Because the levels crowd together near dissociation, the Morse oscillator holds about twice as many states below \( D_e \) as the harmonic oscillator with the same curvature[3](https://qchem.qc-edu.org/HOandMorse.html). This finite bound spectrum is one of the ways the model resembles a real vibrating molecule[3](https://qchem.qc-edu.org/HOandMorse.html).
The spacing formula also shows where the model stops: the level difference
\( E_{n+1} - E_n = h\nu_0 - (n+1)(h\nu_0)^2/(2D_e) \) becomes zero at
\( n^* = 2D_e/(h\nu_0) - 1 \) and negative above it, so levels beyond \( n^* \) are not physically meaningful[10](https://arxiv.org/abs/2206.08975).
## Dissociation energy: De versus D0
\( D_e \) measures the depth of the potential well, the energy difference between the dissociation limit and the bottom of the curve. A real molecule sits at the zero-point level \( v=0 \), not at the minimum, so the energy actually needed to dissociate from the ground vibrational state,
\( D_0 \), is smaller: approximately
\( D_0 \approx D_e - \tfrac{1}{2}\hbar\omega \)[11](https://chem.libretexts.org/Courses/University_of_North_Carolina_Charlotte/CHEM_2141%3A__Survey_of_Physical_Chemistry/09%3A_Quantum_Basics/9.15%3A_Molecular_Spectroscopy). Spectroscopy texts distinguish the two quantities for exactly this reason[12](https://mathabhangacollege.ac.in/wp-content/uploads/2024/02/MOLECULAR-SPECTROSCOPY-AND-PHOTOCHEMISTRY_compressed.pdf).
## Origin and early use
Morse published the potential and its exact wave-mechanical solution in "Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels" (*Physical Review* **34**, 57, received April 8, 1929), showing that the resulting level formula
\( E(n) = E_e + h\omega_0(n+\tfrac{1}{2}) - h\omega_0 x (n+\tfrac{1}{2})^2 \) expresses the experimental vibrational values of diatomic molecules quite accurately[2](https://cds.cern.ch/record/436912?ln=en) • [4](https://jupiter.chem.uoa.gr/thanost/papers/papers4/PR_34%281929%2957.pdf). He tested the function on neutral diatomic molecules and ions in ground and excited electronic states, calculating curves for species including BeO, BO, C₂, CN, CO, F₂, H₂, I₂, N₂, and O₂[5](https://onlinelibrary.wiley.com/doi/full/10.1002/qua.26808). A 2021 review of 50 analytical potential functions for diatomic systems counts the Morse function as the historical reference against which later forms were judged[5](https://onlinelibrary.wiley.com/doi/full/10.1002/qua.26808).
## Applications
Beyond diatomic spectroscopy, the Morse potential is used as an approximate bond-stretching potential energy function in polyatomic molecules and to model interactions between an atom and a surface[7](https://www.sciencedirect.com/science/article/abs/pii/S0003491613001930). In classical molecular dynamics it serves as a bond potential with three parameters per bonded pair, \( D_{ij} \), \( a_{ij} = \sqrt{k_{ij}/(2D_{ij})} \), and \( r_{0,ij} \), and it permits bond dissociation events to be simulated, though only when the bond is specified in advance[6](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/molecular-dynamics/molecular_mechanics.html). At ambient temperatures, where bond energies of interest lie within roughly 40 kJ/mol of the minimum, a Morse potential and a harmonic potential differ little, so most force fields keep the cheaper two-parameter harmonic form[6](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/molecular-dynamics/molecular_mechanics.html).
## Limitations and extensions
The main physical defect is the long-range behavior. Real molecular potentials approach the dissociation limit polynomially, through inverse-power van der Waals terms, whereas the Morse form approaches it exponentially, faster than any power law[5](https://onlinelibrary.wiley.com/doi/full/10.1002/qua.26808) • [13](https://www.sciencemagazinedigital.org/sciencemagazine/11_december_2015?pg=72). The simple form also lacks the flexibility to fit the full shape of accurate potential energy curves[10](https://arxiv.org/abs/2206.08975).
Extensions address these defects. The Morse/Long-Range (MLR) potential of Le Roy and Henderson (2007) builds in the correct long-range asymptotic behavior, which improves extrapolation and allows parameters such as \( D_e \) to be determined from limited data[14](https://www.sciencedirect.com/science/article/abs/pii/S0022285223000760). The Generalized Morse Potential adds one exponential term and one parameter to repair the long-range behavior; it reverts to the original Morse form at special parameter values, but its vibrational levels must be obtained numerically because the added term destroys analytic solvability[10](https://arxiv.org/abs/2206.08975). Comparative fitting studies against experimental Rydberg-Klein-Rees curves also evaluate Morse alongside Manning-Rosen and Rosen-Morse oscillators, with the caution that better goodness-of-fit does not necessarily mean more accurate vibrational levels[15](https://www.sciencedirect.com/science/article/pii/S2211715621001090).
## References
1. [Morse potential, IUPAC Gold Book](https://old.goldbook.iupac.org/html/M/MT07073.html)
2. [Diatomic molecules according to the wave mechanics. II. Vibrational levels, P. M. Morse, Phys. Rev. 34 (1929) pp. 57, CERN Document Server record](https://cds.cern.ch/record/436912?ln=en)
3. [Vibrational Polarizability: The Harmonic Oscillator and the Morse Oscillator, Quantum Chemistry (qc-edu.org)](https://qchem.qc-edu.org/HOandMorse.html)
4. [P. M. Morse (1929), Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels, Physical Review 34, 57 (PDF)](https://jupiter.chem.uoa.gr/thanost/papers/papers4/PR_34%281929%2957.pdf)
5. [Araújo et al. (2021), A comparative review of 50 analytical representations of potential energy interaction for diatomic systems: 100 years of history, International Journal of Quantum Chemistry](https://onlinelibrary.wiley.com/doi/full/10.1002/qua.26808)
6. [Molecular Mechanics, Lecture Notes on Molecular Dynamics, University of Marburg](https://fb15.pages.uni-marburg.de/ag-von-domaros/teaching/molecular-dynamics/molecular_mechanics.html)
7. [Construction of the Barut–Girardello quasi coherent states for the Morse potential, Annals of Physics (ScienceDirect)](https://www.sciencedirect.com/science/article/abs/pii/S0003491613001930)
8. [6.3: Potential Energy Curves, Chemistry LibreTexts (Duke University)](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/06%3A_Molecular_Spectroscopy_of_Iodine/6.03%3A_Potential_Energy_Curves)
9. [Morse Potential - an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/chemistry/morse-potential)
10. [Qadeer et al. (2022), Vibrational Levels of a Generalized Morse Potential, arXiv:2206.08975](https://arxiv.org/abs/2206.08975)
11. [9.15: Molecular Spectroscopy, Chemistry LibreTexts (UNC Charlotte)](https://chem.libretexts.org/Courses/University_of_North_Carolina_Charlotte/CHEM_2141%3A__Survey_of_Physical_Chemistry/09%3A_Quantum_Basics/9.15%3A_Molecular_Spectroscopy)
12. [Molecular Spectroscopy and Photochemistry (course text, Anharmonic Oscillator chapter)](https://mathabhangacollege.ac.in/wp-content/uploads/2024/02/MOLECULAR-SPECTROSCOPY-AND-PHOTOCHEMISTRY_compressed.pdf)
13. [Science Magazine, 11 December 2015 (long-range potential discussion)](https://www.sciencemagazinedigital.org/sciencemagazine/11_december_2015?pg=72)
14. [Extrapolation properties of the Morse-Long Range potential at large internuclear distances, Journal of Molecular Spectroscopy (2023)](https://www.sciencedirect.com/science/article/abs/pii/S0022285223000760)
15. [Accuracy of Morse and Morse-like oscillators for diatomic molecular interaction: A comparative study, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S2211715621001090)
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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Chemical bonding and intermolecular forces*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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