Spectroscopy and physics of atoms and molecules

Approximation of the electronic term of the diatomic molecule by the Morse function. Inversion of anharmonicity

G. S. Denisov, K. G. Tokhadze, R. E. Asfin

Saint Petersburg State University

Abstract: The approximation of interatomic potentials in diatomic molecules using the Morse potential typically leads to an overestimated bond dissociation energy, calculated as $D'_e=\omega_e^2/4\omega_e x_e$, based on known values of $\omega_e$ и $\omega_e x_e$ determined from the first two vibrational transitions, 0–1 and 1–2. This relationship holds true for a wide range of molecules, such as H$_2$, O$_2$, N$_2$, HF, HCl, and many others. However, for some molecules and diatomic ions, the extrapolated value of the dissociation energy $D'_e$ turns out to be lower than the actual value $D_e$. In such molecules, the shape of the potential energy curve deviates significantly from the standard form due to a broadening in the lower part of the potential well, which manifests as a large anharmonicity $\omega_e x_e$. This feature is conveniently analyzed using the difference $\delta(r)=U(r)-M(r)$ between the actual potential and its Morse approximation. This type of approximation yields a Morse solution $M1(r)$ that accurately describes the lower part of the potential for the simple molecules, with a monotonic increase in deviation as it approaches the dissociation asymptote. An alternative solution, $M2(r)$, is constructed based on the known values of $D_e$ and $\omega_e$, while the anharmonicity $\omega_e x'_e$ is computed as $\omega_e x'_e=\omega_e^2/4D_e$. The $M2(r)$ approximation provides a better description of the upper part of the potential and a satisfactory representation of the lower part. The deviation from the actual potential takes the form of a bell-shaped curve, whose maximum is typically located above the midpoint of the potential well. This paper presents several examples of potentials of a special type, for which $D'_e<D_e$ и $\omega_e x_e>\omega_e x'_e$, a behavior that can be described by the term inversion of anharmonicity.

Keywords: diatomic molecules, Morse potential, anharmonicity, vibrational structure, potential function approximation.

Received: 24.06.2025
Revised: 24.06.2025
Accepted: 25.07.2025

DOI: 10.61011/OS.2025.09.61757.8313-25