Eur. Phys. J. Plus (2026) 141: 84
https://doi.org/10.1140/epjp/s13360-025-07252-6

Regular Article

Analytical results for classical stable states of one-electron diatomic molecules: beyond Born–Oppenheimer approximation

Physics Department, Auburn University, 380 Duncan Drive, 36849, Auburn, AL, USA

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Received: 13 September 2025
Accepted: 22 December 2025
Published online: 28 January 2026

Abstract

Classical models of various quantum systems have been successfully used by many authors. Such models provide the physical insight that quantum calculations lack. Classical models of molecules are indispensable in a variety of branches of chemical research. In our previous papers we had presented classical studies of quasimolecules ZeZ₁, i.e., one-electron diatomic molecules consisting of the two nuclei of charges Z and Z₁ ≥ Z, separated by the distance R, and one electron. In plasmas there can be transient one-electron diatomic quasi-molecules containing nuclei of Z₁ ≥ Z >  > 1, which can form during charge exchange between multicharged ions in plasmas. We had shown analytically that the dependence of the scaled energy on the scaled internuclear distance has three branches, i.e., three “classical energy terms”, what was a counterintuitive result. Moreover, two out of these three classical energy terms cross, what allowed constructing a classical model of charge exchange, thus showing that charge exchange actually is not a purely quantum phenomenon, but has classical roots. In the present paper the focus is not on the transient quasimolecules ZeZ₁, but rather on stable molecules ZeZ₁, i.e., pure atomic physics (no plasmas). We perform the classical analytical study of such molecules where, in addition to the conditions of the 3D-equilibrium of the electron, we also take into account the condition of the 1D-equilibrium of the nuclei. The latter condition would be irrelevant in the model, used in the great many works on diatomic molecules, where the nuclei are assumed to have infinite masses, what amounts to the Born–Oppenheimer approximation. Thus, by including the condition of the nuclear equilibrium, we go beyond the Born–Oppenheimer approximation – classically. We show that out of the entire set of 6903 pairs of Z₁ > Z (among the currently known nuclei of charges from 1 to 118), the above equilibrium conditions are met only for 117 pairs, where Z = 1 and Z₁ ranges from 2 to 118. For such stable molecules, we derive analytically parameters of the stable equilibrium orbit of the electron, as well as – for the ground state –the electron total energy and the electron kinetic energy, the latter representing also the ionization energy of these molecules. We provide the corresponding numerical data for the stable ZeZ₁ molecules, where Z = 1 and Z₁ ranges from 2 to 10. We compare our results for the electron total energy in the ground state with quantum calculations in the literature available for only few such molecules, the quantum calculations being performed within the Born–Oppenheimer approximation. The comparison shows a good agreement: the relative difference is just ~ 10%. Such good accuracy of the classical calculations even for the ground state of the molecules ZeZ₁ is a counterintuitive result.