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arXiv:2608.22535

Hydrogen Molecular Ion and Molecule in Classical Electrodynamics with Classical Zero-Point Radiation

Timothy H. Boyer

physics.class-phquant-phSubmitted Aug 23, 2026• Announced Aug 25, 2026
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Abstract

The hydrogen molecular ion and the hydrogen molecule are treated in an approximate calculation based on classical electrodynamics which includes classical zero-point radiation. It is found within the classical theory that a molecular ion is less-well bound than a hydrogen atom plus a distant proton. The smaller binding energy is explained due to the repulsive nature of the force between the proton and the atom when considering the most natural resonant orbit of the electron. The approximate classical electromagnetic calculation gives a binding energy of $2.1eV$ and a proton separation of $0.916A^{o}$ for the hydrogen molecular ion. The hydrogen molecule is formed by adding a single additional electron to the ion. The electrostatic attraction of the electron to the ion gives a binding energy of $4.6eV$ and a inter-proton separation of $0.6A^{o}$ for the hydrogen molecule in this classical electromagnetic approximation.

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14 pages

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The hydrogen molecular ion and the hydrogen molecule are treated in an approximate calculation based on classical electrodynamics which includes classical zero-point radiation. It is found within the classical theory that a molecular ion is less-well bound than a hydrogen atom plus a distant proton. The smaller binding energy is explained due to the repulsive nature of the force between the proton and the atom when considering the most natural resonant orbit of the electron. The approximate classical electromagnetic calculation gives a binding energy of 2.1eV2.1eV2.1eV and a proton separation of 0.916Ao0.916A^{o}0.916Ao for the hydrogen molecular ion. The hydrogen molecule is formed by adding a single additional electron to the ion. The electrostatic attraction of the electron to the ion gives a binding energy of 4.6eV4.6eV4.6eV and a inter-proton separation of 0.6Ao0.6A^{o}0.6Ao for the hydrogen molecule in this classical electromagnetic approximation.
An external magnetic field is applied to the classical hydrogen atom treated in classical electromagnetic theory including classical electromagnetic zero-point radiation. In an earlier article, it was shown that the average value of each of the electron's action variables appears as a discrete value, corresponding to a representation of the rotation group, because of resonance between the periodic orbit of the electron and the random classical zero-point radiation. Here it is shown that, in the presence of a magnetic field and because of resonance, the classical orbital motion of the electron is in resonance with random classical zero-point radiation only for orientations of the orbit which take integer values for the angle made with the direction of the magnetic field, but excluding the m=0 orientation where the magnetic field is parallel to the orbital plane of the electron. Classical electromagnetic explanations are given for the Stern-Gerlach result, and for the Zeeman effect.
Classical electrodynamics including classical electromagnetic zero-point radiation leads to a ground state and resonant excited states for a charged particle in a Coulomb potential. These resonant states correspond to integer values of the action variables analogous to those appearing in the Bohr-Sommerfeld theory of the hydrogen atom. The work on classical zero-point radiation reported here is a continuation of the analysis reported in 1975, but with the addition of the ideas of relativity and resonance between the charged-particle orbit and classical zero-point radiation.
The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are: Nuclear separation 1.0 1.3 1.6 1.8 2.0 2.2 2.4 2.95 Electronic energy 2.896 2.648 2.436 2.309 2.204 2.109 2.025 1.836 The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.
This paper develops and verifies a physics-first description of the hydrogen-atom transition from the L shell (n = 2) to the K shell (n = 1) by combining the historical Bohr model of atomic structure with classical Maxwell electrodynamics. Beginning with the Coulomb force, centripetal dynamics, and Bohr’s angular-momentum quantization condition, the analysis derives the allowed orbital radii, electron velocities, angular momenta, kinetic energies, electrostatic potential energies, and total bound-state energies of the hydrogen atom. For the n = 2 → n = 1 transition, the calculated energy difference is approximately 10.2 eV, corresponding through the Planck–Einstein relation ΔE = hν to a frequency of approximately 2.47 × 10¹⁵ Hz and a wavelength near 121.6 nm, consistent with the hydrogen Lyman-α transition. The emitted radiation is then examined through Maxwell’s equations and the vacuum constitutive relations. The resulting electromagnetic-wave description yields the transverse electric and magnetic fields together with the associated electromagnetic energy density, Poynting energy flux, momentum density, radiation pressure, and Maxwell stress tensor. In this way, the atomic energy difference obtained from the Bohr model is connected quantitatively to the classical propagation properties of the outgoing electromagnetic radiation. The derivation is mathematically consistent as a historical Bohr–Maxwell reconstruction, but it does not constitute a closed dynamical theory of spontaneous emission. The essential bridge between discrete atomic energy levels and the continuous Maxwell field is supplied by the quantum frequency condition ΔE = hν. Neither classical Maxwell electrodynamics nor Newtonian orbital mechanics independently derives this relation, and classical Maxwell theory does not by itself quantize the emitted radiation energy into individual photons. A complete treatment of spontaneous emission, transition probabilities, radiative lifetimes, and photon-field quantization requires quantum mechanics and ultimately quantum electrodynamics. Accordingly, the framework is presented as a verified correspondence between established limiting descriptions rather than as a replacement for Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics. By clearly separating the atomic bound-state calculation, the quantum energy-frequency bridge, and the classical electromagnetic propagation problem, the analysis provides a rigorous conceptual and mathematical foundation for subsequent Maxwell–Schrödinger, Dirac–Maxwell, quantum-electrodynamic, Maxwell–Scretching, and Scretching Quantum Chain formulations of hydrogenic radiative transitions.
A new vibrational level of the H 2 + molecular ion with binding energy of 1.09 × 10 −9 a.u. ≈ 30 neV below the first dissociation limit is predicted, using highly accurate numerical non-relativistic quantum calculations, which go beyond the Born-Oppenheimer approximation. It is the first-excited vibrational level v = 1 of the 2 p σ u electronic state, antisymmetric with respect to the exchange of the two protons, with orbital angular momentum L = 0. It manifests itself as a huge p-H scattering length of a = 750 ± 5 Bohr radii.