# Applications of the Classical Two-Coulomb-Center Systems to Atomic/Molecular Physics

## Abstract

We advanced the classical studies of the two-Coulomb-center (TCC) systems consisting of two nuclei of charges Z and Z' and an electron in a circular or a helical state centered on the internuclear axis. These systems represent diatomic Rydberg quasimolecules encountered, e.g., in plasmas containing more than one kind of multicharged ions. Diatomic Rydberg quasimolecules are one of the most fundamental theoretical playgrounds for studying charge exchange. Charge exchange and crossings of corresponding energy levels that enhance charge exchange are strongly connected with problems of energy losses and of diagnostics in high temperature plasmas; besides, charge exchange is one of the most effective mechanisms for population inversion in the soft x-ray and VUV ranges. The classical approach is well-suited for Rydberg quasimolecules. First, we considered diatomic Rydberg quasimolecules subjected to a static electric field parallel to the internuclear axis. We found the appearance of an additional (fourth) term, which was absent at the zero field, and which had a V-type crossing with the lowest term. We also found X-type crossings (absent at the zero field) which significantly enhance charge exchange. Second, we studied effects of the electron screening in plasmas on diatomic Rydberg quasimolecules. We found that the screening stabilizes the nuclear motion for Z = 1 and destabilizes it for Z > 1. We also found that a so-called continuum lowering in plasmas was impeded by the screening, creating the effect similar to that of the magnetic field and opposite to that of the electric field. The continuum lowering plays a key role in calculations of the equation of state, partition function, bound-free opacities, and other collisional atomic transitions in plasmas. Third, we considered diatomic Rydberg quasimolecules in a laser field. For the situation where the laser field is linearly-polarized along the internuclear axis, we found an analytical solution for the stable helical motion of the electron valid for wide ranges of the laser field strength and frequency. We also found resonances, corresponding to a laser-induced unstable motion of the electron, that result in the destruction of the helical states. For the case of a circularly-polarized field, polarization plane being perpendicular to the internuclear axis, we found an analytical solution for circular Rydberg states valid for wide ranges of the laser field strength and frequency. For this case we demonstrated also that there is a red shift of the primary spectral component. We showed that both under the linearly-polarized laser field and under the circularly-polarized laser field, in the electron radiation spectrum in the addition to the primary spectral component at (or near) the unperturbed revolution frequency of the electron, there appear satellites. Under a laser field of a known strength, in the case of the linear polarization the observation of the satellites would be the confirmation of the helical electronic motion in the Rydberg quasimolecule, while in the case of the circular polarization the observation of the red shift of the primary spectral component would be the confirmation of the specific type of the phase modulation of the electronic motion. Conversely, if the laser field strength is unknown, both the relative intensities of the satellites and the red shift of the primary spectral component could be used for measuring the laser field strength. Fourth, we considered TCC systems consisting of a proton, muon and an electron. We found that a muonic hydrogen atom can attach an electron, with the muon and electron being in circular states. The technique of the separation of rapid and slow subsystems was used, where the muon represented the rapid subsystem and the electron the slow subsystem. We showed that the spectral lines emitted by the muon experience a red shift compared to the corresponding spectral lines in a muonic hydrogen atom. Observing this red shift should be one of the ways to detect the formation of such muonic-electronic negative hydrogen ions. Studies of muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton μ–, have several applications, such as muon-catalyzed fusion, a laser-control of nuclear processes, and a search for strongly interacting massive particles proposed as dark matter candidates and as candidates for the lightest supersymmetric particle. Fifth, we studied fundamental algebraic symmetry of the TCC systems leading to an additional conserved quantity: the projection of a super-generalized Runge-Lenz vector on the internuclear axis. We derived the correct super-generalized Runge-Lenz vector, whose projection on the internuclear axis is conserved, and showed that the corresponding expressions by other authors did not correspond to a conserved quantity and thus were incorrect. The correct super-generalized Runge-Lenz vector for the TCC systems that we derived should be of a general theoretical interest since the TCC systems represent one of the most fundamental problems in physics. It can also have practical applications: e.g., it can be used as a necessary tool while applying to the TCC systems the robust perturbation theory for degenerate states based on the integrals of the motion.