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The effect of spontaneous breaking of initial SO(3) symmetry is shown to be possible for an H-like atom in the ground state, when it is confined in a spherical box under general boundary conditions of "not going out" through the box surface (i.e. third kind or Robin's ones), for a wide range of physically reasonable values of system parameters. The reason is that such boundary conditions could yield a large magnitude of electronic wavefunction in some sector of the box boundary, what in turn promotes atomic displacement from the box center towards this part of the boundary, and so the underlying SO(3) symmetry spontaneously breaks. The emerging Goldstone modes, coinciding with rotations around the box center, restore the symmetry by spreading the atom over a spherical shell localized at some distances from the box center. Atomic confinement inside the cavity proceeds dynamically -due to the boundary condition the deformation of electronic wavefunction near the boundary works as a spring, that returns the atomic nuclei back into the box volume.

Abstract:The ground state properties of the two-electron atom with atomic number 2 Z ≥ in the spherical vacuum cavity with general boundary conditions of "not going out" are studied. It is shown that for certain parameters of the cavity such atom could either decay into the one-electron atom with the same atomic number and an electron or be in stable state with the binding and ionization energies several times bigger than the same energies of the free atom. By analogy with the Wigner-Seitz model of metallic bonding, the possibility of the existence of such effects on the lattice formed by the vacuum cavities filled with the two-electron atoms of the same type is discussed.

The behavior of atomic H in a semi-bounded space z ≥ 0 with the condition of "not going through" the boundary (the surface z = 0) for the electronic wavefunction (WF) is considered. It is shown that in a wide range of "not going through" condition parameters the effective atomic potential, treated as a function of the distance h from H to the boundary plane, reveals a well pronounced minimum at certain finite but nonzero h, which describes the mode of "soaring" of the atom above the plane. In particular cases of Dirichlet and Neumann conditions, the analysis of the soaring effect is based on the exact analytical solutions of the problem in terms of generalized spheroidal Coulomb functions. For h varying between the regions h ) a B and h ( a B , both the deformation of the electronic WF and the atomic state are studied in detail. For a more general case of Robin (third type) condition, the variational estimates and direct numerical tools are used. By means of the latter it is also shown that in the case of a sufficiently large positive affinity of the atom to the boundary plane a significant reconstruction of the lowest levels takes place, including the change of both the asymptotics and the general dependence on h. K E Y W O R D S confined quantum systems, levels reconstruction, soaring effect 1 | INTRODUCTIONConsiderable amount of theoretical and experimental activity has been focused recently on spatially confined atoms and molecules. [1][2][3] The interest is largely due to the nontrivial physical and chemical properties that arise for quantum systems in such a state of complete or incomplete confinement. The interaction of confined particles with the environment, forming the cavity or volume boundary, is usually simulated by means of a suitable boundary condition, imposed on their wavefunctions (WF). The pioneering works on quantum system in a closed cavity are the Wigner-Seitz model [4,5] and the papers [6,7] on atomic H in a spherical cavity. More concretely, in the Wigner-Seitz model the metallic bond formation in alkali metals has been considered in terms of the Neumann condition for the valence electron on the boundary of the corresponding Wigner-Seitz cell. In Ref.[6] the exact solution for atomic H in a spherical cavity with the Dirichlet boundary condition was found, while in Ref.[7] such a model has been used for description of atomic H under high pressure.Atoms in the Euclidean half-space ℜ 3 / 2 have been first explored by Levine, [8] devoted to the properties of the impurity donor atom placed on the plane boundary of the dielectric crystal. Due to a large positive affinity it is energetically favorable for the valence electron to reside inside the crystal that allows to simulate the crystal boundary by means of the Dirichlet condition imposed on the electronic WF. Afterwards, this model All authors contributed equally to this work.

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It is shown that reconstruction of the low lying energy levels of the ion under conditions of confinement in a given spatial volume is substantially more significant as compared to the confinement by simple potential barrier. Depending on the cavity parameters, the ground state binding energy of could be significantly greater than that of the free ion, while the behavior of the lowest molecular term of the ion turns out to be quite different when considered as a function of the internuclear distance. In particular, two minima might occur in , while the relationship between them could be sufficiently different depend ing on the cavity parameters, which is shown on the phase diagram that was obtained in the work. We studied the case of a "mexican hat" structure of the effective ion potential in detail. As a result, the lowest electronic level of ion splits into the ground state level and the first excited one, with difference between them being as small as ~10 -4 eV. As in the NH 3 molecule, in the last case the lowest level gives rise to an effective two level system that is separated from the vibrational and rotational modes by a wide energy gap. More concretely, cal culation using the Neumann conditions that actually reproduce the confinement on a lattice of similar cavi ties shows that the splitting of ~10 -4 eV occurs for the linear sizes of the confinement area of the order of some a B and a shell potential magnitude of ~10 eV.

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