We present a detailed study of the ground state of the Coulomb system $(\ensuremath{\alpha}\ensuremath{\alpha}ee)$ which corresponds to the
He22+ molecular ion in a magnetic field $B=0\ensuremath{-}4.414\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.3em}{0ex}}\mathrm{G}$ in parallel configuration (it is assumed that infinitely massive $\ensuremath{\alpha}$ particles are situated along a magnetic-field line). One uses a variational method, introducing a trial function which includes electronic correlation in the form $\mathrm{exp}(\ensuremath{\gamma}{r}_{12})$, where $\ensuremath{\gamma}$ is a variational parameter. It is shown that the quantum numbers of the lowest total-energy state depend on the magnetic-field strength. This state evolves from the spin-singlet $^{1}{\ensuremath{\Sigma}}_{g}$ metastable state at $0\ensuremath{\le}B\ensuremath{\lesssim}0.85\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ to a repulsive spin-triplet $^{3}{\ensuremath{\Sigma}}_{u}$ state for $0.85\ensuremath{\lesssim}B\ensuremath{\lesssim}1100\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ and, finally, to a stable strongly bound spin-triplet $^{3}{\ensuremath{\Pi}}_{u}$ state at stronger fields $B\ensuremath{\gtrsim}1100\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ The lowest vibrational energy for the latter case is calculated.
The energy levels of hydrogen and helium atoms in strong magnetic fields are calculated in this study. The current work contains estimates of the binding energies of the first few low-lying states of these systems that are improvements upon previous estimates. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these one- and two-electron systems in a self-consistent manner. The method described herein is applicable to calculations of atomic structure in magnetic fields of arbitrary strength as it exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. The method is found to be readily extendable to systems with more than two electrons.
We study the question of existence, i.e., stability with respect to dissociation of the spin-quartet permutation- and reflection-symmetric ${}^{4}{(\ensuremath{-}3)}^{+}{}_{g}$ (${S}_{z}=\ensuremath{-}3/2,M=\ensuremath{-}3$) state of the $(\ensuremath{\alpha}\ensuremath{\alpha}eee)$ Coulomb system: the
He2+ molecular ion, placed in a magnetic field $0\ensuremath{\le}B\ensuremath{\le}10\phantom{\rule{0.16em}{0ex}}000$ a.u. We assume that the $\ensuremath{\alpha}$ particles are infinitely massive (Born-Oppenheimer approximation of zero order) and adopt the parallel configuration, when the molecular axis and the magnetic field direction coincide, as the optimal configuration. The study of the stability is performed variationally with a physically adequate trial function. To achieve this goal, we explore several helium-containing compounds in strong magnetic fields, in particular; we study the spin-quartet ground state of the ${\mathrm{He}}^{\ensuremath{-}}$ ion and the ground (spin-triplet) state of the helium atom, both for a magnetic field in $100\ensuremath{\le}B\ensuremath{\le}10\phantom{\rule{0.16em}{0ex}}000$ a.u. The main result is that the
He2+ molecular ion in the state ${}^{4}{(\ensuremath{-}3)}^{+}{}_{g}$ is stable towards all possible decay modes for magnetic fields $B\ensuremath{\gtrsim}120\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$ and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At
B=1000a.u., the dissociation energy of
He2+ into ${\mathrm{He}}^{\ensuremath{-}}+\ensuremath{\alpha}$ is $\ensuremath{\sim}702\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ and the dissociation energy for the decay channel to $\mathrm{He}+\ensuremath{\alpha}+e$ is $\ensuremath{\sim}729\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$, and both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209.
The existence and stability of the van der Waals
1u state of the molecular ion \ppe with fixed centers (infinitely massive nuclei) in the presence of a magnetic field with arbitrary orientations in the range of strengths
0≤B≤10 a.u. (1 a.u.
≡2.35×109G) is studied within the non-relativistic framework. The study is based on the variational method with physical relevant trial functions. A particular emphasis to the gauge optimization through variational parameters is incorporated. It is shown that, for all the magnetic fields studied, the potential energy curve has a pronounced minimum for finite internuclear distances and the optimal configuration of minimal total energy is realized when the molecular axis is oriented along the magnetic field lines (parallel configuration). We found a domain of magnetic fields
0≤B≲6.6 a.u. where the
1u state in parallel configuration is stable towards dissociation \hbox{\ppe
↛ H + p}. The maximal stability is found to be at magnetic field
B∼2a.u. where the potential well supports one vibrational state below the dissociation limit.
Abstract The variational Monte Carlo method is applied to investigate several properties of the beryllium atom, ions, and its iso-electronic ions. For the ground and the excited states of the beryllium atom and ions, the energy eigenvalues were evaluated freely and under the influence of magnetic field. Furthermore, the iso-electronic ions (B+ and C2+) are also investigated under the influence of the external magnetic field. Suitable trial wave functions including the spin and the correlated parts are used in these investigation. Some new excited states were included in the present work, such as the low-lying states (1s22s3s and 1s22s3p) and the core states (1s2s23s and 1s2s3s2). For the spin functions of these states, we used two different functions for the singlet and the triplet excited states. Moreover, the energies of the beryllium ions (Be+ and Be2+) were evaluated freely and in the presence of magnetic field. The obtained results are in good agreement with the corresponding results of other works. Graphical abstract