On 1/Z expansion for two-electron systems

The $1/Z$-expansion for the Coulomb system of infinitely massive center of charge Z and two electrons is discussed. Numerical deficiency in Baker et al, {\em Phys. Rev. \bf A41}, 1247 (1990) is indicated which continue to raise doubts in correctness of their calculations of the higher order coefficients in $1/Z$-expansion expressed in Refs.[4-5]. It is shown that a minor modification of a few first coefficients found in Ref.[3] allows to calculate the ground state energies at $Z\ =\ 1, 2,\ldots \ 10$ (as well as at $Z > 10$) with a portion of 15th decimal digit in comparison with highly accurate calculations by C. Schwartz and by Nakashima-Nakatsuji. Ground state energies of two-electron ions $Z=11\ (Na^{9+})$ and $Z=12\ (Mg^{10+})$ are found with 14 decimal digits.Comment: 7 pages, 2 Tables, two extra references added, Conclusions extended, two "Note added.I-II." adde

F4F_4 Quantum Integrable, rational and trigonometric models: space-of-orbits view

Algebraic-rational nature of the four-dimensional, $F_4$-invariant integrable quantum Hamiltonians, both rational and trigonometric, is revealed and reviewed. It was shown that being written in $F_4$ Weyl invariants, polynomial and exponential, respectively, both similarity-transformed Hamiltonians are in algebraic form, they are quite similar the second order differential operators with polynomial coefficients; the flat metric in the Laplace-Beltrami operator has polynomial (in invariants) matrix elements. Their potentials are calculated for the first time: they are meromorphic (rational) functions with singularities at the boundaries of the configuration space. Ground state eigenfunctions are algebraic functions in a form of polynomials in some degrees. Both Hamiltonians preserve the same infinite flag of polynomial spaces with characteristic vector $(1, 2, 2, 3)$, it manifests exact solvability. A particular integral common for both models is derived. The first polynomial eigenfunctions are presented explicitly.Comment: 19 pages, Sec.2 rewritten, typos corrected, to be published in Journal of Physics (conference series), talk presented at QTS-8, Mexico-City, August 201

Stable He−^- can exist in a strong magnetic field

The existence of bound states of the system (\al,e,e,e) in a magnetic field $B$ is studied using the variational method. It is shown that for $B \gtrsim 0.13\,{\rm a.u.}$ this system gets bound with total energy below the one of the (\al,e,e) system. It manifests the existence of the stable He$^-$ atomic ion. Its ground state is a spin-doublet $^2(-1)^{+}$ at $0.74\, {\rm a.u.} \gtrsim B \gtrsim 0.13\, {\rm a.u.}$ and it becomes a spin-quartet $^4(-3)^{+}$ for larger magnetic fields. For $0.8\, {\rm a.u.} \gtrsim B \gtrsim 0.7\, {\rm a.u.}$ the He$^-$ ion has two (stable) bound states $^2(-1)^{+}$ and $^4(-3)^{+}$.Comment: 4 pages, 1 figure, 2 tables, one reference added, typos corrected, small modifications done, to be published at Phys Rev Letter

The He2+{}_2^+ molecular ion and the He−{}^- atomic ion in strong magnetic fields

We study the question about existence i.e. stability with respect to dissociation of the spin-quartet, permutation- and reflection-symmetric ${}^4(-3)^+_g$ ($S_z=-3/2, M=-3$) state of the $(\alpha\alpha e e e)$ Coulomb system: the ${\rm He}_2^+$ molecular ion, placed in a magnetic field $0 \le B \le 10000$ a.u. We assume that the $\alpha$-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-contained compounds in strong magnetic fields, in particular, we study the spin-quartet ground state of ${\rm He}^-$ ion, and the ground (spin-triplet) state of the Helium atom, both for a magnetic field in $100 \leq B\leq 10000$ a.u. The main result is that the ${\rm He}_2^+$ molecular ion in the state ${}^4(-3)^+_g$ is stable towards all possible decay modes for magnetic fields $B \gtrsim 120$ a.u. and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At $B=1000$ a.u., the dissociation energy of ${\rm He}_2^+$ into ${\rm He}^- + \alpha$ is $\sim 701.8$ eV and the dissociation energy for the decay channel to ${\rm He} + \alpha + e$ is $\sim 729.1$ eV, latter both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209.Comment: LaTeX revtex4 BibTeX, 34 pages, 7 Tables and 5 Postscript Figures. Typos corrected. Several clarifying sentences were added. List of References was updated. 3 references were removed, 9 new references were added. Some sections were renamed and Section III was adde

Finite Hydrogenic molecular chain H3_3 and ion H2−_2^- exist in a strong magnetic field

The existence and stability of the linear hydrogenic chain H$_3$ and H${}_2^-$ in a strong magnetic field is established. Variational calculations for H$_3$ and H${}_2^-$ are carried out in magnetic fields in the range $10^{11}\leq B \leq 10^{13}\,$G with 17-parametric (13-parametric for H${}_2^-$), physically adequate trial function. Protons are assumed infinitely massive, fixed along the magnetic line. States with total spin projection $S_z=-3/2$ and magnetic quantum numbers $M=-3,-4,-5$ are studied. It is shown that for both H$_3$ and H${}_2^-$ the lowest energy state corresponds to $M=-3$ in the whole range of magnetic fields studied. As for a magnetic field $B \gtrsim 10^{11}\,$G both H$_3$ and H${}_2^-$ exist as metastable states, becoming stable for $B \geq 1.9 \times 10^{11}\,$G and for $B \geq 2.7 \times 10^{11}\,$G, respectively. The excited states $^4(-4)^+$, $^4(-5)^+$ of ${\rm H}_3$ and H${}_2^-$ appear at magnetic fields $B > 7 \times 10^{11}$ and $10^{12}$\,G, respectively.Comment: 4 pages, 1 Figure, 4 Tables, the second half of the paper rewritten, Table split into four Tables, results on first two excited states added, estimate on domain of stability H$_4$ molecule give

Hydrogen atom and one-electron molecular systems in a strong magnetic field: are all of them alike?

Easy physics-inspired approximations of the total and binding energies for the ${\rm H}$ atom and for the molecular ions {\rm H}_2^{(+)} ({\rm ppe}), {\rm H}_3^{(2+)} ({\rm pppe}), ({\rm HeH})^{++} (\al {\rm p e}), {\rm He}_2^{(3+)} (\al \al {\rm e}) as well as quadrupole moment for the ${\rm H}$ atom and the equilibrium distances of the molecular ions in strong magnetic fields $> 10^{9}$ G are proposed. The idea of approximation is based on the assumption that the dynamics of the one-electron Coulomb system in a strong magnetic field is governed by the ratio of transverse to longitudinal sizes of the electronic cloud.Comment: 22 Pages, 22 Figures, 4 Tables, Invited Contribution to appear in Collection of Czechoslovak Chemical Communications, Special Issue in honor of Professor Josef Paldu

The H3+H_3^+ molecular ion in a magnetic field in linear parallel configuration

A first detailed study of the ground state of the H$_3^+$ molecular ion in linear configuration, parallel to a magnetic field direction, and its low-lying \Si,\Pi,\De states is carried out for magnetic fields $B=0-4.414 \times 10^{13}$G in the Born-Oppenheimer approximation. The variational method is employed with a single trial function which includes electronic correlation in the form \exp{(\ga r_{12})}, where \ga is a variational parameter. It is shown that the quantum numbers of the state of the lowest total energy (ground state) depend on the magnetic field strength. The ground state evolves from the spin-singlet {}^1\Si_g state for weak magnetic fields $B \lesssim 5 \times 10^{8}$G to a weakly-bound spin-triplet {}^3\Si_u state for intermediate fields and, eventually, to a spin-triplet $^3\Pi_u$ state for $5 \times 10^{10} \lesssim B \lesssim 4.414 \times 10^{13}$G. Local stability of the linear parallel configuration with respect to possible small deviations is checked.Comment: 27 pages, 6 figures, 12 tables (the text expanded, several new references added, typos corrected

Three-body quantum Coulomb problem: analytic continuation

The second (unphysical) critical charge in the 3-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by F~Stillinger has been calculated to be equal to $Z_{B}^{\infty}\ =\ 0.904854$and$Z_{B}^{m_p}\ =\ 0.905138$for infinite and finite (proton) mass$m_p$, respectively. It is shown that in both cases, the ground state energy$E(Z)$(analytically continued beyond the first critical charge$Z_c$, for which the ionization energy vanishes, to$Re Z < Z_c$) has a square-root branch point with exponent 3/2 at$Z=Z_B$in the complex$Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^-$is predicted to be at -0.51554 a.u. (-0.51531 a.u. for the finite proton mass$m_p$). The first critical charge$Z_c$is found accurately for a finite proton mass$m_p$in the Lagrange mesh method,$Z^{m_p}_{c}\ =\ 0.911\, 069\, 724\, 655$.Comment: 12 pages, 1 figure, 3 tables: title changed and Figure modified, several explanatory sentences added, text improved for better understanding, some typos fixed, to be published at Mod Phys Lett

Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr\"odinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part $S(\theta)$ does not satisfy any linear differential equation. In this case it satisfies a nonlinear ODE that has the Painlev\'e property and its solutions can be expressed in terms of the Painlev\'e transcendent $P_6$. We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.Comment: 43 page

The exotic H32+H_3^{2+} ion in a strong magnetic field. Linear configuration

An accurate study of the lowest $1\sigma_g$ and the low-lying excited $1\sigma_u$, $1\pi_{u,g}$, $1\delta_{g,u}$ electronic states of the exotic molecular ion $H_3^{2+}$ in linear configuration parallel to a magnetic field is carried out. The magnetic field ranges from $10^{10}$ G up to $4.414 \times 10^{13}$ G where non-relativistic considerations are justified. The variational method is exploited and the {\it same} trial function is used for different magnetic fields. It is shown that the states of positive $z$-parity $1\sigma_g, 1\pi_u, 1\delta_{g}$ are bound states of the $H_3^{2+}$ exotic ion for all magnetic fields studied. We also demonstrate that for magnetic fields $B\gtrsim 2.35\times 10^{12}$ G the potential energy surface well corresponding to the $1\sigma_g$ state contains at least one longitudinal vibrational state. It is also shown that the negative $z$-parity states $1\sigma_u, 1\pi_g, 1\delta_{u}$, are purely repulsive in the whole range of magnetic fields studied, $B=10^{10}- 4.414 \times 10^{13}$ G.Comment: 26 pages, 6 figures, 4 table