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The Astrophysical Journal

Hydrogen Atom and Hydrogen Molecule Ion in Homogeneous Magnetic Fields of Arbitrary Strength

Ranjan Bhaduri, Yozo Nogami, Chindhu S. Warke

Published Oct 1, 1977• 36 citations
Publisher

Abstract

A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics

DOI
10.1086/155582

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arXiv:1903.06533•Physical Review A•Jul 11, 2019•4 citations
A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics
Abstract The problem of the hydrogen atom in a magnetic field can be treated satisfactorily by approximate methods which are quite accurate for very low or very high magnetic field strengths. Before this paper the only useful treatments for intermediate field strengths were variational methods giving upper bounds to the energies. As described in the paper the diffusion QMC method was applied in the same way as in earlier solutions for the hydrogen atom, but with additional magnetic terms added to the Hamiltonian expression. These include complex expressions and the required wavefunction includes a corresponding factor. With the magnetic field aligned along one axis, integration over the angle ϕ in (r, θ, ϕ) coordinates reduces the additional terms to real functions of r and z as well as the magnetic quantum number m. In the diffusion/multiplication processes of a random walk procedure these may be simply included with the potential energy term. In this study the energy at each step was determined from the average multiplication factor for the walkers. Importance sampling was not used. Results were obtained for quantum numbers m = 0, 1 at each of three magnetic field strengths. A comparison of results showed good agreement with the lowest values obtained in several of the earlier analytic variational calculations.
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.
In most quantum mechanics textbooks for graduate studies, the hydrogen atom is studied in an approximate way by means of the variational method. The type of trial functions commonly used are the Gaussian and the Lorenzian. In this paper we consider a natural generalization of the Mei method to improve the trial wave functions [1] applied to hydrogen atom. We propose a sequence of functions as trial wave functions to calculate the eigen-energies of the hydrogen atom. These trial wavefunctions are given in terms of three variational parameters, one of them is fixed by means of the normalization condition and the other two are adjustable parameters. One of these parameters can be chosen as being an integer number, then this parameters will define the sequence of functions. We will show that when the integer parameter approaches to infinity, the ground state, first excited state and second excited state converge to the exact results.
Non-relativistic rates for the decay of 2s hydrogen atoms to the ground state by single-photon and two-photon emission in the presence of a homogeneous magnetic field of arbitrary strength (0<or=B<or=4.7*10 6 T) are calculated by variational procedures. Over the whole range of B, two-photon emission is the dominant process. As the magnetic field grows, the two-photon decay rate increases. It is found that the Markov approximation can be applied to the two-photon decay for magnetic fields of strength B>or=4.7*10 3 T.
The ground states of small atoms and molecules, namely those of the hydrogen, helium, and lithium atoms and the hydrogen molecular ion and hydrogen molecule, are studied in the presence of an external magnetic field. For the one-electron systems the ground state is of the same symmetry for arbitrary field strengths, and the corresponding binding energy shows a monotonic increase with increasing field strength. More interesting, the two- and three-electron systems show ground-state transitions which involve both transitions in the spin multiplicity and spatial quantum numbers. An outlook on the expected cascade of transitions in many electron atoms and molecules is provided. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 789–795, 1998
The existence and stability of the linear hydrogenic chain H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ in a strong magnetic field is established. Variational calculations for H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ are carried out in magnetic fields in the range ${10}^{11}\ensuremath{\le}B\ensuremath{\le}{10}^{13}$ G with a 17-parametric (13-parametric for $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$), physically adequate trial function. Protons are assumed to be infinitely massive, fixed along the magnetic line. States with total spin projection ${S}_{z}=\ensuremath{-}3/2$ and magnetic quantum numbers $M=\ensuremath{-}3,\ensuremath{-}4,\ensuremath{-}5$ are studied. It is shown that for both H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$, the lowest-energy state corresponds to $M=\ensuremath{-}3$ in the whole range of magnetic fields studied. As for a magnetic field $B\ensuremath{\gtrsim}{10}^{11}$ G, both H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ exist as metastable states, becoming stable for $B\ensuremath{\ge}1.9\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G and for $B\ensuremath{\ge}2.7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G, respectively. The excited states ${}^{4}{(\ensuremath{-}4)}^{+}$, ${}^{4}{(\ensuremath{-}5)}^{+}$ of H3{\mathrm{H}}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ appear at magnetic fields $B>7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ and 1012{10}^{12}1012 G, respectively.