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Physical Review A

Relativistic effects of the 1s2s2pP4 of He−

Kwong T. Chung

Published Feb 1, 1984• 11 citations
Publisher

Abstract

With the use of an accurate nonrelativistic wave function, the relativistic effects of the ${\mathrm{He}}^{\ensuremath{-}}$ $1s2s2p^{4}P$ state are computed. These include the mass-correction term, Darwin term, and retardation and mass-polarization effects. The calculated result enables us to significantly reduce the uncertainty of the electron affinity given by Bunge and Bunge. To assist future experimental measurement on the fine structure of this ion, the expectation values of the spin-orbit, spin-other-orbit, and spin-spin operators are also calculated.

DOI
10.1103/physreva.29.439

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With the use of an accurate nonrelativistic wave function, the relativistic effects of the He\ensuremath−{\mathrm{He}}^{\ensuremath{-}}He\ensuremath− 1s2s2p4P1s2s2p^{4}P1s2s2p4P state are computed. These include the mass-correction term, Darwin term, and retardation and mass-polarization effects. The calculated result enables us to significantly reduce the uncertainty of the electron affinity given by Bunge and Bunge. To assist future experimental measurement on the fine structure of this ion, the expectation values of the spin-orbit, spin-other-orbit, and spin-spin operators are also calculated.
Large-scale optimization of nonlinear parameters and tighter error-bound estimates yield an accurate value, ${E}_{\mathrm{nr}}=\ensuremath{-}2.1780776(12)$ a.u. (4He^{4}\mathrm{He}4He) for the nonrelativistic energy of ${\mathrm{He}}^{\ensuremath{-}}$ 1s2s2p4Po1s2s2p^{4}P^{o}1s2s2p4Po, in agreement with a previous result of -2.1780740(100). Using Chung's recent calculation of relativistic and mass-polarization effects, an electron affinity $A=77.51\ifmmode\pm\else\textpm\fi{}0.04$ meV is obtained. The calculation and significance of error bars in electronic structure calculations is discussed.
Accurate eigenvalues of Schr\"odinger's nonrelativistic equation for Be (1${s}^{2}2s2p2s2p2s2p{)}^{3}$P and ${\mathrm{Be}}^{\mathrm{\ensuremath{-}}}$ (1${s}^{2}2s22s22s2{p}^{2}$)4{)}^{4})4P have been calculated yielding a corresponding electron affinity A of 276.1\ifmmode\pm\else\textpm\fi{}6.5 meV. The absolute term value of ${\mathrm{Be}}^{\mathrm{\ensuremath{-}}}$ (relative to the Be ground state) obtained as the difference between the experimental term value of Be 3P^{3}\mathrm{P}3P and A is 2.4492\ifmmode\pm\else\textpm\fi{}0.0065 eV, in agreement with a recent measurement of 2.53\ifmmode\pm\else\textpm\fi{}0.11 eV.
The electrostatic fine structure for singly excited lithium is accurately calculated using an effective potential method that has proven to be very useful for helium. Relevant properties of the two-electron Li+{\mathrm{Li}}^{+}Li+ ionic core with finite nuclear mass are determined from variational wave functions. From these, non- relativistic level shifts are obtained analytically, and the leading relativistic correction is applied. A comparison with recent experimental results is made.
Precise numerical calculation of radiofrequency intervals between hyperfine sublevels of the (37,35) state of the antiprotonic helium-4 atom is presented. Theoretical consideration includes the QED corrections of order $m{\ensuremath{\alpha}}^{6}$ to the electron spin-orbit interaction. Only the leading order contribution in the electron-to-antiproton and electron-to-$\ensuremath{\alpha}$ particle mass ratios is considered, so the $\ensuremath{\alpha}$ particle and antiproton are treated nonrelativistically. The effective Hamiltonian is derived using the formalism of the nonrelativistic quantum electrodynamics.
We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He+ ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He+ ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, j2, l2, s2 and jz, where j, l and s are the internal total, orbital and spin angular moments of the He+ ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He+ ion. In addition to the Z2 a2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.
The leading-order relativistic corrections to the ground-state energy of the positronium molecule (Ps2)({\mathrm{Ps}}_{2})(Ps2​) have been computed within the framework of perturbation theory. As the zero-order wave function we used a highly accurate nonrelativistic variational expansion in terms of 6000 explicitly correlated Gaussians that yielded the lowest variational upper bound for this system to date. We also report some expectation values representing the properties of Ps2{\mathrm{Ps}}_{2}Ps2​.