Hylleraas-type Basis Research Articles

2pnp (1,3Pe) states of neutral He and Li+ ions under Debye plasma screening

The energy eigenvalues of doubly excited 2pnp (1Pe) (n = 3–8) and 2pnp (3Pe) (n = 2–7) bound states of a neutral helium atom and singly ionized lithium atom are estimated under weakly coupled plasma screening for the first time using an explicitly correlated Hylleraas-type basis in the framework of the Rayleigh–Ritz variational principle. The Debye screening model has been employed to include the effect of plasma background. Improvement over the existing approximated theoretical results has been achieved for the above-mentioned states of neutral helium under Debye plasma screening. The modified energy eigenvalues of 1,3Pe states arising from two p electrons of a Li+ ion in the Debye plasma environment are being reported for the first time.

Accuracy limits on the description of the lowestSexcitation in the Li atom using explicitly correlated Gaussian basis functions

We have performed very accurate quantum-mechanical calculations for the two lowest $S$ states of the lithium atom in order to determine the transition energy. In the nonrelativistic part of the calculations performed using the variational method, we explicitly included the nuclear motion. The nonrelativistic wave function was expanded in terms of explicitly correlated Gaussian functions. Next, this wave function was used to calculate the leading ${\ensuremath{\alpha}}^{2}$ relativistic correction ($\ensuremath{\alpha}$ is the fine-structure constant) and the ${\ensuremath{\alpha}}^{3}$ QED correction. We also estimated the ${\ensuremath{\alpha}}^{4}$ QED correction by calculating its dominating component. The results obtained with Gaussians are compared with the most accurate results obtained recently with the Hylleraas-type basis functions.

Helium atom in the monochromatic electromagnetic field

We apply a non-perturbative procedure for the calculation of the total photoionization cross-section of two-electron atomic systems. The procedure is based on the Floquet-Fourier representation of the time-dependent Schrodinger equation. With the use of the Hylleraas-type basis functions, the total photoionization cross-sections obtained are within the accuracy of a fraction of a percent, which, we believe, is the most accurate estimate for the cross-sections available. The total photoionization cross-sections for neutral helium deviate notably from the benchmark experimental data [J.A.R. Samson et al., J. Phys. B 27, 887 (1994)].

Examination of quasi-bound states of and the possible existence of a bound state

In this paper, a study is made of the interaction of atomic helium with an antiproton within the Born–Oppenheimer approximation. The electronic energy curve is calculated using the variational method with basis sets containing Hylleraas-type basis functions. The energies and nuclear wave functions for 50 s states are then obtained using this potential by solving the nuclear wave equation numerically using the Cooley–Numerov algorithm. It is possible that the state with the lowest energy is a bound state. The remaining states are all above the lowest continuum threshold for and are thus quasi-bound states.

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three-Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schrödinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

Variational calculation of some S-states of Coulomb three-body systems

A generalized Hylleraas-type basis set with three nonlinear parameters is proposed to study three-body systems interacting via coulomb forces within the framework of non-relativistic quantum mechanics. This basis set improves the rate of convergence with respect to previous ones, specially for non-symmetric systems and excited states of two electron atoms. Accurate binding energies and other properties for S-states of helium-like ions, muonic molecules and the positronium negative ion are reported.

Convergence accelerator approach for the evaluation of some three‐electron integrals containing explicit r ij factors

A simplified analysis is presented for the evaluation of the three-electron one-center integrals of the form {integral}r{sub 1}{sup i}r{sub 2}{sup j}r{sub 3}{sup k}r{sub 23}{sup l}r{sub 31}{sup m}r{sub 12}{sup n}e{sup {minus}{alpha}r{sub 1}{minus}{beta}r{sub 2}{minus}{gamma}r{sub 3}}dr{sub 1}dr{sub 2}dr{sub 3}, for the cases i, j, k, {ge} {minus} 2, l = {minus}2, m {ge} {minus}1, n {ge} {minus}1. These integrals arise in the calculation of lower bounds for energy levels and certain relativistic corrections to the energy when Hylleraas-type basis sets are employed. Convergence accelerator techniques are employed to obtain a reasonable number of digits of precision, without excessive CPU requirements.

Differential scattering and rotational excitation in low-energy positron-hydrogen-molecule collisions

Differential cross sections are calculated for positron-hydrogen-molecule scattering below the positronium formation threshold usingK-matrix elements obtained by the Kohn variational method. The internuclear separation of the hydrogen molecule is fixed at its equilibrium value, 1.4a0. The trial function used in the Kohn calculation is very flexible. It contains separable and Hylleraas-type basis functions and also basis functions to take into account longrange polarization. The cross sections for rotational excitation are also calculated, using the adiabatic-rotation approximation. Preliminary results are presented in this paper; full details will be published elsewhere.

Generalized oscillator-strength calculations for some low-lying excited states of H2 using a high-accuracy configuration-interaction wave function.

The full configuration-interaction (CI) method using elliptical basis functions has been used to calculate the wave function for a few of the lower [sup 1,3][Sigma][sub [ital g]] and [sup 1,3][Sigma][sub [ital u]] excited states. This computational method gives the best wave functions to date for the [ital GK](3 [sup 1][Sigma][sub [ital g]]) and [ital e](2 [sup 3][Sigma][sub [ital u]]) excited states at an internuclear distance [ital R]=1.5 a.u., without using explicitly correlated basis functions, i.e., James-Coolidge or Hylleraas-type basis functions. Using these excited-state wave functions, accurate generalized oscillator strengths for excitation to a few of the lower excited states at [ital R]=1.4 a.u. are calaculated and given for comparison with other calculations reported elsewhere. It is found that the effect of the ground-state electron correlation is important for excitation to the lowest excited states, but it becomes less significant for excitation to still higher excited states. It is concluded that the CI method and the computation technique presented here is the most practical and accurate one for studying the inelastic scattering by H[sub 2] of fast, charged particles or the electron-impact spectroscopy of H[sub 2].

Correlated wave functions and hyperfine splittings of the 2sstate of muonicHe3,4atoms

The correlated wave functions of the 2s state, (1s${)}_{\mathit{e}}$(2s${)}_{\mathrm{\ensuremath{\mu}}}$ of muonic $^{4}\mathrm{He}$ and $^{3}\mathrm{He}$ atoms are calculated by the variational approach, employing an improved set of the Hylleraas-type basis functions to study the 2s state of muonic He atoms. The energy and radial expectation values are calculated. Our proposed wave functions for the muonic $^{4}\mathrm{He}$ and $^{3}\mathrm{He}$ atoms are good enough to describe the electron and muon at small, intermediate, and large values of radial coordinates. The hyperfine splittings of the (1s${)}_{\mathit{e}}$(2s${)}_{\mathrm{\ensuremath{\mu}}}$ muonic $^{4}\mathrm{He}$ and $^{3}\mathrm{He}$ atoms are also presented. Including relativistic, QED, and other corrections up to O(${\mathrm{\ensuremath{\alpha}}}^{2}$), we arrive at the values \ensuremath{\Delta}\ensuremath{\nu}=(4287.01\ifmmode\pm\else\textpm\fi{}0.10 and 4052.64\ifmmode\pm\else\textpm\fi{}0.10 MHz for the total hyperfine splittings of the 2s state of muonic $^{4}\mathrm{He}$ and $^{3}\mathrm{He}$ atoms, respectively.

Hyperfine structure in the muonic 3He atom

In order to examine the recoil effect, the authors calculate the ground-state hyperfine structure of the muonic 3He by the variational approach, employing an improved set of Hylleraas-type basis functions to deal with the non-relativistic three-body wavefunctions of the muon, electron and 3He nucleus. The convergence of the series is very fast, thus eliminating the problem with the propagation of the truncation errors of the computer machine. Using the recoil correction, delta rec= Delta nu F(3 alpha / pi )(me/mmu ) ln(mmu /m$0 they arrive at the value Delta nu =4166.56+or-0.05 MHz for the total hyperfine-structure splitting of the ground state of the muonic 3He atom. But if they follow the treatment of Borie (1979) for the recoil corrections by treating the ( mu -3He)+ system as a pseudonucleus, they will arrive instead at the value Delta nu =4165.91+or-0.05 MHz. The former seems to be in slightly better agreement with the only experimental result, 4166.3+or-0.2 MHz.

Theoretical hyperfine structure of the muonic 4He atom.

We calculate the hyperfine structure of the ground state of the muonic {sup 4}He atom by the variational approach, employing an improved set of the Hylleraas-type basis functions to deal with the nonrelativistic three-body wave functions of the muon, the electron, and the {sup 4}He nucleus. The convergence of the series is very fast, thus eliminating the problem with the propagation of the truncation errors of the computer machine. Using the recoil correction, {delta}{sub rec} ={Delta}{nu}{sub {ital F}}(3{alpha}/{pi})(m{sub e}/m{sub {mu}}) ln({ital m}{sub {mu}}/m{sub e}), we arrived at the value {Delta}{nu}=4464.87{plus minus}0.05 MHz for the total hyperfine-structure splitting of the ground state of the muonic {sup 4}He atom including relativistic and other radiative corrections. But if we follow the treatment of Borie (Z. Phys. A 291, 107 (1979)) for the recoil corrections by treating the ({mu}-{sup 4}He){sup +} system as a pseudonucleus, we will arrive instead at the value {Delta}{nu}=4464.00{plus minus}0.05 MHz. The former seems to be in better agreement with the experimental result of 4465.004{plus minus}0.029 MHz.

Accurate oscillator strengths for neutral helium

With use of variational wave functions, oscillator strengths have been calculated for all the allowed transitions between the states $n^{1}S,n^{1}P,n^{1}D,n^{3}S,n^{3}P$, and $n^{3}D$,$n\ensuremath{\le}9$, in neutral helium, with an estimated accuracy of better than 1% for most of the transitions and better than 0.1% for about a third of the transitions. The error estimate is based on numerical convergence as the number of expansion terms in the wave function is increased. By use of rapidly converging trial functions consisting of two sets of Hylleraas-type basis functions differing in the values of nonlinear parameters, the expansion lengths required are kept moderate (at most 140 terms). A discussion is given about the completeness of the Hylleraas-type basis.

2S autoionizing states of lithium

The energy positions and the wavefunctions of 2S autoionizing states of the neutral lithium atom have been calculated using the derivative method with a Hylleraas-type basis set. Computed energies are in good agreement with experiment and with other theoretical results.