Ground State Of Two-electron Atoms Research Articles

Hartree-Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms.

An implementation of the Hartree-Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be ZHFC=1.031 177 528.This article is part of the theme issue 'Modern theoretical chemistry'.

Double Photoionization of Endohedrally Confined Atoms

We study the double electronic emission by photon impact from ground state of two-electron atoms in the center of a model spherical fullerene, which is described by a square-well shell. Cross-sections for different well depth are computed within a separable model for the final state, and a configuration interaction state for the initial one. Triple differential cross-sections show a strong dependence on the well depth and on the energy of the emitted electrons, due to the delocalization of the electrons in the initial state. The fullerene potential also allows higher angular momenta partial waves to be included in the process, which modifies the well-known two-lobe cross-section from isolated atom.

A simple parameter-free wavefunction for the ground state of two-electron atoms

We propose a simple and pedagogical wavefunction for the ground state of two-electron atoms which (i) is parameter free, (ii) satisfies all two-particle cusp conditions, (iii) yields reasonable ground-state energies, including the prediction of a bound state for H−. The mean energy, and other mean physical quantities, is evaluated analytically. The simplicity of the result can be useful as an easy-to-use wavefunction when testing collision models.

Application of the complex-coordinate rotation to the relativistic Hylleraas-CI method: a case study

It is shown that variational solutions corresponding to the bound states of the Dirac–Coulomb (DC) eigenvalue problem may be obtained using a complex-coordinate rotation method. The discrete eigenvalues of the DC Hamiltonian are treated as resonances coupled to the Brown–Ravenhall continuum (the superposition of the negative- and positive-energy one-electron continua). The method leads to a clean separation of the bound-state energies from the continuum. It is shown that the effects related to the resonance character of the bound states being an artefact of the DC approximation, in particular the instability of the ground state, are proportional to the third power of the fine structure constant α. Thus, the results correctly describe the physical reality up to the terms proportional to α2. The approach has been implemented for the case of the ground state of two-electron atoms described by the Dirac–Coulomb Hamiltonian in a space of the Hylleraas-type trial functions. The variational energies, calculated for Z = 2, 80, 90, correspond to the best available in the literature.

Double photoionization of two-electron atoms based on the explicit separation of dominant ionization mechanisms

Double ionization by a single photon is often discussed in terms of two mechanisms, namely, shakeoff and knockout, dominant at high and low photon energies, respectively. We have developed a model to formulate the explicit but separate contribution of both mechanisms at all energies [Phys. Rev. Lett. 89, 073002 (2002)]. The separation is based on a quasiclassical formulation of knockout which is free from any shakeoff part since the latter is purely quantum mechanical. The relevance of each mechanism from threshold up to several keV photon energy is quantified and discussed in detail for the photoionization from the ground state of two-electron atoms. Photoionization ratios, integral and singly differential cross sections calculated for helium and other members of its isoelectronic sequence are compared to benchmark experimental data and recent theoretical results. A connection to Samson's half-collision model [Phys. Rev. Lett. 65, 2861 (1990)] is also given.

A two-parameter single-particle description of the ground state of two-electron atoms

A simple variational-type model for the ground state of two-electron atoms is here proposed. It is based on a single-particle wavefunction with two free parameters generalizing the screened hydrogenic one, and which holds the Kato cusp condition exactly. The results for the expectation values of different properties including the energy are close to different Hartree-Fock ones reported in the literature.

Perturbative-variational approximation of the ground state of two-electron atoms

Here, we study the applicability of the perturbative variational method (PVM) to obtain the ground-state energy and wavefunction of the two-electron atoms. The convergence of the method has been studied with different bases and we have checked whether it is faster if we work with an orthogonal or quasi-orthogonal basis. We have focused our attention on the polynomial basis and our best results are very close to the best results provided for other calculations that include non-analytical terms in the basis. However, we have to use a number of states in a basis ten times larger than that used in other calculations.

Coulomb hole in the ground state of two-electron atoms

The Coulomb hole Δ(r12) in the ground state of two-electron atoms has been calculated. Except for H-, the function Δ(r12) can be represented by a single curve F(ρ), where ρ = (Z - 0.38)r12.

Logarithmic Terms in the Wave Functions of the Ground State of Two-Electron Atoms

A variational calculation for the ground state of two-electron atoms is carried out with a function containing the nonconventional terms $\mathrm{ln}({r}_{1}+{r}_{2})$, ${[\mathrm{ln}({r}_{1}+{r}_{2})]}^{2}$, and ${({{r}_{1}}^{2}+{{r}_{2}}^{2})}^{\frac{1}{2}}$. The convergence of the energy eigenvalues is very good, lending support to the existence of the logarithmic terms in the exact solution of the wave equation.

Logarithmic Terms in the Wave Functions of the Ground State of Two-Electron Atoms

A variational calculation for the ground state of two-electron atoms is carried out with a function containing the nonconventional terms $\mathrm{ln}({r}_{1}+{r}_{2})$, ${[\mathrm{ln}({r}_{1}+{r}_{2})]}^{2}$, and ${({{r}_{1}}^{2}+{{r}_{2}}^{2})}^{\frac{1}{2}}$. The convergence of the energy eigenvalues is very good, lending support to the existence of the logarithmic terms in the exact solution of the wave equation.

Coulomb hole in the ground state of two-electron atoms

The Coulomb hole Δ(r12) in the ground state of two-electron atoms has been calculated. Except for H-, the function Δ(r12) can be represented by a single curve F(ρ), where ρ = (Z - 0.38)r12.

Further Results Concerning Half-Integral Hylleraas Expansions

Previous calculations concerning the ground state of twoelectron atoms, involving Ritz-Hylleraas expansions with half-integral powers, are continued through expansions involving 31 parameters. As far as can be judged from a comparison of the energies with those obtained with other expansions, the results continue to be favorable. Thus, already with 18 parameters, the computed energies for He and O VII differ from the best published values by 2 and 0.3 parts per million, respectively. Even better results are obtained for He with expansions involving both halfintegral and negative powers. A few initial results of calculations for the excited state 2 /sup 1/S are also presented. (auth)

Ritz-Hylleraas Solutions of the Ground State of Two-Electron Atoms Involving Fractional Powers

Earlier work on the inclusion of half-integral powers in the Ritz-Hylleraas ground-state solutions of the nonrelativistic wave equation for the helium atom is extended through functions involving 18 parameters. Energies that are lower than those found with other published comparable functions are obtained in all cases. Preliminary results are also given of calculations involving more general fractional powers, $Z$ values different from 2, and half-integral-power solutions for which the expectation value of the square of the Hamiltonian is finite. With the latter type of expansions one obtains, at least at an early stage, an additional improvement in the approximation. Thus, with 11 parameters one finds the energy -2.903704 atomic units, which differs by only 0.0007% from the 80-parameter solution of Kinoshita. The initial results found for $Z=8$ indicate an improvement in convergence over that obtained for $Z=2$; the energy -59.156560 atomic units, which was obtained with a 12-parameter function, was only 0.00006% larger than the energy obtained by Pekeris with his 210-term function. The computed mass polarization corrections give also satisfactory results as judged by the similar results obtained with the most extensive solutions available.

Ground State of Two-Electron Atoms

A new method is developed for solving the wave equation for two-electron atoms. The wave function is expanded into a triple orthogonal set in three perimetric coordinates. From the wave equation one obtains an explicit recursion relation for the coefficients in the expansion, and the vanishing of the determinant of these coefficients provides the condition for the energy eigenvalues and for the eigenvectors. The determinant was solved on WEIZAC for $Z=1 \mathrm{to} 10$, using an iteration method. Since the elements of the determinant are integers, and only an average of about 20 per row are nonvanishing, it has been possible to go to an order of 214 before exceeding the capacity of the fast memory of WEIZAC. The nonrelativistic energy eigenvalues obtained for the ground state are lower than any previously published for all $Z$ from 1 to 10. In the case of helium, our nonrelativistic energy value is accurate to within 0.01 ${\mathrm{cm}}^{\ensuremath{-}1}$ and is 0.40 ${\mathrm{cm}}^{\ensuremath{-}1}$ lower than the value computed by Kinoshita. From the wave functions obtained, the mass-polarization and the relativistic corrections were evaluated for $Z=1 \mathrm{to} 10$. Using the values of the Lamb shift computed by Kabir, Salpeter, and Sucher, we obtain an ionization potential for helium of 198 310.67 ${\mathrm{cm}}^{\ensuremath{-}1}$ as against Herzberg's value of 198 ${310.8}_{2}$\ifmmode\pm\else\textpm\fi{}0.15 ${\mathrm{cm}}^{\ensuremath{-}1}$. Comparison is also made with the available experimental data for the other values of $Z$. By the use of our magnetic tape storage, the accuracy of the nonrelativistic energy value for helium could be pushed to about 0.001 ${\mathrm{cm}}^{\ensuremath{-}1}$, should future improvements in the experimental values and in the computed radiative corrections warrant it.