Ground State Energies Of Two-electron Atoms Research Articles

Multiconfiguration Dirac-Fock study on the ground-state energies of two-electron atoms

The multiconfiguration Dirac-Fock method is applied to calculate the ground-state energies of two-electron atoms including radiative corrections and correlation relativistic cross-term contributions. The results are compared with extrapolated experimental data recalibrated using the most recent values of the energy unit conversion factors. The resulting discrepancy does not exceed 1 mHartree, i.e. it is even smaller than an estimated experimental error.

Dimensional expansions for two-electron atoms

Approximate expansions in inverse powers of the dimensionality of space D are obtained for the ground-state energies of two-electron atoms. The method involves fitting polynomials in δ=1/D to accurate eigenvalues of the generalized D-dimensional Schrödinger equation. To the maximum order obtainable from the data, about δ7, the power series for nuclear charges Z=2, 3, and 6 all diverge at D=3. Asymptotic summation yields an energy for the Z=2 atom 1% in excess of the true value at D=3. However, expansions with a shifted origin, i.e., expansions in (δ−δ0), show improved convergence. Of particular interest is the case δ0=1, because the expansion coefficients can in principle be calculated by perturbation theory applied to the one-dimensional atom. Series in powers of (δ−1) appear to converge rapidly. Also the series in (δ−1) can be evaluated even for the hydride ion, with Z=1. For helium, this series is quite comparable to the more familiar expansion in powers of λ=1/Z, with errors in the partial sums decreasing by roughly an order of magnitude per term. Thus, for Z=2 the first four terms of the expansion in (δ−1) yield an energy within 0.02% of the true value at D=3. Similar results are found in an analogous treatment of accurate eigenvalues for the Hartree–Fock approximation. This provides a rapidly convergent dimensional expansion for the correlation energy.

Darboux functions and perturbative theory for the ground-state energy of two-electron atoms

The power series of the lowest eigenvalue epsilon ( lambda ) of the helium isoelectronic sequence, lambda being the coupling constant for the interaction between the two electrons, is summed by the ansatz of a simple analytic function of class D (Darboux). Stationarity criteria are used in constructing such a function, and the basic information needed is only a few terms of the perturbative power series. The ground-state energy is evaluated with great accuracy without using any free parameter. The Darboux method is used to give an estimate of the asymptotic coefficients of the perturbative expansion. The contrast between different empirical procedures and the self-consistency of the present one is remarkable.

Hartree-Fock calculation of the ground state energies of two-electron atoms in an intense magnetic field

Ground-state energies of two-electron atoms H-, He and LiP, embedded in a very strong magnetic field, are calculated using the Hartree-Fock method. The one-particle wavefunction contains a transverse part, which is determined by the magnetic field, and a longitudinal part, which is determined by the Coulomb interactions. The self-consistent field method is used to adjust the longitudinal function. For the strongest fields considered in this study, the improvement over previous variational estimates is between ten and twenty per cent. For lower fields, the author's results contradict those by Mueller et al. (1975) as a somewhat higher ground-state energy for the H- ion is obtained.

Bounds on second-order energies in stationary perturbation theory

Upper and lower bounds are derived for the second-order energy of any state of any atomic system. These bounds require only a knowledge of the unperturbed wave function of the state under consideration and some knowledge of the unperturbed energy spectrum. These bounds have been applied to the calculation of the ground-state energies of two-electron atoms and to the polarizabilities of the ground state of one- and two-electron atoms.

Ground-State Energy of Two-Electron Atoms

We examine the ground-state energy ε(λ) for two electrons bound to an infinite-mass point nucleus regarded as a function of the complex coupling constant λ for the interelectron interaction. In the units used, the ground states of the homologous series H−, He, Li+, Be++, ···, correspond, respectively, to λ=1, ½, ⅓, ¼, ···, so the λ power-series expansion of ε(λ) is equivalent to the familiar expansion in inverse nuclear charge Z−1. It is argued that the power series has a finite radius of convergence imposed by the existence of a branch point on the positive real axis at λ=λ*≅1.1184 with exponent approximately 6/5. Furthermore, ε(λ*) apparently lies above the continuum limit, but still corresponds to a localized wavefunction.

Ground-State Energy of Two-Electron Atoms. Corrective Results

Ground-State Energy of Two-Electron Atoms. Corrective Results