Sten Salomonson

Numerical Many-Body Perturbation Calculations on Be-like Systems Using a Multi-Configurational Model Space

We report here numerical perturbation calculations on Be and C2+ starting from a model space consisting of the two strongly interacting configurations 1s22s2 and 1s22p2. We use numerically represented radial pair functions which are solutions of a system of coupled differential equations obtained by a finite difference method. By iterating the system of pair equations the most important correlation effects are included to all orders. This is demonstrated for C2+, where excitation energies for the 2p2 levels are obtained with an accuracy better than 1% or 0.2 eV. For Be only second-order results are reported. Here the iterative scheme does not converge, probably due to the presence of intruder states of the type 2sns 1S, which lie between the two 1S states originating from the model space. The second-order calculation with the two-configurational model space and orbitals generated in the 1s2 Hertree-Fock core yields 93.6% of the correlation energy, compared to 80.9% for a similar calculation using a model space with only the ground-state configuration 1s22s2 and orbitals generated in the Hartree-Fock potential of that configuration.

A Numerical Coupled-Cluster Procedure Applied to the Closed-Shell Atoms Be and Ne

A coupled-cluster procedure, applicable to closed-shell as well as open-shell atoms, is described, and results from calculations on the closed-shell atoms Be and Ne are presented. The procedure is based on numerical solution of coupled radial pair equations, which can be iterated to self-consistency. The coupling between different pair excitations is considered in two steps. The first step, called intra-shell coupling, includes the interaction between excitations differing in the final states as well as in the mlms values of the electrons being excited. In the second step, the inter-shell coupling, also the residual interaction between excitations involving different nl shells is considered. While the former coupling is quite important, it has been found that the latter contributes only about 1% to the correlation energy. Single and triple excitations are neglected in the present work, but quadrupole excitations are included by means of the exponential ansatz of the coupled-cluster procedure. According to our results for Be, 98.6% of the experimental correlation energy is obtained with complete pair-pair coupling, which is consistent with the estimated value of about 1.5% for the contribution of single and triple excitations. With only intra-shell coupling the result is in almost exact agreement with the experimental value due to cancellations between single/triple excitations and inter-shell couplings. Similar results are obtained for Ne. The results of the present work are compared and discussed in relation to earlier many-body calculations on these systems.

Second order loop after loop self-energy correction for few-electron multicharged ions

Abstract One of the self-energy corrections of second order in α = e 2 h c is calculated for 1 s 1 2 , 2s 1 2 and 2p 1 2 states. The calculations involve the triple summation over the Dirac spectrum that was performed by the numerical method of the space discretization. This was achieved by the rearrangement of the summations in such a way that two of them were included in the corrections to the wave functions.

A new approach to the electron self energy calculation

We present a new practical way to calculate the first order self energy in any model potential (local or non-local). The main idea is to introduce a new straightforward way of renormalization to avoid the usual potential expansion implying a large number of diagrams in higher order QED effects. The renormalization procedure is based on defining the divergent mass term in coordinate space and decomposing it into a divergent sum over finite partial wave contributions. The unrenormalized bound self energy is equally decomposed into a partial wave (l) sum. For each partial wave the difference is taken and the sum becomes convergent. The comparably rapid asymptotic behaviour of the method is l−3. The method is applied to lithium-like uranium, and the self energy in a Coulomb field, the finite nucleus effect and the screened self energy is calculated to an accuracy of at least one tenth of an eV.

Second-order QED corrections for few-electron heavy ions: reducible Breit-Coulomb correction and mixed self-energy-vacuum polarization correction

Some second order in alpha =e2/(h(cross)c) QED corrections are investigated for He-like heavy ions and for the 2p12/-2s shift for the Li-like ions. These corrections correspond to the reducible part of the two-photon exchange Breit-Coulomb graph and to the mixed self-energy-vacuum polarization graph. The calculations show that these corrections are important for the comparison of the recent theoretical and experimental results.

Many-Body Perturbation Theory of the Effective Electron-Electron Interaction for Open-Shell Atoms

The effective-operator form of many-body theory is reviewed and applied to the calculation of the effective interaction of electrons in an open-shell atom. Numerical results are given for the 1s22s22p2 configuration of carbon. The effect of correlation upon the interaction of the 2p electrons of this configuration is represented by effective two-body operators of the form ΣakTk(1) Tk(2). These operators are evaluated using angular-momentum diagrams and solving numerically a two-particle equation for the linear combination of excited states which contribute to the Goldstone diagrams. The effect of the operators of even rank is to depress the values of the two-electron Slater integrals Fk(2p, 2p) below their Hartree-Fock values. The two-body operator of odd rank does not appear in the Hartree-Fock theory. Our second-order values of the Slater integrals agree quite well with experiment but the value which we obtain of the coefficient of odd rank is much too small. This is partly due to a large cancellation which occurs for the contribution of the outer 2s2, 2s2p, 2p2 pair excitations. In order to study the convergence properties of the theory and to obtain more accurate values of the interaction integrals, we consider the higher-order terms in the perturbation expansion. An important family of two-particle effects is included to all orders by solving the pair equations iteratively until self-consistency is achieved. A more accurate description of the electron-electron interaction is obtained in this way. There are three additional families of wave-operator diagrams which can have an important effect. One family has an additional open-shell line which polarizes a closed-, open-, or excited orbital. There are also the coupled-cluster diagrams and a family of diagrams involving two polarizing open-shell lines, which appears first in fourth order. All of these diagrams can be included in our iterative scheme and they include all possible two-particle effects to self-consistency.

Higher-order QED corrections for multi-charged ions

The higher-order QED corrections for one- and two-electron heavy ions are analysed. The different methods of renormalization are discussed: the traditional potential-expansion method and the new one, based on the direct numerical subtraction of divergencies. The extraction of the reference state from the sums over the intermediate states in higher-order corrections is considered. It is shown, that this extraction leads to some special corrections to the energy. The Low theory of the line profile is used for the calculation of the higher-order QED corrections and the nonresonant corrections distorting the Lorentz line profile are also discussed.

Analysis of the hyperfine structure of the 4s 4p configuration in Ca and the nuclear quadrupole moment of43Ca

Hyperfine structure 〈r−3〉 integrals are calculated for the 4s 4p configuration in Ca, including the effects of core polarization. This is done with the use of LS-dependent Hartree-Fock orbitals. The calculated effects describe well the magnetic hyperfine structure for the3P term. The description of the1P term is less good, however, due to the effects of the strong correlation between the valence electrons for this term. The intermediate coupling in the lowestsp configurations of Mg, Ca and Sr is analysed considering also the spin-spin interaction. Using theab initio 〈r−3〉 integrals, the determined intermediate coupling and the experimentally known hyperfine constants, second-order hyperfine corrections are calculated and accurateaij- andbij-parameters evaluated for43Ca. This resolves the previously inconsistent results forb02 determined from the3P1 and3P2b-factors for this isotope. An accurate value forb02 in the3P term of43Ca isb02=−13.27(6) MHz. Combined with the calculated 〈r−3〉02 integral, this gives a more accurate value for the nuclear quadrupole moment of43Ca than given earlier,Q(43Ca)=−49(5) mb, where the uncertainty comes from the uncertainty estimated for the 〈r−3〉02 integral.

Ab-initio calculations of pair-correlation effects on the hyperfine structure of the 4s 4p and 4s 3d configurations of Ca

The hyperfine structure of the 4s 4p and 4s 3d configurations of Ca has been evaluated using many-body perturbation theory. Single excitations and pair-correlation effects have been included by solving coupled one-particle and two-particle differential equations numerically. The correlation between the valence electrons has been treated selfconsistently by solving these equations iteratively for the valence pair. Core-polarisation effects have been evaluated by hyperfine-induced single-particle functions. From the calculated and experimental results of the two configurations,Q=−49(5) mb has been evaluated for the nuclear quadrupole moment of43Ca.

Differentiability in Density-Functional Theory

The differentiability of different functionals used in density-functional theory (DFT) is investigated, and it is shown that the so-called Levy–Lieb functional FLL[ρ] and the Lieb functional FL[ρ] are Gâteaux differentiable at pure-state v-representable and ensemble v-representable densities, respectively. The conditions for the Frechet differentiability of these functionals are also discussed. The Gâteaux differentiability of the Lieb functional has been demonstrated by Englisch and Englisch (Phys. Stat. Solidi 123, 711 and 124, 373 (1984)), but the differentiability of the Levy–Lieb functional has not been shown before.