Multiconfiguration Hartree-Fock Calculations Research Articles

Extreme ultraviolet 4p and 4d excitation spectra in laser-produced Ag plasmas

Experimental and theoretical analyses are reported for the spatiotemporal evolution of extreme ultraviolet spectra in laser-produced silver (Ag) plasmas. The time-resolved spectra were acquired 1.5 mm from the target surface. The 4p and 4d excitations of Ag7+–Ag13+ ions were calculated with the aid of multiconfiguration Hartree-Fock calculations. The spectral line distributions of transitions in the dominant 4p and 4d excitation configurations and their variations with the degree of ionization were analyzed. Complex spectral structures were accurately revealed based on the assumption of a normalized Boltzmann distribution among the excited states and a steady-state collision-radiative model. Experimental peak origins were identified by comparing experimental and simulated spectra. These results will provide data for energy-level structures of highly-charged ions of middle- and high-Z elements and their radiative characteristics in plasmas.

A study for hydrogen-like lawrencium

Studying hydrogenic ions with high Z is an occasion to understand atomic structure. It also provides a reliable test of methods used to determine atomic structures. Many fields and applications require precise atomic data. For this reason, a hydrogen-like study is performed for lawrencium (Lr102+, Z = 103). The energy levels of hydrogen-like lawrencium are calculated with both multiconfiguration Hartree–Fock (MCHF) and multiconfiguration Dirac–Fock (MCDF) methods. The calculations contain Breit–Pauli relativistic corrections in MCHF calculation and the transverse photon and quantum electrodynamics (QED) effects in MCDF calculation along with electron correlations. In addition, some transition parameters (wavelengths, λ, logarithmic weighted oscillator strengths, log(gf) value, and transition probabilities, Aki) for allowed (E1) and forbidden (E2 and M1) transitions are investigated. The results from this study are compared with only a few theoretical works, but there is no available experimental data yet for Lr102+.

Theoretical hyperfine structures of F19 i and O17 i

Multiconfiguration Hartree-Fock (MCHF) and multiconfiguration Dirac-Hartree-Fock (MCDHF) calculations are performed for the $2p^{5}~^{2}P^{o}$, $2p^4(^{3}P)3s~^{4}P$, $2p^4(^{3}P)3s~^{2}P$ and $2p^4(^{3}P)3p~^{4}S^o$ states of $^{19}$F~I to determine their hyperfine constants. Several computing strategies are considered to investigate electron correlation and relativistic effects. High-order correlation contributions are included in MCHF calculations based on single and double multireference (SD-MR) expansions. The largest components of the single reference MCHF wave functions are selected to define the MR sets. In this scheme, relativistic corrections are evaluated in the Breit-Pauli approximation. A similar strategy is used for the calculation of MCDHF relativistic wave functions and hyperfine parameters. While correlation and relativistic corrections are found to be rather small for the ground state, we highlight large relativistic effects on the hyperfine constant $A_{3/2}$ of $2p^4(^{3}P)3p~^{4}S^o$ and, to a lesser extent, on $A_{1/2}$ of $2p^4(^{3}P)3s~^{4}P$. As expected for such a light system, electron correlation effects dominate over relativity in the calculation of the hyperfine interaction of all other levels considered. We also revisit the hyperfine constants of $2p^3(^{4}S)3s~^{5}S^{o}$ and $2p^3(^{4}S)3p~^{5}P$ in $^{17}$O using similar strategies. The results are found to be in excellent agreement with experiment.

Energies and radiative transitions (E1, E2, and M1) for hydrogen-like thorium

Hydrogen-like ions with high Z present unique opportunities to promote the understanding of atomic structure. On the other hand, determining the atomic structure of hydrogen-like atoms provides us the test of relativistic effects as well as accurate values of the spectral characteristics required for many applications. For this reason we have calculated level energies of hydrogen like thorium (Th89+, Z = 90) with both multiconfiguration Hartree–Fock (MCHF) and multiconfiguration Dirac–Fock (MCDF) methods. These calculations have contained the Breit–Pauli relativistic corrections in MCHF calculation and quantum electrodynamics effects in MCDF calculations besides electron correlations. The wavelengths, λ, weighted oscillator strengths, gf-value, and transition probabilities, Aki, have been also represented for allowed (E1) and forbidden (E2 and M1) transitions. We have compared our results with only theoretical work results because there is no available experimental data for Th89+.

Collision rates for electron excitation of Mg V lines

Aims. Transition probabilities and electron impact excitation collision strengths and rates for astrophysically important lines in Mg V are reported. The 86 fine-structure levels of the 2s2p, 2s2p, 2p, 2s2p3s, 2s2p3p and 2s2p3d configurations are included in our calculations. The effective collision strengths are presented as a function of electron temperature for solar and other astrophysical applications. Methods. The collision strengths have been calculated using the B-spline Breit-Pauli R-matrix method for all fine-structure transitions among the 86 levels. The one-body mass, Darwin and spin-orbit relativistic effects are included in the Breit-Pauli Hamiltonian in the scattering calculations. The one-body and two-body relativistic operators are included in the multiconfiguration Hartree-Fock calculations of transition probabilities. Several sets of non-orthogonal spectroscopic and correlation radial orbitals are used to obtain accurate description of Mg V 86 levels and to represent the scattering functions. Results. The calculated excitation energies are in very good agreement with experiment and represents an improvement over the previous calculations. The present collision strengths show good agreement with the previously available R-matrix and distorted-wave calculations. The oscillator strengths for E1 transitions normally compare very well with previous calculations. The thermally averaged collision strengths are obtained by integrating total resonant and non-resonant collision strengths over a Maxwellian distribution of electron energies and these are presented over the temperature range log10 Te = 3.2−6.0 K.

Oscillator strengths and transition probabilities from the Breit–Pauli R-matrix method: Ne IV

The atomic parameters–oscillator strengths, line strengths, radiative decay rates (A), and lifetimes–for fine structure transitions of electric dipole (E1) type for the astrophysically abundant ion Ne IV are presented. The results include 868 fine structure levels with n≤ 10, l≤ 9, and 1/2≤J≤ 19/2 of even and odd parities, and the corresponding 83,767 E1 transitions. The calculations were carried out using the relativistic Breit–Pauli R-matrix method in the close coupling approximation. The transitions have been identified spectroscopically using an algorithm based on quantum defect analysis and other criteria. The calculated energies agree with the 103 observed and identified energies to within 3% or better for most of the levels. Some larger differences are also noted. The A-values show good to fair agreement with the very limited number of available transitions in the table compiled by NIST, but show very good agreement with the latest published multi-configuration Hartree–Fock calculations. The present transitions should be useful for diagnostics as well as for precise and complete spectral modeling in the soft X-ray to infra-red regions of astrophysical and laboratory plasmas.

Relativistic energy-consistent pseudopotentials for superheavy elements 119 and 120 including quantum electrodynamic effects

Relativistic energy-consistent pseudopotentials for the superheavy elements with nuclear charges 119 and 120 replacing 92 electrons of a [Xe]4f(14)5d(10)5f(14) core were adjusted to relativistic multi-configuration Dirac-Coulomb-Breit finite nucleus all-electron reference data including lowest-order quantum electrodynamic effects, i.e., vacuum polarization and electron self-energy. The parameters were fitted by two-component multi-configuration Hartree-Fock calculations in the intermediate coupling scheme to the total valence energies of 131 to 140 relativistic states arising from 31 to 33 nonrelativistic configurations covering also anionic and highly ionized states, with mean absolute errors for the nonrelativistic configurations below 0.01 eV. Primitive basis sets for one- and two-component calculations with errors below 0.02 and 0.03 eV to the Hartree-Fock limit, respectively, as well as general contractions of these basis sets with double- to quadruple-zeta quality were obtained. Atomic highly correlated test calculations using the Fock-space coupled-cluster method yield for valence excitation energies and ionization potentials mean absolute errors of 26 cm(-1) and 59 cm(-1), respectively. Correlated and uncorrelated molecular test calculations show deficiencies below 0.005 Å for the bond lengths and 3 N m(-1) for the force constants.

A partitioned correlation function interaction approach for describing electron correlation in atoms

The traditional multiconfiguration Hartree–Fock (MCHF) and configuration interaction (CI) methods are based on a single orthonormal orbital basis. For atoms with many closed core shells, or complicated shell structures, a large orbital basis is needed to saturate the different electron correlation effects such as valence, core–valence and correlation within the core shells. The large orbital basis leads to massive configuration state function (CSF) expansions that are difficult to handle, even on large computer systems. We show that it is possible to relax the orthonormality restriction on the orbital basis and break down the originally very large calculations into a series of smaller calculations that can be run in parallel. Each calculation determines a partitioned correlation function (PCF) that accounts for a specific correlation effect. The PCFs are built on optimally localized orbital sets and are added to a zero-order multireference (MR) function to form a total wave function. The expansion coefficients of the PCFs are determined from a low dimensional generalized eigenvalue problem. The interaction and overlap matrices are computed using a biorthonormal transformation technique (Verdebout et al 2010 J. Phys. B: At. Mol. Phys. 43 074017). The new method, called partitioned correlation function interaction (PCFI), converges rapidly with respect to the orbital basis and gives total energies that are lower than the ones from ordinary MCHF and CI calculations. The PCFI method is also very flexible when it comes to targeting different electron correlation effects. Focusing our attention on neutral lithium, we show that by dedicating a PCF to the single excitations from the core, spin- and orbital-polarization effects can be captured very efficiently, leading to highly improved convergence patterns for hyperfine parameters compared with MCHF calculations based on a single orthogonal radial orbital basis. By collecting separately optimized PCFs to correct the MR function, the variational degrees of freedom in the relative mixing coefficients of the CSFs building the PCFs are inhibited. The constraints on the mixing coefficients lead to small off-sets in computed properties such as hyperfine structure, isotope shift and transition rates, with respect to the correct values. By (partially) deconstraining the mixing coefficients one converges to the correct limits and keeps the tremendous advantage of improved convergence rates that comes from the use of several orbital sets. Reducing ultimately each PCF to a single CSF with its own orbital basis leads to a non-orthogonal CI approach. Various perspectives of the new method are given.

Accurate relativistic energy-consistent pseudopotentials for the superheavy elements 111 to 118 including quantum electrodynamic effects

Energy-consistent two-component semi-local pseudopotentials for the superheavy elements with atomic numbers 111-118 have been adjusted to fully relativistic multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb Hamiltonian, including perturbative corrections for the frequency-dependent Breit interaction in the Coulomb gauge and lowest-order quantum electrodynamic effects. The pseudopotential core includes 92 electrons corresponding to the configuration [Xe]4f(14)5d(10)5f(14). The parameters for the elements 111-118 were fitted by two-component multi-configuration Hartree-Fock calculations in the intermediate coupling scheme to the total energies of 267 up to 797 J levels arising from 31 up to 62 nonrelativistic configurations, including also anionic and highly ionized states, with mean absolute errors clearly below 0.02 eV for averages corresponding to nonrelativistic configurations. Primitive basis sets for one- and two-component pseudopotential calculations have been optimized for the ground and excited states and exhibit finite basis set errors with respect to the finite-difference Hartree-Fock limit below 0.01 and 0.02 eV, respectively. General contraction schemes have been applied to obtain valence basis sets of polarized valence double- to quadruple-zeta quality. Results of atomic test calculations in the intermediate coupling scheme at the Fock-space coupled-cluster level are in good agreement with those of corresponding fully relativistic all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian. The results demonstrate besides the well-known need of a relativistic treatment at the Dirac-Coulomb level also the necessity to include higher-order corrections for the superheavy elements.

New energy levels and transitions for the 4s4p5p configuration in Kr VI

The spectrum of five-time ionized krypton, Kr VI, was recorded in the 240–2600 Å wavelength range, resulting in 61 new classified lines as transitions between levels of configurations 4p 3, 4s 25p, 4s4p4d, 4s4p5s and 4s4p5p. All the 18 energy levels belonging to 4s4p5p configuration except one were determined. Eight new energy level values corresponding to configurations 4s4p4f and 4p24d, supported by 26 new classified lines were also determined and used in the interpretation of the observed 4s4p5p configuration. The experimental data were obtained from a capillary-discharge tube and theta pinch light sources. Multiconfiguration Hartree–Fock calculations with relativistic corrections and a least-squares fitted parametric calculation has been carried out.

Dependence of the probabilities of the electric-multipole electron transitions inW24+on multipolarity

Usually it is accepted that the probabilities of the electric-multipole electron transitions are rapidly decreasing functions of their multipolarity. Therefore while calculating the probabilities of electronic transitions between the configurations of certain chosen parities, it seems sufficient to take into account the first nonzero term, i.e., to consider the electron transitions of lowest multipolarity permitted by the exact selection rules. This paper aims at verifying this assumption on the example of electric-octupole transitions in ${\mathrm{W}}^{24+}$ ion. For this purpose the large-scale multiconfiguration Hartree-Fock and Dirac-Fock calculations have been performed for the configurations [Kr]$4{d}^{10}4{f}^{4}$ and [Kr]$4{d}^{10}4{f}^{3}5s$ energy levels of ${\mathrm{W}}^{24+}$ ion. The relativistic corrections were taken into account in the quasirelativistic Breit-Pauli and fully relativistic Breit (taking into account QED effects) approximations. The role of correlation, relativistic, and QED corrections is discussed. Line strengths, oscillator strengths, and transition probabilities in Coulomb and Babushkin gauges are presented for $E1$ and $E3$ transitions among these levels.

Unified interpretation of Hund’s first and second rules for 2p and 3p atoms

A unified interpretation of Hund's first and second rules for 2p (C, N, O) and 3p (Si, P, S) atoms is given by Hartree-Fock (HF) and multiconfiguration Hartree-Fock (MCHF) methods. Both methods exactly satisfy the virial theorem, in principle, which enables one to analyze individual components of the total energy E(=T+V(en)+V(ee)), where T, V(en), and V(ee) are the kinetic, the electron-nucleus attraction, and the electron-electron repulsion energies, respectively. The correct interpretation for each of the two rules can only be achieved under the condition of the virial theorem 2T+V=0 by investigating how V(en) and V(ee) interplay to attain the lower total potential energy V(=V(en)+V(ee)). The stabilization of the more stable states for all the 2p and 3p atoms is ascribed to a greater V(en) that is caused by contraction of the valence orbitals accompanied with slight expansion of the core orbitals. The contraction of the valence orbitals for the two rules is a consequence of reducing the Hartree screening of the nucleus at short interelectronic distances. The reduced screening in the first rule is due to a greater amount of Fermi hole contributions in the state with the highest total spin-angular momentum S. The reduced screening in the second rule is due to the fact that two valence electrons are more likely to be on opposite sides of the nucleus in the state with the highest total orbital-angular momentum L. For each of the two rules, the inclusion of correlation does not qualitatively change the HF interpretation, but HF overestimates the energy difference ∣ΔE∣ between two levels being compared. The magnitude of the correlation energy is significantly larger for the lower L states than for the higher L states since two valence electrons in the lower L states are less likely to be on opposite sides of the nucleus. The MCHF evaluation of ∣ΔE∣ is in excellent agreement with experiment. The present HF and MCHF calculations demonstrate the above statements that were originally given by Katriel [Theor. Chem. Acta 23, 309 (1972); 26, 163 (1972)]. We have, for the first time, analyzed the correlation-induced changes in the radial density distribution for the excited LS terms of the 2p and 3p atoms as well as for the ground LS term.

Multiconfigurational time-dependent Hartree-Fock calculations for photoionization of one-dimensional Helium

The multiconfigurational time-dependent Hartree-Fock equations are discussed and solved for a one-dimensional model of the Helium atom. Results for the ground state energy and two-particle density as well as the absorption spectrum are presented and compared to direct solutions of the time-dependent Schrödinger equation.

Exploring biorthonormal transformations of pair-correlation functions in atomic structure variational calculations

Multiconfiguration expansions frequently target valence correlation and correlation between valence electrons and the outermost core electrons. Correlation within the core is often neglected. A large orbital basis is needed to saturate both the valence and core–valence correlation effects. This in turn leads to huge numbers of configuration state functions (CSFs), many of which are unimportant. To avoid the problems inherent to the use of a single common orthonormal orbital basis for all correlation effects in the multiconfiguration Hartree–Fock (MCHF) method, we propose to optimize independent MCHF pair-correlation functions (PCFs), bringing their own orthonormal one-electron basis. Each PCF is generated by allowing single- and double-excitations from a multireference (MR) function. This computational scheme has the advantage of using targeted and optimally localized orbital sets for each PCF. These pair-correlation functions are coupled together and with each component of the MR space through a low dimension generalized eigenvalue problem. Nonorthogonal orbital sets being involved, the interaction and overlap matrices are built using biorthonormal transformation of the coupled basis sets followed by a counter-transformation of the PCF expansions. Applied to the ground state of beryllium, the new method gives total energies that are lower than the ones from traditional complete active space (CAS)-MCHF calculations using large orbital active sets. It is fair to say that we now have the possibility to account for, in a balanced way, correlation deep down in the atomic core in variational calculations.

Linear dichroism in the innershell photoionization of alkaline atoms

The partial photoionization cross section of the 2p ionization of laser-excited aligned as well as ground-state oriented Na atoms were determined by high-resolution photoelectron spectroscopy and compared to multi-configuration Dirac-Fock and Hartree-Fock calculations. The measurements of the linear alignment dichroism (LAD) and the linear magnetic dichroism (LMDAD) are in very good agreement with the theoretical results. Similar studies were undertaken for the 3p photoionization of atomic K.

Energy-consistent pseudopotentials and correlation consistent basis sets for the 5d elements Hf–Pt

New relativistic energy-consistent pseudopotentials have been generated for the 5d transition metals Hf-Pt. The adjustment was done in numerical two-component multiconfiguration Hartree-Fock calculations, using atomic valence-energy spectra from four-component multiconfiguration Dirac-Hartree-Fock calculations as reference data. The resulting two-component pseudopotentials replace the [Kr]4d(10)4f(14) cores of the 5d transition metals and can easily be split into a scalar-relativistic and a spin-orbit part. They reproduce the all-electron reference energy data with deviations of approximately 0.01 eV for configurational averages and approximately 0.05 eV for individual relativistic states. Full series of correlation consistent basis sets from double to quintuple-zeta have also been developed in this work for use with the new pseudopotentials. In addition, all-electron triple-zeta quality correlation consistent basis sets are also reported in order to provide calibration for the pseudopotential treatment. The accuracy of both the pseudopotentials and basis sets are assessed in extensive coupled cluster benchmark calculations of atomic ionization potentials, electron affinities, and selected excitation energies of all the 5d metal atoms, including the effects of spin-orbit coupling.

Theoretical studies of spectroscopic properties ofCm4+andAm3+

In this paper we report on large-scale multiconfiguration Hartree-Fock and multiconfiguration Dirac-Fock calculations of the fine structures of the ${\text{Cm}}^{4+}$ and ${\text{Am}}^{3+}$ ions, both having nominal $5{f}^{6}$ electronic configuration. Correlation effects play an extremely important role for both considered ions, and the single-configuration model is proved to be unfit to account for this complex picture. A comparison of the calculated energy levels with the results of spectroscopic measurements shows that theory and experiment are in quantitatively good agreement; the ground state of both ions is predicted to have nonmagnetic character. The accuracy of the results is estimated and discussed.

4p-inner-shell and double-excitation spectrum ofSrII

We report photoabsorption measurements from the $4p$ inner shell of singly ionized strontium in the spectral region between 26.0 and $37.4\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. More than 60 levels of $\mathrm{Sr}\phantom{\rule{0.2em}{0ex}}II$ are reported which are interpreted as singly excited inner-shell transitions $4{p}^{6}5s$ $^{2}S_{1∕2}\ensuremath{\rightarrow}4{p}^{4}5s(^{1}P,^{3}P)ns$, $md$ and doubly excited transitions $4{p}^{6}5s$ $^{2}S_{1∕2}\ensuremath{\rightarrow}4{p}^{5}4d(^{1}P,^{3}P,^{3}D)nl$. Multiconfiguration Hartree-Fock calculations are performed in $jj$ coupling and the levels are arranged into Rydberg series converging on seven limits, allowing the identification of two levels in $\mathrm{Sr}\phantom{\rule{0.2em}{0ex}}III$.

Theoretical analysis of xuv photoabsorption spectra of laser-produced iodine plasmas

The $4d$ photoabsorption spectra of ${\mathrm{I}}^{2+}$, ${\mathrm{I}}^{3+}$, and ${\mathrm{I}}^{4+}$ have been obtained in the $70--127\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ region with the dual laser-produced plasma technique at time delays ranging from $400\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}520\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$. With decreasing time delay, the dominant contribution to the spectra evolves from the ${\mathrm{I}}^{2+}$ to the ${\mathrm{I}}^{4+}$ ions, and each spectrum contains discrete $4d\text{\ensuremath{-}}nf$ transitions and a broad $4d\text{\ensuremath{-}}\ensuremath{\epsilon}f$ shape resonance, which are identified with the aid of multiconfiguration Hartree-Fock calculations. The excited states decay by direct autoionization involving $5s$ or $5p$ electrons, and rates for the different processes and resulting linewidths were calculated. With increasing ionization, the $4d\text{\ensuremath{-}}\ensuremath{\epsilon}f$ shape resonance become intense and broader in going from ${\mathrm{I}}^{2+}$ to ${\mathrm{I}}^{3+}$, and then vanishes at ${\mathrm{I}}^{5+}$. In addition, the discrete structure of the calculated spectrum of each ion gradually approaches the corresponding shape resonance position. Based on the assumption of a normalized Boltzmann distribution among the excited states and a steady-state collisional-radiative model, we reproduced spectra which are in good agreement with experiment.

Relativistic all-order and multiconfiguration Hartree-Fock calculations of the4d−4fenergy separation in Li I

We present a calculation of the 4d-4f energy separation in Li I using two advanced techniques in atomic structure theory, namely, the relativistic all-order method and the multiconfiguration Hartree-Fock (MCHF) method. The accuracy of our calculations was investigated by conducting a third-order many-body perturbation theory calculation that allowed us to evaluate the importance of fourth- and higher-order corrections. A large-scale MCHF calculation was performed using the active space method and the core-polarization approximation. The obtained results provide an important test of these methods against each other and are shown to agree with the most accurate available experimental data.