Multiconfiguration Hartree-Fock Method Research Articles

Benchmark PhotoIonization Cross-Sections of Neutral Scandium from the Ground and Excited States

The B-spline R-matrix method has been used to investigate cross-sections for photoionization of neutral scandium from the ground and excited states in the energy region from the 3d and 4s valence electron ionization thresholds to 25 eV. The initial bound states of Sc and the final residual Sc+ ionic states have been accurately calculated by combining the multiconfiguration Hartree-Fock method with the frozen-core close-coupling approach. The lowest 20 bound states of Sc I belonging to the ground 3d4s2 and excited 3d24s, 3d24p, 3d4s4p, 4s24p, and 3d3 configurations have been considered as initial states. The 81 LS final ionic states of Sc+ belonging to the terms of 3p63d2, 3p63d4l (l = 0–3), 3p63d5l (l = 0–3), 3p63d6s, 3p64s2, 3p64s4l (l = 0–3), 3p64s5l (l = 0–1), and 3p64p2 configurations have been included in the final-state close-coupling expansion. The cross-sections are dominated by complicated resonance structures in the low energy region converging to several Sc+ ionic thresholds. The inclusion of all these final ionic states has been noted to significantly impact the near-threshold resonance structures and background cross-sections. The important scattering channels for leaving the residual ion in various final states have been identified, and the 3d electron ionization channels have been noted to dominate the cross-sections at higher photon energies.

Partial Photoionization Cross Sections of Chromium from the Ground and Excited States

Partial and total photoionization cross sections of iron-peak elements are important for the determination of abundances in late-type stars and nebular objects. We have investigated photoionization of neutral chromium from the ground and excited states in the low energy region from the first ionization threshold at 6.77 eV to 30 eV. Accurate descriptions of the initial bound states of Cr I and the final residual Cr II ionic states have been obtained in the multiconfiguration Hartree-Fock method together with adjustable configuration expansions and term-dependent non-orthogonal orbitals. The B-spline R-matrix method has been used for the calculation of photoionization cross sections. The 194 LS final ionic states of Cr II 3d44s, 3d34s2, 3d5, 3d44p, and 3d34s4p principal configurations have been included in the close-coupling expansion. The inclusion of all terms of these configurations has significant impact on the near-threshold resonance structures as well as on the nonresonant background cross sections. Total photoionization cross sections from the ground 3d54sa7S and excited 3d54sa5S, 3d44s2a5D, 3d54pz5P, and 3d44s4py5P states of Cr I have been compared with other available R-matrix calculation to estimate the likely uncertainties in photoionization cross sections. We analyzed the partial photoionization cross sections for leaving the residual ion in various states to identify the important scattering channels, and noted that 3d electron ionization channel becomes dominant at higher energies.

Collision and Radiative Rates for Infrared to Extreme Ultraviolet Lines of S iii

Abstract Electron excitation collision strengths for a wide range of transitions giving rise to infrared, optical, ultraviolet, and extreme ultraviolet lines of S iii have been calculated using the B-spline Breit–Pauli R-matrix method. The term-dependent non-orthogonal orbitals have been employed for the accurate representation of target wave functions and the electron plus S iii target scattering system. The multiconfiguration Hartree–Fock method has been utilized for the calculation of 198 S iii fine-structure level energies belonging to the , , , and 3s3p 25s configurations. The transition probabilities between fine-structure levels have also been calculated and compared with available other calculations. The close-coupling expansion includes these 198 fine-structure levels of S iii in the electron collision calculations. The effective collision strengths are calculated at electron temperatures in the range of 103–106 K for all possible transitions between the 198 fine-structure levels. The present calculation includes a larger number of levels in the close-coupling expansion and improved target description than previous calculations and should be useful for the analysis of measured spectra from various astrophysical objects. Comparison with other calculations is used to assess likely uncertainties in the existing collision and radiative rates for S iii. The collision and radiative rates are estimated to be accurate to about 20% or better for most main transitions of astrophysical importance.

ISOTOPE SHIFT EFFECTS ON THE GROUND-STATE LEVELS FOR NEUTRAL TIN

Isotope shift effects including mass shift and field shift on the levels for the levels of 3P0,1,2, 1D2 and 1S0 for the ground-state configuration of the neutral tin (Sn I, Z= 50) have been investigated. In calculations, the multiconfiguration Hartree-Fock method within the frame work of Breit-Pauli Hamiltonian has been used. The calculation of isotope effects for tin has been here performed firstly although there are some works on the atomic structure of tin. The results including isotope shift effects on the levels have been discussed, and new energy values have been compared other available works in literature.

Photoionization of neutral iron from the ground and excited states

The B-spline R-matrix method is used to investigate the photoionization of neutral iron from the ground and excited states in the energy region from the ionization thresholds to 2 Ry. The multiconfiguration Hartree-Fock method in connection with adjustable configuration expansions and term-dependent orbitals is employed for an accurate representation of the initial states of Fe i and the target wave functions of Fe ii. The close-coupling expansion contains 261 LS states of Fe ii and includes all levels of the 3d64s,3d54s2,3d7,3d64p, and 3d54s4p configurations. Full inclusion of all terms from the principal configurations considerably changes both the low-energy resonance structure and the energy dependence of the background cross sections. Partial cross sections are analyzed in detail to clarify the most important scattering channels. Comparison with other calculations is used to place uncertainty bounds on our final photoionization cross sections and to assess the likely uncertainties in the existing data sets.

Electron Scattering from Neutral Fe and Low-energy Photodetachment of Fe−

Abstract Extensive calculations are reported for electron collision strengths and rate coefficients for a wide range of transitions in Fe i. The calculations were carried out with the B-spline R-matrix method. A multiconfiguration Hartree–Fock method with nonorthogonal, term-dependent orbitals was employed to generate accurate initial- and final-state wave functions. The close-coupling expansion contained 221 LS states of Fe and included all levels of the , , , , and configurations. Effective collision strengths were obtained by averaging the electron collision strengths over a Maxwellian speed distribution at electron temperatures ranging from 102 to 105 K. They were tabulated for 24,531 transitions between all LS-terms included in the close-coupling expansion. The present results considerably expand on the few existing data sets for Fe i. They enable more detailed treatments of the available measured spectra from various observatories than previously possible. In particular, nonlocal thermodynamic equilibrium modeling of late-type stars, where large amounts of collisional data for the atomic species of interest are required, can be performed. The same close-coupling expansion was used to study low-energy photodetachment of Fe−, where the cross sections exhibit prominent resonance features. Good agreement with the few existing experimental values for partial cross sections to specific final target states of Fe was obtained.

Electron-impact excitation of forbidden and allowed transitions in Fe ii

Extensive calculations are reported for electron collision strengths, rate coefficients, and transitions probabilities for a wide range of transitions in Fe ii. The collision strengths were calculated in the close-coupling approximation using the B-spline Breit-Pauli R-matrix method. The multiconfiguration Hartree-Fock method in connection with adjustable configuration expansions and a semiempirical fine-tuning procedure is employed for an accurate representation of the target wave functions. The energy correction was also used in the scattering calculations by adding to Hamiltonian matrices prior to transformation to intermediate coupling. The spin-orbit interaction term was added to the final Hamiltonian matrices in scattering calculations. The close-coupling expansion contains 340 fine-structure levels of Fe ii and includes all levels of the $3{d}^{6}4s$, $3{d}^{5}4{s}^{2}$, $3{d}^{7}$, and $3{d}^{6}4p$ configurations, plus a few lowest levels of the $3{d}^{5}4s4p$ configuration. The effective collision strengths are obtained by averaging the electron collision strengths over a Maxwellian distribution of velocities at electron temperatures in the range from ${10}^{2}$ to ${10}^{5}$ K and are reported for all possible inelastic transitions between the 340 fine-structure levels. The present results are more extensive than the previous calculations and considerably expand the existing data sets for Fe ii, allowing a more detailed treatment of the available measured spectra from different space observatories. Comparison with other calculations for collision rates and available experimental radiative rates is used to place uncertainty bounds on our collision strengths and to assess the likely uncertainties in the existing data sets.

The Electric Dipole Transitions of Hydrogen Like Nobelium

It is an occasion to understand and test the methods for determination of atomic structure on hydrogen like ions especially with high atomic number, Z. Many studies such as interpretation of astrophysical spectra and atomic collision studies, the development of X-ray lasers etc. demand exact transition parameters data. This need is a reason of the present paper to support more precise atomic data of electric dipole transitions parameters for hydrogen like nobelium (No 101+ , Z=102). The wavelengths, logarithmic weighted oscillator strengths and transition probabilities of No 101+ have been reproduced with relativistic multiconfiguration Hartree-Fock (MCHF) and fully relativistic multiconfiguration Dirac-Fock (MCDF) methods. The transitions between nl ( n =1-9 and l =0-4) levels have been investigated in this work. The results of MCHF and MCDF methods have been compared with a unique theoretical work, because there has been neither theoretical nor experimental work in available literature.

Ab initio calculations of the hyperfine structure of zinc and evaluation of the nuclear quadrupole moment Q(Zn67)

The relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) and the non-relativistic multiconfiguration Hartree-Fock (MCHF) methods have been employed to calculate the magnetic dipole and electric quadrupole hyperfine structure constants of zinc. The calculated electric field gradients for the $ 4s 4p \,\, ^3 \! P^o_{1} $ and $ 4s 4p \,\, ^3 \! P^o_{2} $ states, together with experimental values of the electric quadrupole hyperfine structure constants, made it possible to extract a nuclear electric quadrupole moment $ Q(^{67}{\rm Zn}) = 0.122(10) $ b. The error bar has been evaluated in a quasi-statistical approach - the calculations had been carried out with eleven different methods, and then the error bar has been estimated from the differences between the results obtained with those methods.

A structure calculation for hydrogen like nobelium

The hydrogen like ion is the simplest structure of an atom, so the studying on it gives a change to test of methods using for determinating of atomic structures. It is also an occasion the studying on hydrogen like ions to understand atomic structure. For this reason, it is performed a study for hydrogen like nobelium (No101+, Z=102). The calculating level structure of No101+ have been used both multiconfiguration Hartree-Fock and multiconfiguration Dirac-Fock method. The calculations have contained some relativistic corrections and quantum electrodynamic effects. There are very few theoretical works in available literature and no experimental one for No101+. The calculated results have been interpreted in comparison with other theoretical results.

Electron-Impact Excitation of 51S – 51Po Resonance Transition in Sr Atom

Main aspects of a new version of the B-spline R-matrix (BSR) method, in which nonorthogonal orbitals are applied, have been described. The BSR approximation is used to calculate the resonance structure of integral cross-sections of the 51S - 51Po transition at the electron scattering by a strontium atom in the energy interval up to 10 eV. The multiconfiguration Hartree–Fock method with a nonorthogonal set of orbitals is employed to accurately represent the target wave functions. The close-coupling expansion included 31 bound states of a neutral strontium atom ranging from the ground state to the 5s5f 1Fo one. The calculated cross-sections are in good agreement with available experimental data and can be exhaustively interpreted. The structure of a resonance feature in the e-Sr scattering cross-sections at about 4 eV is discussed.

Low-energy outer-shell photo-detachment of the negative ion of aluminum

We investigate the photodetachment of the negative ion of aluminum, initially both in the ground state and the excited 1D state. A multiconfiguration Hartree–Fock method with nonorthogonal, term-dependent orbitals is used to generate accurate initial bound-state and final continuum-state wavefunctions. The collision of the ejected electron in the field of the residual ion is described by the B-spline R-matrix method for photon energies ranging from threshold to 12 eV. All principal scattering channels for photodetachment from both the and orbitals are included in the close-coupling expansion. We predict a number of prominent resonance features, whose verification presents new challenges in studies of this highly correlated process. Partial cross sections, together with the contributions of the principal scattering channels, are discussed in order to classify the resonance structure. The predicted cross sections are expected to be the most comprehensive and accurate currently available dataset for photodetachment of Al− at low energies.

Transition and Electron Impact Excitation Collision Rates for O iii

Abstract Transition probabilities, electron excitation collision strengths, and rate coefficients for a large number of O iii lines over a broad wavelength range, from the infrared to ultraviolet, have been reported. The collision strengths have been calculated in the close-coupling approximation using the B-spline Breit–Pauli R-matrix method. The multiconfiguration Hartree–Fock method in combination with B-spline expansions is employed for an accurate representation of the target wave functions. The close-coupling expansion contains 202 O2+ fine-structure levels of the , , , and configurations. The effective collision strengths are obtained by averaging electron excitation collision strengths over a Maxwellian distribution of velocities at electron temperatures ranging from 100 to 100,000 K. The calculated effective collision strengths have been reported for the 20,302 transitions between all 202 fine-structure levels. There is an overall good agreement with the recent R-matrix calculations by Storey et al. for the transitions between all levels of the ground configuration, but significant discrepancies have been found with Palay et al. for transitions to the 1 S 0 level. Line intensity ratios between the optical lines arising from the − 1 D 2 transitions have been compared with other calculations and observations from the photoionized gaseous nebulae, and good agreement is found. The present calculations provide the most complete and accurate data sets, which should allow a more detailed treatment of the available measured spectra from different ground and space observatories.

Optimized pulses for Raman excitation through the continuum: Verification using the multiconfigurational time-dependent Hartree-Fock method

We have verified a mechanism for Raman excitation of atoms through continuum levels previously obtained by quantum optimal control using the multiconfigurational time-dependent Hartree-Fock (MCTDHF) method. For the optimal control, which requires running multiple propagations to determine the optimal pulse sequence, we used the computationally inexpensive time-dependent configuration interaction singles (TDCIS) method. TDCIS captures all of the necessary correlation of the desired processes but assumes that ionization pathways reached via double excitations are not present. MCTDHF includes these pathways and all multiparticle correlations in a set of time-dependent orbitals. The mechanism that was determined to be optimal in the Raman excitation of the Ne $1{s}^{2}2{s}^{2}2{p}^{5}3{p}^{1}$ valence state via the metastable $1{s}^{2}2{s}^{1}2{p}^{6}3{p}^{1}$ resonance state involves a sequential resonance-valence excitation. First, a long pump pulse excites the core-hole state, and then a shorter Stokes pulse transfers the population to the valence state. This process represents the first step in a multidimensional x-ray spectroscopy scheme that will provide a local probe of valence electronic correlations. Although at the optimal pulse intensities at the TDCIS level of theory the MCTDHF method predicts multiple ionization or excitation ionization of the atom, at slightly lower intensities (reduced by a factor of about 4) the TDCIS mechanism is shown to hold qualitatively. Quantitatively, the MCTDHF populations are reduced from the TDCIS calculations by a factor of 4.

Valence-electron correlation in the double K -shell photoionization of atomic beryllium

We extended the multiconfiguration Hartree-Fock (MCHF) + effective charge (EFC) method for double photoionization of atoms and investigated the effect of valence-electron correlation in the double $K$-shell photoionization of atomic beryllium. We used the MCHF method, which accounts for electron correlations, to calculate the initial-state wave function. The wave functions for the two continuum electrons in the final state are calculated in the angle-dependent EFC approximation. The actual interaction potential between the two final-state continuum electrons is approximated by the EFC, which is determined variationally. We studied the effect of valence shell electron correlation on the $K$-shell double photoionization of beryllium for both equal and unequal sharing of excess photon energies at four fixed scattered electron directions in order to study the effects near the double $K$-shell ionization threshold. We found considerable effect of valence shell correlation in the triple-differential cross section for double photoionization of the beryllium atom at each of the excess photon energies.

Electron correlation in beryllium: Effects in the ground state, short-pulse photoionization, and time-delay studies

We apply a three-dimensional (3D) implementation of the time-dependent restricted-active-space self-consistent-field (TD-RASSCF) method to investigate effects of electron correlation in the ground state of Be as well as in its photoionization dynamics by short XUV pulses, including time-delay in photoionization. First, we obtain the ground state by propagation in imaginary time. We show that the flexibility of the TD-RASSCF on the choice of the active orbital space makes it possible to consider only relevant active space orbitals, facilitating the convergence to the ground state compared to the multiconfigurational time-dependent Hartree-Fock method, used as a benchmark to show the accuracy and efficiency of TD-RASSCF. Second, we solve the equations of motion to compute photoelectron spectra of Be after interacting with a short linearly polarized XUV laser pulse. We compare the spectra for different RAS schemes, and in this way we identify the orbital spaces that are relevant for an accurate description of the photoelectron spectra. Finally, we investigate the effects of electron correlation on the magnitude of the relative time-delay in the photoionization process into two different ionic channels. One channel, the ground state channel in the ion, is accessible without electron correlation. The other channel is only accessible when including electron correlation. The time-delay is highly sensitivity to the choice of the active space, and hence to the account of electron-electron correlation.

Time-dependent restricted-active-space self-consistent-field theory with space partition

Aiming at efficient numerical analysis of time-dependent (TD) many-electron dynamics of atoms involving multielectron continua, the TD restricted-active-space self-consistent-field theory with space partition (TD-RASSCF-SP) is presented. The TD-RASSCF-SP wave function is expanded in terms of TD configuration-interaction coefficients with Slater determinants composed of two kinds of TD orbitals: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ orbitals are defined to be nonvanishing in the inner region ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$), a small volume around the atomic nucleus, and $\stackrel{\ifmmode \check{}\else \v{}\fi{}}{M}$ orbitals are nonvanishing in the large outer region ($\stackrel{\ifmmode \check{}\else \v{}\fi{}}{V}$). For detailed discussion of the SP strategy, the equations of motion are derived by two different formalisms for comparison. To ensure continuous differentiability of the wave function across the two regions, one of the formalisms makes use of the property of the finite-element discrete-variable-representation (FEDVR) functions and introduces additional time-independent orbitals. The other formalism is more general and is based on the Bloch operator as in the $R$-matrix theory, but turns out to be less practical for numerical applications. Hence, using the FEDVR-based formalism, the numerical performance is tested by computing double-ionization dynamics of atomic beryllium in intense light fields. To achieve high accuracy, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ should be set large to take into account the strong many-electron correlation around the nucleus. On the other hand, $\stackrel{\ifmmode \check{}\else \v{}\fi{}}{M}$ can be set much smaller than $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ for capturing the weaker correlation between the two outgoing photoelectrons. As a result, compared with more accurate multiconfigurational TD Hartree-Fock (MCTDHF) method, the TD-RASSCF-SP method may achieve comparable accuracy in the description of the double-ionization dynamics. There are, however, difficulties related to the stiffness of the equations of motion of the TD-RASSCF-SP method, which makes the required time step for this method smaller than the one needed for the MCTDHF approach.

Low-energy photodetachment ofGa−and elastic electron scattering from neutral Ga

We present a comprehensive study of the photodetachment of the negative gallium ion and elastic electron scattering from neutral Ga for photon and electron energies ranging from threshold to 12 eV. The calculations are carried out with the $B$-spline $R$-matrix method. A multiconfiguration Hartree-Fock method with nonorthogonal term-dependent orbitals is employed to generate accurate initial- and final-state wave functions. The close-coupling expansions include the $4s{\phantom{\rule{0.16em}{0ex}}}^{2}4pnl(kl)$ bound and continuum states of Ga and the $4s$-excited autoionizing states $4s4{p}^{2}$. The calculated photodetachment and elastic cross sections exhibit prominent resonance features. In order to clarify the origin of these resonances, the contributions of the major ionization channels to the partial cross sections are analyzed in detail.

Electron-impact excitation and ionization of atomic boron at low and intermediate energies

We present a comprehensive study of electron collisions with boron atoms by using the $B$-spline $R$-matrix method for electron energies ranging from threshold to 100 eV. Elastic, excitation, and ionization cross sections were obtained for all transitions between the lowest 11 states of boron. A multiconfiguration Hartree-Fock method with nonorthogonal term-dependent orbitals was employed to generate accurate wave functions for the target states. Close-coupling expansions including 13, 51, and 999 physical and pseudo-target states of boron were used to check the sensitivity of the results to changes in the theoretical model. The cross-section dataset obtained from the large-scale calculations is expected to be sufficiently accurate and comprehensive for most current modeling applications involving neutral boron.

Low-energy outer-shell photodetachment of the negative ion of boron

The photodetachment of the negative ion of boron, B−(2s 22p 2)3P, is investigated by employing the B-spline R-matrixmethod for photon energies ranging from threshold to 12 eV. A multi-configuration Hartree-Fock method with nonorthogonal, term-dependent orbitals is employed to generate accurate initial bound-state and final continuum-state wavefunctions. The close-coupling expansion includes all principal scattering channels for photodetachment from both the 2p and 2s orbitals. The calculated photodetachment cross sections are in good agreement with the available experimental data. Several prominent resonance features are predicted, thereby providing new challenges in the study of this highly correlated process. To classify the resonance structure, both the partial cross sections and the main contributions of the individual scattering channels are discussed. The presented cross sections, along with the asymmetry parameter β for the angular distribution, are believed to be the most comprehensive and accurate dataset currently available for the B− photodetachment process at low energies.