Restricted Variational Method Research Articles

Energy levels and radiative transitions of Mg VII and Si IX

Term energies, line strengths, transition probabilities, and transition wavelengths among the (1s2)2s22p2, 2s22p3p, 2s2p3, 2s22p3s and 2s22p3d 1,3,5L (L = S, P, D, F) configuration states in Mg VII and Si IX are calculated by using the multiconfiguration Rayleigh-Ritz variation method and restricted variation method. The transition line strengths and transition probabilities of the electric dipole transitions are both given in length and velocity gauges. Differences between these two gauge values are discussed. The calculated atomic parameters show good agreement with the observed experimental results and other theoretical data. Furthermore, the uncertainty of each electric dipole transition line strength is estimated. For several transition parameters, uncertainties are decreased compared to the values from the National Institute of Standards and Technology Atomic Spectra Database.

Energy levels and transition probabilities of N+, F3+, and Ne4+ ions

Abstract Term energies, oscillator strengths, transition probabilities, and transition wavelengths among the low-lying states of (1s2)2s22p2, 2s22p3p, 2s2p3, 2s22p3s, and 2s22p3d 1,3,5 L L = S, P, D, F in N+, F3+, and Ne4+ ions were calculated by using the multiconfiguration Rayleigh-Ritz variation method and restricted variation method. The transition oscillator strengths and transition probabilities for the electric dipole transitions are both given in length and velocity gauges. Deviations between these two gauge values are discussed. The calculated atomic parameters are in good agreement with the observed experimental results and other theoretical data. Furthermore, the uncertainty of each electric dipole transition is estimated. Several uncertainties of transition parameters are improved when comparing with values from national institute of standards and technology NIST database. Atomic parameters presented in this paper should be useful for identifying the levels as well as for precise spectral modeling in astrophysical and laboratory plasmas in the future work.

Theoretical calculation of atomic spectrum for 1s23l4l′ states of beryllium-like ions

Using the saddle-point variation and saddle-point complex-rotation methods, energies, radiative, and Auger rates are calculated for the doubly-excited states 1s23l4l′ of beryllium-like ions. The total wave function is expanded by the restricted variation method to saturate the functional space and improve the nonrelativistic energy. Relativistic corrections and mass polarization effects are taken into account by the first-order perturbation theory. The partial Auger rates for these states are studied. The calculated Auger electron energies are compared with the experimental results which are observed in high resolution electron spectroscopy. The total radiative rates and the total Auger rates for these doubly-excited states 1s23l4l′ in the beryllium-like ions are also analyzed along with the increase of atomic number Z. Calculated results show that the total Auger rates are several orders of magnitude larger than the total radiative rates for these low-Z systems.

Quintet States 1s2s2p2 5P and 1s2p3 5So for Be-Like Systems

Abstract By systematic Rayleigh–Ritz variation calculations, the energies are reported for the core-excited states 1s2s2p2 5P and 1s2p3 5So in the Be-like isoelectronic sequence (Z = 11–20). Energy corrections, including the restricted variational method, mass polarisation, and relativistic effect, are considered to improve the accuracy of energy. The oscillator strengths and transition wavelength between these states are also reported. Computational data on hyperfine structures presented in this paper are calculated for the first time.

Radiative and Auger transitions of K-shell excited resonance states in boron-like sulfur ion

<sec>Non-relativistic energy values and wave functions of the K-shell excited resonance states 1s2s<sup>2</sup>2p<sup>2</sup>, 1s2s2p<sup>3</sup>, 1s2p<sup>4 2, 4</sup><i>L</i> (<i>L</i> = S, P, D) in boron-like sulfur ion are calculated in the frame of multi-configuration saddle-point variation method. The electron correlation effects are considered by the expansion of configuration wave function. The wave functions are constructed and optimized by the orbital-spin angular momentum partial waves selected based on the rule of configuration interaction. To saturate the wave functional space and to improve the non-relativistic energy, the restricted variational method is used to calculate the restricted variational energy. Then, the mass polarization effect and relativistic correction are calculated by the perturbation theory. The quantum electrodynamics (QED) effect and higher-order relativistic correction are considered by the screened hydrogenic formula. Furthermore, the energy shift originating from the interaction between closed channel and open channel is also calculated. Finally, the accurate relativistic energy levels for these resonance states are obtained by adding the non-relativistic energy and all corrections.</sec><sec>Using the optimized wave functions, the line strengths, oscillator strengths, radiative transition rates and transition wavelengths of electric-dipole transitions for the K-shell excited resonance states in boron-like sulfur ion are systematically calculated. In this work, the oscillator strengths and transition rates are given in the length, velocity, and acceleration gauges. The good agreement among the three gauges reflects that the calculated wave functions are reasonably accurate. The calculated radiative transition rates and transition wavelengths are compared with other theoretical data. Good agreement is obtained except the transition: 1s2s(<sup>3</sup>S)2p<sup>3</sup> <sup>2</sup>P<sup>o</sup>→1s2<sup>2</sup>s2p<sup>2 2</sup>D. The deviation between our theoretical result and the MCDF theoretical value is about 46%, which needs further verifying. The Auger rates, Auger branching ratios, and Auger electron energy values of the important decay channels of the K-shell excited states are calculated by the saddle-point complex-rotation method. The calculated Auger rates and Auger electron energy values are also in good agreement with the corresponding reference data. For some K-shell states, the related energy levels and Auger branching ratios are reported for the first time. The present calculations results will provide valuable theoretical data for the calibration of spectral lines and Auger electron spectra in the relevant experiments.</sec>

Energy levels and radiative transitions of the core-excited high-spin states in boron atom (ion)

Energy levels of the core-excited high-spin Rydberg states (4,5,6L, L = S, P) in boron atom (ion) are calculated by the Rayleigh-Ritz variation method with using large-scale multi-configuration wave functions. The important orbital-spin angular momentum partial waves are selected based on the rule of configuration interaction. The computational convergence is discussed by the example of the contribution from each partial wave in the non-relativistic energy calculations of the high-spin state 1s2s2p2 5Pe in B+ ion. To saturate the wave functional space and improve the non-relativistic energy, the restricted variational method is used to calculate the restricted variational energy. Furthermore, the mass polarization effect and relativistic energy correction are included by using a first-order perturbation theory. The quantum electrodynamic effects and higher-order relativistic contributions to the energy levels are also calculated by the screened hydrogenic formula. Then, the accurate relativistic energy levels of these high-spin states of B atom (ion) are obtained by adding the non-relativistic energy and all corrections. The fine structure splitting of these high-spin states is also calculated by the Breit-Pauli operators in the first-order perturbation theory. Compared with other theoretical results, our calculation results are in good accordance with the experimental data. The absorption oscillator strengths, emission oscillator strengths, absorption rates, emission rates, and transition wavelengths of the electric-dipole transitions between these high-spin states of B atom (ions) are systematically calculated by using the optimized wave functions. The oscillator strengths and transition rates are obtained in both the length and velocity gauges. By comparing the two gauge results of oscillator strength, we find that there is a good consistency between them when fl 0.3, and a reasonable consistency is obtained when fl 0.3. The accordance between the length and the velocity gauge results reflects that the calculated wave functions in this work are reasonably accurate. The calculated transition data are also compared with the corresponding experimental and other theoretical data. Good agreement is obtained except the wavelengths for two transitions: 1s2p4p 4Se1s2p3d 4P and 1s2p4d 4P1s2p3p 4Pe. The relative differences between our theoretical results and experimental data are 0.7% and 0.3%, respectively. They need to be verified by further theoretical and experimental studies. For some core-excited high-spin states, the related energy levels and transition data are reported for the first time. Our calculation results will provide valuable data for calculating the spectral lines in the relevant experiments.

Relativistic Energies and Auger Widths of High-Lying Doubly-Excited States 1s23lnl′ 1De in Be-Like O4+ Ions

The saddle-point variational method and restricted variational method are used to calculate energies of doubly-excited singlet states 1s23lnl′ (n = 3–5) 1De in Be-like O4+ ions, including the mass polarization and relativistic corrections. The saddle-point complex-rotation method is used to compute the Auger widths and Auger transition rates. These results are compared with other theoretical and experimental data in the literature.

Energies and Auger widths of the high-lying triply excited states for Li-like beryllium

The energies, fine-structure splittings, Auger channel energies, and the Auger widths of the high-lying triply excited 2l2l{sup '}nl{sup ''} (n>=2) states of {sup 2}S, {sup 2}D, and {sup 2}P{sup o} resonances for Be{sup +} are studied using the saddle-point variational method and saddle-point complex-rotation method. The relativistic corrections and mass polarization are included using the first-order perturbation theory. Restricted variational method is carried out to extrapolate better energies. The partial Auger widths are calculated for the individual open channels and the total Auger widths are obtained by coupling the important open channels and summing over the other channels. The configuration mixings of the triply excited resonances are described and differences in the literature on the designation of the intershell states are pointed out. These results are compared with other theoretical and experimental data in the literature.

High‐lying core‐excited quartet states in Li‐like N4+ and F6+ ions

AbstractThe variational method with mutilconfiguration interaction wave function is used to obtain the energies, fine structures, and hyperfine structures of high‐lying core‐excited quartet states 1s2lnl' 4Po(m) (m = 1–5) and 1s2pnp 4P(m) (m = 1–5) in Li‐like N4+ and F6+ ions, including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The oscillator strengths, lifetime, wavelengths, fine structure, and hyperfine structure for this system are also investigated to compare with other theoretical and experimental data in the literature. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Inner-shell excited quartet states in Li-like O5+ and Ne7+ ions

The Rayleigh-Ritz variational method and a configuration-interaction wave function are used to obtain the energies, fine structures and hyperfine structures of high-lying core excited quartet states \(1s2lnl^\prime\,\!^{4}{\rm P}^{0}(n)~(n=1\!-\!5)\) and \(1s2pnp\,\!^{4}{\rm P}(m)~(m=1\!-\!5)\) in O5+ and Ne7+ ions, including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The wavelengths, oscillator strengths, and lifetimes are also explored. The results compared with other theoretical and experimental data in the literature.

Applications of Skyrme energy-density functional to fusion reactions spanning the fusion barriers

The Skyrme energy density functional has been applied to the study of heavy-ion fusion reactions. The barriers for fusion reactions are calculated by the Skyrme energy density functional with proton and neutron density distributions determined by using restricted density variational (RDV) method within the same energy density functional together with semi-classical approach known as the extended semi-classical Thomas–Fermi method. Based on the fusion barrier obtained, we propose a parametrization of the empirical barrier distribution to take into account the multi-dimensional character of real barrier and then apply it to calculate the fusion excitation functions in terms of barrier penetration concept. A large number of measured fusion excitation functions spanning the fusion barriers can be reproduced well. The competition between suppression and enhancement effects on sub-barrier fusion caused by neutron-shell-closure and excess neutron effects is studied.

Energy, fine structure, and hyperfine structure of the core-excited states 1s2s2pnp 5P (n = 2–5) and 1s2p2mp 5S (m = 2–5) for Li- ion

The relativistic energies, the oscillator strength, and the lifetimes of high-lying core-excited states 1s2s2pnp 5P (n=2–5) and 1s2p2mp 5So (m=2–5) of Li- ion are calculated with the saddle-point variational method and restricted variation method. The fine structure and the hyperfine structure of the core-excited states for this system are also explored. The results are compared with other theoretical and experimental data in the literature.

Relativistic energy, fine structure, and hyperfine structure of the high-lying core-excited statesP5(n)(n=1–7) andSo5(m)(m=1–5) for Be-like boron and carbon

The variational method with multiconfiguration interaction wave function is used to obtain the energies of high-lying core-excited states $^{5}P(n)$ ($n=1\char21{}7$) and $^{5}S^{o}(m)$ ($m=1\char21{}5$) in Be-like boron and carbon, including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The oscillator strengths, transition rates, and wavelengths are also calculated. The results are compared with other theoretical and experimental data in the literature. The fine structure and hyperfine structure of core-excited states for this system are also investigated.

ENERGY, OSCILLATOR STRENGTH AND HYPERFINE STRUCTURE OF THE LOW-LYING EXCITED STATES FOR Be-LIKE SYSTEM

The Rayleigh–Ritz variational method is carried out with a multiconfiguration-interaction wave function and restricted variational method to obtain the relativistic energies of the 1s22s2p 1Po , 1s22s2p 3Po , and 1s22p23P states for the beryllium-like isoelectronic sequence (Z=4–10), including the mass polarization and relativistic corrections. The oscillator strengths and transition rates are also calculated. The results are compared with other theoretical and experimental data in the literature. The hyperfine structure of the low-lying excited states for this system is also explored.

Relativistic energy, fine structure and hyperfine structureof the core-excited states 1s2s2pnp 5P(n = 2–6) and 1s2p2mp 5So(m = 2–6)for beryllium

The saddle-point variational method and restricted variational method are used toobtain the relativistic energies of high-lying core-excited states 1s2s2pnp 5P(n = 2–6) and 1s2p2mp 5So(m = 2–6)in beryllium, including the mass polarization and relativistic corrections. Theoscillator strengths and transition rates are also calculated. The results arecompared with other theoretical and experimental data in the literature. The finestructure and hyperfine structure of core-excited states for this system are alsoexplored.

FINE STRUCTURES OF SOME LOW-LYING EXCITED STATES FOR Be-LIKE SYSTEM

The fine structure energies and fine structure splittings of some low-lying excited states of the beryllium isoelectronic sequence (Z = 4–20) are calculated with multi-configuration-interaction method and restricted variation method. The relativistic corrections and mass polarization are included. The results are in good agreement with other theoretical and experimental data.

RADIATIVE TRANSITION OF CORE-EXCITED 2smd4De STATES TO THE 2pnd4Do STATES IN LITHIUM

The saddle-point variational method is used with restricted variation method to obtain the energies of the core-excited states 1s2smd4De(m=3—7) and 1s2pnd4Do(n=3—5) for lithium, including the mass polarization and relativistic corrections. Oscillator strengths, transition rates, lifetimes and wavelengths are also calculated. The results are compared with other theoretical and experiment data in the literature.

RELATIVISTIC ENERGIES AND FINE STRUCTURES OF THE EXCITED STATES FOR BERYLLIUM-LIKE BⅡ

The Rayleigh-Ritz variational method is used with a multiconfiguration-interaction function and restricted variation method to obtain the energies of 1s22s2 1S, 1s22s2p1P0, 1s22s2p3P0 and 1s22p2 3P in beryllium-like boron, including the mass polarization and relativistic corrections. Fine structure, oscillator strengths, transition rates and wavelengths are also calculated. The results are compared with other theoretical and experiment data in the literature.

Energies and lifetimes of the core-excited states of the Li isoelectronic sequence

The energies, fine structures, widths, and radiative rates of core-excited states of a Li isoelectronic sequence $(Z=4\char21{}14)$ are calculated using the saddle point and saddle-point complex-rotation methods. A restricted variational method is used to extrapolate a better nonrelativistic energy. Relativistic effects are included using first-order perturbation theory. The charged effects of the Li isoelectronic sequence are discussed. These results are compared with experimental and theoretical data in the literature.

Energies, fine structures, and transition rates of the core-excited states in BeII

The energies and fine structures of some doubly excited resonances of lithiumlike beryllium are calculated using the saddle-point variational method. A restricted variational method is used to extrapolate a better nonrelativistic energy. Relativistic and mass polarization corrections are included. Oscillator strengths, transition rates, and wavelengths are also calculated. The results are compared with other theoretical and experiment data in the literature. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 656–661, 2000