Evaluating the Accuracy of Theoretical Transition Data for Atoms
|
|
- Magnus Davidson
- 5 years ago
- Views:
Transcription
1 Evaluating the Accuracy of Theoretical Transition Data for Atoms Per Jönsson Group for Materials Science and Applied Mathematics School of Engineering, Malmö University, Sweden 6 maj 2013
2 Code development network Theory and program development Charlotte Froese Fischer, NIST Michel Godefroid, Brussels Gediminas Gaigalas, Vilnius Ian Grant, Oxford Jacek Bieroń, Krakow Chenzong Dong, Lanzhou Stefan Fritzsche, Heidelberg/GSI Tomas Brage, Lund
3 Overview Multiconfiguration methods Strengths and weaknesses of multiconfiguration methods Available multiconfiguration codes Characteristics of codes Methods for evaluating accuracy: internal and external Examples of evaluation: possibilities and problems Summary Future perspectives
4 Dimensions of uncertainty estimates Width of the problem Estimates for a few transitions Estimates for calculations generating massive data sets Estimates for simple systems Estimates for very complex systems including open f - and d-shells
5 Variational multiconfiguration methods Expand electron wave function Ψ(γJM J ) in configuration state functions (CSFs) Φ(γ i JM J ) Ψ(γJM J ) = i c i Φ(γ i JM J ) CSFs are symmetry adapted and anti-symmetrized products of one-electron orbitals Radial part of orbitals represented on a grid, by splines or as a combination of functions The radial parts should (normally) fulfill orthonormality conditions within each symmetry
6 Numerical solution Perform angular integration and express the energy functional as a sum over one- and two-electron radial integrals. Apply the variational principle. Eigenvalue problem for coefficients Hc = Ec where H ij = Φ(γ i JM J ) H Φ(γ j JM J ) Coupled integro differential equations for the radial functions Eigenvalue problem and differential equations solved iteratively until convergence
7 Transition rates Transition parameters such as line strengths can be evaluated in different gauges Limitations S L (γj, γj ) Ψ(γJ) O L Ψ(γ J ) S V (γj, γj ) 1 ( E) 2 Ψ(γJ) O V Ψ(γ J ) Orbital basis building the initial and final state wave function are often required to be the same Full transition operator in velocity form not implemented in Breit-Pauli
8 Strengths of multiconfiguration methods Versatility Can be applied to many states across the periodic table Atoms with several open shells, open f -shells Examples of data bases Iron project MCHF/MCDHF DREAM, Database on Rare Earths At Mons University DESIRE, DatabasE on SIxth Row Elements
9 Strengths of multiconfiguration methods Spectrum calculations Simultaneous optimization of hundreds of states Balanced energy spectra Massive data sets
10 Strengths of multiconfiguration methods 771 fine structure levels in boron-like Fe More than transition rates
11 Strengths of multiconfiguration methods Different correlation effects can be targeted Correlation effects targeted through CSF expansions Close degeneracies often efficiently described Allows a systematic approach Key for evaluating the accuracy
12 Weaknesses of multiconfiguration methods Expansion sizes grows very rapidly with respect to one-electron orbital basis Sometimes not possible to converge properties with respect to orbital basis Performance degrades for spectrum calculations Often impossible to include electron correlation in the core Radial orbitals of the same symmetry should be orthonormal Accuracy strongly dependent on transition: intercombination transitions vs strong allowed transitions, transition influenced by perturbers, transitions that are zero in first approximation two-electron one-photon
13 Properties of computer packages Different codes can be used for different purposes. Documentation that ensures that the code is used in a correct and optimal way? Restrictions on the number of open shells Are semi-empirical corrections available, i.e shifting individual levels of levels belonging to LS term? How large expansions can be handled by the code, does the code support parallel computing? Can model potential be used? Methods for spectrum calculations? Are all relativistic operators implemented? Methods to handle non-orthogonalities?
14 Available computer packages Non-relativistic codes with relativistic corrections in the Breit-Pauli approximation ATSP2K, Froese Fischer et al., CPC, latest release 2007, parallel computing, yes, fine-tuning, yes, non-orthogonalities, yes MCHF BSR, Zatsarinny, Froese Fischer, CPC, latest release 2009, non-orthogonalities, yes SUPERSTRUCTURE, Eissner et al., CPC, not full set Breit-Pauli operators, model potential, non-orthogonalities, no CIV3, Hibbert, CPC, fine-tuning, yes, model potential, non-orthogonalities, yes HFR, Cowan, semi-empirical, model potential, non-orthogonalities, no
15 Available computer packages Fully-relativistic codes GRASP2K, Jönsson et al., CPC, latest release 2013, parallel computing, yes, non-orthogonalities, yes MCDFGME, Indelicato, Desclaux, download from homepage, latest release 2005, non-orthogonalities, yes FAC, Gu, download from home page, latest release 2009, non-orthogonalities, no RATIP, Fritzsche, CPC, latest release 2012, non-orthogonalities, to some extent
16 Non-orthogonalities Many codes can not handle non-orthogonalities. Initial and final state in a transition has to be described by the same orbital set. Different LS terms in a Breit-Pauli calculation need to be described by the same orbital set. Codes that can handle non-orthogonalities have distinct advantages.
17 Example non-orthogonalities It is often desirable to describe the electron distributions of two states with different and non-orthogonal orbitals. 1s 2 2s 2 2p 2 P 1s 2 2s2p 2 2 D transition in B I. Both the 2s and 2p electron distributions for the initial state differ from the corresponding ones in the final states Mixing of 1s 2 2s2p 1 P 1 and 1s 2 2s2p 3 P 1 in Breit-Pauli. 2p electron distribution in 1 P is more diffuse than 2p in 3 P. Generally the case for mixing of LS-terms in Breit-Pauli
18 Internal methods for evaluating accuracy Available methods for estimating accuracy differ strongly between two extremes Isolated transitions where we want to achieve benchmark results Massive spectrum calculations for data production One may argue that one may want to combine these two extremes for internal benchmarking
19 Internal methods for evaluating accuracy Convergence studies of energy differences and transition parameters: with respect to increasing one-electron orbital basis with respect to different models for generating CSFs that account for electron correlation Problems: Rapid increase of CSFs with respect to increasing orbital basis. Often only limited models can be probed. May lead to distorted and unsuitable orbital basis
20 Internal methods for evaluating accuracy Sensitivity test with respect to fine-tuning of energy levels Very efficient for Breit-Pauli calculations, when experimental energies are available Problems: Assumes experimental energy levels Wave functions in jj-coupling are not easily tuned
21 Internal methods for evaluating accuracy Consistency transition parameters evaluated in length and velocity gauges. Efficient way of spotting transitions that are less accurate Problems: Full operators implemented in non-relativistic theory, but not in Breit-Pauli. Can not be used for intercombination transitions For intercombination transition in the fully relativistic theory there are sizeable contributions to parameters in the velocity gauge from the negative continuum and these are often not accounted for Parameters in length and velocity form can agree without the values being close to the correct values.
22 Internal methods for evaluating accuracy Internal benchmarking for spectrum calculations Performance degrades for spectrum calculations since orbital basis needs to span many states. One or more calculations can be performed for individual transitions that serve as benchmarks for the spectrum calculation
23 Internal methods for evaluating accuracy Perturbative analysis of neglected correlation effects, i.e. check correlation effects one by one in smaller calculations Problem: Assumes that effects are additive, which they are not
24 External methods for evaluating accuracy Check computed energy against NIST data Check transition rates against beam-foil and storage ring measurements Possibilities and problems: Some experimental data are very accurate - extremely valuable validation Gives access to accuracy estimates for only part of the theoretical data, accuracy differ strongly dependent on transition Gives only lifetimes Some old beam-foil data are uncertain
25 External methods for evaluating accuracy Check transition rates against values from laser-spectroscopy with branching fraction measurements Possibilities and problems: Some experimental data e.g. from single photon counting, beam-laser techniques are very accurate - extremely valuable validation Gives rates for transition that are connected to the same upper level Accuracy limited by the life-time measurement, around 10 % Available for ions near the neutral end
26 External methods for evaluating accuracy Check transition rates against benchmark calculations Possibilities and problems: Utilize the fact that different methods have different strengths/weaknesses Also accuracy of benchmark calculations are uncertain A benchmark calculation can be fine for some properties, e.g. energies but not for others like transition rates
27 Convergence studies Selection of CSFs guided by Z-dependent perturbation theory Select a set of important CSFs (multireference) Generate CSFs by substitutions of orbitals in the CSFs building the multireference with orbitals in an active space according to some rule Increase active space systematically Monitor convergence of computed properties Monitor convergence with respect to the rule for generating the CSFs
28 Example 1s 2 2s 2 1 S 1s 2 2s2p 1 P in B II M. Godefroid, J. Olsen, P. Jönsson and C. Froese Fischer Astrophysical Journal, 450, 473 (1995). Systematic calculations with different correlation models Valence correlation Valence + core-valence Valence + core-valence + core core
29 Example 1s 2 2s 2 1 S 1s 2 2s2p 1 P in B II
30 Example 1s 2 2s 2 1 S 0 1s 2 2s2p 3,1 P 1 in C III P. Jönsson and C. Froese Fischer Physical Review A 57, 4967 (1998). Problems with gauges for intercombination transitions
31 Example 1s 2 2s 2 1 S 0 1s 2 2s2p 3,1 P 1 in C III
32 Validation with experiment, Be sequence Different correlation models investigated at the start of the calculation Within the chosen model: systematic calculations with increasing active set of orbitals Convergence monitored Comparison between computed and experimental transition rates for 2s2p 1,3 P 1 2s 2 1 S 0 in the Be-sequence
33 Comparing theory and experiment
34 Spectrum calculations C II, N III, O IV Results include levels belonging to 2s 2 2p, 2s2p 2, 2p 3, 2s 2 3s, 2s 2 3p, 2s 2 3d, 2s2p3s in C II, N III, and OIV MR with SD substitutions to n = 10 and l = 6, between and CSFs Good convergence with respect to the increasing active set Odd and even states separately optimized
35 O IV energies, comparison with experiment
36 O IV transition rates, comparison theory Very good agreement between new benchmark calculations for strong transitions Less good agreement for weak (intercombination transitions) For some lines there are large (unexplained) differences
37 Internal validation for spectrum calculations 291 states in 1s 2 2s 2 2p, 1s 2 2s2p 2, 1s 2 2p 3, 1s 2 2s 2 3l, 1s 2 2s2p3l, 1s 2 2p 2 3l, 1s 2 2s 2 4l, 1s 2 2s2p4l, 1s 2 2p 2 4l (l = 0, 1, 2 and l = 0, 1, 2, 3) in boron-like ions from Ti XVIII to Cu XXV. Problems: Performance degrades as more states are spanned by the orbital set Experimental energies often known only for lower states Benchmark results often available for limited number of transitions
38 Internal validation for spectrum calculations Methodology: Main part of the computation is for the spectrum Perform systematic internal benchmark calculations for limited transitions e.g. 1s 2 2s 2 2p, 1s 2 2s2p 2, 1s 2 2p 3 Validate results from spectrum calculations against the internal benchmark
39 Internal and external validation for spectrum calculations Energies in cm 1 for Fe XXII
40 Internal and external validation for spectrum calculations Transition rates s 1 for Fe XXII A(CHI ) FAC-calculations from Chianti database. A(RMBPT ) RMBPT calculations by Safronova et al.
41 Internal and external validation for spectrum calculations In this case: Internal validation: reveals no degradation of accuracy External validation: surprisingly large differences for transition rates for weak transitions. Difference multiconfiguration methods and RMBPT 14% for transitions within n = 2.
42 Separation of certain and uncertain transitions Separation of certain and uncertain transitions in spectrum calculations R = A l /A v ratio of transition rates in length and velocity form. Allowed transitions R very close to 1. Intercombination transitions 0.85 < R < 1.15 (red) Two electron one-photon transitions e.g. 2s 2 3d 2 D 3/2,5/2 2s2p( 3 P)3s 4 P 5/2 R very different from 1 (blue)
43 Accurate and inaccurate transitions
44 Example 3s 2 1 S 3s3p 1 P in Mg I P. Jönsson, C. Froese Fischer and M. Godefroid Journal of Physics B 32, 1233 (1999). Internal methods for evaluating accuracy fails already for 3s 2 1 S 3s3p 1 P. Difficult to estimate contribution from core-core correlation Difference between length and velocity forms gives no indication of the accuracy
45 Example 3s 2 1 S 3s3p 1 P in Mg I
46 Example 3s 2 1 S 3s3p 1 P in Mg I Number of CSFs from SD-excitations from {1s 2 2s 2 2p 6 3s3p 1 P, 1s 2 2s 2 2p 6 3p3d 1 P} to increasing active set AS NCSF n = n = n = n = n = n = 8i The number of CSFs grows much faster for relativistic calculations that need to include 3s3p 3 P 0,1,2
47 Example 3s 2 1 S 3s3p 1 P in Mg I Including core-core correlation gives very large CSF expansions that are difficult to handle Not clear how to build the orbital basis General wisdom that core-core correlation is unimportant is based on validation against experimental data and other benchmarks (no internal validation possible) Methods based on non-orthonormal orbitals are now available that can evaluate contributions also from core-core, Verdebout et al. Journal of Physics B, (2013).
48 Flares, violent eruptions flare
49 Calculations for Fe XVII
50 Benchmark calculations for Fe XVII Benchmark calculations for n = 3 provided by Del Zanna and Ishikawa, A & A 508, (2009) Analysis and reinterpretation of energy levels (different values compared to NIST) Intensities based on R-matrix calculations by Loch, S. D., Pindzola, M. S., Ballance, C. P., & Griffin, D. C., J. Phys. B Atom. Mol. Phys., 39, 85 (2006)
51 Calculations for Fe XVII Systematic MCDHF and RCI calculations for all 2p 6, 2p 5 3s, 2p 5 3p and 2p 5 3d states. All states belonging to a configuration are optimized together The orbital set is systematically increased to n = 7 and l = 6 Convergence monitored, some three-particle effects included Final calculations for 2p 5 3d contains more than CSFs Energies in perfect agreement with the ones given in Del Zanna and Ishikawa, A & A 508, (2009)
52 Comparison with energies State E exp E RCI E MR MP 2p 6 1S p 5 3s 3 P ( 382) ( 599) 2p 5 3s 1 P ( 291) ( 622) 2p 5 3s 3 P ( 475) ( 601) 2p 5 3s 3 P ( 389) ( 612) 2p 5 3p 3 S ( -5) ( 359)... 2p 5 3p 3 P ( 144) ( 472) 2p 5 3p 1 D ( 140) ( 506) 2p 5 3p 1 S (-249) (2220) 2p 5 3d 3 P ( 182) ( 484) 2p 5 3d 3 P (-286) (-84) 2p 5 3d 3 P ( 274) ( 463) 2p 5 3d 3 F ( 255) ( 486) 2p 5 3d 3 F ( 235) ( 537) 2p 5 3d 1 D ( 247) ( 532) 2p 5 3d 3 D ( 203) ( 543) 2p 5 3d 3 D (-476) (-270) 2p 5 3d 3 F ( 357) ( 518) 2p 5 3d 3 D ( 355) ( 522) 2p 5 3d 1 F ( 391) ( 611) 2p 5 3d 1 P (-207) ( 662)
53 Transition rates 2p 5 3p 1 S 0 2p 5 3s 1 P 1 2p 5 3p 1 S 0 2p 5 3s 3 P 1 n λ (Å) gf L gf V λ (Å) gf L gf V Experimental wave length (Del Zanna and Ishikawa) 2 R-matrix, Loch et al. J. Phys. B 39, 85 (2006)
54 Results from validation Calculations gives energies in agreement with Del Zanna and Ishikawa gf values agree with R-matrix calculations to within 10%. gf in length and velocity gauges agrees to within 3 % (large improvement compared to other calculations) Ratios of gf values not in accordance with astrophysical observations
55 Measuremnets of intensity ratios Measured ratio of gf values for 2p 5 3d 1 P 1 2p 6 1 S 0 and 2p 5 3d 3 D 1 2p 6 1 S 0 and
56 External validation Ratio from large scale calculation 3.56
57 Summary Code providers need to supply information/manual that help users to use the code and carry out the calculations Accuracy estimates must be done by transitions Write out transition parameters in length and velocity gauge, or parameters in length together with some ratio for length and velocity Clearly indicate what correlation effects have been accounted for Indicate if fine-tuning has been done or not Mark intercombination transitions, transitions with internal cancellation, two-electron one-photon Include internal benchmarking for spectrum calculations
58 End Thank you for your attention!
59 Electron correlation HF simplest description of the electronic wave function electron correlation, effects beyond the HF approximation. electron correlation divided into static and dynamic correlation
60 Static correlation Arises from near-degeneracies of the HF orbitals. Static correlation accounted for by using an MR expansion 1s 2 2s 2 1 S needs to be described as Ψ = c 1 Φ(1s 2 2s 2 1 S) + c 2 Φ(1s 2 2p 2 1 S) 1s 2 2s 2 2p 6 1 S needs to be described as Ψ = c 1 Φ(1s 2 2s 2 2p 6 1 S) + c 2 Φ(1s 2 2s 2 2p 4 3p 2 1 S)
61 Dynamic correlation Due to the behavior of the wave function in regions close to r ij = 0 Short-range effect. Difficult to account for.
62 SD-MR-MCHF calculations To describe correlation effects expand the wave function in CSFs 1. Start with MR expansion to account for static correlation 2. Generate CSFs by SD excitations from the CSFs in the MR to increasing active set of orbitals. These CSFs account for dynamic correlation 3. The CSFs that account for dynamic correlation build the Correlation Function (CF) space 4. The final wave function Ψ is something built from CSFs in the MR and CF spaces
63 Orthogonality constraints Due to restrictions in Racah algebra orbitals building the CSFs should be orthonormal. Orthonormal orbital basis is inefficient for larger systems with many shells
64 Example ortogonality problems Ground state 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 1 S in Ca I Dynamic correlation in the 1s shell: tailor orbital set where some orbitals have a large overlap with the 1s orbital Dynamic correlation in the 2s shell: needs to be described in terms of the previous orbitals, tailored for describing correlation in the 1s shell, as well as some new orbitals that are overlapping with the 2s orbital etc.
65 Conclusion To capture the dynamic correlation between electrons in all the different shells, the orbital basis needs to be extended to a large number of orbitals for each symmetry. leads to massive CSF expansions performance rapidly degrades with the number of shells scaling-wall
66 Handle non-orthogonalities Orthonality restrictions can be overcome by using a biorthonormal transformation (Verdebout et al J. Phys B , 2013). used to compute transition rates for separately optimized LHS and RHS wave functions transformation can used to evaluate any matrix element
67 Normal SD-MR-MCHF method Normal SD-MR-MCHF method Ψ = Ψ MR + Λ Ψ MR is the multireference CSF expansion Λ is a CSF expansion built from the CF space Everything optimized together in one VERY LARGE expansion Generated orbital basis may be unsuited for describing correlation effects that are not strongly coupled to energy (spin-polarization)
68 Proposed PCFI method Proposed method. Divide the CF space into subspaces and perform separate MCHF calculations Ψ i = Ψ MR i + Λ i, i = 1,..., n where Λ i partitioned correlation functions (PCFs). Normalize Λ i Λ i Expand total wave function Ψ = Ψ MR + i α i Λ i Obtain expansion coefficient by constructing the Hamiltonian and overlap matrices and solving a generalized eigenvalue problem
69 PCFI method Advantages Relies on a divide-and-conquer strategy: many small MCHF calculations Partition of CF space can be done in many ways to capture different effects, spin-polarization can be described with very high accuracy The orbital basis for each PCF optimally located The final expansion is a low-dimensional problem
70 PCFI method Drawbacks Construction of matrix elements between PCFs based on a biorthogonal transformation The expansion coefficients of the CSFs in each PCFs are locked (constraint effect) Constraint effects can now be handled efficiently.
71 PCFI method for 1s 2 2s 2 1 S in Be I 1s 2 2s 2 1S in Be I. Start from MR {1s 2 2s 2, 1s 2 2p 2, 1s 2 3s 2, 1s 2 3p 2, 1s 2 3d 2 } Generate the CF space by SD-excitations from the MR to active sets of orbitals Partition the CF in valence-valence, core-valence, and core-core subspaces
72 PCFI method for 1s 2 2s 2 1 S in Be I Perform three separate MCHF calculations for: Ψ vv = Ψ MR vv Ψ cv = Ψ MR cv Ψ cc = Ψ MR cc Expand the final wave function + Λ vv + Λ cv + Λ cc Ψ = Ψ MR + α vv Λ vv + α cv Λ cv + α cc Λ cc Determine expansion coefficients by solving an eigenvalue problem
73 Radial orbitals Radial orbitals for the different PCFs
74 Results for 1s 2 2s 2 1 S in Be Tabell : Results for the PCFI method. The energies are compared with CAS-MCHF results based on a single orthonormal orbital set. n E PCFI E CAS MCHF CAS-MCHF CSFs, days on a super computer cluster PCFI method, an hour on an ordinary computer.
75 PCFI method for 1s 2 2s 2 2p 2P o 1s 2 2s2p 2 4P in B I The term position of 1s 2 2s2p 2 4 P is not known. Two different positions available from extrapolation Edlen , Kramida
76 PCFI method MR: 1s 2 {2s, 2p, 3s, 3p, 3d} 3 2P o, 1s 2 {2s, 2p, 3s, 3p, 3d} 3 4P Divide the CF space into valence-valence, core-valence, core-core subspaces Run separate MCHF calculations Expand final wave function in the MR and PCFs Add relativistic shift correction
77 Results for B I
78 Results for C II
79 Further validation To say that these calculations are of spectroscopic accuracy we need further validation 1s 2 2s 2 2p 2 P 1s 2 2s2p 2 2 D in B I. Almost finished, results very promising! 1s 2 2s 2 2p 2 P 1s 2 2s 2 3s 2 S in B I. Needs to be evaluated Calculations for Mg I ongoing. Lessons learned so far: the selection of a balanced MR is crucial. Need to improve our methodology for that.
80 Close degeneracies To say that these calculations are of spectroscopic accuracy we need further validation 1s 2 2s 2 2p 2 P 1s 2 2s2p 2 2 D in B I. Almost finished, results very promising! 1s 2 2s 2 2p 2 P 1s 2 2s 2 3s 2 S in B I. Needs to be evaluated Calculations for Mg I ongoing. Lessons learned so far: the selection of a balanced MR is crucial. Need to improve our methodology for that.
Relativistic multiconfiguration methods. Atomic Structure Calculations using GRASP2K. What is needed. Grasp2K manual and references.
Relativistic multiconfiguration methods Atomic Structure Calculations using GRASP2K Jörgen Ekman, Per Jönsson Group for Materials Science and Applied Mathematics Malmö University, Sweden 13 mars 2015 General
More informationMulticonfiguration Dirac-Hartree-Fock Calculations with Spectroscopic Accuracy: Applications to Astrophysics
Review Multiconfiguration Dirac-Hartree-Fock Calculations with Spectroscopic Accuracy: Applications to Astrophysics Per Jönsson 1, *, Gediminas Gaigalas 2, Pavel Rynkun 2, Laima Radžiūtė 2, Jörgen Ekman
More informationAcademic Editor: Joseph Reader Received: 25 November 2016; Accepted: 6 January 2017; Published: 12 January 2017
atoms Article Combining Multiconfiguration and Perturbation Methods: Perturbative Estimates of Core Core Electron Correlation Contributions to Excitation Energies in Mg-Like Iron Stefan Gustafsson 1, Per
More informationAcademic Editor: Joseph Reader Received: 21 December 2016; Accepted:19 January 2017; Published: 27 January 2017
atoms Article JJ2LSJ Transformation and Unique Labeling for Energy Levels Gediminas Gaigalas 1, *, Charlotte Froese Fischer 2, Pavel Rynkun 1 and Per Jönsson 3 1 Institute of Theoretical Physics and Astronomy,
More informationEnergy levels and radiative rates for Ne-like ions from Cu to Ga
Pramana J. Phys. (2017) 89:79 DOI 10.1007/s12043-017-1469-x Indian Academy of Sciences Energy levels and radiative rates for Ne-like ions from Cu to Ga NARENDRA SINGH and SUNNY AGGARWAL Department of Physics,
More informationThe Effect of Correlation on Spectra of the Lanthanides: Pr 3+
Article The Effect of Correlation on Spectra of the Lanthanides: Pr 3+ Charlotte Froese Fischer 1, * and Gediminas Gaigalas 2 1 Department of Computer Science, University of British Columbia, 2366 Main
More informationAtomic structure and dynamics
Atomic structure and dynamics -- need and requirements for accurate atomic calculations Analysis and interpretation of optical and x-ray spectra (astro physics) Isotope shifts and hyperfine structures
More informationAb-initio Calculations for Forbidden M1/E2 Decay Rates in Ti XIX ion
EJTP 3, No. 11 (2006) 111 122 Electronic Journal of Theoretical Physics Ab-initio Calculations for Forbidden M1/E2 Decay Rates in Ti XIX ion A. Farrag Physics Department,Faculty of Science, Cairo University,
More informationAtomic Data for Lowly-Charged Tungsten Ions
Atomic Data for Lowly-Charged Tungsten Ions Patrick Palmeri patrick.palmeri@umons.ac.be ADAS 2012 (Cadarache, France) 1 Outline Introduction HFR+CPOL Method Results: W 0, W 3-5+ Conclusions & Perspectives
More informationEvaluation and Comparison of the Configuration Interaction Calculations for Complex Atoms
Atoms 2014, 2, 1-14; doi:10.3390/atoms2010001 OPEN ACCESS atoms ISSN 2218-2004 www.mdpi.com/journal/atoms Article Evaluation and Comparison of the Configuration Interaction Calculations for Complex Atoms
More informationURL: / /49/18/ Publisher: IOP. This document has been downloaded from MUEP (http://muep.mah.se).
This is an author produced version of a paper published in Journal of Physics B: Atomic, Molecular and Optical Physics. This paper has been peer-reviewed but does not include the final publisher proof-corrections
More informationEnergy level structure of Er 3+ free ion and Er 3+ ion in Er 2 O 3 crystal
Energy level structure of Er 3+ free ion and Er 3+ ion in Er 2 O 3 crystal G. Gaigalas a,b, D. Kato a, P. Jönsson c, P. Rynkun b, L. Radžiūtė b a National Institute for Fusion Science, 322-6 Oroshi-cho,
More informationRelativistic Calculations for Be-like Iron
Commun. Theor. Phys. (Beijing, China) 50 (2008) pp. 468 472 Chinese Physical Society Vol. 50, No. 2, August 15, 2008 Relativistic Calculations for Be-like Iron YANG Jian-Hui, 1 LI Ping, 2, ZHANG Jian-Ping,
More informationURL: Publisher: Elsevier. This document has been downloaded from MUEP (
This is an author produced version of a paper published in Computer Physics Communications. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
More informationOscillator strengths and E1 radiative rates for Ca-like titanium, Ti III
Int. J. New. Hor. Phys. 2, No. 1, 25-31 (2015) 25 International Journal of New Horizons in Physics http://dx.doi.org/10.12785/ijnhp/020105 Oscillator strengths and E1 radiative rates for Ca-like titanium,
More informationTransition Probabilities for the Dipole Allowed Fine Structure Transitions in Sll
Physica Scripta., Vol.55, 200-238, 1997 ISSN: 1402-4896/55/2/012 (print) doi: 10.1088/0031-8949/55/2/012 IOP Publishing http://journals.iop.org/ http://www.iop.org/ej/journal/physscr Transition Probabilities
More informationURL: Publisher: Elsevier. This document has been downloaded from MUEP (
This is an author produced version of a paper published in Atomic Data and Nuclear Data Tables. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
More informationResearch Article Atomic Structure Calculations for Neutral Oxygen
International Spectroscopy Volume 2016, Article ID 1697561, 7 pages http://dx.doi.org/10.1155/2016/1697561 Research Article Atomic Structure Calculations for Neutral Oxygen Norah Alonizan, Rabia Qindeel,
More informationEnergy levels and radiative data for Kr-like W 38+ from MCDHF and RMBPT calculations
Energy levels and radiative data for Kr-like W 8+ from MCDHF and RMBPT calculations XueLing Guo 1,2,, Jon Grumer 1, Tomas Brage 1, Ran Si 2,, ChongYang Chen 2,, Per Jönsson 4, Kai Wang 5,6, Jun Yan 6,7,
More informationRelativistic many-body calculations of transition probabilities for the 2l 1 2l 2 [LSJ] 2l 3 3l 4 [L S J ] lines in Be-like ions
J. Phys. B: At. Mol. Opt. Phys. 32 (999) 3527 3545. Printed in the U PII: S0953-4075(99)0624-7 Relativistic many-body calculations of transition probabilities for the 2l 2l 2 [LSJ] 2l 3 3l 4 [L S J ] lines
More informationAtomic data from the IRON project ABSTRACT
A&A 446, 361 366 (26) DOI: 1.151/4-6361:253631 c ESO 26 Astronomy & Astrophysics Atomic data from the IRON project LX. Electron-impact excitation of n = 3, 4 levels of Fe 17+ M. C. Witthoeft 1,N.R.Badnell
More informationAb initio MCDHF calculations of electron-nucleus interactions
Ab initio MCDHF calculations of electron-nucleus interactions Jacek Bieroń Universitas Iagellonica Cracoviensis Zakład Optyki Atomowej Institute of Physics 1364 Trento 25 Aug 2015 Complete Active Space
More informationFine Structure Calculations of Atomic Data for Ar XVI
Journal of Modern Physics, 2015, 6, 1609-1630 Published Online September 2015 in SciRes. http://www.scirp.org/journal/jmp http://dx.doi.org/10.4236/jmp.2015.611163 Fine Structure Calculations of Atomic
More informationarxiv:physics/ v2 [physics.atom-ph] 31 May 2004
arxiv:physics/0405136v2 [physics.atom-ph] 31 May 2004 Pure spin angular momentum coefficients for non scalar one particle operators in jj coupling G. Gaigalas a and S. Fritzsche b a Institute of Theoretical
More informationarxiv: v1 [physics.atom-ph] 2 Dec 2015
J. Phys. B: At. Mol. Opt. Phys. arxiv:1512.657v1 [physics.atom-ph] 2 Dec 215 Theoretical investigation of spectroscopic properties of W 26+ in EBIT plasma V. Jonauskas, A. Kynienė, P. Rynkun, S. Kučas,
More informationW 25+ More Complex, W 25+, which is In-like. There are now 41 fine structure levels belonging to the ground state,
W 25+ More Complex, W 25+, which is In-like. There are now 41 fine structure levels belonging to the ground state, A partial energy level diagram of In-like tungsten. The energy levels are taken from our
More informationABSTRACT Electron collision excitation strengths of inelastic transitions among the 52 Ðne-structure levels of the
THE ASTROPHYSICAL JOURNAL, 530:1091È1104, 2000 February 20 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. ELECTRON COLLISION EXCITATION OF FINE-STRUCTURE LEVELS IN S
More informationFINE STRUCTURE RADIATIVE TRANSITIONS IN C II AND C III USING THE BREIT PAULI R-MATRIX METHOD SULTANA N. NAHAR
Atomic Data and Nuclear Data Tables, 80, (2002) p.205-234. ISSN: 0092-640x ISSN (electronic): 1090-2090 doi:10.1006/adnd.2002.0879 2002 Elsevier Inc. All rights reserved. http://www.elsevier.com/wps/find/homepage.cws_home
More informationarxiv:physics/ v1 [physics.atom-ph] 4 Nov 2004
1 arxiv:physics/0411043v1 [physics.atom-ph] 4 Nov 2004 S tudies of Lanthanides 6s Ionization Energy G. Gaigalas, Z. Rudzikas and T. Žalandauskas, Vilnius University Research Institute of Theoretical Physics
More informationarxiv: v1 [physics.atm-clus] 11 Jun 2014
Energy levels, radiative rates, and lifetimes for transitions in W LVIII Kanti M. Aggarwal and Francis P. Keenan arxiv:1406.2838v1 [physics.atm-clus] 11 Jun 2014 Astrophysics Research Centre, School of
More informationCanadian Journal of Physics. Energy, fine structure, hyperfine structure, and transitions for the high-lying multi-excited 4Pe,o states of B-like ions
Energy, fine structure, hyperfine structure, and transitions for the high-lying multi-excited 4Pe,o states of B-like ions Journal: Canadian Journal of Physics Manuscript ID cjp-2-69.r3 Manuscript Type:
More informationAllowed and forbidden transition parameters for Fe XXII Sultana N. Nahar *
Atomic Data and Nuclear Data Tables, 96, (2010), p.26 51. ISSN: 0092-640x/0092-640X doi:10.1016/j.adt.2009.09.001 2009 Elsevier Inc. All rights reserved. http://www.elsevier.com/wps/find/homepage.cws_home
More informationEvaluation of electron impact excitation data in RDW calculation
IAEA International Code Centres Network Meeting Evaluation of electron impact excitation data in RDW calculation Chenzhong Dong (C Z Dong) Key Laboratory of Atomic and Molecular Physics & Functional Materials
More informationRadiative rates of transitions from the 2s2p 3 5 S 2 level of neutral carbon
Radiative rates of transitions from the 2s2p 3 5 S 2 level of neutral carbon K Haris 1 and A Kramida National Institute of Standards and Technology, Gaithersburg, MD 20899, USA E-mail: haris.kunari@nist.gov
More informationP. W. Atkins and R. S. Friedman. Molecular Quantum Mechanics THIRD EDITION
P. W. Atkins and R. S. Friedman Molecular Quantum Mechanics THIRD EDITION Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1997 Introduction and orientation 1 Black-body radiation 1 Heat capacities 2 The
More informationOne and Two Photon Ionization along the Fe Isonuclear Sequence
I R A M P 8(), December 017, pp. 81-91 One and Two Photon Ionization along the Fe Isonuclear Sequence International Science Press ISSN: 9-3159 One and Two Photon Ionization along the Fe Isonuclear Sequence
More informationJoint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation August Introduction to Nuclear Physics - 1
2358-19 Joint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation 6-17 August 2012 Introduction to Nuclear Physics - 1 P. Van Isacker GANIL, Grand Accelerateur National d'ions Lourds
More informationSemi-Classical perturbation theory Coulomb only First-order most used
direct reactions Models for breakup Semi-Classical perturbation theory Coulomb only First-order most used TDSE (Time Dependent Schrodinger Equation) Coulomb + Nuclear Semi-classical orbit needed DWBA (Distorted
More informationInvestigation of M1 transitions of the ground-state configuration of In-like Tungsten
Investigation of M1 transitions of the ground-state configuration of In-like Tungsten W Li 1,2,3, J Xiao 1,2, Z Shi 1,2a, Z Fei 1,2b, R Zhao 1,2c, T Brage 3, S Huldt, R Hutton 1, 2 * and Y Zou 1,2 * 1
More informationarxiv: v1 [physics.atom-ph] 21 Dec 2017
Draft version August 3, 208 Typeset using LATEX twocolumn style in AASTeX6 Benchmarking Atomic Data for Astrophysics: Be-like Ions between B II and Ne VII arxiv:72.0834v [physics.atom-ph] 2 Dec 207 Kai
More informationTime Independent Perturbation Theory Contd.
Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n
More informationMulticonfigurational Quantum Chemistry. Björn O. Roos as told by RL Department of Theoretical Chemistry Chemical Center Lund University Sweden
Multiconfigurational Quantum Chemistry Björn O. Roos as told by RL Department of Theoretical Chemistry Chemical Center Lund University Sweden April 20, 2009 1 The Slater determinant Using the spin-orbitals,
More informationQUANTUM CHEMISTRY FOR TRANSITION METALS
QUANTUM CHEMISTRY FOR TRANSITION METALS Outline I Introduction II Correlation Static correlation effects MC methods DFT III Relativity Generalities From 4 to 1 components Effective core potential Outline
More informationEnergy Levels, Oscillator Strengths, and Transition Probabilities of Ni XIX and Cu XX
Optics and Photonics Journal, 2014, 4, 54-89 Published Online March 2014 in SciRes. http://www.scirp.org/journal/opj http://dx.doi.org/10.4236/opj.2014.43008 Energy Levels, Oscillator Strengths, and Transition
More informationarxiv:submit/ [physics.atom-ph] 2 Apr 2018
Ab initio calculations of hyperfine structures of zinc and evaluation of the nuclear quadrupole moment Q( 67 Zn) arxiv:submit/2215204 [physics.atom-ph] 2 Apr 2018 Jacek Bieroń, 1, Livio Filippin, 2 Gediminas
More informationCalculation of the Isotope Shifts on 5S 1/2 4D 3/2,5/2 Transitions of 87,88 Sr +
Commun. Theor. Phys. (Beijing, China) 37 (22) pp 76 7 c International Academic Publishers Vol. 37, No. 6, June 5, 22 Calculation of the Isotope Shifts on 5S /2 4D 3/2,5/2 Transitions of 87,88 Sr + LI Yong,,2
More informationForbidden transitions in B II, C III, 0 V, Ne VII and Mg IX
Mon. Not. R. Astron. Soc. 279, 1289-1293 1996) Forbidden transitions in B II, C III, 0 V, Ne VII and Mg IX J. Fleming,t K. L. Bell,t A. Hibbert,t N. Vaeck 2 * and M. R. Godefroid 2 t ldepartment of Applied
More informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September
More informationJOURNAL OFPHYSICSB: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 36 (2003) PII: S (03)
INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSB: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 3457 3465 PII: S0953-4075(03)60551-1 The influence of core valence electron
More informationPHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8: Solutions. Topics covered: hydrogen fine structure
PHYS85 Quantum Mechanics II, Spring HOMEWORK ASSIGNMENT 8: Solutions Topics covered: hydrogen fine structure. [ pts] Let the Hamiltonian H depend on the parameter λ, so that H = H(λ). The eigenstates and
More informationEnergy Levels and Transition Probabilities for Boron-Like Fe XXII
University of Kentucky UKnowledge Physics and Astronomy Faculty Publications Physics and Astronomy 9-1-2006 Energy Levels and Transition Probabilities for Boron-Like Fe XXII V. Jonauskas The Queen's University
More informationNIST Research on Spectroscopy and Collisional-Radiative Modeling of Highly-Charged Ions of Tungsten
NIST Research on Spectroscopy and Collisional-Radiative Modeling of Highly-Charged Ions of Tungsten Yuri Ralchenko National Institute of Standards and Technology Gaithersburg, USA Vienna, Austria, Dec
More informationarxiv: v1 [astro-ph.sr] 24 Jan 2019
Mon. Not. R. Astron. Soc. 000, 1?? (2016) Printed 25 January 2019 (MN LATEX style file v2.2) Uncertainties on atomic data. A case study: Niv arxiv:1901.08450v1 [astro-ph.sr] 24 Jan 2019 G. Del Zanna 1,
More informationSpin, Isospin and Strong Interaction Dynamics
October, 11 PROGRESS IN PHYSICS Volume 4 Spin, Isospin and Strong Interaction Dynamics Eliahu Comay Charactell Ltd. P.O. Box 3919, Tel Aviv 6139 Israel. E-mail: elicomay@post.tau.ac.il The structure of
More informationECE440 Nanoelectronics. Lecture 07 Atomic Orbitals
ECE44 Nanoelectronics Lecture 7 Atomic Orbitals Atoms and atomic orbitals It is instructive to compare the simple model of a spherically symmetrical potential for r R V ( r) for r R and the simplest hydrogen
More informationCHIANTI An atomic database for astrophysical plasmas
CHIANTI An atomic database for astrophysical plasmas Enrico Landi University of Michigan On behalf of the CHIANTI team: Enrico Landi Ken Dere Peter Young Giulio Del Zanna Helen Mason University of Michigan,
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationQuantum Theory of Many-Particle Systems, Phys. 540
Quantum Theory of Many-Particle Systems, Phys. 540 Questions about organization Second quantization Questions about last class? Comments? Similar strategy N-particles Consider Two-body operators in Fock
More informationPhotoionization of excited states of neon-like Mg III
PRAMANA cfl Indian Academy of Sciences Vol. 58, No. 4 journal of April 2002 physics pp. 639 646 Photoionization of excited states of neon-like Mg III NARENDRA SINGH and MAN MOHAN Department of Physics
More informationTheoretical study on the K α transition properties of F-like ions
J. At. Mol. Sci. doi: 10.4208/jams.013010.022010a Vol. 1, No. 2, pp. 134-142 May 2010 Theoretical study on the K α transition properties of F-like ions X. L. Wang, J. J. Wan, Y. J. Wang, and C. Z. Dong
More informationElectronic, magnetic and spectroscopic properties of free Fe clusters
Electronic, magnetic and spectroscopic properties of free Fe clusters O. Šipr 1, M. Košuth 2, J. Minár 2, S. Polesya 2 and H. Ebert 2 1 Institute of Physics, Academy of Sciences of the Czech Republic,
More informationAtomic data for astrophysics: improved collision strengths for
Astronomy & Astrophysics manuscript no. paper6 c ESO 2014 June 21, 2014 Atomic data for astrophysics: improved collision strengths for Fe VIII G. Del Zanna 1 and N. R. Badnell 2 1 DAMTP, Centre for Mathematical
More informationEffective collision strengths for fine-structure transitions for the electron impact excitation of N II. C. E. Hudson and K. L.
A&A 43, 725 729 (25) DOI: 1.151/4-6361:241969 c ESO 25 Astronomy & Astrophysics Effective collision strengths for fine-structure transitions for the electron impact excitation of N II C. E. Hudson and
More informationThe CHIANTI Atomic Database
The CHIANTI Atomic Database An Overview of Data, Software and Applications Dr Peter Young George Mason University, USA NASA Goddard Space Flight Center, USA Overview 1. Quick guide 2. History of project
More informationR. Clark, D. Humbert, K. Sheikh Nuclear Data Section
Calculation of Atomic Data for Plasma Modeling: Introduction and Atomic Structure Part 1 R. Clark, D. Humbert, K. Sheikh Nuclear Data Section Overview Plasmas in fusion research Data needs for plasma modeling
More informationLamb shift in muonic hydrogen and the proton charge radius puzzle
Lamb shift in muonic hydrogen and the proton charge radius puzzle Krzysztof Pachucki Institute of Theoretical Physics, University of Warsaw Mainz, April 17, 2013 Proton charge radius puzzle global fit
More informationChapter 3. The (L)APW+lo Method. 3.1 Choosing A Basis Set
Chapter 3 The (L)APW+lo Method 3.1 Choosing A Basis Set The Kohn-Sham equations (Eq. (2.17)) provide a formulation of how to practically find a solution to the Hohenberg-Kohn functional (Eq. (2.15)). Nevertheless
More informationarxiv:physics/ v1 [physics.atom-ph] 3 Jun 2004
1 arxiv:physics/0406006v1 [physics.atom-ph] 3 Jun 2004 Spectroscopic LSJ notation for atomic levels obtained from relativistic calculations G. Gaigalas a,b, T. Zalandauskas b and S. Fritzsche a a Fachbereich
More informationStrathprints Institutional Repository
Strathprints Institutional Repository Liang, Guiyun and Whiteford, A.D. and Badnell, N.R. and, UK STFC (Funder) (2009) R- matrix electron-impact excitation data for the Na-like iso-electronic sequence.
More informationThe nuclear shell-model: from single-particle motion to collective effects
The nuclear shell-model: from single-particle motion to collective effects 1. Nuclear forces and very light nuclei 2. Independent-particle shell model and few nucleon correlations 3. Many-nucleon correlations:
More informationDiscrepancies in Atomic Data and Suggestions for Their Resolutions
atoms Article Discrepancies in Atomic Data and Suggestions for Their Resolutions Kanti M. Aggarwal ID Astrophysics Research Centre, School of Mathematics and Physics, Queen s University Belfast, Belfast
More informationShells Orthogonality. Wave functions
Shells Orthogonality Wave functions Effect of other electrons in neutral atoms Consider effect of electrons in closed shells for neutral Na large distances: nuclear charge screened to 1 close to the nucleus:
More informationElectron-ion recombination of Si IV forming Si III: Storage-ring measurement and multiconfiguration Dirac-Fock calculations
Electron-ion recombination of Si IV forming Si III: Storage-ring measurement and multiconfiguration Dirac-Fock calculations E. W. Schmidt, D. Bernhardt, A. Müller, and S. Schippers Institut für Atom- und
More informationTheoretical approaches to electronimpact
Theoretical approaches to electronimpact ionization James Colgan, C. J. Fontes, D. P. Kilcrease, J. Abdallah, Jr., M. E. Sherrill, P. Hakel, and M. S. Pindzola 2 Los Alamos National Laboratory, Los Alamos,
More informationProbing magnetic fields in the solar corona using MITs MITs Magnetic-field Induced Transitions. Jon Grumer
Probing magnetic fields in the solar corona using MITs MITs Magnetic-field Induced Transitions Jon Grumer COMPAS - Division of Mathematical Physics @ Lund University jon.grumer@teorfys.lu.se ICAMDATA /
More informationOscillator strengths and transition probabilities of O II
At. Data Nucl. Data Tables Vol. 96(6):863-877 (2010) ISSN: (print 0092-640X)(online 1090-2090) doi: 10.1016/j.adt.2010.07.002 This is a peer reviewed pre-print version of the following article: Oscillator
More informationAtomic Physics 3 rd year B1
Atomic Physics 3 rd year B1 P. Ewart Lecture notes Lecture slides Problem sets All available on Physics web site: http:www.physics.ox.ac.uk/users/ewart/index.htm Atomic Physics: Astrophysics Plasma Physics
More informationThe Gutzwiller Density Functional Theory
The Gutzwiller Density Functional Theory Jörg Bünemann, BTU Cottbus I) Introduction 1. Model for an H 2 -molecule 2. Transition metals and their compounds II) Gutzwiller variational theory 1. Gutzwiller
More informationPutting a spin on time dependent electronic structure theory! Joshua Goings! General Exam! Thursday, May 14, 10:00am CHB 339
Putting a spin on time dependent electronic structure theory Joshua Goings General Exam Thursday, May 14, 10:00am CHB 339 Thank you to General Exam Committee Xiaosong Li (chair) Jim Pfaendtner (GSR) Matthew
More informationBeta and gamma decays
Beta and gamma decays April 9, 2002 Simple Fermi theory of beta decay ² Beta decay is one of the most easily found kinds of radioactivity. As we have seen, this result of the weak interaction leads to
More informationProblem Set # 4 SOLUTIONS
Wissink P40 Subatomic Physics I Fall 007 Problem Set # 4 SOLUTIONS 1. Gee! Parity is Tough! In lecture, we examined the operator that rotates a system by 180 about the -axis in isospin space. This operator,
More informationActivities at the Atomic Spectroscopy Data Center at the National Institute of Standards. Wolfgang L. Wiese Atomic Spectroscopy Group, NIST
Activities 2007 2009 at the Atomic Spectroscopy Data Center at the National Institute of Standards and Technology (NIST) Wolfgang L. Wiese Atomic Spectroscopy Group, NIST The NIST Atomic Spectroscopy Data
More informationTime Independent Perturbation Theory
apr_0-may_5.nb: 5/4/04::9:56:8 Time Independent Perturbation Theory Note: In producing a "final" vrsion of these notes I decided to change my notation from that used in class and by Sakurai. In class,
More informationPrecision calculations of atoms with few valence electrons
Precision calculations of atoms with few valence electrons arxiv:physics/0306061v1 [physics.atom-ph] 7 Jun 2003 M.G.Kozlov Petersburg Nuclear Physics Institute, Gatchina, 188300, Russia E-mail:mgk@MF1309.spb.edu
More informationISSN : Asian Journal of Engineering and Technology Innovation 02 (03) 2014 (08-13) QR Code for Mobile users
ISSN : 2347-7385 Calculation of the Energy Levels of 25Na-27Na Isotopes S. Mohammadi, Sima Zamani Department of Physics, Payame Noor University, PO BOX 19395-3697 Tehran, Iran. Received on: 09-03-2014
More informationMulti-reference Density Functional Theory. COLUMBUS Workshop Argonne National Laboratory 15 August 2005
Multi-reference Density Functional Theory COLUMBUS Workshop Argonne National Laboratory 15 August 2005 Capt Eric V. Beck Air Force Institute of Technology Department of Engineering Physics 2950 Hobson
More informationAtomic data for opacity calculations: XX. Photoionization cross sections and oscillator strengths for Fe II
J. Phys. B: At. Mol. Opt. Phys., 27, (1994) p.429-446. ISSN: 0953-4075/94/030429 + 18 ISSN (Online): 1361-6455 doi:10.1008/0953-4075/27/3/010 IOP Publishing Ltd This is an un-copyedited version of an article
More information4πε. me 1,2,3,... 1 n. H atom 4. in a.u. atomic units. energy: 1 a.u. = ev distance 1 a.u. = Å
H atom 4 E a me =, n=,,3,... 8ε 0 0 π me e e 0 hn ε h = = 0.59Å E = me (4 πε ) 4 e 0 n n in a.u. atomic units E = r = Z n nao Z = e = me = 4πε = 0 energy: a.u. = 7. ev distance a.u. = 0.59 Å General results
More informationHighly accurate Gaussian basis sets for low-lying excited states of some positive and negative ions
Indian Journal of Chemistry Vol. 46A, September 2007, pp. 1383-1387 Papers Highly accurate Gaussian basis sets for low-lying excited states of some positive and negative ions P J P de Oliveira & F E Jorge*
More informationRelativistic close-coupling calculations for photoionization and recombination of Ne-like Fe XVII
PHYSICAL REVIEW A, VOLUME 64, 032719 Relativistic close-coupling calculations for photoionization and recombination of Ne-like Fe XVII Hong Lin Zhang Applied Theoretical and Computational Physics Division,
More information13 Synthesis of heavier elements. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1
13 Synthesis of heavier elements introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1 The triple α Reaction When hydrogen fusion ends, the core of a star collapses and the temperature can reach
More informationClose-coupling R-matrix calculations for electron ion recombination cross sections
J. Phys. B: At. Mol. Opt. Phys. 32 (1999) 1459 1479. Printed in the UK PII: S0953-4075(99)99518-4 Close-coupling R-matrix calculations for electron ion recombination cross sections Hong Lin Zhang, Sultana
More informationSelected Publications of Prof. Dr. Wenjian Liu
Selected Publications of Prof. Dr. Wenjian Liu College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China 1 Fundamentals of relativistic molecular quantum mechanics 1. Handbook
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationRelativistic effects in Ni II and the search for variation of the fine. structure constant. Abstract
Relativistic effects in Ni II and the search for variation of the fine structure constant. V. A. Dzuba, V. V. Flambaum, M. T. Murphy and J. K. Webb School of Physics, University of New South Wales, UNSW
More informationWorkshop on: ATOMIC STRUCTURE AND TRANSITIONS: COMPUTATION USING SUPERSTRUCTURE PRO- GRAM
Workshop on: ATOMIC STRUCTURE AND TRANSITIONS: COMPUTATION USING SUPERSTRUCTURE PRO- GRAM PROF. SULTANA N. NAHAR Astronomy, Ohio State U, Columbus, Ohio, USA Email: nahar.1@osu.edu http://www.astronomy.ohio-state.edu/
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More informationUTILITIES for the RATIP package
Computer Physics Communications 141 (2001) 163 174 www.elsevier.com/locate/cpc UTILITIES for the RATIP package S. Fritzsche Fachbereich Physik, Universität Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel,
More informationTheory of many-electron atoms in Vilnius. Institute of Theoretical Physics and Astronomy of Vilnius University
Theory of many-electron atoms in Vilnius Institute of Theoretical Physics and Astronomy of Vilnius University About 500 000 inhabitants live in Lithuania VILNIUS is the capital of the Republic of Lithuania.
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpenCourseWare http:ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 008 For information about citing these materials or our Terms of Use, visit: http:ocw.mit.eduterms. Lecture # 8 Supplement
More information