Evaluating the Accuracy of Theoretical Transition Data for Atoms

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.