evaluating the accuracy of theoretical transition data for

Evaluating the Accuracy ofTheoretical Transition Data for Atoms

Per Jonsson

Group for Materials Science and Applied MathematicsSchool of Engineering, Malmo University, Sweden

6 maj 2013

Per Jonsson Group for Materials Science and Applied Mathematics School of Engineering, Malmo University, SwedenEvaluating the Accuracy of Theoretical Transition Data for Atoms

Code development network

Theory and program development

I Charlotte Froese Fischer, NIST

I Michel Godefroid, Brussels

I Gediminas Gaigalas, Vilnius

I Ian Grant, Oxford

I Jacek Bieron, Krakow

I Chenzong Dong, Lanzhou

I Stefan Fritzsche, Heidelberg/GSI

I Tomas Brage, Lund


Overview

I Multiconfiguration methods

I Strengths and weaknesses of multiconfiguration methods

I Available multiconfiguration codes

I Characteristics of codes

I Methods for evaluating accuracy: internal and external

I Examples of evaluation: possibilities and problems

I Summary

I Future perspectives


Dimensions of uncertainty estimates

Width of the problem

I Estimates for a few transitions

I Estimates for calculations generating massive data sets

I Estimates for ”simple” systems

I Estimates for very complex systems including open f - andd-shells


Variational multiconfiguration methods

I Expand electron wave function Ψ(γJMJ) in configurationstate functions (CSFs) Φ(γiJMJ)

Ψ(γJMJ) =∑i

ciΦ(γiJMJ)

I CSFs are symmetry adapted and anti-symmetrized products ofone-electron orbitals

I Radial part of orbitals represented on a grid, by splines or as acombination of functions

I The radial parts should (normally) fulfill orthonormalityconditions within each symmetry


Numerical solution

Perform angular integration and express the energy functional as asum over one- and two-electron radial integrals.

Apply the variational principle.

I Eigenvalue problem for coefficients

Hc = Ec

where Hij = 〈Φ(γiJMJ)|H|Φ(γjJMJ)〉I Coupled integro differential equations for the radial functions

I Eigenvalue problem and differential equations solvediteratively until convergence


Transition rates

Transition parameters such as line strengths can be evaluated indifferent gauges

SL(γJ, γJ ′) ∝ 〈Ψ(γJ)‖OL‖Ψ(γ′J ′)〉

SV (γJ, γJ ′) ∝ 1

(∆E )2〈Ψ(γJ)‖OV ‖Ψ(γ′J ′)〉

Limitations

I Orbital basis building the initial and final state wave functionare often required to be the same

I Full transition operator in velocity form not implemented inBreit-Pauli


Strengths of multiconfiguration methods

Versatility

I Can be applied to many states across the periodic table

I Atoms with several open shells, open f -shells

Examples of data bases

I Iron project

I MCHF/MCDHF

I DREAM, Database on Rare Earths At Mons University

I DESIRE, DatabasE on SIxth Row Elements



Spectrum calculations

I Simultaneous optimization of hundreds of states

I Balanced energy spectra

I Massive data sets



I 771 fine structure levels in boron-like Fe

I More than 100 000 transition rates



Different correlation effects can be targeted

I Correlation effects targeted through CSF expansions

I Close degeneracies often efficiently described

I Allows a systematic approach

I Key for evaluating the accuracy


Weaknesses of multiconfiguration methods

I Expansion sizes grows very rapidly with respect toone-electron orbital basis

I Sometimes not possible to converge properties with respect toorbital basis

I Performance degrades for ”spectrum calculations”

I Often impossible to include electron correlation in the core

I Radial orbitals of the same symmetry should be orthonormal

I Accuracy strongly dependent on transition: intercombinationtransitions vs strong allowed transitions, transition influencedby perturbers, transitions that are zero in first approximationtwo-electron one-photon


Properties of computer packages

Different codes can be used for different purposes.

I Documentation that ensures that the code is used in a correctand optimal way?

I Restrictions on the number of open shells

I Are semi-empirical corrections available, i.e shifting individuallevels of levels belonging to LS term?

I How large expansions can be handled by the code, does thecode support parallel computing?

I Can model potential be used?

I Methods for ”spectrum calculations”?

I Are all relativistic operators implemented?

I Methods to handle non-orthogonalities?


Available computer packages

Non-relativistic codes with relativistic corrections in the Breit-Pauliapproximation

I ATSP2K, Froese Fischer et al., CPC, latest release 2007,parallel computing, yes, fine-tuning, yes, non-orthogonalities,yes

I MCHF BSR, Zatsarinny, Froese Fischer, CPC, latest release2009, non-orthogonalities, yes

I SUPERSTRUCTURE, Eissner et al., CPC, not full setBreit-Pauli operators, model potential, non-orthogonalities, no

I CIV3, Hibbert, CPC, fine-tuning, yes, model potential,non-orthogonalities, yes

I HFR, Cowan, semi-empirical, model potential,non-orthogonalities, no


Available computer packages

Fully-relativistic codes

I GRASP2K, Jonsson et al., CPC, latest release 2013, parallelcomputing, yes, non-orthogonalities, yes

I MCDFGME, Indelicato, Desclaux, download from homepage,latest release 2005, non-orthogonalities, yes

I FAC, Gu, download from home page, latest release 2009,non-orthogonalities, no

I RATIP, Fritzsche, CPC, latest release 2012,non-orthogonalities, to some extent


Non-orthogonalities

Many codes can not handle non-orthogonalities.

I Initial and final state in a transition has to be described by thesame orbital set.

I Different LS terms in a Breit-Pauli calculation need to bedescribed by the same orbital set.

Codes that can handle non-orthogonalities have distinctadvantages.


Example non-orthogonalities

It is often desirable to describe the electron distributions of twostates with different and non-orthogonal orbitals.

I 1s22s22p 2P → 1s22s2p2 2D transition in B I.Both the 2s and 2p electron distributions for the initial statediffer from the corresponding ones in the final states

I Mixing of 1s22s2p 1P1 and 1s22s2p 3P1 in Breit-Pauli.2p electron distribution in 1P is more diffuse than 2p in 3P.

I Generally the case for mixing of LS-terms in Breit-Pauli


Internal methods for evaluating accuracy

Available methods for estimating accuracy differ strongly betweentwo extremes

I Isolated transitions where we want to achieve benchmarkresults

I Massive ”spectrum calculations” for data production

One may argue that one may want to combine these two extremesfor internal benchmarking



Convergence studies of energy differences and transitionparameters:

– with respect to increasing one-electron orbital basis– with respect to different models for generating CSFs thataccount for electron correlation

Problems:

I Rapid increase of CSFs with respect to increasing orbital basis.

I Often only limited models can be probed.

I May lead to distorted and unsuitable orbital basis



Sensitivity test with respect to fine-tuning of energy levels

Very efficient for Breit-Pauli calculations, when experimentalenergies are available

Problems:

I Assumes experimental energy levels

I Wave functions in jj-coupling are not easily tuned



Consistency transition parameters evaluated in length and velocitygauges. Efficient way of spotting transitions that are less accurate

Problems:

I Full operators implemented in non-relativistic theory, but notin Breit-Pauli. Can not be used for intercombinationtransitions

I For intercombination transition in the fully relativistic theorythere are sizeable contributions to parameters in the velocitygauge from the negative continuum and these are often notaccounted for

I Parameters in length and velocity form can agree without thevalues being close to the correct values.



Internal benchmarking for ”spectrum calculations”

I Performance degrades for spectrum calculations since orbitalbasis needs to span many states.

I One or more calculations can be performed for individualtransitions that serve as benchmarks for the spectrumcalculation



Perturbative analysis of neglected correlation effects, i.e. checkcorrelation effects one by one in smaller calculations

Problem:

I Assumes that effects are additive, which they are not


External methods for evaluating accuracy

Check computed energy against NIST data

Check transition rates against beam-foil and storage ringmeasurements

Possibilities and problems:

I Some experimental data are very accurate - extremelyvaluable validation

I Gives access to accuracy estimates for only part of thetheoretical data, accuracy differ strongly dependent ontransition

I Gives only lifetimes

I Some old beam-foil data are uncertain



Check transition rates against values from laser-spectroscopy withbranching fraction measurements


I Some experimental data e.g. from single photon counting,beam-laser techniques are very accurate - extremely valuablevalidation

I Gives rates for transition that are connected to the sameupper level

I Accuracy limited by the life-time measurement, around 10 %

I Available for ions near the neutral end



Check transition rates against benchmark calculations


I Utilize the fact that different methods have differentstrengths/weaknesses

I Also accuracy of benchmark calculations are uncertain

I A benchmark calculation can be fine for some properties, e.g.energies but not for others like transition rates


Convergence studies

Selection of CSFs guided by Z -dependent perturbation theory

I Select a set of important CSFs (multireference)

I Generate CSFs by substitutions of orbitals in the CSFsbuilding the multireference with orbitals in an active spaceaccording to some rule

I Increase active space systematically

I Monitor convergence of computed properties

I Monitor convergence with respect to the rule for generatingthe CSFs


Example 1s22s2 1S − 1s22s2p 1P in B II

M. Godefroid, J. Olsen, P. Jonsson and C. Froese FischerAstrophysical Journal, 450, 473 (1995).

Systematic calculations with different correlation models

I Valence correlation

I Valence + core-valence

I Valence + core-valence + core core


Example 1s22s2 1S − 1s22s2p 1P in B II


Example 1s22s2 1S0 − 1s22s2p 3,1P1 in C III

P. Jonsson and C. Froese FischerPhysical Review A 57, 4967 (1998).

Problems with gauges for intercombination transitions


Example 1s22s2 1S0 − 1s22s2p 3,1P1 in C III


Validation with experiment, Be sequence

I Different correlation models investigated at the start of thecalculation

I Within the chosen model: systematic calculations withincreasing active set of orbitals

I Convergence monitored

I Comparison between computed and experimental transitionrates for 2s2p 1,3P1 → 2s2 1S0 in the Be-sequence


Comparing theory and experiment


Spectrum calculations C II, N III, O IV

I Results include levels belonging to2s22p, 2s2p2, 2p3, 2s23s, 2s23p, 2s23d , 2s2p3sin C II, N III, and OIV

I MR with SD substitutions to n = 10 and l = 6, between800 000 and 1 000 000 CSFs

I Good convergence with respect to the increasing active set

I Odd and even states separately optimized


O IV energies, comparison with experiment


O IV transition rates, comparison theory

I Very good agreement between new benchmark calculations forstrong transitions

I Less good agreement for weak (intercombination transitions)

I For some lines there are large (unexplained) differences


Internal validation for ”spectrum calculations”

291 states in 1s22s22p, 1s22s2p2, 1s22p3, 1s22s23l , 1s22s2p3l ,1s22p23l , 1s22s24l ′, 1s22s2p4l ′, 1s22p24l ′ (l = 0, 1, 2 andl ′ = 0, 1, 2, 3) in boron-like ions from Ti XVIII to Cu XXV.

Problems:

I Performance degrades as more states are spanned by theorbital set

I Experimental energies often known only for lower states

I Benchmark results often available for limited number oftransitions


Internal validation for ”spectrum calculations”

Methodology:

I Main part of the computation is for the ”spectrum”

I Perform systematic internal benchmark calculations for limitedtransitions e.g. 1s22s22p, 1s22s2p2, 1s22p3

I Validate results from spectrum ”calculations” against theinternal benchmark


Internal and external validation for ”spectrum calculations”

Energies in cm−1 for Fe XXII



Transition rates s−1 for Fe XXIIA(CHI ) FAC-calculations from Chianti database.A(RMBPT ) RMBPT calculations by Safronova et al.



In this case:

I Internal validation: reveals no degradation of accuracy

I External validation: surprisingly large differences for transitionrates for weak transitions. Difference multiconfigurationmethods and RMBPT 14% for transitions within n = 2.


Separation of certain and uncertain transitions

Separation of certain and uncertain transitions in spectrumcalculations

R = Al/Av ratio of transition rates in length and velocity form.

I Allowed transitions R very close to 1.

I Intercombination transitions 0.85 < R < 1.15 (red)

I Two electron one-photon transitions e.g.2s23d 2D3/2,5/2 − 2s2p(3P)3s 4P5/2

R very different from 1 (blue)


Accurate and inaccurate transitions


Example 3s2 1S − 3s3p 1P in Mg I

P. Jonsson, C. Froese Fischer and M. GodefroidJournal of Physics B 32, 1233 (1999).

I Internal methods for evaluating accuracy fails already for3s2 1S − 3s3p 1P . Difficult to estimate contribution fromcore-core correlation

I Difference between length and velocity forms gives noindication of the accuracy



Number of CSFs from SD-excitations from{1s22s22p63s3p 1P, 1s22s22p63p3d 1P} to increasing active set

AS NCSF

n = 3 229n = 4 2 899n = 5 10 492n = 6 25 086n = 7 48 590n = 8i 80 584

The number of CSFs grows much faster for relativistic calculationsthat need to include 3s3p 3P0,1,2



I Including core-core correlation gives very large CSF expansionsthat are difficult to handle

I Not clear how to build the orbital basis

I General wisdom that core-core correlation is unimportant isbased on validation against experimental data and otherbenchmarks (no internal validation possible)

I Methods based on non-orthonormal orbitals are now availablethat can evaluate contributions also from core-core,Verdebout et al. Journal of Physics B, 46 085003 (2013).


Flares, violent eruptions

http://en.wikipedia.org/wiki/Solar flare


http://en.wikipedia.org/wiki/Solar_flare

Calculations for Fe XVII


Benchmark calculations for Fe XVII

I Benchmark calculations for n = 3 provided by Del Zanna andIshikawa, A & A 508, 1517-1526 (2009)

I Analysis and reinterpretation of energy levels (different valuescompared to NIST)

I Intensities based on R-matrix calculations by Loch, S. D.,Pindzola, M. S., Ballance, C. P., & Griffin, D. C., J. Phys. BAtom. Mol. Phys., 39, 85 (2006)


Calculations for Fe XVII

I Systematic MCDHF and RCI calculations for all 2p6, 2p53s,2p53p and 2p53d states.

I All states belonging to a configuration are optimized together

I The orbital set is systematically increased to n = 7 and l = 6

I Convergence monitored, some three-particle effects included

I Final calculations for 2p53d contains more than 700 000 CSFs

I Energies in perfect agreement with the ones given inDel Zanna and Ishikawa, A & A 508, 1517-1526 (2009)


Comparison with energies

State Eexp ERCI EMR−MP

2p6 1S0 0 0 0

2p53s 3P2 5 849 490 5 849 108 ( 382) 5 848 891 ( 599)

2p53s 1P1 5 864 760 5 864 469 ( 291) 5 864 138 ( 622)

2p53s 3P0 5 951 478 5 951 003 ( 475) 5 950 877 ( 601)

2p53s 3P1 5 961 022 5 960 633 ( 389) 5 960 410 ( 612)

2p53p 3S1 6 093 568 6 093 573 ( -5) 6 093 209 ( 359). . .

2p53p 3P1 6 245 490 6 245 346 ( 144) 6 245 018 ( 472)

2p53p 1D2 6 248 530 6 248 390 ( 140) 6 248 024 ( 506)

2p53p 1S0 6 353 356 6 353 605 (-249) 6 351 136 (2220)

2p53d 3P0 6 464 095 6 463 913 ( 182) 6 463 611 ( 484)

2p53d 3P1 6 471 233 6 471 519 (-286) 6 471 317 (-84)

2p53d 3P2 6 486 440 6 486 166 ( 274) 6 485 977 ( 463)

2p53d 3F4 6 487 000 6 486 745 ( 255) 6 486 514 ( 486)

2p53d 3F3 6 492 924 6 492 689 ( 235) 6 492 387 ( 537)

2p53d 1D2 6 506 808 6 506 561 ( 247) 6 506 276 ( 532)

2p53d 3D3 6 515 479 6 515 276 ( 203) 6 514 936 ( 543)

2p53d 3D1 6 552 221 6 552 697 (-476) 6 552 491 (-270)

2p53d 3F2 6 594 617 6 594 260 ( 357) 6 594 099 ( 518)

2p53d 3D2 6 601 210 6 600 855 ( 355) 6 600 688 ( 522)

2p53d 1F3 6 605 469 6 605 078 ( 391) 6 604 858 ( 611)

2p53d 1P1 6 660 894 6 661 101 (-207) 6 660 232 ( 662)


Transition rates

2p53p 1S0 → 2p53s 1P1 2p53p 1S0 → 2p53s 3P1

n λ (A) gfL gfV λ (A) gfL gfV3 238.8 0.128 0.076 194.3 0.0789 0.04254 250.2 0.127 0.134 201.7 0.0707 0.06025 252.1 0.126 0.117 202.9 0.0698 0.06296 253.1 0.126 0.123 203.6 0.0695 0.0666

7 254.5 0.127 0.130 204.4 0.0697 0.0712

254.81 0.1332 204.61 0.07932

1 Experimental wave length (Del Zanna and Ishikawa)2 R-matrix, Loch et al. J. Phys. B 39, 85 (2006)


Results from validation

I Calculations gives energies in agreement with Del Zanna andIshikawa

I gf values agree with R-matrix calculations to within 10%.

I gf in length and velocity gauges agrees to within 3 % (largeimprovement compared to other calculations)

I Ratios of gf values not in accordance with astrophysicalobservations


Measuremnets of intensity ratios

Measured ratio of gf values for 2p53d 1P1 → 2p6 1S0 and2p53d 3D1 → 2p6 1S0 and


External validation

Ratio from large scale calculation 3.56


Summary

I Code providers need to supply information/manual that helpusers to use the code and carry out the calculations

I Accuracy estimates must be done by transitions

I Write out transition parameters in length and velocity gauge,or parameters in length together with some ratio for lengthand velocity

I Clearly indicate what correlation effects have been accountedfor

I Indicate if fine-tuning has been done or not

I Mark intercombination transitions, transitions with internalcancellation, two-electron one-photon

I Include internal benchmarking for ”spectrum calculations”


End

Thank you for your attention!


Electron correlation

HF simplest description of the electronic wave function

I electron correlation, effects beyond the HF approximation.

I electron correlation divided into static and dynamic correlation


Static correlation

Arises from near-degeneracies of the HF orbitals.Static correlation accounted for by using an MR expansion

1s22s2 1S needs to be described as

Ψ = c1Φ(1s22s2 1S) + c2Φ(1s22p2 1S)

1s22s22p6 1S needs to be described as

Ψ = c1Φ(1s22s22p6 1S) + c2Φ(1s22s22p43p2 1S)


Dynamic correlation

Due to the behavior of the wave function in regions close to rij = 0

Short-range effect. Difficult to account for.


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 toincreasing active set of orbitals. These CSFs account fordynamic correlation

3. The CSFs that account for dynamic correlation build theCorrelation Function (CF) space

4. The final wave function Ψ is something built from CSFs in theMR and CF spaces


Orthogonality constraints

Due to restrictions in Racah algebra orbitals building the CSFsshould be orthonormal.

Orthonormal orbital basis is inefficient for larger systems withmany shells


Example ortogonality problems

Ground state 1s22s22p63s23p64s2 1S in Ca I

I Dynamic correlation in the 1s shell: tailor orbital set wheresome orbitals have a large overlap with the 1s orbital

I Dynamic correlation in the 2s shell: needs to be described interms of the previous orbitals, tailored for describingcorrelation in the 1s shell, as well as some new orbitals thatare overlapping with the 2s orbital etc.


Conclusion

To capture the dynamic correlation between electrons in all thedifferent shells, the orbital basis needs to be extended to a largenumber of orbitals for each symmetry.

I leads to massive CSF expansions

I performance rapidly degrades with the number of shells

I scaling-wall


Handle non-orthogonalities

Orthonality restrictions can be overcome by using a biorthonormaltransformation (Verdebout et al J. Phys B 46 085003, 2013).

I used to compute transition rates for separately optimized LHSand RHS wave functions

I transformation can used to evaluate any matrix element


Normal SD-MR-MCHF method

Normal SD-MR-MCHF method

Ψ = ΨMR + Λ

ΨMR 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 correlationeffects that are not strongly coupled to energy (spin-polarization)


Proposed PCFI method

Proposed method. Divide the CF space into subspaces and performseparate MCHF calculations

Ψi = ΨMRi + Λi , i = 1, . . . , n

where Λi partitioned correlation functions (PCFs).

Normalize Λi → Λi

Expand total wave function

Ψ = ΨMR +∑i

αiΛi

Obtain expansion coefficient by constructing the Hamiltonian andoverlap matrices and solving a generalized eigenvalue problem


PCFI method

Advantages

I Relies on a divide-and-conquer strategy: many small MCHFcalculations

I Partition of CF space can be done in many ways to capturedifferent effects, spin-polarization can be described with veryhigh accuracy

I The orbital basis for each PCF optimally located

I The final expansion is a low-dimensional problem


PCFI method

Drawbacks

I Construction of matrix elements between PCFs based on abiorthogonal transformation

I The expansion coefficients of the CSFs in each PCFs arelocked (constraint effect)

Constraint effects can now be handled efficiently.


PCFI method for 1s22s2 1S in Be I

1s22s2 1S in Be I.

I Start from MR {1s22s2, 1s22p2, 1s23s2, 1s23p2, 1s23d2}I Generate the CF space by SD-excitations from the MR to

active sets of orbitals

I Partition the CF in valence-valence, core-valence, andcore-core subspaces


PCFI method for 1s22s2 1S in Be I

Perform three separate MCHF calculations for:

Ψvv = ΨMRvv + Λvv

Ψcv = ΨMRcv + Λcv

Ψcc = ΨMRcc + Λcc

Expand the final wave function

Ψ = ΨMR + αvvΛvv + αcvΛcv + αccΛcc

Determine expansion coefficients by solving an eigenvalue problem


Radial orbitals

Radial orbitals for the different PCFs


Results for 1s22s2 1S in Be

Tabell : Results for the PCFI method. The energies are compared withCAS-MCHF results based on a single orthonormal orbital set.

n ≤ EPCFI ECAS−MCHF

4 −14.660 679 48 −14.661 403 175 −14.665 553 46 −14.664 839 936 −14.666 582 83 −14.666 067 327 −14.666 905 87 −14.666 541 148 −14.667 047 86 −14.666 857 419 −14.667 122 76 −14.667 012 75

10 −14.667 168 08 −14.667 114 20

I CAS-MCHF 650 000 CSFs, days on a super computer cluster

I PCFI method, an hour on an ordinary computer.


PCFI method for 1s22s22p 2Po − 1s22s2p2 4P in B I

The term position of 1s22s2p2 4P is not known.

Two different positions available from extrapolation

Edlen 28867.15, Kramida 28644.27


PCFI method

I MR: 1s2{2s, 2p, 3s, 3p, 3d}3 2Po , 1s2{2s, 2p, 3s, 3p, 3d}3 4P

I Divide the CF space into valence-valence, core-valence,core-core subspaces

I Run separate MCHF calculations

I Expand final wave function in the MR and PCFs

I Add relativistic shift correction


Results for B I


Results for C II


Further validation

To say that these calculations are of ”spectroscopic accuracy” weneed further validation

I 1s22s22p 2P − 1s22s2p2 2D in B I. Almost finished, resultsvery promising!

I 1s22s22p 2P − 1s22s23s 2S in B I. Needs to be evaluated

I Calculations for Mg I ongoing.

I Lessons learned so far: the selection of a balanced MR iscrucial. Need to improve our methodology for that.


Close degeneracies

To say that these calculations are of ”spectroscopic accuracy” weneed further validation

I 1s22s22p 2P − 1s22s2p2 2D in B I. Almost finished, resultsvery promising!

I 1s22s22p 2P − 1s22s23s 2S in B I. Needs to be evaluated

I Calculations for Mg I ongoing.

I Lessons learned so far: the selection of a balanced MR iscrucial. Need to improve our methodology for that.


evaluating the accuracy of theoretical transition data for

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