Collision strengths for O2+ + e−

Pete Storey and Taha Sochi

• The planetary nebula NGC7009, showing the [O iii] (green) and [N ii]+Hα

(red) emission.

Introduction• The forbidden lines of [O iii] are probably the most important features in the

spectra of photoionized nebulae (H ii regions and Planetary Nebulae)

• The strongest lines are very bright and are used to determine oxygen abundances

and physical conditions in our and other galaxies out to cosmological distances

(z = 3 at least)

• It has been recently suggested that the abundance/temperature anomalies seen

in PNe might be resolved by using non-Maxwell Boltzmann distributions for the

free electrons

• If electron distributions are generally non-Maxwellian in nebulae it would affect

the analysis of [O iii] lines significantly

• Hence a need for reliable collision strength data to compute Upsilons and

Downsilons

• Surely the collision strengths are well known by now?

• There have been many calculations, the most recent a BP calculation by Palay,

Nahar, Pradhan and Eissner, which shows some significant differences from

careful earlier work (Lennon and Burke, IP ii)

• Checking/confirmation was considered appropriate

• There followed many adventures in stgicf/TCC land - a program bug and

ignorance have slowed progress

• Only interested in forbidden transitions among the 5 lowest levels for

temperatures up to 20000K

• Maximum 2J=15 for N+1 problem is ample for convergence

• Energies up to 5kT at 20000K means ≈ 0.6Ryd above threshold is sufficient

• Very low temperatures might be of importance (T≈100K) when kT≈ 0.001Ryd

• Target comprises n=2 physical orbitals + 3l and 4f correlation orbitals

• Made several calculations with increasing numbers of target states, both with

BP and ICFT method

• Use option in LS RM code to correct for effects of spurious N+1 states causing

non-physical resonances

• Try to put realistic error bars on collision strengths/Upsilons based on

– Convergence as target size is increased

– Effect of correcting for N+1 resonances

– Inclusion of (pseudo) states to represent dipole polarisability of the important

target states

– Effect of Gailitis averaging of Rydberg series of resonances between the

ground 3PJ levels

Table 1: The configurations used to define the scattering target. The 1s2 core is suppressed fromall configurations.

2s2 2p2, 2s 2p3, 2p4

2s2 2p 3l, l=0,1,2; 4f

2s2p2 3l, l=0,1,2; 4f

2p3 3l, l=0,1,2; 4f

2s2 3l 3l’, l,l’=0,1,2; 4f2

2s2p 3l 3l’, l,l’=0,1,2; 4f2

2p2 3l 3l’, l,l’=0,1,2; 4f2

Table 2: The 18 lowest energy levels of O2+ and their experimental (Eexp) and theoretical (Eth1

and Eth2) energies in wavenumbers (cm−1). Eexp are from the NIST database, Eth1 are theenergies with no two-body fine-structure terms and Eth2 include them

Index Level Eexp Eth1 Eth21 2s2 2p2 3Pe

00.00 0.00 0.00

2 2s2 2p2 3Pe1

113.18 115.00 113.00

3 2s2 2p2 3Pe2

306.17 339.00 308.00

4 2s2 2p2 1De2

20273.27 21489.00 21471.00

5 2s2 2p2 1Se0

43185.74 45900.00 45882.00

6 2s 2p3 5So2

60324.79 59600.00 59582.00

7 2s 2p3 3Do3

120058.20 121800.00 121775.00

8 2s 2p3 3Do2

120053.40 121804.00 121799.00

9 2s 2p3 3Do1

120025.20 121812.00 121805.00

10 2s 2p3 3Po2

142393.50 145316.00 145307.00

11 2s 2p3 3Po1

142381.80 145321.00 145307.00

12 2s 2p3 3Po0

142381.00 145331.00 145319.00

13 2s 2p3 1Do2

187054.00 191025.00 191006.00

14 2s 2p3 3So1

197087.70 200423.00 200405.00

15 2s 2p3 1Po1

210461.80 215609.00 215591.00

16 2p4 3Pe2

283759.70 288653.00 288629.00

17 2p4 3Pe1

283977.40 288863.00 288854.00

18 2p4 3Pe0

284071.90 288965.00 288951.00

Table 3: LS gf values within the n=2 complex in the length and velocity formulations

(gf)L (gf)V3Pe - 3Do 1.01 0.99

3Po 1.28 1.313So 1.70 1.61

1De - 1Do 1.57 1.611Po 1.18 1.22

1Se - 1Po 0.27 0.27

The scattering calculations

• ICFT and BP calculations with 9CC, 10CC, 20CC and 72CC.

• 9CC includes all terms of 2s22p2 and 2s2p3

• 10CC also includes 2p4 3P

• 20CC includes 2p4 3P, 1D and 1S plus several non-physical terms

• 72CC also includes all further odd parity terms that contribute significantly to

the polarisability of the three lowest even parity terms. Mainly from 2s22p3d

• Maximum 2J=19 in N+1 problem

• Collision strength vs final electron energy (Ryd)

• ICFT calculations with 9CC, 10CC, 20CC, 0.05Ryd equiv to ≈ 8000K

• The 9CC results are so different to earlier work and BP that considerable time

was spent trying to understand the reason - insufficient target states? bug in

stgicf?

• Collision strength vs final electron energy (Ryd)

• BP calculations with 9CC, 10CC, 20CC, 72CC

• Shows very good convergence

• Comparing 9CC ICFT with 9CC BP

• ICFT with Gailitis averaging (stgicfdamp) generated clearly incorrect results and

a bug was corrected

• Since the results for the 9CC BP calculation are in agreement with earlier work

we concluded (incorrectly) that the 9-state target was not the problem and

looked for a bug in stgicf (which was not found)

• The difference between the 9CC BP and ICFT calculations is that in the BP

run, the entire N+1 LS Hamiltonian is passed to stgjk where the SLJ H is

computed.

• All the channels arising from states that interact with the 9 target states are

included

• This is not the case with a pure LS calculation where the channels are dropped

in stg2

• Hence we need to ensure that when specifying NAST in stg2, that all terms

that interact by CI with the terms of interest are included

• Sounds obvious now

• Comparing ICFT results with and without N+1 state correction

•

•

•

Table 4: Comparison of effective collision strengths, Υ, between Lennon & Burke (LB, IP II) andthe current work. The first row of each temperature is from Table (3) of LB, the second and thirdrows are current work using the 20-term and 72-term targets respectively.

log(T /K) 3P0-3P1 3P0-3P2 3P1-3P2 3P-1D2 3P-1S0 3P-5S2 1D2-1S0

3.0 0.4975 0.2455 1.1730 2.2233 0.2754 0.9760 0.42410.5352 0.2390 1.1540 2.1829 0.2577 0.9770 0.40440.5199 0.2313 1.1218 2.1331 0.2667 0.9110 0.3897

3.2 0.5066 0.2493 1.1930 2.1888 0.2738 0.9673 0.42680.5279 0.2419 1.1715 2.1364 0.2557 0.9677 0.41050.5154 0.2349 1.1430 2.0811 0.2643 0.9044 0.3968

3.4 0.5115 0.2509 1.2030 2.1416 0.2713 0.9712 0.43570.5236 0.2435 1.1823 2.0794 0.2526 0.9702 0.42950.5132 0.2367 1.1558 2.0237 0.2610 0.9126 0.4200

3.6 0.5180 0.2541 1.2180 2.1117 0.2693 1.0224 0.46520.5246 0.2467 1.1991 2.0559 0.2527 1.0208 0.48190.5158 0.2398 1.1736 2.0107 0.2616 0.9732 0.4799

3.8 0.5296 0.2609 1.2480 2.1578 0.2747 1.1196 0.52320.5334 0.2533 1.2304 2.1202 0.2635 1.1215 0.56050.5260 0.2462 1.2057 2.0913 0.2732 1.0814 0.5621

4.0 0.5454 0.2713 1.2910 2.2892 0.2925 1.2074 0.58150.5483 0.2639 1.2762 2.2629 0.2841 1.2172 0.61930.5421 0.2568 1.2526 2.2425 0.2941 1.1769 0.6174

4.2 0.5590 0.2832 1.3350 2.4497 0.3174 1.2574 0.61000.5610 0.2769 1.3228 2.4204 0.3066 1.2731 0.63350.5551 0.2698 1.2994 2.3987 0.3165 1.2295 0.6254

4.4 0.5678 0.2955 1.3730 2.5851 0.3405 1.2720 0.60900.5669 0.2909 1.3619 2.5446 0.3230 1.2792 0.61480.5609 0.2835 1.3378 2.5184 0.3339 1.2380 0.6022

Table 5: Comparison of effective collision strengths, Υ, between Palay et al (PA) and the currentwork as a function of temperature. The first row of each transition is from Table (1) of PA while thesecond and third rows are from the current BP calculation using the 20-term and 72-term targetsrespectively.

T /K 100 500 1000 5000 10000 20000 30000

3P0-3P1 0.5814 0.5005 0.4866 0.5240 0.5648 0.6007 0.61160.6606 0.5628 0.5352 0.5279 0.5483 0.5646 0.56810.6350 0.5430 0.5199 0.5199 0.5421 0.5587 0.5623

3P0-3P2 0.2142 0.2153 0.2234 0.2469 0.2766 0.3106 0.32640.2361 0.2339 0.2390 0.2495 0.2639 0.2838 0.29640.2259 0.2247 0.2313 0.2424 0.2568 0.2766 0.2890

3P1-3P2 1.0360 1.0320 1.0720 1.2100 1.3300 1.4510 1.49901.1562 1.1320 1.1540 1.2125 1.2762 1.3432 1.37581.1121 1.0928 1.1218 1.1873 1.2526 1.3194 1.3518

3P0-1D2 0.1959 0.2088 0.2154 0.2347 0.2693 0.3094 0.32560.2363 0.2433 0.2425 0.2306 0.2514 0.2766 0.28610.2318 0.2389 0.2370 0.2265 0.2492 0.2739 0.2832

3P1-1D2 0.5903 0.6285 0.6483 0.7067 0.8108 0.9313 0.98020.7114 0.7321 0.7300 0.6945 0.7574 0.8328 0.86150.6975 0.7187 0.7132 0.6823 0.7506 0.8247 0.8527

3P2-1D2 0.9934 1.0560 1.0890 1.1880 1.3630 1.5640 1.64501.1943 1.2289 1.2254 1.1676 1.2735 1.3992 1.44641.1702 1.2057 1.1965 1.1474 1.2620 1.3850 1.4310

Table 6: Comparison of effective collision strengths, Υ, between Palay et al (PA) and the currentwork as a function of temperature. The first row of each transition is from Table (1) of PA while thesecond and third rows are from the current work using the 20-term and 72-term targets respectively.

T /K 100 500 1000 5000 10000 20000 30000

1D2-1S0 0.3900 0.3899 0.3899 0.4544 0.5661 0.6230 0.62190.3977 0.4005 0.4044 0.5201 0.6193 0.6271 0.60170.3827 0.3856 0.3897 0.5208 0.6174 0.6163 0.5886

3P0-1S0 0.0597 0.0535 0.0496 0.0409 0.0407 0.0430 0.04420.0289 0.0288 0.0286 0.0285 0.0316 0.0351 0.03620.0299 0.0298 0.0296 0.0295 0.0327 0.0363 0.0375

3P1-1S0 0.1765 0.1590 0.1477 0.1228 0.1223 0.1294 0.13320.0870 0.0867 0.0862 0.0859 0.0951 0.1059 0.10910.0900 0.0897 0.0892 0.0890 0.0985 0.1093 0.1131

3P2-1S0 0.2850 0.2587 0.2421 0.2045 0.2046 0.2170 0.22350.1460 0.1455 0.1447 0.1443 0.1601 0.1782 0.18360.1512 0.1506 0.1497 0.1496 0.1657 0.1839 0.1902

Conclusions• For this kind of calculation, concentrating on a few low levels with a lot of

correlation, BP is much safer (although also much slower) than ICFT

• The ICFT requires that all significant CI is included in the CC expansion in stg2

• This can be very difficult to judge - for example in Co ii - where there is strong

interaction between spectroscopically interesting configurations (3d8, 3d74s) and

those including 4d

• Also for lowly ionized systems resonances can have very low principal quantum

number - ie channels are deeply closed which can cause problems for MQDT.

This difficulty evaporates as Z increases due to Z-scaling

• On O iii our summed collision strength results agree well with the earlier LS

calculation by Lennon & Burke in IP ii

• We do not agree with Palay et al for some transitions and the differences are

spectroscopically significant