IOP PUBLISHING PHYSICA SCRIPTA

Phys. Scr. 79 (2009) 035401 (11pp) doi:10.1088/0031-8949/79/03/035401

A comprehensive set of UV and x-ray

radiative transition rates for Fe XVI

S N Nahar

1

, W Eissner

2

, C Sur

1,3

and A K Pradhan

1

1

Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA

2

Institut für Theoretische Physik, Teilinstitut 1, 70550 Stuttgart, Germany

E-mail: nahar@astronomy.ohio-state.edu, csur@astronomy.ohio-state.edu,

pradhan@astronomy.ohio-state.edu and we@theo1.physik.uni-stuttgart.de

Received 24 July 2008

Accepted for publication 19 November 2008

Published 3 March 2009

Online at stacks.iop.org/PhysScr/79/035401

Abstract

Sodium-like Fe XVI is observed in collisionally ionized plasmas such as stellar coronae and

coronal line regions of active galactic nuclei including black hole-accretion disc environments.

Given its recombination edge from neon-like Fe XVII at ∼25 Å, the Fe XVI bound–bound

transitions lie in the soft x-ray and EUV (extreme ultraviolet) range. We present a

comprehensive set of theoretical transition rates for radiative dipole allowed E1 transitions

including ﬁne structure for levels with n`(SL J ) 6 10, ` 6 9 using the relativistic Breit–Pauli

R-matrix (BPRM) method. In addition, forbidden transitions of electric quadrupole (E2),

electric octupole (E3), magnetic dipole (M1) and magnetic quadrupole (M2) type are

presented for levels up to 5g(SLJ) from relativistic atomic structure calculations in the

Breit–Pauli approximation using code SUPERSTRUCTURE. Some of the computed levels are

autoionizing, and oscillator strengths among those are also provided. BPRM results have been

benchmarked with the relativistic coupled cluster method and the atomic structure Dirac–Fock

code GRASP. Levels computed with the electron collision BPRM codes in bound state mode

were identiﬁed with a procedure based on the analysis of quantum defects and asymptotic

wavefunctions. The total number of Fe XVI levels considered is 96, with 822 E1 transitions.

Tabulated values are presented for the oscillator strengths f , line strengths S and Einstein

radiative decay rates A. This extensive dataset should enable spectral modelings up to highly

excited levels, including recombination-cascade matrices.

PACS numbers: 31.10.+z, 31.15.ag, 31.15.aj

1. Introduction

Highly charged iron ions exist in a variety of high-temperature

astrophysical sources emitting or absorbing radiation from

the optical to the x-ray range. The spectral lines of these

ions provide information about physical conditions and

chemical abundances in many sources. Iron, including Fe

XVI, lines have been recently identiﬁed in XMM observations

of the narrow-line Seyfert 1 galaxy MKN335, currently in a

historically low state of x-ray ﬂux from black hole accretion

[10]. The analysis and modeling of these spectra require

accurate and extensive radiative data. Coronal iron ions are of

special interest because they provide temperature and density

diagnostics. Their presence implies a high degree of ionization

and could also be a discriminant between photoionized and

3

Present address: Indian Institute of Astrophysics, Banglore, India.

collisionally ionized environments. Spectral lines of Ne-like

Fe XVII are signatures of most coronal plasmas. It follows

that in collisional ionization equilibrium adjacent ionization

stages, Na-like Fe XVI and F-like Fe XVIII, would also be

present; indeed, all three ions are often observed. However,

the energetics of the three ions differ signiﬁcantly. Whereas

the strongest x-ray lines of Fe XVII and Fe XVIII lie around

15 Å, the Fe XVI lines are found in the softer x-ray region

below the recombination edge

Fe XVII (2p

6 1

S

0

) + e(k

2

→ 0) −→ Fe XVI (n`; SL J ) + hν,

(1)

at 25.3 Å, hence photo-excitations or bound–bound transitions

at λ > 25 Å may be detected. They span a wavelength region

from soft x-rays to the EUV. For example Fe XVI lines from

solar active regions due to transitions among low n(S L J )

levels are observed around 300 Å on the one hand, and from

0031-8949/09/035401+11$30.00 1 © 2009 The Royal Swedish Academy of Sciences Printed in the UK

Phys. Scr. 79 (2009) 035401 S N Nahar et al

Table 1. Binding energies E

c

calculated by BPRM for Fe XVI (with

magnetic interaction between electrons that can be absorbed into ζ )

and comparison with observed energies E

o

from NIST. I

J

is the

positional level index in the given symmetry J π, K or K + L telling

that closed shells 1s

2

or 1s

2

2s

2

2p

6

are screening all spin–orbit

parameters ζ . Case K + L computed with NRANG2=15 basis

functions while RA=4.03.

Level J I

J

E

c

/Ry E

o

/Ry ν

o

K K + L

2p

6

3s

2

S 0.5 1 35.9570 35.9517 35.9611

3p

2

P

o

1.5 1 33.2342 33.2344 33.2442

3p

2

P

o

0.5 1 33.4269 33.4208 33.4351

3d

2

D 2.5 1 29.7436 29.7494 29.7790 2.93 201

3d

2

D 1.5 1 29.7853 29.7767 29.8055 2.93 070

4s

2

S 0.5 2 18.9363 18.9415 18.9410

4p

2

P

o

1.5 2 17.8566 17.8677 17.8665

4p

2

P

o

0.5 2 17.9324 17.9407 17.9394

4d

2

D 2.5 2 16.5695 16.5836 16.5879 3.92 848

4d

2

D 1.5 2 16.5879 16.5965 16.5992 3.92 714

4f

2

F

o

3.5 1 16.0472 16.0524 16.0462

4f

2

F

o

2.5 1 16.0544 16.0572 16.0503

5s

2

S 0.5 3 11.6932 11.6856 11.7032

5p

2

P

o

1.5 3 11.1648 11.1523 11.1641

5p

2

P

o

0.5 3 11.1997 11.1888 11.2004

5d

2

D 2.5 3 10.5448 10.5335 10.5494 4.92 695

5d

2

D 1.5 3 10.5538 10.5403 10.5545 4.92 495

5f

2

F

o

3.5 2 10.2719 10.2671 10.2734

5f

2

F

o

2.5 2 10.2754 10.2695 10.2761

5g

2

G 4.5 1 10.2407 10.2410 10.2379

5g

2

G 3.5 1 10.2428 10.2424 10.2388

6s

2

S 0.5 4 7.92865 7.92884 7.93052

6p

2

P

o

1.5 4 7.62529 7.62755 7.63071

6p

2

P

o

0.5 4 7.64675 7.64840 7.65350

6d

2

D 2.5 4 7.27514 7.28139 7.28653 5.92 734

6d

2

D 1.5 4 7.28069 7.28539 7.29200 5.92 511

6f

2

F

o

3.5 3 7.12545 7.12931 7.13599

6f

2

F

o

2.5 3 7.12755 7.13072 7.13654

7s

2

S 0.5 5 5.73268 5.73174 5.67969

7p

2

P

o

1.5 5 5.54563 5.54512 5.51567

7d

2

D 2.5 5 5.32979 5.33174 5.33523 6.92 697

7d

2

D 1.5 5 5.33325 5.33426 5.33797 6.92 519

7f

2

F

o

3.5 4 5.23597 5.23697 5.24165

7f

2

F

o

2.5 4 5.23727 5.23783 5.24037

8p

2

P

o

1.5 6 4.21396 4.21377 4.17610

8d

2

D 2.5 6 4.07158 4.07258 4.07586 7.92 520

8d

2

D 1.5 6 4.07386 4.07426 4.07768 7.92 343

8f

2

F

o

3.5 5 4.00876 4.00914 4.01390

8f

2

F

o

2.5 5 4.00965 4.00972 4.01481

9p

2

P

o

1.5 7 3.31016 3.31001 3.27395

9d

2

D 2.5 7 3.21126 3.21171 3.20105 8.94 280

9d

2

D 1.5 7 3.21283 3.21289 3.16460 8.99 416

9f

2

F

o

3.5 6 3.16713 3.16723 3.15548

9f

2

F

o

2.5 6 3.16774 3.16764 3.15548

higher levels between 25 Å < λ < 80 Å on the other hand

(e.g. [6, 7]).

Multi-wavelength spectroscopy of ions such as Fe XVI

requires a reasonably complete set of atomic parameters.

Generally, these are obtained from large-scale calculations of

high accuracy, such as those being carried out under the Iron

Project (IP: [11]), and extensions thereof such as the RmaX

Network (viz. www.astronomy.ohio-state/∼nahar). These

calculations require a great deal of effort and computational

resources. The Ohio State University (OSU) group has

been engaged in calculations of such extensive datasets of

Table 2. 1µ

`

= ν

`+1/2

− ν

`−1/2

for lower n from data in table 1.

n` K K + L Obs

3p 0.00801 0.00775 0.00793

4p 0.00801 0.00770 0.00770

3d 0.00208 0.00134 0.00131

4d 0.00218 0.00154 0.00134

4f 0.00090 0.00059 0.00051

i.e. 7.2 mRy 4.7 mRy 4.1 mRy

radiative transition probabilities for a number of iron ions

as well as astrophysical applications and spectral modeling

(www.astronomy.ohio-state/∼pradhan). With particular

relevance to the present work transition rates for all levels

up to high-n(SL J ) are needed in such models. For example,

recently, we have developed a general spectral modeling

(GSM) code for transient, quasi-steady state, and steady-state

laboratory and astrophysical plasmas [ 18]. One of the features

of GSM is the computation of a collision-less transition

matrix that requires a complete set of transition rates in

order to compute recombination-cascade matrices. The ﬁrst

application of GSM, to interpret hard x-ray spectra from

He-like Ca, Fe and Ni, revealed signiﬁcant and unexpected

discrepancies with previous work for well-known and widely

used line ratios as diagnostics of high-temperature plasmas

[19, 20]. The new models employed BPRM level-speciﬁc

uniﬁed recombination rates for n(SL J ) 6 10, including

radiative and dielectronic recombination in an ab initio and

self-consistent manner, and BPRM radiative transition rates

(e.g. [16]). Similarly, we plan to investigate the accuracy and

completeness of spectral models of other important ionization

stages, namely the Na-, Ne- and F-like iron ions.

The National Institute for Standards and Technology

(NIST; (http://physics.nist.gov/PhysRefData/ASD/index.html)

provides the evaluated and compiled table of transitions for Fe

XVI from earlier calculations. Among forbidden transitions,

it includes only E2 transitions by Tull et al [29], who

calculated them in a frozen-core Hartree–Fock approximation.

Later Charro et al [5] calculated radiative decay rates for

E2 transitions of Fe XVI using a relativistic quantum defect

orbital formalism. In the absence of accurate ﬁne structure

transitions, NIST has carried out ﬁne structure splitting of LS

multiplets for some transitions.

This report provides radiative data obtained from the

ﬁrst R-matrix calculation for Fe XVI-levels. It is part of

a project at OSU for a systematic study of iron and

iron-group atoms and ions by Nahar et al (e.g. Fe V [14,

17], Fe XVII [15], Fe XX [13], Fe XXI [12], Fe XXIV

and Fe XXV [16]). We also benchmark the present results

against other elaborate relativistic calculations using the

multi-conﬁguration Dirac–Fock code (MCDF—GRASP2,

e.g. [21]) and relativistic coupled cluster theory (RCC,

e.g. [27, 28]).

2. Formulation

The relativistic Breit–Pauli R-matrix method with close

coupling (CC) approximation, described in a number of

papers [2, 3, 11, 22, 23, 26], enables calculation of a large

number of ﬁne structure E1 transitions with high accuracy. In

2

Phys. Scr. 79 (2009) 035401 S N Nahar et al

Table 3. Sample table of ﬁne structure energy levels of Fe XVI as sets of L S term components. C

t

is the core conﬁguration, ν is the

effective quantum number.

C

t

(S

t

L

t

π

t

) J

t

nl 2J E/Ry ν SLπ

Nlv = 1,

2

L

e

: S (1)/2

2s

2

2p

6

(

1

S

e

) 0 3s 1 −3.59570E + 01 2.67 2 S e

Nlv(c) = 1:set complete

Nlv = 2,

2

L

o

: P ( 3 1 )/2

2s

2

2p

6

(

1

S

e

) 0 3p 1 −3.34269E + 01 2.77 2 P o

2s

2

2p

6

(

1

S

e

) 0 3p 3 −3.32342E + 01 2.78 2 P o

Nlv(c) = 2:set complete

Nlv = 2,

2

L

e

: D(5 3)/2

2s

2

2p

6

(

1

S

e

) 0 3d 3 −2.97853E + 01 2.93 2 D e

2s

2

2p

6

(

1

S

e

) 0 3d 5 −2.97436E + 01 2.93 2 D e

Nlv(c) = 2:set complete

Table 4. Sample output of ﬁne structure energy levels of Fe XVI.

96 = number of levels, n 6 10, l 6 9

i

e

J i

J

E/Ry Conﬁg

2S+1

L

π

J π index

1 0.5e 1 −3.59611E + 01 2p

6

3s

2

S

e

100001

2 0.5e 2 −1.89410E + 01 2p

6

4s

2

S

e

100002

3 0.5e 3 −1.17032E + 01 2p

6

5s

2

S

e

100003

4 0.5e 4 −7.93052E + 00 2p

6

6s

2

S

e

100004

5 0.5e 5 −5.67969E + 00 2p

6

7s

2

S

e

100005

6 0.5e 6 −4.33725E + 00 2s

2

2p

6 1

S

e

8s

2

S

e

100006

7 0.5e 7 −3.39566E + 00 2s

2

2p

6 1

S

e

9s

2

S

e

100007

8 0.5e 8 −2.73047E + 00 2s

2

2p

6

1S

e

10s

2

S

e

100008

9 0.5o 1 −3.34351E + 01 2p

6

3p

2

P

o

110001

10 0.5o 2 −1.79394E + 01 2p

6

4p

2

P

o

110002

11 0.5o 3 −1.12004E + 01 2p

6

5p

2

P

o

110003

12 0.5o 4 −7.65350E + 00 2p

6

6p

2

P

o

110004

13 0.5o 5 −5.55871E + 00 2s

2

2p

6 1

S

e

7p

2

P

o

110005

14 0.5o 6 −4.22249E + 00 2s

2

2p

6 1

S

e

8p

2

P

o

110006

15 0.5o 7 −3.31599E + 00 2s

2

2p

6 1

S

e

9p

2

P

o

110007

16 0.5o 8 −2.67295E + 00 2s

2

2p

6 1

S

e

10p

2

P

o

110008

17 1.5e 1 −2.98055E + 01 2p

6

3d

2

D

e

300001

18 1.5e 2 −1.65992E + 01 2p

6

4d

2

D

e

300002

19 1.5e 3 −1.05545E + 01 2p

6

5d

2

D

e

300003

20 1.5e 4 −7.29200E + 00 2p

6

6d

2

D

e

300004

21 1.5e 5 −5.33797E + 00 2p

6

7d

2

D

e

300005

22 1.5e 6 −4.07768E + 00 2p

6

8d

2

D

e

300006

the CC approximation the wavefunction expansion 9

E

for a

(N + 1)-electron system with total orbital angular momentum

L, spin multiplicity (2S + 1) and total angular momentum

symmetry Jπ, is expanded in terms of the states of the

N -electron target ion as

9

E

(e + ion) = A

X

i

χ

i

(ion)θ

i

+

X

j

c

j

8

j

(e + ion), (2)

where the multi-conﬁguration wavefunction χ

i

describes the

target in speciﬁc states of S

i

L

i

π

i

or J

i

π

i

, and θ

i

is the suitably

vector coupled wavefunction of the ‘collisional’ electron in

a channel labelled S

i

L

i

(J

i

)π

i

k

2

i

`

i

(S Lπ or J π), at channel

energies k

2

i

taken from the total energy balance

E = E

target

i

(

+k

2

i

if > 0 i.e. an open channel,

−z

2

/ν

2

i

otherwise (and z = Z − N ) .

(3)

The quantities 8

j

are correlation wavefunctions composed

of (N + 1) electrons orbital functions that (a) compensate

for the orthogonality conditions between the continuum and

the bound orbitals and (b) represent additional short-range

correlation that is often of crucial importance in scattering and

radiative CC calculations for each SLπ.

In the Breit–Pauli approximation relativistic terms

of relative magnitude α

2

, in the ﬁne structure constant

α ≡ (e

2

/

¯

hc) ≈ 1/137.036, are added to the non-relativistic

Hamiltonian of say N electrons in the ﬁeld of a nucleus with

electric charge number Z :

H

BP

N

= H

NR

N

+ H

mass

N

+ H

d

N

+ H

so

N

+

1

2

N

X

i6= j

g

i j

(so + so

0

) + g

i j

(ss

0

)

+g

i j

(css

0

) + g

i j

(d) + g

i j

(oo

0

)

, (4)

where H

NR

N

is the non-relativistic Hamiltonian

H

NR

N

=

N

X

i=1

−∇

2

i

−

2Z

r

i

+

N

X

j>i

2

r

i j

, (5)

assuming H := H/Ry and r := r/a

0

, i.e. scaling with the

3

Phys. Scr. 79 (2009) 035401 S N Nahar et al

Table 5. Sample set of f -values, line strengths S and coefﬁcients A for E1 dipole allowed and intercombination transitions in Fe XVI.

26 11

I

i

I

k

λ/Å E

i

/Ry E

k

/Ry f S A

ki

∗ s

2 0 2 1 8 8 64 = gi Pi gf Pf Ni Nf NN

1 1 360.75 −3.5961E + 01 −3.3435E + 01 −1.210E − 01 2.873E − 01 6.199E + 09

1 2 50.56 −3.5961E + 01 −1.7939E + 01 −7.854E − 02 2.615E − 02 2.049E + 11

1 3 36.80 −3.5961E + 01 −1.1200E + 01 −2.276E − 02 5.516E − 03 1.121E + 11

1 4 32.19 −3.5961E + 01 −7.6535E + 00 −1.173E − 02 2.486E − 03 7.549E + 10

1 5 29.97 −3.5961E + 01 −5.5587E + 00 −6.351E − 03 1.253E − 03 4.716E + 10

1 6 28.71 −3.5961E + 01 −4.2225E + 00 −3.817E − 03 7.216E − 04 3.088E + 10

1 7 27.91 −3.5961E + 01 −3.3160E + 00 −2.487E − 03 4.572E − 04 2.129E + 10

1 8 27.38 −3.5961E + 01 −2.6730E + 00 −1.720E − 03 3.100E − 04 1.531E + 10

2 1 62.87 −1.8941E + 01 −3.3435E + 01 6.452E − 02 2.671E − 02 1.089E + 11

2 2 909.81 −1.8941E + 01 −1.7939E + 01 −1.790E − 01 1.072E + 00 1.442E + 09

2 3 117.73 −1.8941E + 01 −1.1200E + 01 −7.449E − 02 5.774E − 02 3.584E + 10

2 4 80.73 −1.8941E + 01 −7.6535E + 00 −3.257E − 02 1.732E − 02 3.334E + 10

2 5 68.10 −1.8941E + 01 −5.5587E + 00 −1.445E − 02 6.476E − 03 2.078E + 10

2 6 61.91 −1.8941E + 01 −4.2225E + 00 −7.427E − 03 3.028E − 03 1.293E + 10

2 7 58.32 −1.8941E + 01 −3.3160E + 00 −4.329E − 03 1.662E − 03 8.490E + 09

2 8 56.02 −1.8941E + 01 −2.6730E + 00 −2.763E − 03 1.019E − 03 5.874E + 09

3 1 41.93 −1.1703E + 01 −3.3435E + 01 1.241E − 02 3.426E − 03 4.707E + 10

3 2 146.13 −1.1703E + 01 −1.7939E + 01 9.760E − 02 9.391E − 02 3.049E + 10

3 3 1812.38 −1.1703E + 01 −1.1200E + 01 −2.409E − 01 2.875E + 00 4.891E + 08

3 4 225.02 −1.1703E + 01 −7.6535E + 00 −8.802E − 02 1.304E − 01 1.160E + 10

3 5 148.31 −1.1703E + 01 −5.5587E + 00 −2.807E − 02 2.741E − 02 8.511E + 09

3 6 121.82 −1.1703E + 01 −4.2225E + 00 −1.308E − 02 1.049E − 02 5.879E + 09

3 7 108.65 −1.1703E + 01 −3.3160E + 00 −7.344E − 03 5.254E − 03 4.150E + 09

3 8 100.91 −1.1703E + 01 −2.6730E + 00 −4.608E − 03 3.061E − 03 3.018E + 09

4 1 35.73 −7.9305E + 00 −3.3435E + 01 5.561E − 03 1.308E − 03 2.907E + 10

4 2 91.05 −7.9305E + 00 −1.7939E + 01 3.136E − 02 1.880E − 02 2.524E + 10

4 3 278.69 −7.9305E + 00 −1.1200E + 01 1.543E − 01 2.831E − 01 1.325E + 10

.

.

.

.

.

. . . . . . .

hydrogenic ionization energy Ry ≡ (α

2

/2)m

0

c

2

≈ 13.59 eV

and Bohr radius a

0

≡ (

¯

h/m

0

c)/α = 0.529 × 10

−8

cm.

The BPRM method includes the three one-body terms:

mass–velocity correction H

mass

N

, the Darwin term H

d

N

and

ordinary spin–orbit interaction H

so

N

. The mutual spin–orbit

and spin–other–orbit effects (so + so

0

) by core electrons

enter only via (Blume–Watson) screening of spin–orbit

parameters, as (Fe XVII+e) is effectively a one body rather

than an (N + 1) electron system, and all spin–spin interaction

(ss

0

) among electrons drops out for an unpolarized Ne-like

core. Complications by L-shell polarization correlation are

discussed in a later section.

This paper is concerned with solutions of

h9|H|9i = E

min

(6)

when all channels are closed, i.e. all k

2

i

< 0 in equation (3),

which is a Hartree–Fock eigenvalue problem with a discreet

spectrum E , because ν = n − µ forms Rydberg series along

integer n (while the target states χ

i

with energies E

i

are

‘frozen cores’). At eigenvalues E for target and (N + 1)

electrons in a bound state, and with ‘frozen cores’ χ

i

in

the particular case of expansion (2), to which it is moreover

applied as a ﬁrst step.

The primary quantity for radiative transitions is the line

strength, for E1-type transitions in the length form the square

of a reduced dipole matrix element:

S =

*

9

f

N +1

X

j=1

C

[1]

r

j

9

i

+

2

, (7)

where 9

i

and 9

f

are the initial and ﬁnal bound wavefunctions

(and r scaled in Bohr units a

0

). Derived from the line strength,

which does not explicitly depend upon the transition energy

E

ji

= E

j

− E

i

, are secondary quantities such as the oscillator

strength f

i j

, which is a pure number going back to classical

physics, and the radiative transition probability or Einstein’s

A-coefﬁcient (E scaled in Ry), an inverse time:

f

i j

=

E

ji

3g

i

S, A

ji

∗ τ

0

= α

3

g

i

g

j

E

2

ji

f

i j

, (8)

where τ

0

≡

¯

h/Ry = 4.8381 × 10

−17

s is the time 2a

0

/(αc) it

takes a hydrogenic ground state electron to cross the Bohr

orbit, and g

i

, g

j

are the statistical weight factors of initial and

ﬁnal state.

Radiative data for forbidden transitions of electric

quadrupole (E2), octupole (E3) and magnetic dipole

(M1), quadrupole (M2) type are computed using

conﬁguration interaction atomic structure calculations

with code SUPERSTRUCTURE (SS), in the low-Z

Breit–Pauli approximation [8, 9, 15]. SS represents the

nuclear and electron–electron potential by a statistical

Thomas–Fermi–Dirac–Amaldi model potential and includes

relativistic one-body terms as well as two-body FS terms

in the Breit interaction of equation (3), and ignores the last

three two-body terms. Radiative probabilities for ‘forbidden’

excitation or de-excitation via various modes of higher

powers in α and E

i j

can be obtained with the generalized line

strength

S

Xλ

(i j ) =

9

j

O

Xλ

9

i

2

, S( ji) = S(i j), (9)

4

Phys. Scr. 79 (2009) 035401 S N Nahar et al

Table 6. Sample table of dipole allowed same-spin and intercombination E1 transitions in Fe XVI, grouped as ﬁne structure components of

L S multiplets. Calculated energies have been replaced by observed energies.

C

i

–C

k

T

i

–T

k

g

i

: I–g

j

: K E

ik

f S A

(Å) (s

−1

)

2p63s–2p63p 2Se–2Po 2: 1- 2: 1 360.75 1.21E − 01 2.87E − 01 6.20E + 09

2p63s–2p63p 2Se–2Po 2: 1- 4: 1 335.41 2.62E − 01 5.80E − 01 7.78E + 09

L S 2Se–2Po 2- 6 3.83E − 01 8.67E − 01 7.20E + 09

2p63s–2p64p 2Se–2Po 2: 1- 2: 2 50.56 7.85E − 02 2.61E − 02 2.05E + 11

2p63s–2p64p 2Se–2Po 2: 1- 4: 2 50.36 1.47E − 01 4.87E − 02 1.93E + 11

L S 2Se–2Po 2- 6 2.26E − 01 7.48E − 02 1.98E + 11

2p63s–2p65p 2Se–2Po 2: 1- 2: 3 36.80 2.28E − 02 5.52E − 03 1.12E + 11

2p63s–2p65p 2Se–2Po 2: 1- 4: 3 36.75 4.33E − 02 1.05E − 02 1.07E + 11

L S 2Se–2Po 2- 6 6.61E − 02 1.60E − 02 1.08E + 11

2p63s–2p66p 2Se–2Po 2: 1- 2: 4 32.19 1.17E − 02 2.49E − 03 7.55E + 10

2p63s–2p66p 2Se–2Po 2: 1- 4: 4 32.17 2.24E − 02 4.75E − 03 7.23E + 10

L S 2Se–2Po 2- 6 3.41E − 02 7.24E − 03 7.29E + 10

2p63p–2p64s 2Po–2Se 2: 1- 2: 2 62.87 6.45E − 02 2.67E − 02 1.09E + 11

2p63p–2p64s 2Po–2Se 4: 1- 2: 2 63.71 6.83E − 02 5.73E − 02 2.24E + 11

L S 2Po–2Se 6- 2 6.70E − 02 8.40E − 02 3.33E + 11

2p64s–2p64p 2Se–2Po 2: 2- 2: 2 909.81 1.79E − 01 1.07E + 00 1.44E + 09

2p64s–2p64p 2Se–2Po 2: 2- 4: 2 848.09 3.86E − 01 2.15E + 00 1.79E + 09

L S 2Se–2Po 2- 6 5.65E − 01 3.22E + 00 1.68E + 09

2p64s–2p65p 2Se–2Po 2: 2- 2: 3 117.73 7.45E − 02 5.77E − 02 3.58E + 10

2p64s–2p65p 2Se–2Po 2: 2- 4: 3 117.18 1.38E − 01 1.07E − 01 3.36E + 10

L S 2Se–2Po 2- 6 2.13E − 01 1.65E − 01 3.41E + 10

2p64s–2p66p 2Se–2Po 2: 2- 2: 4 80.73 3.26E − 02 1.73E − 02 3.33E + 10

2p64s–2p66p 2Se–2Po 2: 2- 4: 4 80.57 6.19E − 02 3.28E − 02 3.18E + 10

L S 2Se–2Po 2- 6 9.45E − 02 5.01E − 02 3.24E + 10

2p63p–2p65s 2Po–2Se 2: 1- 2: 3 41.93 1.24E − 02 3.43E − 03 4.71E + 10

2p63p–2p65s 2Po–2Se 4: 1- 2: 3 42.30 1.30E − 02 7.22E − 03 9.66E + 10

L S 2Po–2Se 6- 2 1.28E − 02 1.06E − 02 1.44E + 11

2p64p–2p65s 2Po–2Se 2: 2- 2: 3 146.13 9.76E − 02 9.39E − 02 3.05E + 10

2p64p–2p65s 2Po–2Se 4: 2- 2: 3 147.85 1.03E − 01 2.01E − 01 6.29E + 10

L S 2Po–2Se 6- 2 1.01E − 01 2.95E − 01 9.30E + 10

2p65s–2p65p 2Se–2Po 2: 3- 2: 3 1812.38 2.41E − 01 2.87E + 00 4.89E + 08

2p65s–2p65p 2Se–2Po 2: 3- 4: 3 1690.35 5.18E − 01 5.76E + 00 6.04E + 08

L S 2Se–2Po 2- 6 7.59E − 01 8.63E + 00 5.66E + 08

2p65s–2p66p 2Se–2Po 2: 3- 2: 4 225.02 8.80E − 02 1.30E − 01 1.16E + 10

2p65s–2p66p 2Se–2Po 2: 3- 4: 4 223.76 1.63E − 01 2.40E − 01 1.09E + 10

L S 2Se–2Po 2- 6 2.51E − 01 3.70E − 01 1.11E + 10

. . . . . .

where Xλ represents the electric or magnetic-type operator

of multipolarity λ, thus generalizing the electric dipole case

O

E1

= C

[1]

r of equation (7) to O

Eλ

= C

[λ]

r

λ

(e.g. [ 15]). Along

the line of the E1 case, for which A of equation (8) reads

g

j

A

E1

ji

= 2.6773 × 10

9

s

−1

(E

j

− E

i

)

3

S

E1

(i, j) (10)

with S of equation (7), higher order transition modes lead to

probabilities

g

j

A

E2

ji

= 2.6733 × 10

3

s

−1

(E

j

− E

i

)

5

S

E2

(i, j), (11)

g

j

A

M1

ji

= 3.5644 × 10

4

s

−1

(E

j

− E

i

)

3

S

M1

(i, j), (12)

for electric quadrupole (E2) and magnetic dipole (M1)

radiation and

g

j

A

E3

ji

= 1.2050 × 10

−3

s

−1

(E

j

− E

i

)

7

S

E3

(i, j), (13)

g

j

A

M2

ji

= 2.3727 × 10

−2

s

−1

(E

j

− E

i

)

5

S

M2

(i, j), (14)

for electric octopole (E3) and magnetic quadrupole (M2)

radiation. Regarding contributions of BP order to M1 see [9].

The lifetime of a level k can be obtained from transition

probabilities as

τ

k

=

1

P

i

A

ki

, (15)

where the sum runs over all levels i to which k can decay.

3. Calculations

3.1. BPRM calculations for E1 transitions

With the BPRM code, the CC calculations proceed in several

stages. The BPRM package requires orbital wavefunction

input for the target or core eigenstates (here provided by

SS). From it STG1 computes one- and two-electron radial

integrals as speciﬁed. STG2 computes angular coefﬁcients

for target and collisional channels in L S-coupling and target

term energies. Stage RECUPD adds the Breit–Pauli algebra,

in intermediate coupling computing target levels and with

the help of their term coupling coefﬁcients the structure of

collisional channel sets of total J π on recoupling the L S

symmetries in a pair-coupling scheme. STGH completes the

inner region (r/a

0

6 RA = 4.0) task computing R-matrices for

the speciﬁed collisional symmetries.

5