for review only...moreover, transition energies for the transitions 3s 2s 1/2 – 3p 2p 3/2,1/2....

For Review Only

Accurate study on the properties of spectral lines for Na-likeCr13+

Journal: Canadian Journal of Physics

Manuscript ID cjp-2018-0218.R2

Manuscript Type: Article

Date Submitted by theAuthor: 06-Jul-2018

Complete List of Authors: Singh, A.K.; University of Delhi, Department of PhysicsDimri, Mayank; University of Delhi, Department of Physics andAstrophysics; Deen Dayal Upadhyaya College, University ofDelhi, DST Research LabDawra, Dishu; University of Delhi, Department of Physics andAstrophysicsJha, Alok; University of Delhi, Department of PhysicsMohan, Man; University of Delhi, Department of Physics

Keyword: Energy levels, transitions wavelength, Extreme Ultraviolet, SoftX-ray, lifetime

Is the invited manuscriptfor consideration in a

Special Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

For Review Only

Accurate study on the properties of spectral lines for Na-

like Cr13+

A. K. Singha,b, Mayank Dimria,b, Dishu Dawrab, Alok K. S. Jhac, Man Mohana,b

a Department of Physics, D.D.U. College, University of Delhi, Delhi-110078, India.

b Department of Physics & Astrophysics, University of Delhi, Delhi-110007, India.

cDepartment of Physics, Kirori Mal College, University of Delhi, Delhi-110007, India.

[email protected]

Abstract

An extended calculation of energy levels, radiative rates and lifetimes are reported for sodium like

chromium. Extensive configuration interaction calculations have been performed using general-

purpose relativistic atomic structure package (GRASP). The radiative rates, oscillator strengths and line

strengths are listed for all electric dipole (E1) transitions. However, for magnetic dipole (M1), electric

quadrupole (E2) and magnetic quadrupole (M2) transitions, only radiative rates are listed. The

importance of valence-valence (VV) and core-valence (CV) correlation effects in the calculation of

energy levels have also been shown. To confirm the accuracy of the present results for energy levels

by GRASP, independent calculations have been performed by using Flexible Atomic Code (FAC) and

configuration interaction method (CIV3). The accuracy of the present levels, wavelengths, transition

rates and lifetimes are assessed by comparing them to available experimental and other theoretical

results. We believe that our extensive results may be beneficial in fusion plasma research and

astrophysical investigations and applications.

PACS. 32.70.Cs Oscillator strengths, lifetimes – 32.10.Fn Fine and hyperfine structure

Keywords: Energy levels, transitions wavelength, Extreme Ultraviolet, Soft X-ray, lifetime

Page 1 of 29



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1. Introduction

During the past few years, the spectra of multiply charged ions have become a subject of Astrophysical

interest as satellite-borne telescopes and instruments expand the range of detectable radiation into the

X-ray region. In particular, the studies of stellar and solar coronae have intensified as a result of quality

data acquisitions by the Extreme-Ultraviolet (EUV) Explorer satellite, Chandra X-ray Observatory, and

the Solar Heliospheric Observatory (SOHO) that recorded superb coronal spectra [1,2]. The Ultraviolet

(UV) through EUV and Soft-X-ray emission lines from multiply charged ions are particularly useful

as they provide detailed knowledge of the coronal atmosphere.

Interest in the line emission from highly ionised atoms in Tokamaks derives, in the first instance, from

the effect of impurities on the overall performance of the Tokamak as a fusion device. In this respect,

the impurities can influence the heating rate, the magnetic field topography, the plasma stability and

equilibrium and the energy and particle confinement. Secondly, the impurity ion line emission may be

used as a diagnostic of the temporal and spatial Plasma conditions [3]. Chromium has been a significant

element that plays an important role in Tokamak as it is present as an impurity in stainless steel which

is a structural material of some Tokamaks [3].

Spectra of sodium like ions are often used to diagnose the hot gases found in Tokamaks, laser produced

plasmas and other devices used for controlled fusion research. A knowledge of the wavelengths of these

spectra is very important for detecting the presence of sodium like ions in the plasmas and for

calibrating spectral measurements [4]. Spectra of different sodium like ions including Cr XIV have

been reported by Peacock et al. [3] by presenting transition wavelengths for D1 and D2 lines

corresponding to the transitions 3s 2S1/2 – 3p 2P3/2,1/2.

In the past, several theoretical calculations and experimental measurements have been done on Na-like

ions by employing various experimental techniques and theoretical methods [5,6]. Edlen [6] has

identified the transitions 3s - np (n = 4,5), 3p -4s, 3p - nd (n = 4,5), and 3d - nf (n = 4-6) in vacuum

spark discharges. The 3d-4p lines at ~101 Å were identified by Fawcett et al. [7]. Reader et al. [4] have

made the measurements for the wavelengths of the 3s-3p, 3p-3d, 3d-4f transitions of the sodium like

ions by photographing laser produced plasmas and tokamak plasmas with grazing incidence

spectrographs. The energies of the transitions were also calculated with Dirac Fock Computer codes.

Cohen et al. [8] have presented the energy levels and term splitting for the Na I isoelectronic sequence

from K X through Mn XV by adopting wavelengths reported by Edlen [6]. Similarly, Fischer et al. [9]

have computed energy levels, lifetimes and transition probabilities for Na-like to Ar-like ions by

applying non-orthogonal spline CI, multiconfiguration Hartree–Fock (MCHF) and multiconfiguration

Dirac–Hartree–Fock (MCDHF) methods. Further, Johnson et al. [10] have presented energy intervals

and transition rates for sodium and lithium like ions with nuclear charge ranging from 3 to 100 using

third-order many-body perturbation theory. Moreover, transition energies for the transitions 3s 2S1/2 –

3p 2P3/2,1/2. have been reported by Kim et al. [11], using Dirac Fock method. Also, Douglas et al. [12]

have used rapid relativistic atomic structure approach to obtain transition energies and electric dipole

oscillator strengths for 71 Na-like ions with 22 ≤ Z ≤ 92. Furthermore, Verner et al. [13] have listed

wavelengths, statistical weights and oscillator strengths for 2249 spectral lines at wavelengths greater

than 228 Å arising from the ground states of various ions including Cr XIV. Also, a comprehensive

study of transition energies of D lines in Na-like ions was presented by Gillaspy et al. [14].

Page 2 of 29



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The main difficulty associated with theoretical studies is electron-electron correlation effects;

moreover, the relativistic corrections have to be taken in to account. The sodium like chromium has a

complete core (1s22s22p6) plus one valence electron, which is the simplest system for studying the

different correlation effects, such as the valence-valence (VV), core-valence (CV), and core-core (CC)

correlations. The energy levels are practically free from effects of configuration mixing and therefore

they are well suited for a theoretical interpretation of line intensities and for diagnostic purposes [15].

Although a lot of work has been done on Na-like ions, only a few works have reported the detailed

structure calculations for Na-like chromium. Therefore, the aim of this work is to perform fully

relativistic calculations in order to upgrade the database for transition energies and radiative properties

such as oscillator strengths, line strengths, wavelengths and transition decay probabilities for E1, E2,

M1 and M2 EUV and SXR transitions for Cr XIV. In the present calculations, we have adopted the

relativistic multi-configuration Dirac–Fock (MCDF) approach. QED corrections due to vacuum

polarization and self-energy effects and Breit correction due to the exchange of virtual photons between

two electrons are fully considered. Further, to test the accuracy of our results for energy levels,

analogous calculations have been performed with CIV3 and Flexible Atomic Code (FAC).

In this work, we have studied with two different configuration sets for MCDF calculations according

to valence-valence correlation (VV) and core-valence correlation (CV) within the framework of

configuration interaction expansion. In our calculations, we have taken in to account the configurations

of 2p6nl with 3 ≤ n ≤ 8 and 0 ≤ l ≤ 4 including one electron excitations from valence to other high

subshells for valence-valence correlation, 2p53l nl' with 3 ≤ n ≤ 5, 2p54l 4l’ including one electron

excitations from 2p subshell to other high subshells and 2s2p6 3l nl' with 3 ≤ n ≤ 5, 2s2p6 4l 4l' including

one electron excitations from 2s subshell to other high subshells for CV correlation. A comparison has

been made wherever possible, and good agreement was achieved. We have also provided a detailed

comparison of our theoretical results with the data available from the NIST database (National Institute

of Standards and Technology, website at http://www.nist.gov/pml/data/asd.cfm). We have theoretically

identified EUV and soft X-ray transitions. An outline of the computational methods is presented in

Section 2 and results are discussed in Sections 3.

2. Theoretical Methods

2.1 Multi Configuration Dirac-Fock Method

To perform the large-scale calculations, fully relativistic MCDF method, revised by Norrington [16],

formerly developed by Grant et al. [17] is applied, which has also been successfully applied in our

previous work [18-22]. QED corrections due to self-energy and vacuum polarization effects and Breit

corrections due to the exchange of virtual photons as a first order perturbation theory have also been

considered. Since the elaborate depiction of this method has been presented elsewhere [16, 22-26], so

only a brief outline is discussed here. The Dirac-Coulomb Hamiltonian in MCDF approach for an N-

electron atom or ion can be written as follows

HDC = ∑Hi

N

i=1

+ ∑ ∑1

|ri − rj| (1)

N

j=i+1

N−1

i=1

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where ��, the one-electron Hamiltonian is given by

Hi = cα i. p i + βmc2 + Vnuc (2)

In equation (2), the first two terms signify kinetic energy of an electron and the last term represents the

Coulomb potential of the nucleus. 𝛼 and 𝛽 are 4x4 Dirac matrices and c is the speed of light.

The N electron wave function constructed from central-field Dirac orbitals is given by

ϕ𝑛𝑘𝑚 =1

𝑟 (

𝑃𝑛𝑘(𝑟) 𝜒𝑘𝑚 (Ɵ, ϕ,σ)

−𝑖 𝑄𝑛𝑘(𝑟) 𝜒−𝑘𝑚 (Ɵ, ϕ, σ)) (3)

where k is the Dirac angular quantum number, k =± (j+1/2) for l = j ± 1/2, so j = k - 1/2, m is the

projection of the angular momentum j, and 𝑃𝑛𝑘 and 𝑄𝑛𝑘 are large and small components of one

electron radial functions. The spin angular momentum 𝜒𝑘𝑚 (Ɵ, ϕ) is a 2 component function defined

by

𝜒𝑘𝑚 (Ɵ, ϕ) = ∑ ⟨𝑙𝑚 − σ 1

2 σ|𝑙

1

2 𝑗𝑚 ⟩ 𝑌𝑙

𝑚−σσ=±

1

2

(Ɵ, ϕ) ϕσ (4)

An atomic state function (ASF) for N electron system constructed by the linear combination of n

electronic configuration state functions (CSFs) is represented by

|ψα(PJM)⟩ = ∑Ci

n

i=1

(α)|γi(PJM)⟩ (5)

where 𝐶𝑖(𝛼) are the expansion mixing coefficients for each CSF and satisfy the relation

(Ci(α))ϯCj(α) = δij (6)

such that ASFs satisfy the orthonormality condition. 𝛼 represents the orbital occupation numbers,

coupling, etc. and γi(PJM) are CSFs which specify a particular state with a given parity and angular

momentum (J,M).

In equation (6), the basis wave-functions are enlarged by considering the important correlations and

relativistic effects. CSFs of particular parity P and symmetry have been generated by taking appropriate

excitations from reference configurations to higher shells.

By taking expectation value of the Dirac–Hamiltonian, we get the energy of N-electron system as

EαPJM = ⟨ψ

α(PJM)|HDC|ψ

α(PJM)⟩

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= ∑Ci∗(α)

ij

Ci(α) ⟨γi(PJM)|HDC|γ

j(PJM)⟩

= (CαDC)ϯ HDC Cα

DC (7)

The elements of Dirac Hamiltonian matrix 𝐻𝐷𝐶 are given by

HrsDC = ⟨γr(PJM)|HDC|γs(PJM)⟩ (8)

Using the condition of normalization

(HDC − EαDC I) Cα

DC = 0 (9)

where I is the (n x n) unit matrix. Thus the predicted atomic energy level EαPJM

can be taken to be

eigenvalues of HDC.

2.2 Configuration Interaction Method

The general configuration interaction program CIV3 of Hibbert [27,28] has been employed to execute

the present calculations. The configuration interaction (CI) atomic state functions (ASFs) are

represented as

Ψi(J) = ∑aij

M

j=1

ϕj(αjLjSjJ) (10)

where {ϕj} denotes a set of single-configuration wave functions, (αj) defines the coupling of the orbital

Lj and spin Sj angular momenta to give the total angular momentum J. The mixing coefficients aij are

obtained by diagonalising the Breit-Pauli Hamiltonian matrix relative to the basis {ϕj}.

The radial functions Pnl(r) are expressed as linear combination of normalized slater-type orbitals in

the form

Pnl(r) = ∑Cjnl

k

j=1

χjnl (r) (11)

where {Cjnl } are the Clementi type [29] coefficients and

χjnl (r) = (2ξjnl)

Ijnl+12

[(2Ijnl)!]12

rIjnl exp(−ξjnl r) (12)

with the integer Ijnl ≥ l+1.

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The parameters Cjnl and ξjnl are determined variationally, while the parameters Ijnl, being integers are

kept fixed in the optimization process. The radial functions are chosen to satisfy the orthonormality

condition

∫ Pnl(r)

∞

0

Pn´l(r)dr = 𝛅nn´ ; l + 1 < n´ ≤ n (13)

The wave functions given by equation (10) are used to calculate the excitation energies of the fine

structure levels.

Substituting the CI expansion of the ASF (10) in a variational principle where the energy of the state

is minimized subject to the normalization condition ⟨Ψ|Ψ⟩ = 1 yields

∑aj

j

( Hij − E δij) = 0 (14)

So that, for a particular Jπ symmetry, the optimum choice of the expansion coefficients { ai } is

identified as the set of eigenvector components of the diagonalized Hamiltonian matrix with typical

element Hij = ⟨Ψi|H|Ψj⟩, for a specific eigenvalue Ei.

The ordered eigenvalues (Ei) of the Hamiltonian matrix are upper bounds to the similarly ordered

energy levels

Ei ≥ Eiexact. (15)

It enables us to optimize different radial functions on different eigenvalues and represent the wave

functions of a variety of states simultaneously to an equivalent degree of accuracy.

In Configuration Interaction calculations using CIV3, we have used 15 orthonormal one-electron

orbitals: 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f and 5g. The 1s, 2s and 2p are chosen to be

Hartree-Fock orbitals of the ground state 1s22s22p63s (2S1/2) of Cr XIV ion, given by Clementi and

Roetti [29]. The spectroscopic orbitals 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f and 5g are optimized on the

excited states 2p63p, 2p63d, 2p6 4s, 2p6 4p, 2p6 4d, 2p64f, 2p6 5s, 2p6 5p, 2p6 5d, 2p6 5f and 2p65g

whereas 3s orbital is optimized on the ground state 2p63s. Moreover, to increase the flexibility of the

radial functions of 3d, 4d, 4f, 5d, 5f and 5g, we have added extra Slater Type Orbital (STO) basis

functions allowing k > n-l in equation (11), so that some of the coefficients 𝐶𝑗𝑛𝑙 could vary freely

while still satisfying orthonormality conditions on 𝑃𝑛𝑙 . The optimized radial function parameters are

listed in Table I. The process of optimizing the radial functions is summarized in Table II, giving a

good representation of valence electron in the excited states for the different LS symmetries. The

configurations included in the CI calculations for both even and odd parity are shown in Table III.

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Table I. Radial function parameters for optimized orbitals of Na-like Cr XIV.

Orbitals Expansion coefficients

(Cjnl)

Power of r

(Ijnl)

Exponents

(ξjnl)

3s 0.19668 1 18.42588

-0.88361 2 7.78541

1.39173 3 5.30220

3p 0.52363 2 9.73570

-1.12187 3 4.86867

3d 0.06275 3 10.63170

0.96139 3 4.87476

4s 0.11751 1 18.86704

-0.80998 2 6.55187

2.00701 3 4.84096

-1.89831 4 3.74207

4p 0.36579 2 9.42200

-2.28188 3 3.82846

2.64893 4 3.68743

4d 0.58736 3 5.07973

0.25990 3 4.99393

-1.29886 4 3.32750

4f 0.85909 4 3.53210

0.14100 4 3.44427

5s 0.08012 1 19.17750

-0.78108 2 5.69738

2.18443 3 4.59778

-4.15885 4 3.28502

3.31909 5 2.95012

5p 0.26629 2 9.28260

-3.84184 3 3.07938

8.02510 4 3.07534

-4.92790 5 2.93884

5d 0.02862 3 9.88394

1.94514 3 3.44615

-5.50036 4 3.25329

4.02585 4 2.53807

5f 1.79671 4 3.01650

0.00764 4 6.93361

0.02264 5 3.96775

-2.25835 5 2.77972

5g 0.99986 5 2.80077

0.00022 5 1.59679

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Table II. Method of Determining the Radial Functions.

Orbital Process of Optimization

1s,2s,2p.............. Hartree-Fock orbitals of 2p63s 2S of Cr XIV (Clementi & Roetti [29])

Orbital Eigenvalues Minimized Configurations

3s....................... 2p63s 2S 2p63s

3p....................... 2p63p 2Po 2p63p

3d....................... 2p63d 2D 2p63d

4s....................... 2p64s 2S 2p63s, 2p64s

4p....................... 2p64p 2Po 2p63p, 2p64p

4d....................... 2p64d 2D 2p63d, 2p64d

4f....................... 2p64f 2Fo 2p64f

5s....................... 2p65s 2S 2p63s, 2p64s, 2p65s

5p....................... 2p65p 2Po 2p63p, 2p64p, 2p65p

5d....................... 2p65d 2D 2p63d, 2p64d, 2p65d

5f....................... 2p65f 2Fo 2p64f, 2p65f

5g....................... 2p65g 2G 2p65g

Table III. Configurations Used. All possible coupling of angular momenta of the orbitals of the

following configurations are included for each LSJ symmetry.

Even Parity

[1s22s22p6] 3s, 3d, 4s, 4d, 5s, 5d, 5g

[1s22s22p5] 3s3p, 3p3d, 3s4p, 3s4f, 3p4s, 3p4d, 3d4p, 3d4f, 4s4p, 4s4f, 4p4d, 4d4f, 3s5p, 3s5f,

3p5s, 3p5d, 3p5g, 3d5p, 3d5f

[1s22s 2p6] 3s3d, 3s2, 3p2, 3d2, 3s4s, 3s4d, 3p4p, 3p4f, 3d4s, 3d4d, 4s4d, 4p4f, 4s2, 4p2, 4d2, 4f2,

3s5s, 3s5d, 3s5g, 3p5p, 3p5f, 3d5s, 3d5d, 3d5g

Odd parity

[1s22s22p6] 3p, 4p, 4f, 5p, 5f

[1s22s22p5] 3s3d, 3s2, 3p2, 3d2, 3s4s, 3s4d, 3p4p, 3p4f, 3d4s, 3d4d, 4s4d, 4p4f, 4s2, 4p2, 4d2, 4f2,

3s5s, 3s5d, 3s5g, 3p5p, 3p5f, 3d5s, 3d5d, 3d5g

[1s22s 2p6] 3s3p, 3p3d, 3s4p, 3s4f, 3p4s, 3p4d, 3d4p, 3d4f, 4s4p, 4s4f, 4p4d, 4d4f, 3s5p, 3s5f,

3p5s, 3p5d, 3p5g, 3d5p, 3d5f

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As this method incorporates two spin independent non-fine structure terms, the Darwin and relativistic

mass correction terms which only affect the overall energy of each LS coupled term and three spin-

dependent fine-structure terms being one body spin-orbit (SO), two body spin-other-orbit (SOO) and

spin-spin (SS) interactions [30], which split each LS state into separate J-dependent levels. However,

in our work, the calculation of SOO and SS matrix elements proved very time consuming due to

extensively employed CSF sets. It has been found [31] that a very little accuracy is lost by neglecting

the SS term and replacing SO terms by a modified spin-orbit operator

H′so = ∝2

2 Z ∑

ξ(l)

ri3

N

i=1

(𝐥𝐢. 𝐬𝐢) (16)

where the parameters {ξ(l)} depend only on the l-value of the interacting electrons in the Breit –Pauli

Hamiltonian matrix element. A reliable approximation to the Breit-Pauli matrix elements was achieved

by taking ξ(s)=0.0, ξ(p)=0.9704 and ξ(d)=0.497.

2.3 FAC Calculations

To illustrate the accuracy of the calculated results from MCDF, parallel calculations have been carried

out using fully relativistic configuration interaction method by using Flexible Atomic Code of Gu [32],

based on self-consistent Dirac-Fock-Slater iteration performed on a selected fictitious mean

configuration in order to derive the local central potential [32,33]. In Flexible Atomic Code (FAC), the

orbitals are optimized self consistently and the average energy of a fictitious mean configuration with

orbital occupation numbers is minimized. We have performed larger calculations up to 3075 fine

structure levels belonging to (2*8) n*1 and (2*7)3*2, 4*2, 3*1 4*1, 3*1 5*1 configurations have been

performed where 3 ≤ n ≤ 20 and n*q represents all possible distributions (without restriction on orbital

angular momentum) of q electrons among the shells specified by their principal quantum numbers.

These results are listed in Table 2.

3. Results and Discussion

3.1 Energy Levels

In the present work, we have performed two sets of calculations using GRASP code. Extensive

configuration interaction (CI) has been incorporated in GRASP and for the optimisation of the orbitals

the option of ‘extended average level’ (EAL), in which a weighted (proportional to 2j+1) trace of the

Hamiltonian matrix is minimized, has been adopted. We have gradually increased the number of

configurations to perform GRASP calculations for up to 2727 levels. In the VV model (MCDF1), we

have included 48 levels generated from the configurations 2p6nl with 3 ≤ n ≤ 8 and l ≤ 4. The CV model

(MCDF2) involves additional configurations, 2p53l nl' with 3 ≤ n ≤ 5, 2p54l 4l', 2s2p6 3l nl' with 3 ≤ n

≤ 5, 2s2p6 4l 4l', obtained by single- and double-excitations from the ground state, generating 2727

levels in total. However, we note that the levels of 2p6nl lie below those of the other configurations.

For this reason, we only list the lowest 21 levels in Table 1, all belonging to 2p6nl. A comparison of

Page 9 of 29



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energies obtained from both the models (MCDF1, MCDF2) is shown in Table 1. Due to the inclusion

of larger CI in MCDF2, one can easily note the convergence of energies for most of the levels. Further,

a point worth mentioning here is that MCDF1 values for the low-lying excited states belonging to the

configurations 1s22s22p63l are in close agreement with NIST values whereas MCDF2 values for higher

excited states belonging to configurations 1s22s22p6nl (n = 4,5) are in close agreement with the NIST.

This is because of the fact that the sodium like chromium system has a complete core 1s22s22p6 plus

one valence electron and hence forms a simpler system for studying the different correlation effects

[15]. In our MCDF1 calculations, the energy levels are practically free from the effects of configuration

mixing and these correlations correct the energies of the low-lying excited levels belonging to

configurations 1s22s22p63l only whereas a deviation from NIST in the energies of higher excited levels

was observed. Therefore, there was a need to expand the configuration interaction set by taking into

account the core-valence correlations as these correlations correct the energies of the high-lying levels.

Also, we have tabulated the energies calculated from Dirac-Coulomb Hamiltonian (without taking into

account Breit and QED corrections) under the column IV and VI in case of MCDF1 and MCDF2

respectively.

0 5 10 15 20

-1

0

1

2

MCDF1

MCDF2

FAC1

CIV3

(E(T

heory

)-E

(NIS

T))

/E(N

IST

)(%

)

E(NIST) (Ryd.)

Figure 1 (online colour). Differences (in Ryd) of various theoretical energies from the NIST compiled values in Cr

XIV.

To assess the effect of further CI on the energy levels, we have adopted the FAC code of Gu

[32], which is also fully relativistic and is available from the website https://wwwamdis.iaea.org/FAC/.

We have performed two sets of calculations with FAC i.e. FAC1 and FAC. FAC1 includes the same

CI as in MCDF2 with additional configurations 2p6nl (6 ≤ n ≤8; 5 ≤ l ≤7) and the results are listed in

column VIII of Table 1, which are confined to the n ≤ 5 levels. For most of the levels, the NIST energies

differ with FAC1 by up to 0.18% and are smaller than 0.13% with those with MCDF2. One can see

that MCDF2 and FAC1 energies are in close agreement with each other. However, smaller

discrepancies between these energies are not because of different levels of CI, but due to

methodological variations. Additionally, the energies obtained from FAC1 are generally lower for most

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of the levels. In FAC, we have performed comparatively larger calculations for up to n = 20 and all

possible values of l generating 3075 levels in total with the same CI as in FAC1 by taking additional

configurations 2p6nl (9 ≤ n ≤20). Since it has not been possible to include higher 2p6nl configurations

in our GRASP calculations due to computational limitation of the code, so in Table 2, the FAC energies

have been listed, for the lowest 396 fine-structure levels, all belonging to 2p6nl configurations with n ≤

20. However, the inclusion of additional CI in the FAC calculations (Table 2) vary the energies up to

0.0002 Ryd (From FAC1) for some of the levels. Therefore, it may be reasonable to say that the

inclusion of CI in our FAC calculations is sufficient to calculate accurate results. Also, there is no

(strong) mixing or ambiguity for the designation of the 2p6nl levels listed in Tables 1 and 2. The

presented data may promote transfer of knowledge between astronomers and experimentalists through

targeted measurements and calculations.

In Table 1, we have also presented our results with CIV3 [27,28]. The fine-tuned [34,35]

excitation energies of the lowest 21 fine-structure levels of Cr XIV are presented under the column

CIV3. The ab-initio calculations (CIV3) have been done with 98 configurations, namely 2p6nl with 3

≤ n ≤ 5 and 0 ≤ l ≤ 4, 2p53l nl' with 3 ≤ n ≤ 5, 2p54l 4l', 2s2p6 3l nl' with 3 ≤ n ≤ 5 and 2s2p6 4l 4l' which

generates 2700 fine-structure levels. For the vast majority of our ab-initio energies, we observe very

satisfactory agreement with NIST.

Comparison of calculated energies by all methods has been made with Fischer et al. [9],

Sampson et al. [12], Johnson et al. [10] and Kim et al. [11]. One can see that our presented results by

all methods are not only in good agreement with all available results but are also very close to NIST

which ensures the reliability of our results and improvement in the energies of these levels due to proper

correlations and optimization. A graphical comparison of the relative differences of the present

MCDF1, MCDF2, FAC1 and CIV3 energies from the NIST is shown in Figure 1. To the best of our

knowledge, the new data presented for energy levels have not been reported elsewhere including NIST.

3.2 Radiative data of EUV and SXR transitions

Apart from energy levels, calculations have been made for absorption oscillator strengths (f-values,

dimensionless), radiative rates (A-values, s−1) and line strengths (S-values, in atomic units, 1a.u. =

6.460 × 10−36 cm2 esu2). Absorption oscillator strength (fij) for a transition i → j is connected to the

radiative rate Aji (in s-1) by the expression given below

fij =mcλji

2

8π2e2

ωj

ωiAji (17)

In the above formula m, c and e are the mass of electron, velocity of light and charge of electron

respectively, and ωj, ωi and λji denotes statistical weights of the upper, lower levels and transition

wavelength in Å respectively. Similarly, f- and A-values are related to S by the following standard

equations –

For the electric dipole (E1) transitions:

Aji =2.0261 × 1018

ωjλji3 Sij and fij =

303.75

λ jiωiSij (18)

Page 11 of 29



For Review Only

for the magnetic dipole (M1) transitions:

Aji =2.6974 × 1013


4.044 × 10−3

λ jiωiSij (19)

for the electric quadrupole (E2) transitions:

Aji =1.1199 × 1018


167.89

λji3ωi

Sij (20)

and for the magnetic quadrupole (M2) transitions:

Aji =1.4910 × 1013


2.236 × 10−3

λji3ωi

Sij (21)

In Table 3, we have tabulated the transition wavelengths, radiative rates, line oscillator strengths

and line strengths for electric dipole (E1) transitions in Na-like Cr, which have been obtained with

MCDF model. For other types of transitions, namely magnetic dipole (M1), electric quadrupole (E2),

and magnetic quadrupole (M2), only the A-values are listed, because the corresponding results for f-

or S-values can be obtained using equations (10–13) given in [36]. Additionally, we have also listed

the ratio (R) of the velocity (Coulomb gauge) and length (Babushkin gauge) forms which often (but

not necessarily) give an indication of the accuracy. In Table 3, indices i and j are the lower and upper

levels of a transition, λij is the transition wavelength (in Å), AjiE1 is the radiative transition probability

(in s−1), fijE1 is the absorption oscillator strength (dimensionless) and SE1 is the line strength in atomic

unit (a.u.) for the E1 transitions. Also AjiE2, Aji

M1 and AjiM2 are the radiative transition probabilities (in

s−1) for E2, M1 and M2 transitions respectively and R(E1) is the ratio of velocity and length forms of

A- (or f- and S-) values for the E1 transitions. From Table 3, we have identified 90 VUV and 64 SXR

spectral lines in dipole and quadrupole transitions. We have theoretically identified many new Extreme

Ultraviolet (EUV) transitions lying in the range 100-1200 Å along with SXR transitions. To the best of

our knowledge, the data presented for the newly identified lines has not been reported elsewhere

including NIST.

Page 12 of 29



For Review Only

0 50 100 150 200 250 300 350 400 450

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

2p63d

2D

3/2-2p

64p

2P

o

1/2

2p63d

2D

5/2-2p

64f

2F

o

7/2

2p63d

2D

3/2-2p

64f

2F

o

5/2

((T

heory

)-(N

IST

))/

(NIS

T)%

(NIST) (Å)

Figure 2 (online colour). Percentage differences of the present MCDF wavelengths from the NIST compiled values in Cr XIV.

In Table 4, we compare our calculated transition wavelengths and transition probabilities for

some transitions with the data compiled by NIST and other available references. To illustrate the

comparison, Figure 2 shows the relative differences of MCDF wavelengths from the NIST. Our results

agree within 0.1-0.3% with the values compiled by NIST for most of the transitions while differing up

to 0.5% for some transitions such as 0.48% for 2p63d 2D3/2 - 2p64f 2F5/2o , 2p63d 2D5/2 - 2p64f 2F7/2

o and

0.49% for 2p63d 2D3/2 - 2p64p 2P1/2o . Also, a good agreement was found between our calculated

transition probabilities and the ones given by NIST and other references.

4. Lifetimes

The lifetime τ of a level j can be determined from the inverse of the sum of transition probabilities of

radiative transitions from level i as

τj(s) =1

∑ Aji(s−1)i

(22)

Since this is a measurable quantity, it helps us to check the accuracy of A-values, particularly when a

single type of transition dominates. In Table 1, we have tabulated lifetimes for the lowest 21 fine-

structure levels for Cr XIV calculated by including all possible E1 (electric dipole), M1 (magnetic

dipole), E2 (electric quadrupole) and M2 (magnetic quadrupole) transitions. Comparison of calculated

lifetimes has also been made with Fischer et al. [9], and there is no significant discrepancy for any

level. We have also reported new data of lifetimes for levels 2p65l. We believe that lifetimes reported

by us in the present calculations will be helpful for future comparisons.

5. Conclusion

Motivated by the need of accurate atomic data, in the present work, energy levels and the radiative data

i.e. radiative rates, transition wavelengths, oscillator strengths and line strengths for E1, E2, M1 and

M2 transitions for the lowest 21 levels have been computed by MCDF method. We have included Breit

and QED corrections in our calculations and found that the order of magnitude is equal for QED and

Page 13 of 29



For Review Only

Breit corrections for high Z ions. Therefore, the importance of QED corrections cannot be avoided for

heavy atoms or ions. We have also reported the energies for 396 fine-structure levels belonging to the

configurations 2p6nl (3 ≤ n ≤ 20; all possible values of l) for Cr XIV computed from FAC method. For

additional accuracy assessments, we have also computed energy levels from CIV3 and our presented

energies are in good agreement with experimentally measured and theoretically calculated energies.

The velocity/length ratio of oscillator strength reaffirms the accuracy of our calculations. In this work,

we have extended the work for Cr XIV and there is no major discrepancy between our calculated and

experimentally observed wavelengths. We have also identified many new EUV and SXR spectral lines

in dipole and quadrupole transitions. Ultimately, we believe that our present work is comprehensive

and may be useful in the diagnosis and classification of EUV and SXR spectral lines. Further, our

presented data may be beneficial in fusion, Astrophysical plasma and plasma modelling.

Acknowledgement

This work was performed under the project No. EMR / 2016 / 001203, sponsored by SERB, the

Department of Science and Technology, Govt. of India, at Deen Dayal Upadhyaya College, University

of Delhi, India. Dr. Avnindra Kumar Singh (PI) is thankful to SERB-DST for providing the financial

support. Mayank Dimri is also thankful to DST for assisting him as a Junior Research Fellow. Prof.

Man Mohan (Co-PI) is also thankful to DST for providing the financial assistance as Co-PI.

References

[1] Chandra X-ray Observatory. Available at http://chandra.harvard.edu/.

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[3] N.J. Peacock, M.F. Stamp and J.D. Silver, Phys. Scr. T8 10 (1984).

[4] J. Reader, V. Kaufman, J. Sugar, J. O. Ekberg, U. Feldman, C.M. Brown, J.F. Seely, W.L. Rowan, J. Opt. Soc. Am

B (1987).

[5] U. Feldman and L. Cohen, J. Opt. Soc. Am. 57 1128 (1967).

[6] B. Edlen, Z. Phys. 100 621 (1936).

[7] B. C. Fawcett, R. D. Cowan, E. Y. Kononov and R. W. Hayes, J. Phys. B 5 (1972).

[8] L. Cohen and W. E. Behring. J. Opt. Soc. Am 66 (1976).

[9] C. F. Fischer, G. I. Tachiev and A. Irimia, At. Data Nucl. Data Tables 92 607 (2006).

[10] W. R. Johnson, Z. W. Liu and J. Sapirstein, At. Data Nucl. Data Tables 64 279 (1996).

[11] Y. K. Kim, D. H. Baik, P. Indelicato and J. P. Desclaux, Phys. Rev. A 44 148 (1991).

[12] D. H. Sampson, H. L. Zhang and C. J. Fontes, At. Data Nucl. Data Tables 44 209 (1990).

[13] D.A. Verner, P.D. Barthel and D. Tyler, Astron. Astrophys. Suppl. Ser. 108 287 (1994).

[14] J. D. Gillaspy, D. Osin, Y. Ralchenko, J. Reader and S. A. Blundell, Phys. Rev. A 87 (2013).

[15] W. O. Younis, S. H. Allam, T. M. El-Sherbini, At. Data Nucl. Data Tables 92 187 (2006).

[16] P. H. Norrington. http://www.am.qub.ac.uk/DARC/ 2009.

[17] I. P. Grant, B. J. McKenzie, P. H. Norrington, D. F. Mayers and N. C. Pyper, Comput. Phys. Commun. 21, 207

(1980).

[18] A. Goyal, N. Singh, S. Aggarwal, A.K. Singh, M. Mohan, Can. J. Phys. 94 712 (2016).

[19] A. Goyal, I. Khatri, A. K. Singh, M. Mohan, R. Sharma and N. Singh, ATOMS 4 22 (2016).

[20] A. Goyal, I. Khatri, S. Aggarwal, A. K. Singh, M. Mohan, Can. J. Phys. 93 487 (2015).

[21] A. Goyal, I. Khatri, S. Aggarwal. A. K. Singh, M. Mohan, J. Quant. Spectrosc. Radiat. Transfer 161 157 (2015).

[22] A. Goyal, I. Khatri, S. Aggarwal, A. K. Singh, M. Mohan, At. Data Nucl. Data Tables 107 406 (2016).

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http://chandra.harvard.edu/

http://sohowww.nascom.nasa.gov/

For Review Only

[23] P. Jönsson, X. He, C. F. Fischer, I. P. Grant, Comput. Phys. Commun. 177 597 (2007).

[24] F. A. Parpia, C. F. Fischer, I. P. Grant, Comput. Phys. Commun. 94 249 (1996).

[25] J. Olsen, M. R. Godefroid, P. Jönsson, P. A. Malmqvist, C. F. Fischer, Phys. Rev. E 52 4499 (1995).

[26] B. Frick, Phys. Scr. T8 129 (1986).

[27] A. Hibbert, Comput. Phys. Commun. 9 141 (1975).

[28] R. Glass and A. Hibbert, Comput. Phys. Commun. 16 19 (1978).

[29] E. Clementi & C. Roetti, At. Data Nucl. Data tables 14 177 (1974).

[30] V. Tayal, G. Gupta, Phys. Scr.75 331 (2007).

[31] A. Hibbert and A. C. Baillie, Phys.Scr. 45 565 (1992).

[32] M. F. Gu, Can. J. Phys. 86 675 (2008).

[33] D.H. Sampson, H.L. Zhang, A.K. Mohanty, Phys. Rev. A 40 604 (1989).

[34] J. Singh, S. Aggarwal, A.K. Singh and M. Mohan, Can. J. Phys. 90 833 (2012).

[35] N. Verma, A. K. S. Jha and M. Mohan, J. Phys. B 38 3185(2005).

[36] I. Khatri, A. Goyal, S. Aggarwal, A.K. Singh, M. Mohan, At. Data Nucl. Data Tables 107 367 (2016).

Page 15 of 29



For Review Only

Table 1. Comparison of threshold energies and lifetimes of our calculated lowest 21 fine structure levels of Cr XIV with other references. (aE±b = a× 10±𝑏).

Index Configuration Level

Energies (in Ryd) Lifetimes (in s)

DC1 MCDF1 DC2 MCDF2 FAC1 NIST and

other

references

CIV3

MCDF1 Fischer et al.

(DC+Breit

+QED) (DC+Breit

+QED)

1 2p63s 2S1/2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ------ ------

2 2p63p 2P1/2o 2.2190 2.2199 2.2273 2.2281 2.2223 2.2116 2.2116 1.828E-10 1.892E-10

2.2313a

2.2175b

2.2088c

2.2115d

3 2p63p 2P3/2o 2.3477 2.3449 2.3564 2.3532 2.3474 2.3370 2.3374 1.543E-10 1.583E-10

2.3626a

2.3431b

2.3358c

2.3373d

4 2p63d 2D3/2 5.3751 5.3688 5.3782 5.3708 5.3651 5.3567 5.3567 6.435E-11 6.564E-11

5.4099a

5.3661b

5 2p63d 2D5/2 5.3920 5.3834 5.3951 5.3855 5.3800 5.3721 5.3721 6.993E-11 7.163E-11

5.4266a

5.3808b

6 2p64s 2S1/2 13.4493 13.4422 13.4711 13.4649 13.4492 13.4729 13.4729 4.875E-12 4.680E-12

13.5138a

13.4649b

7 2p64p 2P1/2o 14.3208 14.3143 14.3477 14.3423 14.3227 14.3419 14.3419 6.163E-12 5.731E-12

14.3919a

14.3322b

8 2p64p 2P3/2o 14.3707 14.3627 14.3977 14.3905 14.3710 14.3905 14.3905 6.405E-12 5.946E-12

14.4432a

14.3837b

9 2p64d 2D3/2 15.4760 15.4668 15.5061 15.4975 15.4692 15.4965 15.4965 4.306E-12 4.135E-12

15.5589a

15.4862b

10 2p64d 2D5/2 15.4838 15.4737 15.5139 15.5043 15.4762 15.5036 15.5036 4.254E-12 4.132E-12

15.5673a

15.4935b

11 2p64f 2F5/2o 15.9180 15.9070 15.9345 15.9244 15.9224 15.9456 15.9456 1.706E-12 1.727E-12

16.0223a

15.9345b

12 2p64f 2F7/2o 15.9207 15.9096 15.9372 15.9270 15.9249 15.9479 15.9479 1.706E-12 1.730E-12

16.0257a

15.9419b

13 2p65s 2S1/2 19.1278 19.1186 19.1659 19.1588 19.1289 19.1619 19.1619 6.476E-12

19.1390b

14 2p65p 2P1/2o 19.5553 19.5465 19.5955 19.5889 19.5573 19.5858 19.5858 7.740E-12

19.5653b

15 2p65p 2P3/2o 19.5798 19.5703 19.6200 19.6125 19.5809 19.6106 19.6107 8.043E-12

19.5947b

16 2p65d 2D3/2 20.1117 20.1015 20.1549 20.1468 20.1100 20.1457 20.1457 5.523E-12

20.1239b

17 2p65d 2D5/2 20.1158 20.1052 20.1590 20.1504 20.1137 20.1488 20.1488 5.483E-12

20.1313b

18 2p65f 2F5/2o 20.3360 20.3251 20.3715 20.3626 20.3492 20.3695 20.3695 3.161E-12

20.3518b

19 2p65f 2F7/2o 20.3374 20.3264 20.3729 20.3639 20.3504 20.3708 20.3708 3.168E-12

20.3591b

20 2p65g 2G7/2 20.3552 20.3441 20.3904 20.3815 20.3563 20.3928 20.3928 6.062E-12

20.3812b

21 2p65g 2G9/2 20.3560 20.3449 20.3913 20.3823 20.3571 20.3929 20.3929 6.094E-12

20.3812b a - Fisher et al. [9], b- Sampson et al. [12], c- Johnson et al. [10], d- Kim et al. [11] and the values without superscript in column IX is the data from NIST.

Page 16 of 29



For Review Only

Table 2. Energies (in Ryd) for the 2p6nl (n ≤ 20) fine structure levels of Cr XIV. (aE±b = a× 10±𝑏).

Index Configuration Level FAC NIST

1 2p63s 2S1/2 0.0000 0

2 2p63p 2P1/2o 2.2221 2.2116

3 2p63p 2P3/2o 2.3473 2.3370

4 2p63d 2D3/2 5.3651 5.3567

5 2p63d 2D5/2 5.3800 5.3721

6 2p64s 2S1/2 13.4491 13.4729

7 2p64p 2P1/2o 14.3226 14.3419

8 2p64p 2P3/2o 14.3709 14.3905

9 2p64d 2D3/2 15.4692 15.4965

10 2p64d 2D5/2 15.4762 15.5036

11 2p64f 2F5/2o 15.9224 15.9456

12 2p64f 2F7/2o 15.9249 15.9479

13 2p65s 2S1/2 19.1289 19.1619

14 2p65p 2P1/2o 19.5572 19.5858

15 2p65p 2P3/2o 19.5807 19.6106

16 2p65d 2D3/2 20.1100 20.1457

17 2p65d 2D5/2 20.1137 20.1488

18 2p65f 2F5/2o 20.3492 20.3695

19 2p65f 2F7/2o 20.3505 20.3708

20 2p65g 2G7/2 20.3563 20.3928

21 2p65g 2G9/2 20.3572 20.3929

22 2p66s 2S1/2 22.0853 22.0934

23 2p66p 2P1/2o 22.3259 22.3350

24 2p66p 2P3/2o 22.3395 22.3448

25 2p66d 2D3/2 22.6366 22.6449

26 2p66d 2D5/2 22.6389 22.6469

27 2p66f 2F5/2o 22.7739 22.7735

28 2p66f 2F7/2o 22.7746 22.7749

29 2p66h 2H9/2o 22.7813

30 2p66h 2H11/2o 22.7816

31 2p66g 2G7/2 22.7822

32 2p66g 2G9/2 22.7827

33 2p67s 2S1/2 23.7958 23.8028

34 2p67p 2P1/2o 23.9445 23.9580

35 2p67p 2P3/2o 23.9529 23.9580

36 2p67d 2D3/2 24.1360 24.1402

37 2p67d 2D5/2 24.1375 24.1443

38 2p67f 2F5/2o 24.2194 24.2234

39 2p67f 2F7/2o 24.2199 24.2240

40 2p67h 2H9/2o 24.2266

41 2p67i 2I11/2 24.2268

42 2p67h 2H11/2o 24.2268

43 2p67i 2I13/2 24.2270

44 2p67g 2G7/2 24.2283

45 2p67g 2G9/2 24.2286

46 2p68s 2S1/2 24.8798

47 2p68p 2P1/2o 24.9780 24.9895

48 2p68p 2P3/2o 24.9835 24.9895

49 2p68d 2D3/2 25.1043 25.1100

50 2p68d 2D5/2 25.1053 25.1089

Page 17 of 29



For Review Only

Table 2 (continued)


51 2p68f 2F5/2o 25.1583

52 2p68f 2F7/2o 25.1586 25.1655

53 2p68h 2H9/2o 25.1648

54 2p68i 2I11/2 25.1648

55 2p68h 2H11/2o 25.1649

56 2p68k 2K13/2o 25.1649

57 2p68i 2I13/2 25.1649

58 2p68k 2K15/2o 25.1650

59 2p68g 2G7/2 25.1668

60 2p68g 2G9/2 25.1670

61 2p69s 2S1/2 25.6097

62 2p69p 2P1/2o 25.6779 25.7057

63 2p69p 2P3/2o 25.6817 25.7057

64 2p69d 2D3/2 25.7655 25.7639

65 2p69d 2D5/2 25.7662 25.7713

66 2p69f 2F5/2o 25.8024

67 2p69f 2F7/2o 25.8026 25.8100

68 2p69i 2I11/2 25.8080

69 2p69h 2H9/2o 25.8080

70 2p69k 2K13/2o 25.8080

71 2p69i 2I13/2 25.8080

72 2p69h 2H11/2o 25.8081

73 2p69k 2K15/2o 25.8081

74 2p69l 2L15/2 25.8081

75 2p69l 2L17/2 25.8081

76 2p69g 2G7/2 25.8099

77 2p69g 2G9/2 25.8101

78 2p610s 2S1/2 26.1246

79 2p610p 2P1/2o 26.1738

80 2p610p 2P3/2o 26.1766

81 2p610d 2D3/2 26.2369

82 2p610d 2D5/2 26.2375 26.2000

83 2p610f 2F5/2o 26.2634

84 2p610f 2F7/2o 26.2636 26.2700

85 2p610i 2I11/2 26.2680

86 2p610k 2K13/2o 26.2681

87 2p610i 2I13/2 26.2681

88 2p610h 2H9/2o 26.2681

89 2p610l 2L15/2 26.2681

90 2p610k 2K15/2o 26.2681

91 2p610l 2L17/2 26.2681

92 2p610m 2M17/2o 26.2681

93 2p610h 2H11/2o 26.2681

94 2p610m 2M19/2o 26.2681

95 2p610g 2G7/2 26.2698

96 2p610g 2G9/2 26.2699

97 2p611s 2S1/2 26.5012

98 2p611p 2P1/2o 26.5379

99 2p611p 2P3/2o 26.5400

100 2p611d 2D3/2 26.5849

101 2p611d 2D5/2 26.5853 26.5700

Page 18 of 29



For Review Only

Table 2 (continued)


102 2p611f 2F5/2o 26.6047

103 2p611f 2F7/2o 26.6048

104 2p611i 2I11/2 26.6084

105 2p611k 2K13/2o 26.6084

106 2p611i 2I13/2 26.6084

107 2p611l 2L15/2 26.6084

108 2p611k 2K15/2o 26.6084

109 2p611l 2L17/2 26.6085

110 2p611m 2M17/2o 26.6085

111 2p611m 2M19/2o 26.6085

112 2p611n 2N19/2 26.6085

113 2p611h 2H9/2o 26.6085

114 2p611n 2N21/2 26.6085

115 2p611h 2H11/2o 26.6085

116 2p611g 2G7/2 26.6099

117 2p611g 2G9/2 26.6100

118 2p612s 2S1/2 26.7851

119 2p612p 2P1/2o 26.8131

120 2p612p 2P3/2o 26.8147

121 2p612d 2D3/2 26.8491

122 2p612d 2D5/2 26.8494

123 2p612f 2F5/2o 26.8642

124 2p612f 2F7/2o 26.8643

125 2p612i 2I11/2 26.8673

126 2p612k 2K13/2o 26.8673

127 2p612i 2I13/2 26.8673

128 2p612l 2L15/2 26.8673

129 2p612k 2K15/2o 26.8673

130 2p612m 2M17/2o 26.8673

131 2p612l 2L17/2 26.8673

132 2p612m 2M19/2o 26.8673

133 2p612n 2N19/2 26.8673

134 2p612n 2N21/2 26.8674

135 2p612o 2O21/2o 26.8674

136 2p612o 2O23/2o 26.8674

137 2p612h 2H9/2o 26.8674

138 2p612h 2H7/2o 26.8674

139 2p612g 2G9/2 26.8685

140 2p612g 2G7/2 26.8686

141 2p613s 2S1/2 27.0043

142 2p613p 2P1/2o 27.0262

143 2p613p 2P3/2o 27.0275

144 2p613d 2D3/2 27.0543

145 2p613d 2D5/2 27.0546

146 2p613f 2F5/2o 27.0663

147 2p613f 2F7/2o 27.0664

148 2p613i 2I11/2 27.0687

149 2p613k 2K13/2o 27.0688

150 2p613i 2I13/2 27.0688

151 2p613l 2L15/2 27.0688

152 2p613k 2K15/2o 27.0688

Page 19 of 29



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Table 2 (continued)


153 2p613m 2M17/2o 27.0688

154 2p613l 2L17/2 27.0688

155 2p613m 2M19/2o 27.0688

156 2p613n 2N19/2 27.0688

157 2p613n 2N21/2 27.0688

158 2p613o 2O21/2o 27.0688

159 2p613o 2O23/2o 27.0688

160 2p613q 2Q23/2 27.0688

161 2p613q 2Q25/2 27.0688

162 2p613h 2H9/2o 27.0689

163 2p613h 2H11/2o 27.0689

164 2p613g 2G7/2 27.0697

165 2p613g 2G9/2 27.0698

166 2p614s 2S1/2 27.1772

167 2p614p 2P1/2o 27.1946

168 2p614p 2P3/2o 27.1956

169 2p614d 2D3/2 27.2170

170 2p614d 2D5/2 27.2172

171 2p614f 2F5/2o 27.2266

172 2p614f 2F7/2o 27.2267

173 2p614i 2I11/2 27.2286

174 2p614k 2K13/2o 27.2286

175 2p614i 2I13/2 27.2286

176 2p614l 2L15/2 27.2286

177 2p614k 2K15/2o 27.2286

178 2p614m 2M17/2o 27.2286

179 2p614l 2L17/2 27.2286

180 2p614n 2N19/2 27.2286

181 2p614m 2M19/2o 27.2286

182 2p614n 2N21/2 27.2286

183 2p614o 2O21/2o 27.2286

184 2p614o 2O23/2o 27.2286

185 2p614q 2Q23/2 27.2287

186 2p614q 2Q25/2 27.2287

187 2p614r 2Ro25/2 27.2287

188 2p614r 2Ro27/2 27.2287

189 2p614h 2H9/2o 27.2288

190 2p614h 2H11/2o 27.2288

191 2p614g 2G7/2 27.2294

192 2p614g 2G9/2 27.2294

193 2p615s 2S1/2 27.3158

194 2p615p 2P1/2o 27.3300

195 2p615p 2P3/2o 27.3308

196 2p615d 2D3/2 27.3481

197 2p615d 2D5/2 27.3482

198 2p615f 2F5/2o 27.3559

199 2p615f 2F7/2o 27.3560

200 2p615i 2I11/2 27.3576

201 2p615k 2K13/2o 27.3576

202 2p615i 2I13/2 27.3576

203 2p615l 2L15/2 27.3576

Page 20 of 29



For Review Only

Table 2 (continued)


204 2p615k 2K15/2o 27.3576

205 2p615m 2M17/2o 27.3576

206 2p615n 2N19/2 27.3576

207 2p615l 2L17/2 27.3576

208 2p615m 2M19/2o 27.3576

209 2p615n 2N21/2 27.3576

210 2p615o 2O21/2o 27.3576

211 2p615o 2O23/2o 27.3576

212 2p615q 2Q23/2 27.3576

213 2p615q 2Q25/2 27.3576

214 2p615r 2Ro25/2 27.3576

215 2p615r 2Ro27/2 27.3576

216 2p615t 2T27/2 27.3576

217 2p615t 2T29/2 27.3576

218 2p615h 2H9/2o 27.3577

219 2p615h 2H11/2o 27.3577

220 2p615g 2G7/2 27.3582

221 2p615g 2G9/2 27.3582

222 2p616s 2S1/2 27.4288

223 2p616p 2P1/2o 27.4404

224 2p616p 2P3/2o 27.4410

225 2p616d 2D3/2 27.4552

226 2p616d 2D5/2 27.4554

227 2p616f 2F5/2o 27.4618

228 2p616f 2F7/2o 27.4618

229 2p616i 2I11/2 27.4631

230 2p616k 2K13/2o 27.4631

231 2p616i 2I13/2 27.4631

232 2p616k 2K15/2o 27.4631

233 2p616l 2L15/2 27.4631

234 2p616m 2M17/2o 27.4631

235 2p616n 2N19/2 27.4631

236 2p616l 2L17/2 27.4631

237 2p616m 2M19/2o 27.4631

238 2p616n 2N21/2 27.4631

239 2p616o 2O21/2o 27.4631

240 2p616o 2O23/2o 27.4631

241 2p616q 2Q23/2 27.4631

242 2p616q 2Q25/2 27.4631

243 2p616r 2Ro25/2 27.4631

244 2p616r 2Ro27/2 27.4631

245 2p616t 2T27/2 27.4631

246 2p616u 2Uo29/2 27.4631

247 2p616t 2T29/2 27.4631

248 2p616u 2Uo31/2 27.4631

249 2p616h 2H9/2o 27.4633

250 2p616h 2H11/2o 27.4633

251 2p616g 2G7/2 27.4636

252 2p616g 2G9/2 27.4636

253 2p617s 2S1/2 27.5220

254 2p617p 2P1/2o 27.5316

Page 21 of 29



For Review Only

Table 2 (continued)


255 2p617p 2P3/2o 27.5322

256 2p617d 2D3/2 27.5440

257 2p617d 2D5/2 27.5441

258 2p617f 2F5/2o 27.5495

259 2p617f 2F7/2o 27.5495

260 2p617i 2I11/2 27.5506

261 2p617k 2K13/2o 27.5506

262 2p617i 2I13/2 27.5506

263 2p617k 2K15/2o 27.5506

264 2p617l 2L15/2 27.5506

265 2p617m 2M17/2o 27.5506

266 2p617n 2N19/2 27.5506

267 2p617o 2O21/2o 27.5506

268 2p617l 2L17/2 27.5506

269 2p617n 2N21/2 27.5506

270 2p617m 2M19/2o 27.5506

271 2p617o 2O23/2o 27.5506

272 2p617q 2Q23/2 27.5506

273 2p617q 2Q25/2 27.5506

274 2p617r 2Ro25/2 27.5506

275 2p617r 2Ro27/2 27.5506

276 2p617t 2T27/2 27.5506

277 2p617t 2T29/2 27.5506

278 2p617u 2Uo29/2 27.5506

279 2p617v 2V31/2 27.5506

280 2p617u 2Uo31/2 27.5506

281 2p617v 2V33/2 27.5506

282 2p617h 2H9/2o 27.5507

283 2p617h 2H11/2o 27.5507

284 2p617g 2G7/2 27.5509

285 2p617g 2G9/2 27.5509

286 2p618s 2S1/2 27.5998

287 2p618p 2P1/2o 27.6079

288 2p618p 2P3/2o 27.6084

289 2p618d 2D3/2 27.6183

290 2p618d 2D5/2 27.6184

291 2p618f 2F5/2o 27.6229

292 2p618f 2F7/2o 27.6230

293 2p618i 2I11/2 27.6239

294 2p618k 2K13/2o 27.6239

295 2p618i 2I13/2 27.6239

296 2p618k 2K15/2o 27.6239

297 2p618l 2L15/2 27.6239

298 2p618n 2N19/2 27.6239

299 2p618m 2M17/2o 27.6239

300 2p618o 2O21/2o 27.6239

301 2p618l 2L17/2 27.6239

302 2p618n 2N21/2 27.6239

303 2p618o 2O23/2o 27.6239

304 2p618m 2M19/2o 27.6239

305 2p618q 2Q23/2 27.6239

Page 22 of 29



For Review Only

Table 2 (continued)


306 2p618q 2Q25/2 27.6239

307 2p618r 2Ro25/2 27.6239

308 2p618r 2Ro27/2 27.6239

309 2p618t 2T27/2 27.6239

310 2p618t 2T29/2 27.6239

311 2p618u 2Uo29/2 27.6239

312 2p618u 2Uo31/2 27.6239

313 2p618v 2V31/2 27.6239

314 2p618w 2Wo33/2 27.6239

315 2p618v 2V33/2 27.6239

316 2p618w 2Wo35/2 27.6239

317 2p618h 2H9/2o 27.6240

318 2p618h 2H11/2o 27.6240

319 2p618g 2G7/2 27.6241

320 2p618g 2G9/2 27.6241

321 2p619s 2S1/2 27.6655

322 2p619p 2P1/2o 27.6723

323 2p619p 2P3/2o 27.6727

324 2p619d 2D3/2 27.6811

325 2p619d 2D5/2 27.6812

326 2p619f 2F5/2o 27.6851

327 2p619f 2F7/2o 27.6851

328 2p619i 2I11/2 27.6859

329 2p619k 2K13/2o 27.6859

330 2p619i 2I13/2 27.6859

331 2p619k 2K15/2o 27.6859

332 2p619l 2L15/2 27.6859

333 2p619o 2O21/2o 27.6859

334 2p619n 2N19/2 27.6859

335 2p619m 2M17/2o 27.6859

336 2p619l 2L17/2 27.6859

337 2p619q 2Q23/2 27.6859

338 2p619o 2O23/2o 27.6859

339 2p619n 2N21/2 27.6859

340 2p619m 2M19/2o 27.6859

341 2p619q 2Q25/2 27.6859

342 2p619r 2Ro25/2 27.6859

343 2p619r 2Ro27/2 27.6859

344 2p619t 2T27/2 27.6859

345 2p619t 2T29/2 27.6859

346 2p619u 2Uo29/2 27.6859

347 2p619u 2Uo31/2 27.6859

348 2p619v 2V31/2 27.6859

349 2p619v 2V33/2 27.6859

350 2p619w 2Wo33/2 27.6859

351 2p619x 2X35/2 27.6859

352 2p619x 2X37/2 27.6859

353 2p619w 2Wo35/2 27.6859

354 2p619h 2H9/2o 27.6860

355 2p619h 2H11/2o 27.6861

356 2p619g 2G7/2 27.6861

Page 23 of 29



For Review Only

Table 2 (continued)


357 2p619g 2G9/2 27.6861

358 2p620s 2S1/2 27.7214

359 2p620p 2P1/2o 27.7272

360 2p620p 2P3/2o 27.7276

361 2p620d 2D3/2 27.7348

362 2p620d 2D5/2 27.7348

363 2p620f 2F5/2o 27.7382

364 2p620f 2F7/2o 27.7382

365 2p620i 2I11/2 27.7388

366 2p620k 2K13/2o 27.7389

367 2p620i 2I13/2 27.7389

368 2p620k 2K15/2o 27.7389

369 2p620l 2L15/2 27.7389

370 2p620o 2O21/2o 27.7389

371 2p620n 2N19/2 27.7389

372 2p620m 2M17/2o 27.7389

373 2p620q 2Q23/2 27.7389

374 2p620l 2L17/2 27.7389

375 2p620o 2O23/2o 27.7389

376 2p620q 2Q25/2 27.7389

377 2p620n 2N21/2 27.7389

378 2p620m 2M19/2o 27.7389

379 2p620r 2Ro25/2 27.7389

380 2p620r 2Ro27/2 27.7389

381 2p620t 2T27/2 27.7389

382 2p620t 2T29/2 27.7389

383 2p620u 2Uo29/2 27.7389

384 2p620u 2Uo31/2 27.7389

385 2p620v 2V31/2 27.7389

386 2p620v 2V33/2 27.7389

387 2p620w 2Wo33/2 27.7389

388 2p620w 2Wo35/2 27.7389

389 2p620y 2Yo37/2 27.7389

390 2p620x 2X35/2 27.7389

391 2p620y 2Yo39/2 27.7389

392 2p620x 2X37/2 27.7389

393 2p620g 2G7/2 27.7390

394 2p620g 2G9/2 27.7390

395 2p620h 2H9/2o 27.7390

396 2p620h 2H11/2o 27.7390

Page 24 of 29



For Review Only

Table 3. Transition wavelengths (λij in Å), radiative rates (Aji in s-1), oscillator strengths (fij, dimensionless), and line strengths (S, in atomic

units) for electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2) and magnetic quadrupole (M2) transitions in Cr XIV. The last

column gives R(E1), the ratio of velocity and length forms of A-values for E1 transitions. (aE±b = a× 𝟏𝟎±𝒃).

i j λij AjiE1 fij

E1 SE1 AjiE2 Aji

M1 AjiM2 R(E1)

1 2 4.105E+02 5.469E+09 1.382E-01 3.734E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00

1 3 3.886E+02 6.484E+09 2.936E-01 7.513E-01 0.000E+00 0.000E+00 4.737E+00 1.0E+00

1 4 1.697E+02 0.000E+00 0.000E+00 0.000E+00 5.435E+05 5.903E-02 0.000E+00 0.0E+00

1 5 1.693E+02 0.000E+00 0.000E+00 0.000E+00 5.520E+05 0.000E+00 0.000E+00 0.0E+00

1 6 6.779E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.082E+01 0.000E+00 0.0E+00

1 7 6.366E+01 1.125E+11 6.832E-02 2.864E-02 0.000E+00 0.000E+00 0.000E+00 9.9E-01

1 8 6.345E+01 1.068E+11 1.289E-01 5.385E-02 0.000E+00 0.000E+00 2.930E+03 9.8E-01

1 9 5.892E+01 0.000E+00 0.000E+00 0.000E+00 6.327E+07 1.784E-01 0.000E+00 0.0E+00

1 10 5.889E+01 0.000E+00 0.000E+00 0.000E+00 6.315E+07 0.000E+00 0.000E+00 0.0E+00

1 11 5.729E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.449E-04 0.0E+00

1 13 4.766E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.522E+01 0.000E+00 0.0E+00

1 14 4.662E+01 6.549E+10 2.134E-02 6.550E-03 0.000E+00 0.000E+00 0.000E+00 9.8E-01

1 15 4.656E+01 6.282E+10 4.084E-02 1.252E-02 0.000E+00 0.000E+00 3.200E+03 9.8E-01

1 16 4.533E+01 0.000E+00 0.000E+00 0.000E+00 3.627E+07 3.213E-01 0.000E+00 0.0E+00

1 17 4.533E+01 0.000E+00 0.000E+00 0.000E+00 3.627E+07 0.000E+00 0.000E+00 0.0E+00

1 18 4.484E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.566E-04 0.0E+00

2 3 7.289E+03 0.000E+00 0.000E+00 0.000E+00 4.543E-03 2.320E+01 0.000E+00 0.0E+00

2 4 2.894E+02 1.319E+10 3.312E-01 6.310E-01 0.000E+00 0.000E+00 6.943E-01 1.1E+00

2 5 2.881E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.067E+00 0.0E+00

2 6 8.120E+01 6.705E+10 6.628E-02 3.544E-02 0.000E+00 0.000E+00 0.000E+00 1.0E+00

2 7 7.535E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 7.719E+00 0.000E+00 0.0E+00

2 8 7.505E+01 0.000E+00 0.000E+00 0.000E+00 1.618E+07 7.841E+02 0.000E+00 0.0E+00

2 9 6.879E+01 1.904E+11 2.701E-01 1.223E-01 0.000E+00 0.000E+00 1.775E+02 9.7E-01

2 10 6.876E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.258E+03 0.0E+00

2 11 6.658E+01 0.000E+00 0.000E+00 0.000E+00 9.973E+07 0.000E+00 0.000E+00 0.0E+00

2 13 5.393E+01 2.964E+10 1.292E-02 4.588E-03 0.000E+00 0.000E+00 0.000E+00 1.0E+00

2 14 5.259E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 8.996E+00 0.000E+00 0.0E+00

2 15 5.252E+01 0.000E+00 0.000E+00 0.000E+00 8.862E+06 5.334E+02 0.000E+00 0.0E+00

2 16 5.096E+01 1.156E+11 9.000E-02 3.020E-02 0.000E+00 0.000E+00 1.963E+02 9.7E-01

2 17 5.095E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.392E+03 0.0E+00

2 18 5.033E+01 0.000E+00 0.000E+00 0.000E+00 1.999E+07 0.000E+00 0.000E+00 0.0E+00

3 4 3.014E+02 2.339E+09 3.185E-02 1.264E-01 0.000E+00 0.000E+00 0.000E+00 1.1E+00

3 5 2.999E+02 1.426E+10 2.885E-01 1.139E+00 0.000E+00 0.000E+00 2.268E+01 1.1E+00

3 6 8.212E+01 1.375E+11 6.952E-02 7.517E-02 0.000E+00 0.000E+00 2.253E+03 1.0E+00

3 7 7.613E+01 0.000E+00 0.000E+00 0.000E+00 3.229E+07 1.833E+03 0.000E+00 0.0E+00

3 8 7.583E+01 0.000E+00 0.000E+00 0.000E+00 1.605E+07 4.779E+01 0.000E+00 0.0E+00

3 9 6.945E+01 3.887E+10 2.810E-02 2.570E-02 0.000E+00 0.000E+00 0.000E+00 9.7E-01

3 10 6.941E+01 2.320E+11 2.514E-01 2.298E-01 0.000E+00 0.000E+00 6.892E+03 9.7E-01

3 11 6.719E+01 0.000E+00 0.000E+00 0.000E+00 2.783E+07 1.439E+00 0.000E+00 0.0E+00

3 12 6.718E+01 0.000E+00 0.000E+00 0.000E+00 1.253E+08 0.000E+00 0.000E+00 0.0E+00

3 13 5.433E+01 6.061E+10 1.341E-02 9.593E-03 0.000E+00 0.000E+00 2.268E+03 1.0E+00

3 14 5.298E+01 0.000E+00 0.000E+00 0.000E+00 1.737E+07 1.315E+03 0.000E+00 0.0E+00

3 15 5.290E+01 0.000E+00 0.000E+00 0.000E+00 8.716E+06 5.496E+01 0.000E+00 0.0E+00

3 16 5.132E+01 2.337E+10 9.225E-03 6.235E-03 0.000E+00 0.000E+00 0.000E+00 9.7E-01

3 17 5.131E+01 1.397E+11 8.270E-02 5.588E-02 0.000E+00 0.000E+00 7.591E+03 9.7E-01

3 18 5.068E+01 0.000E+00 0.000E+00 0.000E+00 5.402E+06 9.745E-01 0.000E+00 0.0E+00

3 19 5.068E+01 0.000E+00 0.000E+00 0.000E+00 2.438E+07 9.745E-01 0.000E+00 0.0E+00

3 20 5.063E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.037E-04 0.0E+00

4 5 6.201E+04 0.000E+00 0.000E+00 0.000E+00 2.193E-08 4.521E-02 0.000E+00 0.0E+00

4 6 1.129E+02 0.000E+00 0.000E+00 0.000E+00 3.869E+06 4.754E-04 0.000E+00 0.0E+00

4 7 1.019E+02 4.896E+10 3.808E-02 5.108E-02 0.000E+00 0.000E+00 2.083E+01 1.1E+00

4 8 1.013E+02 4.746E+09 7.304E-03 9.746E-03 0.000E+00 0.000E+00 0.000E+00 1.1E+00

4 9 9.024E+01 0.000E+00 0.000E+00 0.000E+00 6.925E+06 9.526E+00 0.000E+00 0.0E+00

4 10 9.018E+01 0.000E+00 0.000E+00 0.000E+00 1.979E+06 6.610E+00 0.000E+00 0.0E+00

4 11 8.647E+01 5.472E+11 9.202E-01 1.048E+00 0.000E+00 0.000E+00 1.922E+03 9.9E-01

4 12 8.645E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.462E+03 0.0E+00

4 13 6.628E+01 0.000E+00 0.000E+00 0.000E+00 1.860E+06 4.941E-04 0.000E+00 0.0E+00

4 14 6.428E+01 1.906E+10 5.903E-03 4.996E-03 0.000E+00 0.000E+00 2.038E+01 1.1E+00

Page 25 of 29



For Review Only

Table 3 (continued)

i j λij AjiE1 fij

E1 SE1 AjiE2 Aji

M1 AjiM2 R(E1)

4 15 6.417E+01 1.851E+09 1.143E-03 9.655E-04 0.000E+00 0.000E+00 0.000E+00 1.1E+00

4 16 6.185E+01 0.000E+00 0.000E+00 0.000E+00 3.258E+06 1.024E+01 0.000E+00 0.0E+00

4 17 6.184E+01 0.000E+00 0.000E+00 0.000E+00 9.329E+05 4.188E+00 0.000E+00 0.0E+00

4 18 6.093E+01 2.051E+11 1.712E-01 1.373E-01 0.000E+00 0.000E+00 1.450E+03 9.9E-01

4 19 6.092E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.858E+03 0.0E+00

4 20 6.085E+01 0.000E+00 0.000E+00 0.000E+00 7.844E+07 0.000E+00 0.000E+00 0.0E+00

5 6 1.131E+02 0.000E+00 0.000E+00 0.000E+00 5.785E+06 0.000E+00 0.000E+00 0.0E+00

5 7 1.020E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.223E+02 0.0E+00

5 8 1.015E+02 4.285E+10 4.411E-02 8.843E-02 0.000E+00 0.000E+00 5.957E+02 1.1E+00

5 9 9.038E+01 0.000E+00 0.000E+00 0.000E+00 2.963E+06 2.640E+01 0.000E+00 0.0E+00

5 10 9.031E+01 0.000E+00 0.000E+00 0.000E+00 7.902E+06 3.167E+01 0.000E+00 0.0E+00

5 11 8.659E+01 3.903E+10 4.388E-02 7.505E-02 0.000E+00 0.000E+00 0.000E+00 9.9E-01

5 12 8.657E+01 5.856E+11 8.773E-01 1.500E+00 0.000E+00 0.000E+00 1.690E+04 9.9E-01

5 13 6.635E+01 0.000E+00 0.000E+00 0.000E+00 2.774E+06 0.000E+00 0.000E+00 0.0E+00

5 14 6.434E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.175E+02 0.0E+00

5 15 6.423E+01 1.670E+10 6.887E-03 8.738E-03 0.000E+00 0.000E+00 5.795E+02 1.1E+00

5 16 6.192E+01 0.000E+00 0.000E+00 0.000E+00 1.391E+06 2.080E+01 0.000E+00 0.0E+00

5 17 6.190E+01 0.000E+00 0.000E+00 0.000E+00 3.718E+06 3.387E+01 0.000E+00 0.0E+00

5 18 6.099E+01 1.460E+10 8.142E-03 9.809E-03 0.000E+00 0.000E+00 0.000E+00 9.9E-01

5 19 6.098E+01 2.193E+11 1.630E-01 1.963E-01 0.000E+00 0.000E+00 1.274E+04 9.9E-01

5 20 6.091E+01 0.000E+00 0.000E+00 0.000E+00 8.716E+06 4.269E-01 0.000E+00 0.0E+00

5 21 6.091E+01 0.000E+00 0.000E+00 0.000E+00 8.711E+07 0.000E+00 0.000E+00 0.0E+00

6 7 1.045E+03 1.223E+09 2.002E-01 1.378E+00 0.000E+00 0.000E+00 0.000E+00 1.0E+00

6 8 9.900E+02 1.443E+09 4.240E-01 2.764E+00 0.000E+00 0.000E+00 1.625E-01 1.0E+00

6 9 4.501E+02 0.000E+00 0.000E+00 0.000E+00 6.469E+04 1.446E-03 0.000E+00 0.0E+00

6 10 4.486E+02 0.000E+00 0.000E+00 0.000E+00 6.589E+04 0.000E+00 0.000E+00 0.0E+00

6 11 3.697E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.289E-08 0.0E+00

6 13 1.605E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 7.187E-01 0.000E+00 0.0E+00

6 14 1.493E+02 2.154E+10 7.198E-02 7.075E-02 0.000E+00 0.000E+00 0.000E+00 9.9E-01

6 15 1.487E+02 2.036E+10 1.350E-01 1.322E-01 0.000E+00 0.000E+00 1.017E+02 9.9E-01

6 16 1.368E+02 0.000E+00 0.000E+00 0.000E+00 6.295E+06 5.534E-05 0.000E+00 0.0E+00

6 17 1.368E+02 0.000E+00 0.000E+00 0.000E+00 6.277E+06 0.000E+00 0.000E+00 0.0E+00

6 18 1.324E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.387E-06 0.0E+00

7 8 1.882E+04 0.000E+00 0.000E+00 0.000E+00 5.416E-04 1.348E+00 0.000E+00 0.0E+00

7 9 7.907E+02 2.773E+09 5.199E-01 2.707E+00 0.000E+00 0.000E+00 1.957E-02 1.0E+00

7 10 7.860E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.436E-01 0.0E+00

7 11 5.721E+02 0.000E+00 0.000E+00 0.000E+00 1.298E+04 0.000E+00 0.000E+00 0.0E+00

7 13 1.897E+02 2.106E+10 1.136E-01 1.419E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00

7 14 1.742E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.008E-01 0.000E+00 0.0E+00

7 15 1.734E+02 0.000E+00 0.000E+00 0.000E+00 2.505E+06 6.765E+01 0.000E+00 0.0E+00

7 16 1.575E+02 3.039E+10 2.259E-01 2.342E-01 0.000E+00 0.000E+00 5.411E+00 9.8E-01

7 17 1.574E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.824E+01 0.0E+00

7 18 1.516E+02 0.000E+00 0.000E+00 0.000E+00 1.075E+07 0.000E+00 0.000E+00 0.0E+00

8 9 8.254E+02 4.888E+08 4.992E-02 5.426E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00

8 10 8.202E+02 2.991E+09 4.525E-01 4.888E+00 0.000E+00 0.000E+00 6.360E-01 1.0E+00

8 11 5.901E+02 0.000E+00 0.000E+00 0.000E+00 3.184E+03 2.439E-04 0.000E+00 0.0E+00

8 12 5.891E+02 0.000E+00 0.000E+00 0.000E+00 1.446E+04 0.000E+00 0.000E+00 0.0E+00

8 13 1.916E+02 4.309E+10 1.186E-01 2.992E-01 0.000E+00 0.000E+00 1.296E+02 1.0E+00

8 14 1.758E+02 0.000E+00 0.000E+00 0.000E+00 5.001E+06 1.472E+02 0.000E+00 0.0E+00

8 15 1.750E+02 0.000E+00 0.000E+00 0.000E+00 2.490E+06 1.290E+00 0.000E+00 0.0E+00

8 16 1.588E+02 6.291E+09 2.378E-02 4.973E-02 0.000E+00 0.000E+00 0.000E+00 9.8E-01

8 17 1.587E+02 3.747E+10 2.122E-01 4.434E-01 0.000E+00 0.000E+00 2.129E+02 9.8E-01

8 18 1.528E+02 0.000E+00 0.000E+00 0.000E+00 3.035E+06 1.608E-02 0.000E+00 0.0E+00

8 19 1.528E+02 0.000E+00 0.000E+00 0.000E+00 1.366E+07 0.000E+00 0.000E+00 0.0E+00

8 20 1.524E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.800E-06 0.0E+00

9 10 1.313E+05 0.000E+00 0.000E+00 0.000E+00 9.114E-09 4.770E-03 0.000E+00 0.0E+00

9 11 2.070E+03 1.228E+08 1.183E-01 3.226E+00 0.000E+00 0.000E+00 7.533E-04 1.1E+00

9 12 2.058E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 9.938E-04 0.0E+00

9 13 2.495E+02 0.000E+00 0.000E+00 0.000E+00 1.087E+06 6.936E-05 0.000E+00 0.0E+00

9 14 2.234E+02 2.270E+10 8.491E-02 2.497E-01 0.000E+00 0.000E+00 2.009E+00 1.0E+00

9 15 2.221E+02 2.207E+09 1.632E-02 4.772E-02 0.000E+00 0.000E+00 0.000E+00 1.0E+00

9 16 1.966E+02 0.000E+00 0.000E+00 0.000E+00 1.553E+06 3.969E-01 0.000E+00 0.0E+00

9 17 1.965E+02 0.000E+00 0.000E+00 0.000E+00 4.436E+05 1.175E+00 0.000E+00 0.0E+00

Page 26 of 29



For Review Only

Table 3 (continued)

i j λij AjiE1 fij

E1 SE1 AjiE2 Aji

M1 AjiM2 R(E1)

9 18 1.876E+02 9.016E+10 7.134E-01 1.762E+00 0.000E+00 0.000E+00 6.734E+01 9.9E-01

9 19 1.875E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 8.618E+01 0.0E+00

9 20 1.868E+02 0.000E+00 0.000E+00 0.000E+00 9.665E+06 0.000E+00 0.000E+00 0.0E+00

10 11 2.103E+03 8.361E+06 5.544E-03 2.303E-01 0.000E+00 0.000E+00 0.000E+00 1.1E+00

10 12 2.091E+03 1.278E+08 1.116E-01 4.609E+00 0.000E+00 0.000E+00 6.324E-03 1.1E+00

10 13 2.500E+02 0.000E+00 0.000E+00 0.000E+00 1.624E+06 0.000E+00 0.000E+00 0.0E+00

10 14 2.238E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.144E+01 0.0E+00

10 15 2.225E+02 1.994E+10 9.859E-02 4.332E-01 0.000E+00 0.000E+00 5.766E+01 1.0E+00

10 16 1.969E+02 0.000E+00 0.000E+00 0.000E+00 6.647E+05 2.986E+00 0.000E+00 0.0E+00

10 17 1.968E+02 0.000E+00 0.000E+00 0.000E+00 1.772E+06 1.307E+00 0.000E+00 0.0E+00

10 18 1.878E+02 6.445E+09 3.409E-02 1.265E-01 0.000E+00 0.000E+00 0.000E+00 9.9E-01

10 19 1.878E+02 9.661E+10 6.810E-01 2.526E+00 0.000E+00 0.000E+00 5.927E+02 9.9E-01

10 20 1.871E+02 0.000E+00 0.000E+00 0.000E+00 1.069E+06 2.291E-02 0.000E+00 0.0E+00

10 21 1.871E+02 0.000E+00 0.000E+00 0.000E+00 1.070E+07 0.000E+00 0.000E+00 0.0E+00

11 12 3.536E+05 0.000E+00 0.000E+00 0.000E+00 1.919E-11 2.614E-04 0.000E+00 0.0E+00

11 13 2.838E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.559E-10 0.0E+00

11 14 2.504E+02 0.000E+00 0.000E+00 0.000E+00 4.756E+05 0.000E+00 0.000E+00 0.0E+00

11 15 2.488E+02 0.000E+00 0.000E+00 0.000E+00 6.826E+04 2.364E-05 0.000E+00 0.0E+00

11 16 2.173E+02 3.935E+09 1.856E-02 7.965E-02 0.000E+00 0.000E+00 2.192E+00 1.0E+00

11 17 2.171E+02 1.857E+08 1.312E-03 5.624E-03 0.000E+00 0.000E+00 0.000E+00 1.0E+00

11 18 2.063E+02 0.000E+00 0.000E+00 0.000E+00 9.874E+05 3.872E-01 0.000E+00 0.0E+00

11 19 2.062E+02 0.000E+00 0.000E+00 0.000E+00 1.234E+05 7.478E-02 0.000E+00 0.0E+00

11 20 2.054E+02 1.586E+11 1.337E+00 5.425E+00 0.000E+00 0.000E+00 2.510E+02 1.0E+00

11 21 2.053E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.312E+02 0.0E+00

12 15 2.489E+02 0.000E+00 0.000E+00 0.000E+00 4.092E+05 0.000E+00 0.000E+00 0.0E+00

12 16 2.174E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.747E+00 0.0E+00

12 17 2.172E+02 3.724E+09 1.975E-02 1.130E-01 0.000E+00 0.000E+00 1.709E+01 1.0E+00

12 18 2.064E+02 0.000E+00 0.000E+00 0.000E+00 1.645E+05 3.238E-01 0.000E+00 0.0E+00

12 19 2.063E+02 0.000E+00 0.000E+00 0.000E+00 1.028E+06 9.138E-01 0.000E+00 0.0E+00

12 20 2.055E+02 5.869E+09 3.715E-02 2.011E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00

12 21 2.055E+02 1.644E+11 1.300E+00 7.036E+00 0.000E+00 0.000E+00 1.215E+03 1.0E+00

13 14 2.130E+03 3.818E+08 2.596E-01 3.640E+00 0.000E+00 0.000E+00 0.000E+00 1.0E+00

13 15 2.018E+03 4.496E+08 5.489E-01 7.292E+00 0.000E+00 0.000E+00 1.219E-02 1.0E+00

13 16 9.271E+02 0.000E+00 0.000E+00 0.000E+00 1.251E+04 7.986E-05 0.000E+00 0.0E+00

13 17 9.237E+02 0.000E+00 0.000E+00 0.000E+00 1.275E+04 0.000E+00 0.000E+00 0.0E+00

13 18 7.554E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.547E-10 0.0E+00

14 15 3.837E+04 0.000E+00 0.000E+00 0.000E+00 1.072E-04 1.591E-01 0.000E+00 0.0E+00

14 16 1.642E+03 8.513E+08 6.881E-01 7.438E+00 0.000E+00 0.000E+00 1.394E-03 1.0E+00

14 17 1.631E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.025E-02 0.0E+00

14 18 1.171E+03 0.000E+00 0.000E+00 0.000E+00 3.440E+03 0.000E+00 0.000E+00 0.0E+00

15 16 1.715E+03 1.497E+08 6.604E-02 1.492E+00 0.000E+00 0.000E+00 0.000E+00 1.0E+00

15 17 1.703E+03 9.178E+08 5.989E-01 1.343E+01 0.000E+00 0.000E+00 4.525E-02 1.0E+00

15 18 1.207E+03 0.000E+00 0.000E+00 0.000E+00 8.438E+02 2.063E-05 0.000E+00 0.0E+00

15 19 1.205E+03 0.000E+00 0.000E+00 0.000E+00 3.832E+03 0.000E+00 0.000E+00 0.0E+00

15 20 1.178E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.294E-11 0.0E+00

16 17 2.476E+05 0.000E+00 0.000E+00 0.000E+00 2.854E-09 7.108E-04 0.000E+00 0.0E+00

16 18 4.077E+03 5.652E+07 2.113E-01 1.134E+01 0.000E+00 0.000E+00 8.934E-05 1.0E+00

16 19 4.053E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.178E-04 0.0E+00

16 20 3.757E+03 0.000E+00 0.000E+00 0.000E+00 5.970E+00 0.000E+00 0.000E+00 0.0E+00

17 18 4.145E+03 3.840E+06 9.891E-03 8.099E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00

17 19 4.121E+03 5.866E+07 1.991E-01 1.621E+01 0.000E+00 0.000E+00 7.473E-04 1.0E+00

17 20 3.815E+03 0.000E+00 0.000E+00 0.000E+00 6.141E-01 1.040E-07 0.000E+00 0.0E+00

17 21 3.802E+03 0.000E+00 0.000E+00 0.000E+00 6.250E+00 0.000E+00 0.000E+00 0.0E+00

18 19 6.961E+05 0.000E+00 0.000E+00 0.000E+00 6.356E-12 3.427E-05 0.000E+00 0.0E+00

18 20 4.787E+04 2.055E+04 9.413E-03 8.900E+00 0.000E+00 0.000E+00 5.987E-10 1.0E+00

18 21 4.590E+04 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.861E-10 0.0E+00

19 20 5.141E+04 6.143E+02 2.434E-04 3.295E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00

19 21 4.914E+04 1.970E+04 8.915E-03 1.154E+01 0.000E+00 0.000E+00 2.546E-09 1.0E+00

20 21 1.117E+06 0.000E+00 0.000E+00 0.000E+00 1.976E-13 8.610E-06 0.000E+00 0.0E+00

Page 27 of 29



For Review Only

Table 4. Comparison of computed wavelengths and transition probabilities of Cr XIV with

the data available in NIST and other references. (aE±b = a× 10±𝑏).

i j

λij (in Å) Aji (in s-1)

MCDF NIST and other

references MCDF

NIST and other

references

1 2 4.105E+02 4.120E+02 5.469E+09 5.370E+09

4.121E+02a 5.286E+09d

4.109E+02c 5.139E+09e

4.084E+02d

4.120E+02f

1 3 3.886E+02 3.899E+02 6.484E+09 6.410E+09

3.899E+02a 6.319E+09d

3.889E+02c 6.115E+09e

3.857E+02d

3.898E+02f

1 7 6.366E+01 6.354E+01 1.125E+11 1.130E+11

6.332E+01d 1.222E+11d

1 8 6.345E+01 6.332E+01 1.068E+11 1.070E+11

6.309E+01d 1.170E+11d

1 14 4.662E+01 4.653E+01 6.549E+10 6.700E+10

1 15 4.656E+01 4.647E+01 6.282E+10 6.600E+10

2 4 2.894E+02 2.897E+02 1.319E+10 1.310E+10

2.894E+02c 1.294E+10d

2.867E+02d

2 6 8.120E+01 8.092E+01 6.705E+10 7.000E+10

8.077E+01d 7.099E+10d

6.996E+10e

2 9 6.879E+01 6.859E+01 1.904E+11 1.980E+11

6.837E+01d 1.986E+11d

2 13 5.393E+01 5.376E+01 2.964E+10 3.000E+10

2 16 5.096E+01 5.081E+01 1.156E+11 1.200E+11

3 4 3.014E+02 3.018E+02 2.339E+09 2.300E+09

3.013E+02c 2.292E+09d

2.991E+02d

3 5 2.999E+02 3.003E+02 1.426E+10 1.410E+10

2.999E+02c 1.396E+10d

2.974E+02d

3 6 8.212E+01 8.184E+01 1.375E+11 1.300E+11

8.172E+01d 1.427E+11d

1.435E+11e

3 9 6.945E+01 6.925E+01 3.887E+10 3.800E+10

6.906E+01d 3.984E+10d

3 10 6.941E+01 6.921E+01 2.320E+11 2.310E+11

6.901E+01d 2.390E+11d

3 13 5.433E+01 5.416E+01 6.061E+10 5.900E+10

3 17 5.131E+01 5.116E+01 1.397E+11 1.400E+11

4 7 1.019E+02 1.014E+02 4.896E+10 4.830E+10

1.014E+02b 5.101E+10d

1.015E+02d

4 8 1.013E+02 1.009E+02d 4.746E+09 4.940E+09d

4 11 8.647E+01 8.606E+01 5.472E+11 5.300E+11

8.642E+01c 5.404E+11d

8.587E+01d

4 18 6.093E+01 6.070E+01 2.051E+11 2.050E+11

5 8 1.015E+02 1.011E+02 4.285E+10 4.400E+10

1.011E+02b 4.474E+10d

1.011E+02d

5 11 8.659E+01 8.600E+01d 3.903E+10 3.850E+10d

5 12 8.657E+01 8.616E+01 5.856E+11 5.900E+11

8.652E+01c 5.780E+11d

8.598E+01d

5 15 6.423E+01 6.401E+01 1.670E+10 1.700E+10

Page 28 of 29



For Review Only

Table 4 (continued)

i j

λij (in Å) Aji (in s-1)

MCDF NIST and other

references MCDF

NIST and other

references

5 19 6.098E+01 6.076E+01 2.193E+11 2.190E+11

6 7 1.045E+03 1.038E+03d 1.223E+09 1.245E+09d

6 8 9.900E+02 9.805E+02d 1.443E+09 1.477E+09d

7 9 7.907E+02 7.809E+02d 2.773E+09 2.860E+09d

8 9 8.254E+02 8.168E+02d 4.888E+08 4.995E+08d

8 10 8.202E+02 8.106E+02d 2.991E+09 3.065E+09d

9 18 1.876E+02 1.870E+02 9.016E+10 9.300E+10

10 19 1.878E+02 1.873E+02 9.661E+10 9.600E+10

11 20 2.054E+02 2.049E+02 1.586E+11

12 21 2.055E+02 2.050E+02 1.644E+11

a- Peacock et al. [3], b-Fawcett et al. [7], c-Reader et al. [4], d- Fischer et al. [9], e- Johnson et al.

[10], f- Verner et al. [13] and the values without superscript in columns IV and VI is the data taken

from NIST.

Page 29 of 29



for review only...moreover, transition energies for the transitions 3s 2s 1/2 – 3p 2p 3/2,1/2....

Documents