# Spectroscopic data of Rb-isoelectronic Zr and Nb ions for astrophysical
applications

Jyoti,a Mandeep Kaur,a Bindiya Aroraa and B. K. Sahoob
aDepartment of Physics, Guru Nanak Dev University, Amritsar, Punjab-143005,
India
bAtomic, Molecular and Optical Physics Division, Physical Research Laboratory,
Navrangpura, Ahmedabad-380009, India bindiya.phy<EMAIL_ADDRESS>
###### Abstract

We present high-accuracy spectroscopy data of line strengths, transition
probabilities and oscillator strengths for the allowed transitions among the
$nS_{1/2}$, $nP_{1/2,3/2}$ and $n^{\prime}D_{3/2,5/2}$ states with $n=5$ to
$10$ and $n^{\prime}=4$ to $10$ of the Rb-isoelectronic Zr (Zr IV) and Nb (Nb
V) ions. They can serve to analyse various astrophysical phenomena undergoing
inside the heavenly bodies containing Zr and Nb elements. Since there is a
lack of precise observational and calculated data for the spectroscopic
properties in the above ions, their accurate determinations are of immense
interest. The literature data, that are available only for a few selected low-
lying transitions, have large discrepancies and they cannot be used reliably
for the above purpose. After accounting for electron interactions through
random phase approximation, Brückner orbitals, structural radiations and
normalizations of wave functions in the relativistic many-body methods, we
have evaluated the electric dipole amplitudes precisely. Combining these
values with the observed wavelengths, the above transition properties and
lifetimes of a number of excited states of the Zr IV and Nb V ions are
determined and compared with the literature data.

###### keywords:

Astronomical Data bases,Physical Data and Processes,Astronomical Data bases

##  1 Introduction

The emission spectra of heavenly bodies has been of great interest since last
few decades so as to get the better insight of their atmosphere, chemical
composition and evolution. A wide variety of analysis have been carried out
globally to provide the data for abundance of various spectral lines in these
bodies, however, many of them are still unknown. A diversity of elements and
ions have been observed in these spectra out of which zirconium (Zr) and
niobium (Nb) are of interest to us, as they play important roles in the decay
processes (Nilsson et al., 2010; García-Hernández et al., 2007), which are
generally the combination of slow (s) and rapid (r) neutron-capture
nucleosynthesis processes and are specifically responsible for the detection
of spectral lines of lighter elements. The presence of Nb and its ions has
been observed in the atmosphere of three metal-poor stars HD $209621$, HD
$218732$ and HD $232078$, the standard star Arcturus (Začs et al., 2011), Sun
and AGB M and MS stars (Nilsson et al., 2010; Palme et al., 2014). The
radionuclide of 92Nb decays to 92Zr by neutron-capture process with a half-
life of $37$ Ma (Holden, 1989). The presence of live 92Nb in early Solar
System was first obtained from the iron meteorite Toluca (Harper Jr, 1996),
which further offered a unique opportunity to estimate the timescale of the
early Solar System (Iizuka et al., 2016). Heavy element stars such as ROA
$371$, ROA $5293$, and ROA $3812$ in the globular cluster Omega Centauri
consist of a considerable abundance of s-process elements; i.e. Rb, Y and Zr
(Vanture et al., 1994). Besides, the presence of Zr IV lines has been detected
in the University College London Echelle Spectrograph (UCLES) of a He-rich hot
sdB star, LS IV-$14\degree 116$ on the Anglo-Australian Telescope (AAT)
(Jeffery et al., 2011), in the Far Ultraviolet Spectroscopic Explorer (FUSE)
spectra of hot subdwarf B stars (sdB) (Chayer et al., 2006) and in the UV
spectra of hot white dwarfs, G$191$–B$2$B and RE$0503$–$289$ (Rauch et al.,
2017). The spectral properties obtained from measurements and calculations are
the prerequisites of stellar-atmosphere modeling (Rauch et al., 2016).

The stellar-modeling aspires the determination of chemical abundances and
energy transport through a star, which requires the knowledge of reliable
values of oscillator strengths as well as transition probabilities (Martin et
al., 1992) of the emission spectra from the elements present in the stars.
Accurate values of absorption oscillator strengths as well as transition
probabilities are needed for correctly modeling and analysing the stellar
intensities of the lines so as to infer fundamental stellar parameters
(Ruffoni et al., 2014) such as mass M, radius R and luminosity L of any star
(Wittkowski, 2005). The information regarding the oscillator strengths and
transition probabilities are also useful in the analysis of interstellar and
quasar absorption of lines as well as the photospheric abundance of the
considered element in a star (Rauch et al., 2017), the construction of kinetic
models of plasma processes and for the investigation of processes in
thermonuclear reactor plasma (Glushkov et al., 1996; Tayal, 2012). They are
also needed for the estimation of the electron collisional rate coefficients
and photoionization cross-sections so as to explain various scattering
phenomena (Griem & (Firm), 1974; Zeippen, 1995; Orban et al., 2006). The
precise evaluation of line strengths is also useful for the assessment of the
Stark Broadenings of spectral lines, which is pivotal for the analysis of
astrophysical phenomena (Alonso-Medina & Colón, 2014). Due to this fact, the
determination of radiative properties of lines present in the emission spectra
of stars has become of sheer importance in the astrophysical studies.

To our knowledge, precise observational data of the spectroscopic properties
of Rb-isoelectronic Zr (Zr IV) and Nb (Nb V) ions are not available in the
literature. Limited theoretical results have been reported on the properties
of these ions, but they are only for a few low-lying transitions. The data for
transition probabilities, oscillator strengths, lifetimes, and branching
ratios for Zr IV and Nb V were first provided by Lindgard and Nielsen in 1977
by using numerical Coulomb approximation (Lindgård & Nielsen, 1977). Later on,
in 1979, Migdalek and Baylis studied oscillator strengths for the lowest
$5S_{1/2}\rightarrow 5P_{1/2,3/2}$ transitions for Rb-isoelectronic series,
which included the data for Zr IV and Nb V, using relativistic single-
configuration Hartree-Fock method and investigated the roles of core
polarisation effects for their accurate determination (Migdalek & Baylis,
1979). Karwowski and Szulkin, in 1981, evaluated the ground state energies and
also determined the oscillator strengths for the $5S\rightarrow 5P$
transitions of Zr IV using the modified relativistic Hartree-Fock method
(Karwowski & Szulkin, 1981). Sen and Puri had calculated dipole oscillator
strengths for these transitions in 1989 using the Local Spin Density
approximation and compared them with the then available experimental and
theoretical data (Sen & Puri, 1989). A few years later, in 1996, Glushkov et
al. determined oscillator strengths for the $5S\rightarrow 5P$, $5P\rightarrow
nD$ (n=$4,5$) and $4P\rightarrow 4F$ transitions of Nb V ion using a non-
relativistic method with the Coulomb screened potential approximation
(Glushkov et al., 1996). Zilitis had also evaluated transition probabilities
of the $5S_{1/2}$–$5P_{1/2,3/2}$, $4D_{3/2}$–$5P_{1/2,3/2}$, and
$4D_{3/2}$–$4F_{5/2}$ transitions as well as the lifetimes of the $5P_{1/2}$
and $5P_{3/2}$ states of both the Zr IV and Nb V ions using the Dirac-Fock
(DF) method in 2007 (Zilitis, 2007) and later, in 2009, he extended the
calculations for the oscillator strengths of the $4D_{3/2}$–$nP_{1/2,3/2}$ and
$4D_{3/2}$–$nF_{5/2}$ transitions of both the ions (Zilitis, 2009). In 2016,
transition energies for the $5S$–$5P$, $5P$–$4D$, $4D$–$4F$ and $5D$–$4F$
transitions of these ions were evaluated by Migdalek on the basis of model
potential in the DF method (Migdalek, 2016). Das et.al. studied spectroscopic
properties of a few Rb-like ions including Zr IV and Nb V ions using the
relativistic coupled-cluster method in 2017 (Das et al., 2017). They
calculated the matrix elements due to electric and magnetic multipole
operators, lifetimes and oscillator strengths for different transitions and
studied correlation behaviors in the Rb-isoelectronic series from Y III
through Tc VII ions.

On account of evidence of presence of Zr and Nb in astrophysical bodies, it is
necessary to seek through the radiative data of as many as spectral lines. As
mentioned above, these data are mainly available on the
$ns^{2}S_{1/2}\rightarrow np^{2}P_{1/2,3/2}$, $n=4,5$, $5p^{2}P\rightarrow
nd^{2}D$, $(n=4,5)$, $4p^{2}P\rightarrow 4f^{2}F$, $4D_{3/2}$–$nP_{1/2,3/2}$
and $4D_{3/2}$–$nF_{5/2}$ transitions only in the above ions. In this work, we
extend the calculations on the spectroscopic data for a large number of
transitions including high-lying lines. In particular, we have determined the
allowed electric dipole (E1) matrix elements for the transitions among the
$nS_{1/2},nP_{1/2,3/2}$, and $n^{\prime}D_{3/2,5/2}$ states with $n=5$ to $10$
and $n^{\prime}=4$ to $10$ of the Zr IV and Nb V ions except for a few
transitions which were not gauge invariant. Combining these data with the
observed transition wavelengths, we have estimated the oscillator strengths,
transition probabilities and radiative lifetimes of a number of excited states
precisely. Moreover, we have calculated energies of a few low-lying and
excited states for both the ions and compared them with the values listed in
the National Institute of Science and Technology atomic database (NIST AD)
(Ralchenko et al., 2008). The aforementioned data can be immensely useful for
analysing astrophysical processes; especially in the metal-poor stars HD
$209621$, HD $218732$ and HD $232078$, Arcturus and white dwarfs.

The paper is organized as follows: In Sec. 2, we provide the theoretical
formulae for the E1 matrix elements, transition probabilities, oscillator
strengths and lifetime of the atomic states. Sec. 3 discusses all the acquired
data from present work and compares with the previously reported values, while
findings are concluded in Sec. 4. All the results are given in atomic units
(a.u.) unless stated otherwise.

##  2 Theoretical aspects

###  2.1 Formulae for spectroscopic quantities

Transitions among the atomic states are dominantly driven by the E1 channel
when allowed. Here, we analyze the transition properties only due to the E1
channel as electrons can decay from the excited states of the ions of our
interest mainly through this channel. The transition probability due to the E1
channel ($A^{E1}_{vk}$) from an upper level $|\Psi(v)\rangle$ to a lower level
$|\Psi(k)\rangle$ with the corresponding angular momentum $J_{v}$ and $J_{k}$
respectively is given (in $s^{-1}$) by (Kelleher & Podobedova, 2008)

$A^{E1}_{vk}=\frac{2}{3}\alpha
c\pi\sigma\left(\frac{\alpha\sigma}{R_{\infty}}\right)^{2}\frac{S^{E1}}{g_{v}},$
(1)

where $R_{\infty}=\frac{\alpha^{2}m_{e}c}{2h}$ is the Rydberg constant with
the Planck’s constant $h$ and mass of electron $m_{e}$, $\alpha$ is fine
structure constant $\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}$, $c$ is the
speed of light, $\sigma=E_{v}-E_{k}$ is the excitation energy of the
transition, $S^{E1}$ is the line strength and $g_{v}=2J_{v}+1$. Here,
$S^{E1}=|\langle J_{v}||\textbf{D}||J_{k}\rangle|^{2}$ (Nahar, 1995) with the
E1 operator $\textbf{D}=\Sigma_{j}\textbf{d}_{j}=-e\Sigma_{j}\textbf{r}_{j}$
for the $j^{th}$ electron being at position $\textbf{r}_{j}$. On substituting
the values of fundamental constants as $\alpha=7.297352\times 10^{-3}$,
$R_{\infty}=1.0973731\times 10^{5}$ cm-1 and $c=29979245800$ cm s-1 (Mohr et
al., 2016), the above formula yields (Kelleher & Podobedova, 2008)

$A^{E1}_{vk}=\frac{2.0261269\times 10^{18}}{\lambda^{3}g_{v}}S^{E1},$ (2)

where $\lambda$ is the wavelength of transition in Å.

The absorption oscillator strengths of the allowed transitions ($f^{E1}_{kv}$)
are very useful in the astrophysical analyses, which can be determined from
the lower $|\Psi(k)\rangle$ level to the upper $|\Psi(v)\rangle$ level using
its transition probability which follows as (Kelleher & Podobedova, 2008;
Sobelman, 1979; Kaur et al., 2020)

$\displaystyle f_{kv}^{E1}$ $\displaystyle=$
$\displaystyle\left(\frac{R_{\infty}}{2c\alpha^{3}\pi}\right)\frac{g_{v}}{g_{k}}\times\frac{A_{vk}^{E1}}{\sigma^{2}}$
(3) $\displaystyle=$
$\displaystyle\frac{1}{3\alpha}\left(\frac{\alpha\sigma}{R_{\infty}}\right)\times\frac{S^{E1}}{g_{k}}$
$\displaystyle=$ $\displaystyle\frac{303.756}{g_{k}\lambda}\times S^{E1}.$

The radiative lifetime ($\tau$) of the excited state$|\Psi(v)\rangle$ can also
be obtained by taking reciprocal of the total transition probabilities of all
the lower possible transitions from that level (Qin et al., 2019; Kaur et al.,
2020); i.e.

$\tau_{v}=\frac{1}{\sum_{k}A_{vk}},$ (4)

where sum over $k$ denotes all possible states ($|\Psi(k)\rangle$) to which
allowed transitions from $|\Psi(v)\rangle$ are possible.

###  2.2 Methods for calculations

The presence of the two-body electromagnetic interactions among the electrons
within an atomic system poses challenge to solve atomic wave functions
accurately. In a typical approach, atomic wave functions are obtained by
calculating them using a mean-field approach then incorporating the electron
correlation effects due to the neglected interactions systematically. These
correlation effects are classified into various physical effects such as core-
polarization and pair-correlation effects, which behave differently for
accurate description of wave functions depending upon the electronic
configurations of atomic systems. Large differences among the previous
calculations of transition properties in the considered ions using a variety
of many-body methods (Lindgård & Nielsen, 1977; Migdalek & Baylis, 1979;
Karwowski & Szulkin, 1981; Glushkov et al., 1996; Zilitis, 2007; Das et al.,
2017) suggest that the roles of the electron correlation effects are
significant and they should be accounted for meticulously in order to obtain
reliable results. In view of this, we include the electron correlations due to
the core-polarization effects through random-phase approximation (RPA), pair-
correlation effects through the Brückner orbitals (BOs) and their couplings
through the structural radiations (SRs) in the determination of atomic wave
functions. Corrections in the results due to normalization of the wave
functions (Norms) are also estimated explicitly. A brief description of the
procedures adopted to incorporate the above physical effects in the
calculations is given below.

We first evaluate the mean-field wave function ($|\Phi_{0}\rangle$) of the
$[4p^{6}]$ configuration of the considered ions using the DF method in which
the atomic Hamiltonian is expressed as $H=H_{0}+V_{I}$ with the DF Hamiltonian
$H_{0}$ and residual interaction $V_{I}$, given in atomic units (a.u.) by

$H_{0}=\sum_{i}\epsilon_{i}a_{i}^{{\dagger}}a_{i}$ (5)

and

$V_{I}=\frac{1}{2}\sum_{ijkl}g_{ijkl}a_{i}^{{\dagger}}a_{j}^{{\dagger}}a_{l}a_{k}-\sum_{ij}u_{ij}^{DF}a_{i}^{{\dagger}}a_{j},$
(6)

where the sums over $i,j,k$ and $l$ include all electron orbitals,
$\epsilon_{i}$ are the eigenvalues of the one-electron DF orbitals, $g_{ijkl}$
is the two-body matrix element of the Coulomb interaction ($\frac{1}{r_{ij}}$)
and $u_{ij}^{DF}$ is the DF potential (Blundell et al., 1987; Johnson et al.,
1996; Safronova et al., 2005). The working DF wave functions
($|\Phi_{v}\rangle$) of the interested states of both Zr IV and Nb V are
obtained by appending the respective valence orbital $v$ to $|\Phi_{0}\rangle$
for the configurations $[4p^{6}]v$; i.e.
$|\Phi_{v}\rangle=a_{v}^{{\dagger}}|\Phi_{0}\rangle$. The choice this
$V^{N-1}$ DF potential with ($N$ number of electrons of the system) is to
facilitate for calculating as many as states having common closed core
$[4p^{6}]$. The neglected core-valence effects at this level are included as
corrections in the next level.

The corrections over the DF wave functions due to the electron correlation
effects are estimated using the perturbative analysis of $V_{I}$ by expressing
the exact wave function of the state ($|\Psi_{v}\rangle$) in the relativistic
many-body perturbation theory (RMBPT) analysis as

$\displaystyle|\Psi_{v}\rangle=|\Phi_{v}\rangle+|\Phi_{v}^{(1)}\rangle+|\Phi_{v}^{(2)}\rangle+\cdots$
(7)

and its energy $(E_{v}$) as

$\displaystyle E_{v}=E_{v}^{(0)}+E_{v}^{(1)}+E_{v}^{(2)}+\cdots,$ (8)

where superscripts $k=1,2$ etc. denote order of perturbation and the zeroth-
order energy is $E_{v}^{(0)}=\sum_{k}^{N}\epsilon_{k}$. After obtaining the
wave functions, the E1 matrix element between the states $|\Psi_{v}\rangle$
and $|\Psi_{w}\rangle$ is calculated as

$D_{wv}=\frac{\left\langle\Psi_{w}|D|\Psi_{v}\right\rangle}{\sqrt{\left\langle\Psi_{w}|\Psi_{w}\right\rangle\left\langle\Psi_{v}|\Psi_{v}\right\rangle}}.$
(9)

As mentioned above contributions from the perturbative corrections are
categorized into RPA, BO, SR and Norm contributions by expressing (Blundell et
al., 1987; Johnson et al., 1996)

$\displaystyle
D_{wv}=D^{DF}_{wv}+D^{RPA}_{wv}+D^{BO}_{wv}+D^{SR}_{wv}+D^{Norm}_{wv},$ (10)

where $D^{DF}_{wv}\equiv d_{wv}=\langle\phi_{w}|d|\phi_{v}\rangle$ with the DF
single particle wave functions $|\phi_{v}\rangle$ and $|\phi_{w}\rangle$.
Since core-polarization effects contribute significantly, they are included
through RPA self-consistently to all-orders by defining (Johnson, 2007;
Johnson et al., 1996):

$\displaystyle
D_{wv}^{RPA}=\sum_{\zeta=1}^{\infty}\sum_{na}\left[\frac{d_{an}^{(\zeta)}\tilde{g}_{wnva}}{\epsilon_{v}+\epsilon_{a}-\epsilon_{n}-\epsilon_{w}}+\frac{\tilde{g}_{wavn}d^{(\zeta)}_{na}}{\epsilon_{w}+\epsilon_{a}-\epsilon_{n}-\epsilon_{v}}\right]$
(11)

for $\tilde{g}_{ijkl}=g_{ijkl}-g_{ijlk}$ and superscript $\zeta$ represents
iteration number with

$\displaystyle d_{an}^{(\zeta)}$ $\displaystyle=$ $\displaystyle
d_{an}+\sum_{bm}\left[\frac{d_{bm}^{(\zeta-1)}\tilde{g}_{amnb}}{\epsilon_{v}+\epsilon_{b}-\epsilon_{m}-\epsilon_{w}}+\frac{\tilde{g}_{abnm}d_{mb}^{(\zeta-1)}}{\epsilon_{w}+\epsilon_{b}-\epsilon_{m}-\epsilon_{v}}\right],$

and

$\displaystyle d_{na}^{(\zeta)}$ $\displaystyle=$ $\displaystyle
d_{na}+\sum_{bm}\left[\frac{d_{bm}^{(\zeta-1)}\tilde{g}_{nmab}}{\epsilon_{v}+\epsilon_{b}-\epsilon_{m}-\epsilon_{w}}+\frac{\tilde{g}_{nbam}d_{mb}^{(\zeta-1)}}{\epsilon_{w}+\epsilon_{b}-\epsilon_{m}-\epsilon_{v}}\right].$

The initial values correspond to $d^{(0)}_{an}=0$ and $d^{(1)}_{an}=d_{an}$.
The $a,b,...$, $m,n,...$ and $w,v$ indices in the subscripts denote for the
occupied, virtual and valence orbitals, respectively.

The leading-order electron correlation contributions through BO and SR arise
at the third-order perturbation level. We include these contributions using
the procedure followed by Johnson et al. in (Johnson et al., 1996) in the
present calculations. The Norm contributions have been estimated by adopting
the approach discussed in (Blundell et al., 1987) and approximating the atomic
wave function at the second-order perturbation theory.

It is obvious from the above discussion that our procedure incorporates
various physical effects due to the electron correlation effects that are
complete through the third-order perturbation and core-polarization effects to
all-orders. To verify reliability in the calculations of the E1 matrix
elements and estimate their uncertainties, we use both the length (L) and
velocity gauge (V) expressions in the relativistic form (e.g. see (Kaur et
al., 2020)). The differences in the results from both the gauge forms can be
safely used as the maximum uncertainties associated with our calculated E1
matrix elements. Since calculations with length gauge expression converge
faster with respect to the number of configurations, we believe that results
from the length gauge expression are more reliable. Thus, we consider these
results as the central values. In a few transitions, it is noticed that the
length gauge and velocity gauge results differ by more than 50%. In such
cases, it is possible to improve calculations by considering higher-order
contributions through BO and SR effects, but they are computationally more
expensive. Therefore, we do not endorse the results obtained using above
method for such few cases and do not present them in this work.

Table 1:  Our calculated energy values ($cm^{-1}$) for few low lying and excited states using RMBPT method for Zr IV and Nb V along with their comparison with experimental energies provided in NIST AD (Ralchenko et al., 2008). Ground state is taken at $0$ cm-1. State | Zr IV | Nb V
---|---|---
| Energy ($cm^{-1}$) | NIST ($cm^{-1}$) | Energy ($cm^{-1}$) | NIST ($cm^{-1}$)
$4D_{5/2}$ | $1109.05$ | $1250.70$ | $1741.83$ | $1867.40$
$5S_{1/2}$ | $34572.42$ | $38258.35$ | $72113.24$ | $75929.60$
$5P_{1/2}$ | $76313.25$ | $81976.50$ | $123581.01$ | $129195.20$
$5P_{3/2}$ | $78607.03$ | $84461.35$ | $126972.64$ | $132800.00$
$5D_{3/2}$ | $138820.28$ | $146652.40$ | $203904.11$ | $211694.00$
$5D_{5/2}$ | $139158.53$ | $147002.46$ | $204434.63$ | $212238.40$
$6S_{1/2}$ | $143994.52$ | $152513.00$ | $219451.86$ | $228496.30$
$6P_{1/2}$ | $160898.86$ | $169809.71$ | $241153.11$ | $250506.50$
$6P_{3/2}$ | $161848.63$ | $170815.11$ | $242602.26$ | $252023.3$
$6D_{3/2}$ | $188289.90$ | $197765.10$ | $277261.63$ | $287163.60$
$6D_{5/2}$ | $188450.91$ | $197930.43$ | $277515.77$ | $287425.60$
$7S_{1/2}$ | $190309.74$ | $200123.69$ | $284180.44$ | $294736.00$
$7P_{1/2}$ | $198869.74$ | $208783.36$ | $295423.65$ | $305986.50$
$7P_{3/2}$ | $199358.47$ | $209297.66$ | $296185.80$ | $306788.10$
$7D_{3/2}$ | $213460.26$ |  | $314942.71$ | $325705.70$
$7D_{5/2}$ | $213550.09$ |  | $315085.29$ | $325854.80$
$8S_{1/2}$ | $214494.56$ | $224813.48$ | $318702.16$ | $329884.20$

Table 2: The line strengths ($S_{vk}$) (in a.u.) from both the L and V gauge expressions, wavelengths ($\lambda$) (in Å), transition probabilities ($A_{vk}$) in ($s^{-1}$) and absorption oscillator strengths ($f_{kv}$) for the Zr IV ion through E1 decay channel are presented in this table. Values in square brackets represent the order of 10. Uncertainties are given in parentheses. Upper State(v) | Lower State(k) |  $\lambda$ (in Å) |  $S_{vk}$ (in a.u.) |  $A_{L_{vk}}$(in $s^{-1}$) | $f_{L_{kv}}$
---|---|---|---|---|---
|  |  | L | V |  |
$5P_{1/2}$ | $4D_{3/2}$ | $1219.86$ | $2.04[0]$ | $2.37[0]$ | $1.14(18)[9]$ | $1.27(20)[-1]$
$5P_{1/2}$ | $5S_{1/2}$ | $2287.38$ | $4.69[0]$ | $5.06[0]$ | $3.97(30)[8]$ | $3.12(24)[-1]$
$5P_{3/2}$ | $4D_{3/2}$ | $1183.97$ | $3.93[-1]$ | $4.55[-1]$ | $1.20(18)[8]$ | $2.52(38)[-2]$
$5P_{3/2}$ | $4D_{5/2}$ | $1201.77$ | $3.64[0]$ | $4.21[0]$ | $1.06(16)[9]$ | $1.54(23)[-1]$
$5P_{3/2}$ | $5S_{1/2}$ | $2164.36$ | $9.40[0]$ | $1.02[1]$ | $4.70(39)[8]$ | $6.60(54)[-1]$
$5D_{3/2}$ | $5P_{1/2}$ | $1546.17$ | $1.14[1]$ | $1.20[1]$ | $1.56(8)[9]$ | $1.12(6)[0]$
$5D_{3/2}$ | $5P_{3/2}$ | $1607.95$ | $2.40[0]$ | $2.51[0]$ | $2.92(14)[8]$ | $1.13(5)[-1]$
$5D_{5/2}$ | $5P_{3/2}$ | $1598.95$ | $2.13[1]$ | $2.25[1]$ | $1.76(9)[9]$ | $1.01(5)[0]$
$6S_{1/2}$ | $5P_{1/2}$ | $1417.71$ | $1.82[0]$ | $1.90[0]$ | $6.47(27)[8]$ | $1.95(8)[-1]$
$6S_{1/2}$ | $5P_{3/2}$ | $1469.47$ | $4.06[0]$ | $4.21[0]$ | $1.30(5)[9]$ | $2.10(8)[-1]$
$6P_{1/2}$ | $4D_{3/2}$ | $588.89$ | $6.51[-2]$ | $7.46[-2]$ | $3.23(46)[8]$ | $8.4(1.2)[-3]$
$6P_{1/2}$ | $5S_{1/2}$ | $760.16$ | $4.33[-2]$ | $4.84[-2]$ | $1.00(11)[8]$ | $8.66(10)[-3]$
$6P_{1/2}$ | $5D_{3/2}$ | $4318.29$ | $1.57[1]$ | $1.63[1]$ | $1.97(8)[8]$ | $2.75(11)[-1]$
$6P_{1/2}$ | $6S_{1/2}$ | $5781.45$ | $1.83[1]$ | $1.91[1]$ | $9.61(40)[7]$ | $4.82(20)[-1]$
$6P_{3/2}$ | $4D_{3/2}$ | $585.42$ | $1.42[-2]$ | $1.61[-2]$ | $3.59(46)[7]$ | $1.84(24)[-3]$
$6P_{3/2}$ | $4D_{5/2}$ | $589.75$ | $1.29[-1]$ | $1.47[-1]$ | $3.20(42)[8]$ | $1.11(15)[-2]$
$6P_{3/2}$ | $5S_{1/2}$ | $754.39$ | $5.35[-2]$ | $6.10[-2]$ | $6.31(86)[7]$ | $1.08(15)[-2]$
$6P_{3/2}$ | $5D_{3/2}$ | $4138.61$ | $3.00[0]$ | $3.14[0]$ | $2.15(10)[7]$ | $5.51(25)[-2]$
$6P_{3/2}$ | $5D_{5/2}$ | $4199.45$ | $2.75[1]$ | $2.88[1]$ | $1.88(8)[8]$ | $3.32(15)[-1]$
$6P_{3/2}$ | $6S_{1/2}$ | $5463.85$ | $3.64[1]$ | $3.80[1]$ | $1.13(5)[8]$ | $1.01(5)[0]$
$6D_{3/2}$ | $5P_{1/2}$ | $863.64$ | $2.84[-1]$ | $2.86[-1]$ | $2.24(2)[8]$ | $5.00(4)[-2]$
$6D_{3/2}$ | $5P_{3/2}$ | $882.58$ | $5.01[-2]$ | $5.03[-2]$ | $3.69(1)[7]$ | $4.31(1)[-3]$
$6D_{3/2}$ | $6P_{1/2}$ | $3577.13$ | $3.40[1]$ | $3.50[1]$ | $3.76(11)[8]$ | $1.44(4)[0]$
$6D_{3/2}$ | $6P_{3/2}$ | $3710.58$ | $7.17[0]$ | $7.36[0]$ | $7.11(19)[7]$ | $1.47(4)[-1]$
$6D_{5/2}$ | $5P_{3/2}$ | $881.30$ | $4.73[-1]$ | $4.76]-1]$ | $2.34(1)[8]$ | $4.08(2)[-2]$
$6D_{5/2}$ | $6P_{3/2}$ | $3687.95$ | $6.39[1]$ | $6.56[1]$ | $4.30(12)[8]$ | $1.32(4)[0]$
$7S_{1/2}$ | $5P_{1/2}$ | $846.40$ | $1.73[-1]$ | $1.74[-1]$ | $2.89(2)[8]$ | $3.10(3)[-2]$
$7S_{1/2}$ | $5P_{3/2}$ | $864.59$ | $3.62[-1]$ | $3.63[-1]$ | $5.68(2)[8]$ | $3.18(1)[-2]$
$7S_{1/2}$ | $6P_{1/2}$ | $3298.81$ | $6.32[0]$ | $6.59[0]$ | $1.78(7)[8]$ | $2.91(12)[-1]$
$7S_{1/2}$ | $6P_{3/2}$ | $3411.97$ | $1.40[1]$ | $1.45[1]$ | $3.57(13)[8]$ | $3.12(12)[-1]$
$7P_{1/2}$ | $4D_{3/2}$ | $478.97$ | $1.47[-2]$ | $1.73[-2]$ | $1.35(23)[8]$ | $2.33(39)[-3]$
$7P_{1/2}$ | $5S_{1/2}$ | $586.42$ | $2.22[-2]$ | $2.38[-2]$ | $1.11(8)[8]$ | $5.74(41)[-3]$
$7P_{1/2}$ | $5D_{3/2}$ | $1609.50$ | $2.29[-1]$ | $2.18[-1]$ | $5.57(28)[7]$ | $1.08(5)[-2]$
$7P_{1/2}$ | $6S_{1/2}$ | $1777.13$ | $3.29[-2]$ | $3.84[-2]$ | $5.93(96)[6]$ | $2.81(46)[-3]$
$7P_{1/2}$ | $6D_{3/2}$ | $9075.84$ | $5.01[1]$ | $5.20[1]$ | $6.79(26)[7]$ | $4.19(16)[-1]$
$7P_{1/2}$ | $7S_{1/2}$ | $11547.78$ | $4.69[1]$ | $4.83[1]$ | $3.09(9)[7]$ | $6.17(17)[-1]$
$7P_{3/2}$ | $4D_{3/2}$ | $477.79$ | $3.36[-3]$ | $3.87[-3]$ | $1.56(23)[7]$ | $5.35(77)[-4]$
$7P_{3/2}$ | $4D_{5/2}$ | $480.66$ | $3.08[-2]$ | $3.56[-2]$ | $1.40(21)[8]$ | $3.24(49)[-3]$
$7P_{3/2}$ | $5S_{1/2}$ | $584.66$ | $3.27[-2]$ | $3.55[-2]$ | $8.28(70)[7]$ | $8.48(71)[-3]$
$7P_{3/2}$ | $5D_{3/2}$ | $1596.29$ | $5.21[-2]$ | $5.01[-2]$ | $6.49(26)[6]$ | $2.48(10)[-3]$
$7P_{3/2}$ | $5D_{5/2}$ | $1605.26$ | $4.58[-1]$ | $4.39[-1]$ | $5.61(23)[7]$ | $1.44(6)[-2]$
$7P_{3/2}$ | $6S_{1/2}$ | $1761.04$ | $2.24[-2]$ | $2.88[-2]$ | $2.08(56)[6]$ | $1.94(52)[-3]$
$7P_{3/2}$ | $6D_{3/2}$ | $8671.10$ | $9.62[0]$ | $1.00[1]$ | $7.47(33)[6]$ | $8.42(37)[-2]$
$7P_{3/2}$ | $6D_{5/2}$ | $8797.22$ | $8.79[1]$ | $9.18[1]$ | $6.54(28)[7]$ | $5.06(22)[-1]$
$7P_{3/2}$ | $7S_{1/2}$ | $10900.41$ | $9.27[1]$ | $9.56[1]$ | $3.62(12)[7]$ | $1.29(4)[0]$
$7D_{3/2}$ | $5P_{1/2}$ | $729.14$ | $4.52[-2]$ | $4.39[-2]$ | $5.91(17)[7]$ | $9.42(28)[-3]$
$7D_{3/2}$ | $5P_{3/2}$ | $741.55$ | $7.23[-3]$ | $6.96[-3]$ | $8.97(34)[6]$ | $7.40(28)[-4]$
$7D_{3/2}$ | $6P_{1/2}$ | $1902.54$ | $1.21[0]$ | $1.21[0]$ | $8.88(3)[7]$ | $9.64(3)[-2]$
$7D_{3/2}$ | $6P_{3/2}$ | $1937.55$ | $2.22[-1]$ | $2.22[-1]$ | $1.5449(7)[7]$ | $8.695(4)[-3]$
$7D_{3/2}$ | $7P_{1/2}$ | $6853.77$ | $7.55[1]$ | $7.71[1]$ | $1.19(2)[8]$ | $1.67(3)[0]$
$7D_{3/2}$ | $7P_{3/2}$ | $7091.30$ | $1.60[1]$ | $1.63[1]$ | $2.27(4)[7]$ | $1.71(3)[-1]$
$7D_{5/2}$ | $5P_{3/2}$ | $741.05$ | $7.07[-2]$ | $6.84[-2]$ | $5.87(19)[7]$ | $7.25(24)[-3]$
$7D_{5/2}$ | $6P_{3/2}$ | $1934.18$ | $2.06[0]$ | $2.06[0]$ | $9.59(1)[7]$ | $8.07(1)[-2]$
$7D_{5/2}$ | $7P_{3/2}$ | $7046.41$ | $1.42[2]$ | $1.45[2]$ | $1.37(3)[8]$ | $1.53(3)[0]$
$8S_{1/2}$ | $5P_{1/2}$ | $700.10$ | $5.41[-2]$ | $5.37[-2]$ | $1.60(1)[8]$ | $1.17(1)[-2]$
$8S_{1/2}$ | $5P_{3/2}$ | $712.49$ | $1.12[-1]$ | $1.10[-1]$ | $3.13(4)[8]$ | $1.19(1)[-2]$
$8S_{1/2}$ | $6P_{1/2}$ | $1818.06$ | $5.04[-1]$ | $5.15[-1]$ | $8.50(18)[7]$ | $4.21(9)[-2]$
$8S_{1/2}$ | $6P_{3/2}$ | $1851.91$ | $1.05[0]$ | $1.06[0]$ | $1.67(3)[8]$ | $4.30(7)[-2]$
$8S_{1/2}$ | $7P_{1/2}$ | $6238.26$ | $1.60[1]$ | $1.66[1]$ | $6.66(27)[7]$ | $3.88(16)[-1]$
$8S_{1/2}$ | $7P_{3/2}$ | $6445.03$ | $3.52[1]$ | $3.64[1]$ | $1.33(5)[8]$ | $4.14(15)[-1]$
$8P_{1/2}$ | $4D_{3/2}$ | $455.75$ | $5.34[-3]$ | $6.48[-3]$ | $5.7(1.2)[7]$ | $8.9(1.8)[-4]$
$8P_{1/2}$ | $5S_{1/2}$ | $540.99$ | $1.24[-2]$ | $1.30[-2]$ | $7.91(41)[7]$ | $3.47(18)[-3]$
$8P_{1/2}$ | $5D_{3/2}$ | $1240.73$ | $5.17[-2]$ | $4.77[-2]$ | $2.74(21)[7]$ | $3.16(25)[-3]$
$8P_{1/2}$ | $6S_{1/2}$ | $1325.84$ | $2.14[-2]$ | $2.38[-2]$ | $9.3(1.0)[6]$ | $2.45(27)[-3]$
$8P_{1/2}$ | $6D_{3/2}$ | $3212.52$ | $5.38[-1]$ | $5.14[-1]$ | $1.64(7)[7]$ | $1.27(6)[-2]$
$8P_{1/2}$ | $7S_{1/2}$ | $3435.43$ | $2.81[-2]$ | $3.37[-2]$ | $7.0(1.3)[5]$ | $1.24(24)[-3]$
$8P_{1/2}$ | $7D_{3/2}$ | $16784.45$ | $1.19[2]$ | $1.24[2]$ | $2.55(11)[7]$ | $5.38(23)[-1]$
$8P_{1/2}$ | $8S_{1/2}$ | $20310.35$ | $9.89[1]$ | $1.01[2]$ | $1.20(2)[7]$ | $7.40(15)[-1]$
$8P_{3/2}$ | $4D_{3/2}$ | $455.16$ | $1.28[-3]$ | $1.51[-3]$ | $6.9(1.2)[6]$ | $2.14(37)[-4]$
$8P_{3/2}$ | $4D_{5/2}$ | $457.47$ | $1.17[-2]$ | $1.39[-2]$ | $6.2(1.1)[7]$ | $1.30(23)[-3]$
$8P_{3/2}$ | $5S_{1/2}$ | $540.15$ | $1.92[-2]$ | $2.04[-2]$ | $6.16(39)[7]$ | $5.39(3)[-3]$
$8P_{3/2}$ | $5D_{3/2}$ | $1236.33$ | $1.20[-2]$ | $1.12[-2]$ | $3.21(21)[6]$ | $7.35(48)[-4]$
$8P_{3/2}$ | $5D_{5/2}$ | $1241.52$ | $1.05[-1]$ | $9.78[-2]$ | $2.77(19)[7]$ | $4.27(29)[-3]$
$8P_{3/2}$ | $6S_{1/2}$ | $1320.82$ | $2.46[-2]$ | $2.84[-2]$ | $5.41(80)[6]$ | $2.83(42)[-3]$
$8P_{3/2}$ | $6D_{3/2}$ | $3183.19$ | $1.26[-1]$ | $1.22[-1]$ | $1.98(7)[6]$ | $3.00(10)[-3]$
$8P_{3/2}$ | $6D_{5/2}$ | $3199.59$ | $1.10[0]$ | $1.06[0]$ | $1.69(6)[7]$ | $1.73(6)[-2]$
$8P_{3/2}$ | $7S_{1/2}$ | $3401.92$ | $6.29[-3]$ | $1.02[-2]$ | $8.1(4.4)[4]$ | $2.8(1.5)[-4]$
$8P_{3/2}$ | $7D_{3/2}$ | $16013.68$ | $2.29[1]$ | $2.39[1]$ | $2.82(13)[6]$ | $1.09(5)[-1]$
$8P_{3/2}$ | $7D_{5/2}$ | $16247.40$ | $2.09[2]$ | $2.18[2]$ | $2.47(11)[7]$ | $6.51(29)[-1]$
$8P_{3/2}$ | $8S_{1/2}$ | $19192.52$ | $1.95[2]$ | $1.99[2]$ | $1.40(3)[7]$ | $1.54(4)[0]$
$8D_{3/2}$ | $5P_{1/2}$ | $658.67$ | $1.31[-2]$ | $1.23[-2]$ | $2.32(14)[7]$ | $3.02(18)[-3]$
$8D_{3/2}$ | $5P_{3/2}$ | $668.77$ | $1.93[-3]$ | $1.79[-3]$ | $3.26(24)[6]$ | $2.19(16)[-4]$
$8D_{3/2}$ | $6P_{1/2}$ | $1487.30$ | $2.45[-1]$ | $2.41[-1]$ | $3.78(7)[7]$ | $2.51(4)[-2]$
$8D_{3/2}$ | $6P_{3/2}$ | $1508.61$ | $4.26[-2]$ | $4.17[-2]$ | $6.29(14)[6]$ | $2.15(5)[-3]$
$8D_{3/2}$ | $7P_{1/2}$ | $3417.04$ | $2.97[0]$ | $2.99[0]$ | $3.78(2)[7]$ | $1.32(1)[-1]$
$8D_{3/2}$ | $7P_{3/2}$ | $3475.08$ | $5.56[-1]$ | $5.57[-1]$ | $6.71(2)[6]$ | $1.215(3)[-2]$
$8D_{3/2}$ | $8P_{1/2}$ | $11472.29$ | $1.45[2]$ | $1.47[2]$ | $4.85(7)[7]$ | $1.91(3)[0]$
$8D_{3/2}$ | $8P_{3/2}$ | $11862.56$ | $3.07[1]$ | $3.11[1]$ | $9.31(12)[6]$ | $1.96(2)[-1]$
$8D_{5/2}$ | $5P_{3/2}$ | $668.52$ | $1.95[-2]$ | $1.83[-2]$ | $2.21(15)[7]$ | $2.22(15)[-3]$
$8D_{5/2}$ | $6P_{3/2}$ | $1507.35$ | $4.02[-1]$ | $3.93[-1]$ | $3.96(8)[7]$ | $2.02(4)[-2]$
$8D_{5/2}$ | $7P_{3/2}$ | $3468.39$ | $5.12[0]$ | $5.13[0]$ | $4.14(1)[7]$ | $1.121(3)[-1]$
$8D_{5/2}$ | $8P_{3/2}$ | $11785.04$ | $2.73[2]$ | $2.76[2]$ | $5.63(7)[7]$ | $1.76(2)[0]$
$9S_{1/2}$ | $5P_{1/2}$ | $655.99$ | $2.49[-2]$ | $2.45[-2]$ | $8.94(15)[7]$ | $5.77(10)[-3]$
$9S_{1/2}$ | $5P_{3/2}$ | $666.01$ | $5.10[-2]$ | $5.00[-2]$ | $1.75(4)[8]$ | $5.81(12)[-3]$
$9S_{1/2}$ | $6P_{1/2}$ | $1473.70$ | $1.46[-1]$ | $1.48[-1]$ | $4.64(5)[7]$ | $1.51(2)[-2]$
$9S_{1/2}$ | $6P_{3/2}$ | $1494.62$ | $3.00[-1]$ | $3.01[-1]$ | $9.09(5)[7]$ | $1.52(1)[-2]$
$9S_{1/2}$ | $7P_{1/2}$ | $3346.08$ | $1.16[0]$ | $1.19[0]$ | $3.12(8)[7]$ | $5.24(14)[-2]$
$9S_{1/2}$ | $7P_{3/2}$ | $3401.71$ | $2.39[0]$ | $2.44[0]$ | $6.15(13)[7]$ | $5.33(12)[-2]$
$9S_{1/2}$ | $8P_{1/2}$ | $10709.75$ | $3.34[1]$ | $3.46[1]$ | $2.75(11)[7]$ | $4.73(18)[-1]$
$9S_{1/2}$ | $8P_{3/2}$ | $10813.09$ | $7.34[1]$ | $7.58[1]$ | $5.88(19)[7]$ | $5.15(17)[-1]$
$9P_{1/2}$ | $4D_{3/2}$ | $431.37$ | $2.45[-3]$ | $3.05[-3]$ | $3.09(71)[7]$ | $4.31(99)[-4]$
$9P_{1/2}$ | $5S_{1/2}$ | $506.98$ | $7.59[-3]$ | $7.90[-3]$ | $5.90(24)[7]$ | $2.27(9)[-3]$
$9P_{1/2}$ | $5D_{3/2}$ | $1075.28$ | $2.05[-2]$ | $1.87[-2]$ | $1.67(15)[7]$ | $1.45(13)[-3]$
$9P_{1/2}$ | $6S_{1/2}$ | $1138.63$ | $1.19[-2]$ | $1.30[-2]$ | $8.14(79)[6]$ | $1.58(15)[-3]$
$9P_{1/2}$ | $6D_{3/2}$ | $2297.29$ | $1.12[-1]$ | $1.04[-1]$ | $9.33(69)[6]$ | $3.69(27)[-3]$
$9P_{1/2}$ | $7S_{1/2}$ | $2409.07$ | $2.49[-2]$ | $2.82[-2]$ | $1.80(24)[6]$ | $1.57(21)[-3]$
$9P_{1/2}$ | $7D_{3/2}$ | $5446.84$ | $1.04[0]$ | $1.01[0]$ | $6.51(20)[6]$ | $1.45(4)[-2]$
$9P_{1/2}$ | $8S_{1/2}$ | $5772.02$ | $2.38[-2]$ | $3.00[-2]$ | $1.26(31)[5]$ | $6.3(15)[-4]$
$9P_{1/2}$ | $8D_{3/2}$ | $27139.20$ | $2.40[2]$ | $2.51[2]$ | $1.21(6)[7]$ | $6.70(32)[-1]$
$9P_{1/2}$ | $9S_{1/2}$ | $32636.33$ | $1.85[2]$ | $1.89[2]$ | $5.40(9)[6]$ | $8.63(14)[-1]$
$9P_{3/2}$ | $4D_{3/2}$ | $431.02$ | $6.30[-4]$ | $7.56[-4]$ | $3.99(76)[6]$ | $1.11(21)[-4]$
$9P_{3/2}$ | $4D_{5/2}$ | $433.09$ | $5.49[-3]$ | $6.64[-3]$ | $3.42(68)[7]$ | $6.4(1.3)[-4]$
$9P_{3/2}$ | $5S_{1/2}$ | $506.49$ | $1.21[-2]$ | $1.27[-2]$ | $4.72(25)[7]$ | $3.63(19)[-3]$
$9P_{3/2}$ | $5D_{3/2}$ | $1073.09$ | $4.82[-3]$ | $4.44[-3]$ | $1.97(16)[6]$ | $3.41(28)[-4]$
$9P_{3/2}$ | $5D_{5/2}$ | $1077.00$ | $4.20[-2]$ | $3.87[-2]$ | $1.70(14)[7]$ | $1.98(16)[-3]$
$9P_{3/2}$ | $6S_{1/2}$ | $1136.17$ | $1.53[-2]$ | $1.73[-2]$ | $5.28(66)[6]$ | $2.05(26)[-3]$
$9P_{3/2}$ | $6D_{3/2}$ | $2287.32$ | $2.68[-2]$ | $2.52[-2]$ | $1.13(7)[6]$ | $8.90(54)[-4]$
$9P_{3/2}$ | $6D_{5/2}$ | $2295.78$ | $2.32[-1]$ | $2.17[-1]$ | $9.71(62)[6]$ | $5.11(33)[-3]$
$9P_{3/2}$ | $7S_{1/2}$ | $2398.11$ | $2.27[-2]$ | $2.74[-2]$ | $8.3(1.6)[5]$ | $1.44(28)[-3]$
$9P_{3/2}$ | $7D_{3/2}$ | $5391.15$ | $2.50[-1]$ | $2.45[-1]$ | $8.08(17)[5]$ | $3.52(7)[-3]$
$9P_{3/2}$ | $7D_{5/2}$ | $5417.39$ | $2.16[0]$ | $2.11[0]$ | $6.89(17)[6]$ | $2.02(5)[-2]$
$9P_{3/2}$ | $8D_{3/2}$ | $25810.71$ | $4.61[1]$ | $4.84[1]$ | $1.36(7)[6]$ | $1.36(7)[-1]$
$9P_{3/2}$ | $8D_{5/2}$ | $26185.44$ | $4.20[2]$ | $4.38[2]$ | $1.19(6)[7]$ | $8.13(38)[-1]$
$9P_{3/2}$ | $9S_{1/2}$ | $30734.01$ | $3.64[2]$ | $3.70[2]$ | $6.36(9)[6]$ | $1.80(3)[0]$
$9D_{3/2}$ | $5P_{1/2}$ | $620.51$ | $5.17[-3]$ | $4.72[-3]$ | $1.10(10)[7]$ | $1.27(11)[-3]$
$9D_{3/2}$ | $5P_{3/2}$ | $629.51$ | $7.08[-4]$ | $6.35[-4]$ | $1.44(15)[6]$ | $8.54(90)[-5]$
$9D_{3/2}$ | $6P_{1/2}$ | $1306.15$ | $8.77[-2]$ | $8.46[-2]$ | $1.99(7)[7]$ | $1.02(4)[-2]$
$9D_{3/2}$ | $6P_{3/2}$ | $1322.55$ | $1.47[-2]$ | $1.41[-2]$ | $3.23(14)[6]$ | $8.46(36)[-4]$
$9D_{3/2}$ | $7P_{1/2}$ | $2591.33$ | $6.40[-1]$ | $6.32[-1]$ | $1.86(2)[7]$ | $3.75(5)[-2]$
$9D_{3/2}$ | $7P_{3/2}$ | $2624.57$ | $1.14[-1]$ | $1.13[-1]$ | $3.20(5)[6]$ | $3.31(5)[-3]$
$9D_{3/2}$ | $8P_{1/2}$ | $5542.67$ | $5.96[0]$ | $5.98[0]$ | $1.77(1)[7]$ | $1.63(1)[-1]$
$9D_{3/2}$ | $8P_{3/2}$ | $5632.20$ | $1.12[0]$ | $1.13[0]$ | $3.19(1)[6]$ | $1.516(4)[-2]$
$9D_{3/2}$ | $9P_{1/2}$ | $17729.01$ | $2.51[2]$ | $2.53[2]$ | $2.28(2)[7]$ | $2.15(2)[0]$
$9D_{3/2}$ | $9P_{3/2}$ | $18345.87$ | $5.34[1]$ | $5.38[1]$ | $4.38(4)[6]$ | $2.21(2)[-1]$
$9D_{5/2}$ | $5P_{3/2}$ | $629.37$ | $7.43[-3]$ | $6.72[-3]$ | $1.01(10)[7]$ | $8.97(87)[-4]$
$9D_{5/2}$ | $6P_{3/2}$ | $1321.90$ | $1.40[-1]$ | $1.37[-1]$ | $2.05(8)[7]$ | $8.05(32)[-3]$
$9D_{5/2}$ | $7P_{3/2}$ | $2622.01$ | $1.07[0]$ | $1.05[0]$ | $2.00(3)[7]$ | $3.09(5)[-2]$
$9D_{5/2}$ | $8P_{3/2}$ | $5620.43$ | $1.03[1]$ | $1.03[1]$ | $1.9615(2)[7]$ | $1.3935(2)[-1]$
$9D_{5/2}$ | $9P_{3/2}$ | $18221.68$ | $4.74[2]$ | $4.77[2]$ | $2.65(1)[7]$ | $1.98(1)[0]$
$10S_{1/2}$ | $5P_{1/2}$ | $618.88$ | $1.38[-2]$ | $1.35[-2]$ | $5.90(13)[7]$ | $3.39(8)[-3]$
$10S_{1/2}$ | $5P_{3/2}$ | $627.80$ | $2.81[-2]$ | $2.74[-2]$ | $1.15(3)[8]$ | $3.40(9)[-3]$
$10S_{1/2}$ | $6P_{1/2}$ | $1298.78$ | $6.46[-2]$ | $6.49[-2]$ | $2.99(1)[7]$ | $7.55(4)[-3]$
$10S_{1/2}$ | $6P_{3/2}$ | $1315.00$ | $1.31[-1]$ | $1.31[-1]$ | $5.841(3)[7]$ | $7.572(4)[-3]$
$10S_{1/2}$ | $7P_{1/2}$ | $2562.48$ | $3.18[-1]$ | $3.25[-1]$ | $1.92(4)[7]$ | $1.89(4)[-2]$
$10S_{1/2}$ | $7P_{3/2}$ | $2594.98$ | $6.48[-1]$ | $6.58[-1]$ | $3.76(6)[7]$ | $1.90(3)[-2]$
$10S_{1/2}$ | $8P_{1/2}$ | $5412.36$ | $2.27[0]$ | $2.34[0]$ | $1.45(5)[7]$ | $6.36(20)[-2]$
$10S_{1/2}$ | $8P_{3/2}$ | $5497.69$ | $4.68[0]$ | $4.78[0]$ | $2.85(6)[7]$ | $6.47(14)[-2]$
$10S_{1/2}$ | $9P_{1/2}$ | $16461.27$ | $6.06[1]$ | $6.29[1]$ | $1.38(5)[7]$ | $5.59(20)[-1]$
$10S_{1/2}$ | $9P_{3/2}$ | $16991.74$ | $1.34[2]$ | $1.37[2]$ | $2.76(7)[7]$ | $5.98(16)[-1]$
$10P_{1/2}$ | $4D_{3/2}$ | $416.88$ | $1.45[-3]$ | $1.81[-3]$ | $2.03(48)[7]$ | $2.64(63)[-4]$
$10P_{1/2}$ | $5S_{1/2}$ | $487.08$ | $4.83[-3]$ | $5.00[-3]$ | $4.23(15)[7]$ | $1.51(5)[-3]$
$10P_{1/2}$ | $5D_{3/2}$ | $989.54$ | $1.01[-2]$ | $9.12[-3]$ | $1.06(11)[7]$ | $7.75(77)[-4]$
$10P_{1/2}$ | $6S_{1/2}$ | $1042.94$ | $6.91[-3]$ | $7.53[-3]$ | $6.17(55)[6]$ | $1.01(9)[-3]$
$10P_{1/2}$ | $6D_{3/2}$ | $1938.47$ | $4.17[-2]$ | $3.80[-2]$ | $5.80(53)[6]$ | $1.64(15)[-3]$
$10P_{1/2}$ | $7S_{1/2}$ | $2017.46$ | $1.37[-2]$ | $1.57[-2]$ | $1.69(23)[6]$ | $1.03(14)[-3]$
$10P_{1/2}$ | $7D_{3/2}$ | $3785.49$ | $1.95[-1]$ | $1.84[-1]$ | $3.64(20)[6]$ | $3.91(22)[-3]$
$10P_{1/2}$ | $8S_{1/2}$ | $3939.74$ | $2.86[-2]$ | $3.44[-2]$ | $4.74(91)[5]$ | $1.10(21)[-3]$
$10P_{1/2}$ | $8D_{3/2}$ | $8516.33$ | $1.73[0]$ | $1.71[0]$ | $2.84(4)[6]$ | $1.54(2)[-2]$
$10P_{1/2}$ | $9S_{1/2}$ | $8991.59$ | $2.37[-2]$ | $3.37[-2]$ | $3.3(1.3)[4]$ | $4.0(1.5)[-4]$
$10P_{1/2}$ | $9D_{3/2}$ | $41374.25$ | $4.33[2]$ | $4.58[2]$ | $6.19(36)[6]$ | $7.94(46)[-1]$
$10P_{3/2}$ | $4D_{3/2}$ | $416.63$ | $4.88[-4]$ | $5.76[-4]$ | $3.42(59)[6]$ | $8.9(1.5)[-5]$
$10P_{3/2}$ | $4D_{5/2}$ | $418.56$ | $2.57[-3]$ | $3.21[-3]$ | $1.78(42)[7]$ | $3.11(74)[-4]$
$10P_{3/2}$ | $5S_{1/2}$ | $486.74$ | $7.96[-3]$ | $8.32[-3]$ | $3.50(16)[7]$ | $2.48(11)[-3]$
$10P_{3/2}$ | $5D_{3/2}$ | $988.13$ | $2.42[-3]$ | $2.21[-3]$ | $1.27(11)[6]$ | $1.86(17)[-4]$
$10P_{3/2}$ | $5D_{5/2}$ | $991.44$ | $2.11[-2]$ | $1.93[-2]$ | $1.10(10)[7]$ | $1.08(10)[-3]$
$10P_{3/2}$ | $6S_{1/2}$ | $1041.37$ | $9.51[-3]$ | $1.07[-2]$ | $4.26(50)[6]$ | $1.39(16)[-3]$
$10P_{3/2}$ | $6D_{3/2}$ | $1933.04$ | $1.02[-2]$ | $9.49[-3]$ | $7.18(54)[5]$ | $4.02(30)[-4]$
$10P_{3/2}$ | $6D_{5/2}$ | $1939.07$ | $8.86[-2]$ | $8.17[-2]$ | $6.16(49)[6]$ | $2.31(19)[-3]$
$10P_{3/2}$ | $7S_{1/2}$ | $2011.58$ | $1.53[-2]$ | $1.83[-2]$ | $9.5(1.8)[5]$ | $1.15(22)[-3]$
$10P_{3/2}$ | $7D_{3/2}$ | $3764.81$ | $4.90[-2]$ | $4.69[-2]$ | $4.65(20)[5]$ | $9.89(43)[-4]$
$10P_{3/2}$ | $7D_{5/2}$ | $3777.59$ | $4.21[-1]$ | $3.99[-1]$ | $3.96(21)[6]$ | $5.65(30)[-3]$
$10P_{3/2}$ | $8S_{1/2}$ | $3917.35$ | $1.99[-2]$ | $2.77[-2]$ | $1.68(60)[5]$ | $7.7(2.8)[-4]$
$10P_{3/2}$ | $8D_{3/2}$ | $8412.42$ | $4.35[-1]$ | $4.32[-1]$ | $3.70(2)[5]$ | $3.92(2)[-3]$
$10P_{3/2}$ | $8D_{5/2}$ | $8451.84$ | $3.74[0]$ | $3.66[0]$ | $3.14(7)[6]$ | $2.24(5)[-2]$
$10P_{3/2}$ | $9S_{1/2}$ | $8875.83$ | $7.83[-3]$ | $7.31[-3]$ | $5.3(7.0)[3]$ | $1.3(1.7)[-4]$
$10P_{3/2}$ | $9D_{3/2}$ | $39031.88$ | $8.32[1]$ | $8.79[1]$ | $7.08(40)[5]$ | $1.62(9)[-1]$
$10P_{3/2}$ | $9D_{5/2}$ | $39606.22$ | $7.59[2]$ | $7.93[2]$ | $6.18(28)[6]$ | $9.70(44)[-1]$
$10D_{3/2}$ | $5P_{1/2}$ | $597.21$ | $2.47[-3]$ | $2.20[-3]$ | $5.87(66)[6]$ | $6.28(71)[-4]$
$10D_{3/2}$ | $5P_{3/2}$ | $605.50$ | $3.20[-4]$ | $2.76[-4]$ | $7.3(1.1)[5]$ | $4.02(58)[-5]$
$10D_{3/2}$ | $6P_{1/2}$ | $1206.84$ | $4.17[-2]$ | $3.94[-2]$ | $1.20(7)[7]$ | $5.24(29)[-3]$
$10D_{3/2}$ | $6P_{3/2}$ | $1220.83$ | $6.86[-3]$ | $6.42[-3]$ | $1.91(12)[6]$ | $4.26(28)[-4]$
$10D_{3/2}$ | $7P_{1/2}$ | $2227.66$ | $2.40[-1]$ | $2.31[-1]$ | $1.10(4)[7]$ | $1.63(6)[-2]$
$10D_{3/2}$ | $7P_{3/2}$ | $2252.18$ | $4.18[-2]$ | $4.02[-2]$ | $1.85(7)[6]$ | $1.41(5)[-3]$
$10D_{3/2}$ | $8P_{1/2}$ | $4108.17$ | $1.32[0]$ | $1.29[0]$ | $9.66(21)[6]$ | $4.89(10)[-2]$
$10D_{3/2}$ | $8P_{3/2}$ | $4157.14$ | $2.39[-1]$ | $2.35[-1]$ | $1.69(3)[6]$ | $4.37(8)[-3]$
$10D_{3/2}$ | $9P_{1/2}$ | $8374.94$ | $1.07[1]$ | $1.06[1]$ | $9.26(11)[6]$ | $1.95(2)[-1]$
$10D_{3/2}$ | $9P_{3/2}$ | $8510.11$ | $2.03[0]$ | $2.02[0]$ | $1.67(1)[6]$ | $1.81(1)[-2]$
$10D_{5/2}$ | $5P_{3/2}$ | $605.40$ | $3.43[-3]$ | $3.01[-3]$ | $5.23(66)[6]$ | $4.31(54)[-4]$
$10D_{5/2}$ | $6P_{3/2}$ | $1220.42$ | $6.54[-2]$ | $6.15[-2]$ | $1.22(7)[7]$ | $4.07(25)[-3]$
$10D_{5/2}$ | $7P_{3/2}$ | $2250.78$ | $3.91[-1]$ | $3.76[-1]$ | $1.16(4)[7]$ | $1.32(5)[-2]$
$10D_{5/2}$ | $8P_{3/2}$ | $4152.37$ | $2.21[0]$ | $2.16[0]$ | $1.04(3)[7]$ | $4.05(10)[-2]$
$10D_{5/2}$ | $9P_{3/2}$ | $8490.14$ | $1.86[1]$ | $1.83[1]$ | $1.02(2)[7]$ | $1.66(3)[-1]$
Table 3:  The line strengths ($S_{vk}$) (in a.u.) from both the L and V gauge expressions, wavelengths ($\lambda$) (in Å), transition probabilities ($A_{vk}$) in ($s^{-1}$) and absorption oscillator strengths ($f_{kv}$) for the Nb V ion through E1 decay channel are presented in this table. Values in square brackets represent the order of 10. Uncertainties are given in parentheses. Upper State(v) | Lower State(k) |  $\lambda$ (in Å) |  $S_{vk}$ (in a.u.) |  $A_{L_{vk}}$(in $s^{-1}$) | $f_{L_{kv}}$
---|---|---|---|---|---
|  |  | L | V |  |
$5P_{1/2}$ | $4D_{3/2}$ | $774.02$ | $1.34[0]$ | $1.45[0]$ | $2.94(23)[9]$ | $1.32(10)[-1]$
$5P_{1/2}$ | $5S_{1/2}$ | $1877.38$ | $3.71[0]$ | $3.88[0]$ | $5.67(26)[8]$ | $3.00(14)[-1]$
$5P_{3/2}$ | $4D_{3/2}$ | $753.01$ | $2.56[-1]$ | $2.76[-1]$ | $3.04(23)[8]$ | $2.58(20)[-2]$
$5P_{3/2}$ | $4D_{5/2}$ | $763.75$ | $2.38[0]$ | $2.57[0]$ | $2.71(20)[9]$ | $1.58(12)[-1]$
$5P_{3/2}$ | $5S_{1/2}$ | $1758.38$ | $7.43[0]$ | $7.82[0]$ | $6.92(36)[8]$ | $6.42(33)[-1]$
$5D_{3/2}$ | $5P_{1/2}$ | $1212.14$ | $9.04[0]$ | $9.33[0]$ | $2.57(8)[9]$ | $1.13(4)[0]$
$5D_{3/2}$ | $5P_{3/2}$ | $1267.52$ | $1.90[0]$ | $1.95[0]$ | $4.72(13)[8]$ | $1.14(3)[-1]$
$5D_{5/2}$ | $5P_{3/2}$ | $1258.84$ | $1.70[1]$ | $1.75[1]$ | $2.87(9)[9]$ | $1.02(3)[0]$
$6S_{1/2}$ | $5P_{1/2}$ | $1007.04$ | $1.28[0]$ | $1.32[0]$ | $1.27(4)[9]$ | $1.92(6)[-1]$
$6S_{1/2}$ | $5P_{3/2}$ | $1044.97$ | $2.88[0]$ | $2.95[0]$ | $2.55(6)[9]$ | $2.09(5)[-1]$
$6P_{1/2}$ | $4D_{3/2}$ | $399.19$ | $5.03[-2]$ | $5.55[-2]$ | $8.01(81)[8]$ | $9.57(96)[-3]$
$6P_{1/2}$ | $5S_{1/2}$ | $572.81$ | $5.92[-2]$ | $6.37[-2]$ | $3.19(23)[8]$ | $1.57(12)[-2]$
$6P_{1/2}$ | $5D_{3/2}$ | $2576.49$ | $9.64[0]$ | $9.85[0]$ | $5.71(12)[8]$ | $2.84(6)[-1]$
$6P_{1/2}$ | $6S_{1/2}$ | $4543.35$ | $1.39[1]$ | $1.42[1]$ | $1.50(3)[8]$ | $4.64(10)[-1]$
$6P_{3/2}$ | $4D_{3/2}$ | $396.79$ | $1.10[-2]$ | $1.20[-2]$ | $8.94(78)[7]$ | $2.11(19)[-3]$
$6P_{3/2}$ | $4D_{5/2}$ | $399.75$ | $1.01[-1]$ | $1.11[-1]$ | $8.03(72)[8]$ | $1.28(12)[-2]$
$6P_{3/2}$ | $5S_{1/2}$ | $567.88$ | $8.03[-2]$ | $8.76[-2]$ | $2.22(20)[8]$ | $2.15(19)[-2]$
$6P_{3/2}$ | $5D_{3/2}$ | $2479.59$ | $1.83[0]$ | $1.88[0]$ | $6.09(15)[7]$ | $5.61(14)[-2]$
$6P_{3/2}$ | $5D_{5/2}$ | $2513.52$ | $1.68[1]$ | $1.73[1]$ | $5.37(13)[8]$ | $3.39(8)[-1]$
$6P_{3/2}$ | $6S_{1/2}$ | $4250.44$ | $2.76[1]$ | $2.82[1]$ | $1.82(5)[8]$ | $9.85(25)[-1]$
$6D_{3/2}$ | $5P_{1/2}$ | $633.04$ | $1.05[-1]$ | $1.01[-1]$ | $2.10(9)[8]$ | $2.52(10)[-2]$
$6D_{3/2}$ | $5P_{3/2}$ | $647.82$ | $1.64[-2]$ | $1.56[-2]$ | $3.06(16)[7]$ | $1.92(10)[-3]$
$6D_{3/2}$ | $6P_{1/2}$ | $2727.98$ | $2.72[1]$ | $2.76[1]$ | $6.78(9)[8]$ | $1.51(2)[0]$
$6D_{3/2}$ | $6P_{3/2}$ | $2845.74$ | $5.72[0]$ | $5.78[0]$ | $1.26(1)[8]$ | $1.53(2)[-1]$
$6D_{5/2}$ | $5P_{3/2}$ | $646.72$ | $1.62[-1]$ | $1.54[-1]$ | $2.02(10)[8]$ | $1.90(9)[-2]$
$6D_{5/2}$ | $6P_{3/2}$ | $2824.68$ | $5.11[1]$ | $5.16[1]$ | $7.65(9)[8]$ | $1.37(2)[0]$
$7S_{1/2}$ | $5P_{1/2}$ | $604.08$ | $1.29[-1]$ | $1.29[-1]$ | $5.94(2)[8]$ | $3.25(1)[-2]$
$7S_{1/2}$ | $5P_{3/2}$ | $617.53$ | $2.72[-1]$ | $2.70[-1]$ | $1.17(1)[9]$ | $3.35(3)[-2]$
$7S_{1/2}$ | $6P_{1/2}$ | $2260.93$ | $4.18[0]$ | $4.33[0]$ | $3.66(13)[8]$ | $2.81(10)[-1]$
$7S_{1/2}$ | $6P_{3/2}$ | $2341.22$ | $9.35[0]$ | $9.65[0]$ | $7.38(23)[8]$ | $3.03(10)[-1]$
$7P_{1/2}$ | $4D_{3/2}$ | $326.81$ | $9.14[-3]$ | $1.23[-2]$ | $2.65(84)[8]$ | $2.12(68)[-3]$
$7P_{1/2}$ | $5S_{1/2}$ | $434.68$ | $2.83[-2]$ | $2.93[-2]$ | $3.49(13)[8]$ | $9.87(36)[-3]$
$7P_{1/2}$ | $5D_{3/2}$ | $1060.53$ | $2.23[-1]$ | $2.10[-1]$ | $1.89(11)[8]$ | $1.59(9)[-2]$
$7P_{1/2}$ | $6S_{1/2}$ | $1290.49$ | $6.01[-2]$ | $6.62[-2]$ | $2.83(28)[7]$ | $7.08(69)[-3]$
$7P_{1/2}$ | $6D_{3/2}$ | $5312.68$ | $3.02[1]$ | $3.10[1]$ | $2.04(5)[8]$ | $4.31(11)[-1]$
$7P_{1/2}$ | $7S_{1/2}$ | $8888.49$ | $3.46[1]$ | $3.50[1]$ | $5.00(5)[7]$ | $5.92(6)[-1]$
$7P_{3/2}$ | $4D_{3/2}$ | $325.96$ | $2.27[-3]$ | $2.78[-3]$ | $3.31(72)[7]$ | $5.3(1.2)[-4]$
$7P_{3/2}$ | $4D_{5/2}$ | $327.95$ | $2.01[-2]$ | $2.48[-2]$ | $2.88(64)[8]$ | $3.10(69)[-3]$
$7P_{3/2}$ | $5S_{1/2}$ | $433.17$ | $4.38[-2]$ | $4.58[-2]$ | $2.73(13)[8]$ | $1.54(7)[-2]$
$7P_{3/2}$ | $5D_{3/2}$ | $1051.59$ | $4.86[-2]$ | $4.62[-2]$ | $2.12(11)[7]$ | $3.51(18)[-3]$
$7P_{3/2}$ | $5D_{5/2}$ | $1057.64$ | $4.32[-1]$ | $4.10[-1]$ | $1.85(10)[8]$ | $2.07(11)[-2]$
$7P_{3/2}$ | $6S_{1/2}$ | $1277.27$ | $6.00[-2]$ | $6.88[-2]$ | $1.46(21)[7]$ | $7.1(1.0)[-3]$
$7P_{3/2}$ | $6D_{3/2}$ | $5095.67$ | $5.74[0]$ | $5.92[0]$ | $2.20(7)[7]$ | $8.56(26)[-2]$
$7P_{3/2}$ | $6D_{5/2}$ | $5164.62$ | $5.26[1]$ | $5.42[1]$ | $1.93(6)[8]$ | $5.16(15)[-1]$
$7P_{3/2}$ | $7S_{1/2}$ | $8297.31$ | $6.85[1]$ | $6.94[1]$ | $6.07(8)[7]$ | $1.25(2)[0]$
$7D_{3/2}$ | $5P_{1/2}$ | $508.88$ | $7.89[-3]$ | $6.74[-3]$ | $3.03(46)[7]$ | $2.35(36)[-3]$
$7D_{3/2}$ | $5P_{3/2}$ | $518.39$ | $8.58[-4]$ | $6.86[-4]$ | $3.12(66)[6]$ | $1.26(27)[-4]$
$7D_{3/2}$ | $6P_{1/2}$ | $1329.80$ | $5.85[-1]$ | $5.69[-2]$ | $1.26(3)[8]$ | $6.68(19)[-2]$
$7D_{3/2}$ | $6P_{3/2}$ | $1357.18$ | $1.00[-1]$ | $9.72[-2]$ | $2.03(7)[7]$ | $5.62(18)[-3]$
$7D_{3/2}$ | $7P_{1/2}$ | $5071.20$ | $6.03[1]$ | $6.08[1]$ | $2.34(2)[8]$ | $1.81(1)[0]$
$7D_{3/2}$ | $7P_{3/2}$ | $5286.08$ | $1.27[1]$ | $1.28[1]$ | $4.37(2)[7]$ | $1.83(1)[-1]$
$7D_{5/2}$ | $5P_{3/2}$ | $517.99$ | $9.96[-3]$ | $8.22[-3]$ | $2.42(44)[7]$ | $1.46(27)[-3]$
$7D_{5/2}$ | $6P_{3/2}$ | $1354.44$ | $9.49[-1]$ | $9.20[-1]$ | $1.29(4)[8]$ | $5.32(17)[-2]$
$7D_{5/2}$ | $7P_{3/2}$ | $5244.75$ | $1.13[2]$ | $1.14[2]$ | $2.66(1)[8]$ | $1.64(1)[0]$
$8S_{1/2}$ | $5P_{1/2}$ | $498.28$ | $4.13[-2]$ | $4.06[-2]$ | $3.38(6)[8]$ | $1.26(2)[-2]$
$8S_{1/2}$ | $5P_{3/2}$ | $507.40$ | $8.54[-2]$ | $8.35[-2]$ | $6.63(15)[8]$ | $1.28(3)[-2]$
$8S_{1/2}$ | $6P_{1/2}$ | $1259.80$ | $3.57[-1]$ | $3.63[-1]$ | $1.81(3)[8]$ | $4.31(6)[-2]$
$8S_{1/2}$ | $6P_{3/2}$ | $1284.34$ | $7.50[-1]$ | $7.57[-1]$ | $3.59(3)[8]$ | $4.43(4)[-2]$
$8S_{1/2}$ | $7P_{1/2}$ | $4184.50$ | $1.02[1]$ | $1.06[1]$ | $1.41(5)[8]$ | $3.71(14)[-1]$
$8S_{1/2}$ | $7P_{3/2}$ | $4329.74$ | $2.28[1]$ | $2.36[1]$ | $2.84(9)[8]$ | $4.00(13)[-1]$
$8P_{1/2}$ | $4D_{3/2}$ | $307.43$ | $2.44[-3]$ | $4.19[-3]$ | $8.5(5.3)[7]$ | $6.0(3.7)[-4]$
$8P_{1/2}$ | $5S_{1/2}$ | $394.99$ | $1.69[-2]$ | $1.72[-2]$ | $2.77(5)[8]$ | $6.49(12)[-3]$
$8P_{1/2}$ | $5D_{3/2}$ | $823.88$ | $5.21[-2]$ | $4.76[-2]$ | $9.44(84)[7]$ | $4.80(43)[-3]$
$8P_{1/2}$ | $6S_{1/2}$ | $944.92$ | $2.86[-2]$ | $3.06[-2]$ | $3.43(24)[7]$ | $4.60(32)[-3]$
$8P_{1/2}$ | $6D_{3/2}$ | $2082.48$ | $5.58[-1]$ | $5.32[-1]$ | $6.26(29)[7]$ | $2.03(9)[-2]$
$8P_{1/2}$ | $7S_{1/2}$ | $2433.04$ | $7.52[-2]$ | $8.23[-2]$ | $5.29(48)[6]$ | $4.70(43)[-3]$
$8P_{1/2}$ | $7D_{3/2}$ | $9672.58$ | $7.10[1]$ | $7.34[1]$ | $7.95(26)[7]$ | $5.58(18)[-1]$
$8P_{1/2}$ | $8S_{1/2}$ | $15199.75$ | $7.16[1]$ | $7.17[1]$ | $2.065(4)[7]$ | $7.15(1)[-1]$
$8P_{3/2}$ | $4D_{3/2}$ | $306.99$ | $6.97[-4]$ | $1.01[-3]$ | $1.22(50)[7]$ | $1.72(71)[-4]$
$8P_{3/2}$ | $4D_{5/2}$ | $308.64$ | $6.12[-3]$ | $8.97[-3]$ | $1.05(45)[8]$ | $1.00(42)[-3]$
$8P_{3/2}$ | $5S_{1/2}$ | $394.27$ | $2.65[-2]$ | $2.73[-2]$ | $2.19(6)[8]$ | $1.02(3)[-2]$
$8P_{3/2}$ | $5D_{3/2}$ | $820.74$ | $1.16[-2]$ | $1.07[-2]$ | $1.06(8)[7]$ | $1.08(9)[-3]$
$8P_{3/2}$ | $5D_{5/2}$ | $824.33$ | $1.03[-1]$ | $9.47[-2]$ | $9.29(75)[7]$ | $6.31(51)[-3]$
$8P_{3/2}$ | $6S_{1/2}$ | $940.79$ | $3.72[-2]$ | $4.08[-2]$ | $2.27(21)[7]$ | $6.01(56)[-3]$
$8P_{3/2}$ | $6D_{3/2}$ | $2062.53$ | $1.24[-1]$ | $1.19[-1]$ | $7.15(28)[6]$ | $4.56(18)[-3]$
$8P_{3/2}$ | $6D_{5/2}$ | $2073.40$ | $1.09[0]$ | $1.05[0]$ | $6.21(25)[7]$ | $2.67(11)[-2]$
$8P_{3/2}$ | $7S_{1/2}$ | $2405.86$ | $5.58[-2]$ | $6.53[-2]$ | $2.03(33)[6]$ | $3.52(58)[-3]$
$8P_{3/2}$ | $7D_{3/2}$ | $9256.70$ | $1.35[1]$ | $1.40[1]$ | $8.62(30)[6]$ | $1.11(4)[-1]$
$8P_{3/2}$ | $7D_{5/2}$ | $9380.50$ | $1.24[2]$ | $1.28[2]$ | $7.59(25)[7]$ | $6.67(22)[-1]$
$8P_{3/2}$ | $8S_{1/2}$ | $14197.42$ | $1.41[2]$ | $1.41[2]$ | $2.50(1)[7]$ | $1.510(4)[0]$
$8D_{3/2}$ | $5P_{1/2}$ | $468.43$ | $7.18[-4]$ | $4.58[-4]$ | $3.5(1.4)[6]$ | $2.33(94)[-4]$
$8D_{3/2}$ | $5P_{3/2}$ | $476.00$ | $2.50[-5]$ | $6.25[-6]$ | $1.2(1.2)[5]$ | $4.0(4.0)[-6]$
$8D_{3/2}$ | $6P_{1/2}$ | $1042.70$ | $9.38[-2]$ | $8.78[-2]$ | $4.19(27)[7]$ | $1.37(9)[-2]$
$8D_{3/2}$ | $6P_{3/2}$ | $1058.70$ | $1.45[-2]$ | $1.34[-2]$ | $6.20(48)[6]$ | $1.04(8)[-3]$
$8D_{3/2}$ | $7P_{1/2}$ | $2401.87$ | $1.54[0]$ | $1.52[0]$ | $5.62(8)[7]$ | $9.72(13)[-2]$
$8D_{3/2}$ | $7P_{3/2}$ | $2446.65$ | $2.72[-1]$ | $2.67[-1]$ | $9.42(19)[6]$ | $8.45(17)[-3]$
$8D_{3/2}$ | $8P_{1/2}$ | $8491.33$ | $1.15[2]$ | $1.16[2]$ | $9.52(9)[7]$ | $2.06(2)[0]$
$8D_{3/2}$ | $8P_{3/2}$ | $8839.98$ | $2.44[1]$ | $2.45[1]$ | $1.79(1)[7]$ | $2.10(1)[-1]$
$8D_{5/2}$ | $5P_{3/2}$ | $475.80$ | $5.76[-4]$ | $2.86[-4]$ | $1.8(1.1)[6]$ | $9.2(5.4)[-5]$
$8D_{5/2}$ | $6P_{3/2}$ | $1057.71$ | $1.42[-1]$ | $1.31[-1]$ | $4.04(30)[7]$ | $1.02(7)[-2]$
$8D_{5/2}$ | $7P_{3/2}$ | $2441.38$ | $2.55[0]$ | $2.50[0]$ | $5.91(12)[7]$ | $7.93(16)[-2]$
$8D_{5/2}$ | $8P_{3/2}$ | $8771.50$ | $2.17[2]$ | $2.18[2]$ | $1.1(4.4)[8]$ | $1.9(7.5)[0]$
$9S_{1/2}$ | $5P_{1/2}$ | $463.54$ | $1.92[-2]$ | $1.88[-2]$ | $1.96(5)[8]$ | $6.30(16)[-3]$
$9S_{1/2}$ | $5P_{3/2}$ | $470.95$ | $3.94[-2]$ | $3.82[-2]$ | $3.83(12)[8]$ | $6.36(20)[-3]$
$9S_{1/2}$ | $6P_{1/2}$ | $1018.77$ | $1.05[-1]$ | $1.06[-1]$ | $1.01(1)[8]$ | $1.57(1)[-2]$
$9S_{1/2}$ | $6P_{3/2}$ | $1034.03$ | $2.18[-1]$ | $2.18[-1]$ | $1.997(3)[8]$ | $1.601(2)[-2]$
$9S_{1/2}$ | $7P_{1/2}$ | $2278.56$ | $8.00[-1]$ | $8.20[-1]$ | $6.85(18)[7]$ | $5.33(14)[-2]$
$9S_{1/2}$ | $7P_{3/2}$ | $2318.83$ | $1.68[0]$ | $1.70[0]$ | $1.36(2)[8]$ | $5.49(8)[-2]$
$9S_{1/2}$ | $8P_{1/2}$ | $7127.67$ | $2.11[1]$ | $2.21[1]$ | $5.91(26)[7]$ | $4.51(20)[-1]$
$9S_{1/2}$ | $8P_{3/2}$ | $7371.72$ | $4.73[1]$ | $4.88[1]$ | $1.20(4)[8]$ | $4.88(15)[-1]$
$9P_{1/2}$ | $4D_{3/2}$ | $291.12$ | $6.86[-4]$ | $1.73[-3]$ | $2.8(3.3)[7]$ | $1.8(2.1)[-4]$
$9P_{1/2}$ | $5S_{1/2}$ | $368.48$ | $1.91[-2]$ | $1.91[-2]$ | $3.86(1)[8]$ | $7.86(1)[-3]$
$9P_{1/2}$ | $5D_{3/2}$ | $716.36$ | $2.14[-2]$ | $1.92[-2]$ | $5.89(62)[7]$ | $2.27(24)[-3]$
$9P_{1/2}$ | $6S_{1/2}$ | $806.14$ | $1.53[-2]$ | $1.61[-2]$ | $2.95(16)[7]$ | $2.88(16)[-3]$
$9P_{1/2}$ | $6D_{3/2}$ | $1509.72$ | $1.25[-1]$ | $1.16[-1]$ | $3.67(27)[7]$ | $6.27(46)[-3]$
$9P_{1/2}$ | $7S_{1/2}$ | $1685.81$ | $3.97[-2]$ | $4.24[-2]$ | $8.40(57)[6]$ | $3.58(24)[-3]$
$9P_{1/2}$ | $7D_{3/2}$ | $3501.83$ | $1.14[0]$ | $1.12[0]$ | $2.69(5)[7]$ | $2.47(5)[-2]$
$9P_{1/2}$ | $8S_{1/2}$ | $4032.74$ | $1.04[-1]$ | $1.10[-1]$ | $1.61(9)[6]$ | $3.92(22)[-3]$
$9P_{1/2}$ | $8D_{3/2}$ | $15524.92$ | $1.43[2]$ | $1.49[2]$ | $3.86(17)[7]$ | $6.98(31)[-1]$
$9P_{1/2}$ | $9S_{1/2}$ | $23876.94$ | $1.31[2]$ | $1.31[2]$ | $9.74(2)[6]$ | $8.32(1)[-1]$
$9P_{3/2}$ | $4D_{3/2}$ | $290.83$ | $2.46[-4]$ | $4.49[-4]$ | $5.1(3.6)[6]$ | $6.4(4.5)[-5]$
$9P_{3/2}$ | $4D_{5/2}$ | $292.31$ | $2.13[-3]$ | $3.96[-3]$ | $4.3(3.1)[7]$ | $3.7(2.7)[-4]$
$9P_{3/2}$ | $5S_{1/2}$ | $368.00$ | $1.33[-2]$ | $1.35[-2]$ | $1.35(3)[8]$ | $5.48(11)[-3]$
$9P_{3/2}$ | $5D_{3/2}$ | $714.57$ | $4.82[-3]$ | $4.37[-3[$ | $6.69(64)[6]$ | $5.12(49)[-4]$
$9P_{3/2}$ | $5D_{5/2}$ | $717.29$ | $4.25[-2]$ | $3.85[-2]$ | $5.84(57)[7]$ | $3.00(29)[-3]$
$9P_{3/2}$ | $6S_{1/2}$ | $803.88$ | $2.14[-2]$ | $2.30[-2]$ | $2.08(16)[7]$ | $4.04(30)[-3]$
$9P_{3/2}$ | $6D_{3/2}$ | $1501.79$ | $2.83[-2]$ | $2.65[-2]$ | $4.24(28)[6]$ | $1.43(9)[-3]$
$9P_{3/2}$ | $6D_{5/2}$ | $1507.54$ | $2.49[-1]$ | $2.32[-1]$ | $3.67(25)[7]$ | $8.35(57)[-3]$
$9P_{3/2}$ | $7S_{1/2}$ | $1675.92$ | $4.47[-2]$ | $4.98[-2]$ | $4.81(53)[6]$ | $4.05(45)[-3]$
$9P_{3/2}$ | $7D_{3/2}$ | $3459.45$ | $2.59[-1]$ | $2.54[-1]$ | $3.17(7)[6]$ | $5.69(12)[-3]$
$9P_{3/2}$ | $7D_{5/2}$ | $3476.60$ | $2.28[0]$ | $2.22[0]$ | $2.74(7)[7]$ | $3.31(9)[-2]$
$9P_{3/2}$ | $8S_{1/2}$ | $3976.63$ | $5.18[-2]$ | $6.40[-2]$ | $4.18(92)[5]$ | $1.98(44)[-3]$
$9P_{3/2}$ | $8D_{3/2}$ | $14725.11$ | $2.70[1]$ | $2.80[1]$ | $4.29(16)[6]$ | $1.39(5)[-1]$
$9P_{3/2}$ | $8D_{5/2}$ | $14919.13$ | $2.47[2]$ | $2.55[2]$ | $3.77(13)[7]$ | $8.38(28)[-1]$
$9P_{3/2}$ | $9S_{1/2}$ | $22036.11$ | $2.57[2]$ | $2.53[2]$ | $1.22(2)[7]$ | $1.77(3)[0]$
$9D_{3/2}$ | $5P_{3/2}$ | $446.02$ | $1.16[-5]$ | $2.81[-5]$ | $6.6(7.4)[4]$ | $2.0(2.2)[-6]$
$9D_{3/2}$ | $6P_{1/2}$ | $908.87$ | $2.82[-2]$ | $2.56[-2]$ | $1.90(18)[7]$ | $4.71(44)[-3]$
$9D_{3/2}$ | $6P_{3/2}$ | $921.00$ | $4.03[-3]$ | $3.59[-3]$ | $2.61(30)[6]$ | $3.32(38)[-4]$
$9D_{3/2}$ | $7P_{1/2}$ | $1793.52$ | $2.75[-1]$ | $2.68[-1]$ | $2.42(6)[7]$ | $2.33(6)[-2]$
$9D_{3/2}$ | $7P_{3/2}$ | $1818.37$ | $4.56[-2]$ | $4.38[-2]$ | $3.84(16)[6]$ | $1.91(8)[-3]$
$9D_{3/2}$ | $8P_{1/2}$ | $3861.18$ | $3.13[0]$ | $3.18[0]$ | $2.76(4)[7]$ | $1.23(2)[-1]$
$9D_{3/2}$ | $8P_{3/2}$ | $3931.69$ | $5.67[-1]$ | $5.65[0]$ | $4.73(2)[6]$ | $1.096(4)[-2]$
$9D_{3/2}$ | $9P_{1/2}$ | $13019.37$ | $1.96[2]$ | $2.02[2]$ | $4.50(15)[7]$ | $2.29(7)[0]$
$9D_{3/2}$ | $9P_{3/2}$ | $13640.71$ | $4.25[1]$ | $4.30[1]$ | $8.47(11)[6]$ | $2.36(3)[-1]$
$9D_{5/2}$ | $6P_{3/2}$ | $920.50$ | $4.04[-2]$ | $3.61[-2]$ | $1.75(19)[7]$ | $3.33(37)[-3]$
$9D_{5/2}$ | $7P_{3/2}$ | $1816.42$ | $4.36[-1]$ | $4.17[-1]$ | $2.46(11)[7]$ | $1.82(8)[-2]$
$9D_{5/2}$ | $8P_{3/2}$ | $3922.57$ | $5.28[0]$ | $5.23[0]$ | $2.96(3)[7]$ | $1.02(1)[-1]$
$9D_{5/2}$ | $9P_{3/2}$ | $13531.61$ | $3.78[2]$ | $3.81[2]$ | $5.15(5)[7]$ | $2.12(2)[0]$
$10S_{1/2}$ | $5P_{1/2}$ | $436.74$ | $1.06[-2]$ | $1.02[-2]$ | $1.29(4)[8]$ | $3.68(11)[-3]$
$10S_{1/2}$ | $5P_{3/2}$ | $443.30$ | $2.15[-2]$ | $2.08[-2]$ | $2.50(9)[8]$ | $3.69(13)[-3]$
$10S_{1/2}$ | $6P_{1/2}$ | $897.68$ | $4.59[-2]$ | $4.61[-2]$ | $6.43(2)[7]$ | $7.77(3)[-3]$
$10S_{1/2}$ | $6P_{3/2}$ | $909.51$ | $9.45[-2]$ | $9.38[-2]$ | $1.27(1)[8]$ | $7.89(6)[-3]$
$10S_{1/2}$ | $7P_{1/2}$ | $1750.46$ | $2.20[-1]$ | $2.26[-1]$ | $4.16(11)[7]$ | $1.91(5)[-2]$
$10S_{1/2}$ | $7P_{3/2}$ | $1774.13$ | $4.60[-1]$ | $4.60[-1]$ | $8.34(0)[7]$ | $1.97(0)[-2]$
$10S_{1/2}$ | $8P_{1/2}$ | $3666.98$ | $1.53[0]$ | $1.60[0]$ | $3.14(14)[7]$ | $6.33(27)[-2]$
$10S_{1/2}$ | $8P_{3/2}$ | $3730.52$ | $3.26[0]$ | $3.26[0]$ | $6.35(1)[7]$ | $6.63(1)[-2]$
$10S_{1/2}$ | $9P_{1/2}$ | $11046.79$ | $3.87]1]$ | $4.13[1]$ | $2.91(19)[7]$ | $5.33(35)[-1]$
$10S_{1/2}$ | $9P_{3/2}$ | $11490.90$ | $8.96[1]$ | $9.15[1]$ | $5.98(12)[7]$ | $5.92(12)[-1]$
$10P_{1/2}$ | $5S_{1/2}$ | $353.07$ | $7.48[-3]$ | $5.18[-3]$ | $1.72(58)[8]$ | $3.2(11)[-3]$
$10P_{1/2}$ | $5D_{3/2}$ | $660.34$ | $1.16[-2]$ | $1.03[-2]$ | $4.07(46)[7]$ | $1.33(15)[-3]$
$10P_{1/2}$ | $6S_{1/2}$ | $735.89$ | $9.60[-3]$ | $1.01[-2]$ | $2.44(13)[7]$ | $1.98(11)[-3]$
$10P_{1/2}$ | $6D_{3/2}$ | $1280.73$ | $5.13[-2]$ | $4.72[-2]$ | $2.48(20)[7]$ | $3.04(25)[-3]$
$10P_{1/2}$ | $7S_{1/2}$ | $1405.25$ | $2.21[-2]$ | $2.37[-2]$ | $8.08(54)[6]$ | $2.39(16)[-3]$
$10P_{1/2}$ | $7D_{3/2}$ | $2475.27$ | $2.42[-1]$ | $2.39[-1]$ | $1.61(2)[7]$ | $7.42(9)[-3]$
$10P_{1/2}$ | $8S_{1/2}$ | $2729.25$ | $6.29[-2]$ | $6.54[-2]$ | $3.13(12)[6]$ | $3.50(14)[-3]$
$10P_{1/2}$ | $8D_{3/2}$ | $5469.16$ | $1.94[0]$ | $2.09[0]$ | $1.20(9)[7]$ | $2.70(20)[-2]$
$10P_{1/2}$ | $9S_{1/2}$ | $6237.82$ | $1.86[-1]$ | $1.73[-1]$ | $7.76(54)[5]$ | $4.53(32)[-3]$
$10P_{1/2}$ | $9D_{3/2}$ | $24025.60$ | $2.62[2]$ | $2.88[2]$ | $1.91(19)[7]$ | $8.27(82)[-1]$
$10P_{3/2}$ | $4D_{3/2}$ | $281.10$ | $9.22[-5]$ | $2.28[-4]$ | $2.1(2.4)[6]$ | $2.5(2.9)[-5]$
$10P_{3/2}$ | $4D_{5/2}$ | $282.48$ | $7.84[-4]$ | $2.02[-3]$ | $1.8(2.1)[7]$ | $1.4(1.7)[-4]$
$10P_{3/2}$ | $5S_{1/2}$ | $352.57$ | $1.30[-2]$ | $8.76[-3]$ | $1.50(54)[8]$ | $5.6(2.0)[-3]$
$10P_{3/2}$ | $5D_{3/2}$ | $658.59$ | $2.64[-3]$ | $2.37[-3]$ | $4.68(49)[6]$ | $3.05(32)[-4]$
$10P_{3/2}$ | $5D_{5/2}$ | $660.90$ | $2.33[-2]$ | $2.09[-2]$ | $4.10(44)[7]$ | $1.79(19)[-3]$
$10P_{3/2}$ | $6S_{1/2}$ | $733.72$ | $1.39[-2]$ | $1.50[-2]$ | $1.79(13)[7]$ | $2.89(21)[-3]$
$10P_{3/2}$ | $6D_{3/2}$ | $1274.17$ | $1.19[-2]$ | $1.10[-2]$ | $2.91(22)[6]$ | $7.09(55)[-4]$
$10P_{3/2}$ | $6D_{5/2}$ | $1278.31$ | $1.04[-1]$ | $9.58[-2]$ | $2.53(21)[7]$ | $4.13(35)[-3]$
$10P_{3/2}$ | $7S_{1/2}$ | $1397.36$ | $2.73[-2]$ | $3.05[-2]$ | $5.06(58)[6]$ | $2.96(34)[-3]$
$10P_{3/2}$ | $7D_{3/2}$ | $2450.91$ | $5.83[-2]$ | $5.62[-2]$ | $2.00(7)[6]$ | $1.81(6)[-3]$
$10P_{3/2}$ | $7D_{5/2}$ | $2459.51$ | $5.09[-1]$ | $4.86[-1]$ | $1.73(8)[7]$ | $1.05(5)[-2]$
$10P_{3/2}$ | $8S_{1/2}$ | $2699.66$ | $5.66[-2]$ | $6.68[-2]$ | $1.46(25)[6]$ | $3.18(55)[-3]$
$10P_{3/2}$ | $8D_{3/2}$ | $5351.62$ | $4.86[-1]$ | $4.91[-1]$ | $1.61(2)[6]$ | $6.90(7)[-3]$
$10P_{3/2}$ | $8D_{5/2}$ | $5377.03$ | $4.26[0]$ | $4.23[0]$ | $1.39(1)[7]$ | $4.01(2)[-2]$
$10P_{3/2}$ | $9S_{1/2}$ | $6085.38$ | $3.29[-2]$ | $5.75[-2]$ | $7.4(4.8)[4]$ | $8.2(5.3)[-4]$
$10P_{3/2}$ | $9D_{3/2}$ | $21911.52$ | $4.86[1]$ | $5.11[1]$ | $2.34(12)[6]$ | $1.69(9)[-1]$
$10P_{3/2}$ | $9D_{5/2}$ | $22198.99$ | $4.45[2]$ | $4.61[2]$ | $2.06(7)[7]$ | $1.01(4)[0]$
$10D_{3/2}$ | $5P_{1/2}$ | $421.63$ | $3.36[-5]$ | $9.03[-5]$ | $2.3(2.9)[5]$ | $1.2(1.6)[-5]$
$10D_{3/2}$ | $5P_{3/2}$ | $427.75$ | $4.36[-5]$ | $6.72[-5]$ | $2.8(1.4)[5]$ | $7.7(3.8)[-6]$
$10D_{3/2}$ | $6P_{1/2}$ | $836.10$ | $1.15[-2]$ | $1.03[-2]$ | $10.0(1.1)[6]$ | $2.09(23)[-3]$
$10D_{3/2}$ | $6P_{3/2}$ | $846.35$ | $1.55[-3]$ | $1.33[-3]$ | $1.30(19)[6]$ | $1.39(21)[-4]$
$10D_{3/2}$ | $7P_{1/2}$ | $1530.62$ | $8.92[-2]$ | $8.76[-2]$ | $1.26(2)[7]$ | $8.85(15)[-3]$
$10D_{3/2}$ | $7P_{3/2}$ | $1548.69$ | $1.42[-2]$ | $1.36[-2]$ | $1.93(8)[6]$ | $6.96(28)[-4]$
$10D_{3/2}$ | $8P_{1/2}$ | $2818.87$ | $5.67[-1]$ | $5.99[-1]$ | $1.28(7)[7]$ | $3.06(17)[-2]$
$10D_{3/2}$ | $8P_{3/2}$ | $2856.26$ | $9.80[-2]$ | $9.98[-2]$ | $2.13(4)[6]$ | $2.60(5)[-3]$
$10D_{3/2}$ | $9P_{1/2}$ | $5794.65$ | $5.43[0]$ | $6.08[0]$ | $1.41(17)[7]$ | $1.42(17)[-1]$
$10D_{3/2}$ | $9P_{3/2}$ | $5914.56$ | $9.98[-1]$ | $1.06[0]$ | $2.44(15)[6]$ | $1.28(8)[-2]$
$10D_{5/2}$ | $5P_{3/2}$ | $427.67$ | $1.64[-4]$ | $3.06[-4]$ | $7.1(5.2)[5]$ | $2.9(2.1)[-5]$
$10D_{5/2}$ | $6P_{3/2}$ | $846.04$ | $1.58[-2]$ | $1.37[-2]$ | $8.8(1.2)[6]$ | $1.42(20)[-3]$
$10D_{5/2}$ | $7P_{3/2}$ | $1547.66$ | $1.38[-1]$ | $1.31[-1]$ | $1.25(6)[7]$ | $6.75(34)[-3]$
$10D_{5/2}$ | $8P_{3/2}$ | $2852.75$ | $9.29[-1]$ | $9.27[-1]$ | $1.352(3)[7]$ | $2.47(1)[-2]$
$10D_{5/2}$ | $9P_{3/2}$ | $5899.51$ | $9.28[0]$ | $9.59[0]$ | $1.53(5)[7]$ | $1.19(4)[-1]$

Figure 1: Percentage deviations of the oscillator strengths between the length
(L) and velocity (V) gauge values of the E1 transitions in Zr IV. Dominant and
less-dominant transitions are shown separately.

Figure 2: Percentage deviation of oscillator strengths between the L and
V-gauge calculations in Nb V with dominantly and less dominantly contributing
transitions shown separately.

Table 4:  Comparison of the oscillator strengths for Zr IV and Nb V ions from our calculations with available theoretical data. Uncertainties are given in the parentheses. Transition | Ion | $f_{kv}$
---|---|---
|  | Present | (Migdalek & Baylis, 1979) | (Zilitis, 2007) | (Migdalek, 2016) | (Das et al., 2017)
$4D_{3/2}\rightarrow 5P_{1/2}$ | Zr IV | $0.127(20)$ |  | $0.157$ | $0.264$ | $0.132$
| Nb V | $0.132(10)$ |  | $0.161$ | $0.266$ | $0.137$
$4D_{3/2}\rightarrow 5P_{3/2}$ | Zr IV | $0.0252(38)$ |  | $0.0308$ | $0.0260$ | $0.026$
| Nb V | $0.0258(20)$ |  | $0.0311$ | $0.0258$ | $0.027$
$4D_{5/2}\rightarrow 5P_{3/2}$ | Zr IV | $0.154(23)$ |  |  | $0.236$ | $0.160$
| Nb V | $0.158(12)$ |  |  | $0.235$ | $0.164$
$5S_{1/2}\rightarrow 5P_{1/2}$ | Zr IV | $0.312(24)$ | $0.307$ | $0.393$ | $0.331$ | $0.327$
| Nb V | $0.300(14)$ | $0.297$ | $0.385$ | $0.326$ | $0.314$
$5S_{1/2}\rightarrow 5P_{3/2}$ | Zr IV | $0.660(54)$ | $0.652$ | $0.829$ | $0.699$ | $0.692$
| Nb V | $0.642(33)$ | $0.638$ | $0.821$ | $0.697$ | $0.673$
Table 5:  The estimated lifetimes $\tau$ (in ns) for a few excited states in Zr IV and Nb V ions and their comparisons with the available literature data. Uncertainties are given in the parentheses. State | Zr IV | Nb V
---|---|---
$5P_{1/2}$ | $0.65(9)$ | $0.29(2)$
|  $0.588$ (Zilitis, 2007) |  $0.249$ (Zilitis, 2007)
|  $0.622$ (Das et al., 2017) |  $0.274$ (Das et al., 2017)
$5P_{3/2}$ | $0.61(8)$ | $0.27(2)$
|  $0.550$ (Zilitis, 2007) |  $0.238$ (Zilitis, 2007)
|  $0.5786$ (Das et al., 2017) |  $0.259$ (Das et al., 2017)
$5D_{3/2}$ | $0.54(1)$ | $0.33(9)$
$5D_{5/2}$ | $0.57(3)$ | $0.35(1)$
$6S_{1/2}$ | $0.51(1)$ | $0.26(1)$
$6P_{1/2}$ | $1.40(13)$ | $0.54(4)$
$6P_{3/2}$ | $1.50(16)$ | $0.53(3)$
$7S_{1/2}$ | $0.72(1)$ | $0.35(1)$
$7P_{1/2}$ | $2.46(23)$ | $0.92(10)$
$7P_{3/2}$ | $2.47(23)$ | $0.92(9)$
$8S_{1/2}$ | $0.66(1)$ | $0.51(1)$

##  3 Results and Discussion

The detailed analysis of our results is given in this section along with the
comparison of theoretical and experimental values available in the literature.
First, we discuss the energies, line strengths, transition probabilities and
oscillator strengths of all the transitions in both the ions. The
uncertainties (quoted in the parentheses) for the $S_{vk}$, $A_{vk}$ and
$f_{kv}$ values have been evaluated using the uncertainties in the E1 matrix
elements, which are taken as the differences between the values from both the
gauge expressions. We also depict percentage deviations of the oscillator
strengths obtained using the V-gauge expression with respect to the L-gauge
calculations for the important transitions of both the considered ions for
better understanding of their differences. Then, the lifetimes of the low-
lying states of these ions are estimated by using the above transition
probabilities.

Our RMBPT calculated energy values for a few low-lying and excited states are
given in Table 1 along with their comparison with the experimental values
listed in NIST AD (Ralchenko et al., 2008). It is observed that the deviation
of energy is maximum for the $4D_{3/2}$ state, viz, $\sim 11\%$ and $7\%$,
respectively, in the Zr IV and Nb V ions. However, as we move on to less
penetrating states, the deviation decreases rapidly up to $\sim 5\%$ and $3\%$
for the $8S_{1/2}$ state in Zr IV and Nb V respectively, despite the slow
convergence of perturbation expansion, thereby increasing the accuracy of our
data. Besides, the energy values for the $7D$ states of Zr IV are not provided
in the NIST AD. Therefore, the deviations of the energy values of these states
could not be estimated.

We have listed our results for wavelengths $\lambda$, line strengths $S_{vk}$,
transition probabilities $A_{vk}$ and absorption oscillator strengths $f_{kv}$
for Zr IV in Table LABEL:tab3. The $\lambda$ values are estimated by using the
experimental energies from the NIST AD (Ralchenko et al., 2008) wherever
available. Otherwise, they are evaluated using our RMBPT method. Values from
both the $L$ and $V$ gauge expressions are presented for the line strengths
$S_{vk}$ in the same table.

For the $9P_{3/2}$–$8S$ transition, we found that the reduced E1 matrix
element is very small due to cancellation of large RPA contributions with the
DF value. Correspondingly, the uncertainty in this particular transition is
found to be very large to consider it reliable. Hence, we have not considered
it for further spectroscopic analysis. During the investigation of data, it
has been observed that the uncertainties in the $A_{vk}$ and $f_{kv}$ values
are mostly small for many of the considered transitions except for the
$4D_{3/2}$–$(6,8,9)P_{1/2}$, $4D_{3/2,5/2}$–$8P_{3/2}$,
$5P_{3/2}$–$10D_{3/2}$, $6S_{1/2}$–$8P_{1/2}$, $7S_{1/2}$–$8P_{1/2,3/2}$,
$(7,8)S_{1/2}$–$9P_{3/2}$, $(7,9)S_{1/2}$–$10P_{3/2}$ and
$9S_{1/2}$–$10P_{1/2}$ transitions. The large errors in these transitions are
the consequence of unusually large electron correlation effects exhibited by
the high-lying states.

We have presented the percentage deviations of $f_{kv}$ obtained using the two
gauge expressions for different transitions in Fig. 1. It is seen from these
plots that deviation varies from $0$ to $8.5\%$ for maximum number of
dominantly contributing transitions, however a maximum deviation of $16\%$ is
seen for the $4D_{3/2}$–$5P_{1/2}$ transition. Oscillator strengths for less
dominantly contributing transitions deviate majorly between $0$–$17\%$ and in
the range $21$–$29\%$ for $4D_{3/2}$–$(9,10)P_{3/2}$ and $6S_{1/2}$–$7P_{3/2}$
transitions. However, a maxima of $\sim 39\%$ is observed for the
$8S_{1/2}$–$10P_{3/2}$ transition. It is also analyzed that the
$7S$–$8P_{3/2}$ transition shows an unreasonable percentage deviations of
oscillator strengths for both the gauge expressions.

We have tabulated our results for Nb V in Table LABEL:tab4. The $\lambda$
values are obtained by using energy values available in the NIST database for
many of the transitions; otherwise they are estimated using our RMBPT
calculations. Likewise Zr IV, the RPA and BO correlation contributions are
found to be the dominant corrections towards the final values of dipole
transition amplitudes. On account of our values of the calculated E1 matrix
elements, we have evaluated $S_{vk}$ values using both the $L$ and $V$
expressions and their uncertainties are estimated from the differences in in
these values. The $A_{vk}$ and $f_{kv}$ values along with the uncertainties
are obtained using these line strengths. It has been observed from our
calculations that the uncertainties in the values of transition probabilities
and oscillator strengths are considerably small for maximum transitions except
for the $4D_{3/2}$–$(7,9,10)P_{3/2}$, $4D_{3/2}$–$(8-10)P_{1/2}$,
$4D_{5/2}$–$(9,10)P_{3/2}$, $5S_{1/2}$–$10P_{3/2}$,
$5P_{1/2,3/2}$–$(8-10)D_{3/2}$, $5P_{3/2}$–$(8-10)D_{5/2}$,
$6S_{1/2}$–$7P_{3/2}$, $6P_{3/2}$–$10D_{5/2}$, $8P_{3/2}$–$8D_{5/2}$ and
$9S_{1/2}$–$10P_{3/2}$ transitions. Among them, the results for the
$4D_{3/2}$–$10P_{1/2}$, $5P_{1/2}$–$9D_{3/2}$ and $5P_{3/2}$–$9D_{5/2}$
transitions are found to be highly unreliable. Therefore, we have not provided
further spectroscopic data for these three transitions specifically. We find
that due to strong core-polarization effects arising through RPA causes such
large uncertainties in the $4D_{3/2}$–$(7,9,10)P_{3/2}$,
$4D_{3/2,5/2}$–$(8-10)P_{1/2}$, $4D_{5/2}$–$(9,10)P_{3/2}$ and
$5S_{1/2}$–$10P_{3/2}$ transitions.

We have presented plots for percent deviation of oscillator strengths of Nb V
between the values obtained using both the gauge expressions and are shown in
Fig. 2. In this figure, we separately present the values for dominantly and
less dominantly contributing transitions. We observed that all the dominant
transitions possess deviation less than $12\%$ except for
$5P_{1/2}$–$7S_{1/2}$ transition whose deviation lies at $22\%$. For the other
transitions, deviations of less than $20\%$ are observed for most of the
transitions whereas a countable transitions have deviated from the L-gauge
values in the range $23$–$35\%$. It is also perceived that the
$4D_{3/2,5/2}$–$8P_{3/2}$, $5P_{1/2}$–$6S_{1/2}$ and $5P_{1/2}$–$(6,7)D_{3/2}$
transitions show unusually large percentage deviations in the oscillator
strengths.

We have also compared our results of the oscillator strengths and lifetimes of
a few excited states of Zr IV with those are already available theoretical
data in the literature for both the ions. Comparison of the oscillator
strengths for the $4D_{3/2}\rightarrow 5P_{1/2,3/2}$, $4D_{5/2}\rightarrow
5P_{3/2}$ and $5S_{1/2}\rightarrow 5P_{1/2,3/2}$ transitions are made in Table
4. In this table, it is seen that the results for the transition probabilities
and absorption oscillator strengths agree well with each other within the
quoted error limits. We have found that our results are in perfect accord with
the previous data published by Migdalek and Baylis (Migdalek & Baylis, 1979)
and are in fair agreement with the results reported by Migdalek in (Migdalek,
2016) and Das et.al (Das et al., 2017) except for the $4D_{3/2}\rightarrow
5P_{1/2}$ and $4D_{5/2}\rightarrow 5P_{3/2}$ transitions. However, our results
do not support the values obtained by Zilitis in (Zilitis, 2007). This is so
because, the results published in (Zilitis, 2007) are obtained by employing
the mean-field calculation at the DF level. We have seen in our calculations
that the core-polarization as well as from other effects are contributing
strongly to these transition properties, which were neglected by Zilitis. This
is why disparities between the results from both the works are seen.

Similarly, we have compared the absorption oscillator strengths of various
transitions of Nb V in Table 4, according to which, our results are in good
agreement with the results given by Das et.al. (Das et al., 2017). A deviation
of less than $10\%$ is seen during the comparison of our results with the data
published in (Migdalek, 2016), except for the $4D_{3/2}\rightarrow 5P_{1/2}$
and $4D_{5/2}\rightarrow 5P_{3/2}$ transitions. Our results show a variation
of about $10-22\%$ with respect to the theoretical data reported by Zilitis
(Zilitis, 2007) from the DF calculations.

The estimated lifetime values of the $5P_{1/2,3/2}$, $5D_{3/2,5/2}$,
$6S_{1/2}$, $6P_{1/2,3/2},7S_{1/2},7P_{1/2,3/2}$ and $8S_{1/2}$ states of Zr
IV have been listed in Table 5. Comparison of our calculated values for the
$5P_{1/2,3/2}$ is also made in the same table. We notice that our results are
in better agreement with the relativistic calculations presented by Das et.al.
(Das et al., 2017) as compared to the values given by Zilitis (Zilitis, 2007).
The lifetimes for the
$5P_{1/2,3/2},5D_{3/2,5/2},6S_{1/2},6P_{1/2,3/2},7S_{1/2},7P_{1/2,3/2}$ and
$8S_{1/2}$ states of Nb V are also given in Table 5. These values for the
$5P_{1/2,3/2}$ states show reasonable agreement with available other
theoretical data.

We believe that our aforementioned estimated values for various spectroscopic
data are more reliable. Since the previously reported data do not quote the
uncertainties in their calculations, our reported values will be useful in
analysing various astrophysical processes involving the Zr IV and Nb V ions.
Moreover, our precisely calculated values will be able to guide the future
experiments and astrophysical observations to detect these spectroscopic
properties.

##  4 Conclusion

By evaluating electric dipole matrix elements precisely, we have determined
oscillator strengths, transition probabilities and lifetimes of many atomic
states of the rubidium-isoelectronic zirconium and niobium ions. Calculations
of the matrix elements were performed by accounting electron correlation
effects through random-phase approximation, Brückner orbitals, structural
radiation and normalization of wave functions over the mean-field values from
the Dirac-Fock method. Our transition properties data include $192$
transitions for Zr IV and $190$ transitions for Nb V ion, while lifetimes are
reported only for a few low-lying excited states in both the ions. We have
also compared our results with the previously reported values for a few
selected transition and find a reasonably good agreement among them except in
some cases. These data with the quoted uncertainties can be useful for many
astrophysical applications and for their their observations in the future.

## Acknowledgement

The work of B.A. is supported by SERB-TARE(TAR/2020/000189), New Delhi, India.
The employed relativistic many-body method was developed in the group of
Professor M. S. Safronova of the University of Delaware, USA.

## Data availability

The data underlying this article are available in the article.

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