Extended Calculations of Energy Levels and Transition Rates of Nd ii-iv Ions for Application to Neutron Star Mergers

The GRASP2K package (Jönsson et al. 2013) is based on the MCDHF and relativistic configuration interaction (RCI) methods, and takes into account the transverse photon interaction (Breit interaction) and quantum electrodynamic (QED) corrections (Grant 2007; Fischer et al. 2016).

The MCDHF method used in the present work is based on the Dirac–Coulomb Hamiltonian

$$\begin{eqnarray}&&{H}_{\mathrm{DC}}=\displaystyle \sum _{i=1}^{N}\left(c\,{{\boldsymbol{\alpha }}}_{i}\cdot {{\boldsymbol{p}}}_{i}+({\beta }_{i}-1){c}^{2}+{V}_{i}^{N}\right)+\displaystyle \sum _{i\gt j}^{N}\displaystyle \frac{1}{{r}_{{ij}}},\end{eqnarray} \tag{ 1 }$$

where V^N is the monopole part of the electron–nucleus Coulomb interaction, ${\boldsymbol{\alpha }}$ and β are the 4 × 4 Dirac matrices, and c is the speed of light in atomic units. The atomic state functions (ASF) were obtained as linear combinations of symmetry-adapted configuration state functions (CSFs)

$$\begin{eqnarray}&&{\rm{\Psi }}(\gamma {PJM})=\displaystyle \sum _{j=1}^{\mathrm{NCSFs}}{c}_{j}{\rm{\Phi }}({\gamma }_{j}{PJM}).\end{eqnarray} \tag{ 2 }$$

Here, J and M are the angular quantum numbers and P is parity. ${\gamma }_{j}$ denotes other appropriate labeling of the CSF j such as, for example, the orbital occupancy and coupling scheme. Normally, the label $\gamma$ of the atomic state function is the same as the label of the dominant CSF. The CSFs are built from products of one-electron Dirac orbitals. Based on a weighted energy average of several states, the so-called extended optimal level (EOL) scheme (Dyall et al. 1989), both the radial parts of the Dirac orbitals and the expansion coefficients were optimized to self-consistency in the relativistic self-consistent field procedure. Note that because accurate calculations with the MCDHF method are much more difficult for neutral atoms than for ions (Grant 2007), we focus on ionized Nd.

For these calculations, we used the spin-angular approach (Gaigalas & Rudzikas 1996; Gaigalas et al. 1997), which is based on the second quantization in coupled tensorial form, on the angular momentum theory in three spaces (orbital, spin, and quasispin) and on the reduced coefficients of fractional parentage. It allowed us to study configurations with open f -shells without any restrictions.

In subsequent RCI calculations the Breit interaction

$$\begin{eqnarray}\begin{array}{rcl}{H}_{{\rm{Breit}}} & = & -\displaystyle \sum _{i\lt j}^{N}\left[{{\boldsymbol{\alpha }}}_{i}\cdot {{\boldsymbol{\alpha }}}_{j}\displaystyle \frac{\cos ({\omega }_{{ij}}{r}_{{ij}}/c)}{{r}_{{ij}}}\right.\\ & & +\,\left.({{\boldsymbol{\alpha }}}_{i}\cdot {{\boldsymbol{\nabla }}}_{i})({{\boldsymbol{\alpha }}}_{j}\cdot {{\boldsymbol{\nabla }}}_{j})\displaystyle \frac{\cos ({\omega }_{{ij}}{r}_{{ij}}/c)-1}{{\omega }_{{ij}}^{2}{r}_{{ij}}/{c}^{2}}\right]\end{array}\end{eqnarray} \tag{ 3 }$$

was included in the Hamiltonian. The photon frequencies ${\omega }_{{ij}}$, used for calculating the matrix elements of the transverse photon interaction, were taken as the difference of the diagonal Lagrange multipliers associated with the Dirac orbitals (McKenzie et al. 1980). In the RCI calculation the leading QED corrections, self-interaction and vacuum polarization, were also included.

In the present calculations, the ASFs were obtained as expansions over jj-coupled CSFs. To provide the LSJ labeling system, the ASFs were transformed from a jj-coupled CSF basis into an LSJ-coupled CSF basis using the method provided by Gaigalas et al. (2017).

The evaluation of radiative transition data (transition probabilities, oscillator strengths) between two states, $\gamma ^{\prime} P^{\prime} J^{\prime} M^{\prime}$ and $\gamma {PJM}$, built on different and independently optimized orbital sets, is non-trivial. The transition data can be expressed in terms of the transition moment, which is defined as

$$\begin{eqnarray}&&\begin{array}{l}\langle \,{\rm{\Psi }}(\gamma {PJ})\,\parallel {\boldsymbol{T}}\parallel \,{\rm{\Psi }}(\gamma ^{\prime} P^{\prime} J^{\prime} )\,\rangle \\ =\displaystyle \sum _{j,k}{c}_{j}{c}_{k}^{{\prime} }\,\langle \,{\rm{\Phi }}({\gamma }_{j}{PJ})\,\parallel {\boldsymbol{T}}\parallel \,{\rm{\Phi }}({\gamma }_{k}^{{\prime} }P^{\prime} J^{\prime} )\,\rangle ,\end{array}\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{T}}$ is the transition operator. For electric dipole and quadrupole (E1 and E2) transitions there are two forms of the transition operator: the length (Babushkin) and velocity (Coulomb) forms, which for the exact solutions of the Dirac-equation give the same value of the transition moment (Grant 1974). The quantity dT, characterizing the accuracy of the computed transition rates, is defined as

$$\begin{eqnarray}&&{dT}=\displaystyle \frac{| {A}_{{\rm{l}}}-{A}_{{\rm{v}}}| }{\max ({A}_{{\rm{l}}},{A}_{{\rm{v}}})},\end{eqnarray} \tag{ 5 }$$

where ${A}_{{\rm{l}}}$ and ${A}_{{\rm{v}}}$ are transition rates in length and velocity forms.

The calculation of the transition moment breaks down the task of summing up reduced matrix elements between different CSFs. The reduced matrix elements can be evaluated using standard techniques assuming that both the left and right CSFs formed from the same orthonormal set of spin-orbitals. This constraint is severe, because a high-quality and compact wave function requires orbitals optimized for a specific electronic state; see, for example Fritzsche & Grant (1994). To get around the problem of having a single orthonormal set of spin-orbitals, the wave function representations of the two states $\gamma ^{\prime} P^{\prime} J^{\prime} M^{\prime}$ and $\gamma {PJM}$ were transformed in such a way that the orbital sets became biorthonormal (Olsen et al. 1995). Standard methods were then used to evaluate the matrix elements of the transformed CSFs.

A summary of the MCDHF and RCI calculations for each ion is given in Table 1. The description, which explains how these calculations were done, is given below. As a starting point, DHF calculations were performed in the EOL scheme for the states of the ground configuration. The wave functions from these calculations were taken as the initial ones to calculate even and odd states of multireference (MR) configurations. The set of orbitals belonging to these MR configurations is referred to as 0 layer (0L).

Table 1.  Summary of Active Space Construction

Ion	Ground	MR set		Active Space	Number of Levels		NCSFs
	conf.	Even	Odd		Even	Odd	Even	Odd
	Strategies A, B.1
Nd ii	$4{f}^{4}6s$	$4{f}^{4}6s$, $4{f}^{4}5d$	$4{f}^{3}5{d}^{2}$, $4{f}^{4}6p$	$\{8s,8p,7d,6f,5g\}$	3 890	2 998	24 568	23 966
		$4{f}^{3}5d6p$, $4{f}^{3}6s6p$	$4{f}^{3}5d6s$
	Additional configuration in Strategy B.2
		$4{f}^{4}6d$, $4{f}^{3}5d7p$	$4{f}^{4}7p$, $4{f}^{4}5f$	$\{10s,10p,9d,8f,7g\}$	1 039	1 013	468 652	468 029
		$4{f}^{4}7s$
	Strategy C
		$4{f}^{4}6s$, $4{f}^{4}5d$	$4{f}^{3}5{d}^{2}$, $4{f}^{4}6p$	$\{8s,8p,7d,6f,5g\}$	3 270	2 813	188 357	113 900
		$4{f}^{3}5d6p$, $4{f}^{3}6s6p$	$4{f}^{3}5d6s$
	Strategies A, B
Nd iii	$4{f}^{4}$	$4{f}^{4}$, $4{f}^{3}6p$	$4{f}^{3}5d$, $4{f}^{3}6s$	$\{9s,9p,8d,7f,7g,7h\}$	1 020	468	400 440	259 948
		$4{f}^{2}5{d}^{2}$, $4{f}^{2}5d6s$
	Strategy C
		$4{f}^{4}$, $4{f}^{3}6p$	$4{f}^{3}5d$, $4{f}^{3}6s$	$\{10s,10p,9d,8f,7g,7h\}$	747	706	844 637	559 294
		$4{f}^{2}5{d}^{2}$, $4{f}^{2}5d6s$	$4{f}^{3}6d$, $4{f}^{3}7s$
		$4{f}^{3}5f$, $4{f}^{3}7p$
	Strategy C with 5p,5s
		$4{f}^{4}$, $4{f}^{3}6p$	$4{f}^{3}5d$, $4{f}^{3}6s$	$\{10s,10p,9d,8f,7g,7h\}$	747	706	900 904	586 850
		$4{f}^{2}5{d}^{2}$, $4{f}^{2}5d6s$	$4{f}^{3}6d$, $4{f}^{3}7s$
		$4{f}^{3}5f$, $4{f}^{3}7p$
	Strategy A
Nd iv	$4{f}^{3}$	$4{f}^{2}5d$, $4{f}^{2}6s$	$4{f}^{3}$, $4{f}^{2}6p$	$\{9s,9p,8d,7f,7g,7h,7i\}$	131	110	33 825	26 590
	Strategy B
		$4{f}^{2}5d$, $4{f}^{2}6s$	$4{f}^{3}$, $4{f}^{2}6p$	$\{9s,9p,8d,7f,7g,7h\}$	1 068	465	1 445 481	587 774
		$5{p}^{5}4{f}^{3}5d$	$5{p}^{5}4{f}^{4}$
	Strategy B with 5s
		$4{f}^{2}5d$, $4{f}^{2}6s$	$4{f}^{3}$, $4{f}^{2}6p$	$\{9s,9p,8d,7f,7g,7h\}$	1 068	465	1 474 463	603 827
		$5{p}^{5}4{f}^{3}5d$	$5{p}^{5}4{f}^{4}$

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Unless stated otherwise, the inactive core of each ion used in the present calculations is [Xe]. The CSF expansions for the states of each parity were obtained by allowing single (S) and double (SD) substitutions from the MR configurations up to active orbital sets (see Table 1). The configuration space was increased step-by-step with increasing the number of layers (L). The orbitals of previous layers were held fixed and only the orbitals of the newest layer were allowed to vary. For example, the scheme used to increase the active spaces of the CSFs for Nd iii ion (in Strategy A) is presented below:

• ${\mathrm{AS}}_{0L}$ = {6s, 6p, 5d},
• ${\mathrm{AS}}_{1L}={\mathrm{AS}}_{0L}$ + {7s, 7p, 6d, 5f, 5g},
• ${\mathrm{AS}}_{2L}={\mathrm{AS}}_{1L}$ + {8s, 8p, 7d, 6f, 6g, 6h},
• ${\mathrm{AS}}_{3L}={\mathrm{AS}}_{2L}$ + {9s, 9p, 8d, 7f, 7g, 7h}.

The MCDHF calculations were followed by RCI calculations, including the Breit interaction and leading QED effects. The number of CSFs in the final even and odd state expansions distributed over the different J symmetries is presented in Table 1.

Four strategies were tested for Nd ii ion. All of them were computed in the active space described in the Table 1. For Strategy A, the starting point DHF calculations were performed in the EOL scheme for the states of the ground configuration $4{f}^{4}6s$. The wave functions from these calculations were taken as the initial ones to calculate the even and odd states of MR configurations. The set of orbitals belonging to these MR configurations is referred to as 0 layer (0L). The active spaces were generated as is presented in Table 1.

For Strategy B.1 the starting point was a computation of the wave functions for the core $4{f}^{4}6s$. Wave functions were computed in the neutral system of Nd i - ground state $4{f}^{4}6{s}^{2}$. Then, ${\mathrm{AS}}_{0L}$ was computed: the core shells were frozen and only $5d$ and $6p$ shells of the configurations of MR listed in Table 1 were computed. Even and odd states were computed together. Later, wave functions were optimized separately for states of different parities in the ${\mathrm{AS}}_{1L}$. ${\mathrm{AS}}_{1L}$ and the next active space were generated by SD substitutions from shells $4f,5d,6p,6s$.

In Strategy B.2 the configurations of the Rydberg states listed in Table 1 were added to the MR list; therefore, the first active set included subshells that were larger by one principal quantum number. Then, the first active space of Strategy B.2 was ${\mathrm{AS}}_{1L}={\mathrm{AS}}_{0L}+\{8s,8p,7d,6f,5g\}$.

In Strategy C, computations were performed for each configuration separately. For configurations $4{f}^{4}6s$, $4{f}^{4}6p$ and $4{f}^{4}5d$ SD substitutions were allowed from $4{f}^{4}{nl}$ (where $l=s,p,d$) shells into the ${{AS}}_{0L,1L}$ and S into the ${{AS}}_{2L}$. For configurations $4{f}^{3}5d6s$, $4{f}^{3}5d6p$, $4{f}^{3}6s6p$, and $4{f}^{3}5{d}^{2}$ only S substitutions were allowed. Radial wave functions up to the $4f$ orbital were taken from the ground configuration for these configurations. The Breit interaction and leading QED effects are included in RCI computations.

After ${\mathrm{AS}}_{0L}$ the even and odd states were calculated separately in Strategy A. For the Nd iii ion calculations Strategy B was also applied. Strategy B differs from Strategy A in that virtual orbitals for odd parity were taken from even parity states instead of varying them in layer 1 and higher layers.

In Strategy C, compared to Strategy A additional configurations, $4{f}^{3}6d$, $4{f}^{3}7s$ (odd parity) and $4{f}^{3}5f$, $4{f}^{3}7p$ (even parity), were added to the MR set. In Strategy C with $5p,5s$ just RCI calculations were performed. The wave functions were taken from Strategy C and configurations with S substitutions from $5p$ and $5s$ shells to $\{6s,6p,5d,4f\}$ shells were added additionally in the active space.

In Strategy B, compared to Strategy A, additional configurations, $4{p}^{5}4{f}^{4}$ (odd parity) and $4{p}^{5}4{f}^{3}5d$ (even parity), were added to the MR set. The ASFs for even and odd parities were constructed such that SD substitutions were allowed from the $4f,5d,6s,6p$ shells up to active orbital sets and S substitution from the $5p$ shell to $\{6s,6p,5d,4f\}$ shells. In Strategy B, with 5s just RCI calculations were performed. The wave functions were taken from Strategy B and configurations with S substitutions from $5s$ shells to $\{6s,6p,5d,4f\}$ shells were added additionally in the active space.

A part of all computed excitation energies for Nd ii are listed in Table 2. These data were compared with the NIST database by evaluating the relative difference ${\rm{\Delta }}E/E=({E}_{\mathrm{NIST}}-E)/{E}_{\mathrm{NIST}}$. The energy levels computed with Breit interaction and QED effects are presented in columns marked by *. Levels with changed notations are given in Table 3.

Table 2.  Computations of Energy Values (in cm⁻¹) of Nd ii by Increase of the Active Space Performed, Applying Strategies A, B.1, B.2, and C and Their Comparison with the NIST Database (in %)

				Strategy A	Strategy B.1	Strategy B.2	Strategy C
Config.	Term	J	NIST	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}$	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{2L}* /{\mathrm{AS}}_{3L}$	${\mathrm{AS}}_{1-2L}/{\mathrm{AS}}_{2-3L}/{\mathrm{AS}}_{3-4L}$	${\mathrm{AS}}_{1L}* /{\mathrm{AS}}_{2L}*$
$4{f}^{4}{(}^{5}I)6s$	${}^{6}I$	7/2	0.000
	${}^{6}I$	9/2	513.330	−14/−9	−16/−10/−2/−10	−9/−9/−9	−6/−7
	${}^{6}I$	11/2	1470.105	−5/−3	−6/−3/8/−3	−0/0/−3	5/4
$4{f}^{4}{(}^{5}I)6s$	${}^{4}I$	9/2	1650.205	−13/−8	−15/−9/−3/−8	−8/−7/−8	−8/−8
$4{f}^{4}{(}^{5}I)6s$	${}^{6}I$	13/2	2585.460	−5/−3	−4/−3/9/−3	1/1/−3	6/6
$4{f}^{4}{(}^{5}I)6s$	${}^{4}I$	11/2	3066.755	−9/−6	−10/−6/2/−6	−4/−4/−6	−1/−2
$4{f}^{4}{(}^{5}I)6s$	${}^{6}I$	15/2	3801.930	−5/−4	−5/−4/8/−4	0/0/−4	6/6
$4{f}^{4}{(}^{5}I)5d$	${}^{6}L$	11/2	4437.560	−13/−7	−38/−3/2/−3	−2/−3/1	4/9
$4{f}^{4}{(}^{5}I)6s$	${}^{4}I$	13/2	4512.495	−9/−6	−9/−6/0/−6	−4/−3/−6	0/0
$4{f}^{4}{(}^{5}I)6s$	${}^{6}I$	17/2	5085.640	−6/−5	−6/−5/7/−5	−1/−1/−5	6/6
$4{f}^{4}{(}^{5}I)5d$	${}^{6}L$	13/2	5487.655	−10/−5	−30/−2/1/−2	−1/−1/1	6/10
$4{f}^{4}{(}^{5}I)6s$	${}^{4}I$	15/2	5985.580	−9/−7	−9/−7/4/−7	−4/−4/−7	1/1
$4{f}^{4}{(}^{5}I)5d$	${}^{6}K$	9/2	6005.270	−12/−7	−36/−4/−3/−4	−5/−5/−2	−6/−1
$4{f}^{4}{(}^{5}I)5d$	${}^{6}L$	15/2	6637.430	−8/−4	−24/−1/3/−1	0/0/0	7/10
$4{f}^{4}{(}^{5}I)5d$	${}^{6}K$	11/2	6931.800	−10/−6	−31/−3/−1/−3	−3/−3/−2	−3/1
$4{f}^{4}{(}^{5}I)5d$	${}^{6}I$	7/2	7524.735	−24/−19	−50/−17/−16/−17	−19/−19/−15	−16/−1
$4{f}^{4}{(}^{5}I)5d$	${}^{6}L$	17/2	7868.910	−7/−3	−20/−1/4/−1	1/1/0	7/10
$4{f}^{4}{(}^{5}I)5d$	${}^{6}K$	13/2	7950.075	−9/−5	−27/−3/1/−3	−3/−3/−2	−2/2
$4{f}^{3}{(}^{4}I)5{d}^{2}{(}^{3}F)$	${}^{6}{M}^{o}$	13/2	8009.810	0/−12	−7/−11/−6/−14	−7/−10/−14	32/32
$4{f}^{4}{(}^{5}I)5d$	${}^{6}I$	9/2	8420.320	−21/−16	−43/−14/−12/−14	−15/−16/−13	−13/−9
$4{f}^{4}{(}^{5}I)5d$	${}^{6}G$	3/2	8716.445	−20/−15	−42/−13/−12/−13	−16/−16/−13	−14/−11
	${}^{6}G$	5/2	8796.365	−23/−17	−46/−15/−14/−15	−19/−19/−15	−17/−14
$4{f}^{4}{(}^{5}I)5d$	${}^{6}K$	15/2	9042.760	−8/−5	−24/−3/2/−3	−2/−2/−2	0/3
$4{f}^{4}{(}^{5}I)5d$	${}^{6}L$	19/2	9166.210	−6/−3	−17/−1/5/−1	1/1/−1	7/10
$4{f}^{4}{(}^{5}I)5d$	${}^{6}G$	7/2	9198.395	−24/−18	−46/−16/−15/−16	−19/−19/−15	−17/−14
$4{f}^{4}{(}^{5}I)5d$	${}^{6}I$	11/2	9357.910	−18/−14	−39/−12/−10/−12	−13/−13/−11	−10/−7
$4{f}^{3}{(}^{4}I)5{d}^{2}{(}^{3}F)$	${}^{6}{M}^{o}$	15/2	9448.185	−1/−12	−9/−12/−6/−14	−8/−11/−14	29/29
$4{f}^{4}{(}^{5}I)5d$	${}^{6}H$	5/2	9674.835	−28/−22	−55/−20/−20/−30	−24/−25/−19	−23/−19
$4{f}^{3}{(}^{4}I)5d{(}^{5}L)6s$	${}^{6}{L}^{o}$	11/2	10054.195	−71/−50	−73/−77/73/−79	−73/−75/−79	1/1
$4{f}^{3}{(}^{4}I)5d{(}^{5}K)6s$	${}^{6}{K}^{o}$	9/2	10091.360	−70/−51	−71/−75/−101/−77	−71/−73/−77	−1/0
$4{f}^{4}{(}^{5}I)5d$	${}^{6}K$	17/2	10194.805	−8/−5	−21/−3/2/−3	−2/−2/−3	1/3
$4{f}^{4}{(}^{5}F)6s$	${}^{6}F$	1/2	10256.040	−37/−37	−38/−37/−36/−35	−37/−35/−35	−28/−28
$4{f}^{4}{(}^{5}I)5d$	${}^{6}I$	13/2	10337.100	−17/−13	−35/−11/−8/−11	−12/−12/−11	−8/−6
$4{f}^{4}{(}^{5}F)6s$	${}^{6}F$	3/2	10439.225	−37/−36	−37/−48/−36/−35	−36/−34/−35	−28/−27
$4{f}^{4}{(}^{5}I)5d$	${}^{6}L$	21/2	10516.790	−6/−3	−15/−1/5/−1	0/0/−2	7/9
$4{f}^{4}{(}^{5}I)5d$	${}^{6}H$	7/2	10666.780	−25/−20	−36/−19/−17/−18	−11/−11/−8	−19/−16
$4{f}^{3}{(}^{4}I)5d(5K)6s$	${}^{6}{K}^{o}$	11/2	10720.295	−67/−82	−82/−82/−69/−75	−69/−80/−75	−3/−2
$4{f}^{4}{(}^{5}F)6s$	${}^{6}F$	5/2	10786.775	−36/−36	−37/−50/−35/−34	−35/−47/−34	−27/−27

Note. The "*" energy levels were computed with Breit interaction and QED effects are presented.

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Table 3.  NIST Recommended Energy Level Notations Changed by the Authors for Nd ii

NIST label	Our label
$4{f}^{3}{(}^{4}I* )5d{(}^{5}K* )6s{}^{6}L*$	$4{f}^{3}{(}^{4}I* )5d{(}^{5}L* )6s{}^{6}L*$
$4{f}^{2}{(}^{4}I* )5d{(}^{5}K* )6s{}^{6}I*$	$4{f}^{3}{(}^{4}I* )5d{(}^{5}I* )6s{}^{6}I*$

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Note that the energy levels of Nd ii are also provided by Wyart (2010). They interpreted 596 levels of odd configurations ($4{f}^{3}5d6s$, $4{f}^{3}5{d}^{2}$, $4{f}^{3}6{s}^{2}$, $4{f}^{4}6p$ and $4{f}^{5}$) in a semi-empirical way following the Racah-Slater parametric method, using the Cowan computer codes. In their method, the radial parameters obtained in a least-squares fit were compared with Hartree–Fock (HFR) ab initio integrals. Because of this, the obtained energy levels naturally have very small disagreement with NIST values, and therefore are not presented in this paper.

The energy levels for each configuration are compared with NIST in Figure 1. Among the different strategies, Strategy C ${{AS}}_{2L}$ gives the best agreement with the NIST database. The averaged difference between our computed data and the NIST presented values is 10%. This is a significant improvement compared with Strategy A ${{AS}}_{2L}$ (blue in Figure 1), which was used to compute the opacity of the neutron star mergers in Tanaka et al. (2018). The averaged difference with / NIST is 22% in Strategy A ${{AS}}_{2L}$. For comparison with the NIST, the expression $\overline{{\rm{\Delta }}E/E}=\tfrac{\sum | {\rm{\Delta }}{E}_{i}/{E}_{i}| }{N}$ was used, where N is the number of compared levels.

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Figure 2 shows the distribution of the energy level number over the relative difference compared to NIST for Strategy A in active space ${{AS}}_{2}L$. For strategies A, B.1, and B.2, in all active spaces the distributions look very similar. For Strategy C, in ${{AS}}_{2}$ (see Figure 2) a normal distribution with a smaller ${\rm{\Delta }}E/E$ range is observed.

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The energy data computed in Strategy C at layer 2 are given in machine-readable format in Table 9. This includes number, label, J and P values, and energy value. The transitions data obtained from Strategy C at layer 2 are given in machine-readable form in Table 10. This includes identification of upper and lower levels in LSJ coupling, transition energy, wavelength, line strength, weighted oscillator strength, and transition probabilities in length form.

The results of the energy levels for Nd iii obtained from applying Strategies A, B C, and C with 5p, 5s are compared with the NIST database and presented in Table 4. Among the strategies, Strategy C with 5p, 5s gives the best agreement with the NIST database, although the number of available levels is smaller than that in the case of Nd ii. All the energy levels and transition data obtained from this strategy are listed in machine-readable form in Tables 11 and 12. Figure 3 shows a comparison of the energy levels with the NIST database. The averaged difference between our calculations with Strategy C with 5p, 5s ${{AS}}_{3L}$ and the NIST data is 3%. For comparison, the difference for the case of Strategy B ${{AS}}_{2L}$, which was used by Tanaka et al. (2018), is 5% (blue in Figure 3).

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Table 4.  Comparison of Our Energy Levels to the NIST Database (in %) of Nd iii by Increase of the Active Space Performed, Applying Strategies A, B, C, and C with 5p, 5s

				Strategy A	Strategy B	Strategy C	Strategy C (5p,5s)
Config.	Term	J	NIST	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$
$4{f}^{4}$	${}^{5}I$	4	0.0
		5	1137.8	8.8/8.5/8.4	8.7/8.5/8.4	8.8/8.5/8.4	6.1/5.9/5.7
		6	2387.6	7.9/7.7/7.7	7.9/7.7/7.7	7.9/7.8/7.7	5.4/5.3/5.2
		7	3714.9	7.0/7.0/7.0	7.0/7.0/7.0	7.1/7.1/7.0	4.7/4.7/4.6
		8	5093.3	6.2/6.2/6.3	6.2/6.2/6.3	6.3/6.4/6.3	4.0/4.1/4.1
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{K}^{o}$	5	15262.2	7.2/11.6/7.4	6.8/6.4/6.8	7.5/7.0/6.8	1.4/1.1/0.9
		6	16938.1	7.0/11.0/7.1	6.7/6.3/6.6	7.4/6.8/6.6	1.9/1.5/1.3
		7	18656.3	6.6/10.2/6.7	6.4/6.0/6.3	7.0/6.4/6.2	2.1/1.7/1.5
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	4	18883.7	−2.7/2.1/−1.2	−3.5/−2.8/−1.8	−2.3/−1.7/−1.7	0.5/0.9/0.9
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{H}^{o}$	3	19211.0	−5.3/−0.5/−3.8	−6.2/−5.1/−4.3	−5.0/−4.2/−4.2	−3.7/−2.8/−2.8
		4	20144.3	−3.2/1.3/−1.8	−3.9/−3.1/−2.4	−2.9/−2.3/−2.3	−2.3/−1.6/−1.6
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	5	20388.9	−1.9/2.5/−0.6	−2.6/−2.1/−1.1	−1.5/−1.1/−1.1	0.5/0.9/0.9
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{K}^{o}$	8	20410.9	6.1/9.4/6.2	5.9/5.5/5.8	6.4/5.9/5.7	2.0/1.7/1.5
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{H}^{o}$	5	21886.8	−2.5/1.6/−1.3	−3.1/−2.4/−1.8	−2.2/−1.7/−1.7	−1.6/−1.0/−1.1
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	6	22047.8	−1.3/2.7/−0.2	−1.9/−1.4/−0.6	−0.9/−0.5/−0.6	0.6/0.9/0.9
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{K}^{o}$	9	22197.0	5.5/8.6/5.6	5.4/5.0/5.2	5.8/5.4/5.2	1.9/1.6/1.4
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	7	22702.9	−1.0/2.9/0.2	−1.5/−1.1/−0.3	−0.6/−0.2/−5.8	0.6/0.9/0.9
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{H}^{o}$	6	23819.3	−1.8/1.9/−0.7	−2.3/−1.8/−1.2	−1.5/−1.1/−1.1	−1.2/−0.7/−0.7
$4{f}^{3}{(}^{4}I)5d$	${}^{o* }$	7	24003.2
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	8	24686.4	−1.3/2.3/−0.2	−1.8/−1.4/−0.6	−0.9/−0.6/−0.6	1.4/1.6/1.6
$4{f}^{3}{(}^{4}I)5d$	${}^{o* }$	6	26503.2
$4{f}^{3}{(}^{4}I)5d$	${}^{3}{K}^{o}$	8	27391.4	−0.4/3.0/0.8	−0.8/−0.4/0.4	−0.1/0.4/0.5	−0.9/−0.4/−0.3
$4{f}^{3}{(}^{4}F)5d$	${}^{o* }$	3	27569.8
$4{f}^{3}{(}^{4}F)5d$	${}^{5}{H}^{o}$	3	27788.2	−10.3/−6.9/−8.9	−10.7/−10.2/−9.3	−9.9/−9.4/−9.2	−6.7/−6.2/−6.1
		4	28745.3	−10.1/−6.8/−8.8	−10.5/−10.0/−9.2	−9.7/−9.2/−9.1	−6.6/−6.2/−6.1
$4{f}^{3}{(}^{4}F)5d$	${}^{o* }$	5	29397.3
$4{f}^{3}{(}^{4}F)5d$	${}^{5}{H}^{o}$	5	30232.3	−8.9/−5.7/−7.6	−9.3/−8.8/−8.0	−8.5/−8.0/−7.9	−6.0/−5.5/−5.4
		6	31394.6	−7.5/−4.4/−6.1	−7.8/−7.3/−6.5	−7.2/−6.6/−6.4	−4.8/−4.2/−5.0
		7	32832.6	−7.4/−4.4/−6.1	−7.8/−7.3/−6.5	−7.1/−6.5/−6.4	−5.0/−4.4/−4.3

Note. States marked with the subscript * in the term column are without term identification in the NIST database.

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The results of the energy levels obtained from Strategy C with $5p,5s$ are also compared with those by Dzuba et al. (2003) in Table 5. They evaluated the energy levels and lifetimes of configurations $4{f}^{4}$, $4{f}^{3}5d$ using relativistic Hartree–Fock and configuration interaction (RCI) codes, as well as a set of computer codes written by Cowan (1981). Note that Zhang et al. (2002) also presented low-lying odd energy levels (below 33,000 cm⁻¹) belonging to the configurations: $4{f}^{3}5d$ and $4{f}^{3}6s$. To compute these energy levels, the HFR described and coded by Cowan (1981) but modified with the inclusion of core-polarization effects, was used. It should be mentioned, however, that core–core correlation was not included in the energy level computations.

Table 5.  Comparison of Energy Levels from Present (Strategy C with 5p, 5s) and Other Theoretical Computations with the NIST Database (in %) of Nd iii

				Present	Dzuba et al. (2003)
Config.	Term	J	NIST	${\mathrm{AS}}_{3L}$	Cowan	RCI
$4{f}^{4}$	${}^{5}I$	4	0.0	0	0	0
		5	1137.8	1073/5.7	1137/0.1	1162/−2.1
		6	2387.6	2264/5.2	2397/−0.4	2471/−3.5
		7	3714.9	3543/4.6	3743/−0.8	3898/−4.9
		8	5093.3	4885/4.1	5148/−1.1	5414/−6.3
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{K}^{o}$	5	15262.2	15128/0.9	14742/3.4	15357/−0.6
		6	16938.1	16721/1.3	16338/3.5	17380/−2.6
		7	18656.3	18383/1.5	18000/3.5	19485/−4.4
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	4	18883.7	18714/0.9	18467/2.2	20284/−7.4
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{H}^{o}$	3	19211.0	19753/−2.8	19427/−1.1	20946/−9.0
		4	20144.3	20465/−1.6	20189/−0.2	21926/−8.8
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	5	20388.9	20208/0.9	20006/1.9	21254/−4.2
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{K}^{o}$	8	20410.9	20105/1.5	19725/3.4	21666/−6.1
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{H}^{o}$	5	21886.8	22119/−1.1	21866/0.1	22167/−1.3
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	6	22047.8	21845/0.9	21672/1.7	22664/−2.8
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{K}^{o}$	9	22197.0	21882/1.4	21503/3.1	21919/1.3
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	7	22702.9	22499/0.9	22244/2.0	26537/−16.9
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{H}^{o}$	6	23819.3	23992/−0.7	23733/0.4	24076/−1.1
$4{f}^{3}{(}^{4}I)5d$	${}^{o* }$	7	24003.2
$4{f}^{3}{(}^{4}I)5d$	${}^{5}{I}^{o}$	8	24686.4	24301/1.6	24158/2.1	27396/−11.0
$4{f}^{3}{(}^{4}I)5d$	${}^{o* }$	6	26503.2
$4{f}^{3}{(}^{4}I)5d$	${}^{3}{K}^{o}$	8	27391.4	27465/−0.3
$4{f}^{3}{(}^{4}F)5d$	${}^{o* }$	3	27569.8
$4{f}^{3}{(}^{4}F)5d$	${}^{5}{H}^{o}$	3	27788.2	29494/−6.1	28824/−3.7
		4	28745.3	30506/−6.1	29872/−3.9
$4{f}^{3}{(}^{4}F)5d$	${}^{o* }$	5	29397.3
$4{f}^{3}{(}^{4}F)5d$	${}^{5}{H}^{o}$	5	30232.3	31852/−5.4	31117/−2.9
		6	31394.6	32961/−5.0	32054/−2.1
		7	32832.6	34234/−4.3	33391/−1.7

Note. States marked with the subscript * in the term column are without term identification in the NIST database.

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The disagreement between our data obtained from applying Strategy C and Strategy C with 5p, 5s, compared to the recommended data by NIST, is slightly larger than the disagreement between the data computed by Dzuba et al. (2003) (Cowan) and the recommended data by NIST. In this paper we present the lowest 1453 levels of energy spectra and transitions between these states, whereas Dzuba et al. (2003) presented only a small part of the spectra (88 levels). This paper aims to present a more complete set of atomic data for astrophysics. This is clearly reflected in the Figure 3, where the energy levels for each configuration for different strategies are presented and compared with only a few levels of configurations $4{f}^{4}$ and $4{f}^{3}5d$ available in the NIST.

The results of the energy levels of Strategies A and B are presented and compared with the NIST database in Table 6. The best agreement with the NIST database is obtained for Strategy B with 5s. The energy levels are shown and compared with a few levels of configuration $4{f}^{3}$ available in the NIST in Figure 4. The averaged difference is 11% for Strategy B with 5s, while it is 17% for Strategy A in active space ${{AS}}_{1L}$.

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Table 6.  Comparison of Energy Levels in the NIST Database (in %) of Nd iv by Increase of the Active Space Performed, Applying Strategies A, B, and B with 5s

				Strategy A	Strategy B	Strategy B (5s)
Config.	Term	J	NIST	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$	${\mathrm{AS}}_{1L}/{\mathrm{AS}}_{2L}/{\mathrm{AS}}_{3L}$
$4{f}^{3}$	${}^{4}{I}^{o}$	9/2	0
		11/2	[1880]	6.8/6.5/6.4	6.9/6.8/6.7	7.2/7.1/7.0
		13/2	[3860]	5.6/5.4/5.3	5.8/5.8/5.7	6.1/6.1/6.0
		15/2	[5910]	4.7/4.6/4.6	4.9/5.0/5.0	5.2/5.3/5.3
$4{f}^{3}$	${}^{4}{F}^{o}$	3/2	[11290]	−27.8/−27.0/−26.6	−20.3/−19.2/−18.6	−17.6/−16.4/−15.8
		5/2	[12320]	−24.8/−24.2/−23.8	−17.9/−16.9/−16.4	−15.3/−14.4/−13.8
$4{f}^{3}$	${}^{2}H{2}^{o}$	9/2	[12470]	−14.7/−12.5/−11.3	−12.8/−10.4/−9.4	−12.0/−9.5/−8.5
$4{f}^{3}$	${}^{4}{F}^{o}$	7/2	[13280]	−22.3/−21.6/−21.2	−16.1/−15.2/−14.6	−13.7/−12.8/−12.3
$4{f}^{3}$	${}^{4}{S}^{o}$	3/2	[13370]	−20.2/−16.8/−16.5	−15.6/−12.1/−11.6	−13.3/−9.8/−9.3
$4{f}^{3}$	${}^{4}{F}^{o}$	9/2	[14570]	−18.3/−17.6/−17.1	−13.2/−12.1/−11.6	−11.3/−10.2/−9.7
$4{f}^{3}$	${}^{2}H{2}^{o}$	11/2	[15800]	−9.6/−7.8/−6.8	−8.4/−6.3/−5.5	−7.8/−5.7/−4.9
$4{f}^{3}$	${}^{4}{G}^{o}$	5/2	[16980]	−29.6/−28.1/−27.8	−21.7/−20.2/−19.6	−18.6/−17.1/−16.5
$4{f}^{3}$	${}^{o* }$	7/2	[17100]
$4{f}^{3}$	${}^{4}{G}^{o}$	7/2	[18890]	−23.6/−22.3/−22.0	−16.9/−15.5/−14.9	−14.3/−12.9/−12.3
		9/2	[19290]	−16.6/−27.0/−26.7	−22.1/−20.6/−20.0	−19.7/−18.3/−17.7
$4{f}^{3}$	${}^{2}{K}^{o}$	13/2	[19440]	−17.7/−15.0/−14.2	−15.1/−12.2/−11.4	−13.9/−11.1/−10.3
$4{f}^{3}$	${}^{4}{G}^{o}$	11/2	[21280]	−22.2/−21.0/−20.8	−15.9/−14.6/−14.2	−13.4/−12.2/−11.7
$4{f}^{3}$	${}^{2}{K}^{o}$	15/2	[21430]	−16.1/−13.7/−12.9	−13.6/−11.0/−10.3	−12.5/−9.9/−9.2

Note. States marked with the subscript * in the term column are without a term identification in the NIST database.

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For the Nd iv ion, several experiments and analyses with semi-empirical methods have been performed. The emission spectrum produced by vacuum spark sources was observed in the vacuum ultraviolet on two normal-incidence spectrographs. 550 lines have been identified as transitions from 85 (out of 107 possible) levels of $4{f}^{2}5d$ to 37 (out of 41 possible) levels of $4{f}^{3}$. The method and codes of Cowan were used to predict the spectral ranges of the strong transitions in the spectra Nd iv at the beginning of the series of paper (Wyart et al. 2006).

Later, Wyart et al. (2007) used the same experiment to observe and classify 1426 lines. In total, 41 levels of $4{f}^{3}$ configuration were reported. To derive energy levels with the diagonalization code RCG, the input Hartree–Fock radial integrals including relativistic corrections, treated as parameters (HFR parameters), were scaled according to earlier results on the neighboring ion spectra. Altogether 111 odd parity and 121 even parity configuration levels for $4{f}^{3}$, $4{f}^{2}6p$, $5{p}^{5}4{f}^{4}$, $4{f}^{2}5d$, $4{f}^{2}6s$, and $5{p}^{5}4{f}^{3}5d$ were established. Their optimized values were calculated with the ELCALC code (Radziemski et al. 1970).

Then, Wyart et al. (2008) performed a parametric fit of the level energies for the $4{f}^{3}$ configuration, previously obtained in the experiment (Wyart et al. 2007). Dzuba et al. (2003) used the same computations applied to Nd iii (see Section 4.2). This included only 72 levels of configurations $4{f}^{3}$ and $4{f}^{2}5d$. In Table 7, the energy levels obtained from applying Strategy B with 5s are compared with the experimental values from Wyart et al. (2007) and semi-empirical values from Dzuba et al. (2003).

Table 7.  Comparison of Energy Levels from Present (Strategy B with 5s) and Other Theoretical Computations with the NIST Database (in %) and with the Experimental Wyart et al. (2007; in %) Values of Nd iv

					Present	Dzuba et al. (2003)
Config.	Term	J	NIST	Exp.	${\mathrm{AS}}_{3L}$	Cowan	RCI
$4{f}^{3}$	${}^{4}{I}^{o}$	9/2	0	0	0	0	0
		11/2	[1880]	1897.11	1749/7.0/7.8	1879/0.1/1.0	1945/−3.5/−2.5
		13/2	[3860]	3907.43	3627/6.0/7.2	3890/−0.8/0.4	4049/−4.9/−3.6
		15/2	[5910]	5988.51	5596/5.3/6.6	5989/−1.3/0.0	6267/−6.0/−4.7
$4{f}^{3}$	${}^{4}{F}^{o}$	3/2	[11290]	11698.49	13076/−15.8/−11.8	13294/−17.8/−13.6	12490/−10.6/−6.8
		5/2	[12320]	12747.94	14022/−13.8/−10.0	14333/−16.3/−12.4	13545/−9.9/−6.3
$4{f}^{3}$	${}^{2}H{2}^{o}$	9/2	[12470]	12800.29	13536/−8.5/−5.8	13272/−6.4/−3.7	14522/−16.5/−13.5
$4{f}^{3}$	${}^{4}{F}^{o}$	7/2	[13280]	13719.82	14911/−12.3/−8.7	15249/−14.8/−11.1	14622/−10.1/−6.6
$4{f}^{3}$	${}^{4}{S}^{o}$	3/2	[13370]	13792.49	14617/−9.3/−6.0	15153/−13.3/−9.9	14452/−8.1/−4.8
$4{f}^{3}$	${}^{4}{F}^{o}$	9/2	[14570]	14994.87	15979/−9.7/−6.6	16334/−12.1/−8.9	16183/−11.1/−7.9
$4{f}^{3}$	${}^{2}H{2}^{o}$	11/2	[15800]	16161.53	16581/−4.9/−2.6	16456/−4.2/−1.8	18142/−14.8/−12.3
$4{f}^{3}$	${}^{4}{G}^{o}$	5/2	[16980]	17707.17	19780/−16.5/−11.7
$4{f}^{3}$	${}^{o* }$	7/2	[17100]	17655.11
$4{f}^{3}$	${}^{4}{G}^{o}$	7/2	[18890]	19540.80	21218/−12.3/−8.6
		9/2	[19290]	19969.79	22709/−17.7/−13.7
$4{f}^{3}$	${}^{2}{K}^{o}$	13/2	[19440]	20005.22	21445/−10.3/−7.2
$4{f}^{3}$	${}^{4}{G}^{o}$	11/2	[21280]	22047.39	23768/−11.7/−7.8
$4{f}^{3}$	${}^{2}{K}^{o}$	15/2	[21430]	22043.77	23398/−9.2/−6.1

Note. States marked by the subscript * in the term column are without term identification in the NIST database.

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In addition to the energy levels, transition data can also be compared with experimental data and semi-empirical calculations (Table 8). Our results for the transition wavelengths show good agreement with the experimental data by Wyart et al. (2007). As shown in Figure 5, the agreement in the wavelength is within 20% for most transitions.

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Table 8.  Computed Transition Data of Nd iv from Applying Strategy B with 5s at ${{AS}}_{3L}$, Compared with the Experimental Wavelength ${\lambda }_{\exp }$ (in Å) and Computed Transition Probabilities A_SE (in s⁻¹) from Wyart et al. (2007)

				Strategies B with 5s				Wyart et al. (2007)		Yoca & Quinet (2014)
Upper		Lower		${A}_{{\rm{l}}}$	${A}_{{\rm{v}}}$	dT	λ	ASE	${\lambda }_{\exp }$	ASE
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	4.5	$4{f}^{2}{(}^{3}H)5d{}^{2}H$	4.5	5.26E+8	4.40E+8	0.16	1107.72(15.0)	4.04E+8	1303.32	3.89E+8
$4{f}^{2}{(}^{1}I)5d{}^{2}K$	7.5	$4{f}^{3}{}^{2}L$	8.5	9.32E+7	1.24E+8	0.25	1540.16(−5.1)	1.64E+8	1464.73	1.47E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	5.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	5.5	2.04E+8	1.91E+8	0.06	2412.27(9.5)	1.96E+8	2666.70	1.94E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	4.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	3.5	1.92E+8	1.61E+8	0.16	2410.31(9.6)	1.97E+8	2666.70	1.96E+8
$4{f}^{2}{(}^{1}G)5d{}^{2}I$	6.5	$4{f}^{3}{}^{2}K$	7.5	6.85E+7	8.99E+7	0.24	1559.66(−6.6)	1.28E+8	1463.34	1.14E+8
$4{f}^{2}{(}^{3}F)5d{}^{4}H$	5.5	$4{f}^{3}{}^{4}I$	6.5	3.78E+7	5.67E+7	0.33	1324.01(−3.0)	8.92E+7	1285.61	6.48E+7
$4{f}^{2}{(}^{3}F)5d{}^{4}H$	4.5	$4{f}^{3}{}^{4}I$	5.5	4.11E+7	6.19E+7	0.34	1325.17(−3.1)	1.04E+8	1285.38	7.55E+7
$4{f}^{2}{(}^{3}H)5d{}^{4}H$	3.5	$4{f}^{3}{}^{4}I$	4.5	7.27E+7	1.03E+8	0.29	1401.4 (−4.2)	1.29E+8	1344.74	1.17E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	4.5	$4{f}^{2}{(}^{3}H)5d{}^{4}K$	5.5	1.01E+9	8.05E+8	0.20	1123.17(14.9)	7.61E+8	1319.25	7.29E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}G$	5.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	6.5	1.55E+8	1.65E+8	0.06	2157.57(18.8)	2.17E+8	2656.02	2.16E+8
$4{f}^{2}{(}^{1}I)6p{}^{2}I$	5.5	$4{f}^{2}{(}^{1}I)6s{}^{2}I$	5.5	2.30E+8	2.17E+8	0.06	2443.54(10.3)	1.70E+8	2723.51	1.86E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}H$	4.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	5.5	2.02E+8	2.07E+8	0.02	2416.08(9.5)	2.00E+8	2670.03	2.04E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}H$	3.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	4.5	1.95E+8	2.00E+8	0.02	2423.25(9.5)	1.99E+8	2678.00
$4{f}^{2}{(}^{3}H)5d{}^{4}H$	5.5	$4{f}^{3}{}^{4}I$	6.5	7.27E+7	1.04E+8	0.30	1391.9 (−4.1)	1.29E+8	1336.98	1.00E+8
$4{f}^{2}{(}^{3}H)5d{}^{4}I$	7.5	$4{f}^{3}{}^{4}I$	7.5	4.69E+7	8.18E+7	0.43	1432.37(−4.4)	9.05E+7	1372.20	7.44E+7
$4{f}^{2}{(}^{1}I)6p{}^{2}I$	5.5	$4{f}^{2}{(}^{1}I)6s{}^{2}I$	6.5	1.16E+8	1.20E+8	0.03	2443.59(10.3)	1.17E+8	2723.51	1.22E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}H$	4.5	$4{f}^{2}{(}^{3}H)6s{}^{2}H$	4.5	1.38E+8	1.30E+8	0.05	2436.05(9.6)	1.27E+8	2694.71
$4{f}^{2}{(}^{3}H)5d{}^{4}H$	4.5	$4{f}^{3}{}^{4}I$	5.5	7.02E+7	9.97E+7	0.30	1397.63(−4.1)	1.24E+8	1342.01	1.10E+8
$4{f}^{2}{(}^{3}H)6p{}^{2}I$	6.5	$4{f}^{2}{(}^{3}H)6s{}^{2}H$	5.5	3.29E+8	2.88E+8	0.13	2174 (8.3)	2.84E+8	2370.51	2.94E+8
$4{f}^{2}{(}^{3}F)6p{}^{4}D$	3.5	$4{f}^{2}{(}^{3}F)6s{}^{4}F$	4.5	2.86E+8	2.88E+8	0.01	2394.67(9.4)	2.89E+8	2643.03	2.93E+8
$4{f}^{2}{(}^{1}I)6p{}^{2}K$	6.5	$4{f}^{2}{(}^{1}I)6s{}^{2}I$	6.5	7.68E+7	7.75E+7	0.01	2181.33(8.7)	8.11E+7	2388.79	1.26E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	7.5	$4{f}^{2}{(}^{3}H)5d{}^{4}K$	8.5	1.34E+9	1.07E+9	0.20	1093.02(14.5)	6.16E+8	1278.41	6.63E+8
$4{f}^{2}{(}^{3}H)6p{}^{2}I$	6.5	$4{f}^{2}{(}^{3}H)5d{}^{2}K$	7.5	7.92E+8	6.02E+8	0.24	1182.67(14.6)	4.37E+8	1385.21	4.89E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	6.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	5.5	4.95E+8	4.43E+8	0.10	2132.31(8.1)	3.65E+8	2320.43	4.09E+8
$4{f}^{2}{(}^{3}F)6p{}^{4}G$	5.5	$4{f}^{2}{(}^{3}F)6s{}^{4}F$	4.5	5.00E+8	4.56E+8	0.09	2137.32(7.8)	4.15E+8	2318.07	3.98E+8
$4{f}^{2}{(}^{3}H)6p{}^{2}I$	5.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	4.5	1.90E+7	1.56E+7	0.18	2035.66(11.5)	3.96E+8	2300.68	4.16E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}H$	5.5	$4{f}^{2}{(}^{3}H)5d{}^{4}H$	5.5	2.85E+8	2.39E+8	0.16	1136.61(15.1)	1.54E+8	1338.62	1.03E+8
$4{f}^{2}{(}^{1}I)5d{}^{2}K$	6.5	$4{f}^{3}{}^{2}L$	7.5	6.41E+7	8.42E+7	0.24	1501.86(−5.3)	1.27E+8	1426.05	1.14E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}I$	4.5	$4{f}^{2}{(}^{3}H)6s{}^{4}H$	4.5	1.78E+8	1.65E+8	0.07	2443.44(9.8)	1.71E+8	2708.43	1.70E+8
$4{f}^{2}{(}^{3}H)6p{}^{4}H$	5.5	$4{f}^{2}{(}^{3}H)6s{}^{2}H$	5.5	1.16E+8	1.10E+8	0.06	2462.26(9.6)	1.18E+8	2723.34	1.23E+8
$4{f}^{2}{(}^{3}H)5d{}^{4}I$	6.5	$4{f}^{3}{}^{4}I$	6.5	4.40E+7	7.69E+7	0.43	1440.32(−4.5)	8.54E+7	1378.09	6.95E+7
$4{f}^{2}{(}^{1}I)5d{}^{2}I$	6.5	$4{f}^{3}{}^{2}K$	7.5	3.75E+7	4.99E+7	0.25	1566.59(−5.4)	7.14E+7	1485.64	6.43E+7

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Table 9.  Energy Levels (in cm⁻¹) Relative to the Ground State for the Lowest States of Nd ii

No.	label	J	P	E
1	$4{f}^{4}{(}_{1}^{5}I)6s{}^{6}I$	7/2	+	0.00
2	$4{f}^{4}{(}_{1}^{5}I)6s{}^{6}I$	9/2	+	547.86
3	$4{f}^{4}{(}_{1}^{5}I)6s{}^{6}I$	11/2	+	1404.36
4	$4{f}^{4}{(}_{1}^{5}I)6s{}^{4}I$	9/2	+	1789.52
5	$4{f}^{4}{(}_{1}^{5}I)6s{}^{6}I$	13/2	+	2425.58
6	$4{f}^{4}{(}_{1}^{5}I)6s{}^{4}I$	11/2	+	3120.96
7	$4{f}^{4}{(}_{1}^{5}I)6s{}^{6}I$	15/2	+	3564.21
8	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}L$	11/2	+	4019.24
9	$4{f}^{4}{(}_{1}^{5}I)6s{}^{4}I$	13/2	+	4506.07
10	$4{f}^{4}{(}_{1}^{5}I)6s{}^{6}I$	17/2	+	4789.43
11	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}L$	13/2	+	4941.03
12	$4{f}^{3}{(}_{1}^{4}I)5{d}^{2}{(}_{2}^{3}F){}^{6}M$	13/2	−	5477.69
13	$4{f}^{4}{(}_{1}^{5}I)6s{}^{4}I$	15/2	+	5940.53
14	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}L$	15/2	+	5965.42
15	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}K$	9/2	+	6065.11
16	$4{f}^{3}{(}_{1}^{4}I)5{d}^{2}{(}_{2}^{3}F){}^{6}M$	15/2	−	6750.71
17	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}K$	11/2	+	6881.31
18	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}L$	17/2	+	7078.84
19	$4{f}^{4}{(}_{1}^{5}I)5d{}^{6}K$	13/2	+	7802.57
20	$4{f}^{3}{(}_{1}^{4}I)5{d}^{2}{(}_{2}^{3}F){}^{6}M$	17/2	−	8147.61

Note. Table 9 is published in its entirety in machine-readable format. A portion of the table is shown here for guidance regarding its form and content.

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Table 10.  Transition Energies ${\rm{\Delta }}E$ (in cm⁻¹), Transition Wavelengths λ (in Å), Line Strengths S (in a.u.), Weighted Oscillator Strengths gf, and Transition Rates A (in s⁻¹) for E1 Transitions of the Nd ii ion

upper	lower	${\rm{\Delta }}E$	λ	S	gf	A	dT
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}K\,6p{}^{6}{L}_{11/2}$	22745	4396.48	4.593E+00	3.173E−01	9.127E+06	0.050
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}L\,6p{}^{6}{L}_{11/2}$	23391	4275.02	1.001E+01	7.115E−01	2.164E+07	0.154
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}K\,6p{}^{6}{K}_{11/2}$	24959	4006.51	7.628E−01	5.783E−02	2.002E+06	0.130
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}K\,6p{}^{6}{L}_{11/2}$	25373	3941.14	1.380E+00	1.063E−01	3.807E+06	0.033
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}K\,6p{}^{6}{I}_{11/2}$	26882	3719.95	2.049E−01	1.673E−02	6.722E+05	0.041
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}L\,6p{}^{6}{K}_{11/2}$	27650	3616.62	1.213E−01	1.018E−02	4.330E+05	0.069
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}K\,6p{}^{6}{I}_{11/2}$	28842	3467.14	5.807E−02	5.088E−03	2.352E+05	0.351
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{6}{K}_{11/2}$	29386	3402.93	3.867E−01	3.452E−02	1.657E+06	0.120
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{3}I\,6p{}^{4}{K}_{11/2}$	29450	3395.57	1.159E−02	1.037E−03	5.002E+04	0.124
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{6}{K}_{11/2}$	30233	3307.58	5.567E−02	5.112E−03	2.597E+05	0.164
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{3}I\,6p{}^{4}{H}_{11/2}$	30410	3288.31	3.589E−02	3.316E−03	1.704E+05	0.320
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{6}{I}_{11/2}$	30982	3227.58	2.441E−04	2.297E−05	1.226E+03	0.999
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{6}{I}_{11/2}$	31155	3209.69	5.477E−02	5.184E−03	2.797E+05	0.186
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{4}{K}_{11/2}$	31541	3170.46	3.420E−02	3.276E−03	1.812E+05	0.373
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{4}{H}_{11/2}$	31848	3139.89	2.205E−04	2.133E−05	1.202E+03	0.541
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{3}H\,6p{}^{4}{I}_{11/2}$	32432	3083.34	5.245E−03	5.167E−04	3.021E+04	0.319
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{6}{H}_{11/2}$	33244	3008.02	2.221E−03	2.242E−04	1.377E+04	0.427
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}H\,6p{}^{6}{H}_{11/2}$	33332	3000.04	1.985E−05	2.010E−06	1.241E+02	0.938
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}H\,6p{}^{6}{G}_{11/2}$	33384	2995.44	1.588E−04	1.611E−05	9.980E+02	0.446
$4{f}^{3}{(}_{1}^{4}I)\,5{d}^{2}{(}_{2}^{3}F){}^{6}{L}_{11/2}$	$4{f}^{3}{(}_{1}^{4}I)\,5d{}^{5}I\,6p{}^{6}{I}_{11/2}$	33499	2985.08	8.480E−04	8.629E−05	5.383E+03	0.424

Note. All transition data are in length form. dT is the relative difference of the transition rates in length and velocity form as given by Equation (5). Table 10 is published in its entirety in machine-readable format. A portion of the table is shown here for guidance regarding its form and content.

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We also confirmed a nice agreement in the transition probabilities. Table 8 and Figure 6 show transition probabilities for the strongest transitions computed by Wyart et al. (2007). Our results and theirs agree within a factor of 2. Note that semi-empirical calculations have uncertainties. Using the same HFR method combined with parametric least-squares fits to the same experimental data with Wyart et al. (2007), Yoca & Quinet (2014) computed and presented transition probabilities (only with ${log}\,{gf}\geqslant -1.0$), oscillator strengths, and radiative lifetimes in bigger multiconfiguration expansions than Wyart et al. (2007). Their results are systematically different, and those by Yoca & Quinet (2014) in fact show a slightly better agreement with ours, as shown in the bottom panel of Figure 6.

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Table 11.  Energy Levels (in cm⁻¹) Relative to the Ground State for the Lowest States of Nd iii

No.	label	J	P	E
1	$4{f}^{4}{(}_{1}^{5}I){}^{5}I$	4	+	0.00
2	$4{f}^{4}{(}_{1}^{5}I){}^{5}I$	5	+	1072.58
3	$4{f}^{4}{(}_{1}^{5}I){}^{5}I$	6	+	2263.93
4	$4{f}^{4}{(}_{1}^{5}I){}^{5}I$	7	+	3542.92
5	$4{f}^{4}{(}_{1}^{5}I){}^{5}I$	8	+	4884.66
6	$4{f}^{4}{(}_{1}^{5}F){}^{5}F$	1	+	11739.04
7	$4{f}^{4}{(}_{1}^{5}F){}^{5}F$	2	+	12098.57
8	$4{f}^{4}{(}_{1}^{5}F){}^{5}F$	3	+	12724.33
9	$4{f}^{4}{(}_{0}^{5}S){}^{5}S$	2	+	13433.63
10	$4{f}^{4}{(}_{1}^{5}F){}^{5}F$	4	+	13459.96
11	$4{f}^{4}{(}_{1}^{5}F){}^{5}F$	5	+	14444.26
12	$4{f}^{4}{(}_{2}^{3}K){}^{3}K$	6	+	15065.97
13	$4{f}^{3}{(}_{1}^{4}I)5d{}^{5}K$	5	−	15128.43
14	$4{f}^{3}{(}_{1}^{4}I)5d{}^{5}L$	6	−	15257.69
15	$4{f}^{4}{(}_{4}^{3}H){}^{3}H$	4	+	16151.40
16	$4{f}^{4}{(}_{2}^{3}K){}^{3}K$	7	+	16180.31
17	$4{f}^{3}{(}_{1}^{4}I)5d{}^{5}K$	6	−	16720.95
18	$4{f}^{3}{(}_{1}^{4}I)5d{}^{5}L$	7	−	16985.08
19	$4{f}^{4}{(}_{1}^{5}G){}^{5}G$	2	+	17295.05
20	$4{f}^{4}{(}_{1}^{5}G){}^{5}G$	3	+	17368.09

Note. Table 11 is published in its entirety in machine-readable format. A portion of the table is shown here for guidance regarding its form and content.

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Table 12.  Transition Energies ${\rm{\Delta }}E$ (in cm⁻¹), Transition Wavelengths λ (in Å), Line Strengths S (in a.u.), Weighted Oscillator Strengths gf, and Transition Rates A (in s⁻¹) for E1 Transitions of the Nd iii Ion

upper	lower	${\rm{\Delta }}E$	λ	S	gf	A	dT
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}F)5d{}^{5}{D}_{1}$	6521	15334.41	2.338E−03	4.632E−05	4.380E+02	0.948
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}F)5d{}^{3}{P}_{1}$	9156	10920.99	1.996E−02	5.553E−04	1.035E+04	0.861
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}F)5d{}^{5}{P}_{1}$	10206	9798.04	5.262E−03	1.631E−04	3.778E+03	0.787
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}F)5d{}^{5}{F}_{1}$	10697	9348.19	8.930E−03	2.901E−04	7.382E+03	0.822
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}S)5d{}^{5}{D}_{1}$	11516	8683.29	3.115E−05	1.090E−06	3.214E+01	0.890
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}S)5d{}^{3}{D}_{1}$	12113	8255.57	1.272E−02	4.681E−04	1.527E+04	0.752
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}F)5d{}^{3}{D}_{1}$	14250	7017.52	9.589E−02	4.150E−03	1.874E+05	0.710
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}G)5d{}^{3}{D}_{1}$	17425	5738.84	7.686E−02	4.068E−03	2.746E+05	0.647
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}G)5d{}^{5}{F}_{1}$	18400	5434.62	5.429E−04	3.034E−05	2.284E+03	0.634
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}P)5d{}^{3}{P}_{1}$	19753	5062.36	1.351E−02	8.110E−04	7.036E+04	0.786
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}G)5d{}^{5}{D}_{1}$	20499	4878.06	4.216E−03	2.625E−04	2.453E+04	0.562
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}D)5d{}^{3}{P}_{1}$	20588	4857.14	6.608E−02	4.132E−03	3.894E+05	0.679
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}D)5d{}^{3}{S}_{1}$	21850	4576.58	4.184E−02	2.777E−03	2.948E+05	0.711
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}F)6s{}^{5}{F}_{1}$	22166	4511.37	1.708E−05	1.150E−06	1.257E+02	0.812
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}D)5d{}^{3}{S}_{1}$	22582	4428.21	7.961E−03	5.461E−04	6.192E+04	0.707
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}P)5d{}^{3}{D}_{1}$	23964	4172.76	1.943E−02	1.415E−03	1.806E+05	0.137
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}S)6s{}^{3}{S}_{1}$	24808	4030.93	4.263E−04	3.213E−05	4.396E+03	0.318
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}S)6s{}^{3}{S}_{1}$	25324	3948.70	3.464E−03	2.665E−04	3.800E+04	0.784
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{2}D)5d{}^{1}{P}_{1}$	26223	3813.37	2.834E−03	2.257E−04	3.452E+04	0.389
$4{f}^{4}{(}_{2}^{3}P){}^{3}{P}_{0}$	$4{f}^{3}{(}_{1}^{4}D)5d{}^{5}{F}_{1}$	27992	3572.45	4.279E−09	3.638E−10	6.339E-02	0.999

Note. All transition data are in length form. dT is the relative difference of the transition rates in length and velocity form as given by Equation (5). Table 12 is published in its entirety in machine-readable format. A portion of the table is shown here for guidance regarding its form and content.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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