# Energy Levels and the Landég-Factors for Singly Ionized Lanthanum

Authors: Betül Karaçoban; Leyla Özdemir
Journal: Journal of Atomic and Molecular Physics
 (2013)

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## Abstract

We have calculated the energies and the Landég-factors for 5d2, 5d6s, 6s2, 4f6p, 5d7s, 5d6d, 4f2, 6p2, 6s6d, 6s7s, 4f6s, 4f5d, 5d6p, 6s6p, 4f7s, 4f6d, 5d7p, and 6s7p excited levels of singly ionized lanthanum (La II). These calculations have been carried out by using the multiconfiguration Hartree-Fock method within the framework of the Breit-Pauli Hamiltonian (MCHF+BP) and the relativistic Hartree-Fock (HFR) method. The obtained results have been compared with other works available in the literature. A discussion of these calculations for La II in this study has also been in view of the MCHF+BP and HFR methods.


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## Body

## 1. Introduction

Lanthanides constitute a group of elements characterized by similar chemical and physical properties. The presence of this group in the periodic table is determined by behaviour of the radial part of the wave function for the 4f orbital, which collapses for the values of atomic numberZ>57 and fills in the first (internal) minimum of the electrostatic potential [1]. The rare-earth element lanthanum (Z=57) is a product of neutron-capture fusion reactions that occur in the late stages of stellar evolution. Lanthanum’s abundance relative to other rare earths in stars of different metallicities can lead to insights on the nature of the dominant neutron-capture production sites throughout the Galaxy’s history [2].The lanthanum atom is the first member of the rare-earth elements. It has two naturally occurring isotopes:138La (0.085%) and 139La (99.910%). Meggers viewed first the spectra of singly ionized lanthanum [3, 4]. Later, Russell and Meggers analyzed the spectra of La II [5]. Grevesse and Blanquet determined the abundance of singly ionized lanthanum in the sun [6]. Spector and Gotthelf performed configuration interaction in La II [7]. Xie and coworkers investigated Rydberg and autoionization states of the singly ionized lanthanum [8]. First, ionization potential of lanthanides by laser spectroscopy was studied by Worden et al. [9]. Sugar and Reader obtained by means of a semiempirical calculation ionization potential of singly ionized lanthanum [10]. Eliav et al. reported ionization potential and excitation energies of La II [11]. Theoretical energy levels in La II were calculated by Kułaga-Egger and Migdałek [12]. In the past, different groups [13–26] investigated oscillator strengths, transition probabilities, lifetimes, and the hyperfine structure in singly ionized lanthanum by various experimental and theoretical methods. A list of energy levels for excited states was completed and presented by Sansonetti and Martin [27] and can be found on the NIST website [28].In this work, we have presented the energies and the Landég-factors for 5d2, 5d6s, 6s2, 4f6p, 5d7s, 5d6d, 4f2, 6p2, 6s6d, 6s7s, 4f6s, 4f5d, 5d6p, 6s6p, 4f7s, 4f6d, 5d7p, and 6s7p excited levels of singly ionized lanthanum (La II). Calculations have been carried out by the multiconfiguration Hartree-Fock method within the framework of the Breit-Pauli Hamiltonian (MCHF+BP) [29] and the relativistic Hartree-Fock (HFR) method [30]. These codes consider the correlation effects and relativistic corrections. These effects make an important contribution to understanding physical and chemical properties of atoms or ions, especially lanthanides. The ground-state configuration of La II is [Xe] 5d2F32. We took into account with various configuration sets according to valence-valence and core-valence correlations for correlation effects outside the core [Xe] in La II. These configuration sets used in calculations have been denoted by A, B, C, and D for the MCHF+BP, and A and B for the HFR calculations and given in Table 1. We have performed the atomic structure calculations on lanthanide atoms and ions, systematically. We reported some works related to these atoms and ions using the methods mentioned above [31–42]. In addition, we presented the energies of 5d2, 5d6s, 6s2, and 6p2 excited levels and ionization energy for La II by the MCHF+BP method [34]. In this work, we have considered more levels than in [34] and added the Landé g-factors for these levels.Table 1
Configuration sets taken for La II in MCHF+BP and HFR calculations.
Levels
Configurations
A
B
C
D
For MCHF+BP calculations
Even-parity
nd2
(n = 5, 6), 4f2, 5d6d, 6s6d, 4fnp, 5dns, ns2, np2 (n = 6–9), 6sns, 6pnp, 6dns (n = 7–9), 7sns, 7pnp (n = 8, 9), 8s9s, 8p9p
As in calculation A
5p65d2, 5p65d6s,5p66s2, 5p64f6p,5p64f2, 5p54f6s2,5p55d26p, 5p56s26p
5p6nd2 (n = 5, 6), 5p64f2, 5p65d6d, 5p66s6d, 5p64fnp, 5p65dns, 5p6ns2, 5p6np2 (n = 6–9), 5p66sns, 5p66dns, 5p66pnp (n = 7–9), 5p67sns (n = 8, 9), 5p67p8p, 5p68s9s, 5p68p9p
Odd-parity
4f6s, 4f5d, 5dnp, 6snp (n = 6–9), 6pns, 7snp (n = 7–9), 7pns, 8snp (n = 8, 9), 8p9s, 9s9p
As in calculation A
5p64f6s, 5p64f5d, 5p65d6p, 5p54f26s, 5p55d6s2, 5p55d6p2
For HFR calculations
Even-parity
5d2, 5d6s, 6s2, 4f6p, 4f2
5d2, 5d6s, 6s2, 4f6p, 5d7s, 5d6d, 4f2, 6p2, 6s6d, 6s7s
Odd-parity
4f6s, 4f5d, 5d6p, 6s6p
4f6s, 4f5d, 5d6p, 6s6p, 4f7s, 4f6d, 5d7p, 6s7p


## 2. Calculation Methods: MCHF and HFR

In the MCHF method [29], atomic state functions can be obtained as a linear combination of configuration state functions (CSFs) in LS coupling,
(1)Ψ(γLS)=∑i=1MciΦ(γiLS),∑i=1Mci2=1.
The mixing coefficients {ci} and the radial orbitals are optimized simultaneously, based on the expectation values 〈Ψ|H|Ψ〉.In the MCHF method, the Breit-Pauli Hamiltonian for relativistic corrections is taken as a perturbation with orderα2. The Breit-Pauli Hamiltonian includes relativistic effects. This Hamiltonian can be written as follows:
(2)HBP=HNR+HRS+HFS,
where HNR is the nonrelativistic many-electron Hamiltonian and HRS is the relativistic shift operator including mass correction, one- and two-body Darwin terms, spin-spin contact term, and orbit-orbit term; fine structure Hamiltonian HFS consists of the spin-orbit, spin-other-orbit, and spin-spin terms. Now, the multiconfiguration wave functions are obtained as linear combinations of CSFs in LSJ coupling. Therefore, the radial functions building the CSFs are taken from a previous nonrelativistic MCHF run and only the expansion coefficients are optimized. Therefore, the matrix eigenvalue problem becomes
(3)Hc=Ec,
where H is the Hamiltonian matrix with the following elements
(4)Hij=〈γiLiSiJM|HBP|γjLjSjJM〉.
The Breit-Pauli Hamiltonian is a first order perturbation correction to the nonrelativistic Hamiltonian. The Landé g-factor of an atomic level is related to the energy shift of the sublevels having magnetic number M by
(5)ΔE(γLS)=μBBgγJM,
where B is the magnetic field intensity and μB is the Bohr magneton. In pure LS coupling, the Landé g-factor can be taken given by formula (8) in [43]. The Landé g-factors for energy levels are a valuable aid in the analysis of a spectrum. These factors which are a measure of the magnetic sensitivity of atomic levels can be calculated using the code developed by Jönsson and Gustafsson [43] according to MCHF wave functions.In the HFR method [30], for N electron atom of nuclear charge Z0, the Hamiltonian is expanded as
(6)H=-∑i∇i2-∑i2Z0ri+∑i>j2rij+∑iζi(ri)li·si
in atomic units with the distance ri of the ith electron from the nucleus and rij=|ri-rj|. ζi(R)=(α2/2)(1/r)(∂V/∂r) is the spin-orbit term with α the fine structure constant and V the mean potential field due to the nucleus and other electrons.Wave function|γJM〉 of the M sublevel of a level labeled γJ is expressed in terms of LS basis states |αLSJM〉 by the following formula:
(7)|γJM〉=∑αLS|αLSJM〉〈αLSJ∣γJ〉.Using determinant wave functions for the atom, total binding energy is given by(8)E=∑i(Eki+Eni+∑j<iEij),
where Eki is the kinetic energy, Eni is the electron-nuclear energy, and Eij is the Coulomb interaction energy between electron i and j, averaged over all possible magnetic quantum numbers.This method calculates one-electron radial wave functions for each of any number of specified electron configurations, using the Hartree-Fock or any of several more approximate methods. It obtains the center-of-gravity energy of each configuration and those radial Coulomb and spin-orbit integrals required to calculate the energy levels for the configuration. After the wave functions have been obtained, they are used to calculate the configuration-interaction Coulomb integrals between each pair of interacting configurations. Then, energy matrices are set up for each possible value ofJ, and each matrix is diagonalized to get eigenvalues (energy levels) and eigenvectors (multiconfiguration, intermediate coupling wave functions in various possible angular-momentum coupling representations).Relativistic corrections to total binding energies become quite large for heavy elements; the main contributions come from the tightly bound inner electrons. In the HFR method there have been limited calculations to the mass-velocity and Darwin corrections by using relativistic correction to total binding energy:(9)Er=∑iEri=∑i(Emi+EDi).The Landég-factors in HFR calculations have the same formula as in MCHF calculations. Here, the calculations for the Landé g-factors have been performed according to HFR wave functions.


## 3. Results and Discussion

Here, we have calculated the energies and the Landég-factors for 5d2, 5d6s, 6s2, 4f6p, 5d7s, 5d6d, 4f2, 6p2, 6s6d, 6s7s, 4f6s, 4f5d, 5d6p, 6s6p, 4f7s, 4f6d, 5d7p, and 6s7p excited levels outside the core [Xe] in La II using the MCHF+BP [44] and HFR [45] codes. We have presented the several excited level energies and transition energies for La I and La II [34]. We have performed new various calculations for obtaining configuration state functions (CSFs) according to valence-valence and core-valence correlations. Moreover, it is well known that the Landé g-factors are important in many scientific areas such as astrophysics. Table 1 displays the various configuration sets for considering correlation effects. The results for energy levels and the Landé g-factors of La II have been reported in Table 2. In this table, the calculations for the various configuration sets are represented by A, B, C, and D for the MCHF+BP and by A and B for the HFR calculations. A comparison is also made with other calculations and experiments in the table. References for other comparison values are typed below the table with a superscript lowercase letter. Only odd-parity states in table are indicated by the superscript “o.”Table 2
Energies,E, and the Landé g-factors for some levels in La II.
Levels
E (cm−1)
g-Factors
Conf.
Term
This work
Other works
This work
Other works
MCHF+BP
HFR
MCHF+BP
HFR
For  even-parity
5d2
3F2
0.00A,B,C,D
−0.13A
0.00a
0.728A
0.732A
0.721a
0.02B
−4b
0.709B
0.736B
0.715C
0.722D
3F3
1100.72A
1018.92A
1016.100a
1.083A,B,C,D
1.084A,B
1.038a
816.18B
1046.86B
1028b
743.03C
898.90D
3F4
2131.38A
1935.49A
1970.700a
1.249A,B,C
1.249A,B
1.248a
1612.45B
1974.01B
1963b
1.250D
1456.54C
1741.31D
5d2
1D2
1379.80A
1394.91A
1394.460a
0.993A
0.965A
0.977a
1443.25B
1393.79B
1395b
0.968B,C
0.959B
1236.03C
0.955D
1398.12D
5d2
3P0
5377.78A
5228.71A
5249.700a
5918.77B
5265.77B
5244b
5575.39C
5654.01D
3P1
6000.10A
5769.81A
5718.120a
1.501A,B,C,D
1.501A,B
1.497a
6374.80B
5737.42B
5725b
5867.74C
6132.57D
3P2
6513.01A
6261.60A
6227.420a
1.485A
1.489A
1.481a
6798.78B
6239.70B
6224b
1.494B,D
1.488B
6247.36C
1.497C
6588.07D
5d2
1G4
8739.05A
7445.09A
7473.320a
1.002A
1.002A,B
1.000a
8468.71B
7374.70B
7476b
1.001B,C,D
8941.04C
9027.98D
5d2
1S0
16629.75A
16453.30A
—
18979.63B
13675.30B
18143.98D
5d6s
3D1
1662.59A
1893.01A
1895.150a
0.499A,B,C,D
0.499A,B
0.498a
2481.69B
1898.19B
1902b
1994.95C
2533.26D
3D2
2476.73A
2580.06A
2591.600a
1.119A
1.143A
1.133a
2939.18B
2549.72B
2572b
1.159B
1.144B
2550.88C
1.154C
3010.41D
1.160D
3D3
3094.79A
3311.82A
3250.350a
1.334A,B,C,D
1.334A,B
1.334a
3501.39B
3255.89B
3260b
3204.83C
3614.27D
5d6s
1D2
10064.87A
10088.91A
10094.800a?
1.010A
1.006A,B
1.005a
11480.98B
10095.00B
10096b
1.003B,D
13078.05C
1.001C
11980.57D
6s2
1S0
6428.50A
7371.89A
7394.570a
6664.52B
7393.30B
7395b
9713.71C
6709.28D
4f6p
3F2
—
35544.74A
35787.53a
—
0.669A
0.719a
35756.53B
35771b
0.747B
3F3
—
35687.88A
36954.65a
—
1.050A
1.061a
37062.30B
36953b
1.005B
3F4
—
37018.59A
37790.57a
—
1.212A
1.113a
37733.47B
37779b
1.105B
4f6p
1F3
—
36917.97A
37209.71a
—
0.934A
0.944a
37302.79B
37243b
1.036B
4f6p
3G3
—
37604.82A
35452.66a
—
0.854A
0.876a
35373.81B
35465b
0.856B
3G4
—
37768.59A
37172.79a
—
1.068A
1.127a
37186.27B
37157b
1.098B
3G5
—
39201.02A
39018.74a
—
1.200A,B
1.21a
39035.52B
39007b
4f6p
3D1
—
38123.37A
38534.11a
—
0.499A,B
0.497a
38536.02B
38545b
3D2
—
38214.22A
38221.49a
—
1.131A
1.071a
38070.87B
38210b
1.027B
3D3
—
39512.21A
39402.55a
—
1.329A
1.274a
39535.50B
39403b
1.270B
4f6p
1G4
—
38968.77A
39221.65a
—
1.021A
1.059a
39162.52B
39235b
1.097B
4f6p
1D2
—
40343.10A
40457.71a
—
1.033A
1.036a
40233.91B
40456b
1.059B
5d7s
3D1
—
49703.78B
49733.13a
—
0.500B
0.500a
49714b
3D2
—
49952.52B
49884.35a
—
1.128B
1.117a
49905b
3D3
—
51238.10B
51228.57a
—
1.307B
1.315a
51235b
5d7s
1D2
—
51501.10B
51523.86a
—
1.058B
1.036a
51516b
5d6d
1F3
70728.01A
51978.02B
52137.67a
0.978A
1.048B
0.987a
74919.76B
52216b
1.056B
5d6d
3D1
71108.98A
52220.02B
52169.66a
0.544A
0.557B
0.621a
75022.05B
52148b
0.622B
3D2
71922.34A
52746.60B
52734.81a
1.167A
1.126B
1.154a
75364.62B
52728b
1.033B
3D3
72905.11A
53276.41B
53689.56a
1.302A
1.297B
1.218a
75927.05B
53647b
1.119B
5d6d
1P1
74454.96A
53356.36B
53302.56a
1.048A
1.343B
1.335a
75644.43B
53317b
1.373B
5d6d
3F2
86857.66A
53855.01B
53885.24a
0.687A
0.741B
0.751a
75086.62B
53914b
0.847B
3F3
76042.84B
54392.59B
54840.04a
1.161B
1.050B
1.088a
54755b
3F4
88422.30A
54664.69B
55321.35a
1.196A
1.139B
1.136a
76155.48B
55303b
1.212B
5d6d
3S1
—
54175.01B
54365.80a
—
1.566B
1.455a
54370b
5d6d
1S0
74225.09B
54244.49B
54793.82a
5d6d
1D2
84447.02A
55024.06B
55184.05a
0.996A
1.056B
1.183a
76453.25B
55208b
1.051B
5d6d
3G3
69310.22A
55169.13B
52857.88a
0.803A
0.798B
0.861a
75153.22B
52878b
0.831B
3G4
70592.95A
55607.00B
53333.37a
1.050A
1.069B
1.036a
75339.70B
53368b
1.070B
3G5
71830.19A
56214.97B
54434.65a
1.200A,B
1.200B
1.21a
76224.45B
54435b
5d6d
3P0
90419.13A
55630.81B
54964.19a
76456.13B
54786b
3P1
90863.55A
55411.73B
55230.33a
1.501A
1.535B
1.552a
76858.62B
55352b
1.626B
3P2
91525.97A
55746.81B
56036.60a
1.488A
1.392B
1.203a
76983.87B
56090b
1.385B
5d6d
1G4
76564.89B
59068.02B
56035.70a
1.018B
1.056B
1.027a
55076b
4f2
3H4
—
55201.97A
55107.25a
—
0.839A
0.883a
53676.92B
55079b
0.887B
3H5
—
56092.25A
55982.09a
—
1.033A,B
1.033a
55318.80B
55995b
3H6
—
56863.10A
56837.94a
—
1.166A
1.14a
56547.89B
59845b
1.165B
4f2
1G4
—
56394.91A
59527.60a
—
0.974A
1.046a
55328.98B
59522b
1.016B
4f2
3F2
—
57396.38A
57399.58a
—
0.672A
0.675a
56706.59B
57385b
0.680B
3F3
—
57884.41A
57918.50a
—
1.084A
1.085a
57513.31B
57936b
1.083B
3F4
—
58533.37A
58259.41a
—
1.237A
1.196a
58315.69B
58264b
1.184B
4f2
1I6
—
61521.93A
62408.40a
—
1.001A
1.003a
61759.71B
1.002B
4f2
1D2
—
62065.79A
62026.27a
—
1.028A
1.054a
62417.80B
62029b
1.044B
4f2
3P0
—
63593.22A
63463.95a
—
64001.07B
63496b
3P1
—
63865.10A
63703.18a
—
1.501A,B
1.471a
64480.52B
63736b
3P2
—
64283.97A
64278.92a
—
1.467A
1.414a
65194.71B
64214b
1.449B
4f2
1S0
—
69782.30A
69505.06a
—
71321.50B
6p2
1D2
61118.75A
59830.40B
59900.08a
1.205A
1.061B
1.035a
63903.98B
59899b
1.013B
68314.29D
1.005D
6p2
3P0
57741.75A
60001.21B
60094.84a
57033.69B
60091b
58691.43D
3P1
58647.09A
60514.89B
61128.83a
1.501A,B,D
1.502B
1.528a
57752.71B
61132b
59347.37D
3P2
59269.90A
61418.50B
62506.36a
1.296A
1.435B
1.416a
58826.49B
62504b
1.488B
60398.00D
1.496D
6p2
1S0
70729.58A
66977.30B
66591.91a
83062.12B
82414.06D
6s6d
3D1
79297.74A
63251.92B
64361.28a
0.499A.B
0.499B
0.506a
86162.37B
64374b
3D2
79538.63A
63833.97B
64529.90a
1.133A
1.165B
1.217a
86172.16B
64509b
1.166B
3D3
80684.92A
64853.61B
64692.59a
1.334A.B
1.334B
—
86191.20B
64701b
6s6d
1D2
86532.15B
69906.50B
—
1.001B
1.003B
—
6s7s
3S1
—
61127.80B
60660.18a?
—
2.001B
1.955a
60660b
6s7s
1S0
—
63307.50B
For odd-parity
4f6s
3Fo2
14378.61A
14174.39A
14147.980a
0.666A,C
0.666A,B
0.664a
14264.12B
14267.68B
14184b
0.669B
13925.11C
3Fo3
14756.50A
14283.31A
14375.170a
1.060A,C
1.055A
1.056a
14858.62B
14256.22B
14338b
1.068B
1.047B
14291.70C
3Fo4
16270.43A
15651.91A
15698.740a
1.250A,B
1.250A
1.247a
16250.88B
15747.51B
15682b
1.251C
1.245B
15785.08C
4f6s
1Fo3
16349.78A
15822.29A
15773.770a
1.023A,C
1.028A
1.017a
16448.12B
15725.69B
15790b
1.016B
1.037B
15889.78C
4f5d
1Go4
23958.81A
16965.67A
16559.170a
0.971A,C
0.905A
0.969a
25542.89B
15525.19B
16630b
1.059B
1.005B
23797.47C
4f5d
3Fo2
25007.46A
17448.60A
17211.930a
0.725A
0.713A
0.754a
28148.54B
18658.83B
17196b
0.685B
0.680B
24733.62C
0.735C
3Fo3
25740.72A
18242.30A
18235.560a
1.091A
1.083A
1.086a
29158.71B
18866.08B
18215b
1.083B
1.079B
25479.83C
1.098C
3Fo4
27201.89A
19267.37A
19214.540a
1.225A
1.226A
1.232a
30358.80B
19233.10B
19199b
1.241B
1.241B
26932.00C
1.224C
4f5d
3Ho4
25669.19A
17727.74A
17825.620a
0.855A
0.920A
0.846a
27280.97B
17683.40B
17803b
0.862B
0.806B
25499.67C
0.856C
3Ho5
26756.81A
18250.23A
18580.410a
1.033A,B,C
1.034A,B
1.017a
28223.06B
18122.03B
18573b
26582.89C
3Ho6
28460.49A
19394.18A
19749.620a
1.167A,C
1.167A,B
1.178a
28275.22C
18802.98B
19767b
4f5d
1Do2
31071.25A
19151.58A
18895.410a
0.981A
0.970A
0.923a
32636.97B
21057.91B
18926b
0.997B
1.114B
30916.96C
0.982C
4f5d
3Go3
31665.41A
20315.59A
20402.820a
0.763A
0.757A
0.757a
34021.44B
21064.80B
20405b
0.758B
0.782B
31554.09C
0.764C
3Go4
32976.84A
21247.80A
21331.600a
1.050A,B,C
1.049A
1.049a
35195.62B
21629.90B
21324b
1.054B
32859.39C
3Go5
34349.12A
22107.93A
22282.900a
1.200A,B,C
1.200A
1.197a
36498.47B
21982.01B
22269b
1.199B
34188.45C
4f5d
3Do1
35406.25A
21395.42A
21441.730a
0.503A
0.508A
0.542a
37453.79B
22982.69B
21477b
0.501B
0.591B
35168.59C
0.512C
3Do2
35923.09A
22032.81A
22106.020a
1.170A
1.159A
1.167a
37936.33B
23003.68B
22112b
1.169B,C
1.195B
35808.76C
3Do3
36203.19A
22475.18A
22537.300a
1.264A
1.305A
1.288a
38173.89B
23296.42B
22483b
1.270B
1.251B
36085.58C
1.266C
4f5d
3Po0
39198.12A
22445.78A
22683.700a
40330.43B
22463.20B
22674b
38565.29C
3Po1
39440.20A
22667.40A
22705.150a
1.500A
1.481A
1.431a
40455.15B
22524.24B
22690b
1.498B
1.345B
38560.21C
1.485C
3Po2
40081.21A
23571.92A
23246.930a
1.497A
1.489A
1.459a
40802.14B
21850.66B
23294b
1.492B,C
1.337B
39052.77C
4f5d
1Fo3
38465.61A
24240.21A
24552.700a
1.061A
1.024A
1.034a
40442.67B
22482.21B
24569b
1.057B
1.055B
38356.86C
1.059C
4f5d
1Ho5
46448.76C
28811.47A
28525.710a
1.000C
1.001A,B
1.004a
27193.29B
28529b
4f5d
1Po1
47396.25C
30265.31A
30353.330a
1.004C
1.040A
1.074a
21477.10B
1.060B
5d6p
1Do2
22996.58A
24514.82A
24462.66a
0.969A
1.014A
0.887a
19478.98B
24310.90B
24567b
1.011B
1.017B
22618.34C
0.975C
5d6p
3Do1
24485.41A
25907.37A
25973.37a
0.502A
0.579A
0.782a
20646.65B
25815.48B
25839b
0.500B
0.507B
23978.78C
0.512C
3Do2
25339.58A
26457.83A
27388.11a
1.161A,B
1.133A
1.168a
21467.30B
26411.29B
27362b
1.139C
1.120B
24934.60C
3Do3
26382.52A
26950.18A
28315.25a
1.323A
1.316A
1.308a
22400.30B
26951.83B
28290b
1.332B
1.312B
25902.53C
1.316C
5d6p
1Po1
—
26059.89A
—
—
0.955A
—
30735.20B
1.064B
5d6p
3Fo2
27730.01A
27524.69A
26414.01a
0.702A
0.691A
0.825a
23795.02B
27633.61B
26409b
0.700B
0.705B
27441.41C
0.693C
3Fo3
28937.43A
27941.30A
26837.66a
1.083A,B,C
1.098A
1.088a
24648.21B
28199.61B
26828b
1.103B
28616.27C
3Fo4
30401.18A
28617.50A
28565.40a
1.248A,C
1.251A,B
1.245a
26075.37B
29206.78B
28531b
1.138B
30028.17C
5d6p
3Po0
27489.54A
31804.81A
31785.82a
23324.32B
32009.00B
31797b
27777.75C
3Po1
27767.31A
32243.52A
32160.99a
1.498A
1.497A
1.492a
23622.98B
32711.11B
32134b
1.500B
1.437B
27808.39C
1.473C
3Po2
28480.17A
33269.26A
29593b
1.469A
1.501A
—
24351.70B
33441.59B
1.460B
1.500B
28624.94C
1.482C
5d6p
1Fo3
34048.66A
32144.41A
32201.05a
0.999A
1.002A
1.005a
31107.39B
32749.00B
32273b
1.000B
1.003B
33748.76C
0.997C
6s6p
3Po0
36704.74A
27560.27A
27545.850a
32174.15B
28370.01B
27563b
3Po1
37150.85A
28103.04A
28154.550a?
1.498A
1.438A
1.267a
32612.03B
28403.35B
28147b
1.501B
1.496B
3Po2
37997.38A
29477.41A
29498.050a?
1.494A
1.500A
1.471a
33501.99B
28508.94B
33133b
1.491B
1.501B
6s6p
1Po1
—
45700.40A
45692.170a
—
1.002A
0.999a
45953.40B
45702b
1.121B
4f7s
3Fo2
—
60463.77B
—
—
0.667B
—
3Fo3
—
60470.13B
—
—
1.048B
—
3Fo4
—
61873.10B
—
—
1.250B
—
4f7s
1Fo3
—
61880.70B
—
—
1.036B
—
4f6d
3Fo2
—
57457.37B
—
—
0.798B
—
3Fo3
—
59105.69B
—
—
1.121B
—
3Fo4
—
61261.54B
—
—
1.135B
—
4f6d
1Go4
—
59353.91B
—
—
1.037B
—
4f6d
3Go3
—
57698.08B
—
—
0.854B
—
3Go4
—
60092.42B
—
—
0.998B
—
3Go5
—
61115.66B
—
—
1.082B
—
4f6d
3Do1
—
58144.14B
—
—
0.823B
—
3Do2
—
59943.80B
—
—
1.028B
—
3Do3
—
61591.58B
—
—
1.177B
—
4f6d
3Ho4
—
58359.62B
—
—
0.930B
—
3Ho5
—
60449.51B
—
—
1.081B
—
3Ho6
—
62719.38B
—
—
1.167B
—
4f6d
3Po0
—
60580.29B
—
—
3Po1
—
60380.02B
—
—
1.088B
—
3Po2
—
59703.78B
—
—
1.279B
—
4f6d
1Fo3
—
59802.99B
—
—
1.015B
—
4f6d
1Do2
—
61840.89B
—
—
1.233B
—
4f6d
1Po1
—
62234.11B
—
—
1.106B
—
4f6d
1Ho5
—
62355.62B
—
—
1.071B
—
5d7p
3Do1
—
44866.71B
—
—
0.749B
—
3Do2
—
44866.75B
—
—
1.141B
—
3Do3
—
47740.96B
—
—
1.267B
—
5d7p
3Fo2
—
44842.48B
—
—
0.691B
—
3Fo3
—
44879.84B
—
—
1.068B
—
3Fo4
—
47724.49B
—
—
1.251B
—
5d7p
3Po0
—
44877.71B
—
—
3Po1
—
44424.70B
—
—
1.073B
—
3Po2
—
47745.11B
—
—
1.271B
—
5d7p
1Do2
—
47754.57B
—
—
1.230B
—
5d7p
1Fo3
—
47767.78B
—
—
1.083B
—
5d7p
1Po1
—
48313.70B
—
—
1.057B
—
6s7p
3Po0
—
59907.82B
—
—
3Po1
—
60242.28B
—
—
1.451B
—
3Po2
—
61529.70B
—
—
1.495B
—
6s7p
1Po1
—
63316.70B
—
—
1.033B
—
aReference [28], breference [12], ? reference [28].Electron correlation effects and relativistic effects play an important role in the spectra of heavy elements. Thus, we have to consider these effects for lanthanides. However, it is very difficult to calculate the electron correlation for these atoms because of their complex structures. Although this provides useful information for understanding the correlation effect, computer constraints occur. Therefore, we increasingly varied some parameters in the MCHF atomic structure package (maximum number of eigenpairs, maximum number of configuration state functions, maximum number of terms, and maximum number of coefficients) so that the calculations for the configurations above could reasonably be made. The energies and the Landég-factors for 5d2, 5d6s, 6s2, 4f6p, 5d7s, 5d6d, 4f2, 6p2, 6s6d, 6s7s, 4f6s, 4f5d, 5d6p, 6s6p, 4f7s, 4f6d, 5d7p, and 6s7p excited levels outside the core [Xe] in La II are calculated by the MCHF+BP and HFR methods. The obtained results have been presented as energies (cm−1) relative to 5d2F32 ground state in Table 2.In the MCHF+BP calculations, it is taken into account core [Xe] for calculations A and B and core [Cd] for calculations C and D for La II. The odd- and even-parity levels of calculations A, B, and D are considered the only interaction between the valence electrons, whereas odd- and even-parity levels of calculation C were an included interaction between valence-valence electrons and core-valence electrons. In the MCHF+BP method, firstly, for configurations selected according to valence-valence and core-valence correlations, the non-relativistic wave functions and energies were obtained. In the configuration interaction method, energy levels are performed using wave functions obtained taking into account relativistic corrections. Then the Landég-factors for energy levels are calculated using the Zeeman program developed by Jönsson and Gustafsson [43].In the MCHF+BP calculations, we have reported that the levels belong to 5d2, 5d6s, 6s2, 5d6d, 6p2, and 6s6d for even-parity and 4f6s, 4f5d, 5d6p, and 6s6p for odd-parity in Table 2. We presented the energies of several even-parity levels from obtained calculation A in [34]. Although configurations of calculation B are the same with calculations of calculation A, MCHF run is different. When our MCHF+BP results were compared with others [12, 28], the energies for most of levels are in agreement. 5d2F3 levels obtained from calculations A and D and 5d2P3 levels obtained from calculations A and C are good in agreement with other works. The agreement is good for 5d6s 3D levels obtained from calculation C and 5d6s 1D level obtained from calculation A. The result obtained from calculation D for 6s2 level and the results obtained from calculation A for 6p2 are in agreement when comparing with others. The results obtained from calculations A and B are also somewhat in agreement for 5d6d and 6s6d levels. 4s6s, 5d6p, and 5d6p levels are good in agreement with others. The 4f5d level is somewhat poor according to other works. It can be said that these cases occur due to unfilled d and, in particular, f subshells. The configuration including these subshells complicates the calculations in the MCHF method. Results that obtained the Landé g-factors are in quite good agreement with others.We have studied with two configuration sets (in Table1) for considering correlation effects in the HFR calculations performed using Cowan’s computer code [45]. These configuration sets are also given in Table 1. This approach, although based on the Schrödinger equation, includes the relativistic effects like the mass-velocity corrections and the Darwin contribution beside the spin-orbit effect. In these calculations, the HFR method was combined with a least-squares optimization routine minimizing the discrepancies between observed and calculated energy levels. The scaling factors of the Slater parameters (Fk and Gk) and of configuration interaction integrals (Rk) were chosen equal to 0.75 for calculation A and 0.70 for calculation B, while the spin-orbit parameters were left at their ab initio values. These low values of the scaling factors have been suggested by Cowan for neutral heavy elements [30]. In Table 1, it is taken into account the configuration sets including core [Xe] for calculations A and B of La II. The energies and the Landé g-factors for 5d2, 5d6s, 6s2, 4f6p, 5d7s, 5d6d, 4f2, 6p2, 6s6d, 6s7s, 4f6s, 4f5d, 5d6p, 6s6p, 4f7s, 4f6d, 5d7p, and 6s7p excited levels are presented in Table 2. The agreement between our energies and the Landé g-factors from the HFR calculation obtained according to A and B configuration sets and other works is very good.These energy data and the Landég-factors for La II can be useful in investigations of some radiative properties and interpretation of many levels of La II. Because magnetic fields play a major role in many scientific areas such as astrophysics, the calculation of the Landé g-factor is important. The experimental values for Landé g-factors of rare-earth elements are far from being complete. We hope that the results reported in this work will be useful in the interpretation of atomic spectra of La II. Many feature characteristics of the spectra of neutral atom or ions of lanthanides remain preserved for lanthanide ions implemented in crystals. This is one reason for the wide interest in the application of lanthanides as active media in lasers. In addition, knowledge of electronic levels of lanthanides is important in astrophysics, since it allows precise determination of the abundance of particular elements. Also, the analysis of electronic levels is very important for a description of the interaction in creating chemical bonds or crystalline lattice. Consequently, we hope that our results obtained using the HFR and MCHF methods will be useful for other works in the future for La II spectra.



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*Source: 674242-2013-06-05.xml*