Journal of Atomic and Molecular PhysicsVolume 2013 (2013), Article ID 674242, 12 pageshttp://dx.doi.org/10.1155/2013/674242

Research Article

Energy Levels and the Landé -Factors for Singly Ionized Lanthanum

Betül Karaçoban and Leyla Özdemir

Department of Physics, Sakarya University, 54187 Sakarya, Turkey

Received 25 January 2013; Accepted 20 April 2013

Academic Editor: Gregory Lapicki

Copyright © 2013 Betül Karaçoban and Leyla Özdemir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We have calculated the energies and the Landé -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.

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 number and fills in the first (internal) minimum of the electrostatic potential [1]. The rare-earth element lanthanum () 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 Landé -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] . 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 -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
d² ( = 5, 6), 4f², 5d6d, 6s6d, 4fp, 5ds, s², p² ( = 6–9), 6ss, 6pp, 6ds ( = 7–9), 7ss, 7pp ( = 8, 9), 8s9s, 8p9p
As in calculation A
5p⁶5d², 5p⁶5d6s,5p⁶6s², 5p⁶4f6p,5p⁶4f², 5p⁵4f6s²,5p⁵5d²6p, 5p⁵6s²6p
5p⁶d² ( = 5, 6), 5p⁶4f², 5p⁶5d6d, 5p⁶6s6d, 5p⁶4fp, 5p⁶5ds, 5p⁶s², 5p⁶p² ( = 6–9), 5p⁶6ss, 5p⁶6ds, 5p⁶6pp ( = 7–9), 5p⁶7ss ( = 8, 9), 5p⁶7p8p, 5p⁶8s9s, 5p⁶8p9p

Odd-parity
4f6s, 4f5d, 5dp, 6sp ( = 6–9), 6ps, 7sp ( = 7–9), 7ps, 8sp ( = 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 coupling, The mixing coefficients and the radial orbitals are optimized simultaneously, based on the expectation values .

In the MCHF method, the Breit-Pauli Hamiltonian for relativistic corrections is taken as a perturbation with order . The Breit-Pauli Hamiltonian includes relativistic effects. This Hamiltonian can be written as follows: where is the nonrelativistic many-electron Hamiltonian and 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 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 where is the Hamiltonian matrix with the following elements The Breit-Pauli Hamiltonian is a first order perturbation correction to the nonrelativistic Hamiltonian. The Landé -factor of an atomic level is related to the energy shift of the sublevels having magnetic number by where is the magnetic field intensity and is the Bohr magneton. In pure coupling, the Landé -factor can be taken given by formula (8) in [43]. The Landé -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 electron atom of nuclear charge , the Hamiltonian is expanded as in atomic units with the distance of the th electron from the nucleus and . is the spin-orbit term with the fine structure constant and the mean potential field due to the nucleus and other electrons.

Wave function of the sublevel of a level labeled is expressed in terms of basis states by the following formula:

Using determinant wave functions for the atom, total binding energy is given by where is the kinetic energy, is the electron-nuclear energy, and is the Coulomb interaction energy between electron and , 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 of , 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:

The Landé -factors in HFR calculations have the same formula as in MCHF calculations. Here, the calculations for the Landé -factors have been performed according to HFR wave functions.

3. Results and Discussion

Here, we have calculated the energies and the Landé -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 -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 -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, , and the Landé -factors for some levels in La II.

Levels
(cm⁻¹)
-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
—

Reference [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 Landé -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 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 Landé -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. levels obtained from calculations A and D and 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 and, in particular, subshells. The configuration including these subshells complicates the calculations in the MCHF method. Results that obtained the Landé -factors are in quite good agreement with others.

We have studied with two configuration sets (in Table 1) 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 ( and ) and of configuration interaction integrals () 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 -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 -factors from the HFR calculation obtained according to A and B configuration sets and other works is very good.

These energy data and the Landé -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 -factor is important. The experimental values for Landé -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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