Theoretical study on excited states of ICl<sup>+</sup> molecular ion considering spin-orbit coupling

LI Rui; DOU Ronglong; GAO Ting; LI Qinan; SONG Chaoqun

doi:10.7498/aps.74.20250510

Abstract

The electronic structure of the ICl⁺ molecular ion is investigated by using high-level multireference configuration interaction (MRCI) method. To improve computational accuracy, Davidson corrections, spin-orbit coupling (SOC), and core-valence electron correlations effects are incorporated into the calculations. The potential energy curves (PECs) of 21 Λ-S states associated with the two lowest dissociation limits I⁺(¹D_g)+Cl(²P_u) and I⁺(³P_g)+Cl(²P_u) are obtained. The dipole moments (DMs) of the 21 Λ-S states of ICl⁺ are systematically studied, and the variations of DMs of the identical symmetry state (2²Σ⁺/3²Σ⁺ and 2²Π/3²Π) in the avoided crossing regions are elucidated by analyzing the dominant electronic configuration. When considering the SOC effect, the Λ-S states with the same Ω components may form new avoided crossing point, making the PECs more complex. With the help of calculated SOC matrix element, the interaction between crossing states can be elucidated. Spin-orbit coupling matrix elements involving the 2²Π, 3²Π, 1²Δ and 2²Δ states are calculated. By analyzing potential energy curves of these states and the nearby electronic states, the possible predissociation channels for 2²Π, 3²Π, 1²Δ and 2²Δ states are provided. Based on the computed PECs, the spectroscopic constants of bound Λ-S and Ω states are determined. The comparison of the spectroscopic constants between Λ-S and Ω states indicates that the SOC effect has an obvious correction to the spectroscopic properties of low-lying states. Finally, the transition properties between excited states and the ground state are studied. Based on the computed transition dipole moments and Franck-Condon factors, radiative lifetimes for the low-lying vibrational levels of excited states are evaluated. All the data presented in this paper are openly available at https://doi.org/10.57760/sciencedb.j 00213.00140.

Keywords:

Authors and contacts

Department of Physics, College of Science, Qiqihar University, Qiqihar 161006, China

Corresponding author: LI Rui, ruili06@mails.jlu.edu.cn

Funds: Project supported by the Natural Science Foundation of Heilongjiang Province, China (Grant No. LH2022A026), the Fundamental Research Funds in Heilongjiang Province Universities, China (Grant Nos. 145409330, 145309621), and the National Natural Science Foundation of China (Grant No. 42371113).

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2025年第74卷第12期123102《ICl+分子离子激发态的包含自旋-轨道耦合效应的理论研究》补充材料.rar

References

[1]	Sherwen T, Schmidt J A, Evans M J, Carpenter L J, Großmann K, Eastham S D, Jacob D J, Dix B, Koenig T K, Sinreich R, Ortega I, Volkamer R, Saiz-Lopez A, Prados-Roman C, Mahajan A S, Ordóñez C 2016 Atmos. Chem. Phys. 16 12239 Google Scholar
[2]	Vogt R, Sander R, Glasow R V, Crutzen P J 1999 J. Atmos. Chem. 32 375 Google Scholar
[3]	Calvert J G, Lindberg S E 2004 Atmos. Environ. 38 5087 Google Scholar
[4]	Küpper F C, Feiters M C, Olofsson B, Kaiho T, Yanagida S, Zimmermann M B, Carpenter L J, Luther G W, Lu Z, Jonsson M, Kloo L 2011 Angew. Chem. Int. Ed. 50 11598 Google Scholar
[5]	Evans S, Orchard A F 1971 Inorg. Chim. Acta. 5 81 Google Scholar
[6]	Potts A W, Price W C 1971 Trans. Faraday Soc. 67 1242 Google Scholar
[7]	Eland J H D 1979 J. Chem. Phys. 70 2926 Google Scholar
[8]	Dibeler V H, Walker J A, McCulloh K E, Rosenstock H M 1971 Int. J. Mass Spectro. Ion Phys. 7 209 Google Scholar
[9]	Venkateswarlu P 1975 Can. J. Phys. 53 812 Google Scholar
[10]	Tuckett R P, Castellucci E, Bonneau M 1985 Chem. Phys. 92 43 Google Scholar
[11]	Kaur D, Yencha A J, Donovan R J, Kvaran A, Hopkirk A 1993 Org. Mass Spectrom. 28 327 Google Scholar
[12]	Yencha A J, Lopes M C A, King G C 2000 Chem. Phys. Lett. 325 559 Google Scholar
[13]	Ridley T, Beattie D A, Cockett M C R, Lawley K P, Donovan R J 2002 Phys. Chem. Chem. Phys. 4 1398 Google Scholar
[14]	Straub P A, McLean A D 1974 Theoret. Chim. Acta 32 227 Google Scholar
[15]	Dyke J M, Josland G D, Snijders J G, Boerrigter P M 1984 Chem. Phys. 91 419.Google Scholar
[16]	Balasubramanian K 1985 Chem. Phys. 95 225 Google Scholar
[17]	Werner H, Knowles P J, Knizia G, Manby F R, Schütz M 2012 Wires Comput. Mol. Sci. 2 242 Google Scholar
[18]	Peterson K A, Yousaf K E 2010 J. Chem. Phys. 133 174116 Google Scholar
[19]	Peterson K A, Dunning Jr T H 2002 J. Chem. Phys. 117 10548 Google Scholar
[20]	Knowles P J, Werner H J 1985 Chem. Phys. Lett. 115 259 Google Scholar
[21]	Werner H J, Knowles P J 1985 J. Chem. Phys. 82 5053 Google Scholar
[22]	Knowles P J, Werner H J 1988 Chem. Phys. Lett. 145 514 Google Scholar
[23]	Werner H J, Knowles P J 1988 J. Chem. Phys. 89 5803 Google Scholar
[24]	Langhoff S R, Davidson E R 1974 Int. J. Quantum. Chem. 8 61 Google Scholar
[25]	Berning A, Schweizer M, Werner H J, Knowles P J, Palmieri P 2000 Mol. Phys. 98 1823 Google Scholar
[26]	Le Roy R J 2017 J. Quant. Spectrosc. Ra. 186 167 Google Scholar
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[32]	陈晨, 赵国鹏, 祁月盈, 吴勇, 王建国 2022 物理学报 71 143102 Google Scholar Chen C, Zhao G P, Qi Y Y, Wu Y, Wang J G 2022 Acta Phys. Sin. 71 143102 Google Scholar
[33]	Huber K P, Herzberg G 1979 Molecular Spectra and Molecular Structure (Vol. IV) (New York: Van Nostrand Reinhold) p342
[34]	Moore C E 1971 Atomic Energy Levels (Washington (DC): National Bureau of Standards Publications

Cited By

图 1 ICl⁺分子离子的Λ-S态势能曲线

Figure 1. Potential energy curves of the Λ-S states of ICl⁺ molecular ion.

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图 2 ICl⁺分子离子的Λ-S态偶极矩曲线　(a)双重态的偶极矩; (b)四重态的偶极矩

Figure 2. Dipole moments curves of the Λ-S states of ICl⁺ molecular ion: (a) Dipole moment of the doublet state; (b) dipole moment of the quartet state.

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图 3 2²Π/3²Π (a)和2²Σ⁺/3²Σ⁺ (b)的组态权重 (c²)

Figure 3. The R-dependent weights (c²) of the electronic configurations of 2²Π/3²Π (a) and 2²Σ⁺/3²Σ⁺ (b) states.

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图 4 交叉区域振动能级的放大视图

Figure 4. An amplified view of crossing region with the corresponding vibrational levels.

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图 5 包含2²Π, 1²Δ态(a)和2²Δ, 3²Π态(b)的自旋-轨道耦合矩阵元随核间距的变化

Figure 5. Variation curves of spin-orbit coupling matrix elements of 2²Π, 1²Δ states (a) and 2²Δ, 3²Π states (b) with internuclear distance.

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图 6 ICl⁺分子离子的Ω态势能曲线

Figure 6. Potential energy curves of the Ω states of ICl⁺ molecular ion.

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图 7 ICl⁺分子离子跃迁偶极矩曲线

Figure 7. Transition dipole moments curves of ICl⁺ molecular ion.

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表 1 ICl⁺分子离子R_e附近的组态权重和垂直激发能

Table 1. Electronic configuration and vertical excitation energies of ICl⁺ molecular ion at R_e.

Λ-S 态	R_e 附近的组态权重/%	T/cm^–1
Χ²Π	7σ²8σ²9σ²10σ²11σ⁰3π⁴4π⁴5π³(91)	0
$ {1^4}{\Sigma ^ - } $	7σ²8σ²9σ²10σ²11σ¹3π⁴4π⁴5π²(77) 7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(20)	18885.3
2²Π	7σ²8σ²9σ²10σ²11σ⁰3π⁴4π³5π⁴(87) 7σ²8σ²9σ²10σ¹11σ¹3π⁴4π⁴5π³(6)	23660.8
1²Δ	7σ²8σ²9σ²10σ²11σ¹3π⁴4π⁴5π²(67) 7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(29)	25549.0
1²Σ⁺	7σ²8σ²9σ²10σ²11σ¹3π⁴4π⁴5π²(59) 7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(34)	28700.8
2²Σ⁺	7σ²8σ²9σ²10σ¹11σ⁰3π⁴4π⁴5π⁴(94)	33963.0
1⁴Δ	7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(99)	35589.7
2²Δ	7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(96)	43617.4
3²Σ⁺	7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(95)	44332.0
$ {3^2}{\Sigma ^ - } $	7σ²8σ²9σ²10σ²11σ¹3π⁴4π³5π³(67) 7σ²8σ²9σ²10σ¹11σ²3π⁴4π⁴5π²(22) 7σ²8σ²9σ²10σ²11σ¹3π⁴4π⁴5π²(9)	52169.3
3²Π	7σ²8σ²9σ²10σ²11σ²3π⁴4π⁴5π¹(48) 7σ²8σ²9σ²10σ²11σ²3π⁴4π³5π²(48)	53534.0

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表 2 ICl⁺分子离子Λ-S态的光谱常数

Table 2. Spectroscopic constants of the Λ-S states of ICl⁺ molecular ion.

Λ-S态		T_e/cm^–1	D₀/eV	B_e/cm^–1	ω_e/cm^–1	R_e/Å
Χ²Π	本文	0	2.50	0.1221	432	2.24
Χ²Π	实验^a)	0	2.52
2²Π	本文	18218	0.27	0.0915	259	2.59
	实验^b)	19551
	理论^c)	14352			207	2.78
1²Δ	本文	17516	0.35	0.0737	109	2.89
1⁴Σ^–	本文	13617	0.83	0.0889	218	2.63
1⁴Δ	本文	17183	0.39	0.0670	138	3.03
1²Σ⁺	本文	18015	0.28	0.0654	153	3.05
3²Π	本文	22122	0.80	0.0628	526	3.14
3²Π	实验^a)	22420
2²Δ	本文	24768	0.52	0.0696	171	2.97
2²Σ⁺	本文	25192	0.47	0.0687	164	2.99
3²Σ^–	本文	25241	0.46	0.0630	181	3.13
注: ^a)文献[33]; ^b)文献[12]; ^c)文献[16].

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表 3 ICl⁺分子离子Ω态的解离关系

Table 3. Dissociation relationships of Ω states of ICl⁺ molecular ion.

原子态 (I⁺+Cl)	Ω态	能量/cm^–1
原子态 (I⁺+Cl)	Ω态	本文	实验^a)
I⁺(³Pg₂)+Cl(²Pu_3/2)	7/2, 5/2(2), 3/2(3), 1/2(4)	0	0
I⁺(³Pg₂)+Cl(²Pu_1/2)	5/2, 3/2(2), 1/2(2)	821	882
I⁺(³Pg₀)+Cl(²Pu_3/2)	3/2, 1/2	6433	6448
I⁺(³Pg₁)+Cl(²Pu_3/2)	5/2, 3/2(2), 1/2(3)	6901	7087
I⁺(³Pg₀)+Cl(²Pu_1/2)	1/2	7254	7330
I⁺(³Pg₁)+Cl(²Pu_1/2)	3/2, 1/2(2)	7722	7969
I⁺(¹Dg₂)+Cl(²Pu_3/2)	7/2, 5/2(2), 3/2(3), 1/2(4)	13950	13727
I⁺(¹Dg₂)+Cl(²Pu_1/2)	5/2, 3/2(2), 1/2(2)	14771	14610
注: ^a)文献[34].

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表 4 ICl⁺分子离子Ω态的光谱常数

Table 4. Spectroscopic constants of Ω states of ICl⁺ molecular ion.

Ω态		T_e/cm^–1	D₀/eV	B_e/cm^–1	ω_e/cm^–1	R_e/Å	R_e处主要的Λ-S成分/%
X²Π_3/2	本文	0	2.31	0.1216	419	2.250	X²Π (98.1)
	实验^a)				390
	实验^b)				429	2.240±0.01
	理论^c)				311	2.470
X²Π_1/2	本文	4501	1.75	0.1217	426	2.248	X²Π (97.7)
	实验^d)	4680
	实验^b)	4670±16			437	2.224±0.001
	理论^c)	5424			314	2.460
1/2(II)	本文	14808	0.50	0.0864	244	2.666	$ {1^4}{\Sigma ^ - } $ (79.7) 1²Σ⁺ (12.5)
1/2(III)	本文	17191	0.21	0.0656	78	3.063	$ {1^2}{\Sigma ^ - } $ (38.3) 1⁴Σ⁺ (25.7) 3²Π (10.8) 1²Π (5.8)
3/2(II)	本文	15202	0.44	0.0831	156	1.518	$ {1^4}{\Sigma ^ - } $ (80.7) 1⁴Σ⁺ (6.7)
3/2(III)	本文	16795	0.21	0.0649	107	3.003	1⁴Δ (58.3) 1²Δ (34.8)
注: ^a)文献[6]; ^b)文献[12]; ^c)文献[16]; ^d)文献[33].

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表 5 ICl⁺分子离子跃迁的Frank-Condon因子

Table 5. Frank-Condon factors for the transition of ICl⁺ molecular ion.

		ν" = 0	ν" = 1	ν" = 2	ν" = 3	ν" = 4
1²Δ-Χ²Π
ν' = 0	本文	1.888×10^–17	8.612×10^–16	1.956×10^–14	2.919×10^–13	3.193×10^–12
ν' = 1	本文	3.271×10^–16	1.436×10^–14	3.138×10^–13	4.501×10^–12	4.724×10^–11
ν' = 2	本文	2.958×10^–15	1.252×10^–13	2.635×10^–12	3.635×10^–11	3.662×10^–10
2²Δ-Χ²Π
ν' = 0	本文	4.361×10^–27	5.255×10^–25	2.996×10^–23	1.085×10^–21	2.796×10^–20
ν' = 1	本文	1.373×10^–25	1.547×10^–23	8.325×10^–22	2.864×10^–20	7.046×10^–19
ν' = 2	本文	2.081×10^–24	2.230×10^–22	1.149×10^–20	3.797×10^–19	8.989×10^–18
2²Σ⁺-X²Π
ν' = 0	本文	5.553×10^–27	5.602×10^–25	2.715×10^–23	8.439×10^–22	1.889×10^–20
ν' = 1	本文	2.304×10^–25	2.065×10^–23	9.017×10^–22	2.559×10^–20	5.290×10^–19
3²Σ⁺-X²Π
ν' = 0	本文	4.161×10^–18	2.943×10^–16	1.010×10^–14	2.216×10^–13	3.456×10^–12
ν' = 1	本文	4.071×10^–17	2.814×10^–15	9.431×10^–14	2.019×10^–12	3.067×10^–11
ν' = 2	本文	2.033×10^–16	1.374×10^–14	4.503×10^–13	9.412×10^–12	1.394×10^–10
1/2(II)-Χ²Π_3/2
ν' = 0	本文	1.226×10^–9	3.102×10^–8	3.870×10^–7	3.155×10^–6	1.872×10^–5
ν' = 1	本文	1.923×10^–8	4.431×10^–7	4.996×10^–6	3.649×10^–5	1.921×10^–4
ν' = 2	本文	1.679×10^–7	3.501×10^–6	3.541×10^–5	2.296×10^–4	1.060×10^–3
1/2(III)-Χ²Π_3/2
ν' = 0	本文	1.314×10^–22	9.703×10^–21	3.517×10^–19	8.368×10^–18	1.457×10^–16
ν' = 1	本文	5.532×10^–21	3.879×10^–19	1.346×10^–17	3.083×10^–16	5.184×10^–15
ν' = 2	本文	2.944×10^–20	1.987×10^–18	6.647×10^–17	1.468×10^–15	2.379×10^–14
3/2(III)-Χ²Π_3/2
ν' = 0	本文	1.106×10^–22	5.929×10^–21	1.542×10^–19	2.607×10^–18	3.216×10^–17
ν' = 1	本文	3.736×10^–21	1.836×10^–19	4.435×10^–18	7.052×10^–17	8.256×10^–16

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表 6 ICl⁺分子离子的辐射寿命

Table 6. Radiative lifetimes of ICl⁺ molecular ion.

跃迁		辐射寿命/s
跃迁		ν' = 0	ν' = 1	ν' = 2
1²Δ-Χ²Π	本文	5.69×10^–3	4.92×10^–3	4.40×10^–3
2²Δ-Χ²Π	本文	5.83×10^–5	6.18×10^–5	6.99×10^–5
2²Σ⁺-X²Π	本文	2.91×10^–5	3.28×10^–5
3²Σ⁺-X²Π	本文	1.58×10^–5	1.66×10^–5	1.83×10^–5
1/2(II)-Χ²Π_3/2	本文	3.73×10^–2	3.42×10^–2	3.09×10^–2
1/2(III)-Χ²Π_3/2	本文	1.38×10^–2	5.43×10^–3	7.61×10^–3
3/2(III)-Χ²Π_3/2	本文	3.28×10^–2	2.06×10^–2

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[1]	Sherwen T, Schmidt J A, Evans M J, Carpenter L J, Großmann K, Eastham S D, Jacob D J, Dix B, Koenig T K, Sinreich R, Ortega I, Volkamer R, Saiz-Lopez A, Prados-Roman C, Mahajan A S, Ordóñez C 2016 Atmos. Chem. Phys. 16 12239 Google Scholar
[2]	Vogt R, Sander R, Glasow R V, Crutzen P J 1999 J. Atmos. Chem. 32 375 Google Scholar
[3]	Calvert J G, Lindberg S E 2004 Atmos. Environ. 38 5087 Google Scholar
[4]	Küpper F C, Feiters M C, Olofsson B, Kaiho T, Yanagida S, Zimmermann M B, Carpenter L J, Luther G W, Lu Z, Jonsson M, Kloo L 2011 Angew. Chem. Int. Ed. 50 11598 Google Scholar
[5]	Evans S, Orchard A F 1971 Inorg. Chim. Acta. 5 81 Google Scholar
[6]	Potts A W, Price W C 1971 Trans. Faraday Soc. 67 1242 Google Scholar
[7]	Eland J H D 1979 J. Chem. Phys. 70 2926 Google Scholar
[8]	Dibeler V H, Walker J A, McCulloh K E, Rosenstock H M 1971 Int. J. Mass Spectro. Ion Phys. 7 209 Google Scholar
[9]	Venkateswarlu P 1975 Can. J. Phys. 53 812 Google Scholar
[10]	Tuckett R P, Castellucci E, Bonneau M 1985 Chem. Phys. 92 43 Google Scholar
[11]	Kaur D, Yencha A J, Donovan R J, Kvaran A, Hopkirk A 1993 Org. Mass Spectrom. 28 327 Google Scholar
[12]	Yencha A J, Lopes M C A, King G C 2000 Chem. Phys. Lett. 325 559 Google Scholar
[13]	Ridley T, Beattie D A, Cockett M C R, Lawley K P, Donovan R J 2002 Phys. Chem. Chem. Phys. 4 1398 Google Scholar
[14]	Straub P A, McLean A D 1974 Theoret. Chim. Acta 32 227 Google Scholar
[15]	Dyke J M, Josland G D, Snijders J G, Boerrigter P M 1984 Chem. Phys. 91 419.Google Scholar
[16]	Balasubramanian K 1985 Chem. Phys. 95 225 Google Scholar
[17]	Werner H, Knowles P J, Knizia G, Manby F R, Schütz M 2012 Wires Comput. Mol. Sci. 2 242 Google Scholar
[18]	Peterson K A, Yousaf K E 2010 J. Chem. Phys. 133 174116 Google Scholar
[19]	Peterson K A, Dunning Jr T H 2002 J. Chem. Phys. 117 10548 Google Scholar
[20]	Knowles P J, Werner H J 1985 Chem. Phys. Lett. 115 259 Google Scholar
[21]	Werner H J, Knowles P J 1985 J. Chem. Phys. 82 5053 Google Scholar
[22]	Knowles P J, Werner H J 1988 Chem. Phys. Lett. 145 514 Google Scholar
[23]	Werner H J, Knowles P J 1988 J. Chem. Phys. 89 5803 Google Scholar
[24]	Langhoff S R, Davidson E R 1974 Int. J. Quantum. Chem. 8 61 Google Scholar
[25]	Berning A, Schweizer M, Werner H J, Knowles P J, Palmieri P 2000 Mol. Phys. 98 1823 Google Scholar
[26]	Le Roy R J 2017 J. Quant. Spectrosc. Ra. 186 167 Google Scholar
[27]	Wu D L, Liu B K, Zhou W T, Chen J Y, Lai Z L, Liu B, Yan B 2025 Chin. Phys. B 34 043101 Google Scholar
[28]	刘铭婕, 田亚莉, 王瑜, 李晓筱, 和小虎, 宫廷, 孙小聪, 郭古青, 邱选兵, 李传亮 2025 物理学报 74 023101 Google Scholar Liu M J, Tian Y L, Wang Y, Li X X, He X H, Gong T, Sun X C, Guo G Q, Qiu X B, Li C L 2025 Acta Phys. Sin. 74 023101 Google Scholar
[29]	朱宇豪, 李瑞 2024 物理学报 73 053101 Google Scholar Zhu Y H, Li R 2024 Acta Phys. Sin. 73 053101 Google Scholar
[30]	Li R, Lv H N, Sang J Q, Liu X H, Liang G Y 2024 Chin. Phys. B 33 053101 Google Scholar
[31]	伍冬兰, 郭自依, 周俊杰, 阮文, 曾学锋, 谢安东 2022 物理学报 71 223101 Google Scholar Wu D L, Guo Z Y, Zhou J J, Ruan W, Zeng X F, Xie A D 2022 Acta Phys. Sin. 71 223101 Google Scholar
[32]	陈晨, 赵国鹏, 祁月盈, 吴勇, 王建国 2022 物理学报 71 143102 Google Scholar Chen C, Zhao G P, Qi Y Y, Wu Y, Wang J G 2022 Acta Phys. Sin. 71 143102 Google Scholar
[33]	Huber K P, Herzberg G 1979 Molecular Spectra and Molecular Structure (Vol. IV) (New York: Van Nostrand Reinhold) p342
[34]	Moore C E 1971 Atomic Energy Levels (Washington (DC): National Bureau of Standards Publications

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2025年第74卷第12期123102《ICl+分子离子激发态的包含自旋-轨道耦合效应的理论研究》补充材料.rar

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Theoretical study on excited states of ICl⁺ molecular ion considering spin-orbit coupling

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Theoretical study on excited states of ICl+ molecular ion considering spin-orbit coupling

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Theoretical study on excited states of ICl⁺ molecular ion considering spin-orbit coupling