Fine structure investigation and laser cooling study of the LaH molecule

Abstract

A theoretical feasibility study of the spin–orbit laser cooling of the molecule LaH has been performed based on a complete active space self-consistent field (CASSCF)/MRCI ab initio calculation with Davidson correction in the Λ^(±) and Ω^(±) representations. The adiabatic potential energy curves and spectroscopic constants have been investigated for the considered electronic states. The small value of the equilibrium positions difference ΔR_e between the ground and the electronic states X³Σ₀₊, (1)³Π₀₊, and (1)³Δ₁ predicts the candidacy of the molecule LaH for direct laser cooling between the first two states with the intermediate state (1)³Δ₁. The calculation of the diagonal Franck–Condon factors, the short radiative lifetime, and the experimental parameters (slowing distance, Doppler and recoil temperature, …) suggest that the molecule LaH is a good candidate for Doppler laser cooling, and a corresponding laser cooling scheme is presented.

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1. Introduction

The development of techniques for producing, trapping, and controlling ultracold molecules in the gas phase has marked a major milestone in physics, attracting significant research interest over the years. Ultracold molecules offer a wide range of potential applications, including precision measurements,^1,2 simulation of solid-state systems,^3,4 quantum information processing,⁵ and the control of chemical reactions at ultracold temperatures.⁶ Numerous studies have explored a range of diatomic polar molecules and ions, including hydrogen, as potential candidates for laser cooling. These include alkaline-earth-metal monohydrides⁷ (such as BeH, MgH, CaH, SrH, and BaH), transition metal hydrides (such as AgH,⁸ CuH,⁹ and AuH¹⁰), and other molecular systems (like CH,¹¹ AlH and AlF,¹² BH⁺ and AlH⁺¹³). Our recent work reported HfH¹⁴ as a highly promising candidate for experimental laser cooling.

In general, the study of the electronic structure of molecules with open d and f orbitals (partial occupation), such as transition metals hydrides and Lanthanides hydrides, presents a major challenge for both theorists and experimentalists due to the significant electron–degeneracy correlations^15–17 involved. The formation of chemical bonds arising from d-electrons poses a significant difficulty for theorists, as accurately modeling these molecules requires considering relativistic effects and spin–orbit coupling.¹⁸ Besides their importance in theoretical chemistry, the group of lanthanides hydrides such as LaH plays a critical role in various fields, such as astrophysics (since hydrogen is the most abundant element in the universe, LaH is found in the spectra of sunspots and cool stars),¹⁹ catalysis,²⁰ organometallic chemistry,²¹ and electron's electric dipole moment (EDM) measurements.²² The electronic structure of the LaH molecule has been experimentally examined in the literature,^23–28 with previous theoretical studies provided in ref. 29–31. Recently, Assaf et al.³² conducted a comprehensive theoretical study of the LaH molecule in both Λ^(±) and Ω^(±) representations. Nevertheless, the study of the laser-cooling candidacy of the LaH molecule has never been explored.

This paper presents a spin–orbit coupling theoretical calculation and laser cooling investigation of the LaH molecule based on Assaf et al.'s previous work.³² Additionally, we investigated the diagonal Frank-Condon factors (FCFs) and the radiative lifetimes (τ) for the two transitions X³Σ₀₊–(1)³Π₀₊ and (1)³Δ₁–(1)³Π₀₊, where (1)³Δ₁ is an intermediate state between X³Σ₀₊ and (1)³Π₀₊ states. The branching ratios of the vibrational transitions R_ν′ν have been calculated along with the number of cycles (N) for photon absorption/emission and the slowing distance L, which falls within the practical experimental limits. A laser cooling scheme with an intermediate state is presented.

2. Computational approach

In this paper, we have employed the state-averaged Complete Active Space Self-Consistent Field (CASSCF) method, followed by Multi-Reference Configuration Interaction (MRCI) calculations with Davidson correction (+Q) to study the electronic structure of the LaH molecule. The high-level ab initio computations were conducted with the MOLPRO³³ software, utilizing the GABEDIT³⁴ graphical user interface. Due to the large number of electrons in the lanthanum atom (₅₇La), selecting appropriate basis sets that accurately describe its electronic structure is challenging. Given that this study focuses on the laser cooling of the LaH molecule, we are particularly interested in its ground state and the first two excited states (1)³Δ and (1)³Π. To ensure high-precision calculations while accounting for spin–orbit coupling effects, we examined various basis sets based on previously published data.

In 2014, Mahmoud and Korek³¹ reported theoretical calculations on the low-lying electronic states of the LaH molecule, both with and without spin–orbit coupling, using the CASSCF/MRCI method. They employed the SBKJC-VDZ (ECP46MHF) valence double-zeta³⁵ basis set for lanthanum, which incorporates a relativistic effective core potential, and the augmented correlation-consistent polarized valence quadruple-zeta (aug-cc-pVQZ) basis set for hydrogen.³⁶ With the 12 electrons explicitly considered for the LaH molecule in the C_2v symmetry, the authors performed the calculations with different valence electrons (2, 6, 8, and 10 valence electrons) to check their influence on the values of the transition energy with respect to the ground state minimum (T_e). Their findings indicated that when two or six valence electrons were included, the (1)³Δ state was lower in energy than the (1)³Π state, consistent with the experimentally observed order of states.²⁵ However, for a higher number of valence electrons, the (1)³Π state became lower in energy than the (1)³Δ state. Recently, Assaf et al.³² determined the spectroscopic constants of the LaH molecule using the quasi-relativistic effective core potential (ECP28-MWB) basis set^37,38 for lanthanum. In this approach, lanthanum is described by replacing its 28 inner-core electrons with the effective core potential, while the remaining 29 electrons are explicitly represented by the ANO Gaussian basis set with a contraction scheme of (14s, 13p, 10d, 8f, 6g)/[6s, 6p, 5d, 4f, 3g].^37,38 The hydrogen single electron is treated with the augmented correlation-consistent polarized valence quadruple-zeta (aug-cc-pVQZ) basis set,³⁶ contracted as (7s, 4p, 3d, 2f)/(5s, 4p, 3d, 2f). The authors calculated the low-lying excited states in both Λ^(±) and Ω^(±) representations and determined the corresponding spectroscopic constants. Their results showed strong agreement with experimental data. Motivated by this, we employ the same basis sets (ECP28-MWB in conjunction with ANO Gaussian basis set^37,38 for La and aug-cc-pVQZ³⁶ for H) as well as the all-electron approach described by Assaf et al.³² to carry out our own calculations, both with and without spin–orbit coupling (S.O.C) effects, aiming for a more accurate analysis relevant to laser cooling of the LaH molecule.

To assess the reliability of the employed pseudopotentials and basis sets, we have performed the calculations of our study using a benchmark of basis sets B₁–B₅, along with literature values, all compared with experimental data as shown in Table 1. The B₁ set employs the pseudopotential ECP28MWB^37,38 for lanthanum with a (s, p, d, f, g) ANO Gaussian basis set,^37,38 combined with the aug-cc-pVQZ³⁶ (s, p, d, f) basis for hydrogen. The B₂ set retains the same basis for La as in B₁ but uses a higher-level aug-cc-pV5Z³⁹ (s, p, d, f) basis for H. The B₃ set again uses ECP28MWB^37,38 for La but combines it with the smaller aug-cc-pVTZ³⁹ (s, p, d) basis for H. In contrast, B₄ and B₅ utilize the ECP46MWB^35,40 pseudopotential for La, where Lanthanum is described as a system of 46 inner electrons, and the remaining 11 electrons are represented by the corresponding basis set ECP46MWB-II ((6s6p5d)/[4s4p4d] + 2s1p1d).^35,41,42 The active space of C_2v point group symmetry contains 5σ (La: 5d₊₂, 6p₀, 5d₀; H: 1 s, 2 s), 2π (La: 5d₊₁, 6p₊₁; H: 0), 1δ (La: 5d₋₂; H: 0) molecular orbitals and are distributed into the irreducible representation as 5A₁, 2B₁, 2B₂, and 1A₂, denoted by [5, 2, 2, 1]. A CASSCF calculation was performed with two valence electrons from LaH distributed over the ten active orbitals. B₄ combines ECP46MWB^35,40 with aug-cc-pV5Z³⁹ (s, p, d, f) for H, while B₅ pairs it with aug-cc-pVQZ³⁶ (s, p, d, f) for H. Across all considered electronic states in the Λ^(±) representation X¹Σ⁺, (1)³Δ, and (1)³Π, this benchmarking enables a detailed assessment of the sensitivity of spectroscopic constants: the equilibrium bond length R_e, the transition energy with respect to the ground state minimum T_e, the harmonic frequency ω_e, and the anharmonicity constant ω_ex_e to the choice of basis sets and pseudopotentials, as presented in Table 1.

Table 1 The spectroscopic constants of X¹Σ⁺, (1)³Δ, and (1)³П states of LaH molecule, without spin–orbit coupling using a benchmark of basis sets B₁–B₅, along with literature values, all compared with experimental data

States	Ref.	Te (cm^−1)	Re (Å)	|ΔRe| (Å)	ωe (cm^−1)	|Δωe| (cm^−1)	ωexe (cm^−1)	|Δωexe| (cm^−1)
a Energy corresponding to ν00.b Estimated energy of ^3Λ^(±) state determined by calculating the average of the spin–orbit components' energy. ^B1 This work using the ECP28MWB (s, p, d, f, g) basis set^37,38 for lanthanum and aug-cc-pVQZ (s, p, d, f)^36 for hydrogen. ^B2 This work using the ECP28MWB (s, p, d, f, g) basis set^37,38 for lanthanum and aug-cc-pV5Z (s, p, d, f)^39 for hydrogen. ^B3 This work using the ECP28MWB (s, p, d, f, g) basis set^37,38 for lanthanum and aug-cc-pVTZ (s, p, d)^39 for hydrogen. ^B4 This work using the ECP46MWB (s, p, d, f, g) basis set^35,40 for lanthanum and aug-cc-pV5Z (s, p, d, f)^39 for hydrogen. ^B5 This work using the ECP46MWB (s, p, d, f, g) basis set^35,40 for lanthanum and aug-cc-pVQZ (s, p, d, f)^36 for hydrogen. Theoretical work^31 used the ECP46MHF (s, p, d) basis set^35 for lanthanum and aug-cc-pVQZ (s, p, d, f)^36 for hydrogen.
X^1Σ^+	Exp.^24	0.0	2.032		—		—
	Exp.^25	0.0	—		1418		15.6
	This work^B1	0.0	2.024	0.008	1447.28	29.28	15.43	0.17
	This work^B2	0.0	2.024	0.008	1447.49	29.29	13.81	1.79
	This work^B3	0.0	2.026	0.006	1468.69	50.69	28.79	13.19
	This work^B4	0.0	2.078	0.046	1433.43	15.43	15.74	0.14
	This work^B5	0.0	2.078	0.046	1437.38	19.38	15.45	0.15
	Theo.^29	0.0	2.08	0.048	1433	15	—	—
	Theo.^30	0.0	2.060	0.028	1429	11	20.93	5.33
	Theo.^31	0.0	2.235	0.203	1353.26	64.74	—	—
	Theo.^32	0.0	2.025	0.007	1439.77	21.77	15.242	0.358
(1)^3Δ	Exp.^25	—	—		1355		14.4
	This work^B1	2179.37	2.081		1347.34	7.66	15.22	0.82
	This work^B2	1962.36	2.081		1371.59	16.59	15.89	1.49
	This work^B3	2052.86	2.086		1357.44	2.44	11.02	3.38
	This work^B4	2043.29	2.143		1338.96	16.04	14.49	0.09
	This work^B5	2022.45	2.142		1335.94	19.06	11.03	3.37
	Theo.^29	2805	2.13		1352	3.00	—	—
	Theo.^31	3916	2.272		1314.98	40.02	—	—
	Theo.^32	2232	2.082		1371.65	16.65	15.16	0.76
(1)^3П	Exp.^25	3732^a^,^b	—		—		—
	This work^B1	4222.7	2.065		1358.32		18.39
	This work^B2	4222.74	2.064		1354.91		17.86
	This work^B3	4261.98	2.074		1343.06		18.05
	This work^B4	4764.40	2.145		1337.96		16.97
	This work^B5	4753.36	2.145		1314.65		17.75
	Theo.^29	5147	2.12		1341		—
	Theo.^31	3880	2.235		1341.37		—
	Theo.^32	4263	2.066		1359.20		18.65

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In terms of effective core potential, and for the ground state X¹Σ⁺, B₁ shows excellent agreement in the equilibrium bond length (R_e = 2.024 Å), closely matching the experimental value of 2.032 Å with only a 0.008 Å deviation. Case B₅, however, displays a much more important difference of 0.046 Å. At the same time, the vibrational constant ω_e from B₅ (1437.38 cm⁻¹) is slightly closer to the experimental value (1418 cm⁻¹) than that from B₁ (1447.28 cm⁻¹). For the excited state (1)³Δ, the ω_e and ω_ex_e values from B₁ align well with experimental values, whereas B₅ significantly underestimates this anharmonicity constant. Finally, for the (1)³Π state, B₁ yields excitation with an energy T_e, which is more consistent with available measurements.

These comparisons demonstrate that pseudopotential B₁ is more accurate and reliable for describing the spectroscopic properties of LaH, and it is therefore preferred in our study.

In terms of basis sets, B₃ yields a slightly improved vibrational constant ω_e (1468.69 cm⁻¹) compared to experiment (1418 cm⁻¹); however, it also leads to significantly higher deviations in ω_ex_e. For the (1)³Δ state, B₁ provides an excitation energy T_e = 2179.37 cm⁻¹ and R_e = 2.081 Å, which are closer to experiment than those from B₄ or B₅, which overestimate R_e and show larger deviations in vibrational constants. Concerning the (1)³Π state, B₁ again delivers consistent performance, with T_e = 4222.7 cm⁻¹ and ω_e = 1358.32 cm⁻¹, reasonably close to the experimental values, while B₄ and B₅ significantly overestimate the excitation energy and distort vibrational constants.

Overall, B₁ exhibits the most balanced and accurate agreement across all three electronic states compared to experimental data. Therefore, the B₁ basis set (ECP28-MWB for La and aug-cc-pVQZ for H) is validated as the most suitable choice for describing the spectroscopic properties of LaH molecule.

3. Ab initio results

Our primary focus in this work is the study of the laser cooling of LaH, by considering transitions between the ground and low-lying excited states. The investigated potential energy curves for the three low-lying Λ^(±) and Ω^(±) states: X¹Σ⁺ (X¹Σ₀₊), (1)³Δ {(1)³Δ₁, (1)³Δ₂, (1)³Δ₃}, and (1)³Π {(1)³Π₀₋, (1)³Π₀₊, (1)³Π₁, (1)³Π₂} are displayed as a function of internuclear separation in Fig. 1 and Fig. 2. The spin–orbit coupling splitting of the electronic states (1)³Δ and (1)³Π of the molecule LaH is illustrated in Fig. 3. Due to the dominance of fine structure in the spectra of heavy molecules such as LaH, it is crucial to account for the spin–orbit interaction of lanthanum. This requirement is validated by the notably large splitting energies observed for the ³Δ state (approximately 448 cm⁻¹ for (1)³Δ₂ – (1)³Δ₁) and ³Π state (around 320 cm⁻¹ for (1)³Π₂ – (1)³Π₁) electronic states, as illustrated in Fig. 2 and 3. These results highlight the significant influence of spin–orbit coupling on the electronic states of the LaH molecule. In addition, the transition dipole moment curves (TDMCs) of the allowed transitions X¹Σ₀₊–(1)³Π₀₊ and (1)³Π₀₊–(1)³Δ₁ of the molecule LaH have been computed as a function of internuclear distance and displayed in Fig. 4.

Fig. 1 The potential energy curves of the ground and first two low-lying excited states of the LaH molecule, in the Λ^(±) representation.

Fig. 2 The potential energy curves of the ground and first two low-lying excited states of the LaH molecule, in the Ω^(±) representation.

Fig. 3 Spin–orbit coupling splitting of the electronic states (1)³Δ and (1)³Π of the molecule LaH.

Fig. 4 Transition dipole moments of the transitions (a) X¹Σ₀₊–(1)³Π₀₊ and (b) (1)³Δ₁–(1)³Π₀₊ of the molecule LaH.

The spectroscopic constants such as the equilibrium bond length R_e, the transition energy with respect to the ground state minimum T_e, the harmonic frequency ω_e, and the anharmonicity constant ω_ex_e of all the calculated Λ^(±) (listed as (B₁) in Table 1) and Ω^(±) states are determined and listed in Table 2. As previously mentioned, the ground state has ¹Σ⁺ symmetry, and the first two low-lying excited states are (1)³Δ and (1)³Π. Their spectroscopic constants strongly agree with experimental data.^24,25 In the Λ^(±) representation, the equilibrium internuclear distance R_e of the ground state exhibits a relative error of only 0.4% compared to the experimental value (R_e = 2.032 Å).²⁴ Similarly, the vibrational constants ω_e and ω_ex_e for (1)³Δ state show relative errors of 0.6% and 5.7%, respectively, compared to experimental data.²⁵ The ab initio investigation of LaH conducted in 2014 ³¹ showed that the use of the large-core pseudopotential ECP46MHF³⁵ for lanthanum (La) with 10 valence electrons introduced considerable inaccuracies in the computed equilibrium bond lengths (R_e) of various electronic states, and the transition energies (T_e) for several predicted states were significantly overestimated. For instance, the ground state R_e deviated by approximately 9.4% from the experimental value.²⁴

Table 2 The spectroscopic constants of X¹Σ⁺, (1)³Δ, and (1)³П states with and without spin–orbit coupling of LaH

Spectroscopic constants in the Λ^(±) representation
States	Ref.	Te (cm^−1)	ΔTe/Te%	Re (Å)	ΔRe/Re%	ωe (cm^−1)	Δωe/ωe%	ωexe (cm^−1)	Δωexe/ωexe%
a Energy corresponding to ν00.b Estimated energy of ^3Λ^(±) state determined by calculating the average of the spin–orbit components' energy.
X^1Σ^+	This work	0.0		2.024	0.4	1447.28	—	15.43	—
	Exp.^24	0.0		2.032	—	—	2.1	—	1.1
	Exp.^25	0.0		—	2.7	1418	1.0	15.6	—
	Theo.^29	0.0		2.08	1.7	1433	1.3	—	26.3
	Theo.^30	0.0		2.060	9.4	1429	6.9	20.93	—
	Theo.^31	0.0		2.235	0.0	1353.26	0.5	—	1.2
	Theo.^32	0.0		2.025		1439.77		15.242
(1)^3Δ	This work	2179.37	—	2.081	—	1347.34	0.6	15.22	5.7
	Exp.^25	—	22.3	—	2.3	1355	0.3	14.4	—
	Theo.^29	2805	44.3	2.13	8.4	1352	2.5	—	—
	Theo.^31	3916	2.4	2.272	0.0	1314.98	1.8	—	0.4
	Theo.^32	2232		2.082		1371.65		15.16
(1)^3П	This work	4222.7	—	2.065	—	1358.32	—	18.39	—
	Exp.^25	3732^a^,^b	18.0	—	2.6	—	1.3	—	—
	Theo.^29	5147	8.8	2.12	7.6	1341	1.3	—	—
	Theo.^31	3880	0.9	2.235	0.0	1341.37	0.1	—	1.4
	Theo.^32	4263		2.066		1359.20		18.65

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Spectroscopic constants in the Ω^(±) representation
States	Ref.	Te (cm^−1)	ΔTe/Te%	Re (Å)	ΔRe/Re%	ωe (cm^−1)	Δωe/ωe%	ωexe (cm^−1)	Δωexe/ωexe%
X^1Σ^+0^+	This work	0.0		2.027	—	1443.63	1.8	14.56	6.7
	Exp.^25	0.0		—	0.2	1418	—	15.6	—
	Theo.^29	0.0		2.0319	0.0	—	0.1	—	8.5
	Theo.^32	0.0		2.027		1444.66		15.904
(1)^3Δ1	This work	1664.4	—	2.080	1.4	1359.66	—	13.67	—
	Exp.^23	—	—	2.1102	—	—	0.3	—	5.1
	Exp.^25	1259.5^a	—	—	0.9	1355	—	14.4	—
	Theo.^29	—	0.0	2.099	0.2	—	0.3	—	4.6
	Theo.^32	1665		2.085		1363.659		14.331
(1)^3Δ2	This work	2112.1	—	2.078	0.8	1363.87	—	14.79	—
	Exp.^23	—	—	2.0938	—	—	—	—	—
	Exp.^25	1646^a	—	—	0.2	—	—	—	—
	Theo.^29	—	0.1	2.083	0.3	—	0.2	—	2.4
	Theo.^32	2111		2.084		1366.582		15.15
(1)^3Δ3	This work	2682.8	—	2.085		1371.49		17.81	—
	Exp.^23	—	—	2.0925	0.4	—	—	—	—
	Theo.^29	—	0.0	2.081	0.2	—	0.1	—	0.8
	Theo.^32	2684		2.082	0.1	1373.275		17.960
(1)^3П0−	This work	4011.9	—	2.0637	—	1356.63	—	15.656	—
	Exp.^25	3542^a	0.1	—	0.2	—	0.3	—	2.0
	Theo.^32	4014		2.068		1352.946		15.977
(1)^3П0+	This work	4043.9	—	2.0619	—	1361.47	—	17.899	—
	Exp.^25	3586^a	0.1	—	0.2	—	0.2	—	2.4
	Theo.^32	4047		2.067		1358.629		18.345
(1)^3П1	This work	4239.7	—	2.0646	—	1352.34	—	16.478	—
	Exp.^25	3754^a	0.0	—	0.1	—	0.1	—	1.1
	Theo.^32	4241		2.067		1353.606		16.669
(1)^3П2	This work	4559.7	—	2.0639	—	1369.78	—	15.975	—
	Exp.^27	4048^a	0.1	—	0.1	—	0.1	—	3.6
	Theo.^32	4557		2.065		1367.963		16.579

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The transition energies (T_e) associated with the spin–orbit components of the (1)³Δ and (1)³Π electronic states were not reported in the published experimental studies.²⁵ Instead, it only provided the energies corresponding to ν₀₀, i.e., the T₀ values for these spin–orbit components. As a result, a direct comparison between our calculated T_e values and the experimental T₀ data is not feasible. For a meaningful comparison, we instead consider the spin–orbit splitting energies. However, the calculated splitting energy of the (1)³Δ state is ΔΩ_1–2 (448 cm⁻¹), which is closer to the experimental value²⁵ (ΔΩ_1–2 = 387 cm⁻¹). The calculated spin–orbit splitting for the (1)³Π state demonstrates a remarkably strong agreement with experimental observations reported in references.^25,27 The total splitting energy obtained from our calculations, ΔE_Total = 548 cm⁻¹, aligns closely with the experimental value^25,27 of 506 cm⁻¹, differing by only ∼7%. This level of agreement underscores the accuracy and reliability of the theoretical methods employed, particularly in capturing the fine-structure effects arising from spin–orbit coupling. Such consistency between theory and experiment validates the computational treatment of the (1)³Π state, which plays a key role in our analysis of electronic transitions and laser cooling feasibility.

4. Laser cooling study of the LaH molecule

Laser cooling is a technique that reduces the motion of atoms or molecules by repeatedly scattering photons through fast and controlled optical transitions.⁴³ Each photon scatter imparts a small, directional momentum change, effectively reducing the system's kinetic energy and entropy. Although both direct^44–47 methods (e.g., buffer gas cooling, Stark deceleration) and indirect^48,49 methods (e.g., photoassociation of cold atoms) can be used to achieve molecular cooling and trapping. Laser cooling has uniquely succeeded in reaching the sub-millikelvin range for various diatomic^50–53 and linear triatomic^54–56 species. Several criteria are to be followed when considering cooling transitions among vibronic levels in a diatomic molecule.

(i) A highly diagonal Franck–Condon array for the considered band system, which would ensure a low number of lasers that would be used to retain the closed-loop cycle of the molecule. One could usually recognize band systems with high Franck–Condon arrays when the equilibrium internuclear distance (R_e) among the considered electronic states is minimal.⁵⁷

(ii) No intervening electronic state that would disturb the laser cooling cycle. One should make a distinction in this case between intervening and non-intervening electronic states. An intervening electronic state is usually an intermediate state situated between the excited state and the ground state, forming the cycling loop, and that intervenes with the transition band. This usually takes place if there is a high probability of transition between the upper-level electronic state and this intermediate state. A lower transition probability would render the intermediate state as non-intervening. Recent studies, however, have shown the possible involvement of intervening intermediate states in the cooling process.^58–60

(iii) The transition radiative lifetime among the considered vibrational levels should be very short to ensure high photon scattering rates. Usually, the considered radiative lifetimes are in the range of ns-ms.^14,61,62

The equilibrium positions difference ΔR_e between the ground state X¹Σ₀₊ and the excited state (1)³Π₀₊ (about 0.0349 Å) of the molecule LaH is minimal. This encouraged the authors to consider a closed cycle formed from bands within these states. Fig. (5a) shows highly diagonal Franck–Condon arrays for the transitions X¹Σ₀₊–(1)³Π₀₊ for the vibrational levels 0 ≤ v ≤ 5, obtained using the Level 11 program,⁶³ thus fulfilling criteria (i).

Fig. 5 The Frank–Condon factor of the transitions X¹Σ₀₊–(1)³Π₀ and (1)³Δ₁–(1)³Π₀ of the molecule LaH.

Fig. 2 shows four intermediate states to the considered cycle: (1)³Π₀₋, (1)³Δ₁, (1)³Δ₂, and (1)³Δ₃. Transition among states (1)³Π₀₊ and (1)³Π₀₋ are not allowed due to the rule 0⁺ ↛ 0^− 64 in Hund's case-c. The transitions (1)³Π₀₊ – (1)³Δ₂ and (1)³Π₀₊ – (1)³Δ₂ are not allowed either, since transitions with ΔΩ > 1 are also forbidden⁶⁴ in Hund's case c. As a consequence, the laser cooling of the LaH molecule through the cycle made of the transitions X¹Σ₀₊–(1)³Π₀₊ will necessitate investigating the intervening degree of the intermediate state (1)³Δ₁ only, as required through criteria (ii). The diagonality of the FCF for (1)³Δ₁–(1)³Π₀₊ is shown in Fig. (5b), showing a high transition probability to the first vibrational levels of the intermediate state.

Criteria (iiii) can be evaluated by considering the Transition Dipole Moment curves (TDMCs), i.e., the transition dipole moment variation μ(R) in terms of the internuclear distance R among the involved electronic states. The TDMC for X¹Σ₀₊–(1)³Π₀₊ and (1)³Δ₁–(1)³Π₀₊ transitions are given in Fig. 5, as obtained with the Molpro program.³³ The vibrational radiative lifetime τ_vv′ can be calculated as the inverse of the Einstein coefficient A_vv′ .^65,66 The vibrational Einstein Coefficient among the transition (1)³Δ₁–(1)³Π₀₊ is calculated using the LEVEL 11 program according to the following formula:

A_v′v = (3.1361891)(10⁻⁷)(ΔE)³(〈ψ_ν′|μ(r)|ψ_ν〉)² (1)

where A_νν′ has as units s⁻¹, ΔE is the emission frequency (in cm⁻¹) and μ(r) is the electronic transition dipole moment between the two considered electronic states (in Debye).⁶⁷ For transitions of the nature Σ – Π, such as X¹Σ₀₊–(1)³Π₀₊, the transition dipole moment (TDM) is vertical, as calculated in terms of μ_x, μ_y, μ_z. In this case, the Einstein coefficient expressed in (1) has to be divided by two.⁶⁸

The vibrational branching loss ratio measures how much an intermediate state affects a laser cooling cycle between two other states. In our study, we have to examine how the intermediate state (1)³Δ₁ influences the cycle between the X¹Σ₀₊ and (1)³Π₀₊ states. The vibrational branching loss ratio to this state is approximately equal to:

Given the high degree of interference of the intermediate state with the cycling loop, the laser cooling analysis will be one to include the intermediate state, as done previously in the literature.^69,70 The vibrational branching ratio, which represents the percentage of transition probability between two vibrational levels, is obtained by using the formula:⁷¹

Since we are studying the laser cooling between the two electronic states (1)³Π₀₊ and X¹Σ₀₊ with the intermediate state (1)³Δ₁, the values of the vibrational branching ratio R_v′′v and R_v′′v′ for the first five vibrational levels are given by:⁷²

where A_v′′v and A_v′′v′ are the Einstein coefficients for the transitions (1)³Π₀₊–X¹Σ₀₊ and (1)³Π₀₊–(1)³Δ₁, respectively. The corresponding calculated values of these Einstein coefficients and the vibrational branching ratio are given in Table 3, along with the value of the radiative lifetime, which is within the experimental conditions to realize the laser cooling of the molecule LaH. The experimental parameters needed to realize laser cooling are⁷³where h and k_B are, respectively, the Planck and Boltzmann constants, m is the mass of the molecule, and V, a_max, and L are the speed, the maximum acceleration, and the slowing distance, respectively. In the main cycling transition, N_e is the number of the excited states, while N_tot is the number of the excited states connected to the ground state plus N_e.

Table 3 The radiative lifetimes τ, and the vibrational branching ratio of the vibrational transitions. Between the electronic states (1)³Π₀₊ – X¹Σ₀₊ and (1)³Π₀₊ – (1)³Δ₁ of the molecule LaH

		ν′′ ((1)^3Π0+) = 0	1	2	3	4	5
ν ((^1X0+)) = 0	Avv′′	2458.937631	130.9866596	2.52485 × 10^−5	1.017740631	0.185340271	0.02356491
	Rvv′′	0.123427154	0.01235997	2.52701 × 10^−9	2.52701 × 10^−9	2.1245 × 10^−5	3.29397 × 10^−5
1	Av′′	54.13476602	1988.819523	269.2849008	0.732617602	3.268505736	0.990 576 903
	Rvv′′	0.002717312	0.187666051	0.026951574	8.13052 × 10^−5	0.000374658	0.001384656
2	Avv′′	0.212860911	84.56872455	1574.649772	392.4625508	5.437394721	4.106859785
	Rvv′′	1.06846 × 10^−5	0.007979949	0.15759996	0.043555105	0.000623271	0.005740684
3	Avv′′	0.009726067	0.804036255	99.6551288	1230.988993	502.8191021	14.80275134
	Rvv′′	4.88203 × 10^−7	7.58693 × 10^−5	0.009974056	0.136613938	0.057636543	0.020691701
4	Avv′′	0.000172085	0.019056646	1.710011321	101.2045812	927.9466442	601.3369858
	Rvv′′	8.63788 × 10^−9	1.7982 × 10^−6	0.000171148	0.011231584	0.106367552	0.840565695
ν′ ((1)^3Δ1) = 0	Av'v′′	17353.11041	56.71484895	0.36948472	6.17829E-10	6.17829E-10	0.000120469
	Rv'v′′	0.871044876	0.005351643	3.69801 × 10^−5	6.8566 × 10^−14	7.08197 × 10^−14	1.68395 × 10^−7
1	Av'v′′	43.77651799	8225.202174	52.13234864	0.000400248	0.000400248	2.85186 × 10^−6
	Rv'v′′	0.002197376	0.776134381	0.005217704	4.44191 × 10^−8	4.58792 × 10^−8	3.98641 × 10^−9
2	Av'v′′	10.64881134	78.79191542	7837.876719	1.211771658	1.211771658	0.001 576 422
	Rv'v′′	0.00053452	0.007434846	0.784459554	0.000134481	0.000138902	2.20357 × 10^−6
3	Av'v′′	1.304985207	26.12163032	111.8377067	82.58291984	82.58291984	1.955219988
	Rv'v′′	6.55041 × 10^−5	0.002464851	0.011193358	0.009164971	0.009466216	0.002733061
4	Av'v′′	0.04137685	5.623563619	43.91900286	7200.511483	7200.511483	92.1779253
	Rv'v′′	2.07692 × 10^−6	0.000530642	0.004395665	0.799105624	0.825371568	0.128848888
ΣAvv′′		19922.17726	10597.65213	9991.4351	9010.713057	8723.963561	715.3955837
τ = 1/ΣAvv′		5.01953 × 10^−5	9.43605 × 10^−5	0.000100086	0.000110979	0.000114627	0.001397828
τ (μs)		50.2	94.4	100.1	111.0	114.6	1397.8

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The laser cooling scheme for the molecule LaH for the main transition (1)³Π₀₊–X¹Σ₀₊ with the intermediate state (1)³Δ₁ is given in Fig. 6. The driving and the repumping lasers (of wavelengths λ_0′′0 = 1258.8 nm, λ_0′′2 = 1927.5 nm) are given in solid red lines for the transition (1)³Π₀₊–X¹Σ₀₊ and in solid green lines (of wavelengths λ_0′′0′ = 2147.3 nm, λ_0′′1′ = 1676.1 nm) for the transition (1)³Π₀₊–(1)³Δ₁. These four suggested lasers are in the near-infrared region, a region of the spectrum for which commercial lasers are already available in the market. The spontaneous decays are represented in blue dotted lines for the transition (1)³Π₀₊–X¹Σ₀₊ and in purple dotted lines for the transition (1)³Π₀₊–(1)³Δ₁. The values of the FCF (f_ν′′ν and f_v′′v′) and the vibrational branching ratios R_v′′v and R′_v′′ are specified for the vibrational levels in the laser cooling scheme. The loss to the vibrational level v = 2 is negligible (R_0′′2 = 1.06846 × 10⁻⁵), so that the corresponding vibrational level is not considered in the laser cooling scheme.

Fig. 6 Laser cooling scheme of transitions X¹Σ₀₊–(1)³Π₀ and (1)³Δ₁–(1)³Π₀ of the molecule LaH.

The number of cycles (N) for photon absorption/emission for the vibrational levels is reciprocal to the total loss:

The values of the corresponding experimental parameters are L = 6.05 m, V = 3.73 m s⁻¹, T_ini = 0.117 K, N_e/N_tot = 1/5, and a_max = 1.15 m s⁻². For this cooling scheme, the temperature that can be reached during the process is given by the Doppler limit temperature T_D and the recoil temperature T_r:⁷¹

T_D = h/(4 × π × τ × k_B) = 9.6 nK and T_r = h²/(m × λ²₀₀ × k_B) = 88.1 nK, (6)

5. Conclusion

A theoretical ab initio calculation has been done based on a complete active space self-consistent field (CASSCF)/(MRCI + Q) with Davidson correction in the Λ^(±) and Ω^(±) representations. The calculation of the adiabatic potential energy curves, the spectroscopic constants, the FCFs, and the radiative lifetime for the transition X¹Σ₀₊–(1)³Π₀₊ shows the candidacy of the LaH molecule for direct laser cooling. The calculation of the FCF and the radiative lifetime for the transition (1)³Δ₁–(1)³Π₀₊ shows that the presence of the intermediate state (1)³Δ between X¹Σ₀₊ and (1)³Π₀₊ cannot be ignored. Therefore, a total vibrational branching ratio is calculated, leading to a total radiative lifetime (τ = 50.2 μs). Experimental parameters such as the Doppler and recoil temperatures, the slowing distance, and the number of cycles (N) for photon absorption/emission are proposed. A laser cooling scheme is presented for the transition X¹Σ₀₊–(1)³Π₀₊ with the intermediate state (1)³Δ₁. These results open the way for direct laser cooling of the molecule LaH.

Conflicts of interest

The authors declare no competing interests.

Data availability

All data generated or analysed during this study are included in this published article.

Acknowledgements

This publication is based upon work supported by the Khalifa University of Science and Technology under Award No. RIG-2024-053 and the Abu-Dhabi Department of Education and Knowledge, under award number AARE20-031. Al MESBAR High Power Computer was used for the completion of this work.

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Footnote

† Current affiliation: Department of Physical and Theoretical Chemistry at Comenius University in Bratislava- Slovakia.

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