Laser cooling of InF, InCl and InH with an ab initio study.

The feasibility of laser cooling InF, InCl and InH is investigated based on ab initio quantum chemistry. To determine their suitability for laser cooling molecules, we have calculated the electronic structures, spectroscopic parameters, transition dipole moments (TDMs), radiative lifetimes, Franck-Condon factors (FCFs) and diode laser excitation wavelengths of InF, InCl and InH. Calculated spectroscopic constants of the first three electronic states for InF, InCl and InH show good agreement with available theoretical and experimental results. InF has highly diagonally distributed FCFs (f 00 = 0.961, f 11 = 0.909) for the C1Π → X1Σ+ transition, and the rather short lifetime of the state C1Π is computed to be 2.77 ns at the lowest vibrational level. Notable is that the 3Π → X1Σ+ transition of InF also has large diagonal FCFs and short lifetimes. Therefore, InCl and InH are not potential laser-cooling candidates because the FCFs of the 1Π → X1Σ+ transition are off-diagonal. We further propose laser cooling schemes for InF. The present results could provide a promising theoretical reference for further theoretical and experimental research on InF, InCl and InH. This journal is © The Royal Society of Chemistry.

Introduction

The production of ultracold diatomic molecules has drawn considerable interest for several years because of their wide range of prospective applications, for example, quantum computations,[1] controlling chemistry[2] and precision measurement.[3-5] Recently, laser cooling of diatomic molecules SrF,[6,7] YO,[8] CaF[9] and KRb[10] have been experimentally performed. These successful laser cooling experiments have initiated a search for more molecules since more than 90 elements in the periodic table possibly form more than 4000 different diatomic molecules. The potential laser cooling candidates must have highly diagonal Franck–Condon factors (FCFs) and shorter lifetimes (i.e. high spontaneous-emission rates). Highly diagonal FCFs limit the number of lasers required to keep the molecule in a quasi-closed-loop cooling cycle and rapid laser cooling demands a shorter lifetime.

The A2Π → X2Σ+ transition suggested by the experiments[6-9] for SrF, YO and CaF has been theoretically used to establish the laser cooling schemes for RaF,[4] BeF,[11] MgF[12] and MH[13] (M = Be, Mg, Ca, Sr, and Ba). In 2015, laser cooling of LiBe molecule based on the B2Π → X2Σ+ transition has been reported by You et al.[14] Interestingly, AlX (X = H,[15] F,[15] Cl[16] and Br[16]) and BBr[17]/BCl[17] molecules with an intervening state a3Π existed between the A1Π and X1Σ+ states have also been identified as potential laser-cooling candidates. The laser cooling scheme using the A1Π → X1Σ+ and a3Π → X1Σ+ transitions was theoretically suggested. In our previous works, theoretical optical schemes have been predicted for AlCl[16]/AlBr[16] and BBr[17]/BCl.[17] Li et al.[18] propose to utilize the 1Σ–1Σ electronic transition system for direct laser cooling of AgH and AgD molecules.

The boron group molecules are of persistent interest in the investigation of cold molecules. BH[19]/BCl[17]/BBr,[17] AlH[15]/AlF[15]/AlCl[16]/AlBr[16] and GaF[19] were identified as the promising laser-cooling targets theoretically. However, the study on laser cooling of indium diatomics is very limited. On the basis of previous studies, we believe InF, InCl and InH molecules are possible laser-cooling candidates. In this paper, we focus on the theoretical studies of the laser cooling of InF, InCl and InH. The electronic structure and spectroscopic properties of the low-lying electronic states of InF were given by Banerjee et al.[20] from the relativistic configuration interaction calculations. They also determined the lifetime of the state C1Π was about 2.3 ns at the lowest vibrational level. Zou et al.[21] obtained the potential energy curves and the spectroscopic constants on the ground and low-lying excited states of InCl using all-electron relativistic calculations. The theoretical calculations have been performed on the InH molecule based on multi-reference configuration interaction plus Davidson corrections method.[22] Leininger et al.[23] performed the calculations for spectroscopic constants of InH, InF, and InCl at the self-consistent-field and correlated levels. Several experimental results on the spectroscopic parameters were available in the literature for InF, InCl and InH.

Though InF, InCl and InH have been extensively investigated in previous works, as mentioned above, a systematic study of laser cooling of InF, InCl and InH is not available in the literature to the best of our knowledge. The main objective of this paper is to identify whether InF, InCl and InH are potential laser-cooling candidates and briefly design laser-cooling schemes. To determine their suitability for laser cooling molecules, we have calculated the electronic structures, spectroscopic parameters, TDMs, radiative lifetimes, FCFs and diode laser excitation wavelengths of InF, InCl and InH. Since in this work we mean to focus on the vibrational cooling, the rotational degrees of freedom can be neglected in the first approximation. Here, the C1Π → X1Σ+ transition for InF/InCl and A1Π → X1Σ+ transition for InH are the main transitions of laser cooling and the transition from intervening state 3Π to the ground state X1Σ+ is spin forbidden.

The paper is organized as follows. Section 2 details the theoretical methods and basis sets used in the calculations. Section 3 presents the results and discussions of the data, outlining the schemes for direct laser cooling. Section 4 provides a conclusion for this work.

Computational details

All the ab initio calculations are performed with the MOLPRO package.[24] It should be noted that only the subgroup C2v point group symmetry can be adopted for InF, InCl and InH in the package. The electronic structures for the X1Σ+, C1Π and 3Π states of InF/InCl and the X1Σ+, A1Π and a3Π states of InH are calculated using the complete active space self-consistent field method (CASSCF),[25,26] which is followed by MRCI plus Davidson corrections (MRCI+Q)[27-29] calculations. The scalar relativistic effect is taken into account using the Douglas–Kroll–Hess (DHK)[30,31] transformation of the relativistic Hamiltonian. The spectroscopic constants, including equilibrium bond distance (Re), dissociation energy (De), harmonic frequency (we), anharmonic vibrational frequency (weχe), rotational constant (Be) and electronic transition energy (Te) are evaluated by LEVEL 8.0 program.[32] The TDMs of InF, InCl and InH are computed by taking the expectation and transition values using the MRCI wave functions. With the potential energy curves (PECs) and TDMs of different electronic states, we have obtained the FCFs and radiative lifetimes of the various vibrational levels from LEVEL 8.0 program.

For the indium atom, being considerably heavy, we use the small-core scalar relativistic effective core potential ECP28MDF[33] together with the corresponding valence basis sets. The 4s24p64d105s25p1 electrons of the In atom are kept in the valence space, while the remaining core electrons are replaced by the above mentioned pseudopotentials. The correlation consistent polarized valence quintuple zeta aug-cc-pV5Z (=AV5Z)[34] is chosen for H. Meanwhile, the AVQZ[35] all-electron basis sets are employed for F and Cl. That is to say, for InF, nineteen molecule orbitals (MOs) are put into active space, including 9a1, 4b1, 4b2, 1a2 symmetry (9,4,4,1), which correspond to the 1s2s2p shells of the F atom and 4s4p4d5s5p shells of the In atom; for InCl, twenty two MOs are selected as the active space, including 11a1, 5b1, 5b2, 1a2 symmetry (11,5,5,1), which correspond to the Cl 1s2s2p3s3p and In 4s4p4d5s5p shells; for InH, the active space is (7,3,3,1), which correspond with the H 1s and In 4s4p4d5s5p.

The main transitions of laser cooling for InF, InCl and InH are the 1Π → X1Σ+ transitions which are singlet states. Moreover, the excited state decays exclusively on the 1Π → X1Σ+ transition because of the selection rules for the change in parity and angular momentum in an electric dipole transition. Besides, the transition from intervening state 3Π to the ground state X1Σ+ is spin forbidden. Therefore, spin–orbit coupling effects are not considered in the calculations. Previous works on laser cooling of MH[13] (M = Be, Mg, Ca, Sr, and Ba) and GaF[19] also indicate that the influence of spin–orbit coupling on the spectroscopic properties is generally weak.

Results and discussions

PECs and spectroscopic constants

The calculated PECs of X1Σ+, 1Π and 3Π states of InF, InCl and InH acquired at the MRCI level are shown in Fig. 1. The PECs of X1Σ+ and 3Π states look surprisingly similar, implying that highly diagonal FCFs for 3Π → X1Σ+ transitions are possible. In Table 1, we report the corresponding spectroscopic constants of InF/InCl for the X1Σ+, C1Π and 3Π states and InH for the X1Σ+, A1Π and a3Π states. The experimental[36] and theoretical[20-23] values for these states are also listed in Table 1 for comparison.

Fig. 1

Potential energy curves of the first three electronic states of InF (a), InCl (b) and InH (c) at the MRCI level of theory.

Spectroscopic constants of X1Σ+, C1Π and 3Π states for InF/InCl and X1Σ+, A1Π and a3Π states for InH calculated at the MRCI level of theory

Molecule	States	T e	R e (Å)	w e (cm−1)	w e χ e (cm−1)	B e	D e	Ref.
InF	X1Σ+	0	1.987	550.04	2.36	0.2619	5.45	This work
		0	1.985	535.35	—	—	5.48	Exp.[36]
		0	2.010	564.00	—	—	5.37	Ref. 20
	C1Π	43 050	1.964	467.05	16.51	0.2673	0.44	This work
		42 809	1.966	463.90	—	—	—	Exp.[36]
		42 255	2.000	460.00	—	—	—	Ref. 20
	3Π	30 463	1.953	588.93	3.13	0.2727	1.94	This work
		—	1.946	575.25	—	—	—	Exp.[36]
		28 145	1.980	590.00	—	—	—	Ref. 20
InCl	X1Σ+	0	2.398	325.51	1.03	0.1094	4.70	This work
		—	2.401	317.40	—	—	4.68	Exp.[36]
		0	2.431	313.70	1.38	—	—	Ref. 21
	C1Π	37 478	2.455	210.06	14.07	0.1334	0.06	This work
		37 484	2.473	177.30	12.60	—	—	Exp.[36]
		37 757	2.577	106.90	2.33	—	—	Ref. 21
	3Π	27 802	2.335	350.00	1.90	0.1153	1.23	This work
		—	2.333	340.30	—	—	—	Exp.[36]
		27 871	2.340	325.20	1.16	—	—	Ref. 21
InH	X1Σ+	0	1.821	1537.00	24.83	4.8703	2.88	This work
		0	1.838	1476.00	25.61	4.9945	2.75	Exp.[36]
		—	1.867	1434.00	—	—	2.65	Ref. 23
	A1Π	22 500	1.862	284.30	—	—	—	This work
		22 570	1.954	—	—	—	—	Ref. 22
	a3Π	16 766	1.752	1523.74	61.55	5.389	—	This work
		16 303	1.793	1495.85	63.62	5.270	—	Ref. 22

For the ground state X1Σ+ of InF, the calculated Re and we results are only 0.002 Å and 14.69 cm−1 larger, while the De result is only 0.03 cm−1 smaller than the experimental data.[36] The corresponding percentage errors in Re, we and De are 0.10%, 3.08% and 0.55%, respectively. Banerjee et al.[20] provided even larger Re (1.26%) and we (5.35%), an even smaller De (2.00%) for the X1Σ+ of InF compared to the experimental data.[36] The first excited state of InF is 3Π which lies well above the ground state. Our calculated Re and we of the 3Π state are in remarkably good agreement with experiment,[36] and the relative errors are only 0.36% and 2.38%, respectively. While the percentage errors in Re and we given by Banerjee et al.[20] are 1.74% and 2.56%, respectively. The present calculation yields a Te value for the 3Π of 30 463 cm−1, and the observed A3Π0+, B3Π1–X1Σ+ transitions[36] take place in the range 30 400–31 300 cm−1. The relativistic configuration interaction calculations by Banerjee et al.[20] yielded a Te value for the 3Π of 28 145 cm−1 which was not within the observed range. It is also encouraging to see that the present values of Re, we and Te for the C1Π state are 1.964 Å, 467.05 cm−1 and 43 050 cm−1, which are in very good agreement with the experimental data 1.966 Å, 463.90 cm−1 and 42 809 cm−1 and the theoretical data 2000 Å, 460.00 cm−1 and 42 255 cm−1.

For InCl, good agreement is found between our results and experimental data. The percentage errors of Re, we and De for the ground state X1Σ+ are 0.12%, 2.56% and 0.43%, respectively. The equilibrium bond distance Re by the available all-electron relativistic calculations[21] is 2.431 Å, and the corresponding percentage error is 1.25%. The present weχe of the X1Σ+ state is 1.03 cm−1, which is slightly smaller than the previous theoretical value[21] 1.38 cm−1. For the C1Π, our present results compare very well with the experimental data[36] (Te ∼ 6 cm−1, Re ∼ 0.018 Å, we ∼ 32.76 cm−1, weχe ∼ 1.47 cm−1). While the differences between the previous theoretical value[21] and experimental data are bigger (Te ∼ 273 cm−1, Re ∼ 0.104 Å, we ∼ 70.40 cm−1, weχe ∼ 9.27 cm−1). Whereas for the 3Π, our spectroscopic constants calculated at MRCI level are in reasonable agreement with the available all-electron relativistic calculations[21] (Te ∼ 69 cm−1, Re ∼ 0.005 Å, we ∼ 24.8 cm−1, weχe ∼ 0.74 cm−1). In the case of Re and we, our results (2.335 Å, 350.0 cm−1) are in good agreement with experiment[36] (2.333 Å, 340.3 cm−1). The complete active space self-consistent field method (CASSCF) is used in both the previous work[21] and our work. So the effects of the different active spaces (this work: 11,5,5,1) and (ref. 20: 6,3,3,1) on the spectroscopic constants of InCl can be seen from Table 1. It is obvious that the calculated spectroscopic constants of InCl for active space (11,5,5,1) are more close to the experimental values.

Concerning the X1Σ+ of InH, our calculated parameters Re, we, weχe, Be and De are also in very good agreement with experiment,[36] and the relative errors are 0.92%, 4.13%, 3.05%, 2.49% and 4.73%, respectively. While the percentage errors in Re, we and De given by Leininger et al.[23] are 1.58%, 2.85% and 3.64%, respectively. For the A1Π and a3Π states of InH, Table 1 shows that our present results accord with existing theoretical calculations.[22,23] For example, the A1Π and a3Π states are located at 22 500 cm−1 and 16 766 cm−1 in our calculations, 22 570 cm−1 and 16 303 cm−1 were reported by Zhang et al.[22]

In summary, our present work yield spectroscopic parameters of InF, InCl and InH for the first three electronic states that agree well with previous experimental and theoretical results.

TDM curves

Because the transition 3Π → X1Σ+ is spin forbidden at spin-free level, we only obtain the TDMs of the 1Π → X1Σ+ transition. Fig. 2 shows the computed TDMs of the dipole-allowed transitions C1Π → X1Σ+ for InF/InCl and A1Π → X1Σ+ for InH as functions of the internuclear distance. We note that the 1Π → X1Σ+ transitions for InF, InCl and InH are strong. As seen in Fig. 2, the TDM curves of InF, InCl and InH demonstrate similar behavior. The magnitude gradually increases as internuclear distance R increases, reaches a maximum (InF: 1.80 a.u., InCl: 1.40 a.u., InH: 0.92 a.u.), and drops thereafter. The TDMs of InF, InCl and InH all trend to zero at around 4.0 Å, 3.5 Å and 3.0 Å, respectively.

Fig. 2

TDMs for the C1Π → X1Σ+ transitions of InF/InCl and A1Π → X1Σ+ transitions of InH at MRCI level.

FCFs, spontaneous radiative lifetime and radiative width

InF molecule

The potential laser cooling candidates must have highly diagonal Franck–Condon factors (FCFs). The calculated FCF data of the C1Π → X1Σ+ transition have been tabulated in Table 2. It is clear that InF has highly diagonal FCFs f (f00 = 0.961 and f11 = 0.909) for the C1Π(ν′) → X1Σ+(ν) transition. Most FCFs of experimental cooling molecules are greater than 0.9, as listed in the previous literature.[6-10] The predicted f00 of 0.961 for InF is greater than 0.9, thus it is sufficiently large to be potentially viable for cooling. Of course, a larger FCF has significant benefits for limiting the number of lasers required to keep the molecule in a quasi-closed-loop cooling cycle. To visually demonstrate the distributions of FCFs f for the different vibrational states of the C1Π(ν′) → X1Σ+(ν) transition, we have plotted the cooling transition C1Π(ν′) → X1Σ+(ν) between 0 ≦ ν′ ≦ 6 and 0 ≦ ν ≦ 6 in Fig. 3. The FCFs for the 3Π → X1Σ+ transition are also presented in Table 2. The 3Π → X1Σ+ transition also have large diagonal FCFs, although the calculated f00 of 0.807 for the 3Π → X1Σ+ transition is smaller than that predicted for the C1Π → X1Σ+ transition (f00 = 0.961).

The calculated FCFs f for C1Π(ν′) → X1Σ+(ν) and 3Π(ν′) → X1Σ+(ν) transitions of InF

Molecule	Transition	f 00, f10	f 01, f11	f 02, f12	f 03, f13
InF	C1Π → X1Σ+	0.9606, 0.0377	0.0326, 0.9085	0.0062, 0.0369	0.0004, 0.0155
	3Π → X1Σ+	0.8068, 0.1788	0.1670, 0.4997	0.0231, 0.2542	0.0027, 0.0565

Fig. 3

The calculated FCFs of InF for the lowest vibrational levels of the cooling transition C1Π → X1Σ+.

Except for large FCFs, the potential laser cooling candidates must also have sufficiently short lifetimes. In Table 3, we have reported the computed lifetimes for the C1Π → X1Σ+ transition. Our computed lifetimes can be compared with those of the results predicted by Banerjee et al.[20] The radiative lifetimes of the C1Π → X1Σ+ transition are computed to be 2.77–3.41 ns for the first five vibrational levels of InF, which shows that the C1Π of InF is a rather short lifetime state. At the same time, the radiative width for the C1Π → X1Σ+ transition are computed to be 19.15–15.53 cm−1 for the first five vibrational levels of InF. The radiative lifetimes (7.58–8.34 ns) of the 3Π are a little longer lived than these of the electronic state C1Π, which shows that the 3Π of InF is also a rather short lifetime state. As shown in Table 3, the radiative lifetimes show a slight increase with increasing vibrational level, and the radiative widths show a slight decrease with increasing vibrational level.

Estimated radiative lifetimes (ns) and radiative width (cm−1) (in italics) (theoretical values obtained in brackets)

Molecule	Transition	ν′ = 0	1	2	3	4
InF	C1Π → X1Σ+	2.77	2.88	3.02	3.20	3.41
		(2.32)a	(2.44)a	(2.61)a	—	—
		19.15(−4)	18.38(−4)	17.53(−4)	16.55(−4)	15.53(−4)
	3Π → X1Σ+	7.58	7.74	7.94	8.12	8.34
		6.99(−4)	6.85(−4)	6.68(−4)	6.52(−4)	6.35(−4)

Characteristic base 10 given parenthetically.

InCl molecule

For the C1Π(ν′) → X1Σ+(ν) transition of InCl, the diagonal term f11(0.07) is obtained, and the off-diagonal term f10(0.29), f12(0.16), f13(0.18) are also calculated (see Table 4). For the 3Π → X1Σ+ transition of InCl, off-diagonal term f10(0.29), f12(0.39), f13(0.15) are larger than the diagonal term f11(0.14) (see Table 4). It is clear that InCl does not have highly diagonal FCFs for the C1Π → X1Σ+ and 3Π → X1Σ+ transitions. Besides, the present FCFs f00(0.61) for the C1Π → X1Σ+ transition and f00(0.60) for the 3Π → X1Σ+ transition are not large enough to be potentially viable for cooling. The relative probabilities from C1Π(ν′ = 0) to X1Σ+(ν) are governed by the FCFs which are approximately 61% for ν = 0, 23% for ν = 1, 10% for ν = 2, 4% for ν = 3, 1.5% for ν = 4, 0.5% for ν = 5, 0.2% for ν = 6, 0.07% for ν = 7, and negligibly small for all ν > 7. To limit the inefficiency, or loss of InCl molecules in the cooling cycle, eight cooling lasers are required. Experimentally, cycling transitions requiring one or two repump lasers are common in atomic systems. Here the number of lasers required is not practical. There are similarities to the laser cooling of InCl on the 3Π → X1Σ+ transition. So we identify that InCl is not a potential laser-cooling candidate. Thus the radiative lifetimes and diode laser excitation wavelengths of InCl do not need to be discussed in the following section.

The calculated FCFs f for C1Π(ν′) → X1Σ+(ν) and 3Π(ν′) → X1Σ+(ν) transitions of InCl

Molecule	Transition	f 00, f10	f 01, f11	f 02, f12	f 03, f13
InCl	C1Π → X1Σ+	0.6079, 0.2912	0.2293, 0.0716	0.1010, 0.1620	0.0385, 0.1766
	3Π → X1Σ+	0.5957, 0.2907	0.3282, 0.1437	0.0691, 0.3913	0.0067, 0.1521

InH molecule

As listed in Table 5, the A1Π state is not suitable for laser cooling because of the off-diagonal FCFs and the small diagonal FCF f00(0.790) for the A1Π → X1Σ+ transition. Fig. 4 shows the FCFs of the a3Π → X1Σ+ transition are highly diagonal. The present FCF of 0.918 for the a3Π → X1Σ+ transition of InH is still large enough for laser cooling. However, this a3Π → X1Σ+ transition is very weak because it is spin-forbidden. Because of the poor linestrength of the a3Π → X1Σ+ transition, the InH molecule must have been precooled using the A1Π → X1Σ+ transition before the a3Π → X1Σ+ transition could be adopted for laser cooling of InH. It is similar to the laser cooling of Ca, where cooling on the weak 1S → 1P transition must followed by the strong 1S → 3P transition. The problem with the laser cooling scheme using the a3Π → X1Σ+ transition is that the InH molecule could not be precooled using the A1Π → X1Σ+ transition. So InH is not a promising laser-cooling candidate. Thus the radiative lifetimes and diode laser excitation wavelengths of InH do not need to be discussed in the following section.

The calculated FCFs f for A1Π(ν′) → X1Σ+(ν) and a3Π(ν′) → X1Σ+(ν) transitions of InCl

Molecule	Transition	f 00, f10	f 01, f11	f 02, f12	f 03, f13
InH	A1Π → X1Σ+	0.7897, 0.0573	0.1172, 0.0169	0.0604, 0.0372	0.0181, 0.0715
	a3Π → X1Σ+	0.9176, 0.0672	0.0809, 0.8217	0.0012, 0.0969	0.0001, 0.0014

Fig. 4

The calculated FCFs of InH for the lowest vibrational levels of the cooling transition a3Π → X1Σ+.

Laser cooling scheme

We have established the laser cooling schemes for InF, and Fig. 5 shows the proposed scheme. The corresponding computed radiative wavelengths are collected in Table 6. For InF, the main cycle is the X1Σ+(ν = 0) → C1Π(ν′ = 0) transition at wavelength λ00 = 232.4 nm. As shown in Fig. 5(a), there is a small probability of decay to the X1Σ+(ν = 1) state (≈3%), an even smaller probability of decay to the X1Σ+(ν = 2) state (≈0.6%), and a negligible probability of decay to the X1Σ+(ν ≥ 3) state (<10−4). To enhance the cooling effect, we add the X1Σ+(ν = 1) → C1Π(ν′ = 0) transition as the first vibrational pump and the X1Σ+(ν = 2) → C1Π(ν′ = 1) transition for the second pump. With this scheme, two repumping lasers λ10 = 229.8 nm and λ21 = 230.2 nm are required. The required cooling wavelengths of λ00 = 232.4 nm, λ10 = 229.8 nm and λ21 = 230.2 nm can be generated by a frequency quadrupled Ti:sapphire laser (189–235 nm). One problem with the above laser cooling scheme, is the presence of the intervening 3Π state between the two states (X1Σ+ and C1Π). Notable, the 3Π → X1Σ+ transition also have large diagonal FCFs and short lifetimes. Thus, this state 3Π could also be another option for laser cooling. We propose a four laser cyclic system involving the ν = 0–3 of the X1Σ+ state and ν′ = 0, 1 of the 3Π state. The main cycling laser at the wavelength of 328.1 nm can drive the X1Σ+(ν = 0) → 3Π(ν′ = 0) transition. Similar to the case in Fig. 5(a), a pair of repumping lasers with λ10 = 321.9 nm and λ21 = 321.6 nm are required to enhance the cooling effect. A third repumping laser with λ31 = 315.8 nm may be required on this scheme due to the non-negligible ν′ = 1 → ν = 3 transition (FCF = 0.0565). The required cooling wavelengths on this scheme are located in the ultraviolet light (UVA) range (400–320 (or 315) nm). There are similarities here to the laser cooling of AlF. Wells and Lane[15] have studied on laser cooling of AlF molecules, and the X1Σ+ → C1Π and X1Σ+ → 3Π transitions of AlF had strongly FCFs. So for InF, the cooling on a strong C1Π → X1Σ+ transition may be followed by further cooling on the X1Σ+ → 3Π transition, which is similar to the laser cooling of AlF. The relatively long lifetimes of the 3Π state can be exploited to reach a much lower Doppler temperature than possible on the X1Σ+ → C1Π transition.

Fig. 5

Proposed laser cooling schemes for InF using (a) the C1Π(ν′) → X1Σ+(ν) (solid red) and (b) the 3Π(ν′) → X1Σ+(ν) (solid red) transition. The decay pathways with calculated f are shown as dotted line.

The calculated wavelength λ

Molecule	λ 00 (nm)	λ 10 (nm)	λ 21 (nm)	λ 31 (nm)	Transition
InF	232.4	229.8	230.2	227.8	C1Π → X1Σ+
	328.1	321.9	321.6	315.8	3Π → X1Σ+
InH	596.9	549.5	556.3	525.0	a3Π → X1Σ+

Compared with other potential laser-cooling candidates, diagonal FCFs f00 of InF (f00 = 0.961) is slightly larger than that predicted for BeF (f00 = 0.897), LiRb (f00 = 0.872), InH (f00 = 0.918) and KRb (f00 = 0.947). Experimentally, KRb is a typical laser-cooled molecule. Radiative lifetime τ(ν′ = 0) = 2.8 ns of InF is shorter than the one of BeF (7.9 ns), LiRb (102.6 ns), InH (205.7 ns) and KRb (229.3 ns). A shorter τ of the transition could produce a strong Doppler force, which will produce rapid laser cooling. We have identified InF is a more promising laser-cooling candidate. Not only it has bigger FCFs and shorter radiative lifetime, it also appears to attain a lower Doppler temperature on the X1Σ+ → C1Π transition followed by the weak X1Σ+ → 3Π transition. Generally, it is true that the Doppler-limit temperature is inversely proportional to the excited state lifetime. As shown in Table 3, the lifetime (7.58 ns) of the 3Π state is about three times larger than that of the C1Π state. However, the lifetime (7.58 ns) of InF is still shorter than that of KRb (229.3 ns). The short lifetime is sufficient to produce large spontaneous scattering forces.

Conclusions

The PECs of X1Σ+, 1Π and 3Π states of InF, InCl and InH are acquired at the MRCI level. Our present work yield spectroscopic constants (Re, De, we, weχe, Be, Te) of InF, InCl and InH for the first three electronic states that agree well with previous experimental and theoretical results. The TDMs of the dipole-allowed transitions C1Π → X1Σ+ for InF/InCl and A1Π → X1Σ+ for InH are obtained. Using the FCFs and radiative lifetimes obtained, we have identified InF is a promising candidate for laser cooling. Because of its off-diagonal FCFs, InCl is not a potential laser-cooling candidate. For InH, the A1Π state has off-diagonal FCFs and the FCFs of the a3Π → X1Σ+ transition are highly diagonal. However, this a3Π → X1Σ+ transition is very weak because it is spin-forbidden. The problem with the laser cooling scheme using the a3Π → X1Σ+ transition is that the InH molecule could not be precooled using the A1Π → X1Σ+ transition. So InH is not a promising laser-cooling candidate. The proposed laser cooling schemes of InF that drive the X1Σ+ → C1Π transition use three laser wavelength. Furthermore, the 3Π → X1Σ+ transition of InF is also strongly diagonal and the lifetimes of the excited 3Π state of InF is relatively long. So for InF, the X1Σ+ → C1Π transition may be followed by the weak X1Σ+ → 3Π transition to obtain a lower Doppler temperature. All of these three required laser-cooling wavelengths are in the visible range.

Conflicts of interest

There are no conflicts to declare.