Rong
Yang
*a,
Bin
Tang
b and
XiangYu
Han
ac
aSchool of Mathematics and Physics, Chongqing Jiaotong University, Chongqing 400074, PR China. E-mail: cqyr88@126.com
bInstitute of Finance & Trade, Chongqing City Management College, Chongqing 401331, PR China
cInstitute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, PR China
First published on 4th October 2019
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 (f00 = 0.961, f11 = 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.
The A2Π → X2Σ+ transition suggested by the experiments6–9 for SrF, YO and CaF has been theoretically used to establish the laser cooling schemes for RaF,4 BeF,11 MgF12 and MH13 (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 Cl16 and Br16) and BBr17/BCl17 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 AlCl16/AlBr16 and BBr17/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. BH19/BCl17/BBr,17 AlH15/AlF15/AlCl16/AlBr16 and GaF19 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.
For the indium atom, being considerably heavy, we use the small-core scalar relativistic effective core potential ECP28MDF33 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 AVQZ35 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 MH13 (M = Be, Mg, Ca, Sr, and Ba) and GaF19 also indicate that the influence of spin–orbit coupling on the spectroscopic properties is generally weak.
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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. |
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![]() |
1.964 | 467.05 | 16.51 | 0.2673 | 0.44 | This work | |
42![]() |
1.966 | 463.90 | — | — | — | Exp.36 | ||
42![]() |
2.000 | 460.00 | — | — | — | Ref. 20 | ||
3Π | 30![]() |
1.953 | 588.93 | 3.13 | 0.2727 | 1.94 | This work | |
— | 1.946 | 575.25 | — | — | — | Exp.36 | ||
28![]() |
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![]() |
2.455 | 210.06 | 14.07 | 0.1334 | 0.06 | This work | |
37![]() |
2.473 | 177.30 | 12.60 | — | — | Exp.36 | ||
37![]() |
2.577 | 106.90 | 2.33 | — | — | Ref. 21 | ||
3Π | 27![]() |
2.335 | 350.00 | 1.90 | 0.1153 | 1.23 | This work | |
— | 2.333 | 340.30 | — | — | — | Exp.36 | ||
27![]() |
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![]() |
1.862 | 284.30 | — | — | — | This work | |
22![]() |
1.954 | — | — | — | — | Ref. 22 | ||
a3Π | 16![]() |
1.752 | 1523.74 | 61.55 | 5.389 | — | This work | |
16![]() |
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 30463 cm−1, and the observed A3Π0+, B3Π1–X1Σ+ transitions36 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 calculations21 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 value21 1.38 cm−1. For the C1Π, our present results compare very well with the experimental data36 (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 value21 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 calculations21 (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 experiment36 (2.333 Å, 340.3 cm−1). The complete active space self-consistent field method (CASSCF) is used in both the previous work21 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 22500 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.
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Fig. 2 TDMs for the C1Π → X1Σ+ transitions of InF/InCl and A1Π → X1Σ+ transitions of InH at MRCI level. |
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 |
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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.
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 |
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 |
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.
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