School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
† Corresponding author. E-mail:
zygsr2010@163.com
1. IntroductionLaser cooling to create unique ultracold molecules has aroused considerable interest[1–4] because of their promising applications, for example, new platforms for quantum computing,[5] quantum controlled chemistry,[6,7] precision measurement,[8–10] and quantum simulation.[11,12] Naturally the search for potential laser cooling candidates is attracting more and more interest. The direct cooling of a diatomic molecule to the order of microkelvin was firstly achieved in SrF using only three laser beams in 2010,[1] which has initiated a search for more molecules that may be controlled in a similar way. Later, successful laser cooling experiments have been performed for YO,[13] CaF,[14] BH,[15] and BaH.[16] Besides the experimental studies on laser cooling candidates, theoretical optical schemes have been suggested for MgCl,[17] MgBr,[17] BeI,[18] MgI,[18] BeCl,[19] BeBr,[19] BeF,[20] MgF,[21] LiBe,[22] as well as for MH (, Mg, Ca, Sr, and Ba).[23] According to the previous investigations, the Doppler laser-cooling process must meet these significant criteria: highly diagonal Franck–Condon factors (FCFs), limiting number of lasers required to keep the molecule in a closed-loop cooling cycle, and short radiative lifetime τ describing rapid laser cooling. In theory, most of the studies have focused on spin-allowed transitions ( or ). While Kobayashi et al. experimentally confirmed that the spin-forbidden transition is well suited for laser cooling of the KRb[3] molecule in 2014. In addition, the laser cooling scheme using the spin-forbidden transition has also been theoretically studied on AlF,[24] BBr,[25] BCl,[25] and LiRb.[26] These results open the door to all-optical production of polar molecules at sub-microkelvin temperatures and highlight the possibility of finding similar molecules that can be laser cooled.
Indium monohalides have been attracting interest for a long time because of their applications in the etching process of semiconductor devices. Experimental studies and theoretical calculations have been performed for InH over the past several decades. In 1939, the electronic spectrum of InH was reported, and bands of two transitions (i.e., Σ–Σ and Π–Σ) were rotationally analyzed by Grundström.[27,28] Garton[29] presented a band system in the wavelength region of 233–250 nm in 1951, but the system was not assigned. Subsequently, Neuhaus et al.[30,31] measured the – and – absorption spectra of InH and obtained the spectroscopic constants. Ginter[32–34] fitted the Rydberg–Klein–Rees curves of , , and states, and found – and –X transitions. Ogilvie[35] later determined the potential energy and coefficients of radial functions of InH in the ground state from the rovibrational spectrum. In 1993, Rajamanickam et al.[36] refitted the Rydberg–Klein–Rees curves of the and states, and obtained the FCFs of the – system. In 2003, Zou et al.[37] investigated the potential energy curves (PECs) and the spectroscopic constants of the , , , , , and states using the second order CI and relativistic CI methods, and predicted the transition properties of the excited states. Most recently, analytic potential energy functions were computed for the and states of InH using a dict-potenial-fit analysis by Alireza Shayesteh et al.[38] However, a systematic study of laser cooling of InH is, to the best of our knowledge, is so far unavailable. We focus in the present work on the theoretical study of the laser cooling of InH molecule employing different active spaces. The electronic structures and transition properties associated with the laser cooling of InH are calculated, including FCFs, transition dipole moments (TDMs), and radiative lifetimes of the , , and transitions of the InH molecule. A scheme for a feasible laser cooling cycle of InH is designed briefly.
Section 2 describes the ab initio methods and basis sets used in the calculations of the electronic states of InH. Section 3 presents the results and discussion of the data, outlining laser cooling schemes for InH. We draw a conclusion for this work in Section 4.
2. Computational detailsAll the ab initio calculations are performed using the MOLPRO package.[39] The electronic states , , , and of InH molecule have been calculated with multi-reference configuration interaction (MRCI) plus Davidson corrections (MRCI+Q) method,[40–42] which is based on the complete active space self-consistent-field (CASSCF)[43,44] wave functions. Scalar relativistic effects are included throughout the Douglas–Kroll–Hess[45,46] transformation of the relativistic Hamiltonian. The spin–orbit coupling (SOC) effects are also taken into account following the MRCI+Q calculations for the InH molecule. Due to the limitation of the symmetry of the MOLPRO program package, the computations are performed within the point group symmetry, which has four irreducible representations (A1, B1, B2, and A2). In the CASSCF and MRCI+Q calculations, 23 or 14 molecular orbitals are chosen as the active space, including eleven or seven a1, five or three b1, five or three b2, and two or one a2 symmetry. Fourteen electrons are distributed in (11552) or (7331) active space. The aug-ccpV5Z (AV5Z) basis set is used for the H atom. For the In atom, we take the small-core scalar relativistic effective core potential (ECP) ECP28MDF-AV5Z with the corresponding valence basis sets.
The nuclear Schrödinger equation is solved using the LEVEL 8.2 program[47] to evaluate the spectroscopic constants, including the equilibrium bond length (), harmonic and inharmonic vibrational constants ( and ), rotational constant (), adiabatic relative electronic energy referred to the ground state (), and dissociation energy () for the ground and the low-lying states of InH. The states are repulsive; therefore, they are not discussed in detail. All PECs are calculated with an interval of 0.05 Å over the distance from 1.1 Å to 9 Å. To obtain accurate results, the interval value is reduced to 0.02 Å near the equilibrium bond distance. The permanent dipole moments (PDMs) and TDMs are computed by the MRCI+Q method. The FCFs and radiative lifetimes of the various vibrational levels for –, – and – transitions of InH are also determined from the LEVEL 8.2 program with the PECs and TDMs of different electronic states.
3. Results and discussion3.1. PECs and spectroscopic constantsIn order to understand the possibility of laser-cooling InH molecule, we investigate the , , , and states of InH with the lowest dissociation channel at the MRCI+Q level. Because InH is a heavy nuclear molecule, the SOC effects are considered in calculations. The dissociation limit splits into two asymptotes, namely, and . These two dissociation limits produce eight states. The PECs for the , , , , , , , and states of InH are plotted in Fig. 1. The corresponding spectroscopic constants are tabulated in Table 1 along with available experimental data[48] for comparison.
Table 1.
Table 1.
| Table 1.
Spectroscopic constants for InH molecule at the MRCI+Q level. . |
As can be seen from Table 1, the influence of the SOC effect on the spectroscopic constants for the ground state seems to be weak. For example, the vibrational frequency and anharmonic vibrational constants of the state are computed to be 1439.7738 cm−1 and 21.6642 cm−1, which differ from the values of by 0.4825 cm−1 and 0.101 cm−1, respectively. The SOC effects are considered in the calculations for the state, which splits into four states (, , , and . In addition, to obtain accurate results, the effects of the active space are also considered for the InH molecule. The study of spectroscopic constants mainly focuses on two active spaces (11552) and (7331). The active space has a weak influence on the spectroscopic constants of the and states. For the state, the values of and from active spaces (11552) and (7331) are very close. For the state, the calculated and results for active space (11552) are only 1.462 cm−1 and 0.030 cm−1 larger than those for active space (7331). Meanwhile, the effects of the different active spaces (11552) and (7331) on the spectroscopic constants of the sub-states , , , , and are considered. The results indicate that the effects of the active space on the spectroscopic constants of these states are insignificant. The values of the and sub-states for active space (11552) are 1496.8152 cm−1 and 1511.6053 cm−1, which are in excellent agreement with the values of 1494.6673 cm−1 and 1505.7961 cm−1 for active space (7331). Our values of of the and states for active space (11552) are slightly larger than those for active space (7331). Thus the following analyses mainly focus on the results on the basis of active space (11552). For , , and states, the equilibrium bond distances are calculated to be 1.8683 Å, 1.7871 Å, and 1.7881 Å, and the corresponding percentage errors are only 1.65%, 0.43%, and 1.15% with respect to the experimental values, respectively; the differences in the rotational constant for the three states are 0.1613 cm−1, 0.0289 cm−1, 0.1076 cm−1, respectively; and notably, the calculated for the , , and states are 0, 15374.570 cm−1, and 15956.470 cm−1, which match well with the experimental values of 0, 16278.15 cm−1, and 16941.61 cm−1. The spectroscopic constants for the and states are also predicted: cm−1 and 1511.6053 cm−1, cm−1 and 70.9483 cm−1. However, the equilibrium distances of the , , , , and states for active space (11552) are respectively 1.8683 Å, 1.7871 Å, 1.7871 Å, 1.7881 Å, and 1.7878 Å, which are closer to the experimental data[48] than those for active space (7331). Because the equilibrium separation has an important influence on FCFs, the results from active space (11552) are more suitable to evaluate the effect of laser cooling.
As shown in Fig. 2, the influence of the SOC effects and active space on the PECs of the state is significant. The dissociation limit of the state is greater than that of the Λ–S state. The PECs of the active space (11552) are smoother than those of the active space (7331), which implies that active space (11552) is more conducive to research laser cooling. The PECs of the and states have two potential wells. Moreover, the second potential well and the second equilibrium bond length have a distinct change with the SOC effects. It can be seen from Table 1, for the state, the second well is placed at 22057.855 cm−1 above the ground state and the depth is 44.360 cm−1. For the state, the second well is placed at 22646.205 cm−1 above the ground state, and the depth is 126.628 cm−1. The equilibrium internuclear distances of the second potential well for the and states are 3.9783 cm−1 and 3.1884 cm−1, respectively. In addition, the , , and of the first potential well and of the second potential well cannot be obtained because the state has two so shallow potential wells that there is only one vibrational level in them. So no experimental data were available for the and states before.
3.2. PDMs and TDMsThe PDMs for the , , , , , , , , and states and the TDMs for the , , and transitions as a function of the internuclear distance at the MRCI+Q level of InH are plotted in Figs. 3 and 4, respectively. Figure 3(a) exhibits that the maxima of PDMs of the and states are much larger than those of the and states, and the reason is that the potential well of the ground state is the deepest. The PDMs of the and states exhibit a similar behavior with respect to the internuclear distance, the magnitudes of the PDMs reach a maximum (: 1.86080 a.u. and : 1.84172 a.u.) and drop thereafter. The PDMs of the and states also demonstrate a similar behavior, the magnitude increases as the internuclear distance increases, reaching a maximum (: 0.14758 a.u. and : 0.14762 a.u.). There are minima of PDMs of the and states at 2.6 Å and 2.55 Å, respectively. Figure 3(b) shows that the PDMs for the , , , , and states at are 0.23545 a.u., 0.23496 a.u., 0.23506 a.u., 0.23426 a.u., and 0.23545 a.u., respectively. The PDM shows a linear behavior up to approximately 2.00 Å for the , , , , and states and then drops, and the maxima of PDMs are 0.27961 a.u., 0.27831 a.u., 0.28025 a.u., 0.27589 a.u., and 0.27961 a.u., respectively. The PDMs of the and states overlap. The PDM of the state increases from 3.2 Å to 4.4 Å and then declines. The PDMs of these states all tend to zero at about 5.3 Å.
Three transitions , , and are calculated in the present work. We omit the and transitions because they are forbidden according to the selection rules. Figure 4 shows that the TDMs of the and transitions tend to zero at large distances owing to the orbit-forbidden transitions at the atomic limits from to . Nevertheless, the TDM of the transition tends to zero at large distances because the dissociation limits of the and states are the same. That is, there is no transition at the atomic limits from to . The maxima of TDMs for these three transitions are less than 1.50 a.u., and the and transitions are much weaker than the transition because the and transitions are spin-forbidden. The TDMs for the and transitions increase as the internuclear distance R increases, reach maxima (1.46031 a.u. and 0.44178 a.u.), and drop thereafter. The TDM for the transition is reducing. Finally, the TDMs of InH all tend to zero at around 6.6 Å.
3.3. FCFs and spontaneous radiative lifetimesThe calculated FCFs can be used to describe the overlap of the vibrational wave functions for the cooling transition. To demonstrate the distributions of FCFs () for the different vibrational states of , , and transitions, we have sketched all possible transitions between and in Table 2 to show the obvious characteristic that the transitions of have the largest probabilities. Highly diagonal FCFs obey the first criterion to be a potential laser-cooling candidate which could limit the number of lasers required to keep the molecule in a closed-loop cooling cycle. Unfortunately, the FCFs of the transition are very small. Therefore, it is not possible to cool the InH molecule based on the spin-allowed transition.
As listed in Table 2, the present FCFs of the and transitions reach 0.9100 and 0.9151, respectively. The state is prohibitive to the cooling cycle because of the small diagonal FCF () for the transition. By comparing with other molecules, our calculated value is slightly larger than that predicted for BeF (),[20] LiRb (),[26] and smaller than that for SrF ().[1] Therefore, the FCFs of InH are sufficiently large for laser cooling. Aside from the large diagonal FCFs, short spontaneous radiative lifetime τ (10−8–10−5 s)[23] is another criterion for the laser cooling of molecules, which can provide a significant rate of rapid cycling. The corresponding computed radiative lifetimes are collected in Table 2. The spontaneous radiative lifetime τ of the transition is 1.9649 × 10−4 s, which is too long. Yet, the spontaneous radiative lifetime τ of the transition is 1.0119 × 10−6 s, which is suitable for laser cooling InH molecule.
We also evaluate the branching ratios , which can be expressed as . Branching ratios of the diagonal terms and for the transition are obtained; and branching ratios of the off-diagonal terms , , , , , and are also calculated. Due to the branching to states of InH are expressed as , we propose a cyclic system with three lasers involving , 1, 2, 3 of the state and , 1 of the state based on the calculated . The laser driven transitions (solid red) and spontaneous decays (dash line) in the proposed scheme are plotted in Fig 5. The calculated wavelength of the principal laser-driven cycling of the transition is the main pump in Fig. 5, whose wavelength λ00 is 626.7 nm. To augment the cooling effect, we add two cycles that the transition is the first vibrational pump and the transition is the second vibrational pump. Therefore, two additional lasers with and are required.
4. ConclusionThe PECs of , , , , , , , and states of InH molecule are investigated at the MRCI+Q level. The AV5Z-DK all-electron basis set for H and ECP28MDF-AV5Z for In atom are chosen. Scalar relativistic corrections are performed using Douglas–Kroll–Hess. The spectroscopic constants (, , , , , of the , , , , , , , , and states are calculated by solving the radial Schrödinger equation using the LEVEL8.2 program from two active spaces. In addition, the calculated result shows that the state has two potential wells, and the influence of the SOC effects and active space on the PECs is obvious. The comparison between our present work and available experimental data in the literature shows a good agreement.
The PDMs for the , , , , , , , , and states and the TDMs for the , , and transitions are obtained based on the MRCI+Q method. On the basis of accurate PECs and TDMs, the theoretical results indicate that the transition has highly diagonal branching ratios () and suitable radiative lifetime (). These two conditions can ensure rapid and efficient laser cooling of InH molecule. The transition is prohibitive to laser cooling because the transition has a long radiative lifetime. Concurrently, an optical scheme of three laser cycles with the transition to create an ultracold InH molecule is proposed. The main cycling laser wavelength nm is determined for the transition, and two repumping lasers and are chosen for and . These results imply the feasibility of laser cooling of InH.