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.
