Introduction

Researchers have been interested in the spectroscopic studies of the alkali and alkaline earth halides1,2 because of their relevance to astrophysics. These have been detected in the interstellar medium3 and the upper atmosphere4. In this view, MgF, SrF1, and MgCl molecules are predicted to appear in S-stars2, on the sun’s surface, and in the sunspot’s spectrum. Moreover, these alkaline-earth mono-halide molecules are highly interesting for high-temperature reactions in catalysis and corrosion processes5.

From the perspective of laser cooling experiments, the compounds of alkaline-earth metals have been proposed as promising candidates for laser cooling and controlling the preparation of many-body entangled states6,7,8. SrF and YO9,10 molecules have been cooled using transverse cooling methods, while CaF has been cooled by longitudinal laser cooling11. Extensive theoretical studies have also been performed for molecules that possess similar electronic structures, such as BeF12 and MgF13

The electronic structure of the alkaline-earth halide molecules, including MgAt, has been studied in the literature. The first few low-lying excited electronic states of the molecules MgCl, MgBr, and MgI have already been investigated14,15,16,17,18,19. In 2015, Wan et al.20 presented for BeI and MgI an ab initio investigation for the effect of spin–orbit coupling on laser cooling, where they calculated the spectroscopic properties and the cooling wavelength of these molecules in the ultra-violet region. The suitability of laser cooling of alkaline earth mono halides BaX (X = F, Cl, Br, I) and MgX(X = Br, At, I) has been verified respectively by Yang et al.21 and Yang and Tao22.

We present a theoretical study by using the ab initio method (CASSCF/MRCI + Q) for the molecules SrAt and BaAt to test the candidacy of alkaline-earth astatine species for laser cooling. Section “Computational approach” includes the computational approach followed for the pursued computations. The adiabatic potential energy curves, the dipole moment curves of the low-lying doublet and quartet electronic states, and their spectroscopic constants in the 2S+1Λ+/− and Ω(±) representations are presented in Section “Potential energy curves, spectroscopic parameters, and permanent dipole moment curves”. In addition, the vibrational energy Ev, the ro-vibrational constants Bv, Dv, the abscissa of the turning points Rmin, and Rmax of the ground, and the bound excited electronic states are displayed in Section “The ro-vibrational parameters”. Section “Laser cooling study of SrAt and BaAt molecules” includes a laser cooling investigation of the molecules SrAt and BaAt, done by calculating the Franck–Condon Factors (FCF), the Einstein Coefficients, the radiative lifetime, and the branching ratio among specific vibrational levels. Experimental parameters are presented, including the minimum slowing distance, the Doppler and recoil temperatures, and the maximal deceleration of the molecules. Laser cooling schemes for the molecules SrAt and BaAt are presented with three and four lasers in the visible and near-infrared regions, respectively.

Computational approach

The Complete Active Space Self Consistent Field (CASSCF) has been used as a reference for generating the multiconfiguration wavefunctions of the considered two molecules. It is followed by the Multireference configuration interaction (MRCI) method, with Davidson correction (+ Q)23. The current calculations are done by employing the MOLPRO program package24, taking advantage of the graphical user interface GABEDIT25 to study the electronic structure of the electronic states of SrAt and BaAt in the doublet and quartet multiplicities with and without considering the spin–orbit coupling effect. For the BaAt molecule, the electronic wavefunctions of seventy-eight core electrons of At are described by the quasi-relativistic effective core potential ECP78MWB26 for s, p, d functions, while for the SrAt, they are described by ECP60MDF. Thirty-six electrons of Sr were frozen using the ECP36SDF for s, p functions, and 46 electrons of Ba were frozen using the ECP46MWB for s, p, d functions. It is worth noting that the Cꝏv group was decomposed into C2v sub-group because of the limitations of the MOLPRO software. Table 1 reports the active space orbitals for the two considered molecules. Thus, the molecular orbitals are labeled in the irreducible representation as 4a1, 1b1, 1b2, and 0a2 for SrAt denoted by [4,1,1,0], and 6a1, 3b1, 3b2, and 1a2 denoted by1,3,3,6 for BaAt. Also, the molecules SrAt and BaAt have been investigated in spin–orbit Ω(±) representation where Sr is treated as a system of 10 electrons using ECP28MDF27, Ba is treated by ECP46MDF 28, and At is treated by ECP60MDF for SrAt molecule and ECP78MDF for BaAt molecule28. Then the active space in the spin–orbit calculations of SrAt becomes 4σ (Sr: 5s; At:6p0, 6s, 7s), 1π (Sr:0; At: 6p1±), 0δ and that of BaAt is 6σ (Ba: 5d0, 5d+2,6p0, 6 s; At:7 s, 6p0), 3π (Ba:5d±1, 5p±1; At: 6p±1),1δ (Ba: 5d-2) and the molecular orbitals are labeled as [4,1,1,0] for SrAt and [1,3,36] for BaAt.

Table 1 The active space orbitals for the SrAt and BaAt molecules.
Full size table

Additionally, the potential energy curves of the molecules BeAt, MgAt, and CaAt have been computed for a spectroscopic trend comparison (see Section “Potential energy curves, spectroscopic parameters, and permanent dipole moment curves”). The basis set is cc-pV5Z29 for Be and Mg atoms and ECP10MWM30 for Ca atom. For At atom, the same basis set (ECP78MDF) was used among all molecules. The corresponding potential energy curves of the doublet and quartet electronic states for the considered five molecules are given in Figs. FS1-FS9 along with their static dipole moments (Figs. FS10–FS15) in the supplementary materials.

Results and discussion

Potential energy curves, spectroscopic parameters, and permanent dipole moment curves

The ab initio method employed in the present work allowed the investigation of the adiabatic potential energy curves (PECs) of the electronic states of the alkaline earth astatine molecules SrAt and BaAt in their doublet and quartet multiplicities. The PECs of thirty-five electronic states (eight doublet and five quartet states of SrAt molecule) and (seven doublet and 15 quartet states of BaAt molecule), taking into account spin–orbit coupling, in the representation Ω(±) are provided in Figs. 1, 2 as a function of the internuclear distance R, while the PECs of 28 states (eight doublet and ten quartet states of SrAt) and (five doublet and five quartet states of BaAT) calculated without considering this effect are given in the supplementary material in Figs. FS5–FS8. One can notice that the two molecules have deep potential wells reflecting a dominancy of the attractive forces within the molecule’s constituents and shallower ones reflecting the dominancy of repulsive forces. Additionally, many unbound repulsive states are observed. The ground state is X2∑+, which has a deep potential well for the two molecules. The spectroscopic parameters Te, Re, \({\upomega }_{{\text{e}}}\), and Be have been calculated for the bound states upon fitting their potential energy curves into a polynomial around the equilibrium position Re. The calculated spectroscopic parameters of BaAt and SrAt molecules with and without spin–orbit coupling effects are listed in Tables 2 and 3. The data we present here has been calculated for the first time, so comparing it with the literature is not possible. Still, the validity of the spectroscopic constants can be confirmed in Table 4 through the homogeneous trend of Te, Re, and ωe of the ground and some of the low-lying electronic states of the molecules BeAt, MgAt, CaAt, SrAt, and BaAt, as in previously published work31. The correct trend of the spectroscopic constants is evident for all the investigated electronic states: an increase in the atomic mass of the alkaline earth atom corresponds to a decrease in the electronegativity, which leads to an increase in the equilibrium bond length Re, a decrease in the transition energy Te, and the harmonic frequency ωe. The spectroscopic constants are not calculated for the remainder of the excited states because they are either unbound states, have very shallow potential wells, or present an avoided crossing behavior near their minimum.

Figure 1
figure 1

Potential energy curves of the lowest Ω(±) doublet and quartet states of the SrAt molecule.

Full size image
Figure 2
figure 2

Potential energy curves of the lowest Ω(±) doublet and quartet states of the BaAt molecule.

Full size image
Table 2 Spectroscopic constants of the molecules SrAt and BaAt without spin–orbit coupling calculated by using the multireference configuration interaction technique.
Full size table
Table 3 Spectroscopic constants of the molecules SrAt and BaAt with spin–orbit coupling effects taken into consideration, calculated by using the multireference configuration interaction technique.
Full size table
Table 4 Spectroscopic constants trends among different electronic states of the molecules BeAt, MgAt, CaAt, SrAt, BaAt.
Full size table

Moreover, By using the basis set ECP60MDF For At atom, the comparison of our spectroscopic constants for MgAt (Table 4) with those given by Yang and Gao22 shows a very-good agreement with relative differences of 0.6%, 1.8%, and 2.5%, respectively, for ΔRe/Re, Δωe/ωe, ΔBe/Be for the ground state X2∑+. For (2)2Π states, these relative differences are 1.63%, 1.60%, 3.70%, and 3.6% for ΔTe/Te, ΔRe/Re, Δωe/ωe, and ΔBe/Be, respectively.

Given the correct trend and the very good agreement of our spectroscopic constants with those available in the literature22, we may confirm the accuracy of our results for the two molecules, SrAt and BaAt.

The permanent dipole moment curves (PDMCs) are an effective tool for understanding the polarity and the strength of the long-range dipole–dipole forces in diatomic molecules. The permanent dipole moment curves (PDMCs) of the five molecules, BeAt, MgAt, CaAt, SrAt, and BaAt (without including the spin–orbit coupling effects), are represented in Figs. FS10–FS15 of the supplementary material. The electrons’ density distribution can be understood according to the polarity of the dipole moments ranging from − µ to + µ. The dipole moment usually exhibits positive values when the electrons’ density is closer to the alkaline earth metal considered at the origin. On the contrary, flipping in the polarity occurs when the dipole moment becomes negative as the electrons’ density becomes closer to the At atom. Consequently, the positive values of dipole moments can be denoted by Srδ−Atδ+ and Baδ−Atδ+. The values of dipole moment, which tend to be zero at large internuclear distances, are evidence of the molecule’s dissociation into neutral fragments. In contrast, those with constant values indicate dissociation into ionic fragments.

The ro-vibrational parameters

The theoretical determination of a given level’s rovibrational constants is effective in the prediction process of absorption/emission line positions. These are useful in guiding experimental investigations that facilitate the detection of unknown molecules. In the conventional approach of the Rayleigh-Schrödinger perturbation theory (RSPT), the first analytical expressions of the centrifugal distortion constants (CDC) have been derived by Albritton et al.32. To overcome the complexity of the computation of such expressions, Hutson derived an algorithm33 by using the Numerov difference equation for the determination of the constants \({D}_{\upnu }\), \({H}_{\upnu }\), \({L}_{\upnu }\), and \({M}_{\upnu }\) in terms of the vibrational wave function \({\Psi }_{\upnu }\). But in this algorithm, some difficulties had appeared for some potentials (like the Lennard–Jones potential), such as the problem of treating high vibrational levels near the dissociation limit. An improvement has then been introduced to the Huston algorithm Tellinghuisen34, but it is still insufficient to reach larger orders of centrifugal distortion constants. For this purpose, the quantum mechanical canonical function method35,36,37 was developed to calculate the rotation–vibration constants for highly excited electronic states with many centrifugal distortion constants.

This approach is used in the present work to determine the rovibrational parameters of the BaAt molecules, including the vibrational energy Ev, the rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning point Rmin and Rmax. These values, including the spin–orbit coupling effects, are given in (Tables 5, 6). Since most states are unbound, the spectroscopic constants and the ro-vibrational parameter of the quartet spin–orbit potential energy curves have not been calculated. There are no comparisons with other results because these constants are calculated here for the first time.

Table 5 The rovibrational constants for the different vibrational levels of the ground state X2\(\sum_{1/2}^{ + }\) of BaAt molecule calculated with the spin–orbit coupling effects taken into account.
Full size table
Table 6 The rovibrational constants of different vibrational levels of some excited states of BaAt molecule calculated with the spin–orbit coupling effect taken into account.
Full size table

Laser cooling study of SrAt and BaAt molecules

The difference in equilibrium positions ΔRe between the ground state X2∑+ and the two excites states (1)2Π and (2)2∑+ states of SrAt and BaAt are minimal; this directed our attention to verify the laser cooling suitability for these molecules through cycles involving the aforementioned states, in the Ω(±) representation. However, an experimental confirmation of the presented electronic structure calculation is highly recommended before such step is taken.

The main criterion for keeping a molecule in a closed-loop cycle is a highly diagonal Franck–Condon factor (FCF) among the lowest vibrational levels of a bound excited state and those of the ground state 38. The vibrational FCF of the transition X2∑+1/2—(1)2\(\sum_{1/2}^{ + }\) of the molecule SrAt (calculated by using the LEVEL 11 program39) is plotted in Fig. 3. One can notice that the transition among the vibrational levels v′ = v = 0 has a higher probability than the remaining ones. At the same time, the deexcitation of the vibrational level v′ = 0 takes place mainly through the channel v′0v1, v′0v2, and v′0v3 with the following FCF, respectively f0′0 = 0.812067, f0′1 = 0.161978, f0′2 = 0.022776 and f0′3 = 0.002822. The deexcitation through the remaining channels is minimal.

Figure 3
figure 3

Franck–Condon factor for the transitions X2\(\sum_{1/2}^{ + }\) − (2)2∑1/2 and X2\(\sum_{1/2}^{ + }\) − (1)2Π1/2 of the molecules SrAt and BaAt, respectively.

Full size image

A short radiative lifetime among vibrational levels involved in the cooling cycle is the second criterion for a successful laser cooling process, as it maximizes the cooling rate and produces a strong Doppler force. This can be done by calculating the vibrational Einstein coefficient Aν′ν given by40

$$A_{{v^{\prime}v}} = \frac{{\left( {3.1361891} \right)\left( {10^{ - 7} } \right)(\Delta E)^{3} \left( {\left\langle {\left. {\psi_{{\nu^{\prime}}} } \right|M\left( r \right)\left| {\psi_{\nu } } \right.} \right\rangle } \right)^{2} }}{2}$$
(1)

where M(r) is the electronic transition dipole moment (in Debye), and ΔE is the energy difference between the two studied electronic states. The computed X2\(\sum_{1/2}^{ + }\) − (1)2\(\sum_{1/2}^{ + }\) transition dipole moment is represented in Fig. 4. The radiative lifetimes (given by \(\tau_{{v^{\prime}}} = \frac{1}{{\mathop \sum \nolimits_{v} A_{{v^{\prime}v}} }}\)) of six considered vibrational levels (v′), and the vibrational branching ratio (given by Rv′v = \(\frac{{A_{{v^{\prime}v}} }}{{\mathop \sum \nolimits_{v} A_{{v^{\prime}v}} }}\)41,42) among the vibrational transitions between different levels (v′) and (v) are displayed in Table 7. The transition X2\(\sum_{1/2}^{ + }\) − (1)2\(\sum_{1/2}^{ + }\) of SrAt molecule satisfies this condition, given the short radiative lifetimes that vary as 92.50 ns ≤ τ ≤ 101.9 ns among different values of v′.

Figure 4
figure 4

Transition dipole moments for the transitions X2\(\sum_{1/2}^{ + }\) − (2)2\(\sum_{1/2}^{ + }\) and X2\(\sum_{1/2}^{ + }\) − (1)2Π1/2 of the molecules SrAt and BaAt, respectively.

Full size image
Table 7 The radiative lifetimes τ, and the vibrational branching ratio of the vibrational transitions between the electronic states (2)2\(\sum_{1/2}^{ + }\) − X2\(\sum_{1/2}^{ + }\) of the molecule SrAt.
Full size table

Finally, the number of cycles (N) for photon absorption/emission should be maximized to decelerate the molecule sufficiently43,44. One can define N in terms of total decay channels involved (ɳ) as the following:

$${\text{N}} = \frac{1}{1 - \eta }$$
(2)

In our case, we propose ɳ = R0′0 + R0′1 + R0′2 + R0′3, for which N = 1786. The corresponding laser cooling scheme is given in Fig. 5. The solid red lines represent the cycling lasers, while the dotted lines represent the spontaneous decay. The values of the vibrational transitions FCF (fν′ν) and the vibrational branching ratios Rν′ν are annotated under the ground state vibrational level involved in the corresponding transition. The proposed laser wavelengths are in the visible domain, with the primary pumping laser at λ0′0 = 666.8 nm, and the three repumping lasers used to close the leaks from higher vibrational levels at wavelengths λ0′1 = 673.3 nm, λ0′2 = 679.8 nm, λ0′3 = 686.4 nm.

Figure 5
figure 5

Laser cooling scheme with the transition X2\(\sum_{1/2}^{ + }\) − (2)2\(\sum_{1/2}^{ + }\) of the molecule SrAt.

Full size image

The lowest SrAt temperature that can be reached through the Doppler and Sysphus laser cooling processes are in the order of the μK, as shown with the following corresponding experimental parameters needed below43,45:

$$V_{rms} = \frac{hN}{{m\lambda_{00} }} = {3}.{66}\;{\text{m}}/{\text{s}}$$
(3)
$${\text{T}}_{{{\text{ini}}}} = \frac{{mV^{2} }}{{2k_{B} }} = 0.{24}\;{\text{K}}$$
(4)
$${\text{a}}_{{{\text{max}}}} = \frac{{hN_{e} }}{{N_{tot} m\lambda_{00} \tau }} = {1}.{91} \times {1}0^{{3}} \;{\text{m}}/{\text{s}}^{{2}}$$
(5)
$$L_{min} = \frac{{k_{B} T_{ini} }}{{ma_{max} }}{3}.{\text{5 mm}}$$
(6)
$${\text{T}}_{{\text{D}}} = {\text{h}}/\left( {{4} \times \pi \times \tau \times {\text{k}}_{{\text{B}}} } \right) = {17}.{8}\;\mu {\text{K}}\;{\text{and}}\;{\text{T}}_{{\text{r}}} = {\text{h}}^{{2}} /\left( {{\text{m}} \times \lambda^{{2}}_{00} \times {\text{k}}_{{\text{B}}} } \right) = 0.{15}\;\mu {\text{K}}$$
(7)

where amax and Tini are the molecule’s maximum deceleration and initial temperature, respectively, and Vrms is the rms velocity. The parameters m and Lmin are the molecule’s mass and minimum slowing distance, respectively. Ne is the number of excited states in the main cycling transition, and Ntot is the number of the excited states connected to the ground state plus Ne. According to the SrAt laser cooling scheme, the ratio Ne/Ntot equals 1/5, considering the vibrational ground and excited states. T.D. and Tr are, respectively, the Doppler and recoil temperatures. The slowing distance Lmin is relatively small; however, a close scale (12 mm) stopping length has been proposed to slow down hydrogen atoms44.

Following the same investigation type, we considered the transition X2\(\sum_{1/2}^{ + }\) − (1)2Π1/2 for the molecule BaAt. The FCF and the transition dipole moment for this transition are represented respectively in Figs. 3 and 4. This system shows a more evident FCF scheme diagonal feature compared to that of SrAt, where (v′,v) transitions among (0,0), (1,1) and (2,2) vibrational states have a higher probability compared to non-diagonal ones. The corresponding branching ratio values and radiative lifetime are given in Table 8. Several laser cooling loops can be built up for this molecule, with different numbers of cycles (N) for photon absorption/emission (Eq. 3). The number of cycles (N) and the corresponding schemes are given in Table 9, along with the corresponding experimental parameters (L, Vrms, amax, and Ntot). Ne was considered equal to one for all schemes.

Table 8 The vibrational Einstein Coefficients Av′v, the vibrational branching ratios Rv′v, and the radiative lifetimes τ for transitions between the electronic states (1)2Π1/2—X2\(\sum_{1/2}^{ + }\) of the molecules BaAt.
Full size table
Table 9 Variation of the laser slowing distance (L) in function of the number of the lasers needed (Laser N°), the number of cycles (N) for photon absorption/emission and the total decay channels involved (ɳ) for cooling BaAt and SrAt molecular beam.
Full size table

The slowing distances of the three schemes are within the experimental conditions for the cooling of a molecule, as they range between 2.52 mm and 1.36 m. The laser cooling scheme (A) is represented in Fig. 6a, where the solid red lines represent the driven lasers, and the dotted lines represent the spontaneous decays. Their main pumping and repumping laser wavelengths, in addition to the FCF (fν′ν) and the vibrational branching ratios Rν′ν among different transitions, are also represented. The wavelength of the primary pumping laser is λ0′0 = 1041.2 nm, and those of the repumping laser are λ0′1 = 1053.7 nm, λ0′2 = 1068.2 nm, and λ0′3 = 1083.3 nm in the near-infrared region. The graphical representation of the scheme (B) (by using three lasers) is given in Fig. 6b. Scheme (C) represents another suggested scheme with four lasers for the molecule BaAt, given in Fig. 6c. This last scheme presents new pumping lasers whose wavelengths are λ1′2 = 1055.3 nm and λ1′3 = 1070.0 nm. The lowest attainable Doppler and recoil temperatures for BaAt are TD = 104.0 μK, and Tr = 51.9 nK among all three schemes as they only depend on the value of τ and λ00 for a given molecule.

Figure 6
figure 6

Laser cooling schemes with the transition X2\(\sum_{1/2}^{ + }\) − (1)2Π1/2 of the molecule BaAt.

Full size image

Conclusion

The MRCI + Q technique allowed the investigation of 63 electronic states with and without considering the spin–orbit coupling effect of the doublet and quartet electronic states of SrAt and BaAt molecules. The adiabatic potential energy curves and the static dipole moment curves have been plotted for these electronic states. The spectroscopic constants \({{\text{T}}}_{{\text{e}}}\), Re, \({\upomega }_{{\text{e}}}\), Be were deduced here for the first time to the best of our knowledge. The results are compatible with our previously published work of molecules containing alkaline earth metals and halogens, obtained using the same calculation method25,26. Based on the canonical function approach, the values of the ro-vibrational constants Ev, Bv, Dv, with the abscissas of turning points Rmin and Rmax, have been calculated for the ground and some low-lying excited states of the BaAt molecule. Transition parameters such as the FCFs, the radiative lifetime, the branching ratio, and the experimental parameters for the molecules SrAt and BaAt confirm their candidacy for Doppler and Sysphus laser cooling. The proposed laser cooling schemes may open the way for new laser cooling experiments.