*Ab initio*study on the electronic states and laser cooling of AlCl and AlBr

*Ab initio*study on the electronic states and laser cooling of AlCl and AlBr

† Corresponding author. E-mail:

We investigate whether AlCl and AlBr are promising candidates for laser cooling. We report new *ab initio* calculations on the ground state X^{1}Σ^{+} and two low-lying states (A^{1}Π and a^{3}Π) of AlCl and AlBr. The calculated spectroscopic constants show good agreement with available theoretical and experimental results. We also obtain the permanent dipole moments (PDMs) curve at multi-reference configuration interaction (MRCI) level of theory. The transition properties of A^{1}Π and a^{3}Π states are predicted, including the transition dipole moments (TDMs), Franck–Condon factors (FCFs), radiative times and radiative width. The calculated radiative lifetimes are of the order of a nanosecond, implying that they are sufficiently short for rapid laser cooling. Both AlCl and AlBr have highly diagonally distributed FCFs which are crucial requirement for molecular laser cooling. The results demonstrate the feasibility of laser cooling AlCl and AlBr, and we propose laser cooling schemes for AlCl and AlBr.

Ultracold molecules offer a unique opportunity for studying chemical reactions at the quantum state level. Recently, the laser cooling of diatomic molecules has been demonstrated in the laboratory.^{[1–3]} It is notable that molecule SrF^{1} has been directly cooled by Doppler and Sisyphus cooling for the first time. Later YO^{2} and CaF^{3} have also been experimentally cooled by laser cooling. It would be good to note that similar schemes have now been proposed for molecular ions (eg the molecular hydrogen ions^{[4]} HD^{+}). The researches on the laser cooling in our country are also motivated by all sorts of prospective applications.^{[5–7]} Since there are more than 90 elements in the periodic table, it may form more than 4000 different diatomic molecules. Naturally the search for potential laser cooling candidates is getting people to pay attention. For laser cooling of a molecule, it is very difficult to keep a closed optical pumping cycle with repeated optical spontaneous emissions due to the molecule’s complex internal structure. SrF, YO, and CaF all demonstrate highly-diagonal Franck–Condon factors (FCFs) which suppress decays to unwanted sublevels. So in order to be a possible source for laser cooling, the molecule must meet the first criteria: highly-diagonal FCFs. The highly-diagonal FCFs would limit the number of lasers required to keep the molecule in a closed-loop cooling cycle. A second criteria is that rapid laser cooling calls for shorter lifetimes. SrF, YO, and CaF all have short lifetimes. The calculations of FCFs and lifetimes would help us to identify promising candidates for laser cooling theoretically. Thus, we use *ab initio* calculations to investigate the possibility of laser cooling AlCl and AlBr in this paper.

In 2004, a brief survey of laser cooling candidates was presented by Di Rosa.^{[8]} The list of candidates included aluminum compounds. The laser cooling of the related diatomics AlH and AlF has been discussed in detail by Wells and Lane.^{[9]} They identified AlH and AlF as suitable laser cooling candidates. Now we would like to choose the two molecules AlCl and AlBr, since they have large emission probability (thus high scattering rates), very low scattering probabilities into off-diagonal bands 0 → 1 and 0 → 2, the desirable ^{1}Π ← ^{1}Σ^{+} cycling transitions and extremely short excited state lifetimes.

Aluminum monohalides (AlCl and AlBr) are readily produced in the vapor phase by high temperature pyrolysis^{[10]} and from chemiluminescent reactions^{[11]} of Al atoms with the halogens. A high-resolution emission spectrum of AlCl at 20 μm was conducted by Hedderich *et al.*^{[12]} They have found that the ground state X^{1}Σ^{+} of AlCl possesses a deep potential. The previous experimental studies of electronic excited states for AlCl consisted mainly of absorption and emission spectra from the first singlet–singlet (i.e. A–X), triplet–triplet (b–a) and triplet–singlet (a–X) systems.^{[12–18]} Theoretical calculations have also been performed for AlCl over the past several decades. For AlCl, the potential curves for the X^{1}Σ^{+} and A^{1}Π states, and A^{1}Π vibrational lifetimes were given by Langhoff *et al.*^{[19]} Brites *et al.*^{[20]} obtained the lifetime of AlCl for the A^{1}Π *ν′* = 0 level of 5.9 ns. These authors all agreed that AlCl has highly-diagonal FCFs and short radiative lifetimes. Several experimental and theoretical studies of the AlBr molecule are also available. Bredohl *et al*.^{[21]} have provided an extensive high-resolution band-by-band analysis of 28 bands of the A–X transition of AlBr. The A^{1}Π –X^{1}Σ^{+} transition of AlBr was recorded by Fleming *et al*.^{[22]} using a Bruker IFS 120 HR Fourier transform spectrometer. Their results showed that the 0–0 band was the most intense and the Δ*ν* = 0 sequence dominated the observed spectrum. Recent theoretical calculations have been performed for 12 electronic states for AlBr by Hamade *et al*.^{[23]} The radiative lifetime for the *ν′* = 0 level of the A^{1}Π state was computed by Langhoff *et al*.^{[19]} to be about 8.5 ns for AlBr using averaged complete active space self-consistent field (CASSCF)^{[24,25]} calculations.

As stated above, AlCl and AlBr have been researched systematically. Nevertheless, systematic studies of laser cooling of AlCl and AlBr are not available in the literature. So our work focuses on identifying whether AlCl and AlBr are possible candidates for laser cooling with *ab initio* calculations. Because AlCl and AlBr both possess ^{1}Π ← ^{1}Σ^{+} transition, the FCFs, transition dipole moments (TDMs), radiative lifetimes and radiative widths of the A^{1}Π –X^{1}Σ^{+} transtions of AlCl and AlBr are predicted. Also accurate determinations of the spectroscopic constants, potential energy curves (PECs) and permanent dipole moments (PDMs) of the X^{1}Σ^{+} and A^{1}Π states of AlCl and AlBr have been given. Additionally, there is a low lying a^{3}Π state between the A^{1}Π and ground states for AlCl and AlBr. The PECs, spectroscopic constants of a^{3}Π state, and the transition properties of the a^{3}Π state to the ground state are also obtained. Finally, we briefly design laser cooling schemes for AlCl and AlBr.

The present paper is organized as follows. In Section 2 we give the details of our calculations. Section 3 presents the results and discussion, outlining laser cooling schemes for AlCl and AlBr. The conclusions are summarized in Section 4.

All our *ab initio* calculations are performed using the MOLPRO package.^{[26]} The ground state (X^{1}Σ^{+}) and two low-lying excited states (A^{1}Π and a^{3}Π) for AlCl and AlBr have been conducted with multi-reference configuration interaction (MRCI) plus Davidson corrections^{[27–29]} which is based on CASSCF^{[24,25]} wave functions. For AlBr, the scalar relativistic effects are taken into account using the Douglas–Kroll–Hess^{[30,31]} transformation of the relativistic Hamiltonian. The spectroscopic constants (*R*_{e}, *D*_{e}, *w*_{e}, *w*_{e}*χ*_{e}, *B*_{e}, *T*_{e}) are derived by using Le Roy’s LEVEL 8.0 program.^{[32]} The PDMs and TDMs of AlCl and AlBr are evaluated by taking the expectation and transition values using the MRCI wave functions. Spin–orbit coupling is ignored in our calculations, since the principal transitions under investigation are singlet states.

In our calculations, the basis sets aug-cc-PVQZ^{[33]} are used for Al and Cl. As for Br, we choose the small-core scalar relativistic effective core potential ECP10MDF^{[34]} together with the corresponding valence basis sets.^{[35]} Owing to the limitation of the MOLPRO software package, *C*_{2ν} point group symmetry has been considered for AlCl and AlBr in all computations, which holds (*a*_{1}, *b*_{1}, *b*_{2}, *a*_{2}) irreducible representations. For AlCl, eight molecule obitals (MOs) are put into active space, including four *a*_{1}, two *b*_{1}, two *b*_{2} symmetry MOs (4220), which correspond to the 3s3p shells of the Al atom and 3s3p shells of the Cl atom. The active space of AlBr consists of eight MOs 4*a*_{1}, 2*b*_{1}, 2*b*_{2}, 0*a*_{2} (4220) which correspond to Al 3s3p and Br 4s4p.

Figure ^{1}Σ^{+}) and two low-lying excited states (A^{1}Π and a^{3}Π) of AlCl and AlBr obtained at the MRCI level of theory. We summarize the corresponding spectroscopic constants in Table

For AlCl, our results are close to the experimental data. The percentage error in *R*_{e}, *w*_{e}, and *w*_{e}*χ*_{e} are 0.70%, 0.69%, and 5.80% for the ground state X^{1}Σ^{+} of AlCl. Findings for the excited state a^{3}Π of AlCl are similar; the percentage error in *R*_{e}, *w*_{e}, and *w*_{e}*χ*_{e} are 0.57%, 0.19%, and 19.72% compared with the experimental data.^{[17]} Although the deviation of the *w*_{e}*χ*_{e} seems to be a little large, our calculated values are closer to the experimental data^{[17,35]} than the theoretical results obtained by Brites *et al.*^{[20]} The percentage error in *w*_{e}*χ*_{e} calculated by Brites *et al.*^{[20]} are 212.56% and 76.09% for the X^{1}Σ^{+} and a^{3}Π states of AlCl. The computed ground state dissociation energy *D*_{e} of 5.22 eV compares well with the experimental value of (5.25±0.01) eV.^{[14,36]} Our calculated *T*_{e} result is only 49 cm^{−1} larger than the observed data^{[37]} for the A^{1}Π state of AlCl.

We turn now to the discussion of AlBr. Our calculated spectroscopic constants of AlBr are also close to the experimental values, compared with previous theoretical data obtained by Langhoff *et al.*^{[19]} The calculated *R*_{e}, *w*_{e}, and B_{e} for X^{1}Σ^{+} state of AlBr are 2.306 Å, 378.72 cm^{−1} and 0.1577 cm^{−1} respectively, which are in good agreement with the experimental data 2.295 Å, 378.11 cm^{−1}, and 0.1592 cm^{−1}. For the A^{1}Π state of AlBr, the percentage error in *T*_{e}, *R*_{e}, and *w*_{e} are 1.50%, 0.13%, and 6.22% compared with the experimental data,^{[22]} and for the a^{3}Π states of AlBr, the percentage error in *T*_{e}, *w*_{e}, and *w*_{e}*χ*_{e} are 1.98%, 0.20%, and 0.29%.

Overall, our computed spectroscopic constants agree well with the experimental data as well as other available theoretical results. We expect that our methods are accurate for the two molecules.

Figures ^{1}Σ^{+} and A^{1}Π states of AlCl and AlBr as functions of the internuclear distance at the MRCI level, respectively. A study of Fig. ^{1}Π state demonstrate similar behavior as the PDMs of the X^{1}Σ^{+} state. The peaks of the PDMs are −1.38 a.u. and −1.17 a.u. for AlCl and AlBr, respectively.

The relationship between the TDMs of the A^{1}Π –X^{1}Σ^{+} transitions and the internuclear distance is depicted in Fig. *R* increases, reaches a maximum (AlCl: 1.66 a.u., AlBr: 1.47 a.u.), and drops thereafter.

Rapid laser cooling requires sufficiently short lifetimes, which can provide a significant rate (10^{5} s^{−1}–10^{8} s^{−1}) of optical cycling. The radiative lifetimes and radiative width for the transitions from the A^{1}Π state to the ground state X^{1}Σ^{+} are collected in Table ^{[19]} are also listed in Table ^{1}Π *ν′* = 0 level of 5.04 ns is close to the experimental value of (6.4±2.5) ns determined by Rogowski and Fontijn.^{[39]} For AlBr, our value for the *ν′* = 0 level of 8.53 ns also agrees with the theoretical value of 8.5 ns estimated by Langhoff *et al.*^{[19]}

The radiative lifetimes of the A^{1}Π (*ν′*) vibrational states are computed to be 5.04 ns–5.76 ns and 8.53 ns–10.83 ns for the first five vibrational levels (*ν′* = 0–4) of AlCl and AlBr. So these lifetimes are sufficiently short that AlCl and AlBr meet the criteria as suitable laser cooling candidates. Taking the *ν′* = 0 level as an example, the line strengths (Einstein A coefficients) of the A^{1}Π (*ν′* = 0) vibrational states for AlCl and AlBr are 1.98×10^{8} s^{−1} and 1.17×10^{8} s^{−1}, giving lifetimes of 5.04 ns and 8.53 ns (about 1/5 and 2/5 that of the transition in SrF). The radiative widths of the a^{3}Π and A^{1}Π vibrational states are predicted in Table *ν′*.

The calculated FCFs in AlCl for the transition A^{1}Π → X^{1}Σ^{+} are tabulated in Table *et al*.^{[19]} did not give FCFs for A^{1}Π → X^{1}Σ^{+} transition in AlCl. But Einstein coefficients (A) were given (*A*_{00} = 1.933×10^{8} s^{−1}, *A*_{11} = 1.891×10^{8} s^{−1}, *A*_{22} = 1.832×10^{8} s^{−1}, *A*_{33} = 1.735×10^{8} s^{−1}) by them. Our calculated Einstein coefficients are the following: *A*_{00} = 1.979×10^{8} s^{−1}, *A*_{11} = 1.921×10^{8} s^{−1}, *A*_{22} = 1.841×10^{8} s^{−1} and *A*_{33} = 1.716×10^{8} s^{−1}. Thus, our results can be compared with the results obtained by Langhoff *et al.*^{[19]} Of course, our calculated FCFs need to be confirmed by later experimental data. It can be seen that the determined FCFs of AlCl are highly diagonally (*f*_{00} = 0.9993, *f*_{11} = 0.9960, *f*_{22} = 0.9819, *f*_{33} = 0.9419). Highly-diagonal FCFs are desirable for laser cooling. The calculated *ν′* = 0 → *ν* = 0 FCF (*f*_{00}) of 0.9993 (Table *f*_{00} = 0.98), thus it is sufficiently large to be potentially viable for cooling. On the basis of the calculated FCFs, we propose a two color laser cooling scheme. The main cooling transition is the transition A^{1}Π (*ν′* = 0) ← X^{1}Σ^{+} (*ν* = 0) using wavelength *λ*_{00} = 261.2 nm (Fig. ^{[37]} is 261.5 nm. Because there is a non-negligible probability of decay to the X^{1}Σ^{+} (*ν* = 2) state (≈ 0.07%), a repumping laser can be tied to the *ν′* = 1 ← *ν* = 2 transition with *λ*_{21} = 258.6 nm. Decays to the X^{1}Σ^{+} (*ν* = 3) state occur with probability ≈ 10^{−6}. Using the repump should result in *N*_{scat} = 1/*f*_{03 +} > 10^{6} photons scattered before higher vibrational levels are populated. The required cooling wavelength is deep into the UVC range (UVC: 280–190 nm), where it is hard to produce continuous wave laser radiation. However, a frequency tripled Ti: sapphire laser should be capable of generating useful quantities of light at the wavelength (261.2 nm). It is also important to note that there is a low lying a^{3}Π state between the two states (A^{1}Π and X^{1}Σ^{+}) for AlCl. The intervening a^{3}Π state could kill the cooling scheme based on the transition A^{1}Π ← X^{1}Σ^{+}, if the a^{3}Π ← A^{1}Π transition depopulates the excited state. So far there has been no experimental observation of the a^{3}Π ← A^{1}Π transition. The rate for the A^{1}Π → a^{3}Π transition should be very small. Also we have computed the FCFs for the a^{3}Π → X^{1}Σ^{+} transition, in order to see whether the a^{3}Π → X^{1}Σ^{+} transition is another option for laser cooling.

Table ^{3}Π → X^{1}Σ^{+} transition are still highly diagonal. Thus, a four colour laser cooling scheme for AlCl using the a^{3}Π → X^{1}Σ^{+} transition is shown in Fig. ^{3}Π (*ν′* = 0) ← X^{1}Σ^{+} (*ν* = 0) transition at length 415.4 nm (visible violet region, the corresponding experimental value is 407.36 nm^{[18]}). The a^{3}Π (*ν′* = 0) ← X^{1}Σ^{+} (*ν* = 1) transition as the first vibrational repump and the a^{3}Π (*ν′* = 1) ← X^{1}Σ^{+} (*ν* = 2) transition for the second repump are required to reclaim molecules falling from *ν′* = 0 to *ν* = 1 (FCF = 0.1077) and *ν′* = 1 to *ν* = 2 (FCF = 0.1895). Owing to the non-negligible a^{3}Π (*ν′* = 1) to X^{1}Σ^{+} (*ν* = 3) transition (FCF = 0.0211), a third repumping laser may be required on that transition. The relatively long lifetimes of the a^{3}Π state can be exploited to reach a much lower Doppler temperature than possible on the A^{1}Π ← X^{1}Σ^{+} transition. Considering that the 0–0 band of the a^{3}Π → X^{1}Σ^{+} transition of AlCl at 407.36 nm has been observed by Saksena *et al.*,^{[18]} AlCl may first be cooled on the strong A^{1}Π ← X^{1}Σ^{+} transition, then be further cooled on the weak a^{3}Π ← X^{1}Σ^{+} transition. This is similar to the laser cooling of alkaline earth atoms.

Turning to AlBr, the FCFs follow the same trend as that of AlCl (see Table ^{1}Π –X^{1}Σ^{+} and a^{3}Π –X^{1}Σ^{+} transitions were observed near 280 nm^{[38]} and 420 nm.^{[38]} As in AlCl, there is an a^{3}Π state between A^{1}Π and X^{1}Σ^{+} states for AlBr. For simplicity, here we only propose laser cooling schemes of AlBr in Fig.

On the whole, we have identified AlCl and AlBr as promising laser cooling candidates. Compared with AlBr (*f*_{00} = 0.9371), AlCl (*f*_{00} = 0.9993) is more promising. Because of the hyperfine structure present in Al diatomics, sub- Doppler sisyphus cooling can take place in AlCl and AlBr just like alkali atoms. For AlCl and AlBr, the A^{1}Π ← X^{1}Σ^{+} transition may be followed by the weak a^{3}Π ← X^{1}Σ^{+} transition. The recoil temperature (*T*_{recoil} = (*h*/*λ*)^{2}/2*mk*_{B}) is the temperature limitation. AlCl and AlBr have greater mass and cooling wavelengths of lower energy (AlCl: 415.4 nm; AlBr: 428.8 nm) than AlF investigated by Wells and Lane.^{[9]} So the recoil temperature for AlCl and AlBr will be smaller than AlF.

We have calculated the X^{1}Σ^{+}, A^{1}Π, and a^{3}Π states of AlCl and AlBr in order to explore the possibility of laser cooling these molecules. First, we investigated the PECs and spectroscopic constants (*R*_{e}, *D*_{e}, *w*_{e}, *w*_{e}*χ*_{e}, *B*_{e}, *T*_{e}). The calculated spectroscopic constants are in good agreement with previous theoretical and experimental data. Second, PDMs and TDMs have been calculated for the X^{1}Σ^{+} and A^{1}Π states. Finally, we obtained FCFs, radiative lifetimes and radiative widths. The results demonstrate that both molecules have highly diagonally distributed FCFs and short lifetimes. It means that AlCl and AlBr meet the criteria as promising candidates for direct laser cooling. Additionally, we have briefly designed laser cooling schemes for AlCl and AlBr. Since the a^{3}Π → X^{1}Σ^{+} transition is also strongly diagonal and the lifetimes of the excited a^{3}Π state are relatively long, the A^{1}Π ← X^{1}Σ^{+} transition may be followed by the weak a^{3}Π ← X^{1}Σ^{+} transition to obtain a lower Doppler temperature.

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