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Xu Jian-Gang, Zhang Cong-Ying, Zhang Yun-Guang. Vibronic spectra of aluminium monochloride relevant to circumstellar molecule. Chinese Physics B, 2020, 29(3): 033102  
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Vibronic spectra of aluminium monochloride relevant to circumstellar molecule
Xu Jian-Gang, Zhang Cong-Ying, Zhang Yun-Guang †
School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

 

† Corresponding author. E-mail: zygsr2010@163.com

Project supported by the National Natural Science Foundation of China (Grant No. 61705182).

Abstract

The A1Π → X1 Σ+ transition system of aluminium monochloride is determined by using ab initio quantum chemistry. Based on the multi-reference configuration interaction method in conjugate to the Davidson correction (MRCI + Q), the potential energy curves (PECs) of the three electronic states are obtained. Transition dipole moments (TDMs) and the vibrational energy levels are studied by employing the aug-cc-pwCV5Z-DK basis set with 4220-active space. The rovibrational constants are first determined from the analytic potential by solving the rovibrational Schrödinger equation, and then the spectroscopic constants are determined by fitting the vibrational levels, and these values are well consistent with the experimental data. The effect of spin–orbit coupling (SOC) on the spectra and vibrational properties are evaluated. The results show that the SOC effect has almost no influence on the spectroscopic constants of AlCl molecules. For the A1Π → X1 Σ+ transition, the highly diagonalized Frank–Condon factor (FCF) is f00 = 0.9988. Additionally, Einstein coefficients and radiative lifetimes are studied, where the vibrational bands include ν″ = 0–19 → ν′ = 0–9. The ro-vibrational intensity is calculated at a temperature of 296 K, which can have certain astrophysical applications. At present, there is no report on the calculation of AlCl ro-vibrational intensity, so we hope that our results will be useful in analyzing the interstellar AlCl based on the absorption from A1Π → X1 Σ+.

PACS: ;31.15.A-;;37.10.Mn;;87.80.Cc;
Keyword:;spectroscopic constants;;radiative lifetime;;Franck-Condon factor;;transition intensity;
1. Introduction

Aluminium monochloride (AlCl) molecule is very important in astrophysics. So far, several research groups have explored it in deep space. In 1973, Tsuji[1] first predicted the existence of AlCl in the atmosphere of the carbon-rich and oxygen-rich stellar by themal equilibrium calculations. In 1987, the molecule at outer circumstellar envelope of carbon-rich star IRC + 10216 was successfully detected through microwave spectroscopy by Cernicharo and Guelin.[2] Ziurys[3] also positively confirmed it in 2006. In 2012, Agúndez et al.[4] not only found a large amount of IRC + 10216 in the cold outer shell, but also studied the abundance of IRC + 10216 in inner circumstellar molecular layers, and reported that the abundance of AlCl relative to H2 was 7 × 10− 8. In O-rich stars, AlCl has also been observed by using ALMA submillimeter telescope array by Decin et al.[5] in 2017, who reported their findings in the circumstellar envelope of the red asymptotic giant branch stars IK Tau and R Dor. In addition, AlCl may also be detected in the photosphere of the Sun. The solar abundance of Al and Cl element have been estimated by Asplund et al.,[6] which makes it possible to detect AlCl in the Sun. In the various studies of the astronomy, the accurate spectroscopic constants, transition probabilities of emissions, and vibronic spectra of this molecule are necessary. Therefore a lot of experiments and theoretical researches have been carried out on this molecule.

Bhaduri and Fowler,[7] and Mahanti[8] first studied AlCl in experiment. After these pioneering observations, several experimental studies were carried out on AlCl. The rotational transitions of the millimeter region were gauged by Wyse and Gordy,[9] and a high-resolution emission spectrum at 20 μm was reported by Hedderich et al.[10] The dissociation energy D0 = 5.25 ± 0.01 eV was assessed from the thermochemical measurements by Hildenbrand and Theard[11] and optical experimental data by Ram et al.,[12] which shows that the ground state has a well depth. In addition, there were some experimental researches of the AlCl excited states from singlet–singlet and triplet–singlet systems. Ram et al.[12] have measured a rotational structure of 18 bands for the A1Π → X1 Σ+ system containing 0 ≤ ν′ ≤ 10 and 0 ≤ ν″ ≤ 15. The A1Π → X1 Σ+ system was reported between 2500 Å and 2850 Å by Mahieu et al.,[13] which includes 29 bands with 2 ≤ ν′ ≤ 10 and 0 ≤ ν″ ≤ 16. The 0–0 band of the a3 Π → X1Σ+ transition has also been obtained at high resolution, and its rotational structure has been explicitly analyzed.[12–15]

Some theoretical calculations have also been carried out for AlCl molecule. Langhoff et al.[16] obtained the potential energy curves (PECs) of A1Π and X1Σ+ states and the moment functions of A1Π → X1Σ+ transition. Brites et al.[17] calculated the spectroscopic constants of X1Σ+, A1Π, and a3Π states by using the multireference configuration interaction (MRCI) approach. Moreover, the values of transition energy and radiative lifetime for A1Π → X1 Σ+ system were also acquired by using ab initio method.[16,17] Recently, Yousefi and Bernath[18] have combined the AWCV5Z basis set with active space (6330) to compute spectroscopic constants and vibrational levels of the ground state. In addition, the PECs and transition dipole moments (TDMs) for X1Σ+, A1Π, and a3Π states were studied with ACVQZ basis set, and the spin–orbit coupling (SOC) effects were considered at the MRCI level.[19] The Franck–Condon factors (FCFs) and emission rates were generated from v = 0, 1 levels of high state and v = 0–3 levels of the low state for , A1Π1 → a3Π0+, 1, and transitions.

As mentioned above, some emission bands of A1Π → X1 Σ+ transition were gauged. So far, although many spectroscopic constants have been calculated and some bands of this molecule have been studied. However, the systematic and accurate study of AlCl molecule is scarce. The research of SOC effect and transition properties is very important for molecular spectroscopy. It is necessary to consider this effect in the calculation. For these reasons, the transition probabilities and radiative lifetimes between A1Π and X1Σ+ states are studied in detail by using highly accurate MRCI approach with aug-cc-pwCV5Z-DK (AWCV5Z-DK) basis set. And finally, line intensity is obtained based on these results. In this paper, we depict the computational theory and approach in Section 2. In Section 3, the results and discussion are reported and lines intensity of the A1Π → X1 Σ+ transition is outlined. Finally, the conclusion is presented in Section 4.

2. Theory and method

The ab initio calculations are implemented with the MOLPRO 2018 program package.[20] First of all, the spin-restricted Hartree–Fock method is used to calculate the X1Σ+ state energy. Based on these results, the initial values for orbital optimization of complete active space self-consistent-field (CASSCF) are obtained. Then, the multi-reference wave function[21,22] is carried out by the CASSCF method. Finally, the high-level MRCI together includes the Davidson correction (MRCI + Q), and the CASSCF wave functions are used to calculate the energy as a zero-order function.[23–25] The scalar relativistic corrections are performed by the third-order Douglas–Kroll Hamilton.[26,27] On the MRCI + Q level,[28] the SOC effect is assessed by Breit–Pauli operators. The above method has been well applied to several kinds of interstellar molecules.[29–32]

All calculations are performed in the C2ν point group, which has four irreducible representations (A1, B1, B2, and A2). The A1 irreducible representation yields Σ+ state and a component of Δ state, the B1 and B2 provide the Π state, and the A2 yields the Σ− state and the other component of Δ state. In the CASSCF calculation, eight molecular orbitals are put into active space (7σ 8σ 3 πx 3πy 9σ 4πx 4πy 10σ), including four A1, two B1, and two B2 molecular symmetry orbitals, which correspond to the 3s3p shells of the Al and Cl atom. In other words, ten electrons are distributed in one (4, 2, 2, 0) active space. In the following MRCI + Q step, the 1s22s22p6 shell of Al and Cl are used for core-valence correlation, i.e., there are a total of 30 electrons in the calculations of correlation energy. The AWCV5Z-DK basis set is chosen for Al and Cl[32–35] because the core–valence correlation is essential for the accurate calculations.

The adiabatic PEC of the electronic state for AlCl molecule is composed of 146 single points in energy, with internuclear distances ranging from 1 Å to 8 Å. In order to obtain precise results, the interval value close to the equilibrium bond distance is reduced to 0.02 Å. Transition dipole moments (TDMs) and permanent dipole moments (PDMs) are also calculated in the same way. On this basis, spectroscopic constants are confirmed by solving the radial Schrödinger equation with the LEVEL program.[36]

The Einstein spontaneous emission coefficient Aν′ν″ from upper electronic state level (ν′,J′) to lower state level (ν″,J″) is estimated by[37]

where M(r) denotes the dipole moment in units of D, S(J′,J″) is the Hönl–London rotational intensity factor, v is the wavenumber of transition in units of cm− 1, Ψv′,J′ and Ψv″,J″ are normalized radical wave function for upper and lower states, respectively. The radiative lifetime can be computed by the reciprocal of the Einstein coefficient of total spontaneous emission as follows:

The conventional FCF is the square of the matrix element of the zeroth power of radial variable and expressed as[36]

The transition intensity of vibrational band spectrum is proportional to FCF as follows:[38]

And that is, the relative transition intensity can also be calculated by the FCF for spontaneous emission.

3. Results and discussion
3.1. The PECs and spectroscopic constants of Λ–S states

In this paper, the PECs of three lowest Λ–S states (X1Σ+, a3Π, and A1Π) are studied and the results are shown in Fig. 1. It can be seen that the three states are typical bound states, and the dissociation asymptote is Al(2Pu) + Cl(2Pu). Through the LEVEL8.2 program, the spectroscopic constants containing the electronic transition energy (Te), harmonic frequency (ωe), equilibrium bond distance (Re), anharmonic vibrational frequency (ωe χe), rotational constants (Be), and dissociation energy (De) are calculated for AlCl as shown in Table 1. For comparison, some other relevant calculated and experimental data of these three states are also listed in the table.

Fig. 1. Potential energy curves of X1Σ+, a3Π, and A1Π states.
Table 1.

Spectroscopic constants of X1Σ+, a3Π, and A1Π states.

.

So far, the spectroscopic constants of AlCl have been studied extensively both theoretically[12,16–19,39] and experimentally.[9,10,13,40] For the ground state, Yousefi and Bernath[18] calculated the equilibrium bond distance Re to be 2.1283 Å by using AWCV5Z basis set with 6330-active space, which is better consistent with the experimental value,[9,10,13,40] Obviously, the larger active space can gain more accurate results on the same basis set. Moreover, it shows that the AWCV5Z basis set can better reflect the core–valence correlation. Our harmonic frequency ωe is 481.83 cm− 1, the absolute errors are only 0.16 cm− 1, 0.53 cm− 1, 0.43 cm− 1, and 0.0553 cm− 1 obtained by separately comparing the experimental results,[9,10,13,40] indicating that our result is closer to the experimental value than others. The value of ωe χe = 2.0112 cm− 1 is also in good accordance with experimental data,[9,10,13,40] which has an effect on the vibrational levels. That is, the more reliable vibrational levels are obtained from the precise value of ωe χe.

For ωe and Re of the a3Π state, these calculated results,[12,17,19,39] including ours, are basically consistent with experimental value.[40] However, for X1Σ+ and a3Π states, the calculated ωe χe by Brites et al.[39] is far from the experimental one.[40] Our result of ωe χe is closer to Yang et al.’s[39] value for a3Π, but the error with respect to the experimental value is more than 20%. This is because in this paper three lowest states of AlCl are calculated, and those states have cross interaction with each other. In fact, if other higher electron states are considered, there will be more cross interactions.

As for the A1Π state, it can be seen that the difference of ωe is large. Our simulation shows that this state has a deep potential well and a very small potential barrier. The potential well is located at 38436.3652 cm− 1 above the ground state, and the potential barrier is located at 42901.400 cm− 1 above the ground state at 2.95 Å. The ωe is 453.43 cm− 1 obtained by fitting the data of near equilibrium bond distance, which is consistent with the experimental result.[40]

3.2. The PECs and spectroscopic constants of Ω States

After considering the effect of SOC, three Λ–S states is divided into six Ω states, which include one 2 state, two 1 state, two 0+ states, and one 0− state. PECs of six Ω states are shown in Fig. 2. The initial dissociation channel Al(2Pu) + Cl(2Pu) produces four new dissociation channels, i.e., Al(2P3/2) + Cl(2P3/2), Al(2P1/2) + Cl(2P1/2), Al(2P1/2) + Cl(2P3/2), and Al(2P3/2) + Cl(2P1/2).

Fig. 2. Potential energy curves of Ω state.

The and A1Π1 states are composed of X1Σ+ and A1Π in the Franck–Condon region. That is to say, when considering the SOC effect, X1Σ+ and A1Π states are not divided. The spectroscopic constants of Ω state are listed in Table 2. For the state, the anharmonic vibrational frequency ωe χe is calculated to be 2.0110 cm− 1, and the corresponding difference is 0.0002 cm− 1 with respect to the X1Σ+ state. The and X1Σ+ states have the same values of equilibrium bond distance Re, the harmonic frequency ωe, and rotational constants Be. The value of ωe = 453.42 cm− 1 is also in good accordance with the datum of 455.6 cm− 1 obtained by Wan et al.[19] Additionally, Wan et al.[19] calculated the values of Re, Be, and De for and A1Π1 states by using the ACVQZ basis set, which are in good agreement with our result. Nonetheless, our basis set (AWCV5Z) can better reflect the core–valence correlation and describe the molecular orbitals. Therefore, our calculated values are more reliable.

Table 2.

Spectroscopic constants of Ω state.

.

The Λ–S state 3Π is split into four states under the SOC effect (3Π0−, 3Π0+, 3Π1, and 3Π2). The energy sequence of these four Ω states from high to low is 2, 1, 0+, and 0−. The energy interval of a3Π0+ − a3Π1 and a3Π1 − a3Π2 are 58.8453 cm− 1 and 55.6518 cm− 1, respectively. And yet, as seen in Table 2, the excitation energy interval of a3Π0− − a3Π0+ is close to 0 cm− 1. For a3Π0−, a3Π0+, a3Π1, and a3Π2 electronic states, their potential well depths are calculated to be 2.2636 eV, 2.2705 eV, 2.2253 eV, and 2.2576 eV, respectively. Moreover, comparing with the a3Π state, the spectroscopic constants of four states do not change significantly, considering the SOC effect. For example, the spectroscopic constants of a3Π and a3Π1 states are almost the same.

Overall, the differences in spectroscopic constant between X1Σ+, a3Π, A1Π, and , a3Π0−, 0+,1,2, A1Π1 are very small, which imples that the SOC effect has a weak influence on spectroscopic constant. So none of them is to be considered in the next calculations.

3.3. PDMs and TDMs

The PDMs for Λ–S states as a function of equilibrium bond distance at MRCI + Q level are depicted in Fig. 3. The PDM for X1Σ+, a3Π, and A1Π states at Re are 0.3543 a.u. (atomic unit), 0.6966 a.u., and 0.5263 a.u., respectively. For the ground state, the curve drops to a minimum value of −4.1222 a.u. at about 4.1 Å, then rises to 0. The PDM function of a3Π state reaches a minimum value of about −1.4223 a.u. at 2.7 Å, and then drops to zero at approximately 6.00 Å for this state. There are minimum and maximum value of PDM for A1Π state at 2.26 Å and 3.05 Å.

Fig. 3. PDM of X1Σ+, a3Π, and A1Π states.

The spin-allowed A1Π → X1 Σ+ transition as a function of internuclear distance R is described in Fig. 4, and some TDM data are listed in Table 3. The TDM of A1Π → X1Σ+ transition first decreases to a minimum value of 1.6705 a.u. at 1.3 Å, then rises quickly and reaches 0 at about 4.35 Å. Then there is a small increase of 4.8 Å to a maximum value of 0.0074 a.u., and finally tends to 0. The TDM of the A1Π → X1 Σ+ transition system has been calculated with ab initio method by Yang et al.[39] at an MRCI level in a range of 1 Å ≤ R ≤ 11 Å and by Wan et al.[19] in a valid range of 1 Å ≤ R ≤ 8 Å at an MRCI + Q level. These results are compared with our values of TDM in the range of 1Å ≤ R ≤ 8 Å at the MRCI + Q/AWCV5Z-DK/4220 level. Our results are close to those calculations. In addition, it is clear that the TDM function of the A1Π → X1 Σ+ transition tends to zero at large internuclear distance, because of the orbit-forbidden transition at the atomic limit.

Fig. 4. TDM of A1Π → X1Σ+ transition.
Table 3.

TDM function of A1Π → X1Σ+ transition.

.

The X1Σ+, a3Π, and A1Π states have well depth of 42439.744 cm− 1, 18235.2380 cm− 1, and 4000.4776 cm− 1, which possess 159, 53, and 10 vibrational levels, respectively. The values of vibrational level Gv and rotational constant Bv of these three states are evaluated in this paper. For clarity, some yalues of Gv and Bv are listed in Table 4. The Gv values of the first 10 vibrational levels of X1Σ+ and A1Π states were reported by Langhoff et al.[16] through using the ANO basis set. Brites et al.[17] also calculated the Gv values for A1Π state (v ≤ 9) with cc-pVQZ basis set. Until recently, the accurate Gv (v ≤ 10) was computed by Yousefi and Bernath[18] by using a larger basis set (AWCV5Z) and active space (6330). It is clear from Table 4 that our results are fairly close to the calculated values of Yousefi and Bernath. It may imply that the predicted values of a3Π and A1Π states are reliable.

Table 4.

Values of vibrational level Gv and rotational constant Bv of X1Σ+, a3Π, and A1Π states.

.

The electronic configuration is primarily characterized by 8σ23 π49 σ2 close to the equilibrium bond distance for the X1Σ+ state. And the dominant electronic configurations of states a3Π and A1Π are both 8 σ23π49σ14π1. Thus, the dominant electronic transition from X1Σ+ to A1Π state is 9σ2–9σ14π1. Additionally, we report that the Re of the A1Π state is 2.1324 Å, while that of the X1Σ+ state is 2.1366 Å. Obviously, their values are very close to each other. Furthermore, the PECs of the two states are similar to each other, and both states have deep wells. According to these results, the FC principle predicts that the emissions of the A1Π → X1Σ+ system should be strong. To prove this, Einstein coefficients of emissions and FCFs are calculated at vibrational energy levels (v′ = 0–9 → v″ = 0–19) subsequently.

3.4. FCFs and radiative lifetimes

FCFs can be used to describe the overlap degree of vibrational wave functions for the transition. The FCFs fν′ν″ and Einstein coefficients Aν′ν″ of the A1Π → X1Σ+ transition are calculated via the LEVEL8.2 program.[36] Tables 5 and 6 list the values of FCFs and Aν′ν″ including ν″ = 0–19 → ν′ = 0–9 transition bands. These results confirm the expectations from the FC principle. It also shows that the A1Π state is not difficult to measure via spectroscopy. The A1Π → X1Σ+ transition has highly diagonal FCF (f00 = 0.9988). It can be seen from Table 6 that the larger Aν′ν″ are located in the Frank–Condon region, and the maximum value (2.0101 × 108 s−1) corresponds to the strong transition v″ = 0 → v′ = 0. The Aν′ν″ values of A1Π → X1 Σ+ system were determined by Langhoff et al.[16] over v″ = v′ = 0–9 bands. Since Langhoff et al.[16] used a different basis set and did not consider the Davidson correction in calculations, leading their results to be a little different from ours.

Table 5.

Values of FCF fν′ν″ of A1Π → X1Σ+ system. The presentation 4.83E-04 represents 4.83 × 10− 4.

.
Table 6.

Values of Einstein coefficient Aν′ν″ of A1Π(ν′ = 0–9)–X1Σ+ (ν″ = 0–19) transition bands (in unit s− 1). The presentation 2.01E+08 represents 2.01 × 108.

.

The radiative lifetimes for all vibrational levels of the A1Π state are estimated by using Aν′ν″ and are summarized in Table 7. Rogowski and Fontijn[41] determined the lifetime of A1Π v′ = 0 band to be about 6.4 ± 2.5 ns in experiment. According to Table 7, the radiative lifetimes of all vibrational levels are about 10− 9 s. The magnitude of lifetime of ν′ = 0 band is the same as the experimental value,[41] both of which are 10−9 s. Langhoff et al.[16] calculated the lifetimes of the ν′ = 0–9 at the MRCI/ANO level. The maximum difference between our result and their result is 5.2856% for v′ = 9 energy level. In addition, some theoretical values were obtained only for lower vibrational levels by Yang et al.[39] at the MRCI + Q/ACVQZ level. Our values are of the same order of magnitude as theirs, but we believe that our results are more accurate because the larger basis set (AWCV5Z) is used. Based on Aν′ν″ of the A1Π → X1Σ+ transition and radiative lifetime, the A1Π state should be easily detectable in the spectrum.

Table 7.

Values of radiative lifetime (in unit ns) for vibrational levels (ν′ = 0–9) of A1Π state.

.

The relationship between line intensities and wavenumber of the A1Π → X1Σ+ transition is determined based on FCFs, and the results are shown in Fig. 5. For atmospheric applications, the computational values are divided into R and P branches up to J″ = 16 at 296 K for the A1Π → X1 Σ+ transition. Figure 5 shows that the absolute line intensities change with the decrease of wavenumber, and the curve shows a trend of first increase and then decrease. The difference between the highest point and other points is large, which causes the highest point to be prominent. That is, when Δν = + 1, the A1Π → X1 Σ+ transition f10, f21, f32, and f43 are dramatically higher than other FCFs. These results show that ν′ = 2 − ν″ = 1 band is the strongest excitation. For 2–1 band of A1Π → X1Σ+ system, the maximum value of line intensities is 5.4807 × 10− 37 cm− 1/(molecule · cm−2) corresponding to the position 38822.2939 cm− 1 (P(1)). So far, these line intensities have not been reported yet. But we hope that our results will be useful for analyzing interstellar AlCl based on emission from A1Π → X1 Σ+.

Fig. 5. Absolute line intensities of Δν = + 1 transition bands of A1Π → X1 Σ+ system.
4. Conclusions

Based on the MRCI + Q method with aug-cc-pwCV5Z-DK basis set and 4220 active space, the PECs, spectroscopic constants, PDMs, TDM, and vibrational levels for the three Λ–S states of AlCl are studied. In addition, the PECs and spectroscopic constants of six Ω states, i.e., , a3Π0m -, a3Π0+, a3Π1, a3Π2, and A1Π1, are determined. The results show that the SOC effect has little influence on spectroscopic constants of these states. According to TDMs combined with PECs, Einstein coefficients Aν′ν″, FCFs, and radiative lifetime are determined, and these results are close to the existing theoretical and experimental data.

The radiative lifetime of A1Π state is about 10−9 s, which is very short. That is, the spontaneous emissions should be very strong, and the emissions of the A1Π → X1 Σ+ system should not be difficult to detect. Moreover, the line intensity of the A1Π → X1 Σ+ system Δν = + 1 transition band is predicted, and the results show that 2–1 band is the strongest excitation. We expect that our theoretical calculations will be helpful in the further study of interstellar AlCl and even other system transitions. It is also useful to analyze other theoretical spectrum properties, especially ro-vibrational transitions or high overtone bands.

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