Configuration interaction calculations on the spectroscopic and transition properties of magnesium chloride
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Wu Dong-lan, Lin Cheng-quan, Wen Yu-feng, Xie An-dong, Yan Bing. Configuration interaction calculations on the spectroscopic and transition properties of magnesium chloride. Chinese Physics B, 2018, 27(8): 083101 
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Configuration interaction calculations on the spectroscopic and transition properties of magnesium chloride
† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grand Nos. 11564019, 11147158, 11264020, and 11574114) and Jiangxi Provincial Education Department Project, China (Grand No. GJJ170654).
Abstract
The potential energy curves (PECs) of 14 Λ–S states for magnesium chloride (MgCl) have been calculated by using multi-reference configuration interaction method with Davidson correction (MRCI + Q). The core-valence correlation (CV), scalar relativistic effect, and spin–orbit coupling (SOC) effect are considered in the electronic structure computations. The spectroscopic constants of X2Σ+ and A2Π states have been obtained, which are in good agreement with the existing theoretical and experimental results. Furthermore, other higher electronic states are also characterized. The permanent dipole moments (PDMs) of Λ–S states and the spinorbit (SO) matrix elements between Λ–S states are also computed. The results indicate that the abrupt changes of PDMs and the SO matrix elements are attributed to the avoided crossing between the states with the same symmetry. The SOC effect is taken into account with Breit–Pauli operator, which makes the 14 Λ–S states split into 30 Ω states, and leads to a double-well potential of the Ω = (3)1/2 state. The energy splitting for the A2Π is calculated to be 53.61 cm−1 and in good agreement with the experimental result 54.47 cm−1. The transition dipole moments (TDMs), Franck–Condon factors (FCFs), and the corresponding radiative lifetimes of the selected transitions from excited Ω states to the ground state X2Σ + 1/2 have been reported. The computed radiative lifetimes τν′ of low-lying excites Ω states are all on the order of 10 ns. Finally, the feasibility of laser cooling of MgCl molecule has been analyzed.
1. Introduction
The spectroscopic investigations of alkaline-earth monohalides (MgX, X = F, Cl, Br, I) have attracted much theoretical and experimental research interest for a long time.[1–6] Since the alkaline-earth monohalides are ionic molecules, these compounds can be treated with ionic bonding models,in which the Mg+ and X− are described as two closed-shell ion cores surrounding by an unpaired electron. The previous spectroscopic studies of ionic bonding models of MgX mainly focused on the low-lying electronic states, i.e., ground state X2Σ+, the first excited state A2Π, and B2Σ+ state.[7–9] Very recently, the MgX were theoretically regarded as candidates for laser cooling molecules.[10,11] The accurate spectroscopic and transition properties for MgX are very helpful for designing the cooling scheme.
The previous experimental spectroscopic studies of the MgCl molecule mainly focused on two band systems A2Π–X2Σ+ and B2Σ+–A2Π. The vibrationally resolved absorption bands were detected by Morgan,[5] which was ascribed to 2Π–X2Σ+ transition. The rotational structure of B2Σ+–A2Π spectral band was discussed by Singh et al.,[12] and the rotational constants of B2Σ+ and A2Π were evaluated. Recently, the emission spectrum of the MgCl A2Π–X2Σ+ band system was studied with high-resolution Fourier transform spectroscopy by Hiroa et al.,[13] and accurate spectroscopic constants of low-lying states were determined. Ohshima et al.[14] used a Fourier-transform microwave spectrometer to detect the rotational spectrum of X2Σ+ (v = 0, 1), and obtained hyperfine coupling constants. The B2Σ+–A2Π band system in the 4500 Å–5000 Å region was observed by Rao and Rao,[15] and A2Π was assigned by the vibrational analysis of the spectral band system. Darji et al.[16] confirmed the vibrational analysis given by Rao and Rao[15] using the isotope-shifts studies. Later, Singh et al. observed the B2Σ+ – A2Π (0, 0) band,[17] from which the perturbations from 2Πi state to the v = 0 of B2Σ+ state were discussed.
In the aspect of theoretical investigations, Parlant et al.[18] carried out ab initio calculations by using first and secondorder configuration interaction (SOCI) on MgCl. They investigated the predissociation mechanisms of A2Π state via the interstate coupling and spin–orbit coupling (SOC) interaction. Bogey et al.[19] analyzed the rotational spectrum of the ground state and obtained the rotationally resolved molecular constants. Parlant et al.[18] analyzed the SOC splits of A2Π state. Our previous report[20] investigated radiative transition probabilities of A2Π–X2Σ+ transition by employing the MRCI + Q method. Recently, Wan et al.[4] utilized MRCI method to calculate the potential energy curves of X2Σ+, A2Π, and 22Π of MgCl, and the laser cooling mechanism based on A2Π–X2Σ+ transition was discussed. As mentioned above, the spectroscopic and transition properties of low-lying electronic states in MgCl were previously reported, while in these computational studies, the core-valence correlation and scalar relativistic effect were not fully considered. To the best of our knowledge, the studies of spectroscopic parameters and transition properties of MgCl have been focused on the ground state X2Σ+ and excited states A2Π, B2Σ+, and C2Π.[10,13,17] However, other higher excited states of MgCl have not been studied in previous work, and it may provide extensive and accurate information for the electronic structure of MgCl system.
In this work, we perform high-level ab initio calculations on the MgCl molecule. The potential energy curves (PECs) of the 14 Λ–S states for the MgCl molecule, as well as those of 30 Ω states generated from the Λ–S states are computed with multireference configuration interaction method including Davidson correction (MRCI + Q). Based on the computed Λ–S compositions of the Ω states, the configurations mixings of some Λ–S states are analyzed. The transition properties are also predicted including transition dipole moments (TDMs), Franck–Condon factors, and the radiative lifetimes.
2. Computational method
All the ab initio calculations of MgCl are carried out by employing the MOLPRO 2012 quantum chemistry package.[21,22] The C2υ point group is used for MgCl in the calculations, which includes A1/B1/B2/A2 irreducible representations. For MgCl, 4a1, 2b1, and 2b2 symmetry molecular orbitals corresponding to the 3s3p shells for Mg and Cl are selected as the active space, Hence, the 3s2 electrons for Mg and 3s23p5 electrons for Cl are put into the active space. All 2s22p6 electrons in the inner shell of Mg are put into the closed-shell, and these orbitals are doubly occupied in all reference configuration state functions, but correlated through singly and doubly excitations in MRCI computations. The remaining inner electrons are put into the core orbitals. That is to say, there are a total of 17 electrons in the correlation energy calculations.
In order to obtain accurate PECs of MgCl, the potential energies at a set of bond lengths are calculated using the state-averaged complete active space self-consistent field (SA-CASSCF)[23,24] approach followed by the internally contracted multireference configuration interaction method including the Davidson correction (icMRCI + Q).[25–28] The third-order Douglas–Kroll integrals have been utilized to calculate the scalar relativity effect.[29] As for the basis set, the correlation-consistent aug-cc-pwCV5Z-DK basis set[30] and aug-cc-pV5Z-DK basis set[31] are employed for Mg and Cl, respectively. The PECs of these 14 Λ–S electronic states are drawn with the help of the avoided crossing rule of the same symmetry.
Moreover, the SOC effect is treated by the state interaction method with the full Breit–Pauli Hamiltonian[32] via a two-step perturbative procedure. The off-diagonal SO matrix elements are determined from the MRCI wavefunctions, while the diagonal SO matrix elements are substituted by the ic-MRCI + Q energies. The SOC effect makes the 14 Λ–S states split into 30 Ω states. The PECs of Ω states are constructed by our calculated energies with the aid of the avoided crossing rule.
Based on the calculated PECs of the Λ–S and Ω states, the spectroscopic constants are determined by solving the radial Schrödinger equations of nuclear motion using the LEVEL8.0 program,[33] which include equilibrium internuclear distance Re, excitation energy Te, vibrational constants ωe and ωeχe, balanced rotation constant Be, and vibration-rotation coupling constant αe. The dissociation energy De is obtained by subtracting the molecular energy at Re from the energy at a large separation. The SO matrix elements between electronic states were computed and used to analyze the coupling between the interacting Λ–S states. The TDMs are calculated at the MRCI level. The Franck–Condon factors (FCFs) are evaluated with the LEVEL8.0 program.[33] And the radiative lifetimes of the several lowest transitions are predicted based on the calculated TDMs and Franck–Condon (FC) factors.
3. Results and discussion
3.1. The PECs of the 14 Λ–S states
All 14 Λ–S states of MgCl molecule have been studied by using the ic-MRCI + Q method with aug-cc-pwCV5Z-DK for Mg and aug-cc-pV5Z-DK for Cl, respectively. The single-point energies at a series of internuclear distances from 1.5 Å to 5 Å are calculated. The corresponding PECs are depicted in Fig.
 | Fig. 1. (color online) The PECs of the Λ–S states of MgCl molecule. |
 | Table 1. The spectroscopic constants of the bound Λ–S states of MgCl molecule. . |
As shown in Fig.
The first excited state A2Π also correlates with Mg (1Sg) + Cl (2Pu). In Franck–Condon region, the A2Π state is mainly described by the electronic configuration 7σ28σ29σ010σ03π44πα, which corresponds to one-electron excitation from X2Σ+ state, namely, 9σ → 4π. The calculated Te of 26580 cm−1 differs from the experimental value of 26739.91 cm−1 by 159.91 cm−1 (0.5%).[13] There is an obvious barrier owing to an avoided crossing near 2.55 Å, which locates 31030 cm−1 above the ground state. The calculated dissociation energy De is 0.552 eV, which is much smaller than that of the ground state. Our computed ωeχe result differs from the experiment[13] by 0.002 cm−1. The other spectroscopic constants Be, Re, De, and ωe are generally consistent with the recent theoretical results,[10] with percent errors of 0.65%, 0.36%, 3.0%, and 2.78%, respectively. The C2Π state correlates to the second neutral dissociation limit Mg (3Pu) + Cl (2Pu). From Fig.
Of particular note is the B2Σ+ state, which is the only one state correlating to the ion-pair Mg+(2Sg) + Cl−(1Sg). The PECs of B2Σ+ state is characterized by a double-well structure, with the equilibrium internuclear distance of 2.1660 Å and 3.860 Å, and the calculated Te of 38274 cm−1 and 29453 cm−1, respectively. The wavefunction of the B2Σ+ state exhibits the obvious multi-configuration character, which is made up of two main electronic configurations 7σ28 σα9σ210σ03π44π0 (41.7%) and 7σ28σ29σα10σ03π44π0 (34.3%) at Re, indicating the mixture of covalence bond and ionic bond. As for other excited electronic states, there are very few theoretical or experimental studies reported in the literatures. So, our computed spectroscopic constants with ic-MRCI + Q method will provide accurate spectroscopic information of the MgCl molecule.
Compared with other homologues, the calculated ground state De (3.325 eV) of MgCl is smaller than that of MgF (4.67 eV),[37] but larger than 3.157 eV of MgBr[34] for X2Σ+ state. The excitation energy Te for A2Π, B2Σ+, and C2Π excited states of MgCl is 26580 cm−1, 29453 cm−1, and 31894 cm−1, respectively, which is a little smaller than that of MgF (27935 cm−1, 38431 cm−1, and 47969 cm−1),[37] and larger than that of MgBr (25891 cm−1, 26539 cm−1, and 29096 cm−1).[34] The calculated Re values are 2.2140 Å, 2.1855 Å, 3.8595 Å, and 2.5605 Å, which are larger than those of MgF (1.7611 Å, 1.7470 Å, 3.5848 Å, and 2.3000 Å),[37] while smaller than those of MgBr (2.3710 Å, 2.3365 Å, 3.8950 Å, and 2.6275 Å).[34] The ωe values are 460.44 cm−1, 506.02 cm−1, 171.77 cm−1, and 705.16 cm−1, which are smaller than those of MgF (555.00 cm−1, 653.10 cm−1, 263.60 cm−1, and 1001.30 cm−1)[37] and larger than those of MgBr (369.21 cm−1, 407.69 cm−1, 146.06 cm−1, and 605.46 cm−1).[34] As can be seen from the above, all the spectroscopic constants show an obvious trend of change as the atomic number of the halogen atoms increases from F element to Br element.
The permanent dipole moments (PDMs) as the internuclear distance R changes from 1.5 Å to 5.0 Å are calculated and depicted in Fig.
 | Fig. 2. (color online) The permanent dipole moments of the Λ–S states of MgCl as a function of the bond distance. |
Another interesting feature to note in Fig.
3.2. The PECs of the 30 Ω states
The SOC effect could lead to the strong interaction between the adjacent electronic states, especially the states involving the PECs’ intersection. And the interaction could be quantitatively evaluated by the spin–orbit (SO) matrix element. The larger spin–orbit matrix element indicates great interaction between different electronic states. These phenomena in molecules have attracted much research interest in the previous investigations.[38–40] Figure
 | Fig. 3. (color online) The R-dependent spin–orbit matrix elements involving the A2Π, B2Σ+, C2Π, 32Σ+, and 14Σ+ states. |
The SO matrix elements have the largest values 298 cm−1 for the B2Σ+–C2Π system after the avoided crossing point of the B2Σ+ and 32Σ+ states, indicating the strong interaction between the B2Σ+ and C2Π state. The system B2Σ+–A2Π have larger SO matrix elements 340 cm−1, which also shows the strong interaction between the B2Σ+ and A2Π states. Therefore, the perturbation of B2Σ+ state is mainly arising from the A2Π and C2Π states, which is consistent with the analysis in the literature.[17] The matrix element is only about 79 cm−1 for the crossing point of the B2Σ+ and 14Σ+ states, indicating weak interaction induced between the B2Σ+ and 14Σ+ states. As discussed above, four pairs of the complementary sudden changes have been exhibited, all of which result from the PECs’ avoided crossings of B2Σ+ and 32Σ+ states, A2Π and C2Π states, and the crossing of B2Σ+ and C2Π states.
When the SOC effects are taken into account, 30 Ω states, including fifteen Ω = 1/2 states, ten Ω = 3/2 states, four Ω = 5/2 states, and one Ω = 7/2 state, are generated from the 14 Λ–S states of MgCl. Eight dissociation limits Mg (1S0g) + Cl (2P3/2u), Mg (1S0g) + Cl (2P1/2u), Mg (3P0u) + Cl (2P3/2u), Mg (3P1u) + Cl (2P3/2u), Mg (3P2u) + Cl (2P3/2u), Mg (3P0u) + Cl (2P1/2u), Mg (3P1u) + Cl (2P1/2u), and Mg (3P2u) + Cl (2P1/2u) are generated from the original Λ–S atomic limits Mg (1Sg) + Cl (2Pu) and Mg (3Pu) + Cl (2Pu). The detailed dissociation relationships are listed in Table
| Table 2. The dissociation relationships of the Ω states of MgCl molecule. . |
The calculated atomic energy intervals of Mg and Cl are 21682.361 cm−1 for Mg (1S0g)–Mg (3P0u), 18.292 cm−1 for Mg (3P1u)–Mg (3Pu), 46.847 cm−1 for Mg (3P2u)–Mg (3P1u), and 836.325 cm−1 for Cl (2P1/2u)–Cl (2P3/2u), which are in good accordance with the corresponding observed values 21850.405, 20.059, 40.714, and 882.350 cm−1.[41] The PECs of 30 Ω states are depicted in Fig.
 | Fig. 4. (color online) The PECs of the Ω states (a) Ω = 1/2, (b) Ω = 3/2, (c) Ω = 5/2, and Ω = 7/2. |
| Table 3. The spectroscopic constants of the bound Ω states of MgCl molecule. . |
The ground state X2Σ + 1/2 is almost completely composed of X2Σ+ Λ–S state, hence, the spectroscopic constants are hardly influenced by the SOC effect. However, slight change (only 0.0028 eV) can be found for the dissociation energy De after considering the SOC effect, which mainly comes from the SO splitting of Cl atom. The A2Π state splits into two bound Ω states of 1/2 and 3/2 state. The wave functions of these two states are mainly a mixture of B and a small part of C states, and the effect of SOC is mainly reflected in Te and De. We also find that the calculated SO splitting is 53.61 cm−1, which is in good agreement with the experimental observation[13] 54.47 cm−1 and the recent theoretical result 51.24 cm−1.[10] Considering the SOC effect and the avoided crossing rule, the PECs will become complicated. From Fig.
From the above analysis, it can be seen that the change of the wavefunction lead to the change of spectroscopic constant for Ω state. In order to illustrate the Λ–S component mixture in the wavefunction of the Ω state, we provide the R-dependent Λ–S components in the wavefunctions of the Ω = (2)1/2, (3)1/2, (1)3/2, and (2)3/2 states. As shown in Fig.
 | Fig. 5. (color online) The Λ–S component of 1/2 and 3/2 (a) Ω = (2)1/2, (b) Ω = (1)3/2, (c) Ω = (3)1/2, and (d) Ω = (2)3/2. |
3.3. The transition properties and the radiative lifetimes
where ν′ and ν″ correspond to the upper and lower vibrational states, the energy difference ΔEν′,ν″ is in unit cm−1, the averaged electronic transition dipole moment TDM is in atomic units, qν′,ν″ is the FC factor, and the radiative lifetime τν′ is in second.
Figure
 | Fig. 6. (color online) The TDMs of the transitions (2)1/2–X2Σ + 1/2, (3)1/2–X2Σ + 1/2, (4)1/2–X2Σ + 1/2, (1)3/2–X2Σ + 1/2, and (2)3/2–X2Σ + 1/2. |
The Franck–Condon factors of all the transitions mentioned have been evaluated with the aid of the LEVEL program and listed in Table
| Table 4. The FC factors of the transitions (2)1/2–X2Σ + 1/2, (3)1/2–X2Σ + 1/2, (1)3/2–X2Σ + 1/2, and (2)3/2–X2Σ + 1/2. . |
The radiative lifetimes of the selected vibrational level ν′ have been calculated by the following formula:[42,43]
 |
Aside from the larger diagonal FC factors, the shorter spontontaneous radiative lifetime τ (10−9 s ∼ 10−6 s) can be applied for rapid laser cooling, which can provide a signifcant rate of rapid cycling. The radiative lifetimes τν′ of ν′ = 0–3 for the lowest transitions are determined and listed in Table
| Table 5. The radiative lifetimes of the transitions (2)1/2–X2Σ + 1/2, (3)1/2–X2Σ + 1/2, (1)3/2–X2Σ + 1/2, and (2)3/2–X2Σ + 1/2. . |
4. Conclusions
High-level ab initio calculations on the ground and low-lying excited states of MgCl have been carried out by employing the MRCI + Q method with relativistic corrections. The PECs of 14 Λ–S states are calculated and plotted with the aid of the avoided crossing rule. From the PECs of the bound states, the corresponding spectroscopic constants are obtained, and the constants of the X2Σ+ and A2Π states are in good agreement with previous measurements. The B2Σ+ state is characterized by the shallow potential well with large equilibrium internuclear distance, which is associated with the ion-pair dissociation limit Mg+(2Sg) + Cl−(1Sg). The calculated PDMs exhibit two asymptotic limits because of the different dissociation product channels of Mg + Cl and Mg+ + Cl− at large distance. The abrupt changes of PDMs and SO reveal the sharp changes of the electron configurations in the wavefunctions of the A2Π and C2Π states at their avoided crossing point. The SOC effect splits 14 Λ–S states into 30 Ω states. The spectroscopic results indicate that the SOC effect brings little effect on the ground state X2Σ+, except for the dissociation energy. The computed spliting between the Ω = 1/2 and 3/2 components of the A2Π state is 53.61 cm−1, which agrees well with the experimental observation 54.47 cm−1. Under the influence of the SOC effect, many avoided crossings of the PECs are generated. The strong coupling between the A2Π and C2Π states at their crossing region are analyzed with the help of the Λ–S compositions for the Ω states. The TDMs, FC factors, and radiative lifetimes are determined. The radiative lifetimes for the four transitions are all evaluated on the order of 10 ns. The present computational work should shed light on the spectroscopy of low-lying electronic state and pave further experimental studies on the laser cooling of MgCl molecule.
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