Electronic Structure with Rovibrational Calculations of the Magnesium Monohalides MgX and Their Cations MgX+ (X = Cl, Br, and I)

Nariman Abu el kher; Nayla El-Kork; Mahmoud Korek

doi:10.1021/acsomega.9b02486

. 2019 Dec 11;4(26):21741–21760. doi: 10.1021/acsomega.9b02486

Electronic Structure with Rovibrational Calculations of the Magnesium Monohalides MgX and Their Cations MgX⁺ (X = Cl, Br, and I)

Nariman Abu el kher ^†, Nayla El-Kork ^‡,^*, Mahmoud Korek ^†

PMCID: PMC6933579 PMID: 31891053

Abstract

graphic file with name ao9b02486_0002.jpg

Alkaline-earth monohalides are popular compounds that are used in various applications. Little is known, however, in terms of electronic structure, about their cations and their low-lying electronic states. We present in this work electronic structure ab-initio calculations based on multireference configuration interaction plus Davidson correction of three magnesium monohalides and their cations (MgCl, MgBr, MgI, MgCl⁺, MgBr⁺, and MgI⁺). We determine the spectroscopic constants T_e, R_e, ω_e, B_e, and α_e and the dissociation energies D_e for their bound states. Additionally, we investigate their vibrational properties by calculating the vibrational eigenvalue E_v, the rotational constant B_v, and the centrifugal distortion constant D_v. We additionally study the electric charge distribution of several states by determining their permanent dipole moment and transition dipole moment curves. Finally, we calculate the Franck–Condon factors and the radiative lifetimes as precursors for laser cooling experiments.

1. Introduction

Metal-containing molecules such as alkaline-earth monohalides are of high interest for scientists working in different types of disciplines such as astrophysics, high- and low-temperature physics, etc. They have been detected in the interstellar medium,¹ in the upper atmosphere,² and in high-temperature reactions that occur in flames, catalysis, and corrosion processes.³ Moreover, alkaline-earth halides can be used as scintillators⁴ and utilized in laser window materials.⁵ In recent years, different laser cooling schemes have been proposed for the production of cold and ultracold diatomic molecules. Ultracold molecules, compared with ultracold atoms, have a more complex structure due to their rotational and vibrational motions. One can take advantage of this manifold configuration to propose new cooling techniques. The alkaline-earth materials have potential for laser cooling and are promising candidates for the controlled preparation of many-body entangled states.⁶ These molecules are consequently attractive for the fabrication of fundamentally new condensed-matter phases, which may be later used for state of the art applications such as qubit encoding and quantum memory engineering.⁷ The alkaline-earth halides SrF⁸ and CaF⁹ have been successfully cooled experimentally. In addition, RaF¹⁰ and BeF¹¹ molecules are suggested as good candidates for direct laser cooling. Ultracold molecules are largely used, for example, in quantum information processing,¹² chemical dynamics,¹³ and controlling chemistry.¹⁴ In addition, they can be used in Bose−Einstein condensate materials.¹⁵ Moreover, trapped cold ions¹⁶ can be exploited in a wide range of applications such as quantum computing,¹⁷ atom-ion sympathetic cooling¹⁸⁻²⁰ ultracold quantum and superchemistry,^14,21−25 precision measurements,^26,27 and local probing of quantum degenerate gases.²⁸

The first few low-lying excited electronics states of the molecules MgCl, MgBr, and MgI have already been examined.²⁹⁻⁴⁶ A study on the MgCl⁺ molecule has already been published;⁴⁷ however, it only considered the two low-lying singlet states of the molecule. MgBr⁺ and MgI⁺ remain uninvestigated until now. Given the lack of information on the electronic structure of MgX and MgX⁺ molecules in the literature (X = Cl, Br, and I), we were strongly motivated to perform an accurate analysis of the electronic states of these molecules and their corresponding cations.

Therefore, we investigate in this work 127 electronic states for MgCl, MgBr, MgI, MgCl⁺, MgBr⁺, and MgI⁺ molecules and molecular ions by using an ab initio method (CASSCF/MRCI+Q). A full spectroscopic analysis was carried out for these electronic states in order to calculate the spectroscopic constants T_e, R_e, ω_e, B_e, α_e, and D_e, the permanent and transition dipole moments, the rovibrational parameters E_v, B_v, and D_v, the abscissas of turning points R_min and R_max, and their Franck–Condon factors (FCFs).

2. Results and Discussion

2.1. Potential Energy Curves (PECs)

We investigated in this work 127 electronic states for MgCl, MgBr, MgI, MgCl⁺, MgBr⁺, and MgI⁺ molecules. The potential energy curves of these states are plotted as a function of the internuclear distance in Figures 1–12. All the studied electronic states correlate with the molecular dissociation asymptotes as reported in Table 1. Notably, the (2)²Σ⁺ state in the MgCl molecule, (4)²Σ⁺ state in the MgBr molecule, and (4)²Σ⁺ state in the MgI molecule are not given in Table 1 since they are polarized states. At the dissociation limit, the three molecules dissociate into the ionic fragments Mg⁺(²S) + Cl^–(¹S), Mg⁺(²S) + Br^–(¹S), and Mg⁺(²S) + I^–(¹S), respectively. To check the precision of our calculations, a comparison between our calculated asymptotic energies and those available in the NIST experimental atomic spectra database⁴⁸ is carried out in the same table. This comparison shows an overall good agreement in which the percentage relative difference ranges between 0.0 and 5.44% for MgCl and MgCl⁺, 0.0 and 3.09% for MgBr and MgBr⁺, and 0 and 4.30% for MgI and MgI⁺. The dissociation limits of higher excited states are missing due to the breakdown of the Born–Oppenheimer approximation, which lead to the undulations in the potential energy curves for these electronic states.

Potential energy curves of the lowest ²Σ^(+/−), ²Π, and ²Δ electronic states of the MgCl molecule.

Potential energy curves of the lowest ³Σ^(+/−), ³Π, and ³Δ electronic states of the MgI⁺ molecule.

Table 1. Lowest Dissociation Limits of MgCl, MgCl⁺, MgBr, MgBr⁺, MgI, and MgI⁺ Molecules.

dissociation of atomic levels Mg + Cl	dissociation energy limit of MgCl levels (cm^–1)	molecular states of MgCl	total dissociation energy limit of Mg + Cl atoms (cm^–1)	relative error (%)
Mg (2p⁶3s², ¹S) + Cl (3s²3p⁵, ²P⁰)	0.00^a	X²Σ⁺, (1)²Π	0.00^b	0.00
Mg (3s3p, ³P⁰) + Cl (3s²2p⁵, ²P⁰)	21,501.40^a	(3)²Σ⁺, (4)²Σ⁺, (2)²Π, (3)²Π, (1)²Δ, (1)²Σ^–, (1)⁴Σ⁺, (2)⁴Σ⁺, (1)⁴Π, (2)⁴Π, (1)⁴Δ, (1)⁴Σ^–	21,850.41^b	1.62
Mg (3s3p, ¹P⁰) + Cl (3s²2p⁵, ²P⁰)	34,750.80^a	(5)²Σ⁺, (4)²Π	35,051.26^b	0.86
Mg (3s4s, ³S) + Cl (3s²2p⁵, ²P⁰)	43,565.69^a	(3)⁴Σ⁺, (3)⁴Π	41,197.40^b	5.44

dissociation limit of atomic levels Mg + Cl	dissociation energy limit of MgCl⁺ levels (cm^–1)	molecular states of MgCl⁺	total dissociation energy limit of Mg + Cl atoms (cm^–1)	relative error (%)
Mg⁺ (2p⁶3s, ²S) + Cl (3s²3p⁵, ²P⁰)	0.00^a	X¹Σ⁺, (1)¹Π, (1)³Σ⁺, (1)³Π	0.00^b	0.00
Mg⁺ (2p⁶3p, ²P⁰) + Cl (3s²3p⁵, ²P⁰)	34,635.36^a	(2)¹Σ⁺, (1)¹Δ, (3)¹Σ⁺, (2)¹Π, (3)¹Π, (1)¹Σ^–, (2)³Σ⁺, (1)³Δ, (3)³Σ⁺, (2)³Π, (3)³Π, (1)³Σ^–	35,669.31^b	2.90

dissociation limit of atomic levels Mg + Br	dissociation energy limit of MgBr levels (cm^–1)	molecular states of MgBr	total dissociation energy limit of Mg + Br atoms (cm^–1)	relative error (%)
Mg (2p⁶3s², ¹S) + Br (4s²4p⁵, ²P⁰)	0.00^a	X²Σ⁺, (1)²Π	0.00^b	0.00
Mg (3s3p, ³P⁰) + Br (4s²4p⁵, ²P⁰)	21,174.32^a	(2)²Σ⁺, (3)²Σ⁺, (2)²Π, (3)²Π, (1)²Δ, (1)²Σ^–, (1)⁴Σ⁺, (2)⁴Σ⁺, (1)⁴Π, (2)⁴Π, (1)⁴Δ, (1)⁴Σ^–	21,850.41^b	3.09
Mg (3s3p, ¹P⁰) + Br (4s²4p⁵, ²P⁰)	35,161.26^a	(2)²Δ, (2)²Σ^–	35,051.264^b	0.31

dissociation of atomic levels (Mg⁺ + Br)	dissociation energy limit of MgBr⁺ levels (cm^–1)	molecular states of MgBr⁺	total dissociation energy limit of Mg + Br atoms (cm^–1)	relative error (%)
Mg⁺(2p⁶3s, ²S) + Br (4s²4p⁵, ²P⁰)	0.00^a	X¹Σ⁺, (1)¹Π, (1)³Σ⁺, (1)³Π	0.00^b	0.00
Mg⁺ (2p⁶3p, ²P⁰) + Br (4s²4p⁵, ²P⁰)	34,918.37^a	(2)¹Σ⁺, (1)¹Δ, (3)¹Σ⁺, (2)¹Π, (3)¹Π, (1)¹Σ^–, (2)³Σ⁺, (1)³Δ, (3)³Σ⁺, (3)³Π	35,669.31^b	2.11

dissociation of atomic levels Mg + I	dissociation energy limit of MgI levels (cm^–1)	molecular states of MgI	total dissociation energy limit of Mg + I atoms (cm^–1)	relative error (%)
Mg (2p⁶3s², ¹S) + I (5s²5p⁵, ²P⁰)	0.00^a	X²Σ⁺, (1)²Π	0.00^b	0.00
Mg (3s3p, ³P⁰) + I (5s²5p⁵, ²P⁰)	20,909.765^a	(2)²Σ⁺, (3)²Σ⁺, (2)²Π, (3)²Π, (1)²Δ, (1)²Σ^–, (1)⁴Σ⁺, (2)⁴Σ⁺, (1)⁴Π, (2)⁴Π, (1)⁴Δ, (1)⁴Σ^–	21,850.41^b	4.30
Mg (3 s,3p, ¹P⁰) + I (5s²5p⁵, ²P⁰)	34,538.218^a	(2)²Δ, (2)²Σ^–	35,051.26^b	1.46
Mg (3s4s, ³S) + I (5s²5p⁵, ²P⁰)	40,392.704^a	(3)⁴Σ⁺, (3)⁴Π	41,197.40^b	1.95
Mg (3s3d, ³D) + I (5s²5p⁵, ²P⁰)	47,010.609	(2)⁴Δ, (4)⁴Σ⁺, (4)⁴Π, (2)⁴Σ^–	47,841.12^b	1.74

dissociation of atomic levels Mg + I	dissociation energy limit of MgI⁺ levels (cm^–1)	molecular states of MgI⁺	total dissociation energy limit of Mg + I atoms (cm^–1)	relative error (%)
Mg⁺ (2p⁶3s, ²S) + I (5s²5p⁵, ²P⁰)	0.00^a	X¹Σ⁺, (1)¹Π, (1)³Σ⁺, (1)³Π	0.00^b	0.00
Mg⁺ (2p⁶3p, ²P⁰) + I (5s²5p⁵, ²P⁰)	35,135.58^a	(3)¹Σ⁺, (2)¹Δ, (4)¹Σ⁺, (3)¹Π, (1)¹Σ^–, (2)³Σ⁺, (1)³Δ, (3)³Σ⁺, (3)³Π, (2)³Σ^–	35,669.31^b	1.50

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Present work.

Experimental values from the NIST atomic spectra database.

Potential energy curves of the lowest ⁴Σ^(+/−), ⁴Π, and ⁴Δ electronic states of the MgCl molecule.

Potential energy curves of the lowest ¹Σ^(+/−), ¹Π, and ¹Δ electronic states of the MgCl⁺ molecule.

Potential energy curves of the lowest ³Σ^(+/−), ³Π, and ³Δ electronic states of the MgCl⁺ molecule.

Potential energy curves of the lowest ⁴Σ^(+/−), ⁴Π, and⁴Δ electronic states of the MgBr molecule.

Potential energy curves of the lowest ¹Σ^(+/−), ¹Π and¹Δ electronic states of the MgBr⁺ molecule.

Potential energy curves of the lowest ³Σ^(+/−), ³Π, and ³Δ electronic states of the MgBr⁺ molecule.

Potential energy curves of the lowest ⁴Σ^(+/−), ⁴Π, and ⁴Δ electronic states of the MgI molecule.

Potential energy curves of the lowest ¹Σ^(+/−), ¹Π, and ¹Δ electronic states of the MgI⁺ molecule.

Depth of potential wells can be an indicator of the strength of the binding forces linking two atoms of a diatomic molecule. A shallow potential usually suggests the dominancy of the forces of repulsion over the forces of attraction. Obviously, as shown in Figures 1–12, the low doublet and singlet states have deep potential wells, which indicates that the molecules are more stable on lower levels, while the higher excited states have shallower wells. In contrast, the low quartet and triplet states are shallow around the equilibrium positions.

A detailed analysis of the potential energy curves reveals some crossings and avoided crossings between them. Their positions are given in Table S1 in the Supporting Information, where R_c is the position of crossing between two electronic states, R_AC is the position of avoided crossing, and ΔE is the energy gap separation. In Figures 1, 5, and 9, the PECs of the lowest two ²Π states show avoided crossing at about 2.60, 2.54, and 2.64 Å for MgCl, MgBr, and MgI molecules, respectively. However, it is clear that the avoided crossings are more abundant in the magnesium monohalide molecules MgCl, MgBr, and MgI compared with their molecular cations MgCl⁺, MgBr⁺, and MgI⁺.

Potential energy curves of the lowest ²Σ^(+/−), ²Π and²Δ electronic states of the MgBr molecule.

Potential energy curves of the lowest ²Σ^(+/−), ²Π, and ²Δ electronic states of the MgI molecule.

2.2. The Spectroscopic Constants

The spectroscopic constants T_e, R_e, ω_e, B_e, and α_e of the bound electronic states have been calculated for the three magnesium monohalide molecules (MgCl, MgBr, and MgI) and their molecular cations (MgCl⁺, MgBr⁺, and MgI⁺) by fitting the energy data for these states around their equilibrium position R_e into a polynomial in terms of the internuclear distance. The calculated spectroscopic constants are reported in Tables 2–7 in addition to the dissociation energies D_e and the dipole moments of the considered electronic states at their equilibrium position R_e. An acceptable agreement is achieved upon comparison of these values with the available experimental and theoretical data in the literature, which confirms the reliability of our calculations. The absence of the spectroscopic constants of some electronic states is due to the presence of crossing and avoided crossing near their minima.

Table 2. Spectroscopic Parameters for the X²Σ⁺ and 14 Excited States of the MgCl Molecule (Experimental Values Are Indicated in Bold).

states (^2S+1Λ)	T_e (cm^–1)	R_e (Å)	ω_e (cm^–1)	B_e (cm^–1)	D_e (eV)	α_e (cm^–1)	\|μ_e\| (a.u.)
X²Σ⁺	0.0^a	2.202^a	466.44^a	0.241^a	3.523^a	0.0018^a	1.37^a
	0.0^b	2.216^b	461.90^b	0.241^b	3.293^b	0.0015^b
	0.0^c	2.199^c	462.12^c	0.245^c	3.291^c	0.0016^c
	0.0^d	2.196^d	466.00^d	0.246^d	3.370^d
	0.0^e	2.229^e	450.30^e
	0.0^f	2.203^f	467.53^f	0.241^f	3.302^f
	0.0^g		462.10^g	0.246^g
	0.0^h	2.190^h	483.20^h	0.250^h	3.420^h
(1)²Π	26,427.43^a	2.181^a	540.00^a	0.246^a	0.221^a	0.0015^a	1.76^a
	26,442.30^b	2.190^b	443.95^b	0.191^b	0.549^b	0.0019^b
	26,469.40^c	2.181^c	491.60^c	0.249^c		0.0018^c
	26,496.40^d	2.169^d	490.80^d
	26,143.90^e	2.220^e	482.00^e
	26,062.04^f	2.178^f	492.33^f	0.247^f	0.536^f
	26,739.91^g			0.251^g
	26,958.71^h	2.17^h	515.92^h	0.250^h	0.55^h
(2)²Σ ⁺ (ext)	30,673.14^a	4.013^a	136.16^a	0.073^a	1.813^a	0.0009^a	2.64^a
(2)²Σ ⁺ (ext)	30,867.66^h	3.660^h	179.32^h	0.090^h	2.260^h
(2)²Π	32,869.02^a	2.611^a	772.24^a	0.172^a	2.040^a		1.26^a
	31,945.56^f	2.554^f	622.72^f	0.179^f	2.013^f
	32,363.35^h	2.520^h	681.20^h	0.190^h	2.07^h
(2)²Σ ⁺ (int)	37,562.76^a	2.161^a	498.74^a	0.250^a	0.958^a	0.0011^a	1.98^a
(2)²Σ ⁺ (int)	38,613.09^h	2.150^h	540.39^h	0.260^h	0.10^h
(3)²Σ⁺ (ext)	41,859.09^a	2.477^a	680.02^a	0.191^a	0.906^a	0.0013^a	0.60^a
(3)²Σ⁺ (ext)	42,918.81^h	2.370^h	705.16^h	0.210^h	0.77^h
(3)²Σ⁺ (int)	43,102.32^a	2.124^a	550.02^a	0.259^a	0.751^a	0.0013^a	0.99^a
(1)⁴Σ ⁺	48,152.87^a	2.871^a	124.60^a	0.138^a	0.174^a	0.039^a	0.72^a
(1)⁴Δ	48,833.67^a	3.043^a	120.67^a	0.124^a	0.09^a	0.043^a	0.60^a
(1)⁴Σ^–	49,240.68^a	3.299^a	78.61^a	0.106^a	0.041^a	0.018^a	0.45^a
(3)⁴Π	68,785.22^a	2.756^a	235.50^a	0.154^a	0.373^a	0.0053^a	0.31^a
(3)⁴Σ ⁺	71,834.18^a	6.887^a	18.30^a	0.025^a	0.001^a	0.0014^a	0.12^a
(4)⁴Π	75,248.12^a	2.884^a	124.65^a	0.130^a	0.267^a	0.1667^a	1.67^a
(4)⁴Σ⁺	77,409.02^a	4.943^a	21.65^a	0.048^a	0.016^a	0.0014^a	0.09^a
(5)⁴Π	79,591.18^a	2.529^a	257.36^a	0.183^a	0.240^a	0.0024^a	0.79^a

Open in a new tab

Present work.

Ref (42).

Ref (33).

Ref (34).

Ref (40).

Ref (43).

Ref (36).

Ref (45).

Table 7. Spectroscopic Parameters for the X¹Σ⁺ and 14 Excited States of the MgI⁺ Molecule.

states (^2S+1Λ)	T_e (cm^–1)	R_e (Å)	ω_e (cm^–1)	B_e (cm^–1)	D_e (eV)	α_e (cm^–1)	\|μ_e\| (a.u.)
X¹Σ⁺	0.0^a	2.478^a	369.68^a	0.135^a	2.133^a	0.00065^a	4.68^a
(1)³Π	11,673.57^a	3.009^a	161.88^a	0.091^a	0.606^a	0.00104^a	2.79^a
(1)¹Π	12,653.15^a	3.022^a	168.23^a	0.090^a	0.461^a	0.00089^a	2.86^a
(2)¹Σ⁺	31,740.14^a	2.848^a	224.17^a	0.102^a	1.814^a	0.00023^a	3.23^a
(1)³Σ^–	34,711.03^a	4.709^a	50.80^a	0.037^a	0.074^a	0.01494^a	0.62^a
(2)³Π	34,737.44^a	5.018^a	40.74^a	0.033^a	0.065^a	0.00151^a	0.61^a
(2)³Σ⁺	37,635.82^a	2.755^a	259.27^a	0.109^a	1.658^a	0.00068^a	2.58^a
(1)³Δ	38,553.28^a	2.780^a	238.56^a	0.107^a	1.547^a	0.00058^a	2.56^a
(1)¹Δ	38,802.35^a	2.816^a	256.08^a	0.104^a	0.957^a	0.00053^a	2.55^a
(1)¹Σ^–	39,201.07^a	2.819^a	239.79^a	0.103^a	1.432^a	0.00079^a	2.53^a
(3)¹Σ⁺	43,104.35^a	2.952^a	201.68^a	0.095^a	0.936^a	0.00082^a	2.97^a
(2)¹Π	46,096.60^a	4.841^a	44.33^a	0.035^a	0.039^a	0.06990^a	0.53^a
(2)¹Δ	46,856.47^a	4.069^a	251.49^a	0.049^a	0.479^a	–0.05451^a	0.80^a
(3)¹Π	48,725.56^a	3.938^a	142.28^a	0.053^a	0.255^a	0.00081^a	4.27^a
(3)³Σ^–	54,455.42^a	5.278^a	54.42^a	0.029^a	0.085^a	0.00005^a	0.63^a

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Present work.

Table 3. Spectroscopic Parameters for the X¹Σ⁺ and 13 Excited States of the MgCl⁺ Molecule.

states (^2S+1Λ)	T_e (cm^–1)	R_e (Å)	ω_e (cm^–1)	B_e (cm^–1)	D_e (eV)	α_e (cm^–1)	\|μ_e\| (a.u.)
X¹Σ⁺	0.0^a	2.111^a	562.45^a	0.263^a	3.363^a	0.00161^a	1.77^a
X¹Σ⁺		2.101^b	583.0^b		3.20^b
(1)³Π	24,042.15^a	2.699^a	168.05^a	0.161^a	0.397^a	0.00344^a	1.30^a
(1)¹Π	24,474.31^a	2.754^a	164.16^a	0.152^a	0.322^a	0.00183^a	1.30^a
(1)¹Π	22,514.0^b	2.746^b	167.0^b		0.41^b
(2)¹Σ⁺	38,157.51^a	2.642^a	226.06^a	0.167^a	2.886^a	0.00041^a	0.68^a
(2)³Σ⁺	51,888.97^a	2.426^a	335.34^a	0.199^a	1.220^a	0.00162^a	0.96^a
(1)³Δ	52,790.17^a	2.446^a	327.63^a	0.195^a	1.094^a	0.001819^a	0.99^a
(1)¹Δ	53,183.47^a	2.484^a	288.65^a	0.189^a	1.013^a	0.00177^a	1.09^a
(1)¹Σ^–	53,516.58^a	2.489^a	277.33^a	0.188^a	0.980^a	0.00199^a	1.10^a
(1)³Σ^–	53,676.89^a	2.464^a	350.90^a	0.193^a	0.990^a	0.00193^a	1.04^a
(3)¹Σ⁺	56,127.71^a	2.614^a	217.21^a	0.172^a	0.681^a	0.08764^a	1.01^a
(2)³Π	60,291.31^a	2.778^a	66.97^a	0.146^a	0.155^a	–0.52311^a	0.84^a
(3)³Π	61,503.71^a	3.184^a	140.45^a	0.115^a	0.021^a	0.00168^a	0.70^a
(4)¹Π	82,152.33^a	2.445^a	338.97^a	0.196^a	0.206^a	0.00144^a	1.21^a
(5)¹Π	91,788.59^a	2.353^a	346.58^a	0.211^a	0.088^a	0.00125^a	0.75^a

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Present work.

Ref (47).

Table 4. Spectroscopic Parameters for the X²Σ⁺ and 16 Excited States of the MgBr Molecule (Experimental Values Are Indicated in Bold).

states (^2S+1Λ)	T_e (cm^–1)	R_e (Å)	ω_e (cm^–1)	B_e (cm^–1)	D_e (eV)	α_e (cm^–1)	\|μ_e\| (a.u.)
X²Σ⁺	0.0^a	2.378^a	367.44^a	0.160^a	2.241^a	0.00098^a	1.29^a
	0.0^b	2.371^b	370.52^b	0.163^b	3.221^b	0.00087^b
		2.360^c	373.80^c	0.165^c	3.351^c
	0.0^f	2.356^f	369.18^f	0.163^f	2.781^f
	0.0^j	2.396^j	365.90^j
	0.0^k		374.23^k	0.166^k
	0.0^l	2.371^l	369.21^l	0.163^l	3.157^l	0.00086^l
	0.0^m	2.347^m		0.166^m		0.00092^m
(1)²Π	25,564.30^a	2.381^a	323.63^a	0.159^a	–0.906^a	0.00350^a	1.39^a
	25,726.22^b	2.340^b	407.93^b	0.167^b	0.353^b	1.22345^b
	25,766.90^c	2.332^c	393.90^c	0.169^c
	25,414.31^f	2.328^f	391.89^f	0.167^f	0.235^f
	25,362.80^j	2.354^j	397.40^j
	25,824.31^k		392.76^k	0.169^k
	25,890.69^l	2.337^l	407.69^l	0.167^l	0.346^l	0.01271^l
(2)²Σ ⁺	24,043.15^a	3.654^a	163.78^a	0.068^a	1.868^a	–0.02812	1.61^a
(2)²Σ ⁺	26,539.35^l	3.895^l	146.06^l	0.060^l		–0.02823
(2)²Π	27,211.88^a	2.564^a	580.33^a	0.138^a	1.481^a	0.00041^a	0.69^a
	28,720.01^f	2.607^f	561.13^f	0.133^f	1.928^f
	29,096.07^l	2.628^l	605.46^l	0.132^l	1.864^l	0.00032^l
(1)⁴Σ⁺	38,089.29^a	3.179^a	136.96^a	0.089^a	0.109^a	–0.05099^a	0.67^a
(1)⁴Δ	38,748.25^a	3.423^a	88.89^a	0.079^a	0.042^a	0.01133^a	0.50^a
(1)²Δ	38,959.95^a	3.942^a	24.23^a	0.058^a	0.021^a	0.00347^a	0.23^a
(3)²Σ⁺	39,061.39^a	2.474^a	387.59^a	0.148^a	0.009^a	0.00414^a	0.85^a
(3)²Σ⁺	39,820.55^l	2.505^l	575.91^l	0.146^l	0.538^l	0.00564^l
(1)⁴Σ^–	39,066.40^a	3.949^a	29.29^a	0.058	0.006^a	0.00749^a	0.24^a
(1)²Σ^–	39,113.69^a	4.567^a	411.41^a	0.086^a	0.003^a	0.00351^a	0.09^a
(2)²Δ	52,388.82^a	3.138^a	126.99^a	0.093^a	0.109^a	0.00480^a	0.93^a
(2)²Σ^–	52,790.59^a	3.366^a	86.63^a	0.079^a	0.055^a	0.00215^a	0.74^a
(3)⁴Σ⁺	66,882.09^a	2.695^a	206.59^a	0.125^a	–0.130^a	0.00109^a	3.69^a
(2)⁴Δ	68,821.61^a	2.684^a	208.18^a	0.126^a	0.790^a	0.00124^a	3.41^a
(3)⁴Π	63,469.59^a	2.910^a	186.42^a	0.107^a	0.298^a	0.00078^a	0.90^a
(4)⁴Π	69,076.35^a	2.716^a	202.15^a	0.123^a	0.668^a	0.00159^a	3.89^a
(2)⁴Σ^–	70,290.34^a	2.685^a	227.13^a	0.125^a	0.538^a	0.00202^a	3.25^a

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Present work.

Ref (42).

Ref (33).

Ref (46).

Ref (41).

Ref (37).

Ref (44).

Ref (35).

Table 5. Spectroscopic Parameters for the X¹Σ⁺ and 10 Excited States of the MgBr⁺ Molecule (Experimental Values Are Indicated in Bold).

states (^2S+1Λ)	T_e (cm^–1)	R_e (Å)	ω_e (cm^–1)	B_e (cm^–1)	D_e (eV)	α_e (cm^–1)	\|μ_e\| (a.u.)
X¹Σ⁺	0.0^a	2.276^a	467.79^a	0.175^a	2.267^a	0.00028^a	4.65^a
(1)³Π	15,199.53^a	2.898^a	149.66^a	0.109^a	0.385^a	0.00411^a	2.79^a
(1)¹Π	15,989.60^a	2.958^a	115.81^a	0.103^a	0.290^a	0.00668^a	2.95^a
(2)¹Σ⁺	33,729.84^a	2.701^a	260.07^a	0.124^a	2.369^a	0.00038^a	3.16^a
(2)³Σ⁺	42,913.11^a	2.592^a	299.90^a	0.134^a	1.222^a	0.00105^a	2.43^a
(1)³Δ	43,798.28^a	2.622^a	264.77^a	0.132^a	1.115^a	0.00122^a	2.45^a
(1)¹Δ	44,043.71^a	2.651^a	255.38^a	0.129^a	1.089^a	0.001860^a	2.43^a
(1)¹Σ^–	44,432.83^a	2.656^a	240.27^a	0.129^a	1.042^a	0.00300^a	2.44^a
(1)³Σ^–	44,595.67^a	2.652^a	244.09^a	0.129^a	0.242^a	0.00269^a	2.45^a
(2)¹Δ	58,638.68^a	4.501^a	55.55^a	0.044^a	0.099^a	0.00679^a	1.17^a
(3)¹Σ⁺	47,776.13^a	2.826^a	259.33^a	0.113^a	0.673^a	0.00872^a	2.83^a

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Present work.

Table 6. Spectroscopic Parameters for the X¹Σ⁺ and 10 Excited States of the MgI Molecule (Experimental Values Are Indicated in Bold).

states (^2S+1Λ)	T_e (cm^–1)	R_e (Å)	ω_e (cm^–1)	B_e (cm^–1)	D_e (eV)	α_e (cm^–1)	\|μ_e\| (a.u.)
X²Σ⁺	0.0^a	2.5887^a	317.38^a	0.123^a	1.88^a	0.00057^a	1.11^a
	0.0^b	2.6005^b	315.92^b	0.124^a	2.82^b	0.00057^b
			316.00^c		2.90^c
	0.0^d	2.5975^d	314.27^d	0.123^d	2.27^d
(1)²Π	23,654.78^a	2.5489^a	301.27^a	0.128^a	1.04^a	0.00932^a	0.86^a
	24,354.58^b	2.5540^b	329.33^b	0.127^b	0.08^b	0.00704^b
	24,319.00^c		323.00^c
	23,919.88^d	2.5640^d	319.40^d	0.125^d	0.06^d
(2)²Π	24,615.45^a	2.7046^a	515.28^a	0.112^a	1.41^a	0.00046^a	0.76^a
(2)²Π	25,554.47^d	2.7554^d	523.32^d	0.109^d	1.71^d
(1)⁴Σ⁺	34,183.96^a	3.1986^a	105.09^a	0.081^a	0.213^a	0.00169^a	0.95^a
(1)²Σ^–	35,758.61^a	3.907^a	44.16^a	0.053^a	0.027^a	0.01877^a	0.44^a
(1)⁴Σ^–	35,553.21^a	3.5630^a	89.02^a	0.06370^a	0.050^a	0.00128^a	0.69^a
(1)⁴Δ	35,034.45^a	3.3455^a	69.36^a	0.074^a	0.114^a	0.00622^a	0.86^a
(3)²Π	44,426.37^a	2.4723^a	363.55^a	0.135^a	1.049^a	0.00030^a	0.83^a
(2)²Σ^–	47,475.30^a	3.137^a	123.43^a	0.084^a	0.214^a	0.00172^a	1.22^a
(3)⁴Π	51,163.41^a	3.0594^a	189.57^a	0.088^a	0.530^a	0.00048^a	0.68^a
(2)⁴Σ^–	57,514.93^a	2.981^a	200.28^a	0.093^a	0.556^a	0.00011^a	1.23^a

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Present work.

Ref (42).

Ref (33).

Ref (46).

Our calculated values of the equilibrium bond length R_e of the ground state X²Σ⁺ are relatively consistent with the theoretical data in the literature where the relative differences are as follows: 0.1%⁴³ ≤ ΔR_e/R_e ≤ 1.2%,⁴⁰ 0.3%^42,44 ≤ ΔR_e/R_e ≤ 0.9%,⁴⁶ and 0.5%⁴² ≤ ΔR_e/R_e ≤ 1.5%⁴⁶ for MgCl, MgBr, and MgI, respectively. Also, they are in accordance with the experimental data with average relative differences ΔR_e/R_e = 0.2% for MgCl and ΔR_e/R_e = 1.0% for MgBr. The harmonic frequencies ω_e calculated in the present work are also in a very good agreement with those given in the literature where the average relative differences are 1.4% for MgCl, 1.0% for MgBr, and 0.6% for MgI. There is additionally good conformity in the values of the rotational constant B_e between our data and those in the literature, where the average relative errors are ΔB_e/B_e = 1.5%, ΔB_e/B_e = 2.8%, and ΔB_e/B_e = 0.4% for MgCl, MgBr, and MgI respectively. For the higher excited electronic states, one can find that the calculated values of spectroscopic constants are generally compatible with those available in the literature.

Concerning the investigated cations, the spectroscopic constants of MgCl⁺ are compatible with available theoretical data. However, those of the ions MgBr⁺ and MgI⁺ are reported here for the first time to our knowledge.

In terms of the trend among the different neutral molecules and anions, it is noted that, as the halogens and their cations vary from Cl to I, the equilibrium internuclear distances R_e of X²Σ⁺ and A²Π states increase. This tendency can be explained by the decreasing value of the electronegativity of the halide elements as we go down through the periodic table. Also, the vibrational force constant ω_e is much smaller for the ground state of the neutral molecules compared to their corresponding ions. For example, for MgCl, ω_e = 466.44 cm^–1 for the X²Σ⁺ state, while for the ground state of MgCl⁺, ω_e = 562.45 cm^–1. This is most probably attributed to a higher bond in the ions consistent with the removal of an extra electron. A similar behavior applies to MgBr/MgBr⁺ and for MgI/MgI⁺.

2.3. Electric Dipole Moments

2.3.1. The Permanent Dipole Moment Curves (PDMCs)

The permanent dipole moment curves play an essential role in the representation of the charge distribution and the types of bonds (ionic or covalent) of diatomic molecules. The dipole moment curves (DMCs) of the investigated doublet and singlet electronic states for the six molecules as a function of internuclear separation R have been plotted in Figures 13–15, while those of the quartet and triplet states are given in Figures S1–S6 in the Supporting Information, where Mg is taken at the origin in the molecules. One can notice the agreement between the position of the avoided crossing of the PECs and the positions of the crossing of the DMCs of these states, which confirm the accuracy of the present work.

(a) Dipole moment curves of the lowest ²Σ^(+/−), ²Π, and ²Δ electronic states of the MgCl molecule. (b) Dipole moment curves of the lowest ¹Σ^(+/−), ¹Π, and ¹Δ electronic states of the MgCl⁺ molecule.

(a) Dipole moment curves of the lowest ²Σ^(+/−), ²Π, and ²Δ electronic states of the MgI molecule. (b) Dipole moment curves of the lowest ¹Σ^(+/−), ¹Π, and ¹Δ electronic states of the MgI⁺ molecule.

(a) Dipole moment curves of the lowest ²Σ^(+/−), ²Π, and ²Δ electronic states of the MgBr molecule. (b) Dipole moment curves of the lowest ¹Σ^(+/−), ¹Π, and ¹Δ electronic states of the MgBr⁺ molecule.

The majority of electronic states for MgCl, MgBr, and MgI molecules dissociate into neutral atoms at the asymptotic limit of dissociation over the range R > 8 Å (the permanent dipole moment curve tends to zero). However, for the states, (2)²Σ⁺ state in the MgCl molecule, (4)²Σ⁺ state in the MgBr molecule, and (4)²Σ⁺ state in the MgI molecule, the bond character is of covalent character at small internuclear distances, and the dipole moments increase to a constant value at the asymptotic limit of dissociation, where these states become ionic. The dipole moment of the ground states X²Σ⁺ of MgCl, MgBr, and MgI molecules presents negative values with maximum magnitudes |μ| = 3.77 a.u. at R = 3.66 Å, |μ| = 2.70 a.u. at R = 3.38 Å, and |μ| = 2.29 a.u. at R = 3.48 Å, respectively. This indicates partially ionic bonds for Mg^δ+Cl^δ−, Mg^δ+Br^δ−, and Mg^δ+I^δ− at small internuclear distances. The dipole moment values then decrease to zero at large internuclear distances, which is an indicator of covalent character near dissociation. The PDMCs of the molecular cations present many crossings between their different electronic states, which correlate to the corresponding potential energy curves avoided crossing.

Concerning the ionic molecules, the ¹Π curves for MgCl⁺, MgBr⁺, and MgI⁺ molecules show a significant number of crossings, especially between the two states (2)¹Π and (3)¹Π at small distances (about 3.18, 2.74, and 3.72 Å, respectively). The PDMCs of singlet ion MgCl⁺ are plotted in Figure 13, and the triplet states are given in Figure S2 in the Supporting Information, where the interatomic distance R is extended between 1.4 and 6.4 Å. As shown, several maxima with high amplitude for most of the states are observed at small distances, where the ionic character dominates. At large distances, all the states tend to zero, except states (4)¹Π and (5)¹Π, which are correlated to (Mg^δ− + Cl^δ+) as they tend toward ( + μ). The PDMCs of ions MgBr⁺ and MgI⁺ have two different directions at large distances. States that dissociate to Mg^δ+ tend to ( – μ), while those dissociating to Br^δ+ (MgBr⁺) and I^δ+ (MgI⁺) progressively go toward ( + μ).

2.3.2. The Transition Dipole Moments Curves (TDMCs)

The TDMCs of the allowed transitions from the lowest-lying excited states to the ground state (X)Σ⁺ have been investigated for the molecules MgCl, MgBr, and MgI and their ionic systems MgCl⁺, MgBr⁺, and MgI⁺ and are plotted in Figures 16 and 17. All the TDMCs of the (X)Σ⁺–(1)Π transition tend to zero at the asymptotic limit of dissociation (R ≈ 5.2 Å) in the six magnesium monohalide molecules.

Transition dipole moment curves of X²Σ⁺–²Σ⁺ and X²Σ⁺–²Π transitions for MgCl, MgBr, and MgI.

Transition dipole moment curves of X¹Σ⁺–¹Σ⁺ and X¹Σ⁺–¹Π transitions for MgCl⁺, MgBr⁺, and MgI⁺.

On the basis of the calculated TDMs values, the radiative lifetimes have been computed using the following formula⁴⁹

where σ_ν′ν is the wavenumber of the transition between the upper vibrational level ν′ and lower vibrational level ν (in cm^–1), Λ′ and Λ are the projections of electronic orbital angular momentum on the internuclear axis for the upper and lower electronic levels, R_e^ν′ν is the electronic-vibrational transition moment expectation value, which can be obtained from the vibrational wave functions (ν and ν′) and electronic transition dipole moment (in atomic units), and τ_ν′ν is the radiative lifetimes, which are evaluated as the inverse of the Einstein coefficients A_ν′ν.

The radiative lifetimes τ_ν′ν for the bound states are calculated between 0 ≤ ν′ ≤ 6 of the upper state and 0 ≤ ν ≤ 6 of the lower state for the investigated transitions corresponding to MgCl⁺, MgBr⁺, and MgI⁺. These values are given in Tables S2–S4 in the Supporting Information.

It can be seen from Tables S2–S4 that the range of the radiative lifetime of the vibrational transitions between the electronic states (X¹Σ⁺–2¹Σ⁺) is 30.7 ns ≤ τ ≤ 21.6 μs, 24.9 ns ≤ τ ≤ 596 ns, and 289 ns ≤ τ ≤ 1250 μs for MgCl⁺, MgBr⁺, and MgI⁺, respectively. We attribute the large difference between the radiative lifetimes of the vibrational levels for the same electronic state transition of a given molecule to two factors: (i) the large variation of the transition dipole moment function with internuclear distance for the (X¹Σ⁺–2¹Σ⁺) transition in MgCl⁺, MgBr⁺, and MgI⁺ (as shown in Figure 17) and (ii) the remarkable difference between FCF values of the vibrational levels of one given electronic transition, as shown in Tables S11–S13 in the Supporting Information. Such a difference probably emanates from the large shift between the ground state and the excited state for the investigated molecules.

2.4. The Rovibrational Calculations

We calculated, using the canonical function approach^50,51 and cubic spline interpolation method between each two consecutive points of the potential energy curves, the vibrational energy E_v, the rotational constant B_v, the centrifugal distortion constant D_v, and the abscissas of the turning points R_min and R_max for the vibrational levels of the ground state of the investigated monohalides and their cations. These constants are given in Tables 8 and 9, and those of some excited electronic states are provided in Tables S5–S10 in the Supporting Information. The rovibrational values are missing for some electronic states due to their shallow potential wells and/or the presence of avoided crossing within their potential energy curves. The comparison of our results with the experimental data reported by Rostas et al.³⁴ for the ground state of the three vibrational levels for the MgCl molecule shows a good agreement with an average relative difference ΔB_v/B_v = 1.8% and ΔD_v/D_v = 5.9%. No comparison for the values of other vibrational levels is available since they are given here for the first time.

Table 8. Rovibrational Constants for the Different Vibrational Levels of X²Σ⁺ of MgCl, MgBr, and MgI Molecules.

MgCl
state	ν	E_v (cm^–1)	B_v (cm^–1)	D_v × 10⁷ (cm^–1)	R_min (Å)	R_max (Å)
X²Σ⁺	0	233.82^a	0.2403^a	2.56^a	2.1347	2.2763
	0		0.2448^d	2.72^d
	1	698.00^a	0.2388^a	2.56^a	2.0896	2.3357
	1		0.2432^d	2.72^d
	2	1157.75^a	0.2372^a	2.56^a	2.0603	2.3794
	2		0.2416^d	2.72^d
	3	1613.05	0.2356	2.57	2.0376	2.4167
	4	2063.92	0.2340	2.57	2.0187	2.4503
	5	2510.36	0.2324	2.58	2.0024	2.4815
	6	2952.38	0.2308	2.59	1.9879	2.5109
	7	3389.95	0.2292	2.59	1.9749	2.5391
	8	3823.13	0.2276	2.59	1.9631	2.5662
	9	4251.92	0.2260	2.59	1.952	2.5925
	10	4676.35	0.2244	2.60	1.9421	2.6181
	11	5096.42	0.2228	2.60	1.9327	2.6432
	12	5512.20	0.2212	2.60	1.9239	2.6678
	13	5923.79	0.2196	2.59	1.9156	2.6919
	14	6331.30	0.2181	2.58	1.9077	2.7157
	15	6734.90	0.2166	2.56	1.9003	2.7391
	16	7134.73	0.2151	2.57	1.8932	2.7621
	17	7530.74	0.2135	2.62	1.8865	2.7848
	18	7922.53	0.2119	2.79	1.8801	2.8066
	19	8308.88	0.2100	3.03	1.8740	2.8341
	20	8688.44	0.2079	2.96	1.8681	2.8573
	21	9435.28	0.2059	1.45	1.8572	2.8988
	22	9811.26	0.2054	2.21	1.8519	2.9198
	23	10185.48	0.2036	3.31	1.8468	2.9409
	24	10552.26	0.2016	2.37	1.8419	2.9617
	25	10916.43	0.2010	1.57	1.8371	2.9824
	26	11281.48	0.1998	3.11	1.8325	3.0032
	27	11640.56	0.1977	2.53	1.8280	3.0239
	28	11996.21	0.1970	1.65	1.8237	3.0445
	29	12351.79	0.1956	3.26	1.8195	3.0652
	30	12701.25	0.1938	1.86	1.8155	3.0858
	31	13049.52	0.1932	2.42	1.8116	3.1064
	32	13394.70	0.1912	2.75	1.8079	3.1270
	33	13735.69	0.1903	1.73	1.8042	3.1476
	34	14075.62	0.1888	3.10	1.8006	3.1682

MgBr
state	ν	E_v (cm^–1)	B_v (cm^–1)	D_v × 10⁷ (cm^–1)	R_min (Å)	R_max (Å)
X²Σ⁺	0	181.27	0.1595	1.26	2.3104	2.4517
	1	538.79	0.1584	1.31	2.2649	2.5124
	2	889.76	0.1575	1.25	2.2355	2.5561
	3	1239.45	0.1565	1.31	2.2125	2.5933
	4	1585.00	0.1556	1.25	2.1933	2.6266
	5	1928.49	0.1546	1.30	2.1767	2.6575
	6	2268.59	0.1537	1.26	2.1619	2.6864
	7	2606.09	0.1528	1.27	2.1486	2.7140
	8	2940.90	0.1519	1.30	2.1365	2.7404
	9	3272.60	0.1509	1.23	2.1253	2.7660
	10	3602.04	0.1500	1.31	2.1149	2.7908
	11	3928.39	0.1492	1.26	2.1051	2.8149
	12	4252.22	0.1482	1.26	2.0960	2.8386
	13	4573.52	0.1474	1.30	2.0874	2.8618
	14	4891.91	0.1465	1.24	2.0793	2.8846
	15	5207.93	0.1455	1.27	2.0716	2.9070
	16	5521.30	0.1447	1.29	2.0642	2.9291
	17	5831.98	0.1438	1.24	2.0572	2.9510
	18	6140.27	0.1429	1.28	2.0504	2.9726
	19	6445.94	0.1421	1.28	2.0440	2.9940
	20	6749.01	0.1412	1.24	2.0379	3.0153
	21	7049.68	0.1403	1.27	2.0319	3.0364
	22	7347.79	0.1395	1.27	2.0262	3.0573
	23	7643.36	0.1386	1.25	2.0207	3.0782
	24	7936.51	0.1377	1.26	2.0153	3.0989
	25	8227.21	0.1369	1.27	2.0102	3.1195
	26	8515.48	0.1360	1.24	2.0053	3.1399
	27	8801.39	0.2390	7.08	2.0005	3.1602
	28	9084.91	0.2386	10.7	1.9958	3.1808
	29	9365.82	0.2381	16.1	1.9913	3.2013
	30	9643.98	0.2373	23.8	1.9869	3.2219
	31	10192.34	0.2342	47.7	1.9785	3.2629
	32	10462.81	0.2440	3.71	1.9745	3.2835
	33	10730.63	0.2437	5.74	1.9706	3.3041
	34	10995.77	0.2433	8.66	1.9668	3.3248
	35	11258.38	0.2426	12.7	1.9631	3.3456
	36	11518.33	0.2496	0.69	1.9595	3.3664
	37	11775.53	0.2399	24.8	1.9560	3.3873
	38	12030.12	0.2494	1.60	1.9526	3.4084
	39	12281.91	0.2491	2.35	1.9492	3.4296
	40	12530.90	0.2551	0.089	1.9460	3.451
	41	12777.10	0.2483	4.52	1.9429	3.4726
	42	13020.35	0.2476	5.85	1.9398	3.4943
	43	13260.72	0.2466	7.06	1.9368	3.5164
	44	13498.01	0.2450	7.62	1.9339	3.5387
	45	13732.24	0.2548	0.33	1.9310	3.5613
	46	13963.28	0.2546	0.26	1.9283	3.5842

MgI
state	ν	E_v (cm^–1)	B_v (cm^–1)	D_v × 10⁸ (cm^–1)	R_min (Å)	R_max (Å)
X²Σ⁺	0	156.30	0.1230	7.68	2.5197	2.6657
	1	466.51	0.1224	7.68	2.4722	2.7252
	2	774.64	0.1217	7.80	2.4413	2.7692
	3	1079.77	0.1211	7.66	2.4171	2.8063
	4	1383.13	0.1205	7.52	2.3969	2.8396
	5	1685.37	0.1200	7.12	2.3793	2.8712
	6	1987.94	0.1197	6.52	2.3636	2.8933
	7	2292.48	0.1194	7.08	2.3492	2.9193
	8	2596.16	0.1187	9.09	2.3359	2.9469
	9	2892.64	0.1175	9.83	2.3240	2.9732
	10	3181.36	0.1168	6.10	2.3130	2.9980
	11	3471.64	0.1166	7.51	2.3027	3.0225
	12	3760.11	0.1156	9.92	2.2930	3.0465
	13	4042.29	0.1150	5.75	2.2839	3.0697
	14	4325.71	0.1146	9.11	2.2752	3.0928
	15	4604.86	0.1136	7.87	2.2670	3.1153
	16	4881.67	0.1133	6.96	2.2592	3.1376
	17	5157.44	0.1124	9.55	2.2517	3.1597

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Present work.

Ref (34).

Table 9. Rovibrational Constants for the Different Vibrational Levels of X¹Σ⁺ of MgCl⁺, MgBr⁺, and MgI⁺ Cations.

MgCl⁺
state	ν	E_v (cm^–1)	B_v (cm^–1)	D_v × 10⁷ (cm^–1)	R_min (Å)	R_max (Å)
X¹Σ⁺	0	281.06	0.2619	2.28	2.0483	2.1773
	1	840.70	0.2605	2.28	2.0066	2.2307
	2	1396.23	0.2590	2.28	1.9794	2.2697
	3	1947.58	0.2576	2.28	1.9582	2.3027
	4	2494.84	0.2562	2.28	1.9405	2.3323
	5	3037.96	0.2548	2.28	1.9251	2.3597
	6	3576.96	0.2533	2.27	1.9114	2.3853
	7	4111.88	0.2519	2.28	1.8991	2.4098
	8	4642.67	0.2505	2.27	1.8878	2.4332
	9	5169.37	0.2491	2.28	1.8774	2.4559
	10	5691.98	0.2477	2.28	1.8678	2.4779
	11	6210.48	0.2463	2.28	1.8587	2.4993
	12	6724.89	0.2448	2.28	1.8503	2.5203
	13	7235.18	0.2434	2.28	1.8423	2.5409
	14	7741.34	0.2420	2.28	1.8347	2.5612
	15	8243.40	0.2406	2.29	1.8275	2.5812
	16	8741.31	0.2392	2.29	1.8207	2.6009
	17	9235.06	0.2377	2.29	1.8141	2.6204
	18	9724.65	0.2363	2.30	1.8079	2.6398
	19	10210.05	0.2349	2.31	1.8019	2.6590
	20	10691.18	0.2334	2.33	1.7962	2.6781
	21	11167.98	0.2319	2.37	1.7906	2.6970
	22	11640.10	0.2304	2.48	1.7853	2.7161

MgBr⁺
state	ν	E_v (cm^–1)	B_v (cm^–1)	D_v × 10⁷ (cm^–1)	R_min (Å)	R_max (Å)
X¹Σ⁺	0	230.12	0.1744	1.02	2.2149	2.3387
	1	684.74	0.1739	1.15	2.1715	2.3920
	2	1122.59	0.1727	1.12	2.1440	2.4312
	3	1554.82	0.1719	1.17	2.1225	2.4647
	4	1980.15	0.1708	1.13	2.1047	2.4949
	5	2401.60	0.1699	1.19	2.0892	2.5228
	6	2817.63	0.1690	1.12	2.0754	2.5489
	7	3230.74	0.1681	1.16	2.0629	2.5739
	8	3640.22	0.1673	1.07	2.0516	2.5974
	9	4048.59	0.1665	1.07	2.0410	2.6180
	10	4455.67	0.1659	1.14	2.0311	2.6389
	11	4859.52	0.1647	1.33	2.0219	2.6621
	12	5255.90	0.1633	1.47	2.0133	2.6846
	13	5643.15	0.1621	1.17	2.0053	2.7065
	14	6027.12	0.1613	1.12	1.9977	2.7280
	15	6408.78	0.1603	1.37	1.9905	2.7492
	16	6784.64	0.1591	1.21	1.9837	2.7699
	17	7156.83	0.1582	1.22	1.9772	2.7908
	18	7525.46	0.1571	1.36	1.9709	2.8115
	19	7888.94	0.1560	1.21	1.9650	2.8320
	20	8248.98	0.1551	1.29	1.9593	2.8524
	21	8605.09	0.1541	1.25	1.9538	2.8726
	22	8957.70	0.1531	1.28	1.9485	2.8922
	23	9306.48	0.1520	1.35	1.9435	2.9125
	24	9650.74	0.1509	1.43	1.9385	2.9334
	25	9989.84	0.1496	1.50	1.9338	2.9543
	26	10323.33	0.1484	1.38	1.9292	2.9752
	27	10652.47	0.1473	1.38	1.9248	2.9962
	28	10977.55	0.1462	1.48	1.9205	3.0174
	29	11297.72	0.1449	1.52	1.9164	3.0387
	30	11612.85	0.1437	1.40	1.9125	3.0602
	31	11923.84	0.1426	1.58	1.9087	3.0819
	32	12229.69	0.1412	1.52	1.9051	3.1039
	33	12530.73	0.1400	1.51	1.9015	3.1261
	34	12827.13	0.1387	1.63	1.8981	3.1487
	35	13118.26	0.1374	1.56	1.8948	3.1716
	36	13404.62	0.1361	1.71	1.8916	3.1949
	37	13685.58	0.1347	1.62	1.8885	3.2186
	38	13961.52	0.1333	1.78	1.8855	3.2427
	39	14231.93	0.1318	1.75	1.8825	3.2674
	40	14496.98	0.1304	1.88	1.8797	3.2926
	41	14756.31	0.2651	2.52	1.8770	3.3184
	42	15010.00	0.2643	2.44	1.8744	3.3449
	43	15257.71	0.2630	1.39	1.8719	3.3721

MgI⁺
state	ν	E_v (cm^–1)	B_v (cm^–1)	D_v × 10⁸ (cm^–1)	R_min (Å)	R_max (Å)
X¹Σ⁺	0	184.32	0.1344	7.20	2.4128	2.5469
	1	550.40	0.1338	7.20	2.3691	2.6020
	2	914.18	0.1331	7.18	2.3405	2.6421
	3	1275.84	0.1325	7.10	2.3181	2.6758
	4	1635.92	0.1320	7.08	2.2993	2.7051
	5	1994.28	0.1314	7.28	2.2829	2.7330
	6	2349.62	0.1307	7.63	2.2684	2.7597
	7	2700.60	0.1299	7.45	2.2553	2.7850
	8	3048.41	0.1293	7.15	2.2433	2.8091
	9	3394.22	0.1287	7.57	2.2323	2.8326
	10	3736.74	0.1280	7.52	2.2220	2.8554
	11	4076.20	0.1273	7.41	2.2124	2.8776
	12	4413.02	0.1266	7.64	2.2034	2.8994
	13	4746.73	0.1259	7.49	2.1949	2.9208
	14	5077.71	0.1253	7.70	2.1869	2.9418

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Besides, we calculated the Franck–Condon factors (FCFs) for transitions between the ground and excited states of the cations MgCl⁺, MgBr⁺, and MgI⁺ by using the LEVEL8.2 program.⁵² The FCF study does not include the neutral molecules MgCl, MgBr, and MgI due to the failure of this approach in the presence of avoided crossings. The Franck–Condon factors, f_ν′ν, are tabulated in Tables S11–S13 in the Supporting Information, where the level ν′ of the upper state and ν for the lower state ranges between 0 ≤ ν′ ≤ 9 and 0 ≤ ν ≤ 9, respectively. Additionally, for the three cations (MgCl⁺, MgBr⁺, and MgI⁺), the Franck–Condon factors of the ¹Σ⁺–¹Σ⁺ and ¹Σ⁺–¹Π transitions are given in Figure 18. The obtained FCFs have a very small value for ν ≥ 0 in the considered transitions for these cations; thus, for these transitions, the FCF array is off-diagonal. Consequently, for the magnesium monohalide cations, the condition for the feasibility of laser cooling is not attained.

Plotting of the calculated FCFs of the MgCl⁺, MgBr⁺, and MgI⁺ molecules for the lowest nine vibrational levels of the transitions of ¹Σ⁺–¹Σ⁺ and ¹Σ⁺–¹Π.

3. Conclusions

In the present work, the PECs and PDMCs for the ground and excited doublet and quartet electronic states of the magnesium monohalide molecules MgCl, MgBr, and MgI, in addition to the excited singlet and triplet states of their molecular cations MgCl⁺, MgBr⁺, and MgI⁺, were investigated via ab initio CASSCF/(MRCI+Q) calculations. The spectroscopic constants T_e, R_e, ω_e, B_e, α_e, the dipole moment μ_e, and the dissociation energies D_e have been calculated for the bound states. A comparison between our calculated spectroscopic constants and previous data in the literature shows good agreement. A similar type of agreement has been achieved in our previously published works.^53,54 Also, the TDMCs of the (X)Σ⁺–Σ⁺ and (X)Σ⁺–Π transitions have been investigated for the six molecules. These calculations were followed by a study in which the rovibrational constants for different vibrational levels of low-lying electronic states are calculated. Finally, the Franck–Condon factors of the magnesium monohalide cations were found to be off-diagonal and therefore cannot be used in laser cooling applications.

4. Computational Approach

The electronic structure calculations of the three magnesium monohalides MgCl, MgBr, and MgI, in addition to their molecular cations MgCl⁺, MgBr⁺, and MgI⁺, were performed by using the quantum computational program package MOLPRO⁵⁵ taking the advantage of the graphical user interface GABEDIT.⁵⁶ High-level potential energy curves (PECs) have been investigated by employing the state-averaged complete active space self-consistent field (CASSCF) followed by the multireference single and double configuration interaction (MRCI) method with Davidson correction (+Q). The symmetry point group of MgX and MgX⁺ is C_∞v, but all the calculations are done in the C_2v subgroup of the C_∞v point group due to the restriction of the Molpro program. The basis set used for the six entire molecules including their corresponding orbitals are given in Table 10 with the active space of C_2v symmetry. The orbitals are distributed into the irreducible representation as follows: 5a₁, 2b₁, 2b₂, and 0a₂ for MgCl and MgCl⁺, 7a₁, 3b₁, 3b₂, and 1a₂ for MgBr and MgBr⁺, and 6a₁, 3b₁, 3b₂, and 1a₂ for MgI and MgI⁺ symbolized by [5,2,2,0], [7,3,3,1], and [6,3,3,1], respectively. The basis sets cc-pwCV5Z, cc-pVTZ, and aug-cc-PVQZ-DK were given by Prascher et al.,⁵⁷ while aug-cc-pwCV5Z was given by Peterson et al.⁵⁸ The basis sets ECP28MWB and ECP46MWB known as the quasi-relativistic energy consistent pseudo-potential were given by Dolg et al.⁵⁹

Table 10. Employed Basis Set and the Active Space Orbitals for the Magnesium Monohalides and Their Cations.

molecule	atom	basis	orbital	orbitals of active space
MgCl, MgCl⁺	Mg	cc-pwCV5Z	s, p, d, f	5σ (Mg: 3s, 3p₀, 4s; Cl: 3p₀, 4s), 2π (Mg: 3p ± 1; Cl: 3p ± 1)
MgCl, MgCl⁺	Cl	aug-cc-pwCV5Z	s, p, d, f
MgBr, MgBr⁺	Mg	cc-pVTZ	s, p, d	7σ (Mg: 3s, 3p₀, 3d₀, 3d₊₂, 4s; Br: 4p₀, 5s), 3π (Mg: 3p_±1, 3d_±1; Br: 4p_±1),1δ (Mg: 3d_–2)
MgBr, MgBr⁺	Br	ECP28MWB	s, p
MgI, MgI⁺	Mg	aug-cc-pVQZ-DK	s, p, d	6σ (Mg: 3s, 3p₀, 3d₀, 3d₊₂, 4s; I: 5p₀), 3π (Mg: 3p_±1, 3d_±1; I: 5p_±1),1δ (Mg: 3d_–2)
MgI, MgI⁺	I	ECP46MWB	s, p

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Acknowledgments

This publication is based on the work supported by the Khalifa University of Science and Technology under award no. CIRA-2019-054. The authors would like to acknowledge the use of MASDAR high power computer, Khalifa University Nuclear Engineering Department high power computer, and Ankabut high power computer for the completion of their work.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.9b02486.

The static dipole moment curves of the quartet electronic states of MgCl, MgBr, and MgI in addition to the triplet states for MgCl⁺, MgBr⁺, and MgI⁺ molecules (Figures S1–S6), the positions of crossing and avoided crossing among the electronic states (Table S1), the values of the vibrational energy E_v, the rotational constant B_v, the centrifugal distortion constant D_v, and the turning points R_min and R_max for some excited electronic states of the considered molecules ( Tables S2–S7), and the FCF values for the considered cations (Tables S8–S10) (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao9b02486_si_001.pdf^{(777.8KB, pdf)}

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Associated Data

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PERMALINK

Electronic Structure with Rovibrational Calculations of the Magnesium Monohalides MgX and Their Cations MgX+ (X = Cl, Br, and I)

Nariman Abu el kher

Nayla El-Kork

Mahmoud Korek

Abstract

1. Introduction

2. Results and Discussion

2.1. Potential Energy Curves (PECs)

Figure 1.

Figure 12.

Table 1. Lowest Dissociation Limits of MgCl, MgCl+, MgBr, MgBr+, MgI, and MgI+ Molecules.

Figure 2.

Figure 3.

Figure 4.

Figure 6.

Figure 7.

Figure 8.

Figure 10.

Figure 11.

Figure 5.

Figure 9.

2.2. The Spectroscopic Constants

Table 2. Spectroscopic Parameters for the X2Σ+ and 14 Excited States of the MgCl Molecule (Experimental Values Are Indicated in Bold).

Table 7. Spectroscopic Parameters for the X1Σ+ and 14 Excited States of the MgI+ Molecule.

Table 3. Spectroscopic Parameters for the X1Σ+ and 13 Excited States of the MgCl+ Molecule.

Table 4. Spectroscopic Parameters for the X2Σ+ and 16 Excited States of the MgBr Molecule (Experimental Values Are Indicated in Bold).

Table 5. Spectroscopic Parameters for the X1Σ+ and 10 Excited States of the MgBr+ Molecule (Experimental Values Are Indicated in Bold).

Table 6. Spectroscopic Parameters for the X1Σ+ and 10 Excited States of the MgI Molecule (Experimental Values Are Indicated in Bold).

2.3. Electric Dipole Moments

2.3.1. The Permanent Dipole Moment Curves (PDMCs)

Figure 13.

Figure 15.

Figure 14.

2.3.2. The Transition Dipole Moments Curves (TDMCs)

Figure 16.

Figure 17.

2.4. The Rovibrational Calculations

Table 8. Rovibrational Constants for the Different Vibrational Levels of X2Σ+ of MgCl, MgBr, and MgI Molecules.

Table 9. Rovibrational Constants for the Different Vibrational Levels of X1Σ+ of MgCl+, MgBr+, and MgI+ Cations.

Figure 18.

3. Conclusions

4. Computational Approach

Table 10. Employed Basis Set and the Active Space Orbitals for the Magnesium Monohalides and Their Cations.

Acknowledgments

Supporting Information Available

Supplementary Material

References

Associated Data

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Electronic Structure with Rovibrational Calculations of the Magnesium Monohalides MgX and Their Cations MgX⁺ (X = Cl, Br, and I)

Table 1. Lowest Dissociation Limits of MgCl, MgCl⁺, MgBr, MgBr⁺, MgI, and MgI⁺ Molecules.

Table 2. Spectroscopic Parameters for the X²Σ⁺ and 14 Excited States of the MgCl Molecule (Experimental Values Are Indicated in Bold).

Table 7. Spectroscopic Parameters for the X¹Σ⁺ and 14 Excited States of the MgI⁺ Molecule.

Table 3. Spectroscopic Parameters for the X¹Σ⁺ and 13 Excited States of the MgCl⁺ Molecule.

Table 4. Spectroscopic Parameters for the X²Σ⁺ and 16 Excited States of the MgBr Molecule (Experimental Values Are Indicated in Bold).

Table 5. Spectroscopic Parameters for the X¹Σ⁺ and 10 Excited States of the MgBr⁺ Molecule (Experimental Values Are Indicated in Bold).

Table 6. Spectroscopic Parameters for the X¹Σ⁺ and 10 Excited States of the MgI Molecule (Experimental Values Are Indicated in Bold).

Table 8. Rovibrational Constants for the Different Vibrational Levels of X²Σ⁺ of MgCl, MgBr, and MgI Molecules.

Table 9. Rovibrational Constants for the Different Vibrational Levels of X¹Σ⁺ of MgCl⁺, MgBr⁺, and MgI⁺ Cations.