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

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, ωe, B e, and αe 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.

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 scintillators[4] 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 SrF[8] and CaF[9] have been successfully cooled experimentally. In addition, RaF[10] and BeF[11] 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 ions[16] can be exploited in a wide range of applications such as quantum computing,[17] atom-ion sympathetic cooling[18−20] 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 Te, Re, ωe, Be, αe, and De, the permanent and transition dipole moments, the rovibrational parameters Ev, Bv, and Dv, the abscissas of turning points Rmin and Rmax, and their Franck–Condon factors (FCFs).

Results and Discussion

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 –12. All the studied electronic states correlate with the molecular dissociation asymptotes as reported in Table . 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 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 database[48] 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.

Figure 1

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

Figure 12

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)	0.00a	X2Σ+, (1)2Π	0.00b	0.00
Mg (3s3p, 3P0) + Cl (3s22p5, 2P0)	21,501.40a	(3)2Σ+, (4)2Σ+, (2)2Π, (3)2Π, (1)2Δ, (1)2Σ–, (1)4Σ+, (2)4Σ+, (1)4Π, (2)4Π, (1)4Δ, (1)4Σ–	21,850.41b	1.62
Mg (3s3p, 1P0) + Cl (3s22p5, 2P0)	34,750.80a	(5)2Σ+, (4)2Π	35,051.26b	0.86
Mg (3s4s, 3S) + Cl (3s22p5, 2P0)	43,565.69a	(3)4Σ+, (3)4Π	41,197.40b	5.44

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 –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 Rc is the position of crossing between two electronic states, RAC is the position of avoided crossing, and ΔE is the energy gap separation. In Figures , 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+.

Figure 5

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

Figure 9

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

The Spectroscopic Constants

The spectroscopic constants Te, Re, ωe, Be, and αe 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 Re into a polynomial in terms of the internuclear distance. The calculated spectroscopic constants are reported in Tables –7 in addition to the dissociation energies De and the dipole moments of the considered electronic states at their equilibrium position Re. 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)	Te (cm–1)	Re (Å)	ωe (cm–1)	Be (cm–1)	De (eV)	αe (cm–1)	|μe| (a.u.)
X2Σ+	0.0a	2.202a	466.44a	0.241a	3.523a	0.0018a	1.37a
	0.0b	2.216b	461.90b	0.241b	3.293b	0.0015b
	0.0c	2.199c	462.12c	0.245c	3.291c	0.0016c
	0.0d	2.196d	466.00d	0.246d	3.370d
	0.0e	2.229e	450.30e
	0.0f	2.203f	467.53f	0.241f	3.302f
	0.0g		462.10g	0.246g
	0.0h	2.190h	483.20h	0.250h	3.420h
(1)2Π	26,427.43a	2.181a	540.00a	0.246a	0.221a	0.0015a	1.76a
	26,442.30b	2.190b	443.95b	0.191b	0.549b	0.0019b
	26,469.40c	2.181c	491.60c	0.249c		0.0018c
	26,496.40d	2.169d	490.80d
	26,143.90e	2.220e	482.00e
	26,062.04f	2.178f	492.33f	0.247f	0.536f
	26,739.91g			0.251g
	26,958.71h	2.17h	515.92h	0.250h	0.55h
(2)2Σ + (ext)	30,673.14a	4.013a	136.16a	0.073a	1.813a	0.0009a	2.64a
	30,867.66h	3.660h	179.32h	0.090h	2.260h
(2)2Π	32,869.02a	2.611a	772.24a	0.172a	2.040a		1.26a
	31,945.56f	2.554f	622.72f	0.179f	2.013f
	32,363.35h	2.520h	681.20h	0.190h	2.07h
(2)2Σ + (int)	37,562.76a	2.161a	498.74a	0.250a	0.958a	0.0011a	1.98a
	38,613.09h	2.150h	540.39h	0.260h	0.10h
(3)2Σ+ (ext)	41,859.09a	2.477a	680.02a	0.191a	0.906a	0.0013a	0.60a
	42,918.81h	2.370h	705.16h	0.210h	0.77h
(3)2Σ+ (int)	43,102.32a	2.124a	550.02a	0.259a	0.751a	0.0013a	0.99a
(1)4Σ +	48,152.87a	2.871a	124.60a	0.138a	0.174a	0.039a	0.72a
(1)4Δ	48,833.67a	3.043a	120.67a	0.124a	0.09a	0.043a	0.60a
(1)4Σ–	49,240.68a	3.299a	78.61a	0.106a	0.041a	0.018a	0.45a
(3)4Π	68,785.22a	2.756a	235.50a	0.154a	0.373a	0.0053a	0.31a
(3)4Σ +	71,834.18a	6.887a	18.30a	0.025a	0.001a	0.0014a	0.12a
(4)4Π	75,248.12a	2.884a	124.65a	0.130a	0.267a	0.1667a	1.67a
(4)4Σ+	77,409.02a	4.943a	21.65a	0.048a	0.016a	0.0014a	0.09a
(5)4Π	79,591.18a	2.529a	257.36a	0.183a	0.240a	0.0024a	0.79a

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

Present work.

Our calculated values of the equilibrium bond length Re 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] ≤ ΔRe/Re ≤ 1.2%,[40] 0.3%[42,44] ≤ ΔRe/Re ≤ 0.9%,[46] and 0.5%[42] ≤ ΔRe/Re ≤ 1.5%[46] for MgCl, MgBr, and MgI, respectively. Also, they are in accordance with the experimental data with average relative differences ΔRe/Re = 0.2% for MgCl and ΔRe/Re = 1.0% for MgBr. The harmonic frequencies ωe 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 Be between our data and those in the literature, where the average relative errors are ΔBe/Be = 1.5%, ΔBe/Be = 2.8%, and ΔBe/Be = 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 Re 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 ωe is much smaller for the ground state of the neutral molecules compared to their corresponding ions. For example, for MgCl, ωe = 466.44 cm–1 for the X2Σ+ state, while for the ground state of MgCl+, ωe = 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+.

Electric Dipole Moments

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 –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.

Figure 13

(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.

Figure 15

(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 Mgδ+Clδ−, Mgδ+Brδ−, and Mgδ+Iδ− 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 , 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 (Mgδ− + Clδ+) as they tend toward ( + μ). The PDMCs of ions MgBr+ and MgI+ have two different directions at large distances. States that dissociate to Mgδ+ tend to ( – μ), while those dissociating to Brδ+ (MgBr+) and Iδ+ (MgI+) progressively go toward ( + μ).

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 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.

Figure 16

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

Figure 17

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 formula[49]where σν′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, Reν′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 τν′ν is the radiative lifetimes, which are evaluated as the inverse of the Einstein coefficients Aν′ν.

The radiative lifetimes τν′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 ) 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.

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 Ev, the rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning points Rmin and Rmax for the vibrational levels of the ground state of the investigated monohalides and their cations. These constants are given in Tables 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 ΔBv/Bv = 1.8% and ΔDv/Dv = 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	Ev (cm–1)	Bv (cm–1)	Dv × 107 (cm–1)	Rmin (Å)	Rmax (Å)
X2Σ+	0	233.82a	0.2403a	2.56a	2.1347	2.2763
			0.2448d	2.72d
	1	698.00a	0.2388a	2.56a	2.0896	2.3357
			0.2432d	2.72d
	2	1157.75a	0.2372a	2.56a	2.0603	2.3794
			0.2416d	2.72d
	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

Present work.

Ref (34).

Table 9

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

MgCl+
state	ν	Ev (cm–1)	Bv (cm–1)	Dv × 107 (cm–1)	Rmin (Å)	Rmax (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

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, fν′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 . 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.

Figure 18

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.

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 Te, Re, ωe, Be, αe, the dipole moment μe, and the dissociation energies De 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.

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 MOLPRO[55] 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∞, but all the calculations are done in the C2 subgroup of the C∞ 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 with the active space of C2 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)
	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)
	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)
	I	ECP46MWB	s, p

Table 3

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

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

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

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

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

Present work.

Ref (42).

Ref (33).

Ref (46).