Electronic Structure, Spectroscopy, Cold Ion–Atom Elastic Collision Properties, and Photoassociation Formation Prediction of the (MgCs)⁺ Molecular Ion

Abstract

In this paper, we extensively study the electronic structure, interactions, and dynamics of the (MgCs)⁺ molecular ion. The exchanges between the alkaline atom and the low-energy cationic alkaline earths, which are important in the field of cold and ultracold quantum chemistry, are studied. We use an ab initio approach based on the formalism of non-empirical pseudo-potential for Mg²⁺ and Cs⁺ cores, large Gaussian basis sets, and full-valence configuration interaction. In this context, the (MgCs)⁺ cation is treated as an effective two-electron system. Adiabatic potential energy curves and their spectroscopic constants for the ground and the first 20 excited states of ^1,3Σ⁺ symmetries are determined. Furthermore, we identify the avoided crossings between the electronic states of ^1,3Σ⁺ symmetries. These crossings are related to the charge transfer process between the two ionic limits, Mg/Cs⁺ and Mg⁺/Cs. Therefore, vibrational-level spacings and the transition and permanent dipole moments are presented and analyzed. Using the produced potential energy data, the ground-state scattering wave functions and elastic cross-sections are calculated for a wide range of energies. In addition, we predict the formation of a translationally and rotationally cold molecular ion (MgCs)⁺ in the ground-state electronic potential energy through a stimulated Raman-type process aided by ion–atom cold collision. In the low-energy limit (<1 mK), elastic scattering cross-sections exhibit Wigner law threshold behavior, while in the high-energy limit, the cross-sections act as a function of energy E go as E^−1/3. A qualitative discussion about the possibilities of forming cold (MgCs)⁺ molecular ions by photoassociative spectroscopy is presented.

1. Introduction

In recent years, cold and ultracold molecules [1] have received considerable attention owing to successful demonstrations of cooling and trapping of molecules [2] and their importance in high-resolution spectroscopy [3]. Due to the characteristics of long-range ion–atom potential, ion–neutral collisions [4,5,6] are different from neutral–neutral and ion–ion collisions. The long-range ion–atom potential is described by −C₄/r⁴, where C ₄ = α q² /(8πε₀), α is the static electric polarizability of the atom, r is the ion–atom separation, q is the charge of the ion, and ε₀ is the permittivity of free space. The scattering caused by this potential offers the main channel for the exchange of energy between the ions and the atoms. Accordingly, research on low-energy ion–neutral-atom collisions is very useful for the study of the sympathetic cooling of the translational motion of atomic ions [7,8] and for obtaining precise control of the internal degrees of freedom and external molecular ions [9]. In addition, interactions between the alkali atoms and alkaline earth ions (Be⁺, Mg⁺, Sr⁺, and Ca⁺) offer opportunities for new developments in the field of ultracold quantum matter, with the benefit of simpler and more reliable trapping compared to neutral molecules. Furthermore, a multitude of cold molecular ion species could be created, paving the way for rich chemistry at temperatures of a few mK or less [10]. The collision between cold atoms and ions is also relevant for important applications related to molecule formation in Bose–Einstein condensates [11] and quantum information. Photoassociation [12] and Feshbach resonance tuning [13] are two main experimental techniques for the coherent production of ultracold molecules from atoms.

Radiative emission during cold collisions between trapped laser-cooled Rb atoms and the alkaline earth ions Ca⁺, Sr⁺, and Ba⁺ has been studied theoretically by Aymar et al. [14] using an effective core potential based on quantum chemistry calculations of potential energy curves and transition dipole moments of the related molecular ions. Furthermore, many experimental groups have carried out experiments with various combinations of alkali atoms and alkaline earth atomic ions: Rb atoms with Ca⁺ [15,16] and Ba⁺ [17] ions. Moreover, recently, other groups have studied the energy scaling of the cold-atom–atom–ion three-body recombination Ba⁺ + Rb + Rb in the mK regime where a single ¹³⁸Ba⁺ ion in a Paul trap is immersed in a cloud of ultracold ⁸⁷Rb atoms [18], as well as the life and death of a cold BaRb⁺ molecule inside an ultracold cloud of Rb atoms [18]. In addition, an optical dipole trap of Rb atoms has also been merged in a Paul trap containing a few Ba⁺ atoms [18].

Concerning MgH⁺, the first evidence of the formation of molecular MgH⁺ ions in a laser-cooled ion trap was reported by Baba and Waki [19], who introduced air into an Mg⁺ trap. Drewsen and colleagues [20] carried out a more controlled experiment, introducing a thermal gas H₂ or D₂ into a laser-cooled Mg⁺ trap, thereby creating trapped MgH⁺ or MgD⁺ ions. Many other studies concerning magnesium hydride investigated its photodissociation [21], branching ratios [21], etc. More recently, Aymar et al. [22] carried out a study of the electron structure of the MgH⁺ ionic molecule containing potential energy curves, transition and permanent dipole moment spectra, and static polarizabilities. Using an ab initio approach, Khemeri et al. [23] performed a study of the electronic properties of this molecule in the adiabatic representation.

Concerning the MgLi⁺ ionic molecule, Boldyrev et al. [24] determined its ground-state spectroscopic constants using the split-valence basis MP2/6-311⁺G*. Pyykkö [25] employs two methods—Hartree–Fock (HF/6-31G*) and Møller–Plesset (MP)—to calculate the equilibrium distance, well depth, and vibration frequency for the same states. More recently, Yufeng Gao and Tao Gao [26] applied the ab initio program package MOLPRO to determine the potential energy curves and the permanent and transition dipole moments of the same molecules. They used the multireference configuration interaction and valence full-configuration interaction with large aug-cc-pCVQZ basis sets, taking into account the core–valence and scalar relativistic corrections.

A review of the scientific literature shows that experimental studies on the (MgCs)⁺ [27] cationic molecule are practically absent. We only noticed the existence of one recent theoretical study of (MgCs)⁺, in which the authors Smialkowski and Tomza [28] only reported spectroscopic constants for the ground state. Therefore, a more detailed and refined investigation of this molecular system is desirable. Methods fundamentally similar to those used for the treatment of the (MgLi)⁺ molecule [29,30] will be employed in the present study.

Several groups in the world have experimentally investigated atom–ion and cold molecular diatomic mixtures. The list of such mixtures includes Yb₂⁺ [31], RbCa⁺ [16], CaBa⁺ [32], CaYb⁺ [33], RbYb⁺ [34], LiCa⁺ [35], RbCa⁺ [15], NaCa⁺ [36], RbSr⁺ [37], LiYb⁺ [38], KCa⁺ [39], RbBa⁺ [40], Rb₂⁺ [41,42], and BaRb⁺ [43], as well as homonuclear mixtures [44]. Alkaline earth ions are a common choice as advanced methods for manipulation and detection of such ions have been developed over the years, creating prospects for their applications in quantum simulation and computations [45,46]. Atom–ion diatomic molecular systems can be produced via cold collision-induced charge-transfer radiative association, light-induced photoassociation [47,48,49,50,51], or photoionization of ultracold neutral molecules [42,44]. The effective core potential methods followed by CI and quantum chemistry calculations are employed to calculate the potential energy curves and dipole moments. In a very recent work [52] by our group, we investigated the electronic structure of the five diatomic molecular ions composed of a Ca⁺ ion interacting with an alkali metal atom: CaX⁺ (X =Li, Na, K, Rb, Cs). Also, the molecular ion structure of BeX⁺ (X = Na, K, Rb) systems has been theoretically studied [53], and the forming of a cold ion–atom mixture by stimulated Raman adiabatic process has been investigated [54]. In addition, an extensive study of the electronic structure, the cold ion–atom elastic collision properties, and the possibility of laser cooling BeCs⁺ molecular ions has recently been investigated [55].

Very recently, Smialkowski et al. theoretically investigated the ground-state electronic structure of single-charged molecular ions formed from two or three interacting alkali metal and alkaline earth metal atoms. They calculated the ground-state electronic properties of all diatomic AB⁺ and most of the triatomic A₂B⁺ molecular ions consisting of Li, Na, K, Rb, Cs, Mg, Ca, Sr, Ba, and Yb atoms. Smialkowski and Tomza [28] carried out a systematic study of the electronic structure of MgX⁺ (X = Li, Na, K, Rb and Cs) ionic systems and presented the potential energy curves and spectroscopic constants for their ground states.

Here, we investigate the adiabatic potential energy curves for many electronic states of ^1,3Σ⁺, ^1,3Π, and ^1,3Δ symmetries below the Mg²⁺Cs⁻ asymptotic limit. Then, we extract the spectroscopic constants from these curves (equilibrium distance R_e, well depth D_e, electronic excitation energy T_e, frequency ω_e, harmonicity constant ω_ex_e, and rotational constant B_e). In addition, vibrational-level spacings and the dipole moment function are also presented and analyzed. By exploiting the ab- initio potential energy data, we study elastic collisions between an alkali ion Cs⁺ and a neutral alkaline earth atom Mg for the ground-state collisional threshold. Besides that, we predict and analyze the formation of translationally and rotationally cold molecular ions by a two-photon incoherent Raman process in the presence of two applied lasers.

This paper is organized as follows. In Section 2, the methodology used is briefly presented. We report our results for (MgCs)⁺ and discuss them in Section 3.

2. Theory and Computational Details

In this section, we give an overview of the technical details of the calculation of the (MgCs)⁺ molecular ion. Numerous studies on heteronuclear alkaline earth dimers such as (MgK)⁺ [49], (BeLi)⁺ [56], (BeH)⁺ [57], (CaRb)⁺, (SrRb)⁺, and (BaRb)⁺ [15] have been performed. We used the CIPSI package (Configuration Interaction by Perturbation of a Multi-configuration Wave Function Selected Iteratively) of the Laboratoire de Physique et Chimie Quantique of Toulouse in France [58]. The ionic system (MgCs)⁺ has 67 electrons. Therefore, we replaced the cores of the Mg²⁺ (10 electrons) and Cs⁺ (54 electrons) with non-empirical pseudo-potentials proposed by Barthelat and Durand [59,60,61]. Consequently, (MgCs)⁺ is treated as an effective two-electron system, where the two valence electrons are moving in the field of the two cores. The pseudo-potential is complemented by the corrections of core–core and core–valence correlations according to the operator formalism of Muller et al. [62]. In addition, the cut-off function reported by the formulation of Foucrault et al. [63] is taken to be a function of l, the orbital angular momentum, in order to treat separately the interaction of valence electrons of different spatial symmetry and the core electrons. The cut-off radii for the lowest valence s, p, d, and f one-electron states are reported in

Table 1. Polarizabilities, cut-off radii, and pseudo-potential parameter (in a.u.) atoms.

Atom	α	ρs	ρp	ρd
Mg(Z = 12)	0.46904	0.9	1.25499	1.500
Cs(Z = 55)	15.117	2.69	1.85	2.810

.

In the present work, the core polarizabilities are taken to be 0.46904 a₀³ [63] for Mg²⁺ and 15.117a₀³ [64,65], respectively. For magnesium, we used a large Gaussian basis set (9s, 7p, 5d, and 4f) composed of 83 functions. The diffuse orbital exponents have been optimized to reproduce (with good accuracy) all atomic states dissociating into the following: Mg⁺(3s, 3p, 4s, 3d, 4p, 5s, 4d, 4f, 5p, and 6s) and Mg (3s² (¹S), 3s3p (³P), 3s3p (¹P), 3s4s (³S), 3s4s (¹S), 3s3d (¹D), 3s4p (³P), 3s3d (³D), 3s4p (¹P), 3s5s (³S), 3s5s (¹S), 3s4d (¹D), 3s4d (³D), and 3s5p (³P)). After contraction, this basis was reduced to 7s/7p/4d/4f, and the function number decreased to only 76. Aymar et al. [22] used a (7s/5p/4d/2f) basis set contracted to (6s/5p/2d/2f); therefore, the function number decreases from 56 to 45, while for Cs, we used a (7s4p5d1f/6s4p4d) Gaussian basis set taken from [22,66]. Molecular energies for (MgCs)⁺, at the dissociation limits, are given in Table 2.

The quality of the used basis sets and cut-off radii is confirmed by the good agreement between our values and the experimental [67] and theoretical [68,69] atomic energy levels. The difference between our results and the experimental values for almost all energy levels is lower than 179 cm⁻¹, which is found for the Mg (3s4s) atomic level. However, a difference of 1445 cm⁻¹ is found for the Mg (3s3d) atomic level. Nevertheless, the agreement between the energy levels of Mg⁺ and the experimental ones is much better, seeing that we have reproduced Mg⁺ experimental binding energies exactly for 3s, 3p, 3d, and 4p. The difference with experimental values for the Mg⁺(4s) atomic level is equal to 1.53 cm⁻¹, and for the other highly excited states 5s, 4d, 4f, 5p, and 6s, the discrepancy is between 22 and 105 cm⁻¹.

Table 2. Asymptotic energy of the alkali earth (MgCs)⁺ electronic states (in cm⁻¹): comparison between our calculated energy, Moore et al.’s [70] calculated energy, and the corresponding experimental dissociation.

State	Asymptotic Molecular State	This Work	Experiment [68,70]
X (1) 1Σ+	Mg (3s2) + Cs+	0	0
A (2) 1Σ+	Mg+ (3s) + Cs (6s)	30,263.62	30,398.60
C (3) 1Σ+	Mg (3s3p) + Cs+	35,050.59	35,319.67
D (4) 1Σ+	Mg+ (3s) + Cs (6p)	41,811.30	41,946.72
E (5) 1Σ+	Mg (3s4s) + Cs+	43,502.57	43,472.29
F (6) 1Σ+	Mg+ (3s) + Cs (5d)	44,821.40	44,956.60
G (7) 1Σ+	Mg (3s3d) + Cs+	46,402.50	47,967.79
H (8) 1Σ+	Mg+ (3s) + Cs (7s)	48,802.02	48,934.58
I (9) 1Σ+	Mg (3s4p) + Cs+	49,346.10	49,487.00
J (10) 1Σ+	Mg+ (3s) + Cs (7p)	52,143.74	52,284.65
a (1) 3Σ+	Mg (3s3p) + Cs+	21,876.60	21,994.46
c (2) 3Σ+	Mg+ (3s) + Cs (6s)	30,263.62	30,398.60
d (3) 3Σ+	Mg (3s4s) + Cs+	41,196.77	41,154.85
e (4) 3Σ+	Mg+ (3s) + Cs (6p)	41,811.30	41,946.72
f (5) 3Σ+	Mg+ (3s) + Cs (5d)	44,821.40	44,956.60
g (6) 3Σ+	Mg (3s4p) + Cs+	47,857.84	48,197.80
h (7) 3Σ+	Mg (3s3d) + Cs+	47,956.38	46,647.87
I (8) 3Σ+	Mg+ (3s) + Cs (7s)	48,802.02	48,934.58
J (9) 3Σ+	Mg (3s5s) + Cs+	51,871.81	51,976.72
k (10) 3Σ+	Mg+ (3s) + Cs (7p)	52,143.74	52,284.65

Table 2. Asymptotic energy of the alkali earth (MgCs)⁺ electronic states (in cm⁻¹): comparison between our calculated energy, Moore et al.’s [70] calculated energy, and the corresponding experimental dissociation.

State	Asymptotic Molecular State	This Work	Experiment [68,70]
X (1) 1Σ+	Mg (3s2) + Cs+	0	0
A (2) 1Σ+	Mg+ (3s) + Cs (6s)	30,263.62	30,398.60
C (3) 1Σ+	Mg (3s3p) + Cs+	35,050.59	35,319.67
D (4) 1Σ+	Mg+ (3s) + Cs (6p)	41,811.30	41,946.72
E (5) 1Σ+	Mg (3s4s) + Cs+	43,502.57	43,472.29
F (6) 1Σ+	Mg+ (3s) + Cs (5d)	44,821.40	44,956.60
G (7) 1Σ+	Mg (3s3d) + Cs+	46,402.50	47,967.79
H (8) 1Σ+	Mg+ (3s) + Cs (7s)	48,802.02	48,934.58
I (9) 1Σ+	Mg (3s4p) + Cs+	49,346.10	49,487.00
J (10) 1Σ+	Mg+ (3s) + Cs (7p)	52,143.74	52,284.65
a (1) 3Σ+	Mg (3s3p) + Cs+	21,876.60	21,994.46
c (2) 3Σ+	Mg+ (3s) + Cs (6s)	30,263.62	30,398.60
d (3) 3Σ+	Mg (3s4s) + Cs+	41,196.77	41,154.85
e (4) 3Σ+	Mg+ (3s) + Cs (6p)	41,811.30	41,946.72
f (5) 3Σ+	Mg+ (3s) + Cs (5d)	44,821.40	44,956.60
g (6) 3Σ+	Mg (3s4p) + Cs+	47,857.84	48,197.80
h (7) 3Σ+	Mg (3s3d) + Cs+	47,956.38	46,647.87
I (8) 3Σ+	Mg+ (3s) + Cs (7s)	48,802.02	48,934.58
J (9) 3Σ+	Mg (3s5s) + Cs+	51,871.81	51,976.72
k (10) 3Σ+	Mg+ (3s) + Cs (7p)	52,143.74	52,284.65

2.1. Results and Discussions

2.1.1. Adiabatic Potential Energy and Spectroscopic Constants

Using the method reported in the previous section, a full and expanded calculation was performed for many electronic states of ^1,3Σ⁺, symmetries for the (MgCs)⁺ cationic molecule. These states dissociate into Mg⁺ (3s, 3p, 4s, 3d, 4p, 5s, 4d, 4f, 5p, and 6s) + Cs (6s, 6p, 4d, 5d, 7s, and 7p) and Mg {(3s² (¹S), 3s3p (³P), 3s3p (¹P), 3s4s (³S), 3s4s (¹S), 3s3d (¹D), 3s4p (³P), 3s3d (³D), (¹P), 3s5s (³S), 3s5s (¹S), 3s4d (¹D), 3s4d (³D), and 3s5p (³P))} + Cs⁺. The adiabatic potential energy is performed for an interval of intermolecular distances from 3.50 to 200.00 a₀ with a step of 0.1 a₀ around the avoided crossings. The potential energy curves as functions of the internuclear distance R for the ^1,3Σ⁺ electronic states are drawn, respectively, in Figure 1 and Figure 2.

In order to assess the precision of our calculated potential energy curves for the states previously studied, we have extracted the spectroscopic constants (equilibrium distance R_e, well depth D_e, electronic excitation energy T_e, frequency ω_e, harmonicity constant ω_ex_e, and rotational constant B_e for the equilibrium separation R_e). These constants were determined by least-squares interpolation of the vibrational energy levels. Unfortunately, no experimental spectroscopic information for (MgCs⁺) has been published yet. However, Smialkowski and Tomza [28] reported a theoretical calculation for the ground state. They found R_e = 7.85 a₀, D_e = 1861 cm⁻¹, ω_e = 73.2cm⁻¹, and B_e = 0.0481 cm⁻¹, which are close to our values (R_e = 7.70 a₀, D_e = 2047 cm⁻¹, ω_e = 73.2cm⁻¹, and B_e = 0.049364 cm⁻¹) (see

Table 3. Spectroscopic constants of the ground and excited singlet and triplet electronic states of (MgCs)⁺ molecule.

State	Re (a.u.)	De (cm−1)	Te (cm−1)	ωe (cm−1)	ωexe (cm−1)	Be (cm−1)
X1Σ+	7.70	2047	0	73.2	0.63	0.049364
	7.85 [28]	1861 [28]		73.2 [28]	0.0481 [28]
21Σ+	11.45	2662	29,782	44.64	0.25	0.022354
31Σ+	16.88	1054	36,391	21.92	0.16	0.010280
41Σ+	9.33	−2750
	19.46	1891	42,101	21.73	0.06	0.007700
51Σ+	28.24	263	45,256	9.12	0.07	0.003674
61Σ+	58.89	4	46,990	−0.48	14.61	0.001160
71Σ+	8.68	−3301
	27.83	509	48,507	8.92	26.51	0.038919
81Σ+	8.68	−8647
	31.99	923	50,059	10.13	29.95	0.038864
91Σ+	8.69	−10,460
	42.75	300	51,240	5.44	30.84	0.038839
101Σ+	8.71	−10,297
	42.27	1105	53,289	8.07	40.10	0.038614
13Σ+	8.58	4504	19,556	74.78	0.52	0.039826
23Σ+	14.98	813	31,630	25.56	0.17	0.013050
33Σ+	17.20	2658	40,542	26.85	0.05	0.009905
43Σ+	28.15	506	43,486	10.12	0.04	0.003698
53Σ+	58.89	4	46,990	−0.48	14.61	0.001160
63Σ+	28.13	1600	48,413	13.19	0.02	0.003704
73Σ+	38.20	31	50,667	6.81	0.03	0.002008
83Σ+	49.32	131	50,850	4.59	0.02	0.001205
93Σ+	40.09	1055	52,967	8.79	0.01	0.001823
103Σ+	56.76	395	53,999	3.95	0.01	0.000910

). Therefore, the spectroscopic constants for the excited states are presented here for the first time. Consequently, we think that these potential energies and spectroscopic data can initiate future experimental and theoretical investigations on this molecular ionic system. In Table 3, we report the spectroscopic constants for the ground and the excited states for all symmetries.

The potential energy curves for the 20 first ^1,3Σ⁺ states of (MgCs)⁺ molecular ion are plotted in Figure 1 and Figure 2. These states are labelled by increasing numbers from 1 to 10 for both singlet and triplet states

The ground (X¹Σ⁺) and the first excited (2¹Σ⁺) states are well separated from the highest states of the same symmetry, and they maintain an energy gap of about 42,012 cm⁻¹. Both are found with unique minimums at 7.7 and 11.45 a₀ and well depths of 2047 and 2662 cm⁻¹, respectively. The first three excited triplet states (1–3)³Σ also exhibit unique minimums at 8.58, 14.98, and 17.20 a₀, with significant well depths of 4504, 813, and 2658 cm⁻¹, respectively. An accurate analysis of these curves shows that the most strongly bound states are the first excited states of both singlet and triplet states with well depths of 2662 and 4504 cm⁻¹ located at 11.45 and 8.58 a₀, respectively.

An interesting feature of the higher excited states of all symmetries of ^1,3Σ⁺ states has been noticed. The shape becomes more involved, with double wells and potential barriers. This feature is due to the presence of many avoided crossings at short and large values of internuclear distances between excited states of the same symmetries. The positions of such avoided crossings are presented in Figure 1 and Figure 2. These series of avoided crossings are related to underlying charge transfer between the ionic configurations Mg⁺/Cs and Mg/Cs⁺. Later, these avoided crossings will be discussed in light of the behavior of the permanent dipole moments.

The PECs of excited states starting from 4¹Σ⁺ present double wells and barriers, which are the consequence of avoiding crossings that happen at specific internuclear distances. These avoided crossings are related to underlying charge transfer, as can be seen from the change of location of the positive charge at dissociation limits between two neighbor states. The positive charge moves from one atom to the second one. The existence of these avoided crossings will generate large nondiabetic couplings. This same behavior has been seen for other symmetries, such as ³Σ⁺ states.

2.1.2. Vibrational Levels

For a better understanding of the potential energy curve structure, the vibrational levels for all attractive states are computed using the Numerov algorithm. The interpolation of the potential curves is carried out with 12,000 points and a grid ranging from 3.5 to 180.0 a₀.

The vibrational spacings (E_v+1–E_v) as a function of the vibrational quantum number v are reported in Figure 3 for the ^1,3Σ⁺ states. As usual, these spacings are not constant due to the important anharmonicity observed in the potential energy curves. The number of the vibration levels for the ground and the first excited states are 80 and 180, respectively. The vibrational energy spacing of these two potentials shows a linear behavior for low vibrational levels, and then it vanishes at the dissociation limits.

The spacing between vibrational levels decreases gradually as vibrational energy increases. This behavior is particularly prominent for vibrational levels of the PECs that reflect the strong anharmonicity. The overall pattern of energy spacings of different electronic states of the molecular ion is similar. However, for some states, irregularities related to the avoided crossings are visible, e.g., for the ¹Σ⁺ states.

2.1.3. Permanent and Transition Dipole Moments

In addition to the potential energy curves, the practical implementation of cold molecule formation via photoassociation requires knowledge of their electronic properties, such as the radial variation of permanent or transition dipole moments. Furthermore, the knowledge of the dipole function of molecular systems can be considered a sensitive test for the accuracy of the calculated electronic wave functions and energies.

Keeping this broad perspective of the ion–atom systems in mind, we have determined the permanent and transition dipole moments for the same large and dense grid of internuclear distances used for the potential energy curves.

Permanent Dipole Moments

The permanent and transition dipole moments are determined by considering the origin at atom Mg, and where the distance R between Mg/Mg⁺ and Cs/Cs⁺ stands for the internuclear distance between them. We calculated the permanent dipole moments as a function of R; the intermolecular distance for all studied states (1−10¹Σ⁺, 1−10³ Σ⁺) are displayed in Figure 4 and Figure 5.

At a short and intermediate internuclear distance, the permanent dipole moments present undulations with abrupt variation between neighbor electronic states. For example, there is an abrupt variation between the 8¹Σ⁺ and 9¹Σ⁺ states located at 33.5 a₀. We remark that this particular distance represents approximately the position of the avoided crossing between the two potential energy curves that we noticed before. We can conclude that the significant changes in permanent dipole moment at a small internuclear distance are due to the change in the polarity of the molecule. In addition, the positions of the irregularities in the R-dependence of the permanent dipole are correlated to the avoided crossings between potential energy curves, which are both manifestations of abrupt changes in the character of the electronic wave functions.

For the large internuclear distance, the permanent dipole moment of the X¹Σ⁺, 3¹Σ⁺, 5¹Σ⁺, 7¹Σ⁺, and 9¹Σ⁺ states dissociating into Mg (3s², 3s3p, 3s3p, 3s4s, 3s4s, 3s3d, 3s4p, 3s3d, 3s5s, 3s5s, 3s4d, 3s4d, and 3s5p) + Cs⁺ are significant and yield a pure linear behavior function of R, especially at intermediate and large distances. For the remaining states 2¹Σ⁺, 4¹ Σ⁺, 6 ¹Σ⁺, and 10 ¹Σ⁺ dissociating into Mg⁺ (3s, 3p, 4s, 3d, 4p, 5s, 4d, 4f, 5p, and 6s) + Cs (6s, 6p, 4d, 5d,7s and 7p), we note significant permanent dipole moments in specific regions and then they vanish swiftly at large distances.

For the permanent dipole moments of the first ten states of ³Σ⁺ symmetry, similar observations were noticed as for the ¹Σ⁺ states. They exhibit undulations with abrupt variations at a short and intermediate internuclear distance R, such as between the two states 8³Σ⁺ and 9³Σ around 27.7 a₀. We remark that the permanent dipole moments of these states, one after another, behave in a straight line on the same curve and then drop to zero at particular distances corresponding to the avoided crossings between neighboring electronic states.

Transition Dipole Moments

In addition, we have determined the transition dipole moment functions for the (MgCs)⁺ ionic molecular system between states of similar symmetries. As the transition dipole moments between adjacent states are the most significant, here, we only present the transition between neighbor states. They are displayed in Figure 6.

Theoretically, Fedorov et al. [27] reported dipole moments for the molecular ions (MgLi)⁺ and (MgNa)⁺. They have calculated the transition dipole moments for the latter systems; however, no data for the (MgCs)⁺ ionic molecule are available, and these important physical quantities are presented here for the first time. We note a meaningful variation with numerous extremum, which can be assigned to avoid crossing positions between adjacent states. We also note a change in the sign of some dipole transitions, such as between 3¹Σ⁺->4¹Σ⁺ states. This change in sign corresponds to a sudden change in electronic wave functions, which is usually related to the avoided crossings between the potential energy curves.

For the triplet sigma states, the transition dipole moment curves between the neighbor states are displayed in Figure 6. We note similar behaviors observed previously for the singlet states. For both multiplicities,^1,3Σ states, the weakly avoided crossing series already found between neighboring states lead to abrupt variations in the transition dipole moments.

2.1.4. Radiative Lifetime

In this section, we present the vibrational state lifetimes for A¹ Σ⁺ (2¹ Σ⁺) excited state. Given a vibrational level of the first excited electronic state, two possible transitions can occur: bound–bound and bound-free transitions. The radiative lifetime of a vibrational level corresponding to only bound–bound transitions is given by

$${\tau }_{\upsilon \prime }=\frac{1}{{\Gamma }_{\upsilon \prime }} \mathrm{where} {\Gamma }_{\upsilon ’}={{\displaystyle \sum }}_{\upsilon =0}^{nvt}{A}_{\upsilon {\upsilon }^{\prime }}$$

(1)

A_vv’ is the Einstein coefficient linking, for example, the A¹Σ⁺(v′) and X¹Σ⁺(v) levels. Zemke et al. [71] have previously shown that there is a missing contribution in the radiative lifetime. It corresponds to the bound-free transitions, which are more significant for the higher vibrational levels close to the dissociation limit of the excited electronic state, found in the form of the continuum radiation to states above the dissociation limit of the ground state. It corresponds to the contribution of the bound-free transition missed in Equation (1). It is related to the transition between the vibrational level v′, which belongs to the excited electronic state, to the continuum of the ground state or the lower state in general. Such contribution is not negligible, as was demonstrated by Partridge et al. [72], Zemke et al. [71], and Berriche and Gadea [73,74]. It is more important for the higher vibrational levels due to the difference in the location of the potential wells. We calculate this contribution using two different approximations, “Franck-Condon” [72,73] and the “sum rule method” [72,75] approximation. In the Franck–Condon (FC) approximation proposed by Zemke et al. [71], the bound-free contribution is given by

$$A_{\upsilon \prime }\left(bound-free\right)=2.14198639\times {10}^{10}{\left|\mu \left(R_{\upsilon \prime +}\right)\right|}^{2}F{C}_{\upsilon \prime ,cont}{(\Delta E)}^3{}_{\upsilon \prime ,cont}$$

(2)

where ${\Delta E}_{\upsilon \prime ,cont}=E_{\upsilon \prime }-E_{as}$ is the energy difference between the vibrational level v and the energy of the asymptotic limit of the lower electronic state to which the continuum belongs. The quantity $\mu (R_{\upsilon \prime +})$ corresponds to the transition dipole moment at the right external turning point of the vibrational level v′.

$$F{C}_{\upsilon \prime ,cont}=\int {|\langle {\chi }_{\upsilon \prime }|{\chi }_E\rangle |}^{2}dE=1-{{\displaystyle \sum }}_{\upsilon =0}^{nvt}{|\langle {\chi }_{\upsilon \prime }|{\chi }_{\upsilon }\rangle |}^2$$

(3)

The approximate sum rule approach, implemented by Pazyuk et al. [76], allows the production of the radiative lifetime components for diatomic vibronic states, including the bound-free contribution. In addition, this approximation has a high efficiency for non-diagonal systems, particularly for those with significant continuum contributions. The radiative lifetime using the sum rule method is given by

$$\frac{1}{\tau }={\displaystyle \underset{R_{\min }}{\overset{R_{\max }}{\int }}{\varphi }_{{\nu }^{\prime }}}\left(R\right)\left(D\left(R^2\right)\right)\left(\Delta \left(R^3\right)\right){\varphi }_{{\nu }^{\prime }}\left(R\right)dR$$

(4)

${\phi }_{\upsilon \prime }\left(R\right)$ is the wave function of the vibrational level belonging to the A¹Σ⁺ excited electronic state. D(R) is the transition dipole moment between the ground X ¹Σ⁺ and the first excited A¹Σ⁺ states. ∆U(R) is the energy difference between the ground X¹Σ⁺ and the first excited A¹Σ⁺ states. The total radiative lifetime, taking into account the bound–bound and bound-free contributions, using the two approaches, FC approximation and the approximate sum approach, are presented in

Table 4. Radiative lifetimes and vibrational-level spacing $E_{\nu }-E_{v-1}$ of the 2¹Σ⁺ state of the (MgCs)⁺ ion.

Vibrational Level	Ev − Ev−1 (cm−1)	Franck–Condon (ns)	Sum Rule (ns)
0		10.579	10.582
1	44.110	10.658	10.661
2	43.878	10.739	10.742
3	43.599	10.825	10.828
4	43.367	10.912	10.915
5	43.080	11.003	11.006
6	42.848	11.096	11.098
7	42.550	11.191	11.194
8	42.278	11.290	11.294
9	42.016	11.394	11.399
10	41.744	11.502	11.511
11	41.477	11.634	11.666
12	41.213	11.770	11.824
13	40.941	12.055	12.253
14	40.666	12.294	12.589
15	40.389	12.989	13.787
16	40.109	13.549	14.649
17	39.832	14.486	16.450
18	39.534	15.301	17.708
19	39.237	15.689	18.283
20	38.926	15.967	18.465
21	38.604	16.259	18.689
22	38.293	17.042	19.857
23	37.992	17.532	20.454
24	37.668	18.321	21.644
25	37.352	18.660	21.795
26	37.041	19.142	22.251
27	36.707	19.728	22.916
28	36.381	20.504	23.969
29	36.044	21.158	24.555
30	35.699	21.574	24.792
31	35.361	22.446	25.859
32	35.012	23.075	26.408
33	34.666	23.816	27.190
34	34.312	24.277	27.467
35	33.959	25.217	28.468
36	33.599	26.400	29.687
37	33.235	26.976	30.140
38	32.868	27.668	30.647
39	32.493	28.386	31.252
40	32.114	29.689	32.700
41	31.726	30.554	33.375
42	31.333	31.446	34.124
43	30.932	32.939	35.617
44	30.526	33.864	36.390
45	30.118	34.807	37.156
46	29.705	35.796	37.999
47	29.287	37.175	39.363
48	28.863	38.664	40.724
49	28.435	39.788	41.692
50	28.000	41.625	43.490

.

To the best of our knowledge, this is the first time that the radiative lifetimes of the vibrational levels of the A¹Σ⁺ state of the (MgCs)⁺ molecules have been presented. These radiative lifetimes increase with the vibrational level v in a range of 10–43 ns.

2.1.5. Ion–Atom Elastic Collisions

In this section, we study cold elastic collisions between an alkaline earth atom and an alkali ion in the ground electronic state (X¹Σ⁺) of the (MgCs)⁺ molecular system. The ground-state potential asymptotically correlates to the heavier alkali element in the ionic state (Cs⁺) and the alkaline element in its neutral ground state (Mg). The first singlet excited state 2¹Σ⁺ asymptotically corresponds to the charge-exchanged state of the X¹Σ⁺ potential. We use Mg + Cs⁺ as a prototype system, and several results equally apply to other alkaline-earth–alkali atom–ion collisional systems. The elastic collisions between the alkaline earth atom, Mg, and alkali ion, Cs⁺, for the ground-state potential (X¹Σ⁺) can be represented as

$$Mg+C{s}^{+}\to \left\{MgC{s}^{+}\left({\mathrm{X}\Sigma }^{+}\right)\right\}\to Mg+C{s}^{+}$$

(5)

The partial-wave Schrödinger equation for the ion–atom elastic collision can be expressed as

$$\left[\frac{d^2}{d{r}^2}+k^2-\frac{2\mu }{{\hslash }^2}V\left(r\right)-\frac{l\left(l+1\right)}{r^2}\right]{\psi }_l\left(kr\right)=0$$

(6)

where r is the internuclear separation between an atom and ion, $\mu$ is the reduced mass of an ion–atom colliding pair, and ${\psi }_l\left(kr\right)$ is the wave function for the l^th partial wave. The wave function satisfies the asymptotic boundary condition ${\psi }_l\left(kr\right) ~ sin\left[kr-\frac{l\pi }{2}+{\eta }_l\right]$. The above second-order differential Equation 6 can be solved by the Numerov–Cooley algorithm [77] using three-point recursion relation. Initially, we set the values of the wave function at two initial consecutive points at small internuclear separation $r~0$. Thereafter, the code calculates the wave function at the third point by utilizing the three-point recursion relation. Then, the second and third points are reset to the initial point, and the wave function is calculated for the fourth point. This process is continued until the asymptotic boundary is attained. The initial boundary conditions are included by expanding the interaction potential at small r and solving the Schrödinger equation at this limit. The boundary condition at the asymptotic limit is incorporated by setting the condition that the effective long-range part of potential V(r) is much less than the collisional energy E (at least one-tenth of E). We considered a constant step size of 0.01 during the propagation of the code.

The total elastic scattering cross-section is given by

$${\sigma }_{tot}=\frac{4\pi }{k^2}{\displaystyle \sum }_{l=0}^{\infty }\left(2l+1\right)si{n}^2\left({\eta }_l\right)$$

(7)

where k is the wave number related to the collisional energy $E=\frac{{ħ}^2{k}^2}{2\mu }$ and η_l being the phase shift for the l-th partial wave. As with an increase in energy, a large number of partial waves start to contribute to elastic scattering cross-section. Under this condition, the scattering cross-section is approximated as [4]

$${\sigma }_{tot}\approx \pi {\left(\frac{\mu C_4^2}{{\hslash }^2}\right)}^{\frac{1}{3}}\left(1+\frac{{\pi }^2}{16}\right)E^{-\frac{1}{3}}$$

(8)

The coefficient C₄ is related to the static electric dipole polarizability of the concerned atom.

But, at a very low energy limit, i.e., $E\to 0$, the phase shift follows the Wigner threshold laws. At this energy regime, the phase shift is related to the s-wave scattering length $a_s=-\underset{k\to 0}{\lim }\frac{tan{\eta }_0}{k}$. The scattering length carries the information about the interaction potential of a system. A positive (negative) scattering length is associated with repulsive (attractive) interaction.

In Figure 7, we show the logarithm of total elastic scattering cross-sections (σ_tot) in a.u. as a function of the logarithm of energy E in K for ground-state collisions between neutral alkaline earth atom Mg and alkali ion Cs⁺. In order to calculate the convergence results in σ_tot, we need 59 partial waves.

In our calculations, we have included 70 partial waves, since, with the increase in energy, large numbers of partial waves start to contribute to the total elastic scattering cross-section. Considering the logarithm on both sides of Equation (8), a straight line is obtained with a slope equal to −1/3. From the linear fitting of log σ_tot vs. log E of Figure 7, we have verified the numerically calculated slope of the straight line is quite close to the theoretical value.

2.1.6. Two-Photon Photoassociation: Molecular Ion Formation

In this section, we explore the molecular ion formation in the ground-state electronic potential X¹Σ⁺ by a two-photon incoherent Raman process in the presence of two applied lasers, L₁ and L₂, respectively. The formation of the molecular ion needs two steps. Initially, the atom–ion pair present in the ground-state scattering continuum (X¹Σ⁺) is irradiated with laser L₁ with a suitable frequency to form the molecular ion in one of the bound levels of the first excited potential 2¹Σ⁺. Once the molecular ion is formed, laser L₂ is turned on to de-excite the molecular ion to the ground electronic potential followed by bound–bound transition.

Here, the rate coefficient of the first step concerned, which is known as single-photon photoassociation (PA) at a temperature T, is given by [78]

$$K_{PA}=\langle \frac{\pi v_{rel}}{k^2}{\displaystyle \sum }_{l=0}^{\infty }\left(2l+1\right){\left|S_{PA}\left(E,l,{\omega }_l\right)\right|}^2\rangle$$

(9)

where v_rel is the relative velocity of the interacting particles, S_PA is the scattering matrix element, and $\langle \dots \rangle$ implies an averaging over thermal velocity distribution. At a very low-temperature limit, the energy of the interacting particle is such that their dynamics may be described by the s-wave scattering. If one considers a Maxwellian velocity distribution of an ultracold mixture of gas, Equation (9) can be represented as

$$K_{PA}\left(T,{\omega }_L\right)=\frac{1}{h{Q}_T}{\displaystyle \sum }_{l=0}^{\infty }\left(2l+1\right){{\displaystyle \int }}_0^{\infty }{\left|S_{PA}\left(E,l,{\omega }_L\right)\right|}^2{e}^{-\beta E}dE$$

(10)

Here, $Q_T={\left(2\pi \mu k_{B}T/h^2\right)}^{3/2}$ is the translational partition function, β = 1/k_bT, and k_B is the Boltzmann constant. The scattering S-matrix element is given by

$${\left|S_{PA}\right|}^2=\frac{{\gamma }_s{\Gamma }_l}{{\delta }_E^2+{(\gamma /2)}^2}$$

(11)

The scattering phase shift (η) is related to the scattering T-matrix as $T=\frac{1-e^{2i\mathsf{\eta }}}{2i}=-e^{i\mathsf{\eta }}\sin \mathsf{\eta }$, while the quantity ${\left|T\right|}^2={\sin }^2\mathsf{\eta }$. Again, the scattering S-matrix is related to the T-matrix as $S=1-2iT$ and the quantity ${\left|S\right|}^2=1+4{\sin }^2\mathsf{\eta }$. Therefore, in terms of phase shift and T-matrix, Equation (11) becomes

$${\left|T\right|}^2={\sin }^2\mathsf{\eta }=\frac{1}{4}\left[\frac{{\gamma }_s{\Gamma }_l}{{\delta }_E^2+{(\gamma /2)}^2}-1\right]$$

(12)

where detuning parameters are defined as ${\delta }_E=\frac{E}{ħ}+{\delta }_{v,j}$, ${\delta }_{v,j}={\omega }_L-{\omega }_{v,j}$ with $E_{v,j}=ħ{\omega }_{v,j}$ is the binding energy of the excited co-vibrational state, ${\omega }_L$ is the frequency of the applied laser L₁. Here, γ_s is the spontaneous line width, and Γ_l is the stimulated line width. We consider that other decay processes, such as molecular predissociation, are either negligible or not present in our system.

The PA rate is determined, basically, by stimulated line width and can be expressed by the Golden rule as

$${\Gamma }_l=\frac{\pi I}{{\epsilon }_{0}c}{|\langle {\varphi }_{v,j}\left|D\left(r\right)\right|{\psi }_l\left(kr\right)\rangle |}^2$$

(13)

We termed the quantity $D_{v,j,l}=\langle {\varphi }_{v,j}\left|D(r)\right|{\psi }_l(kr)\rangle$ as the radial transition dipole matrix element between the unit normalized bound state wave function ϕ_v,j and energy normalized continuum wave function ψ_l (kr), and D(r) is the transition dipole moment. Here, I is the intensity of the laser L₁, c is the velocity of light, and ε₀ is the vacuum permittivity.

In order to calculate the quantity D_v,j,l for the optimal production of molecular ions by the PA process, we have to calculate the scattering state and suitable bound state wave functions. We solve the partial-wave Schrödinger Equation (6) by standard renormalized Numerov–Cooley method [78] for the determination of the wave functions that we discussed. In general, molecular dipole transitions between two bound states or between continuum and bound states are controlled by the FC principle. The free-bound FC value is highest for the bound state ${\varphi }_{v,j}$ (v = 17, j = 1) of the first excited potential 2¹${\Sigma }^{+}$ of (MgCs)⁺ system. The PA rate coefficient is calculated according to Equation (9) to form the molecular ion in the excited electronic state 2¹${\Sigma }^{+}$. In Figure 8, we have shown the rate coefficient of PA as a function of energy E (in Kelvin), considering the atom–ion pairs Mg-Cs⁺ are initially lying in the ground-state scattering continuum. We considered the laser intensity L₁ = 10 W/cm² tuned at the PA resonance. We find that our calculated PA rate coefficient is consistent with the results of Li⁺-Be [79] and K⁺-Mg [49] systems. Finally, in Figure 9, we have shown the variation of the rate of ion–atom PA as a function of the intensity of the applied laser L₁ considering the temperature at 10 μK. We have observed that a saturation effect on the ion–atom PA rate occurs near the laser intensity I = 35 kW/cm².

Now we discuss the formation of a ground-state molecular ion by stimulated Raman-type process in the presence of a second applied laser L₂ tuned near a bound–bound transition between the potentials 2¹${\Sigma }^{+}$ and X¹${\Sigma }^{+}$, respectively. To find the efficacy of the coherent laser coupling between the two bound states, we calculate the Rabi coupling Ω given by

$$ħ\Omega ={\left(\frac{I}{4\pi c{\epsilon }_0}\right)}^{1/2}|\langle {\varphi }_{v,j}\left|D\left(r\right).ϵ\right|{\varphi }_{v^{\prime }{j}^{\prime }}\rangle |$$

(14)

where I is the intensity of the second applied laser L₂ tuned to the transition frequency, $ϵ$ is the unit vector of laser polarization, and ${\varphi }_{v,j}$ and ${\varphi }_{v^{\prime }{j}^{\prime }}$ are the two bound states of excited (2¹${\Sigma }^{+})$ and ground (X¹${\Sigma }^{+}$) state electronic potentials, respectively. We have calculated the Rabi frequencies between the selected bound state (v = 15, j = 1) of the first excited molecular potential and different bound states of the ground electronic potential, considering an intensity of laser L₂ = 10 kW/cm⁻². We observed that the bound–bound Rabi coupling is maximum (Ω = 150 MHz) for the bound state ${\varphi }_{v^{\prime }{j}^{\prime }}$($v^{\prime }$=16, $j^{\prime }=0$) of X¹${\Sigma }^{+}$ potential. Comparing this value with the spontaneous line width (γ = 120 kHz) of the excited bound state calculated using the formula [80], we found that bound–bound Rabi coupling (Ω) is several orders of magnitude higher. Therefore, we may infer that the ground-state molecular ion formation is likely to be possible by a stimulated Raman-type process [49,80,81].

3. Conclusions

In this work, an extended and complete study devoted to the (MgCs)⁺ molecular ion was presented. We have performed precise ab initio calculations for the potential energy curves and their related spectroscopic constants for the ground and the 20 low-lying excited states of ^1,3Σ⁺ symmetries below the Mg²⁺Cs⁻ asymptotic limit.

The computational method used is based on a non-empirical pseudo-potential approach for Mg²⁺ and Cs⁺ cores, which allows treating the ionic molecule as a two-electron system where full configuration interaction FCI calculations can be easily performed. An interesting series of avoided crossings have been observed between the same symmetry of the ^1,3Σ⁺ states. These avoided crossings often lead to relatively irregular dipole moment curves, which are both manifestations of abrupt changes in the character of the electronic wave functions with underlying strong interactions between the electronic states. These interactions could give rise to important charge transfer for collisions between the atom–ion combinations. In addition, the vibrational levels and their spacings were investigated for all attractive states. Finally, electric permanent and transition dipole moments were calculated. As expected, the permanent dipole moments of the electronic states dissociating into Mg {(3s² (¹S), 3s3p (³P), 3s3p (¹P), 3s4s (³S), 3s4s (¹S), 3s3d (¹D), 3s4p (³P), 3s3d (³D), (¹P), 3s5s (³S), 3s5s (¹S), 3s4d (¹D), 3s4d (³D), and 3s5p (³P))}+Cs⁺ have shown an almost linear features function of R, especially for intermediate and large internuclear distances. Moreover, the abrupt changes in the permanent dipole moment are localized at particular distances corresponding to the avoided crossings between the two neighboring electronic states.

In this study, the ab initio calculations were carefully performed to produce accurate data, even though there is a lack of experimental studies for this molecular ion. However, theoretically, only the ground state of (MgCs)⁺ was studied in a recent work published by Smialkowski and Tomza [28]. We expect that our data can serve as a solid foundation for future and advanced dynamic studies for the (MgCs)⁺ molecular ion.

These precise data were then used for elastic diffusion calculations between Mg and Cs⁺ at low temperatures. In fact, a theoretical understanding of low-energy atom–ion scattering can help probe the dynamics of quantum gases. Furthermore, we theoretically predict the formation of a cold molecular ion in translation and rotation by an incoherent two-photon Raman process. The formation of such cold molecular ions is very important in the research domain of ultracold chemistry and precision spectroscopy. Our data can be used to understand several phenomena, like charge transfer, diffusion, and partial-wave phase locking, and is comparable with the recently studied Li-Ba⁺ system [81]. We believe that our analysis, including the ab initio data, may serve as a solid platform for future study on (MgCs)⁺ molecular ionic systems.