Theoretical Investigation of the Electronic Structure and Spectra ofMg2+He and Mg+HeM. Bejaoui,† J. Dhiflaoui,† N. Mabrouk,† R. El Ouelhazi,† and H. Berriche*,†,‡

†Laboratory of Interfaces and Advanced Materials, Physics Department, Faculty of Science, University of Monastir, Avenue deL’Environnment, 5019 Monastir, Tunisia‡Department of Mathematics and Natural Sciences, School of Arts and Sciences, American University of Ras Al Khaimah, Ras AlKhaimah, P.O. Box, RAK, United Arab Emirates

ABSTRACT: The ground and many excited states of the Mg+He vander Waals molecular system have been explored using a one-electronpseudopotential approach. In this approach, effective potentials are usedto consider the Mg2+ core and the electron−He effects. Furthermore, acore−core interaction is included. This has reduced the number of activeelectrons of the Mg+He, to be considered in the calculation, to a singlevalence electron. This has permitted to use extended Gaussian basis setsfor Mg and He. Therefore, the potentianl energy and dipole momentscalculations are carried out at the Hartree−Fock level of theory, and thespin−orbit effect is included using a semiclassical approach. The core−core interaction for the Mg2+He ground state is included using accurateCCSD(T) calculations. The spectroscopic constants of the Mg+He electronic states are extracted and compared with the existingtheoretical works, where very good agreement is observed. Moreover, the transition dipole function has been determined for alarge and dense grid of internuclear distances including the spin−orbit effect.

■ INTRODUCTION

Great attention at both experimental and theoretical levels hasbeen emphasized in recent times on the alkaline earth andalkali−rare gas dimers. They present remarkable and interestingtraits, which can be used for several technology applications.Due to the closed shells of atoms and/or ions that form thesesystems, their ground state presents very weak interactions. Incontrast, considerable bonding can exist in the excited states.Several alkaline earth or alkali−rare gas systems were studied inthe past using different techniques and methods. The mostused methods are semiempirical,1−6 model potential,7−15 andpseudopotential16−22 approaches. The shape of the potentialsurface due to different types of intermolecular interactions isimportant for many problems in molecular dynamics, such asthe modeling of physical adsorption and desorption processesand collisional processes in general.van der Waals interaction between neutral molecules has

been extensively studied in the past few decades and is oftendescribed by a simple model that combines an empiricalrepulsive inner wall and a long-range attractive interaction. Thisattractive part of the potential is due to the interaction betweenthe dipole, induced dipole, quadrupole, and sometimes higher-order electric multipole moments of the two atoms. A secondimportant kind of intermolecular interaction is that between aneutral atom and an ion. Such an interaction may lead to theformation of a relatively strongly bound complex with a bondthat approaches a chemical bond, particularly in cases when oneof the participating particles is an open-shell species

On the experimental side, alkaline earth−rare gas open-shellionic complexes were produced in a pulsed free-jet expansionand detected by a time-of-flight (TOF) mass spectrometer orvia laser-induced fluorescence (LIF).23,24 Unfortunately, thereare no experimental studies for Mg+He; however, severaltheoretical investigations have been performed.25−29 Partridgeet al.25 made the first theoretical study. They determined thespectroscopic constants for the ground and some exited states.In their work, they compared many metal−noble gas ions bymodifying a coupled functional level of electronic correlationtreatment. Leung et al.26 have estimated the potential energycurves of Mg+He by MP2, MP3, and MP4 ab initio calculations.In a recent study, Bu et al.27 have investigated the physical andchemical proprieties and the geometric structures of Mg+Henand Mg2+He (n = 1−10) using the MP2 method. Adrian et al.28

have presented high-level calculation of the potential energycurves of the Mn+−Rg (n = 1 and 2) complexes, and they haveextracted their spectroscopic constants.This paper presents a theoretical study of the Mg+He

structure (potential energy, spectroscopic constant, and dipolefunctions) for many electronic states of different symmetries.The calculation method is based on a pseudopotentialtechnique, which reduces the Mg+He ionic dimer to only oneactive electron system. Therefore, Mg2+ and He are consideredas closed-shell cores interacting with the alkali earth valence

Received: October 19, 2015Revised: January 19, 2016Published: January 19, 2016

Article

pubs.acs.org/JPCA

© 2016 American Chemical Society 747 DOI: 10.1021/acs.jpca.5b10089J. Phys. Chem. A 2016, 120, 747−753

electron through semilocal pseudopotentials. This has reducedMg+He to one electron−core and core−core interactions. Theground state Mg2+−He core−core potential energy wasperformed at the CCSD(T) level using the MOLPRO programsuite.30 We produced accurate ground-state potential inter-actions that we interpolated to an analytical function for abetter description of the potential at short, intermediate, andlong-range internuclear distances. Section 2 is devoted to thepresentation of the theoretical method used in our calculations.The results are presented and discussed in section 3. Finally, weconclude.

2. METHOD OF CALCULATION

Our approach follows closely the model used in our previousworks21,22 for the alkali−rare gas systems. In this model, thetotal potential is the sum of three contributions: the core−coreinteractions, the interaction between the valence electron andthe ionic system Mg2+He, and finally the spin−orbit interaction.However, the Mg2+He ground-state potential energy inter-actions are calculated using the coupled cluster theory asincorporated in the Molpro program suite.30

2.1. Mg2+He Energy Potential. The Mg2+He ground-statepotential energy is performed using the coupled cluster single,double and triple excitation (CCSD(T)) theory as imple-mented in the Molpro program suite.30 Standard aug-cc-pvqzand aug-cc-pv5z basis sets are used for the Mg and the Heatoms, respectively. The ground-state spectroscopic constantsare extracted and summarized in Table 1. The better agreementis obtained with the spectroscopic constants found by ref 26.This good agreement is observed for the equilibrium distanceas well as for the well depth. Our equilibrium distance, Re, of1.921 au compares well with 1.910 au of ref 43. Similarly, ourbinding energy of 3104.56 × 10−4 eV is very close to that of ref43 of 3161.6 × 10−4 eV. In contrast, the spectroscopicconstants of ref 28 are underestimated for Re and overestimatedfor De. It is clearly stated that the doubly charged Mg2+Hesystem presents a real chemical bond. The accuracy of theMg2+He ground-state potential energy is crucial for the study ofthe Mg+He and MgHe. The accuracy of all electronic states ofthe latter will depend on it. For a better representation of thedoubly charged ground state, we calculated the potential energyfor a wide and dense range of internuclear distances. Inaddition, to improve the representation of the potential in theregion of interest for the Mg+He simply charged system, theMg2+He potential energy curve was fitted using the analyticalform of Tang and Toennies.31 This potential is the sum of threeterms: (I) an exponential short-range repulsion Aeff exp(−bR)(with A = 61.5944 au and b = 2.18934 au), (II) a long-range−D6R

−6 − D8R−8 − D10R

−10 attractive term (with D6 = 94.0358au, D8 = −355.28 au, and D10 = 423.507 au), and (III) thehelium polarization contribution −(1/2)αHeR

−4 (with αHe =1.3834 a0

332).

The parameters Aeff, b, D6, D8, and D10 were obtained byleast-squares fitting using the numerical Mg2+He potentialcalculated at the CCSD(T) level. Figure 1 shows the numerical

potential of Mg2+He compared to the analytical potential. Thedifference between the analytical and the numerical potentialsfor all internuclear distances is less than 10−5 Hartree (<2.2cm−1).

2. Electron−Mg2+He Interaction. For the Ve−Mg2+He

interaction, we have performed a one-electron ab initio SCFcalculation where the [Mg2+] core and the electron−He effectshave been replaced by a semilocal pseudopotential. In this case,we have interpolated the numerical pseudopotential given byref 33 using the analytical form according to

∑α= −=

W r r C r( ) exp( )li

n

in2

1

i

(1)

In Figures 2 and 3, we compare our pseudopotential resultswith ref 33 for each symmetry. A good agreement is observedbetween our pseudopotential and that of ref 33 for s and psymmetries.The core polarization pseudopotentials are incorporated

using the l-dependent formulation of Foucrault et al.34 that

Table 1. Spectroscopic Constants for Mg2+He Calculated at the CCSD(T)

Re(Å) De(cm−1) D0(cm

−1) ωe(cm−1) ωeχe(cm

−1) Be(cm−1) ref

1.921 2504 436.65 30.73 1.3297191.885 2752.7 2526.8 462.3 21.1 1.38 281.910 2550 441 261.910 2144 434 272.054 441.909 44

Figure 1. Comparison between the analytical and CCSD(T)potentials for Mg2+He.

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generalizes the initial method of Muller and Meyer.35 Theyaccount for the polarization of the alkali earth ionic cores aswell as that of the helium atom considered as a whole.36 Foreach atom (λ = Mg2+ or He), the core polarization effects aredescribed by the following effective potential

∑ α= − λ

λ λ λW f f12CPP

(2)

where αλ is the electric dipole polarizability of the core λ and fλis the electric field created at the center λ produced by thevalence electrons and all other cores. The latter is modified bythe l-dependent cutoff function F as defined in ref 34 by

∑ ∑ρ ρ λ λ= | ⟩⟨ |λ λ λ λ=

∞

=−

+

F r F r lm lm( , ) ( , )il m l

l

l il

0 (3)

where |lmλ⟩ is the spherical harmonic centered on λ and thecutoff Fl function is written as

ρρ

ρ=

<

>λ λλ λ

λ λ⎪

⎪⎧⎨⎩

F rr

r( , )

0

1l il i

i (4)

The electric dipole polarizabilities are taken to be equal to1.3834 a0

3 for He32 and 0.46904 a03 for Mg(2+)37 cores.

For the magnesium atom, the cutoff radii (see Table 2) wereoptimized in order to reproduce the IPs and the lowest valence

s, p, and d one-electron states obtained from atomic datatables.37 We have also used a large 7s5p4d3f uncontractedGaussian basis set optimized previously in our group.38 Theionization potential (IP) and the energy differences betweenthe ground state and lowest excited energy levels for the Mg+

ionic atom are determined using the optimized pseudopoten-tials and basis set. They are listed in Table 3 and compared with

the experimental data.39 Good agreement is observed between

our theoretical values and the experimental ones. For the He,

we used an uncontracted 3s/3p basis set.40 The use of a basis

set on the He atom is necessary to treat correctly the steric

distortion of the Mg+; valence electron orbitals resulting from

their orthogonality to the rare gas closed shells are represented

via the pseudopotential. Because there is not an active electron

on the He atom, the exponents were determined in order to

provide correct overlaps with the 3s and 3p orbitals of He and

to extend toward the diffuse range.41

2.3. Spin−Orbit Coupling. The spin−orbit interaction

Hamiltonian is given by

ξ= · V L SS.O. (5)

It is evaluated in this study using the semiempirical scheme of

Cohen and Schneider.42 The spin−orbit coupling for the

electronic states dissociating into 3p and 4p is given by the

eigenvalues of the matrix

Figure 2. Comparison between pseudopotentials of Czuchaj et al.33

(red straight line) and this work (black dashed line) for s symmetry ofthe e−He interaction.

Figure 3. Comparison between pseudopotentials of Czuchaj et al.33

(red straight line) and this work (black dashed line) for p symmetry ofthe e−He interaction.

Table 2. Cutoff Radii for Mg and He Atoms (a0 units)

Mg He

ρλ0 1.355 3.2914

ρλ1 1.389 0.8312

ρλ2 1.393 1.4

Table 3. Calculated and Observed IPs and AtomicTransitions for Atomic Mg+

Mg+ this work (cm−1) exptl (cm−1) ΔE (cm−1)

IP(3s) 121267.62 121267.62 0.003s−3p 35715.17 35715.17 0.003s−4s 69806.66 69805.12 1.533s−3d 71490.69 71490.69 0.003s−4p 80634.90 80634.90 0.003s−5s 92819.05 92790.52 28.533s−4d 93229.47 93310.90 81.43

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749

ξ ξ

ξ

ξ

Π −

Σ

Π +

+

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

E

E

E

( )12

22

0

22

( ) 0

0 0 ( )12

p2

p2

p2

(6)

where Ep(2Π) and Ep(2Σ) are the spin-free energies and ξ is the

spin−orbit coupling constant. The solutions provide thesplitting energies related to the atomic limits S1/2, P1/2, andP3/2. The following values of the spin−orbit coupling constantsare used in the present calculation:43 ξ3p(Mg+) = 91.6 cm−1,ξ4p(Mg+) = 30.51 cm−1. The spin−orbit interaction was notconsidered for the Mg+(3d) dissociation limit due to the smallspin−orbit constant associated with this configuration.

3. RESULTS

3.1. Potential Energy Curves and SpectroscopicConstant. There are no experimental results for the Mg+Heionic dimer, but several theoretical studies have beenperformed.25−30 The potential energy curves of the Mg+Hedissociating into Mg+(3s, 3p, 4s, 3d, 4p, 5s, 4d) + He are shownin Figure 4. The spectroscopic constants are presented andcompared with the available theoretical works25−30 in Table 4.To verify the accuracy of our calculated potential energy

curves, we have derived their spectroscopic constants Re, De, Te,ωe, ωeχe, and Be and compared them with the availabletheoretical works25−30 (see Table 4). A very good agreement isobserved between our spectroscopic constants and most of thetheoretical studies. For example, we found for the ground statean equilibrium distance of Re = 6.55 Å, in excellent agreementwith that of Leung et al.26 of 6.58 Å. The RCCD(T)/daVQZcalculation (restricted coupled cluster single, double and tripleexcitations using a dAug-cc-pVQZ basis set) of Adrian et al.28

also yielded values in very good agreement with the presentones for Re, De, and ωe. We found De = 48.42 cm−1, ωe = 7.34cm−1, and Be= 0.407 cm−1, and they found De = 45.8 cm−1, ωe= 7.68 cm−1, and Be = 0.412 cm−1.Re= 3.61 a. u., De= 2233 cm−1, Te = 33537 cm−1, ωe = 438.64

cm−1 and Be = 1.34 cm−1 are the molecular constants for theA2Π state when spin−orbit coupling is not included. Thesevalues are in good agreement with the theoretical ab initiocalculations of Leung et al.26 (Re= 3.51 a. u, De= 2263 cm−1 andωe = 468 cm−1).When spin−orbit effect is incorporated using the spin−orbit

constant ξ3p = 91.6 cm−1, a significant splitting in well depth De(45 cm−1) and excitation energy Te (92 cm

−1) are observed. Incontrast, the equilibrium distance and the vibrational constantsare slightly affected. The new De for the A2Π1/2 and A2Π3/2molecular states are respectively 2188 and 2233 cm−1,respectively. Our spectroscopic data for the A2Π3/2 state arein good agreement with the theoretical work of Leung et al.26

We obtained Re = 3.61 Å, De = 2233 cm−1 and ωe = 438.64cm−1 to be compared with those of Leung et al.26 Re = 3.51 Å,De = 2263 cm−1 and ωe= 468 cm−1). This excellent accordprovides assurance in our used theoretical approach.The spectroscopic constants of the B2Σ+ state are also

affected by the spin−orbit coupling through its interaction withthe A2Π state (Re= 3.61 a. u., De = 2233 cm−1, Te= 33537 cm−1,ωe = 438.64 cm−1, ωe χe = 21.31 cm−1 and Be = 1.340 cm−1).Figure 5 presents the potential energy of the Mg+He ion with

spin−orbit interactions for states dissociating into Mg+ (32P1/2and 32P3/2) + He. The spin−orbit effect is also included, for thefirst time, for the higher excited states of dissociation limit Mg+

(4p) + He. We used the spin−orbit constant ξ4p = 30.51 cm−1.The potential interactions with spin−orbit effect, for the states32Π1/2 (4

2p1/2) and 32Π3/2 (42p3/2), are presented in Figure 6.

The spin orbit coupling has introduced two electronic states32Π1/2 and 32Π3/2, for which the molecular energy andtransition energy splitting are 15 and 31 cm−1, and their welldepths are, respectively, 2594 and 2609 cm−1. A slight change isobserved for the vibrational constant and the equilibriumdistance.We observe from the derived spectroscopic constants and the

figure presenting the potential energy curves that the excited2Σ+ states are mostly repulsive and some of them have a barriersuch as 42Σ+ and 52Σ+ states. This comportment can be linkedto the interaction between the neutral helium and the Rydbergelectron. It can be explained, for the 2Σ+ symmetry, by thesignificant penetration of the helium potential in the Mg+ pRydberg orbital projected on the Z molecular axis. However,for the 2Π and 2Δsymmetries, this penetration is insignificant asit happens with the off-axis Rydberg orbitals. For that reason,their potential energy curves are similar to that of Mg2+He (Re= 3.56 a. u and De = 2752 cm−1) ground state. The 42Σ+ statepresents a very interesting profile. It has a low minimumfollowed by an extended barrier that will trap metastable energy

Figure 4. Potential energy curves of the electronic states dissociatinginto Mg+(3s, 3p, 4s, 3d, 4p, 5s, 4d) + He.

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levels. It is expected that these levels or states will exhibit longlifetimes due to the slow process of tunneling from side to sideof such barrier. This unusual shape is associated with an

Table 4. Spectroscopic Constants of the Electronic States of the Mg+He System

state Re(au) De(cm−1) Te(cm

−1) ωe(cm−1) ωexe(cm

−1) Be(cm−1) ref

X2Σ+ (3s) 6.55 54 48.42 7.34 0.4076.58 73.2 45.8 7.68 0.412 286.74 70 21 256.72 65 44 296.96 76.6 42 306.82 30 44 27

A2Π(3p) 3.61 2233 33537 438.64 21.31 1.340A2Π1/2(3p) 3.61 2188 33505 442.2 24.58 1.342A2Π3/2(3p) 3.61 2233 33537 438.64 21.31 1.340

3.51 2263 468 26B2Σ+(3p) repulsiveB2Σ1/2

+(3p) repulsive32Σ+(4s) 3.52 3070 66791 501.16 17.56 1.41542Σ+(3d) 3.37 −4988 76533 598.24 29.86 1.54522Π(3d) 3.47 3781 67764 536.32 18.79 1.45512Δ(3d) 3.63 2332 69213 421.20 20.04 1.33052Σ+(4p) 4.25 −3483 84172 246.28 5.21 0.969152Σ1/2

+(4p) 4.24 −3468 84187 245.11 7.34 0.973232Π(4p) 3.58 2609 78080 456.76 20.91 1.36832Π1/2(4p) 3.58 2594 78079 460.28 24.41 1.36932Π3/2(4p) 3.58 2609 78110 460.33 24.54 1.369162Σ+(5s) 3.57 2984 89889 472.28 19.9 1.37772Σ+(4d) 5.17 642 92642 305.87 36.43 0.18342Π(4d) 3.50 3252 90031 510.38 19.99 1.43222Δ(4d) 3.60 2557 90727 453.46 31.87 1.347

Figure 5. Mg+He potential energy curves including spin−orbitcoupling dissociating into Mg+(3 2P1/2) + He and Mg+ (32P3/2) +He: Full lines, states including spin−orbit coupling, dashed line, A2Πand B2Σ+ states without spin−orbit coupling.

Figure 6. Mg+He potential energy curves including spin−orbitcoupling dissociating into Mg+(4 2P1/2) + He and Mg+(42P3/2): Fulllines, states including spin−orbit coupling, dashed line, 32Π and 52Σ+

states without spin−orbit coupling.

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avoided crossing between the state dissociating into 3d, whichis repulsive, and the state dissociating into 4p, which is intenselybound. The equilibrium distance of the 52Σ+ state is shifted tolarger distance due to this avoided crossing.3.2. Transition Dipole Moments. The transition dipole

functions from the X2Σ+ state to several excited states and fromlower excited states to higher excited states were performed fora better understanding and interpretation. In Figure 7, we

present the transition dipoles from the ground state to the A2Πand B2Σ+ excited states. We note that the X2Σ+−B2Σ+ transitiondipole exhibits a maximum of 1.558 au at an internuclearlocation of 7 au. At the asymptotic limits, the X2Σ+−A2Π andX2Σ+−B2Σ+ transitions tend to the Mg+(3s)−Mg+(3p) pureatomic transition. The transition dipole function including thespin−orbit interaction is also performed and presented in thesame figure (Figure 7). This effect is introduced using therotational matrix issued from the diagonalization of the energydiscussed previously. As we can see, the X2Σ+− A2Π transitionmoment is split into two curves according to the X2Σ+−A2Π1/2and X2Σ+−A2Π3/2 transitions. The difference between the splittransition dipole moments is remarkable at intermediatedistances. However, at large internuclear distances, the X2Σ+−B2Σ+, X2Σ+−A2Π1/2, and X2Σ+−A2Π3/2 transition dipolesconverge to the same limit constant, which is the pure atomictransition between Mg+(3s)−Mg+(3p).

■ CONCLUSIONA one-electron + CPP (core pseudopotential) calculations forthe Mg+He van der Waals system was performed using theeffective potential approach. The Mg2+ core and the electron−He interactions were treated using semilocal pseudopotentials,which reduced the ion to only one valence electron, for which

an SCF calculation is sufficient. In addition and due to the onlyone valence electron, large Gaussian basis sets for Mg and Heatoms were used. The Mg2+He core−core interactions weredetermined at the CCSD(T) level of theory using the Molproprogram. The spectroscopic constants of the Mg2+He groundstate were derived and compared with the available studies tovalidate its accuracy and its use in the modeling of the Mg+Hesystem. For a better description of the Mg2+He interactions atintermediate distances where the minimum of several states ofMg+He are located, this potential was fitted to the Tang andToennies31 analytical expression. In addition, the spin−orbiteffect was incorporated using the semiempirical approach ofCohen and Schneider.42 As a result, a splitting in energy,transition energy, and transition dipole moment and aninsignificant shift in the equilibrium distances of some stateswere observed. A comparison between our results and theprevious studies,25−30 especially for the X2Σ+ and the A2Πstates, has shown good agreement. The accurate data producedfor the ground state of the Mg2+He and Mg+He ionic dimerswill be used to generalize structural and dynamical studies oflarge Mg2+Hen and Mg+He clusters and immersion of the Mg2+

and Mg+ ions into helium nanodroplets. However, the results ofthe excited states of the Mg+He dimer will be used toinvestigate the broadening effect of the Mg+ spectrum bycollision with the He gas. This will stimulate experimentalinvestigations for these dimers and clusters.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: hamidberriche@yahoo.fr or hamid.berriche@aurak.ac.ae. Tel: +971561566046.NotesThe authors declare no competing financial interest.

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