Article
pubs.acs.org/JPCA
Theoretical Investigation of the Electronic Structure and Spectra of
Mg2+He and Mg+He
M. 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 de
L’Environnment, 5019 Monastir, Tunisia
‡
Department of Mathematics and Natural Sciences, School of Arts and Sciences, American University of Ras Al Khaimah, Ras Al
Khaimah, P.O. Box, RAK, United Arab Emirates
ABSTRACT: The ground and many excited states of the Mg+He van
der Waals molecular system have been explored using a one-electron
pseudopotential approach. In this approach, effective potentials are used
to consider the Mg2+ core and the electron−He effects. Furthermore, a
core−core interaction is included. This has reduced the number of active
electrons of the Mg+He, to be considered in the calculation, to a single
valence electron. This has permitted to use extended Gaussian basis sets
for Mg and He. Therefore, the potentianl energy and dipole moments
calculations are carried out at the Hartree−Fock level of theory, and the
spin−orbit effect is included using a semiclassical approach. The core−
core interaction for the Mg2+He ground state is included using accurate
CCSD(T) calculations. The spectroscopic constants of the Mg+He electronic states are extracted and compared with the existing
theoretical works, where very good agreement is observed. Moreover, the transition dipole function has been determined for a
large and dense grid of internuclear distances including the spin−orbit effect.
■
INTRODUCTION
On the experimental side, alkaline earth−rare gas open-shell
ionic complexes were produced in a pulsed free-jet expansion
and detected by a time-of-flight (TOF) mass spectrometer or
via laser-induced fluorescence (LIF).23,24 Unfortunately, there
are no experimental studies for Mg+He; however, several
theoretical investigations have been performed.25−29 Partridge
et al.25 made the first theoretical study. They determined the
spectroscopic constants for the ground and some exited states.
In their work, they compared many metal−noble gas ions by
modifying a coupled functional level of electronic correlation
treatment. Leung et al.26 have estimated the potential energy
curves of Mg+He by MP2, MP3, and MP4 ab initio calculations.
In a recent study, Bu et al.27 have investigated the physical and
chemical proprieties and the geometric structures of Mg+Hen
and Mg2+He (n = 1−10) using the MP2 method. Adrian et al.28
have presented high-level calculation of the potential energy
curves of the Mn+−Rg (n = 1 and 2) complexes, and they have
extracted their spectroscopic constants.
This paper presents a theoretical study of the Mg+He
structure (potential energy, spectroscopic constant, and dipole
functions) for many electronic states of different symmetries.
The calculation method is based on a pseudopotential
technique, which reduces the Mg+He ionic dimer to only one
active electron system. Therefore, Mg2+ and He are considered
as closed-shell cores interacting with the alkali earth valence
Great attention at both experimental and theoretical levels has
been emphasized in recent times on the alkaline earth and
alkali−rare gas dimers. They present remarkable and interesting
traits, which can be used for several technology applications.
Due to the closed shells of atoms and/or ions that form these
systems, their ground state presents very weak interactions. In
contrast, considerable bonding can exist in the excited states.
Several alkaline earth or alkali−rare gas systems were studied in
the past using different techniques and methods. The most
used methods are semiempirical,1−6 model potential,7−15 and
pseudopotential16−22 approaches. The shape of the potential
surface due to different types of intermolecular interactions is
important for many problems in molecular dynamics, such as
the modeling of physical adsorption and desorption processes
and collisional processes in general.
van der Waals interaction between neutral molecules has
been extensively studied in the past few decades and is often
described by a simple model that combines an empirical
repulsive inner wall and a long-range attractive interaction. This
attractive part of the potential is due to the interaction between
the dipole, induced dipole, quadrupole, and sometimes higherorder electric multipole moments of the two atoms. A second
important kind of intermolecular interaction is that between a
neutral atom and an ion. Such an interaction may lead to the
formation of a relatively strongly bound complex with a bond
that approaches a chemical bond, particularly in cases when one
of the participating particles is an open-shell species
© 2016 American Chemical Society
Received: October 19, 2015
Revised: January 19, 2016
Published: January 19, 2016
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DOI: 10.1021/acs.jpca.5b10089
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The Journal of Physical Chemistry A
Table 1. Spectroscopic Constants for Mg2+He Calculated at the CCSD(T)
Re(Å)
De(cm−1)
1.921
1.885
1.910
1.910
2.054
1.909
2504
2752.7
2550
2144
D0(cm−1)
ωe(cm−1)
ωeχe(cm−1)
Be(cm−1)
436.65
462.3
441
434
30.73
21.1
1.329719
1.38
2526.8
ref
28
26
27
44
44
The parameters Aeff, b, D6, D8, and D10 were obtained by
least-squares fitting using the numerical Mg2+He potential
calculated at the CCSD(T) level. Figure 1 shows the numerical
electron through semilocal pseudopotentials. This has reduced
Mg+He to one electron−core and core−core interactions. The
ground state Mg2+−He core−core potential energy was
performed at the CCSD(T) level using the MOLPRO program
suite.30 We produced accurate ground-state potential interactions that we interpolated to an analytical function for a
better description of the potential at short, intermediate, and
long-range internuclear distances. Section 2 is devoted to the
presentation of the theoretical method used in our calculations.
The results are presented and discussed in section 3. Finally, we
conclude.
2. METHOD OF CALCULATION
Our approach follows closely the model used in our previous
works21,22 for the alkali−rare gas systems. In this model, the
total potential is the sum of three contributions: the core−core
interactions, the interaction between the valence electron and
the ionic system Mg2+He, and finally the spin−orbit interaction.
However, the Mg2+He ground-state potential energy interactions are calculated using the coupled cluster theory as
incorporated in the Molpro program suite.30
2.1. Mg2+He Energy Potential. The Mg2+He ground-state
potential energy is performed using the coupled cluster single,
double and triple excitation (CCSD(T)) theory as implemented in the Molpro program suite.30 Standard aug-cc-pvqz
and aug-cc-pv5z basis sets are used for the Mg and the He
atoms, respectively. The ground-state spectroscopic constants
are extracted and summarized in Table 1. The better agreement
is obtained with the spectroscopic constants found by ref 26.
This good agreement is observed for the equilibrium distance
as well as for the well depth. Our equilibrium distance, Re, of
1.921 au compares well with 1.910 au of ref 43. Similarly, our
binding energy of 3104.56 × 10−4 eV is very close to that of ref
43 of 3161.6 × 10−4 eV. In contrast, the spectroscopic
constants of ref 28 are underestimated for Re and overestimated
for De. It is clearly stated that the doubly charged Mg2+He
system presents a real chemical bond. The accuracy of the
Mg2+He ground-state potential energy is crucial for the study of
the Mg+He and MgHe. The accuracy of all electronic states of
the latter will depend on it. For a better representation of the
doubly charged ground state, we calculated the potential energy
for a wide and dense range of internuclear distances. In
addition, to improve the representation of the potential in the
region of interest for the Mg+He simply charged system, the
Mg2+He potential energy curve was fitted using the analytical
form of Tang and Toennies.31 This potential is the sum of three
terms: (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 = 94.0358
au, D8 = −355.28 au, and D10 = 423.507 au), and (III) the
helium polarization contribution −(1/2)αHeR−4 (with αHe =
1.3834 a3032).
Figure 1. Comparison between the analytical and CCSD(T)
potentials for Mg2+He.
potential of Mg2+He compared to the analytical potential. The
difference between the analytical and the numerical potentials
for all internuclear distances is less than 10−5 Hartree (<2.2
cm−1).
2. Electron−Mg2+He Interaction. For the Ve−Mg2+He
interaction, we have performed a one-electron ab initio SCF
calculation where the [Mg2+] core and the electron−He effects
have been replaced by a semilocal pseudopotential. In this case,
we have interpolated the numerical pseudopotential given by
ref 33 using the analytical form according to
n
Wl (r ) = exp( −αr 2)∑ Cir ni
i=1
(1)
In Figures 2 and 3, we compare our pseudopotential results
with ref 33 for each symmetry. A good agreement is observed
between our pseudopotential and that of ref 33 for s and p
symmetries.
The core polarization pseudopotentials are incorporated
using the l-dependent formulation of Foucrault et al.34 that
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The Journal of Physical Chemistry A
where |lmλ⟩ is the spherical harmonic centered on λ and the
cutoff Fl function is written as
⎧
0
⎪
Fl(riλ , ρλl ) = ⎨
⎪
⎩1
riλ < ρλ
riλ > ρλ
(4)
The electric dipole polarizabilities are taken to be equal to
1.3834 a30 for He32 and 0.46904 a30 for Mg(2+)37 cores.
For the magnesium atom, the cutoff radii (see Table 2) were
optimized in order to reproduce the IPs and the lowest valence
Table 2. Cutoff Radii for Mg and He Atoms (a0 units)
ρ0λ
ρ1λ
ρ2λ
Mg
He
1.355
1.389
1.393
3.2914
0.8312
1.4
s, p, and d one-electron states obtained from atomic data
tables.37 We have also used a large 7s5p4d3f uncontracted
Gaussian basis set optimized previously in our group.38 The
ionization potential (IP) and the energy differences between
the ground state and lowest excited energy levels for the Mg+
ionic atom are determined using the optimized pseudopotentials and basis set. They are listed in Table 3 and compared with
Figure 2. Comparison between pseudopotentials of Czuchaj et al.33
(red straight line) and this work (black dashed line) for s symmetry of
the e−He interaction.
Table 3. Calculated and Observed IPs and Atomic
Transitions for Atomic Mg+
generalizes the initial method of Müller and Meyer.35 They
account for the polarization of the alkali earth ionic cores as
well as that of the helium atom considered as a whole.36 For
each atom (λ = Mg2+ or He), the core polarization effects are
described by the following effective potential
1
WCPP = − ∑ αλfλ⃗ fλ⃗
2 λ
(2)
where αλ is the electric dipole polarizability of the core λ and fλ⃗
is the electric field created at the center λ produced by the
valence electrons and all other cores. The latter is modified by
the l-dependent cutoff function F as defined in ref 34 by
∞
l = 0 m =−l
Fl(riλ , ρλl )|lmλ⟩⟨lmλ|
exptl (cm−1)
ΔE (cm−1)
IP(3s)
3s−3p
3s−4s
3s−3d
3s−4p
3s−5s
3s−4d
121267.62
35715.17
69806.66
71490.69
80634.90
92819.05
93229.47
121267.62
35715.17
69805.12
71490.69
80634.90
92790.52
93310.90
0.00
0.00
1.53
0.00
0.00
28.53
81.43
VS.O. = ξL⃗ ·S ⃗
(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
+l
∑∑
this work (cm−1)
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
Figure 3. Comparison between pseudopotentials of Czuchaj et al.33
(red straight line) and this work (black dashed line) for p symmetry of
the e−He interaction.
F(riλ , ρλ ) =
Mg+
(3)
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⎡
⎤
1
2
ξ
0
⎢ Ep(2Π) − ξ
⎥
2
2
⎢
⎥
⎢
⎥
2
+
⎢
⎥
0
ξ
Ep(2Σ )
2
⎢
⎥
⎢
1 ⎥
⎢
0
0
Ep(2Π) + ξ ⎥
⎣
2 ⎦
(6)
where Ep(2Π) and Ep(2Σ) are the spin-free energies and ξ is the
spin−orbit coupling constant. The solutions provide the
splitting energies related to the atomic limits S1/2, P1/2, and
P3/2. The following values of the spin−orbit coupling constants
are used in the present calculation:43 ξ3p(Mg+) = 91.6 cm−1,
ξ4p(Mg+) = 30.51 cm−1. The spin−orbit interaction was not
considered for the Mg+(3d) dissociation limit due to the small
spin−orbit constant associated with this configuration.
3. RESULTS
3.1. Potential Energy Curves and Spectroscopic
Constant. There are no experimental results for the Mg+He
ionic dimer, but several theoretical studies have been
performed.25−30 The potential energy curves of the Mg+He
dissociating into Mg+(3s, 3p, 4s, 3d, 4p, 5s, 4d) + He are shown
in Figure 4. The spectroscopic constants are presented and
compared 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 available
theoretical works25−30 (see Table 4). A very good agreement is
observed between our spectroscopic constants and most of the
theoretical studies. For example, we found for the ground state
an equilibrium distance of Re = 6.55 Å, in excellent agreement
with that of Leung et al.26 of 6.58 Å. The RCCD(T)/daVQZ
calculation (restricted coupled cluster single, double and triple
excitations using a dAug-cc-pVQZ basis set) of Adrian et al.28
also yielded values in very good agreement with the present
ones for Re, De, and ωe. We found De = 48.42 cm−1, ωe = 7.34
cm−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 the
A2Π state when spin−orbit coupling is not included. These
values are in good agreement with the theoretical ab initio
calculations of Leung et al.26 (Re= 3.51 a. u, De= 2263 cm−1 and
ωe = 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. In
contrast, the equilibrium distance and the vibrational constants
are slightly affected. The new De for the A2Π1/2 and A2Π3/2
molecular states are respectively 2188 and 2233 cm−1,
respectively. Our spectroscopic data for the A2Π3/2 state are
in good agreement with the theoretical work of Leung et al.26
We obtained Re = 3.61 Å, De = 2233 cm−1 and ωe = 438.64
cm−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 accord
provides assurance in our used theoretical approach.
The spectroscopic constants of the B2Σ+ state are also
affected by the spin−orbit coupling through its interaction with
the 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
Figure 4. Potential energy curves of the electronic states dissociating
into Mg+(3s, 3p, 4s, 3d, 4p, 5s, 4d) + He.
spin−orbit interactions for states dissociating into Mg+ (32P1/2
and 32P3/2) + He. The spin−orbit effect is also included, for the
first 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 states
32Π1/2 (42p1/2) and 32Π3/2 (42p3/2), are presented in Figure 6.
The spin orbit coupling has introduced two electronic states
32Π1/2 and 32Π3/2, for which the molecular energy and
transition energy splitting are 15 and 31 cm−1, and their well
depths are, respectively, 2594 and 2609 cm−1. A slight change is
observed for the vibrational constant and the equilibrium
distance.
We observe from the derived spectroscopic constants and the
figure presenting the potential energy curves that the excited
2 +
Σ states are mostly repulsive and some of them have a barrier
such as 42Σ+ and 52Σ+ states. This comportment can be linked
to the interaction between the neutral helium and the Rydberg
electron. It can be explained, for the 2Σ+ symmetry, by the
significant penetration of the helium potential in the Mg+ p
Rydberg orbital projected on the Z molecular axis. However,
for the 2Π and 2Δsymmetries, this penetration is insignificant as
it 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+ state
presents a very interesting profile. It has a low minimum
followed by an extended barrier that will trap metastable energy
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Table 4. Spectroscopic Constants of the Electronic States of the Mg+He System
Re(au)
state
X Σ (3s)
2 +
A2Π(3p)
A2Π1/2(3p)
A2Π3/2(3p)
B2Σ+(3p)
B2Σ1/2+(3p)
32Σ+(4s)
42Σ+(3d)
22Π(3d)
12Δ(3d)
52Σ+(4p)
52Σ1/2+(4p)
32Π(4p)
32Π1/2(4p)
32Π3/2(4p)
62Σ+(5s)
72Σ+(4d)
42Π(4d)
22Δ(4d)
6.55
6.58
6.74
6.72
6.96
6.82
3.61
3.61
3.61
3.51
repulsive
repulsive
3.52
3.37
3.47
3.63
4.25
4.24
3.58
3.58
3.58
3.57
5.17
3.50
3.60
De(cm−1)
54
73.2
70
65
76.6
30
2233
2188
2233
2263
3070
−4988
3781
2332
−3483
−3468
2609
2594
2609
2984
642
3252
2557
Te(cm−1)
33537
33505
33537
66791
76533
67764
69213
84172
84187
78080
78079
78110
89889
92642
90031
90727
ωe(cm−1)
ωexe(cm−1)
Be(cm−1)
48.42
45.8
21
44
42
44
438.64
442.2
438.64
468
7.34
7.68
0.407
0.412
21.31
24.58
21.31
1.340
1.342
1.340
501.16
598.24
536.32
421.20
246.28
245.11
456.76
460.28
460.33
472.28
305.87
510.38
453.46
17.56
29.86
18.79
20.04
5.21
7.34
20.91
24.41
24.54
19.9
36.43
19.99
31.87
ref
28
25
29
30
27
26
1.415
1.545
1.455
1.330
0.9691
0.9732
1.368
1.369
1.3691
1.377
0.183
1.432
1.347
Figure 6. Mg+He potential energy curves including spin−orbit
coupling dissociating into Mg+(4 2P1/2) + He and Mg+(42P3/2): Full
lines, states including spin−orbit coupling, dashed line, 32Π and 52Σ+
states without spin−orbit coupling.
+
Figure 5. Mg He potential energy curves including spin−orbit
coupling dissociating into Mg+(3 2P1/2) + He and Mg+ (32P3/2) +
He: Full lines, states including spin−orbit coupling, dashed line, A2Π
and B2Σ+ states without spin−orbit coupling.
levels. It is expected that these levels or states will exhibit long
lifetimes due to the slow process of tunneling from side to side
of such barrier. This unusual shape is associated with an
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an SCF calculation is sufficient. In addition and due to the only
one valence electron, large Gaussian basis sets for Mg and He
atoms were used. The Mg2+He core−core interactions were
determined at the CCSD(T) level of theory using the Molpro
program. The spectroscopic constants of the Mg2+He ground
state were derived and compared with the available studies to
validate its accuracy and its use in the modeling of the Mg+He
system. For a better description of the Mg2+He interactions at
intermediate distances where the minimum of several states of
Mg+He are located, this potential was fitted to the Tang and
Toennies31 analytical expression. In addition, the spin−orbit
effect was incorporated using the semiempirical approach of
Cohen and Schneider.42 As a result, a splitting in energy,
transition energy, and transition dipole moment and an
insignificant shift in the equilibrium distances of some states
were observed. A comparison between our results and the
previous studies,25−30 especially for the X2Σ+ and the A2Π
states, has shown good agreement. The accurate data produced
for the ground state of the Mg2+He and Mg+He ionic dimers
will be used to generalize structural and dynamical studies of
large Mg2+Hen and Mg+He clusters and immersion of the Mg2+
and Mg+ ions into helium nanodroplets. However, the results of
the excited states of the Mg+He dimer will be used to
investigate the broadening effect of the Mg+ spectrum by
collision with the He gas. This will stimulate experimental
investigations for these dimers and clusters.
avoided crossing between the state dissociating into 3d, which
is repulsive, and the state dissociating into 4p, which is intensely
bound. The equilibrium distance of the 52Σ+ state is shifted to
larger distance due to this avoided crossing.
3.2. Transition Dipole Moments. The transition dipole
functions from the X2Σ+ state to several excited states and from
lower excited states to higher excited states were performed for
a better understanding and interpretation. In Figure 7, we
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected] or [email protected].
ae. Tel: +971561566046.
Notes
The authors declare no competing financial interest.
■
Figure 7. Transition dipole moments X2Σ+−A2Π, X2Σ+−B2Σ+, X2Σ+−
A2Π1/2, X2Σ+−A2Π3/2, and X2Σ+−B X2Σ+1/2 as a function of the
internuclear distance R. The inset shows the behavior of all of the
TDMs around D∞= 1.65 au.
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present the transition dipoles from the ground state to the A2Π
and B2Σ+ excited states. We note that the X2Σ+−B2Σ+ transition
dipole exhibits a maximum of 1.558 au at an internuclear
location of 7 au. At the asymptotic limits, the X2Σ+−A2Π and
X2Σ+−B2Σ+ transitions tend to the Mg+(3s)−Mg+(3p) pure
atomic transition. The transition dipole function including the
spin−orbit interaction is also performed and presented in the
same figure (Figure 7). This effect is introduced using the
rotational matrix issued from the diagonalization of the energy
discussed previously. As we can see, the X2Σ+− A2Π transition
moment is split into two curves according to the X2Σ+−A2Π1/2
and X2Σ+−A2Π3/2 transitions. The difference between the split
transition dipole moments is remarkable at intermediate
distances. However, at large internuclear distances, the X2Σ+−
B2Σ+, X2Σ+−A2Π1/2, and X2Σ+−A2Π3/2 transition dipoles
converge to the same limit constant, which is the pure atomic
transition between Mg+(3s)−Mg+(3p).
■
CONCLUSION
A one-electron + CPP (core pseudopotential) calculations for
the Mg+He van der Waals system was performed using the
effective 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
752
DOI: 10.1021/acs.jpca.5b10089
J. Phys. Chem. A 2016, 120, 747−753
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