791
ARTICLE
Adiabatic and quasi-diabatic investigation of the strontium
hydride cation SrH+: structure, spectroscopy, and dipole
moments
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Sana Belayouni, Chedli Ghanmi, and Hamid Berriche
Abstract: Ab initio investigation has been performed for the strontium hydride cation SrH + using a standard quantum chemistry
approach. It is based on the pseudopotentials for atomic core representations, Gaussian basis sets, as well as with full configuration
interaction calculations. A diabatisation procedure based on the effective hamiltonian theory and an effective metric is used to
produce the quasi-diabatic potential energy. Adiabatic and quasi-diabatic potential energy curves and their spectroscopic
parameters for the ground and many excited electronic states of 1,3兺+, 1,3⌸, and 1,3⌬ symmetries have been determined. Their
predicted accuracy is discussed by comparing our well depths and equilibrium positions with the available experimental and
theoretical results. Moreover, we localized and analyzed numerous avoided crossings between the electronic states of 1,3兺+ and
1,3⌸ symmetries. The correction of the electron affinity of the H atom is also considered, for the 1–101兺+ electronic states, to
improve the accuracy of the adiabatic potential energies of these states. In addition, we calculated the dipole moments, for a
wide range of internuclear distances in both diabatic and quasi-diabatic representations. The adiabatic permanent dipole
moments for the 101兺+ electronic states revealed ionic characters related to electron transfer and yields both SrH(+) and Sr(+)H
arrangements. The transition dipole moments between neighbor electronic states revealed many peaks around the avoided
crossing positions.
Key words: pseudopotentials, configuration interaction, potential energy curves, spectroscopic parameters, dipole moments.
Résumé : Nous avons complété une étude ab initio du cation hydride du strontium SrH+ en utilisant une approche de chimie
quantique standard. Elle est basée sur les pseudo-potentiels pour représenter le cœur atomique, sur des ensembles de base gaussiens
et des calculs avec interaction à pleine configuration. Nous utilisons une procédure de diabatisation basée sur la théorie du Hamiltonien effectif et une métrique effective est utilisée pour produire l’énergie potentielle quasi-diabatique. Nous déterminons les
courbes d’énergie potentielle adiabatique et quasi-diabatique pour le fondamental et plusieurs états excités des symétries 1,3兺+, 1,3⌸ et
1,3⌬. La précision de nos prédictions est analysée en comparant nos profondeurs de puits et nos positions d’équilibre avec les valeurs
théoriques et expérimentales disponibles. De plus, nous localisons et analysons les nombreux croisements évités entre les états
électroniques des symétries 1,3兺+ et 1,3⌸. Nous considérons aussi la correction de l’affinité électronique de l’atome H pour les états
1–101兺+, de façon à améliorer la précision des énergies potentielles adiabatiques de ces états. Les moments dipolaires sont calculés pour
un large domaine des distances internucléaires, dans les deux représentations, diabatique et quasi-diabatique. Les moments dipolaires
adiabatiques permanents pour les états électroniques 101兺+ révèlent des caractères ioniques reliés au transfert d’électron et donnent
les deux arrangements SrH(+) et Sr(+)H. Les moments dipolaires de transition entre états électroniques voisins révèlent plusieurs pics
autour des positions des croisements évités. [Traduit par la Rédaction]
Mots-clés : pseudo-potentiel, interaction de configuration, courbes d’énergie potentielle, paramètres spectroscopiques, moments
dipolaires.
1. Introduction
In the recent past, there has been considerable interest in the
creation of ultracold molecules at temperatures below 1 ␮K by
magneto-association [1, 2] or photo-association [3, 4], especially by
using heteronuclear molecules, such as alkali dimers, alkaline earth
hydrides, and their corresponding ions. This opened an exciting prospect to test the variation of fundamental constants on both
the experimental and theoretical scales. The precise knowledge of
the long-range interactions between two different types of alkali
atoms is necessary for the understanding and realization of cold
collision processes and formation of cold and ultra-cold heteronuclear molecules. These molecules can be produced through photoassociation of atoms or by a laser-cooled atomic vapor. For example,
several cold and ultra-cold diatomic molecule have been formed,
such as RbCs [5], KRb [6, 7], NaCs [8], and NaCs+ [9].
The literature reveals that the structural and spectroscopic
properties of the alkaline earth hydrides cations XH+ (X = Be, Mg,
Received 19 December 2015. Accepted 13 May 2016.
S. Belayouni. Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue de
l’Environnement, 5019 Monastir, Tunisia.
C. Ghanmi. Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue de
l’Environnement, 5019 Monastir, Tunisia; Physics Department, Faculty of Sciences, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia.
H. Berriche. Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue de
l’Environnement, 5019 Monastir, Tunisia; Mathematics and Natural Sciences Department, School of Arts and Sciences, American University of
Ras Al Khaimah, Ras Al Khaimah, UAE.
Corresponding author: Hamid Berriche (email: [email protected]).
Copyright remains with the author(s) or their institution(s). Permission for reuse (free in most cases) can be obtained from RightsLink.
Can. J. Phys. 94: 791–802 (2016) dx.doi.org/10.1139/cjp-2015-0801
Published at www.nrcresearchpress.com/cjp on 9 June 2016.
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792
Ca, Sr, Ba, and Ra) have been developed rapidly on both the experimental and theoretical sides. In this context, the potential energy
curves, the spectroscopic constants, and the permanent and transition dipole moments of BeH + have been studied [10–12]. Moreover,
many theoretical and experimental studies [13–17] have determined
the structure and electronic properties of the magnesium hydride
cation MgH+. Mølhave et al. [17] produced and cooled the molecular
ions MgH+ and MgD+ in a linear Paul trap. Other works studied the
molecular properties of the CaH+ ionic system [18–23]. In addition,
Allouche et al. [24] and Mejrissi et al. [25] performed a theoretical
study of the low-lying electronic states of the BaH+ ion. The strontium hydride cation SrH+ is also investigated extensively [26–32].
Experimentally, only the well depth of the ground state has been
evaluated by Dalleska et al. [26]. Theoretically, the structure and spectroscopic properties of the ground state of the strontium hydride
cation SrH+ was studied for the first time by Fuentealba et al. [27].
After that, Schilling et al. [28] successfully completed a theoretical
study for the strontium hydride cation SrH+ using a generalized valence bond plus configuration interaction calculations. They have
analyzed the trends in bond energies, equilibrium geometries,
vibrational frequencies, and metal orbital hybridizations. Recently, Abe et al. [29] realized an ab initio study on vibrational
dipole moments of XH+ where X = Mg, Ca, Zn, Sr, Cd, Ba, Yb, and
Hg. Also, in 2011 Katija et al. [30] evaluated a precise measurement of
the ⱍX1⌺nv⫽0,J⫽0,F⫽1/2,M⫽±1/2典 ¡ ⱍX1⌺nv⫽1,J⫽0,F⫽1/2,M⫽±1/2典 transition frequencies of the same molecular ion XH+. Aymar et al. [31] determined the
potential energy curves, permanent and transition dipole moments,
and the static dipole polarizabilities of SrH+, CaH+, and BaH+ systems.
They used a full configuration interaction performed in a twovalence electronic configuration space built from a large Gaussian
basis set. The structure and spectroscopic properties of the strontium hydride cation SrH+ have been calculated by Habli et al. [32].
They used an ab initio approach based on large basis sets, nonempirical atomic pseudopotential for strontium core, correlation treatment for core valence through the effective core polarization
potentials, and for the valence through full valence configuration
interaction.
Recently, the accurate data, such the potential energy curves,
the spectroscopic parameters, and the transition and permanent dipole moments, have been produced in our research group for several
diatomic systems like LiH [33], LiNa [34], CsLi [35], CsNa [36] BeH+ [12],
LiRb [37] and LiX (X = Na, K, Rb, Cs, and Fr) [38]. Our goal in this work
is to extend our previous studies [12, 33–38] and determine the adiabatic and quasi-diabatic potential energy curves, their spectroscopic
parameters, and the permanent as well as transition dipole moments of the strontium hydride cation SrH+. We hope that this study
may help to explore further the photoassociation processes. In the
following section we briefly present the computational methods. Section 3 is devoted to presenting our results, where we indicate
the adiabatic potential energy curves and their spectroscopic parameters for the 1–101,3兺+, 1–61,3⌸, and 1–61,3⌬ electronic states, the quasidiabatic potential energy curves, and finally the permanent and
transition dipole moments. Section 4 contains our conclusions
and a brief summary.
2. Methods of calculation
We used the non-empirical pseudopotential in its semi-local
form proposed by Barthelat and Durand [39], where the strontium
hydride cation SrH+ is modeled as a system with two valence electrons. In addition, we also considered the self-consistent field calculation, which is followed by a full valence configuration interaction
calculation. For the simulation of the interaction between the polarizable Sr2+ core with the valence electrons and the hydrogen nucleus,
1
Can. J. Phys. Vol. 94, 2016
a core polarization potential VCPP is used, according to the operator
formulation of Müller et al. [40]
VCPP ⫽ ⫺
1
2
兺 ␣ f ·f
␭ ␭
␭
␭
where ␣␭ represents the dipole polarizability of the core ␭ and f␭
represent the electric field created by valence electrons and all
other cores on the core ␭.
f␭ ⫽
兺 r F(r , ␳ ) ⫺ 兺 R
ri␭
i
3
i␭
R␭␭
i␭
␭
␭≠␭
3
Z␭
␭␭
where ri␭is a core–electron vector and R␭␭ is a core–core vector.
As reported by the formulation of Foucrault et al. [41], the cutoff
function F(ri␭, ␳␭) is taken to be a function of l to treat separately
the interaction of valence electrons of different spatial symmetry
and the core electrons. It has a physical meaning of excluding the
valence electrons from the core region for calculating the electric
field. In the Müller et al. [40] formalism, the cutoff function is unique
for a given atom and is generally adjusted to reproduce the atomic
energy levels for the lowest states of each symmetry.
For the strontium and hydrogen atoms, we have used a (5s5p6d1f/
5s5p3d1f) [42] and (9s5p3d/7s5p3d) [33] basis set of Gaussian-type orbitals, where diffuse orbital exponents have been optimized to
reproduce the atomic levels 1s, 2s, and 2p; 5s, 4d, 5p, 6s, 5d, 6p, 7s, 6d,
and 7p; and 5s2, 5s5p, 5s4d, 5s4d, and 5s5p for H, Sr+, and Sr species,
respectively. Following the formulation of Foucrault et al. [41], cutoff
functions with l-dependent adjustable parameters are fitted to reproduce not only the first experimental ionization potential but also the
lowest excited states of each l for H, Sr+, and Sr. In the present work,
the core polarizability of Sr2+ is taken as ␣Sr2⫹ = 5.51 a30 [40] and the
optimized cutoff parameters for the lowest valence s, p, and d oneelectron states of the Sr atom are 2.08205, 1.91905, and 1.64474 a.u.,
respectively. To produce the energy levels of the neutral Sr atom, we
have performed a full configuration interaction. Table S1, summarize the data about the atomic energy levels of Sr+ (5s, 4d, 5p, 6s, 5d,
6p, 7s, 6d, and 7p) and Sr(5s2, 5s5p, 5s4d, 5s4d, and 5s5p) and compares them with the available theoretical [42] and experimental [43]
data. This table is given as supplementary material1.
For the quasi-diabatic study, our idea is to construct a unitary
transformation matrix that cancels the non-adiabatic coupling elements. In this context, the quasi-diabatic wave function can be written as a linear combination of the adiabatic ones. The diabatisation
method has been published previously in several works [44–52]. We
mention here only the most important features of the diabatisation
method, which is based on effective Hamiltonian theory [53] and the
effective metric method [47]. The main purpose of this method is to
evaluate the nonadiabatic coupling between the considered adiabatic states and to cancel it by an appropriate unitary transformation
matrix. This matrix gives us the quasi-diabatic energies and wave
functions. Then the quasi-diabatic wave functions can be written
as a linear combination of the adiabatic ones. This non adiabatic
coupling estimation is closely related to an overlap matrix between the
adiabatic multiconfigurational states and the reference states corresponding to a fixed large distance equal to 105.00 a.u. The quasi-diabatic
states are deduced from the symmetrical orthonormalization of the
projection of the model space wave functions onto the selected adiabatic wave functions. The recovery matrix constructed by the projection
is clearly a recovery matrix over nonorthogonal functions seeing
that the two sets are related to different interatomic distances. The
reference states corresponding to the adiabatic ones are taken at an
Supplementary data are available with the article through the journal Web site at http://nrcresearchpress.com/doi/suppl/10.1139/cjp-2015-0801.
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Fig. 1. Adiabatic potential energy curves without (solid lines) and
with (dashed lines) the electron affinity of the H atom for the 1–31兺+
(a) and 4–101兺+ (b) electronic states of the strontium hydride cation
SrH+. [Colour online.]
infinite distance, taken to be equal to 105.00 a.u. At this distance all
adiabatic states have reached their asymptotic limits, while adiabatic
and quasi-diabatic states coincide. We take the origin on the strontium atom, which is the heavier atom. This diabatisation scheme is
based on the recovery matrix between the reference and the adiabatic
states, which correspond to a numerical estimation of the nonadiabatic
coupling, but do not involve the electric dipole matrix at all.
3. Results and discussion
3.1. Adiabatic potential energy curves
Computation of the potential energy curves is the first required
step to characterize any molecular systems. An accurate potential
energy curve opens a window to investigate and interpret the behaviour of the molecule at any experimental conditions. Using the
method of calculation reported in the previous sections, we have
investigated the adiabatic potential energy curves of 44 low-lying
electronic states of 1,3兺+, 1,3⌸, and 1,3⌬ symmetries for the strontium
hydride cation SrH + dissociating into Sr+ (5s, 4d, 5p, 6s, 5d, 6p, 7s, 6d,
and 7p) + H(1s, 2s, and 2p) and Sr(5s2, 5s5p, 5s4d, 5s4d, 5s5p) + H+ . The
adiabatic potential energy is performed for an interval of intermolecular distances ranging from 2.40 to 105.00 a.u. The 1兺+ and 3兺+
electronic states are depicted in Figs. 1 and 2, respectively, whereas
the 1,3⌸, 1,3⌬ states are displayed in Figs. 3 and 4, respectively. In Fig. 1,
we present the adiabatic potential energy curves of the 1–31兺+ (Fig. 1a)
793
Fig. 2. Adiabatic potential energy curves for the 10 lowest 3兺+
electronic states of the strontium hydride cation SrH+. [Colour online.]
and 4–101兺+ (Fig. 1b) electronic states without (solid lines) and with
(dashed lines) the electron affinity for the strontium hydride cation
SrH+. This Figure shows that the 1–51兺+ electronic states are found
with a unique well depth located respectively at 3.73, 4.64, 5.68, 8.32,
and 5.82 a.u. In addition, we found that the ground state 11兺+ dissociating into Sr+(5s) + H(1s) presents a well depth of 17 405 cm−1. This
value is in excellent agreement with the available experimental result (17 502 ± 480 cm−1) found by Dalleska et al. [26]. In a similar way
to the alkali dimers LiH [33], LiNa [34], LiCs [35], and CsNa [36], the
imprint of a state behaving as (–2/R) can be clearly seen in Fig. 1,
which is formed between the 4–101兺+ electronic states and corresponding to the Sr2+H− structure. The same behavior has been observed previously for the CaH+ [23], SrH+ [32], and BaH+ [25] alkaline
earth ions. In Fig. 2, we found that the 53兺+ and 63兺+ electronic states
dissociating, respectively, into Sr+(5d) + H(1s) and Sr+(6p) + H(1s) have
a particular shape and present a high potential barrier of 636 and
1550 cm−1 located at 8.49 and 9.05 a.u., respectively. This feature can
be explained by the interaction with the upper excited state.
In addition, Fig. 1 presents the correction due to the hydrogen
electron affinity. It is the difference between the electron affinity
calculated in our basis set and the known experimental value. In
fact, the presence of the Sr base makes this correction dependent
on R whereas it is constant for the atom. It is a question of calculating the energy of the two systems H and the ion H− with a basis
set on the ion Sr+, which makes this quantity dependent on the
interatomic distance. The calculation of this quantity is limited by
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794
Can. J. Phys. Vol. 94, 2016
Fig. 3. Adiabatic potential energy curves for the 12 lowest 3⌸ (solid
line) and 1⌸ (dashed line) of the strontium hydride cation SrH+.
Fig. 4. Adiabatic potential energy curves for the six lowest 1⌬ (solid
line) and 3⌬ (dashed line) of the strontium hydride cation SrH+.
the incomplete basis set we use. In the diabatic representation, this
correction concerns only the ionic state dissociating into Sr2+ + H−. In
this representation, it induces a general shift of crossings between
the ionic curve and the neutral curve to larger internuclear distances. The shift of the crossings is more important when the crossing takes place at long distance. It is about 0.1 a.u. for the first state
and increases quickly to be of an order of 10 a.u. for the higher
excited states. This can be explained by the behaviour in 2/R of the
ionic state. In the adiabatic representation, the diagonalisation of the
diabatic matrix containing this correction in the ionic curve will
distribute it to all the adiabatic states that result from this diagonalisation. This will imply thereafter: (i) a light change in the equilibrium
distance for the first electronic states, which will be increasingly
important for the excited states; and (ii) a change in the depth of the
potential wells, which can reach 100 cm−1 for some states. By taking
into account the electron affinity of the H atom, we observe that
there is no change in the feature of the potential energy curves.
However, they are shifted towards lower energy.
Our potential energy curves of the strontium hydride cation SrH+
present a similar shape to those obtained by Aymar et al. [31] and
Habli et al. [32]. This is not surprising because we used the same
procedure (exchange core-polarization and core polarization potential), but a different cutoff radius. The differences between our results and those of Aymar et al. [31] and Habli et al. [32] will be
discussed in detail in the section accorded to the spectroscopic parameters. In addition, we observe that the potential energy curves
present many avoided crossings at short and large values of internuclear distance between many excited states of 1,3兺+ and 1,3⌸ symmetries located at intermediate and large values of internuclear
distance. In several cases, these avoided crossings are responsible for
the presence of particular forms in the potential energy curves, such
as the appearance of the double wells and the barriers of potential.
Here, we can say that the 6–101兺+ electronic states presented double
wells and 8–103兺+ electronic states have a barrier of potential. Most
of the avoided crossings can be explained by the interaction between
the electronic states of Sr(+)H and SrH(+) structures. We have localized
the positions of the avoided crossings between the neighbor electronic states of 1兺+, 3兺+, and 3⌸ symmetries. These positions are
shown in Table 1. ⌬E represents the difference of energies at the
positions of the avoided crossing. For example, we quote the avoided
crossings between 71兺+ and 81兺+ at 9.08 and 22.83 a.u., between 81兺+
and 91兺+ at 6.44 and 25.07 a.u., between 91兺+ and 101兺+ at 6.28 and
29.00 a.u., and between 93兺+ and 103兺+ at 11.34 a.u.
3.2. Quasi-diabatic potential energy curves
In addition to the adiabatic potential energy curves, we have
calculated the quasi-diabatic potential curve below the ionic limit
Sr2+H− related to all the electronic states of 1,3兺+, 1,3⌸, and 1,3⌬ symmetries for the strontium hydride cation SrH+. In recent past, the
quasi-diabatic method was applied successfully in our group on
many diatomic molecules like the mixed alkali diatomic molecule
and the quasi-diabatic potential energy curves for the LiH [33], LiCs
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795
Table 1. Avoided crossing positions
(in a.u.) between the neighbour
electronic states of the strontium
hydride cation SrH+.
State
5 兺 /6 兺
With EA
71兺+/81兺+
With EA
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1
+
1
+
With EA
81兺+/91兺+
With EA
With EA
91兺+/101兺+
With EA
33兺+/43兺+
83兺+/93兺+
93兺+/103兺+
63⌸/73⌸
Rc (a.u.)
⌬E (a.u.)
15.92
15.93
9.08
9.10
22.83
22.87
6.46
6.51
25.07
25.10
6.28
6.30
29.00
29.03
4.05
9.33
14.36
11.34
12.01
0.00863
0.00841
0.00332
0.00302
0.00028
0.00018
0.00401
0.00403
0.00258
0.00248
0.00323
0.00324
0.00288
0.00294
0.00136
0.00045
0.00220
0.00080
0.00199
Fig. 5. Quasi-diabatic potential energy curves D1–10 related to the
ten lowest 1兺+ electronic states of the strontium hydride cation SrH+.
[Colour online.]
Note: ⌬E, difference of energies at
the positions of the avoided crossing;
EA, H electron affinity.
[35], and CsNa [36] have been calculated. We extend here the same
quasi-diabatic approach for the strontium hydride cation SrH+. The
origin is taken at the strontium atom. We fixed the reference states
as the adiabatic ones in the larger internuclear distance equal
105.00 a.u. Then we calculated the quasi-diabatic potential energy
curves related to the 1,3兺+, 1,3⌸, and 1,3⌬ symmetries in the adiabatic
representation. In this section, we present only the quasi-diabatic
potential energy curves related to the 1兺+ adiabatic representation.
Figure 5 presents the quasi-diabatic potential energy curves related
to the 1–101兺+ adiabatic electronic states and named D1–10. We remark that the ionic quasi-diabatic curve noted as D1, dissociating
into Sr2+H−, behaves as –2/R at intermediate and large internuclear
distances. This ionic quasi-diabatic curve crosses the quasi-diabatic
curves D2–9 at different interatomic distances. The avoided crossing
between the 1兺+ electronic states discussed previously in the adiabatic representation are transformed in the quasi-diabatic one into
real crossings. The lowest real crossings occur with the quasi-diabatic
state D2 dissociating into Sr+(4d) + H(1s) at 6.90 a.u., with the quasidiabatic state D3 dissociating into Sr+(5p) + H(1s) at 5.96 a.u., with the
quasi-diabatic state D4 dissociating into Sr+(5s) + H(1s) at 7.43 a.u., and
with the quasi-diabatic state D5 dissociating into Sr+(6s) + H(1s) at
10.49 a.u. The ionic quasi-diabatic state D1 crosses the higher quasidiabatic excited states D6–9 at much larger internuclear distances.
For these crossings, we found, respectively, 10.71, 10.87, 10.42, and
11.28 a.u. The second ionic quasi-diabatic curve noted in our work D8,
dissociating into SrH+, is expected to cross the neutral excited states
D5–7 and D9–10 at 13.66, 15.35, 16.38, 23.13, and 26.63 a.u., respectively.
Moreover, we show the presence of clear undulations in the shape of
the quasi-diabatic potentials of D8–10 states. These undulations are
related to the electron density and can be interpreted as resulting
from the repulsive and attractive effects. The same undulations have
been observed previously by Dickinson et al. [54, 55] in the quasidiabtic study of several diatomic systems.
3.3. The spectroscopic parameters
The adiabatic potential energy curves have been used to extract
the spectroscopic parameters, such as the equilibrium distance
(Re), the well depth (De), the electronic transition energy (Te), the
harmonicity frequency (␻e), the anharmonicity constant (␻e␹e), and
the rotational constant (Be). The spectroscopic parameters of the
ground and the low-lying electronic states of the different symmetries 1,3兺+, 1,3⌸, and 1,3⌬ are collected in Table 2. These spectroscopic
parameters are compared with the available experimental [26] and
theoretical [28, 29, 31, 32] results. To the best of our knowledge, the
experimental information on strontium hydride SrH+ are still limited, except the ground (X1兺+) state. As it seems from Table 2, the
agreement between our well depth and the experimental value
found by Dalleska et al. [26] is very good. In fact, we found a well
depth of 17 405 cm−1, while Dalleska et al. [26] present a well depth of
17 502 ± 480 cm−1. The difference is about 98 cm−1. The experimental
result presents a rather large uncertainty of ±480 cm−1.
There have been a few theoretical studies realized on the ground
state (X1兺+). A general good agreement is observed between our spectroscopic parameters and other theoretical [28, 29, 31, 32] values. We
found the following spectroscopic parameters Re = 3.73 a.u.,
De = 17 405 cm−1, ␻e = 1387.75 cm−1, ␻e␹e = 19.88 cm−1, and Be =
4.345 cm−1 for the ground state. These values are in very good agreement with the theoretical results of Habli et al. [32], who reported Re,
De, ␻e, ␻e␹e, and Be as 3.72 a.u., 17 588 cm−1, 1351.2 cm−1, 19.31 cm−1,
and 4.356 cm−1, respectively. This is not surprising because we used
the same method of calculation, but a different basis set of Gaussiantype orbital and different cutoff radius. Good agreement is observed
between our spectroscopic parameters and those found by Aymar
et al. [31] (Re = 3.73 a.u, De = 18 078 cm−1, and ␻e = 1429 cm−1). We also
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796
Can. J. Phys. Vol. 94, 2016
Table 2. Spectroscopic parameters of the ground and the low-lying electronic states of the
different symmetries 1,3兺+, 1,3⌸, and 1,3⌬ of the strontium hydride cation SrH+.
State
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X1兺+
With EA
21兺+
With EA
31兺+
With EA
41兺+
With EA
51兺+
With EA
61兺+
With EA
2nd min
With EA
71兺+
With EA
2nd min
With EA
81兺+
With EA
2nd min
With EA
91兺+
With EA
2nd min
With EA
3rd min
With EA
101兺+
With EA
2nd min
With EA
3rd min
With EA
4th min
With EA
13兺+
Re
(a.u.)
3.73
3.74
3.96
3.87
3.73
3.72
4.64
4.65
4.64
4.67
5.65
5.67
5.77
5.81
8.32
8.34
8.29
8.29
5.78
5.79
5.70
5.72
4.65
4.67
4.64
13.67
13.80
13.62
5.42
5.43
5.46
15.54
15.56
15.51
5.24
5.26
5.29
8.31
8.34
8.25
5.23
5.24
5.21
6.59
6.61
6.54
21.25
21.34
21.95
4.91
4.93
4.89
6.34
6.36
6.26
11.51
11.57
11.51
26.84
26.98
8.00
8.04
8.45
De
(cm−1)
Te
(cm−1)
␻e
(cm−1)
␻e␹e
(cm−1)
Be
(cm−1)
References
17405
16948
17502±480
15950.82
16374.53
18078
17588
9030
8681
8843
8830
6714
6397
6462
6098
9909
9668
10064
10140
6133
5818
6135
6229
2815
2471
2930
1556
1473
1725
7075
6754
6804
5751
5628
1725
2711
2267
2409
5419
5177
5219
3937
3578
3612
4306
4026
4096
2579
2494
2455
2677
2341
2422
4221
3923
4007
1723
1552
1671
1153
1090
170
155
88
—
—
—
—
—
—
—
23071
22963
19845
—
34806
34667
29135
—
55281
55065
54497
—
64756
64613
58605
—
70430
70405
—
65566
71403
71689
74394
—
68297
75719
74970
—
80402
—
74099
77694
—
—
81371
—
—
81001
—
—
82728
—
—
84297
83762
77130
82339
—
—
84837
—
—
85821
85012
17234
11493
—
1384.75
—
—
1346
1397.3
1429
1351.2
726.21
733.58
728.8
712
395.43
393.44
419.3
359
513.48
512.96
505.1
510
540.44
532.77
551.6
599
699.18
689.69
287.9
101.22
108.46
—
555.38
—
181.4
195.86
197.43
—
399.91
—
393.6
340.61
—
—
136.24
—
172.3
59.69
—
—
50.10
—
—
310.03
—
312.3
1090.52
—
—
147.98
—
—
43.75
51.02
91.190
88.3
78
19.88
18.43
—
—
—
—
19.31
5.44
7.17
5.248
—
5.82
5.43
2.263
—
4.19
3.12
4.509
—
51.90
62.83
56.095
—
47.40
44.08
16.142
—
—
—
16.210
—
2.638
—
—
—
11.49
—
11.137
—
—
—
4.82
—
5.207
—
—
—
—
—
—
13.81
—
10.675
—
—
—
—
—
—
—
—
12.22
13.600
—
4.345
4.324
—
—
4.072
—
4.356
2.811
2.794
2.806
—
1.903
1.883
1.814
—
0.872
0.869
0.879
—
1.812
1.802
1.859
—
2.785
2.772
2.805
—
—
—
2.058
—
2.026
—
—
—
2.158
—
2.159
—
—
—
2.151
—
2.225
—
—
—
—
—
—
2.151
—
2.526
—
—
—
—
—
—
—
—
0.951
0.934
—
This work
This work
[26]
[28]
[29]
[31]
[32]
This work
This work
[32]
[31]
This work
This work
[32]
[31]
This work
This work
[32]
[31]
This work
This work
[32]
[31]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
[32]
This work
This work
This work
[32]
[31]
Published by NRC Research Press
Belayouni et al.
797
Table 2 (continued).
State
2 兺
3
+
33兺+
Can. J. Phys. Downloaded from www.nrcresearchpress.com by Univ of Georgia Libraries on 10/05/16
For personal use only.
43兺+
Hump
53兺+
Hump
63兺+
Hump
73兺+
83兺+
2nd min
Hump
93兺+
2nd min
3rd min
103兺+
2nd min
11⌸
21⌸
31⌸
41⌸
51⌸
61⌸
13⌸
23⌸
33⌸
43⌸
53⌸
63⌸
11⌬
Re
(a.u.)
De
(cm−1)
Te
(cm−1)
␻e
(cm−1)
␻e␹e
(cm−1)
Be
(cm−1)
References
8.03
8.91
9.30
11.77
11.77
12.00
4.44
4.63
4.65
9.50
4.60
4.59
4.57
8.49
5.60
5.41
9.05
5.22
5.15
4.90
4.90
10.63
10.45
15.50
4.83
4.78
9.27
8.91
13.256
12.307
5.051
5.019
10.84
10.31
5.85
5.85
5.77
5.19
5.14
5.13
4.96
4.97
4.95
5.08
5.07
4.87
4.82
4.99
5.00
4.58
4.94
4.52
4.93
4.95
4.92
5.19
5.20
5.18
5.02
5.00
5.01
4.91
5.04
5.06
5.35
5.33
5.32
311
155
44
36
19
19
2777
2849
2838
107
2717
2647
2801
754
1785
1855
1550
3122
2994
5053
4764
1453
1211
215
3546
3296
2229
1866
752
348
6202
6432
4850
4517
3546
3522
3476
1142
1150
1239
2332
2350
2383
2162
2160
7549
7384
3655
3588
2277
2125
2182
2203
2095
2196
3051
3087
3143
2214
2218
9034
9052
3996
3837
850
761
856
31789
26201
—
41484
39630
—
62412
57656
—
—
68172
64259
—
—
71548
65693
—
79576
73720
80253
74958
83853
—
—
83013
77781
84330
—
85808
—
84558
78821
85914
—
5158
—
5704
33981
—
34330
62159
—
62289
64774
65125
71360
71597
56507
76513
29824
24163
—
39316
—
33405
67838
—
61512
71118
65074
76273
69870
82.563
76.215
31250
—
25193
89.460
62.4
46
42.900
32
38
683.23
6.83.7
876
—
784.370
752.0
584
—
858.590
456.8
—
725.100
473.6
681.200
541.7
235.360
—
—
577.580
541.7
490.180
—
158.700
—
236.010
393.1
310.380
—
337.57
207
219.9
369.030
410
404.8
540.670
559
563.2
409.29
484.3
566.87
625.6
427.70
516.7
603.080
607.8
619
553.340
541
552.3
486.820
495
497.5
502.530
496.2
572.080
599.2
526.400
529.6
237.21
339.0
335.8
6.433
10.260
—
12.780
12.006
—
42.020
31.480
—
—
56.66
18.485
—
—
73.616
–9.335
—
42.100
16.939
22.958
15.886
9.531
—
—
23.519
15.886
26.948
—
8.372
—
16.295
12.395
4.965
—
52.170
—
28.570
37.230
—
35.277
31.52
—
32.179
25.00
26.910
10.858
11.533
25.420
29.264
39.81
—
43.264
31.99
—
34.543
18.58
—
18.277
30.93
29.624
15.56
11.533
23.6
26.3893
16.54
—
35.990
0.938
0.834
—
0.440
0.436
—
3.066
2.818
—
—
2.660
2.867
—
—
1.930
2.064
—
1.985
2.277
2.565
2.516
0.535
—
—
2.589
2.516
0.704
—
0.344
—
2.463
2.313
0.514
—
0.322
—
1.814
2.413
—
2.295
2.453
—
2.465
2.432
2.350
2.469
2.600
1.848
2.416
2.901
—
2.957
2.492
—
2.496
2.246
—
2.251
2.401
2.416
2.407
2.505
2.377
2.359
2.114
—
2.134
This work
[32]
[31]
This work
[32]
[31]
This work
[32]
[31]
This work
This work
[32]
[31]
This work
This work
[32]
This work
This work
[32]
This work
[32]
This work
[32]
This work
This work
[32]
This work
[32]
This work
[32]
This work
[32]
This work
[32]
This work
[31]
[32]
This work
[31]
[32]
This work
[31]
[32]
This work
[32]
This work
[32]
This work
[32]
This work
[31]
[32]
This work
[31]
[32]
This work
[31]
[32]
This work
[32]
This work
[32]
This work
[32]
This work
[31]
[32]
Published by NRC Research Press
798
Can. J. Phys. Vol. 94, 2016
Table 2 (concluded).
State
2⌬
1
31⌬
13⌬
Can. J. Phys. Downloaded from www.nrcresearchpress.com by Univ of Georgia Libraries on 10/05/16
For personal use only.
23⌬
33⌬
Re
(a.u.)
De
(cm−1)
Te
(cm−1)
␻e
(cm−1)
␻e␹e
(cm−1)
Be
(cm−1)
References
5.15
5.07
5.05
4.86
5.04
5.34
5.30
5.29
5.15
5.06
5.05
4.69
5.04
1930
1902
1973
2369
2235
892
794
885
1935
1908
1978
2377
2238
68958
—
62678
79938
76635
31208
—
25157
68954
—
62672
79930
76635
382.99
487
508.7
479.57
497.2
239.93
345
594.6
382.71
486
487.8
1065.67
552.2
19.00
—
34.395
23.294
28.279
16.134
—
−203.474
18.92
—
30.623
52.801
37.536
2.282
—
2.368
2.739
2.378
2.106
—
2.158
2.282
—
2.368
2.750
2.378
This work
[31]
[32]
This work
[32]
This work
[31]
[32]
This work
[31]
[32]
This work
[32]
Note: EA, H electron affinity.
Fig. 6. Permanent dipole moments of the 1–31兺+ (a) and 4–101兺+ (b)
electronic states of the strontium hydride cation SrH+. [Colour online.]
Fig. 7. Permanent dipole moments of the 1–43兺+ (a) and 5–103兺+ (b)
electronic states of the strontium hydride cation SrH+. [Colour online.]
observe that our spectroscopic parameters agree well with the
values obtained by Abe et al. [28] using the complete active space
second-order perturbation theory method. However, we remark
that the well depth of Schilling [28] is largely underestimated (De =
15 951 cm−1) when compared with the experimental value of Dallesta et al. [26] (De = 17 502 ± 480 cm−1) or the other theoretical
results. For the first excited state, there is a very good agreement
between our equilibrium position as well as the well depth (Re =
4.64 a.u. and De = 9030 cm−1) and those of Aymar et al. [31] (Re =
4.67 a.u. and De = 8830 cm−1). However, our theoretical vibrational
constant (␻e = 726.21 cm−1) and the rotational constant (Be =
2.811 cm−1) are in good agreement with the theoretical values of
Habli et al. [32] (␻e = 728.8 cm−1 and Be = 2.806 cm−1). From the
comparison between our spectroscopic parameters, we remark
Published by NRC Research Press
Belayouni et al.
Can. J. Phys. Downloaded from www.nrcresearchpress.com by Univ of Georgia Libraries on 10/05/16
For personal use only.
Fig. 8. Permanent dipole moments of the first six 1⌸ electronic
states of the strontium hydride cation SrH+. [Colour online.]
first, that the correction of the electron affinity of the H atom
increases the equilibrium distances of all states 1兺+. They are increased by 0.01–0.04 a.u. compared to the initial values without
the electron affinity of the H atom. Second, the potential well
depths are decreased by several tens of inverse centimetres.
The spectroscopic parameters of the higher excited states of
1,3兺+, 1,3⌸, and 1,3⌬ symmetries are presented in Table 2 to compare with the available theoretical results of Aymar et al. [31] and
Habli et al. [32]. In general, our spectroscopic parameters are close
to the values found by Aymar et al. [31] and those found by Habli
et al. [32]. This is not surprising because we used the same procedure of calculation. In addition, several excited states exhibit double potential wells and sometimes triple wells, as in the case of the
6–101兺+ and 8–103兺+ electronic states. The double or triple well
depths are due to the avoided crossing between many electronic
states. Their existences have generated substantial non-adiabatic
coupling, and have led to an undulating behaviour of the higher
excited states at large internuclear distances.
3.4. Permanent and transition dipole moments
The knowledge of the dipole moment of a molecular system is
considered as a sensitive test for the precision of the calculated
electronic wave functions and energies. In fact, the dipole moments
of the dipolar molecules have a great number of applications, such as
the control of ultracold chemical reactions [56], the creation of a
platform for quantum information processing [57, 58], and the ex-
799
Fig. 9. Permanent dipole moments of the first six 3⌸ electronic
states of the strontium hydride cation SrH+. [Colour online.]
amination of fundamental theories like the measurement of the
electron dipole moments [59, 60].
To understand the ionic behaviour of the excited electronic states,
we have calculated the adiabatic permanent dipole moments for
all the electronic states of 1,3兺+, 1,3⌸, and 1,3⌬ symmetries of the
strontium hydride cation SrH+. These adiabatic permanent dipole
moments are presented in Figs. 6–10. Figure 6 shows the adiabatic
permanent dipole moments of the 1–31兺+ (Fig. 6a) and 4–101兺+
(Fig. 6b) electronic states. In the Fig. 6a, we remark that, at short
distances, the adiabatic permanent dipole moments of the 1–31兺+
states exhibit positive and negative values with small amplitudes. At
large distances they approach zero and vanish. Figure 6b shows that
the adiabatic permanent dipole moments of the 4–51兺+ and 7–101兺+
electronic states, one after another, behave as linear functions of
R, although there is some numerical noise around some abrupt
changes. In addition, they drop to zero at particular distances corresponding to the avoided crossings between the neighbour electronic
states. The linear interpolation between the adiabatic permanent
dipole moments of the 7–81兺+ shows a positive linear variation because of the ionic character of the SrH(+) structure. In the same way,
we can observe that the linear interpolation between the adiabatic
permanent dipole moments of the 4–51兺+ and 7–101兺+ shows a negative linear variation because of the ionic character of the Sr(2+)H(−)
structure. In addition, we remark that 7–81兺+ states exhibit positive
and negative dipole moments. This significant change of sign in the
adiabatic permanent dipole moments can be explained by the
change of the polarity in the SrH+ system, going from the Sr(2+)H(−)
Published by NRC Research Press
800
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For personal use only.
Fig. 10. Permanent dipole moments of the first six 1⌬ and 3⌬ electronic
states of the strontium hydride cation SrH+. [Colour online.]
structure for the negative sign to the SrH(+) structure for the positive
sign. This interpretation confirmed the sign convention as it is demonstrated in many previous studies [12, 34–36, 61–64].
The discontinuities between consecutive portions are due to the
avoided crossings. Moreover, the positions of these abrupt changes
are correlated to the avoided crossings between the adiabatic potential energy curves. The abrupt changes and the avoided crossings are
both results of the manifestations of abrupt changes of the character
of the electronic wave functions. For example, we mention the positions of the abrupt changes in the adiabatic permanent dipole moments distribution between 71兺+ and 81兺+ at 22.94 a.u., between 81兺+
and 91兺+ at 25.55 a.u., and between 91兺+ and 101兺+ at 39.92 a.u. These
positions of real crossings between the adiabatic permanent dipole
moments are identical to the regions of internuclear distances exhibiting avoided crossings between the adiabatic potential energy
curves of 1兺+ electronic states. It is clear that the positions of the
irregularities in the R-dependence of the adiabatic permanent dipole
moments are correlated to the avoided crossings between the potential energy curves, which are both manifestations of abrupt changes
of the character of the electronic wave functions.
In Fig. 7, we depict the adiabatic permanent dipole moments of
the 1–43兺+ (Fig. 7a) and 4–103兺+ (Fig. 7b) electronic states of the
strontium hydride cation SrH+. We remark that these dipole moments pass through several maximums located at intermediate and
large distances. In addition, there are many abrupt changes between
the adiabatic permanent dipole moments of the electronic states;
33兺+ and 43兺+, 83兺+ and 93兺+, and 93兺+ and 103兺+ located at 4.06,
9.28, and 11.24 a.u., respectively. We conclude that the positions of
these abrupt changes coincide with the crossing positions previously
seen in the adiabatic potential energy curves of the 3兺+ electronic
states. The adiabatic permanent dipole moments have also been determined for the electronic states of the 1⌸, 3⌸, and 1,3⌬ remaining
symmetries. Their adiabatic permanent dipole moments are displayed in Figs. 8–10. As expected, these dipole moments are not
Can. J. Phys. Vol. 94, 2016
Fig. 11. Quasi-diabatic permanent dipole moments of the 1–61兺+
and 7–101兺+ electronic states of the strontium hydride cation SrH+.
[Colour online.]
negligible and they become more significant for the higher excited
states. The exception is the permanent dipole moments of 61,3⌸
states, which tend to be a constants. The other permanent dipole
moments of 1–51,3⌸ states vanish at large distance from zero.
The quasi-diabatic permanent dipole moments of the 1–61兺+
and 7–101兺+ electronic states of the strontium hydride cation SrH+
are depicted in Figs. 11a and 11b, respectively. These quasi-diabatic
permanent dipole moments are obtained using a simple rotation of
the adiabatic permanent dipole moments matrix. From Fig. 11b, we
can see that the quasi-diabatic permanent dipole moments of the
7–101兺+ states present the same behaviour as in the adiabatic case
with abrupt variations. These abrupt variations are situated nearly at
different positions corresponding to the avoided crossings between
the neighbour electronic states mentioned in Table 1. For example,
we can quote the crosses between the quasi-diabatic permanent dipole moments of the 71兺+ and 81兺+ states localized at 21.14 and
22.88 a.u. and accompanied by abrupt variations. At large distances,
it is clearly observed that the quasi-diabatic dipole moments of the
71兺+ state are characterized by a linear divergence. Similar behaviours are obtained between the quasi-diabatic permanent dipole moments of 71兺+ and 81兺+ states located at 34.26 and 39.76 a.u.
In addition to the permanent dipole moments, we also calculated
the transition dipole moments between neighbour electronic states
for the electronic states of 1,3兺+, 1,3⌸, and 1,3⌬ symmetries. Here, we
present only the transition dipole moments between the neighbour
Published by NRC Research Press
Belayouni et al.
Can. J. Phys. Downloaded from www.nrcresearchpress.com by Univ of Georgia Libraries on 10/05/16
For personal use only.
Fig. 12. Transition dipole moments between neighbor electronic
states of 1兺+ symmetry of the strontium hydride cation SrH+. [Colour
online.]
801
a standard quantum chemistry approach based on pseudopoentials,
Gaussian basis sets, effective core polarization potentials, and full
configuration interaction calculations. The adiabatic potential energy curves and their spectroscopic parameters for the ground and
many excited electronic states of 1,3兺+, 1,3⌸, and 1,3⌬ symmetries have
been computed for a large and dense grid of internuclear distances
varying from 2.40 to 105.00 a.u. The higher excited states have shown
undulations related to avoided crossings or undulating orbitals of
the atomic Rydberg states. They led to multiple potential barriers
and wells in the potential energy curves. The agreement between the
spectroscopic parameters obtained in our work and those of
the previous studies [26, 28, 29, 31, 32] for the ground (X1兺+) and the
high excited states are shown to be satisfactory. The correction of
the electron affinity of the H atom shows that there is no change in
the feature of the potential energy curves. Compared to the initial
values without the electron affinity of the H atom, there are small
changes in the equilibrium distances accompanied by an increase in
the potential well depths. In the quasi-diabatic representation, two
ionic quasi-diabatic states are clearly observed. The first represents
that the Sr(2+)H(−) structure crosses the quasi-diabatic curves D2–9 at
different distances. The second ionic quasi-diabatic state dissociating
into SrH(+), is expected to cross only the D5–7 and D9–10 states at
large internuclear distances. Such avoided crossings became real
crossings in the quasi-diabatic representation.
For a better understanding of the ionic character of the electronic
states of the strontium hydride cation SrH+, we have calculated the
permanent and transition dipole moments. The permanent dipole
moments of the 1–101兺+ electronic states have shown the presence of
the ionic state, corresponding to the SrH+ structure, which is almost
a linear feature function of R, especially for the higher excited states
and at intermediate and large distances. Moreover, the abrupt
changes in the adiabatic permanent dipole moments are localized at
particular distances corresponding to the avoided crossings between
the neighbor electronic states.
References
electronic states of the 1兺+ symmetry. They are displayed in Figs. 12a
and 12b. The transition between the ground state (11兺+) and the excited state (21兺+), dissociating into Sr+(5s) + H(1s) and Sr+(4d) + H(1s),
respectively, is very large. It presents a maximum of 2.53 a.u. located
at 7.44 a.u. We can conclude that around this distance there is an
important overlap between the corresponding molecular wave functions. At large distances, the 11兺+–21兺+ transition becomes a constant
equal to 2.39 a.u., which is related to the atomic transition between
Sr+(5s) and Sr+(4d). Moreover, we remark that the 21兺+–31兺+ and 31兺+–
41兺+ transitions, Sr+(4d) + H(1s) and Sr+(5p) + H(1s), and Sr+(5p) + H(1s)
and Sr+(5d) + H(1s), present the same behavior. They decrease at small
distances, then they pass by a minimums located at 7.67 and 8.13 a.u.,
respectively, and finally they go to the absolute values of 1.63 and
1.65 a.u., respectively. These values are in good agreement with the
transition dipole moments (1.69 and 1.84 a.u.) deduced from the theoretical oscillator strength of Mitroy et al. [65]. The other transitions
between the high excited states present many peaks located at particular distances very close to the avoided crossings in adiabatic
representation. We mention here the peaks observed in the 81兺+–
91兺+ and 91兺+–101兺+ transitions located at 25.60 and 39.90 a.u.,
respectively.
4. Conclusion
This work is focused on the structure and electronic properties
of the strontium hydride cation SrH+ dissociating into Sr+(5s, 4d,
5p, 6s, 5d, 6p, 7s, 6d, and 7p) + H(1s, 2s, and 2p) and Sr(5s2, 5s5p,
5s4d, 5s4d, 5s5p) + H+ It has been systematically investigated using
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