Theor Chem Acc (2016) 135:142
DOI 10.1007/s00214-016-1897-6
REGULAR ARTICLE
Adiabatic and quasi‑diabatic study of FrRb: structure,
spectroscopy and dipole moments
Ibtissem Jendoubi1 · Chedli Ghanmi1,2 · Hamid. Berriche1,3 Received: 7 October 2015 / Accepted: 21 April 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract Ab initio investigations are performed for the
FrRb using a standard quantum chemistry approach based
on pseudopotentials for atomic core representations, Gaussian basis sets and 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. Firstly, we determine
the adiabatic and quasi-diabatic potential energies and the
spectroscopic constants for the ground and several excited
states of 1,3Σ+, 1,3Π and 1,3Δ symmetries. Their predicted
accuracy is discussed by comparing our well depths and
equilibrium positions with the available theoretical results.
Secondly, we calculate the permanent dipole moments for a
wide range of internuclear distances. The adiabatic permanent dipole moments for the 151Σ+ electronic states have
revealed ionic characters related to electron transfer and
yielding both Fr−Rb+ and Fr+Rb− arrangements. Finally,
the transition dipole moments between neighbor electronic
states revealed many peaks around the avoided crossing
positions.
* Hamid. Berriche
[email protected]; [email protected]
1
Laboratoire des Interfaces et Matériaux Avancés,
Département de Physique, Faculté des Sciences, Université
de Monastir, Avenue de l’Environnement, 5019 Monastir,
Tunisia
2
Physics Department, Faculty of Sciences, King Khalid
University, P. O. Box 9004, Abha, Saudi Arabia
3
Mathematics and Natural Sciences Department, School
of Arts and Sciences, American University of Ras Al
Khaimah, Ras Al Khaimah, UAE
Keywords FrRb · Pseudopotential · Configuration
interaction · Potential energy · Spectroscopic constants ·
Dipole moment
1 Introduction
Over the last decade, research on cold and ultra-cold molecules whether neutral or ionic species has been receiving
a growing attention and has become a major research topic
in atomic and molecular physics on both experimental and
theoretical sides [1, 2]. Special intense interest is put on the
study of the ground state and especially near the asymptotic limit, since 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. Nowadays, cold and ultra-cold heteronuclear molecules can be produced by photoassociation
or by a laser-cooled atomic vapor. Recently, such ultra-cold
polar dimers have been formed by photoassociation from
ultra-cold samples of their atomic constituents as demonstrated for RbCs [1], KRb [3, 4] and NaCs [5]. At the
same time, a formation of ultra-cold NaCs+ ion has been
observed in a novel two-species magneto-optical trap [6].
More recently, the first steps toward the implementation of
a simple scheme for rotational cooling of the magnesium
hydride molecular ion MgH+ have been reported by Højbjerre et al. [7]. These molecular activities stimulate theoretical developments to compute relevant potential energy
and transition dipole moments, especially within the framework of pseudopotential and/or model potential methods.
In recent times, and by using the pseudopotential method
developed by Foucrault et al. [8], calculations of the electronic structure of the molecules MgH+ [9], LiH [10], LiNa
13
142 Page 2 of 19
[11], CsLi [12], CsNa [13] and BeH+ [14] have been performed. Using a basically similar approach, we extend our
calculations to the potential energy curves and permanent
and transition dipole moments of the FrRb molecule.
To our knowledge, the first work corresponding to
mixed species involving francium and rubidium or cesium
was performed by Lim et al. [15] who have determined
the spectroscopic properties for the neutral and positively
charged alkali dimers from K2 to Fr2. In their calculation,
they have used various approaches based on a relativistic coupled-cluster method or functional density theory.
Moreover, the van der Waals coefficients for the heteronuclear alkali-metal dimers of Li, Na, K, Rb, Cs and Fr have
been computed by Derevianko et al. [16] using relativistic
ab initio methods augmented by high-precision experimental data. In recent times and using a similar method,
Aymar et al. [17] have performed the first calculations of
the electronic structure of francium diatomic compounds
Fr2, FrRb and FrCs, yielding to the potential energy curves,
permanent and transition dipole moments. In addition, the
same group of Aymar et al. [18] has performed careful calculations of transition and permanent dipole moments of
alkali diatomics involving sodium atom NaX (X = K, Rb
and Cs). Moreover, the potential energy curves and the permanent dipole moments structure of the RbCs molecule
have been determined by Lim et al. [19]. They employed
large-scale multireference configuration interaction calculations using both spin-averaged and two-component
spin–orbit energy-consistent effective core potentials. In
addition, several theoretical studies interested in the electronic properties have been conducted for the heteronuclear
alkali dimers using various quantum chemistry methods.
These studies determined the potential energy curves of the
ground and exited states of RbCs [20–22], NaRb and LiRb
[23], NaCs, LiCs and CsK [24], KRb [25], LiK [26], NaK
[27, 28] and LiNa [29, 30]. We extend here our previous
studies on the structure and electronic properties of many
neutral and ionic diatomic systems such as LiH [10], LiNa
[11], CsLi [12], CsNa [13], BeH+ [14], LiRb [31] and LiX
(X = Na, K, Rb, Cs and Fr) [32] molecules. Our aim is to
determine the adiabatic and quasi-diabatic potential energy
curves and their spectroscopic constants and the permanent
and transition dipole moments. This study represents the
necessary initial steps toward investigating the possibility
of creating a cold FrRb molecule.
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 constants for the 1–151Σ+,
1–103Σ+, 1–71,3Π and 1–31,3Δ electronic states, the quasidiabatic potential energy curves and finally the permanent
and transition dipole moments. Section 4 contains our conclusions and a brief summary.
13
Theor Chem Acc (2016) 135:142
2 Methods of calculations
The FrRb molecule is treated as a two-electron system
using the nonempirical pseudopotential in its semi-local
form for the Rb+ core and a semi-local l-dependent pseudopotential (PP) for the Fr+ core proposed by Barthelat
and Durand [33]. Furthermore, the self-consistent field
(SCF) calculation is followed by a full valence configuration interaction (FCI) calculation using the CIPSI algorithm of the standard chain of programs of Laboratoire
de Physique Quantique de Toulouse. For the simulation
of the interaction between the polarizable Fr+ and Rb+
cores with the valence electrons, a core polarization potential VCPP is used, according to the operator formulation of
Müller et al. [34]:
VCPP = −
1 � �
α f .f
2
where αλ is the dipole polarizability of the core λ and f is
the electric field created by valence electrons and all other
cores on the core λ.
f� =
�ri
i
3
ri
F(�ri , ρ ) −
R
� ′ Z
R3′ ′
� =
� ′ is a core–core
where ri is a core-electron vector and R
vector.
As reported by the formulation of Foucrault et al. [8],
the cutoff function F(ri , ρ ) is taken to be a function of
l in order 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 Müller et al. [34] 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 francium atom, we have used a (9s9p9d) basis
set of Gaussian-type orbital, taken from Ref. [17], and
while for the rubidium atom, we have used (6s5p5d2f/5s4p
Table 1 Polarizabilities and l-dependent cutoff for francium and
rubidium atoms
Atom
α
ρs
ρp
ρp
Fr(Z = 87)
A
B
Opt.
20.38
23.2
19.19
3.16372
3.343
3.007
3.045
3.435
2.891
3.1343
3.292
3.006
2.5213
2.279
2.511
Rb(Z = 37)
9.245
Page 3 of 19 Theor Chem Acc (2016) 135:142 142
Table 2 Binding energies of the external electron of the francium atom (in cm−1): comparison between our results, Aymar et al. [17] energies
and the corresponding experimental [35, 36] values
Atomic
level
Fr(7s)
Fr(7p)
Fr(6d)
Fr(8s)
Fr(8p)
Fr(7d)
Fr(9s)
Fr(9p)
Fr(8d)
Our work
Opt. basis
−32,848.38
−19,467.21
−16,475.33
−13,172.67
−9451.47
−8406.77
−7167.17
−5622.73
−4989.98
Fig. 1 Adiabatic potential
energy curves for the 15 lowest
1 +
Σ electronic states of the
FrRb molecule
Our work
A basis
−32,848.82
−19,486.96
−16,668.46
−13,161.03
−9448.17
−8394.69
−7161.90
−5619.65
−4980.98
Our work
B basis
−32,848.82
−19,487.18
−16,470.94
−13,136.80
−9424.91
Aymar et al. [17].
A basis
Aymar et al. [17].
B basis
−32,848.87
−32,848.87
−19,487.07
−19,487.18
−16,471.07
−16,470.94
−13,116.62
−13,136.97
−9372.35
−8371.65
−9424.96
−8550.96
−7151.37
−8371.59
−7177.85
−5608.46
−7151.28
−5573.18
−4962.98
−5605.27
−5221.39
−4963.01
Experimental
[35, 36]
−32,848.87
∆E
0.49
−19,487.07
19.86
−13,116.62
56.05
−16,471.07
−9372.35
4.26
79.12
−8550.96
144.19
−5573.18
49.55
−7177.85
−5221.39
10.68
231.41
Fr(9p)+Rb(5s)
151Σ+
131Σ+
141Σ+
121Σ+
111Σ+
101Σ+
Fr(7s)+Rb(7s)
Fr(7p)+Rb(5p)
Fr(7s)+Rb(5d)
Fr(9s)+Rb(5s)
Fr(7d)+Rb(5s)
Fr(7s)+Rb(6p)
Fr(8p)+Rb(5s)
1 +
8Σ
91Σ+
71Σ+
Fr(7s)+Rb(6s)
Fr(8s)+Rb(5s)
51Σ+
Fr(7s)+Rb(4d)
61Σ+
41Σ+
31Σ+
Fr(7p)+Rb(5s)
1 +
Fr(7s)+Rb(5p)
2Σ
11Σ+
4d1f) the Gaussian basis sets taken from Ref. [20]. For the
Fr+ core, we have used three different values of polarizability: αA = 20.38 a30 [17], αB = 23.2 a30 [17] and α = 19.19
Fr(6d)+Rb(5s)
Fr(7s)+Rb(5s)
a30 [19]. In addition to the cutoff functions, noted A and B,
taken from Ref. [17] and following to the formulation of
Foucrault et al. [8], we have fitted new cutoff radii values
13
142 Page 4 of 19
Fig. 2 Adiabatic potential
energy curves for the 10 lowest
3 +
Σ electronic states of the
FrRb molecule
Theor Chem Acc (2016) 135:142
Fr(7d)+Rb(5s)
103Σ+
93Σ+
63Σ+
Fr(8s)+Rb(6p)
83Σ+
Fr(8p)+Rb(5s)
73Σ+
Fr(7s)+Rb(6s)
Fr(8s)+Rb(5s)
53Σ+
Fr(7s)+Rb(4d)
43Σ+
33Σ+
23Σ+
13Σ+
using the polarizability α = 19.19 a30 for the Fr atom. The
different values of polarizability and the cutoff radii for the
lowest s, p and d one-electron levels, reported in Table 1,
are adjusted to reproduce the experimental energies of
the lowest s, p and d levels for the Fr and Rb atoms. We
present in Table 2 our theoretical atomic energies for the
Fr(7s, 7p, 6d, 8s, 8p, 7d, 9s and 9p) atom and the corresponding energy differences ∆E using the three Fr+ polarizabilities. These values are compared with the available
theoretical [17] and experimental [35, 36] data. Therefore,
our atomic energies obtained using our optimal basis and
calculated with the polarizability α = 19.19 a30 are in good
agreement with the experimental values. The difference
between our results and the experimental values does not
exceed 232 cm−1, which is found in Fr(8d) atomic limit.
This represents the accuracy of the present calculations at
the atomic level, which will be transmitted to the molecular
energy.
For the quasi-diabatic study, the diabatisation method
used in our work has been described in detail previously
[37–40] and only the most important features are given
here. The diabatisation method is based on effective Hamiltonian theory [41–45] and an effective metric [40]. The
13
Fr(6d)+Rb(5s)
Fr(7p)+Rb(5s)
Fr(7s)+Rb(5p)
Fr(7s)+Rb(5s)
idea 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
will provide us with 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
nonadiabatic 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 90.00 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 references
states corresponding to the adiabatic ones are taken at infinite distance equal to 90.00 a.u. At this distance all adiabatic states have reached their asymptotic limits, and the adiabatic and quasi-diabatic states coincide. We take the origin
on the francium 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
Page 5 of 19 Theor Chem Acc (2016) 135:142 Fig. 3 Adiabatic potential
energy curves for the 14 lowest
3
Π (solid line) and 1Π (dashed
line) of the FrRb molecule
142
Fr(7s)+Rb(5d)
71Π
Fr(7d)+Rb(5s)
Fr(7s)+Rb(6p)
3
7Π
61Π
51Π
53Π
63Π
Fr(7s)+Rb(4d)
41Π
43Π
Fr(6d)+Rb(5s)
31Π
33Π
23Π
1
2Π
Fr(7p)+Rb(5s)
Fr(7s)+Rb(5p)
11Π
13Π
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 and spectroscopic
constants
There are various interesting aspects in the potential energy
curves of the alkali dimers and their ions. Using the pseudopotential technique, we have computed the adiabatic
potential energy curves of 45 low-lying electronic states
of 1,3Σ+, 1,3Π and 1,3Δ symmetries for the FrRb molecule
dissociating into Fr (7s, 7p, 6d, 8s, 8p, 7d, 9s and 9p) and
Rb (5s, 5p, 4d, 6s, 6p, 5d and 7s). The adiabatic potential
energy values are calculated for an interval of intermolecular distances from 5.00 to 90.00 a.u. The 1Σ+ and 3Σ+
electronic states are displayed, respectively, in Figs. 1 and
2, whereas the 1,3Π, 1,3Δ states are displayed, respectively,
in Figs. 3 and 4. Figure 1 shows that the electronic states
1–51Σ+ are found with a unique well depth located, respectively, at 8.17, 9.51, 10.68, 10.32 and 9.24 a.u. Similarly
to the LiH [10], LiNa [11], LiCs [12] and CsNa [13] systems, the imprint of a state behaving as (−1/R) can be
clearly seen in Fig. 1. There are two ionic characters in
the higher excited states of 1Σ+ symmetry. The first corresponding to the ionic structure Fr−Rb+ which is formed
between the 5–151Σ+ electronic states, while the second is
composed of the six electronic states 9–141Σ+ corresponding to the Fr+Rb− structure. In Fig. 2, we remark that the
33Σ+ electronic state dissociating into Fr(7p) + Rb(5s) has
a particular shape and is thermodynamically unstable and
presents a minimum located at 8.58 a.u. and a high potential barrier of 1006 cm−1 located at 12.03 a.u. This feature is due to the interaction with the upper excited state.
In addition, a very interesting behavior is observed in the
adiabatic potential energy curves for the highly Rydberg
excited electronic states. It corresponds to the spectacular
undulations including multiple barriers and potential wells.
13
142 Page 6 of 19
Theor Chem Acc (2016) 135:142
Fig. 4 Adiabatic potential
energy curves for the 6 lowest
1
Δ (solid line) and 3Δ (dashed
line) of the FrRb molecule
Fr(7d)+Rb(5s)
33∆
31∆
Fr(7s)+Rb(4d)
23∆
21∆
Fr(6d)+Rb(5s)
13∆
11∆
It has been noted that the 61Σ+, 8–121Σ+, 151Σ+, 3–43Σ+,
8–103Σ+ and 73Π electronic states present a specific form
and exhibit double and triple potential wells. These undulations are closely correlated with the oscillations of atomic
radial density in the Rydberg states. Moreover, we observe
that the higher adiabatic potential energy curves present
many avoided crossings at short and large values of internuclear distance between many excited states of Σ+ and
Π symmetries. Their existence can be explained by the
interaction between electronic states and to the chargetransfer process between the former and two ionic systems
Fr+Rb− and Fr−Rb+. The internuclear distances Rc and the
avoided crossing positions between the electronic states are
given in Table 3. In particular, it is noted the avoided crossings among 1Σ+ and 3Σ+ electronic states, namely those
between 51Σ+ and 61Σ+ at 20.41 a.u., between 81Σ+ and
91Σ+ at 40.23 a.u., between 91Σ+ and 101Σ+ at 35.63 a.u.
and at 42.38 a.u., between 101Σ+ and 111Σ+ at 48.65 a. u,
between 111Σ+ and 121Σ+ at 41.16 a.u. and at 66.11 a.u.,
between 121Σ+ and 131Σ+ at 52.73 a.u. and at 66.62 a.u.,
between 63Σ+ and 73Σ+ at 14.56 a.u. and between 93Σ+
and 103Σ+ at 19.33 a.u.
13
To enrich our study on the FrRb molecule, we have
investigated the potential energy curve for the ground
state 11Σ+ using the multireference configuration interaction (MRCI) [46, 47] method, as implanted in the Molpro
program suite, for dealing with the electron correlation
calculation. The state averaged molecular orbitals used
in the MRCI calculations are generated with a multiconfigurational self-consistent field (MCSCF) calculation
[48, 49]. Two calculations were carried out. In these two
calculations, the outer electrons of Fr and Rb atoms are
described with the (24s17p13d10f7g)/[16s13p11d10f7g]
and (24s17p13d10f7g)/[16s13p11d10f7g] basis sets,
respectively. In the first one, we have performed a MRCI
calculation where the 78 and 28 inner electrons of Fr and
Rb atoms are described by the small core pseudopotentials
ECP78MDF and ECP28MDF, respectively [50]. However,
in the second one, we used the same pseudopotential for
Rb and the core pseudopotential CRENBLECP78 for the
Fr. Figure 5 presents a comparison between our potential
energy curves for the FrRb ground state obtained by the
CIPSI and the MRCI methods. As can be seen from this
figure, the well depth of the ground state 11Σ+ of the FrRb
Page 7 of 19 Theor Chem Acc (2016) 135:142 Table 3 Avoided crossing positions (in a.u.) between the electronic
states of FrRb molecule
States
Rc (a.u.)
51Σ+/61Σ+
10.20
20.41
24.43
40.23
13.71
35.63
42.38
48.65
6.40
41.16
66.11
6.49
19.68
52.73
66.62
6.82
52.47
75.97
7.15
9.63
14.56
8.20
15.99
12.14
19.33
81Σ+/91Σ+
91Σ+/101Σ+
101Σ+/111Σ+
111Σ+/121Σ+
121Σ+/131Σ+
131Σ+/141Σ+
23Σ+/33Σ+
53Σ+/63Σ+
63Σ+/73Σ+
83Σ+/93Σ+
93Σ+/103Σ+
63Π/73Π
21.23
molecule obtained using the ab initio calculation is acceptable agreement when compared to the well depth using the
CRENBLECP78 basis set for the rubidium atom. However, the full electron calculations equilibrium distance is
slightly larger than that found using the pseudopotential
approach.
The spectroscopic constants: the equilibrium distances
(Re), the well depths (De), the vertical transition energies
(Te), the harmonicity frequencies (ωe), the anharmonicity frequencies (ωexe) and the rotational constants (Be) are
extracted for all studied states from their numerical potential energy curve data. These spectroscopic constants are
calculated for the 223Fr85Rb isotope using the reduced mass
61.787063 a.u. The spectroscopic constants for the ground
and the low-lying states are collected in Tables 4, 5 and 6.
As no experimental data are available for the FrRb molecule, we discuss our results by comparison with the only
theoretical work performed by Aymar et al. [17]. In Table 4,
we report the spectroscopic constants for the ground
X1Σ+ and the first 13Σ+ excited states compared with the
142
theoretical work of Aymar et al. [17]. These authors have
presented only the equilibrium distance (Re) and the well
depth (De) for the ground X1Σ+ and the first 13Σ+ excited
states. They have been using the same formalism used in
our work and the two data sets A and B. As it can be seen
from Table 4, the agreement between our spectroscopic
constants and those of Aymar et al. [17] is very good.
This good agreement is confirmed by the excellent accord
between our well depth and the equilibrium distance with
Aymar et al. [17] values. For the ground state, we found a
well depth of 3625, 3639 and 3671 cm−1 located at 8.17,
8.18 and 8.19 a. u, using, respectively, our optimal basis
and the two data sets A and B of Aymar et al. [17]. These
values are in good agreement with the spectroscopic constants of Aymar et al. [17]. They found a well depth of
3654 and 3690 cm−1 located at 8.20 a.u. with the data sets
A and B, respectively. For the 13Σ+ state and using our
optimal basis, we found a well depth of 209 cm−1 located
at 12.10 a.u., which are in good agreement with the work
of Aymar et al. [17] (207.20 cm−1 located at 12.0 a.u. and
209.80 cm−1 located at 12.00 a.u. obtained with the two
basis sets A and B, respectively). For the two electronic
states 11Σ+ and 13Σ+, we remark that the influence of the
Fr+ polarizability is clearly larger than in the well depth,
while the equilibrium distances are very close.
Moreover, we have realized a comparison between our
spectroscopic, for the ground state 11Σ+, obtained by the
ab initio method and the calculations using an all electron approach. This comparison shows that the well depth
obtained by the CIPSI package (De = 3625 cm−1) is in
good agreement with the well depth calculated with the
small core pseudopotential ECP78MDF of the Fr atom
(De = 3505 cm−1). However, the equilibrium positions
(Re = 9.03 and 8.99 a.u., respectively) calculated using
the small core pseudopotentials ECP78MDF and CRENBLECP78 are overestimated compared to the equilibrium
position (Re = 8.17 a.u.) obtained by the CIPSI calculation. This discrepancy between our results using the CIPSI
and MRCI methods might be explained by the core–core
repulsion. Tables 5 and 6 are devoted only to our spectroscopic constants for the higher excited states of 1,3Σ+, 1,3Π
and 1,3Δ symmetries obtained using our optimal basis. For
our best knowledge, neither ab initio nor experimental data
exist for the higher excited states of 1,3Σ+, 1,3Π and 1,3Δ
symmetries. They are studied here for the first time. Table 5
presents the spectroscopic constants of the 2–151Σ+,
1–71Π and 1–31Δ electronic states. The spectroscopic constants of the 2–103Σ+, 1–73Π and 1–33Δ electronic states
are presented in Table 6. We remark that several excited
states exhibit double potential wells and sometime triple
wells as in the case of the 9–111Σ+ electronic states. The
double or triple well depths are due to the avoided crossing
13
142 Page 8 of 19
Theor Chem Acc (2016) 135:142
Fig. 5 Comparison between
our potential energy curves of
the ground state of the FrRb
molecule obtained by the ab initio calculations and the MRCI
method
Table 4 Spectroscopic constants for the X1Σ+ and 11Σ+ electronic states of FrRb molecule
State
Re (a.u.)
De (cm−1)
X1Σ+
A
B
A
B
ECP78MDF
CRENBLECP78
13Σ+
A
B
A
8.17
8.18
8.19
8.20
8.20
9.03
8.99
12.10
12.09
12.08
12.00
3625
3639
3671
3654
3690
3505
3352
209
211
217
207.2
B
12.00
209.8
Te (cm−1)
3416
3428
3453
ωe (cm−1)
ωexe (cm−1)
Be (cm−1)
Refs.
Type of ab initio calculation
44.59
44.85
44.69
0.13
0.11
0.08
0.0146
0.0146
0.0146
9.25
10.65
11.06
0.11
0.13
0.14
0.0066
0.0066
0.0066
This work
This work
This work
[17]
[17]
This work
This work
This work
This work
This work
[17]
CIPSI
CIPSI
CIPSI
CIPSI
CIPSI
MRCI
MRCI
CIPSI
CIPSI
CIPSI
CIPSI
[17]
CIPSI
between many electronic states. Their existences have generated large nonadiabatic coupling and have led to an undulating behavior of the higher excited states at large internuclear distances. Moreover, several humps are also found
for the 33Σ+, 21Π and 23Π electronic states with energy
barriers of 1006, 51 and 128 cm−1 located at 12.06, 17.45
and 13.58 a.u., respectively.
13
3.2 Quasi‑diabatic potential energy curves
Recently, we have determined the quasi-diabatic potential energy curves for the LiH [10], LiCs [12] and CsNa
[13]. We extended here the quasi-diabatic study for the
FrRb diatomic molecule. For the quasi-diabatic calculation, the origin is taken at the francium atom. We fixed the
Page 9 of 19 Theor Chem Acc (2016) 135:142 Table 5 Spectroscopic
constants of the excited singlet
electronic states of FrRb
molecule
ωe (cm−1)
ωexe (cm−1)
Be (cm−1)
10,888
13,520
18,021
19,275
35.12
25.92
32.32
31.29
0.05
0.04
0.08
0.29
0.0108
0.0086
0.0092
0.0114
3371
1905
1716
19,956
214,211
22,024
37.70
13.52
30.09
0.07
0.07
0.11
0.0102
0.0030
0.0105
9.27
12.71
3279
2879
23,760
24,198
35.98
20.58
0.13
0.04
0.0113
0.0017
9.26
21.92
36.59
3073
2940
337
24,364
24,497
27,101
23.78
10.69
3.36
0.05
0.01
0.01
0.0114
0.0006
0.0002
10.29
13.49
34.80
3124
2688
986
24,968
25,403
27,105
18.77
10.16
10.46
0.08
0.45
0.03
0.0088
0.0015
0.0003
10.42
33.83
45.29
4091
1763
1239
25,234
27,562
28,086
43.36
4.15
2.33
0.83
0.002
0.001
0.0010
0.0002
0.0001
10.60
39.68
10.31
9.61
3562
1259
3209
3081
25,786
28,089
26,530
26,657
25.96
7.50
31.99
41.00
1.22
0.01
0.19
0.12
0.0087
0.0002
0.0092
0.0106
9.45
23.16
8.74
2889
119
1727
26,849
29,620
14,649
38.37
25.68
36.68
0.19
1.39
0.17
0.0109
0.0005
0.0127
10.03
17.45
9.58
9.86
9.44
9.43
9.23
8.39
10.05
1273
51
756
1418
4439
3412
3299
2590
1570
15,727
27.86
0.03
0.0100
19,261
21,576
22,600
24,026
24,794
17,427
21,423
17.70
26.29
32.33
31.59
17.70
36.52
25.34
0.09
0.17
0.08
0.04
0.10
0.09
0.08
0.0106
0.0100
0.0109
0.0110
0.0120
0.0140
0.0100
9.01
4344
23,748
37.18
0.12
0.0120
State
Re (a.u.)
De (cm−1)
21Σ+
3 1Σ+
4 1Σ+
5 1Σ+
6 1Σ+
First min
Second min
7 1Σ+
8 1Σ+
First min
Second min
9 1Σ+
First min
Second min
Third min
101Σ+
First min
Second min
Third min
111Σ+
First min
Second min
Third min
121Σ+
First min
Second min
131Σ+
141Σ+
151Σ+
First min
Second min
1 1Π
2 1Π
First min
First hump
3 1Π
4 1Π
5 1Π
6 1Π
7 1Π
1 1Δ
2 1Δ
9.51
10.68
10.32
9.24
5487
3481
1996
3719
9.76
17.82
9.64
3 1Δ
142
reference states as the adiabatic ones in the larger internuclear distance equal to 90.00 a.u. We have calculated the
quasi-diabatic potential energy curves related to the 1,3Σ+,
1,3
Π and 1,3Δ symmetries in the adiabatic representation.
Te (cm−1)
In this section, we present only the quasi-diabatic potential energy curves related to the 1Σ+ adiabatic representation. Figure 5 presents the quasi-diabatic potential energy
curves related to the 1–151Σ+ adiabatic electronic states
13
142 Page 10 of 19
Table 6 Spectroscopic
constants of the excited triplet
electronic states of FrRb
molecule
Theor Chem Acc (2016) 135:142
State
Re (a.u.)
De (cm−1)
Te (cm−1)
ωe (cm−1)
ωexe (cm−1)
Be (cm−1)
23Σ+
33Σ+
First min
First hump
4 3Σ+
First min
Second min
5 3Σ+
63Σ+
73Σ+
83Σ+
First min
Second min
9 3Σ+
First min
Second min
103Σ+
First min
Second min
1 3Π
2 3Π
First hump
33Π
4 3Π
5 3Π
6 3Π
7 3Π
First min
Second min
1 3Δ
2 3Δ
10.15
2664
13,712
27.22
0.09
0.0100
8.58
12.03
308
1006
16,692
37.13
0.10
0.0132
9.89
11.87
11.32
9.98
10.18
1088
1031
2647
1735
1259
18,929
18,986
20,347
21,591
22,480
24.55
22.58
41.82
42.77
33.39
0.35
0.12
0.08
0.79
0.56
0.0100
0.0019
0.0076
0.0080
0.0094
9.34
20.55
3166
430
23,873
26,609
40.05
5.80
0.23
0.02
0.0111
0.0006
9.39
22.25
2675
235
24,762
27,202
35.20
5.80
0.10
0.04
0.0106
0.0005
9.58
18.82
8.11
9.42
13.58
9.13
10.35
9.41
9.31
2912
795
6509
374
128
1539
2773
4719
3542
25,180
27,296
9867
16,626
30.52
11.86
46.09
26.54
0.10
0.04
0.04
0.14
0.0106
0.0007
0.0148
0.0110
18,477
20,220
22,343
23,896
26.68
29.65
33.36
34.76
0.33
0.04
0.08
0.08
0.0117
0.0091
0.0110
0.0113
9.20
11.26
8.47
10.00
3370
2984
2139
1427
24,746
25,132
17,877
21,566
36.34
43.32
34.36
25.53
0.10
0.16
0.12
0.02
0.0115
0.0021
0.0136
0.0098
9.13
4140
23,952
35.28
0.12
0.0117
3 3Δ
and named D1–15. We remark that the ionic quasi-diabatic
curve noted D1, dissociating into Fr+Rb−, nicely behaves
as 1/R at intermediate and large internuclear distances. In
addition, we observed that the same quasi-diabatic state D1
crosses the quasi-diabatic curves D2–11 and D13–14 at different interatomic distances. The avoided crossing between
the 1Σ+ electronic states discussed previously in the adiabatic representation is transformed here in the quasidiabatic one into real crossings. The lowest real crossings
occur with the quasi-diabatic state D2 dissociating into
Fr(7s) + Rb(5s) at 12.35 a.u., with the quasi-diabatic state
D3 dissociating into Fr(7s) + Rb(5p) at 14.33 a.u., with the
quasi-diabatic state D4 dissociating into Fr(7p) + Rb(5s) at
17.09 a.u. and with the quasi-diabatic state D5 dissociating
13
into Fr(6d) + Rb(5s) at 18.39.0 a.u. The ionic D1 quasidiabatic state is crossing, the higher quasi-diabatic excited
states, D6–11 and D13–14 at much larger internuclear distances. For these crossings, we found, respectively, 23.18,
24.53, 24.96, 41.02, 47.10, 50.45, 69.69 and 70.04 a.u. The
second ionic quasi-diabatic curve noted in our work D12,
dissociating into Fr−Rb+, is expected to cross the neutral
excited states D9–11 and D13, respectively, in 35.15, 38.32,
41.74 and 58.63 a.u.
3.3 Permanent and transition dipole moments
Obtaining ultra-cold samples of dipolar molecules is a
current challenge which requires an accurate knowledge
Theor Chem Acc (2016) 135:142 Page 11 of 19 142
Fig. 6 Quasi-diabatic potential
energy curves D1–15 related to
the 15 lowest 1Σ+ electronic
states of the FrRb molecule
of their electronic properties to guide the ongoing experiments. Furthermore, their dipole moments can be considered as a sensitive test for the accuracy of the calculated
electronic wave functions and energies. In this context,
the dipole moment is one of the most important parameters determining electric and optical properties of molecules. For this aim, we have calculated the permanent
dipole moments for all the electronic states of 1,3Σ+, 1,3Π
and 1,3Δ symmetries for the FrRb molecule. For our best
knowledge, only the theoretical data of Aymar et al. [17]
are actually available for the permanent dipole moments
of the FrRb molecule. They present only permanent dipole
moments for the ground X1Σ+ and the lowest triple 13Σ+
electronic states. This work has determined the permanent
and transition dipole moments for the large and a dense
grid of internuclear distances, from 5.70 to 76.00 a.u. We
note that our permanent dipole moments for the X1Σ+
and 13Σ+ states are very similar to those of Aymar et al.
[17]. To understand the ionic behavior of the excited electronic states, we have presented in Figs. 6, 7, 8, 9, 10
and 11 the permanent dipole moments for the 1Σ+, 3Σ+,
1
Π, 3Π, 1Δ and 3Δ symmetries, respectively. Figure 6
shows the permanent dipole moments of the 1–41Σ+ (a)
and 5–151Σ+ (b) electronic states. Therefore, here we
observed two ionic quasi-diabatic states. The first one is
the Fr+Rb− state, which is formed between the 8–111Σ+
13
142 Page 12 of 19
Theor Chem Acc (2016) 135:142
Fig. 7 Permanent dipole
moments of the 1–41Σ+ (a) and
5–151Σ+ (b) electronic states of
the FrRb molecule
(a)
41Σ+
21Σ+
11Σ+
31Σ+
141Σ+
151Σ+
(b)
1 +
12 Σ
91Σ+
51Σ+
61Σ+
81Σ+
101Σ+
111Σ+
131Σ+
and 131Σ+ electronic states, and as shown in Fig. 6,
its permanent dipole moments nicely behave as −R, as
expected, although there is some numerical noise around
some avoided crossings. The second one which is formed
between the 9–151Σ+ electronic states is the Fr−Rb+ state
and should present a permanent dipole moment of opposite sign, which is positive. It is clear that the significant
changes of the sign of the permanent dipole moment at
smaller and larger internuclear distances are due to the
change of the polarity in the molecule, going from the
13
Fr+Rb− structure for the negative sign to the Fr−Rb+ structure for the positive sign. This interpretation confirmed the
sign convention as it is demonstrated in many previous
studies [17–19] for the heteronuclear alkali dimers and in
Ref. [14] for the beryllium hydride ion BeH+. In addition,
we remark that the adiabatic permanent dipole moments of
the 5–151Σ+ electronic states, one after another, behaves
as a linear function of R and then drops to zero at particular internuclear distances corresponding to the avoided
crossings between the two neighbor electronic states.
Page 13 of 19 Theor Chem Acc (2016) 135:142 Fig. 8 Permanent dipole
moments of the 1–43Σ+ (a) and
5–103Σ+ (b) electronic states of
the FrRb molecule
142
(a)
3 +
3Σ
13Σ+
23Σ+
43Σ+
103Σ+
(b)
73Σ+
83Σ+
53Σ+
63Σ+
Therefore, we notice that the permanent dipole
moments of the higher excited states present many
abrupt changes corresponding to the avoided crossings
between neighboring electronic states. The positions
of these abrupt changes observed between permanent
dipole moments are correlated with the avoided crossings
between potential energy curves, which are both manifestations of abrupt changes of the character of the electronic
wave functions. Many abrupt changes can be related to
the charge-transfer process between the two ionic systems
93Σ+
Fr+Rb− and Fr−Rb+. We mention here the abrupt changes
between the 91Σ+ and 101Σ+ electronic states located
at 35.51 and 42.85 a.u. However, the avoided crossings
among 1Σ+ electronic states mentioned earlier are also
well indicated by the crossing of R-dependent permanent dipole moments occurring between 51Σ+ and 61Σ+
at 20.33 a.u., between 81Σ+ and 91Σ+ at 39.80 a.u.,
between 101Σ+ and 111Σ+ at 48.40 a.u., between 111Σ+
and 121Σ+ at 40.47 a.u., between 121Σ+ and 131Σ+ at
51.52 a.u. and between 131Σ+ and 141Σ+ at 52.53 a.u.
13
142 Page 14 of 19
Theor Chem Acc (2016) 135:142
Fig. 9 Permanent dipole
moments of the first seven 1Π
electronic states of the FrRb
molecule
61Π
71Π
21Π
51Π
31Π
11Π
41Π
These positions of real crossings between permanent
dipole moments are identical to the regions of internuclear
distances exhibiting avoided crossings. It is clear that the
positions of the irregularities in the R-dependence of permanent dipole moments are correlated with the avoided
crossings between the potential energy curves, which are
both manifestations of abrupt changes of the character of
the electronic wave functions.
The permanent dipole moments have been also determined for the electronic states of the 3Σ+, 1Π, 3Π and 1,3Δ
remaining symmetries. Their permanent dipole moments
are displayed, respectively, in Figs. 7, 8, 9 and 10. As
expected, these permanent dipole moments are not negligible and they become more significant for the higher excited
states. At large distances, they decrease and vanish to zero.
The transition dipole moments between neighbor electronic states have been also determined for the electronic
13
states of 1,3Σ+, 1,3Π and 1,3Δ symmetries. Here, we present
only the transition dipole moments between the neighbor
electronic states of the 1Σ+ symmetry. They are displayed
in Fig. 11. The transition between the ground state 11Σ+
and the excited state 21Σ+, dissociating, respectively, into
Fr(7s) + Rb(5s) and Fr(7s) + Rb(5p), is very large. It
presents a maximum of 5.17 a.u. located at 10.86 a.u. So
it is expected that around this distance there is an important overlap between the corresponding molecular orbitals
(wave functions). At large distances, the 11Σ+–21Σ+ transition becomes a constant equal to 3.15 a.u., corresponding exactly to the atomic transition dipole moment between
Rb(5s) and Fr(7s) to be compared with the experimental
value of 2.99 a.u. found by Sansonetti [51]. In addition,
we observe that all other transitions present many peaks
located at particular distances very close to the avoided
crossings in adiabatic representation (Fig. 12).
Page 15 of 19 Theor Chem Acc (2016) 135:142 Fig. 10 Permanent dipole
moments of the first seven 3Π
electronic states of the FrRb
molecule
142
73Π
23Π
43Π
13Π
33Π
53Π
63Π
4 Conclusion
In the present paper, we have performed an accurate
investigation for several electronic states of the FrRb molecule dissociating into Fr (7s, 7p, 6d, 8s, 8p, 7d, 9s and
9p) and Rb (5s, 5p, 4d, 6s, 6p, 5d and 7s). We have used
a standard quantum chemistry approach based on pseudopotentials, Gaussian basis sets, effective core polarization
potentials and full configuration interaction calculations.
The potential energy curves and their spectroscopic constants for the ground and many excited electronic states
of 1,3Σ+, 1,3Π and 1,3Δ symmetries have been computed
for a large and dense grid of internuclear distances varying from 5.00 to 90.00 a.u. The higher excited states
are studied here for the first time. 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. Our equilibrium distance and the well
depth for the ground X1Σ+ and the first excited 13Σ+
states are in very good agreement with the best theoretical work of Aymar et al. [17]. In addition, a diabatisation study is performed for the first time for the FrRb
molecule in order to determine the quasi-diabatic states.
Their features have revealed the imprint of the two ionic
states Fr+Rb− and Fr−Rb+. In the quasi-diabatic representation, two ionic quasi-diabatic structures Fr+Rb− and
Fr+Rb− are observed. The first noted D1, dissociating
into Fr+Rb−, crosses the quasi-diabatic curves D2–11 and
D13–14 at different interatomic distances. The second ionic
13
142 Page 16 of 19
Theor Chem Acc (2016) 135:142
Fig. 11 Permanent dipole
moments of the first six 1Δ and
3
Δ of the FrRb molecule
33∆
31∆
13∆
11∆
21∆
23∆
quasi-diabatic curve noted in our work D12, dissociating
into Fr−Rb+, is expected to cross only the neutral excited
states D9–11 and D13 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 FrRb molecule, we have performed calculations of the permanent and transition dipole
13
moments. As expected the permanent dipole moments have
shown the presence of two ionic states, corresponding to
the Fr+Rb− and Fr−Rb+ structures, which are an almost
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
Page 17 of 19 Theor Chem Acc (2016) 135:142 Fig. 12 Transition dipole
moments between neighbor
electronic states of 1Σ+ symmetry
142
11Σ+→21Σ+
21Σ+→ 31Σ+
41Σ+→51Σ+
31Σ+→41Σ+
71Σ+→81Σ+
81Σ+→91Σ+
51Σ+→61Σ+
61Σ+→71Σ+
electronic states. The accurate data produced in this work
constitute an important support for both theoretical and
experimental further use.
References
1. Kerman AJ, Sage JM, Sainis S, Bergeman T, DeMille D (2004)
Production and state-selective detection of ultracold RbCs molecules. Phys Rev Lett 92:153001
2. Doyle J, Friedrich B, Krems RV, Masnou-Seeuws F (2004) Topical issue on ultracold polar molecules: formation and collisions.
Eur Phys J D 31:149
3. Wang D, Qi J, Stone MF, Nikolayeva O, Wang H, Hattaway
B, Gensemer SD, Gould PL, Eyler EF, Stwalley WC (2004)
91Σ+→101Σ+
Photoassociative production and trapping of ultracold KRb molecules. Phys Rev Lett 93:243005
4. Mancini MW, Telles GD, Caires ARL, Bagnato VS, Marcassa
LG (2004) Observation of ultracold ground-state heteronuclear
molecules. Phys Rev Lett 92:133203
5. Haimberger C, Kleinert J, Bhattacharya M, Bigelow NP (2004)
Formation and detection of ultracold ground-state polar molecules. Phys Rev A 70:021402
6. Shaffer JP, Chalupczak W, Bigelow NP (1999) Photoassociative ionization of heteronuclear molecules in a novel two-species
magneto-optical trap. Phys Rev Lett 82:1124
7. Højbjerre K, Hansen AK, Skyt PS, Staanum PF, Drewsen M
(2009) Rotational state resolved photodissociation spectroscopy
of translationally and vibrationally cold MgH+ ions: toward rotational cooling of molecular ions. New J Phys 11:055026
8.Foucrault M, Millie Ph, Daudey JP (1992) Nonperturbative method for core–valence correlation in pseudopotential
13
142 Page 18 of 19
calculations: application to the Rb2 and Cs2 molecules. J Chem
Phys 96:1257
9. Aymar M, Guérout R, Sahlaoui M, Dulieu O (2009) Electronic
structure of the magnesium hydride molecular ion. J Phys B: At
Mol Opt Phys 42:154025
10. Berriche H (1994) Ab-initio spectroscopic study of the LiH and
LiH+ molecules beyond the Born–Oppenheimer approximation.
Ph.D. Thesis, Paul Sabatier University, France
11. Mabrouk N, Berriche H (2008) Theoretical study of the NaLi
molecule: potential energy curves, spectroscopic constants, dipole
moments and radiative lifetime. J Phys B: At Mol Opt Phys
41:155101
12. Mabrouk N, Berriche H, Ben Ouada H, Gadea FX (2010) Theoretical study of the LiCs molecule: adiabatic and diabatic potential energy and dipole moment. J Phys Chem A 114:6657
13. Mabrouk N, Berriche H (2014) Theoretical study of the CsNa
molecule: adiabatic and diabatic potential energy and dipole
moment. J Phys Chem A 118:8828
14. Farjallah M, Ghanmi C, Berriche H (2013) Structure and spectroscopic properties of the beryllium hydride ion BeH+: potential energy curves, spectroscopic constants, vibrational levels and
permanent dipole moments. Eur Phys J D 67:245
15. Lim IS, Schwerdtfeger P, Söhnel T, Stoll H (2005) Ground-state
properties and static dipole polarizabilities of the alkali dimers
from Kn2 to Frn2 (n = 0, + 1) from scalar relativistic pseudopotential coupled cluster and density functional studies. J Chem Phys
122:134307
16. Derevianko A, Babb JF, Dalgarno A (2001) High-precision calculations of van der Waals coefficients for heteronuclear alkalimetal dimers. Phys Rev A 63:052704
17. Aymar M, Dulieu O, Spiegelman F (2006) Electronic properties
of francium diatomic compounds and prospects for cold molecule formation. J Phys B: At Mol Opt Phys 39:S905
18. Aymar M, Dulieu O (2005) Calculation of accurate permanent
dipole moments of the lowest 1,3Σ+ states of heteronuclear alkali
dimers using extended basis sets. J Chem Phys 122:204302
19. Lim IS, Lee WC, Lee YS, Jeung GH (2006) Theoretical investigation of RbCs via two-component spin-orbit pseudopotentials:
spectroscopic constants and permanent dipole moment functions.
J Chem Phys 124:234307
20. Pavolini D, Gustavsson T, Spigelmann F, Daudey JP (1989) Theoretical study of the excited states of the heavier alkali dimers. I.
The RbCs molecule. J Phys B: At Mol Opt Phys 22:1721
21. Allouche AR, Korek M, Fakherddin K, Chalaan A, Dagher M,
Taher F, Aubert-Fécon M (2000) Theoretical electronic structure
of RbCs revisited. J Phys B: At Mol Opt Phys 33:2307
22. Fahs H, Allouche AR, Korek M, Aubert-Frécon M (2002) The
theoretical spin-orbit structure of the RbCs molecule. J Phys B:
At Mol Opt Phys 35:1501
23. Korek M, Allouche AR, Kobeissi M, Chaalan A, Dagher M,
Fakherddin K, Aubert-Frécon M (2000) Theoretical study of
the electronic structure of the LiRb and NaRb molecules. Chem
Phys 256:1
24. Korek M, Allouche AR, Fakhreddine K, Chaalan A (2000) Theoretical study of the electronic structure of LiCs, NaCs and KCs
molecules. Can J Phys 78:977
25. Rousseau S, Allouche AR, Aubert-Frécon M (2000) Theoretical
study of electronic structure of the KRb molecule. J Mol Spectrosc 203:235
26. Rousseau S, Allouche AR, Aubert-Frécon M, Magnier S, Kowalczyk P, Jastrzebski W (1999) Theoretical study of the electronic
structure of KLi and comparison with experiments. Chem Phys
247:193
27. Magnier S, Aubert-Frécon M, Millié Ph (2000) Potential energies, permanent and transition dipole moments for numerous
electronic excited states of NaK. J Mol Spectrosc 200:96
13
Theor Chem Acc (2016) 135:142
28. Magnier S, Millié P (1996) Potential curves for selected electronic states of K2 and NaK. Phys Rev A 54:204
29. Schmidt-Mink I, Müller M, Meyer W (1984) Potential energy
curves for ground and excited states of NaLi from ab initio calculations with effective core polarization potentials. Chem Phys
Lett 112:120
30. Petsalakis ID, Tzeli D, Theodorakopoulos G (2008) Theoretical
study on the electronic states of NaLi. J Chem Phys 129:054306
31. Jendoubi I, Berriche H, Ben Ouada HB, Gadea FX (2012) Structural and spectroscopic study of the LiRb molecule beyond the
Born–Oppenheimer approximation. J Phys Chem A 116:2945
32. Bellayouni S, Jendoubi I, Mabrouk N, Berriche H (2014) Systematic study of the electronic proprieties and trends in the LiX
(X = Na, K, Rb, Cs and Fr) molecules. Adv Quant Chem 68:203
33. Barthelat JC, Durand Ph (1975) Theoretical method to determine
atomic pseudopotentials for electronic structure calculations of
molecules and solids. Theor Chem Acta 38:283
34. Müller W, Flesh J, Meyer W (1984) Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. Method and application to alkali and alkaline
earth atoms. J Chem Phys 80:3297
35. Arnold E, Borchers W, Duong HT, Juncar P, Lerme J, Lievens
P, Neu W, Neugart R, Pellarin M, Pinard J, Vialle JL, Wendt K,
ISOLDE (1990) Optical laser spectroscopy and hyperfine structure investigation of the 102S, 112S, 82D and 92D excited levels
in francium. J Phys B: At Mol Opt Phys 23:3511
36. Grossman JM, Fliller RP, Mehlstäubler TE, Orozoco LA, Pearson MR, Sprouse GD, Zaho WZ (2000) Energies and hyperfine splittings of the 7D levels of atomic francium. Phys Rev A
62:052507
37. Gadea FX, Boutalib A (1993) Computation and assignment of
radial couplings using accurate diabatic data for the LiH molecule. J Phys B: At Mol Opt Phys 26:61
38. Boutalib A, Gadea FX (1992) Ab initio adiabatic and diabatic potential-energy curves of the LiH molecule. J Chem Phys
97:1144
39. Gadea FX (1987) Ph.D. Théorie des hamiltoniens effectifs,
applications aux problèmes de diabatisation et de collision réactive. Université Paul Sabatier, Toulouse, France
40. Gadea FX, Pélissier M (1990) Approximately diabatic states: a
relation between effective Hamiltonian techniques and explicit
cancellation of the derivative coupling. J Chem Phys 93:545
41. Berriche H, Gadea FX (1995) Ab initio adiabatic and diabatic permanent dipoles for the low-lying states of the LiH molecule. A direct illustration of the ionic character. Chem Phys Lett
247:85
42. Khelifi N, Zrafi W, Oujia B, Gadea FX (2002) Ab initio adiabatic
and diabatic energies and dipole moments of the RbH molecule.
Phys Rev A 65:042513
43. Zrafi W, Oujia B, Gadea FX (2006) Theoretical study of the CsH
molecule: adiabatic and diabatic potential energy curves and
dipole moments. J Phys B: At Mol Opt Phys 39:3815
44. Romero T, Aguilar A, Gadea FX (1999) Towards the ab initio
determination of strictly diabatic states, study for (NaRb)+. J
Chem Phys 110:6219
45. Gadea FX (1991) Variational effective hamiltonians as a generalization of the Rayleigh quotient. Phys Rev A 43:1160
46. Werner HJ, Knowles PJ (1988) An efficient internally contracted
multiconfiguration–reference configuration interaction method. J
Chem Phys 89:5803
47. Knowles PJ, Werner HJ (1988) An efficient method for the evaluation of coupling coefficients in configuration interaction calculations. Chem Phys Lett 145:514
48. Werner HJ, Knowles PJ (1985) A second order multiconfiguration
SCF procedure with optimum convergence. J Chem Phys 82:5053
Theor Chem Acc (2016) 135:142 49. Knowles PJ, Werner HJ (1985) An efficient second-order MC
SCF method for long configuration expansions. Chem Phys Lett
115:259
50. Lim IS, Schwerdtfeger P, Bernhard M, Hermann S (2005) Allelectron and relativistic pseudopotential studies for the group 1
element polarizabilities from K to element 119. J Chem Phys
122:104103
Page 19 of 19 142
51. Sansonetti JE (2006) Wavelengths, transition probabilities, and
energy levels for the spectra of rubidium (Rb I through RbXXXVII). J Phys Chem Ref Data 35:301
13