Structural and Spectroscopic Study of the LiRb Molecule beyond the

Article pubs.acs.org/JPCA

Structural and Spectroscopic Study of the LiRb Molecule beyond the Born−Oppenheimer Approximation I. Jendoubi,† H. Berriche,*,†,‡ H. Ben Ouada,† and F. X. Gadea§ †

Laboratoire des Interfaces et Matériaux avancés, Département de Physique, Faculté des Sciences de Monastir Université de Monastir, Avenue de l’Environnement 5019 Monastir, Tunisia ‡ Physics Department, College of Science, King Khalid University, P.O.B. 9004, Abha, Saudi Arabia § Laboratoire de Chimie et Physique Quantique, UMR5626 du CNRS Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex 4, France ABSTRACT: Adiabatic and diabatic potential energy curves and the permanent and transition dipole moments of the lowlying electronic states of the LiRb molecule dissociating into Rb(5s, 5p, 4d, 6s, 6p, 5d, 7s, 6d) + Li(2s, 2p) have been investigated. The molecular calculations are performed with an ab initio approach based on nonempirical pseudopotentials for Rb+ and Li+ cores, parametrized l-dependent core polarization potentials and full configuration interaction calculations. The derived spectroscopic constants (Re, De, Te, ωe, ωexe, and Be) of the ground state and lower excited states are in good agreement with the available theoretical works. However, the 8−101Σ+, 8−103Σ+, 61,3Π, and 31,3Δ excited states are studied for the first time. In addition, to the potential energy, accurate permanent and transition dipole moments have been determined for a wide interval of internuclear distances. The permanent dipole moment of LiRb has revealed ionic characters both relating to electron transfer and yielding Li−Rb+ and Li+Rb− arrangements. The diabatic potential energy for the 1,3Σ+, 1,3Π, and 1,3Δ symmetries has been performed for this molecule for the first time. The diabatization method is based on variational effective Hamiltonian theory and effective metric, where the adiabatic and diabatic states are connected by an appropriate unitary transformation.

1. INTRODUCTION During the past decade, an important effort has been devoted to the development of ultrashort and intense laser pulses opening a new research domain in molecular dynamics and molecular spectroscopy. Such an important theoretical and experimental effort is recently motivated by the possible formation, manipulation, and controlling of ultracold molecule by photoassociation.1−5 The synthesis of alkali metal dimers by photoassociation needs a very good knowledge of the electronic structure of these dimers. Furthermore, the new development of cold and ultracold atomic trapping techniques has increased the interest in experimental and theoretical investigation on homonuclear and also heteronuclear alkali metal molecular systems. This important theoretical and experimental effort is motivated by possible applications such as manipulation and controlling of ultracold chemical reactions,6−8 ultracold molecular collision dynamics,9−12 quantum computing,13,14 and experimental preparation of few-body quantum effects15 (such as Efimov states) where the aim is to prepare molecules in definite quantum states with respect to the center of mass and electronic, rotational, and vibrational motions.16 Moreover, it is suspected that the long-range dipole−dipole interaction will introduce new important physical phenomena. The growing development of the laser cooling and optical trapping technique demands accurate knowledge of the spectroscopic © 2012 American Chemical Society

proprieties of the alkali metal dimers. In this context, several theoretical17−20 and experimental studies12,16,21 have been performed for the alkali metal dimers.These studies are related to the homonuclear dimers as well as to the heteronuclear ones. Müller and Meyer22 have studied all molecules XY (X, Y = Li, Na, K). They have determined the spectroscopic constants, ionization energies, and dipole moments for the ground state only. Recently, an extensive ab initio study on heteronuclear mixed alkali metal pairs LiNa, LiK, LiRb, LiCs, NaK, NaRb, NaCs, KRb, KCs, and RbCs has been performed by Aymar and Dulieu23,24 to produce accurate potential energy and permanent dipole moments for the ground state and the lower triplet states. These authors determined permanent dipole moments for each system by using two basis sets noted A and B. Their method of calculation uses the technique of pseudopotential, an SCF calculation followed by an interaction of configuration (CI). Recently, in our group Mabrouk and Berriche have studied extensively the molecules LiCs,25 LiNa,26,27 and NaCs.28They identified the potential energy curves of a large number of states and their spectroscopic constants. They also determined the permanent and transition dipole moments and the radiative Received: September 21, 2011 Revised: February 20, 2012 Published: February 23, 2012 2945

dx.doi.org/10.1021/jp209106w | J. Phys. Chem. A 2012, 116, 2945−2960

The Journal of Physical Chemistry A

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Table 1. Asymptotic Energy of LiRb Electronic States (cm−1) asymptotic molecule state Li(2s) + Rb(5s) Li(2s) + Rb(5p) Li(2p) + Rb(5s) Li(2s) + Rb(4d) Li(2s) + Rb(6s) Li(2s) + Rb(6p) Li(2s) + Rb(5d) Li(2s) + Rb(7s) Li(2p) + Rb(5p) Li(2s) + Rb(6d)

( Σ) 1,3 +

this work 0.0

expt.61,62 0.0

theo.32 0.0

|E − Eexpt| 0.0

Σ,1

1,3

Π)

12737.23

12737.36

12737.4

0.130

1,3 +

Σ,2

1,3

Π)

14905.42

14903.89

14905.4

1.530

(4 1,3Σ+, 3 1 1,3Δ) (5 1,3Σ+)

1,3

Π,

19355.06

19355.27

19355.0

0.210

20101.27

20133.60

20101.5

32.33

Σ , 41,3Π)

23799.43

23766.86

23799.6

32.57

(7 1,3Σ+, 51,3Π, 21,3Δ) (8 1,3Σ+)

25709.52

25702.336

25709.7

26301.00

26311.437

10.437

(9 1,3Σ+, 6 1,3Π, 3 1,3Δ) (10 1,3Σ+, 7 1,3 Π, 41,3Δ)

27642.65

27601.416

41.234

29011.30

28688.258

323.042

(2

1,3 +

(3

(6

1,3 +

and electric properties of the ground state. They have used the coupled cluster method which can be applied in the framework of the spin-free Dirac formalism. Several theoretical32−43 studies have been performed for the LiRb molecule. The first theoretical study of the LiRb molecule is made by Igel-Mann et al.40 on alkali metal homonuclear and heteronuclear dimers XY (X, Y = Li to Cs). These authors used the method of the pseudopotential and CI calculations. They have determined the spectroscopic constants for the ground state. On the other hand, Urban et al.41 have calculated the dipole moment and dipole polarizabilities of a series of the alkali metal atom involving Li, Na, K, and Rb. These authors have used different coupled cluster approximations as well as complete active space self-consistent field (SCF) approaches followed by perturbative treatment for the dynamical correlation effects.The aim of their work wasthe interpretation of the electric property of the alkali metal dimers, which is based on the analysis of the valence and core-polarization contributions. Korek et al.32 have made an ab initio calculation for the LiRb molecule, where they have determined the potential energy curvesfor 28 electronic states. The spectroscopic constants (Te, Re, ωe, Be) for 7, 5, and 2 states of, respectively, 1,3Σ+, 1, 3Π, 1,3Δ symmetries were presented. The LiRb molecule was also treated by many authors such as Smirnov,42 who has calculated vibrational, rotational, and centrifugal spectroscopic constants for the ground state by introducing an analytical form for the potential energy curves of Korek et al. 33 His calculation was performed by the perturbation method on the basis of semiempirical potential curves constructed in a wide range of internuclear distance.

7.184

lifetimes for the vibrational levels of the 2 and 31Σ+ excited states. Other authors were interested in studying the heteronuclear alkali metal dimers with various quantum chemistry methods to determine the potential energy curves of the ground and exited states of RbCs,29−31 NaRb and LiRb,32 NaCs, LiCs, and KCs,33 KRb,34 LiK,35 NaK,36,37 and LiNa.38 The LiCs molecule was also treated by many authors; recently, Sørensenet al.39 have determined the spectroscopic constants Re, De, and ωe. They have presented a systematic study of electron correlation and relativistic effects on the spectroscopic

Figure 1. Adiabatic potential energy curves of the 1−10 1Σ+ electronic states of LiRb. 2946



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Figure 2. Adiabatic potential energy curves of the 1−10 3Σ+ electronic states of LiRb.

Table 2. Avoided Crossing Positions state 4 Σ /5 Σ 61Σ+/71Σ+ 71Σ+/81Σ+ 81Σ+/91Σ+ 91Σ+/101Σ+ 43Σ+/53Σ+ 63Σ+/73Σ+ 83Σ+/93Σ+ 93Σ+/103Σ+ 31Π/41Π 41Π/51Π 1 +

1 +

position (au)

state

position (au)

24.8 46.4 14.9 18.6 15.9 8.90 7.10 20.6 25.6 5.89 3.60 8.74

5 Π/6 Π

3.75 8.0 5.56 7.25 15.25 9.75 19.95 3.7 6.75 25 4.25 8.80 6.50

1

1

33Π/43Π

43Π/53Π 53Π/63Π

21Δ/31Δ 23Δ/33Δ

the spin−orbit effect within the range 3.0−34.0 au of the internuclear distance R. The spin−orbit effects have been taken into account through a semiempirical spin−orbit pseudopotential added to the electrostatic Hamiltonian with Gaussian basis sets for both atoms. Recently, Ivanova et al.44 have investigated the ground sate, X 1 + Σ , of LiRb by using the Fourier transform LIF spectroscopy. More recently, Dutta et al.45 have studied the X 1Σ+ and 1 1Π states of 7Li85Rb, by laser induced fluorescence (LIF) spectroscopy and fluorescence excitation spectroscopy (FES). These authors have extracted the molecular constants for levels v = 0−2 of the X 1Σ+ state and the levels v = 0−20 of the 11Π state.

Figure 3. Adiabatic potential energy curves of the seven lowest 1Π states of LiRb.

Recently, Korek et al.43 have calculated potential energy for the 58 lowest electronic states of the molecule LiRb including 2947



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Figure 4. Adiabatic potential energy curves of the seven lowest 3Π states of LiRb.

Figure 5. Adiabatic potential energy curves of the three lowest

1,3

Δ states of LiRb. 2948



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Table 3. Spectroscopic Constants of the 1Σ+ States of the LiRb Molecule state

Re (Å)

De (cm−1)

3.428 3.466 3.43 3.428B 3.449A 3.45 3.497

5968 5927.9(40) 5959a

Te (cm−1)

ωe (cm−1)

ωeχe (cm−1)

Be (cm−1)

ref

196.02

1.44

0.223

0.220

this work expb CIc,d CIe,f CIe,f CIg CCSD(T)-SFh PMi

X Σ

1 +

194.0

0.220

195.0 6180.4 ± 170

194.0

(2) 1Σ+ 4.137 4.138

7053 7057a

11654 11639

118.78 119.6

1.04

0.153 0.152

this work CIc,d

4.243 4.257

3494 3516a

17382 17348

114.24 113.0

1.22

0.145 0.144

this work CIc,d

7.671 7.629a

3998 3995a

21326 21318a

41.27 41.68a

0.10

0.044 0.044a

this work CIc,d

3.962 3.952a 11.650 11.68a

3360 3372a 598 595a

22702 22688a 25473 25465a

211.81 212.27a 13.10 14.70a

7.07

0.167 0.167a 0.019 0.019a

this work CIc,d this work CIc,d

4.021 4.026a 12.666 12.624a

4438 4434a 3502 3498a

25330 25325a 26266 26260a

132.28 132.34a 22.38 21.66a

0.162 0.162a 0.016 0.162a

this work CIc,d this work CIc,d

4.301 4.306a 21.470 22.269a 4.116 3.947

4535 4509a 1979 1910a 3803 3907

27143 27159a 29700 29758a 27874 28362

98.04 103.52a 11.10 8.70a 137.00 141.10

1.91 1.70

0.141 0.141a 0.005 0.002a 0.154 0.168

this work CIc,d this work CIc,d this work this work

4.021 5.772

3668 2754

29512 30426

134.67 119.09

1.61 1.61

0.162 0.078

this work this work

(3) 1Σ+

(4) 1Σ+

(5) 1Σ+ first min second min (6) 1Σ+ first min second min (7) 1Σ+ first min second min (8) 1Σ+ (9) 1Σ+ (10) 1Σ+ first min second min

7.07

0.86 0.86

0.33 0.33

a

These values are extracted from the Korek et al.32 potential energy curves available in their Web site.71 bReference 44. cReference 32. dReference 71. eReference 23. fReference 24. gReference 40. hReference 41. iPM (perturbation method) ref 42.

parts, is devoted to present our results: the adiabatic and diabatic potential energy curves, their spectroscopic constants, and the permanent and transition dipole functions. Finally, we will summarize and conclude in section 4.

The present work completes our theoretical study on the LiRb alkali metal molecule published as an extended abstract in the AIP Conf. Proc.46 We have presented the adiabatic potential energy curves and derived their spectroscopic constants for the 1−10 1Σ+ states. In this paper we generalize our study to the remaining symmetries, to the diabatic states, and to dipole moment functions. Therefore, we have performed the adiabatic and diabtic potential energies curves for 38 electronic states of different symmetries 1,3Σ+, 1,3Π, and 1,3Δ and we have extracted their spectroscopic properties (Re, De, Te, ωe, ωexe, and Be), which will be compared with the available theoretical and experimental works. In addition, we have performed the adiabatic and diabatic permanent and transition dipole functions for a better understanding of the ionic character of the LiRb molecule. We present in section 2 a summary of the ab initio calculation, the numerical method based on CI calculation, and the diabatization method. Section 3, which is divided into three

2. METHOD OF CALCULATION 2.1. Adiabatic Calculation. The Li and Rb atoms are treated through the nonempirical pseudopotential proposed by Durand and Barthelat47,48 and used in many previous works.49−55 For the interaction between the polarizable Li+ and Rb+ cores with the valence electrons, a core polarization potential VCPP is used according to the operatorial formulation of Müller et al.56 VCPP = − 2949

1 2

∑ αλfλ⃗ . fλ⃗ λ dx.doi.org/10.1021/jp209106w | J. Phys. Chem. A 2012, 116, 2945−2960


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Table 4. Spectroscopic Constants of the 3Σ+ States of the LiRb Molecule state

Re (Å)

De (cm−1)

5.126 5.126 5.140

276 280a 277.6(40)

Te (cm−1)

ωe (cm−1)

ωeχe (cm−1)

Be (cm−1)

ref

5693 5678

40.13 41.2

17.46

0.098 0.100

this work CIb,c expd

(1) Σ

3 +

(2) 3Σ+ 4.058 4.064

3969 3977a

14737 14719

128.63 129.9

1.09

0.159 0.157

this work CIb,c

3.904 3.915a

362 365a

20513 20499a

136.61 136.46a

1.82

0.171 0.171a

this work CIb,c

4.137 4.142a 5.433 5.433a

1527 1529a 1983 1997a

23797 23784a 23340

117.85 117.5a 109.58 109.44a

1.74

0.153 0.153a 0.088 0.088a

this work CIb,c this work CIb,c

(3) 3Σ+

(4) 3Σ+ first min second min

1.74

23317a (5) Σ

3 +

(6) 3Σ+ first min second min

4.740 4.672a

1964 1964a

24106 24101a

215.11 215.93a

43.55

0.116 0.120a

this work CIb,c

4.343 4.343a 9.560 9.560a

3598 3596a 444 433a

26170 26162a 29324 29325a

175.31 175.65a 19.35 19.36a

6.02

0.139 0.139a 0.028 0.028a

this work CIb,c this work CIb,c

4.158 4.074a 3.989

4373 4357a 4243

27306 27314a 28026

130.50 128.02a 147.88

1.24 2.10

0.159 0.159a 0.165

this work CIb,c this work

4.037 10.555

4093 908

29087 32272

132.91 36.87

1.08 1.08

0.160 0.023

this work this work

3.936 7.798

3628 961

29984 32650

146.90 34.41

1.59 1.59

0.169 0.043

this work this work

6.02

(7) 3Σ+

(8) 3Σ+ (9) 3Σ+ first min second min (10) 3Σ+ first min second min a

These values are extracted from the Korek et al.32 potential energy curves available in their Web site.71 bReference 32. cReference 71. Reference 44.

d

where αλ is the dipole polarizability of the core λ and fλ⃗ is the electric field produced by valence electrons and all other cores on the core λ. fλ⃗ =

∑ i

ri ⃗λ ri λ 3

F( ri ⃗λ ,ρλ) −

∑

⃗ Rλ′λ

3 λ′≠λ Rλ′λ

9s8p4d1f/8s6p3d1f and 7s5p5d2f/6s4p4d1f orbitals as used in previous works of some of our colleagues.23,55 Table 1 presents a comparison between our calculation and the experimental61,62 and theoretical32 values of ionization potential (IP) and atomic energy levels. The difference with the experimental levels61,62 is ranging between 0.13 and 323.042 cm−1. To conclude, a very good agreement between our theoretical atomic energy levels and the experimental ones is observed. Such accuracy is important for the molecular calculation because it ensures correctly positioned asymptotic limits. 2.2. Diabatic Calculation. Although the diabatization method used in this work was applied previously with success for several diatomic molecules,63−68 its application for the LiRb can be considered as the first. This method is based on variational Effective Hamiltonian theory65 and effective metric.67 We briefly describe this method in this section as it was described in detail in previous studies. The idea is to cancel the nonadiabatic coupling between the considered adiabatic states by an appropriate unitary transformation. This unitary transformation matrix U, will provide us with the quasi-diabatic energies and wave functions. The quasi-diabatic wave functions are therefore written as a linear combination of the adiabatic

Zλ

where ri⃗ λ is a core−electron vector and R⃗ λ′λ is a core−core vector. According to the formulation of Foucrault et al.,57 the cutoff function F(ri⃗ λ,ρλ) is taken to be a function of l to consider differently the interaction of valence electrons of different spatial symmetry with core electrons. In this context, the LiRb molecule is considered as a two valence electrons system. Furthermore, the self-consistent field calculation (SCF) is followed by a full valence configuration interaction (CI) calculation using the Toulouse package (CIPSI, MOYEN, BDAV).58,59 The core dipole polarizability of Rb and Li are respectively 9.245 and 0.1915 a03.57,60 The cutoff radii for the lowest one-electron valence s, p, d, and f are respectively 1.434, 0.982, 0.600, and 0.400 au for lithium55 and 2.5213, 2.2790, 2.5110, and 2.5110 au for the rubidium.30 For the Li and Rb atoms, we have used Gaussian basis sets of, respectively, 2950



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Table 5. Spectroscopic Constants of the 1Π, 3Π, 1Δ, and 3Δ states of the LiRb Molecule state

Re (Å)

De (cm−1)

Te (cm−1)

ωe (cm−1)

ωeχe (cm−1)

Be (cm−1)

ref

(1) Π

3.873

1461 1634.5(45) 1480a

17245 17110 17205

116.12 122.3(3) 140.5

2.92

0.175 0.179

this work expb CIb,c

19235 19201 24335 24295 25850 25839 27415 27447 28992 10249 10233 19464 19484 23269 23262a 25285 25272a 27115 27125a 27580 22277 22275 27913 27936 28522 23046 23029 28091 28142 29336

122.08 141.7 81 116.8 127.0 129.8 122.9 115.0 126.7 192.0 191.7 103.3 39.4 74.29 71.94a 198.74 193.31a 108.10 120.55a 138.9 143.2 141.7 138.9 142.0 137.3 133.5 133.5 131.3 129.7 153.15

1.83

0.157 0.155 0.125 0.124 0.169 0.165 0.161 0.164 0.157 0.235 0.232 0.156 0.160 0.081 0.082a 0.168 0.169a 0.144 0.146a 0.167 0.19 0.188 0.177 0.175 0.163 0.174 0.173 0.164 0.162 0.172

this work CIb,c this work CIb,c this work CIb,c this work CIb,c this work this work CIb,c this work CIb,c this work CIb,c this work CIb,c this work CIb,c this work this work CIb,c this work CIb,c this work this work CIb this work CIb this work

1

(2) 1Π (3) 1Π (4) 1Π (5) 1Π (6) 1Π (1) 3Π (2) 3Π (3) 3Π (4) 3Π (5) 3Π (6) 3Π (1) 1Δ (2) 1Δ (3) 1Δ (1) 3Δ (2) 3Δ (3) 3Δ

3.813 4.084 4.101 4.570 4.582 3.936 3.971 4.037 3.983 4.084 3.338 3.348 4.100 4.115 5.661 5.650a 3.941 3.941a 4.259 4.238a 3.962 3.714 3.720 3.851 3.857 4.150 3.878 3.877 3.994 4.004 3.899

1639 1638a 989 997a 3918 3911a 4263 4239a 4618 8457 8464a 1411 1412a 2055 2052a 4483 4486a 4562 4543a 6030 3063 3035a 3781 3737a 5107 2294 2289a 3604 3530a 4292

a c

These values are extracted from the Korek et al. Reference 71.

32

1.34 1.52 1.78 1.16 0.860 1.820 1.09 14.44 2.19 1.09 1.6 1.42 0.84 1.67 1.78 1.51

potential energy curves available in their Web site.71 bReference 45. bReference 32.

space). The overlap matrix involved in the projection is obviously an overlap matrix over nonorthogonal functions because the two sets are related to different interatomic distances. The diabatization method based on effective Hamiltonian theory uses reference states usually taken at infinite distance corresponding to the dissociation limit. In our study the reference states correspond to the adiabatic ones taken at R = 96.8 au. Thus, at this distance the adiabatic and diabatic states coincide and all adiabatic states have reached their asymptotic limits, except the ionic one. We take the origin on the heavier atom, which corresponds here to the rubidium atom. To summarize, this diabatization scheme is based on the overlap matrix between the reference states and the adiabatic ones, which corresponds to a crude numerical estimation of the nonadiabatic coupling but does not involve the electric dipole matrix at all. This overlap matrix is made unitary thanks to a symmetric orthogonalization procedure, similarly to the des Cloizeaux alternative effective Hamiltonian approach.70

Table 6. Permanent Dipole Moments (Debye) at Equilibrium Distance of the Ground State of the RbLi Molecule state

Re (Å)

μ (Debye)

ref

X 1Σ+

3.428 3.39 3.45 3.50 3.45A 3.43B

4.78 4.46 4.13 4.34 4.168 4.142 4.0 4.05

our work CIa CIb CCSD(T)-SFc CId,e CId,e expf estimatedg

a

Reference 43. bReference 40. cReference 41. dReference 23. Reference 24. fReference 79. gReference 8 (empirical estimation).

e

ones. This crude nonadiabatic coupling estimation is closely related to an overlap matrix between the R-dependent adiabatic multiconfigurational states and the reference states corresponding to a set of adiabatic states at a fixed large distance. The quasi-diabatic states result from the symmetrical orthonormalization of the projection of the model space wave functions (references) onto the selected adiabatic wave functions (target

3. RESULTS AND DISCUSSION 3.1. Adiabatic Potential Energy and Spectroscopic Constants. The adiabatic potential energy curves of the 2951



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Figure 6. Diabatic potential energy curves of the ten lowest 1Σ+ states of LiRb.

electronic states of 1,3Σ+, 1,3Π, and 1,3Δ symmetries dissociating into Rb(5s, 5p, 4d, 6s, 6p, 5d, 7s, 6d) + Li(2s, 2p) have been investigated for a dense and wide grid of internuclear distance ranging from 5.40 to 96.80 au. The 1Σ+ and 3Σ+ electronic states are displayed, respectively, in Figures 1 and 2, whereas the 1,3Π and 1,3Δ electronic states are displayed, respectively, in Figures 3−5. It is worthwhile to note that the 8−10 1Σ+, 9−10 3 + Σ , 6 1,3Π, and 3 1,3Δ excited states are studied here for the first time. An interesting general behavior can be observed for the higher excited states. It corresponds to the existence of undulations, which can provide double wells and sometimes triple wells. For example, the 6−7 1Σ+ and 10 1Σ+ states have a specific form; they exhibit two large potential wells. Many avoided crossings for the higher excited states are observed (Table 2), which could be explained by the interaction between the two ionic states Rb+ Li− and Rb− Li+. To evaluate the quality of our calculated potential energy curves for the nearly 38 states studied, we have extracted their spectroscopic constants (Re, De, Te, ωe, ωexe, and Be) and compared them with the available theoretical and experimental works (Tables 3−6). Our spectroscopic data, especially for the ground state, will be compared with the theoretical works of Korek et al.,32 IgelMann et al.,40 Urban et al.41 and the experimental works of Ivanova et al.44 and Dutta et al.45 However, for the excited states, our data will be compared only with those of Korek et al. In addition, as they did not present in their paper the well depth, we have extracted this value from the potential energy presented on their Web site.61 For some other higher excited states all spectroscopic data were extracted from the data in their Web site. Our Re for the ground state (Re = 3.428 Å) is in

very good agreement with the one of Korek et al.,32 Igel-Mann et al.,40 Urban et al.,41 and Aymar and Dulieu,23,24 who found respectively 3.43, 3.45, 3.497, and 3.428 Å. An excellent agreement is observed between our well depth for this state of 5968 cm−1 and the value found by Korek et al.32 of 5959 cm−1. Smirnov42 has redetermined the spectroscopic constants (De = 6180 ± 170 cm−1, ωe = 194.0 cm−1, and Be = 0.220 cm−1) for the ground states by introducing an analytical form using the data of Korek et al.32 We remark that he has almost reproduced the Korek et al. data, except De, which is overestimated. The comparison with the experimental results for the ground state, X 1Σ+, is made for the equilibrium distance Re and the dissociation energy De of ref 44. Our Re and De are in very good accord with the values found by Ivanova et al.44 (Re = 3.466 Å, De = 5927.9 cm−1). Dutta et al.45 did not report spectroscopic constants for the ground state as they found them in excellent agreement with those of Ivanova et al. Our results for the higher excited states are compared especially with those of Korek et al. presented in their paper or extracted from their data presented in their Web site.71 Our spectroscopic constants for the first 2 1Σ+ excited state (Re = 4.137 Å, De = 7053 cm−1, Te = 11654 cm−1, ωe = 118.78 cm−1, and Be = 0.153 cm−1) are in good agreement with the work of Korek et al.32 (Re = 4.138 Å, De = 7057 cm−1, Te = 11639 cm−1, ωe = 119.6 cm−1, and Be = 0.152 cm−1). It is important to note the existence of double wells for the 5, 6, and 7 1Σ+ states. Their first minima are located at, respectively, 3.962, 4.021, and 4.301 Å, which are very close to that of the ionic system LiRb+ 72 (Re = 4.074 Å). However, their second minima are located respectively, at 11.65, 12.666, and 21.47 Å. The 8−10 1Σ+ higher excited states are studied here for the first time. The 2952



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Figure 7. Diabatic potential energy curves of the ten lowest 3Σ+ states of LiRb.

Figure 8. Diabatic potential energy curves of the seven lowest 1Π states of LiRb.

same good agreement is observed for the 1Σ+ higher excited states as well as for the other symmetries. Ivanova et al have

also investigated the 1 3Σ+ (a 3Σ+) excited state. They found Re = 5.140 Å and De = 277.6 cm−1, which are in excellent accord with 2953



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Figure 9. Diabatic potential energy curves of the seven lowest 3Π states of LiRb.

Figure 10. Diabatic potential energy curves of the three lowest 1Δ states of LiRb. 2954



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Figure 11. Potential energy diabatic curves of the three lowest 3Δ states of LiRb.

Figure 12. Adiabatic permanent dipole moment of the 1, 2, and 3 1Σ+ states of LiRb.

Figure 13. Adiabatic permanent dipole moment of the first ten 1Σ+ states of LiRb. 2955



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Figure 16. Adiabatic permanent dipole moment of the seven 1Π states of LiRb. Figure 14. Adiabatic permanent dipole moment of the 1, 2, 3, and 4 3 + Σ states of LiRb.

Figure 15. Adiabatic permanent dipole moment of the first ten 3Σ+ states of LiRb.

Figure 17. Adiabatic permanent dipole moment of the seven 3Π states of LiRb.

our results (Re = 5.126 Å, De = 276 cm−1). The spectroscopic constants for all remaining states are gathered in details in Tables 3−5. 3.2. Diabatic Potential Energy. The diabatization method has been presented in details previously,63−68,73,74 and its

application has shown its efficiency on several molecules such as LiH.63,69,70 However, we can consider the present work as the first application of this method for the LiRb heteronuclear alkali metal diatomic system. The diabatization method was tested, first, for the CsH molecule75 and applied later for LiH63,65,69,75 and RbH,66 and 2956



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Figure 20. Diabatic permanent dipole moment of the first ten 1Σ+ states of LiRb. Figure 18. Adiabatic permanent dipole moment of the three 1Δ states of LiRb.

Figure 21. Transition dipole moment: X 1Σ+−2 1Σ+, 2 1Σ+−3 1Σ+, and 3 1Σ+−1Σ+.

more recently for the LiCs,25 LiNa,25−27 and NaCs28 molecules determined in our group. This diabatization method can be considered as among the most effectives for molecular ab initio

Figure 19. Adiabatic permanent dipole moment of the three Δ states of LiRb. 3

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Figure 23. Transition dipole moment: 8 1Σ+−9 1Σ+ and 9 1Σ+−10 1Σ+. Figure 22. Transition dipole moment: 4 1Σ+−5 1Σ+, 5 1Σ+−6 1Σ+, 6 1 + Σ −7 1Σ+, and 7 1Σ+−8 1Σ+.

respectively, with the Li atom in its ground state 2s, and the diabatic potentials simply reflect the amplitude of these Rydberg states. However, this qualitative analysis is difficult to be pursued because there are also interactions between the diabatic states, and when both atoms are in Rydberg states the Fermi model cannot be easily applied. However, huge basis sets are used here for both atoms and we found some numerical difficulties in the diabatization procedure related to quasi linear dependence. The effective metric method may found here some limitations. This was not the case in the previous application, for the alkali metal hydride series, for example, where the H atom needed only a limited basis set because it was involved mainly in its ground state and a large basis set was used only for the alkali metal atom. 3.2. Permanent and Transition Dipole Moments. Obtaining ultracold dipolar heteronuclear alkali metal molecules is currently a challenge, which requires the knowledge of accurate electronic properties. To complete this work, the adiabatic permanent and transition dipole moments are determined for the same large and dense grid of intermolecular distances, from 5.4 to 96.8 au. To shed light on the ionic behavior of the excited electronic states, we have presented the permanent dipole moments of the first ten 1Σ+ states as a function of the intermolecular distance, R. In Table 6, we have determined the permanent dipole moment (in Debye) at equilibrium distance of the ground state. We remark that our calculation is very close the results found by Korek et al.43 Indeed, we found the permanent dipole moment μ = 4.78 D, whereas Korek et al.43 found μ = 4.46 D; this difference could be due to spin−orbit effects explicitly considered by Korek et al.43 Figure 12 presents the permanent dipole moment of 1, 2, and 3 1Σ+ states. We remark that the dipole moment of the 2 1 + Σ state is important mainly in the region of their Li−Rb+ ionic character.

calculations as shown in ref 76. We take the origin on the heavier atom, which corresponds here to the rubidium atom, and we fixed the reference states as the adiabatic ones at the largest internuclear distance, 96.8 au. Figure 6 presents the first ten diabatic potential energy curves related to the 1−10 1Σ+ adiabatic ones; they are determined here for the first time. We notice that the ionic state D1, which nicely behaves as 1/R at large internuclear distances and dissociates into Li−Rb+, crosses all the other D2−10 curves that possess neutral character. Such crossings occur with the electronic states dissociating into Li(2s) + Rb(5s), Li(2s) + Rb(5p), Li(2p) + Rb(5s), Li(2s) + Rb(4d), Li(2s) + Rb(6s), Li(2s) + Rb(6p), Li(2s) + Rb(5d), and Li(2s) + Rb(7s) states at internuclear distance around 10.66, 16.17, 16.70, 24.16, 26.48, 44.77, 72.34, and 91.36 au. Such crossings become avoided crossings in the adiabatic scheme. A higher ionic state, dissociating into Li+Rb−, which crosses higher neutral excited states at shorter distances is also observed. These real crossings in the diabatic representation are transformed into avoided crossings in the adiabatic one; the positions of such avoided crossings are presented in Table 2. A diabatization is also realized for the remaining symmetries. Figures 7−11 present the diabatic potential energy curves for, respectively, 1−10 3Σ+,1−6 1,3Π, and 1−3 1,3Δ states. We observe various crossings and undulations with repulsive barriers at short distances. Interestingly, these barriers are similar to the ones found in the corresponding singlet states. This effect was related previously with the Fermi model77 where an atom is colliding with a Rydberg electron, and the potential is then proportional to the scattering length and to the Rydberg amplitude. For D2, D4, and D5, the Rydberg wave functions of Rb involved are expected to be 5s, 5p, and 4d, 2958



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observed with the results of Korek et al. This is not surprising because, for the adiabatic results, we used similar theoretical techniques and the some different basis sets. In general, we have a good agreement with the one found by Korek et al.32 However, this study is extended to higher excited states and a study beyond the Born−Oppenheimer approximation, diabatic potential energy, and dipole moment. These data have been already exploited to evaluate the radial coupling (first and second derivative terms of the coupling between the adiabatic wave functions), the adiabatic correction, and the vibronic shift.78,80 Furthermore, the diabatic potential energy curves are presented for the different symmetries and for the first time. The diabatic curve of the1Σ+ state named D1, related to the ionic state Li−Rb+, is observed to cross almost all the quasi neutral states, leading in 1Σ+ symmetry to a series of avoiding crossings located at short, intermediate, and large distances. Such crossings became avoided crossings in the adiabatic scheme and gave undulating behaviors for several adiabatic states. In addition, we performed the permanent and transition dipole moments. The permanent dipole moment of the 1−3 1 + Σ states is found to be small; however, it is more significant for higher excited states 4−10 1Σ+. Furthermore, it behaves as a linear function of R giving significant dipole moments for the higher excited states corresponding to a long-range ionic character. Our accurate data are made available for experimental and theoretical use to perform simulation on the possible formation of ultracold LiRb molecule through photoassociation and radiative process.

Figure 13 displayed the permanent dipole moment as a function of R, the intermolecular distance, for the 4 1Σ+−10 1Σ+ states. We observe that some states, such as the 6−9 1Σ+ states, have also a positive permanent dipole moment at short and intermediate distance in addition to the negative one at larger distances. This could be related to the influence of the inverse ionic state Li+Rb−, in this region. The feature of the permanent dipole moment is similar to that observed at a larger distance: when it vanishes for one state, it increases for the next one and so one. This is related to the avoided crossings between neighbor states in the short and intermediate region. Moreover, we remark that the dipole moment of these states, one after other, behaves as a −R function and then drops to zero at particular distances corresponding to the avoided crossings between the two neighbor electronic states. If these curves are combined, they produce piecewise the whole −R function due to the ionic character of these states. The discontinuities between the consecutive parts are due to the avoided crossings and such nodes are expected to disappear in the diabatic representation and they would lead to a full linear curve corresponding to the dipole moment of the pure ionic state as it was observed previously in the LiH, RbH, CsH, LiNa, and LiCs molecules.63,68,70,27,25 The permanent dipole moment has been also determined for the electronic states of the 3Σ+, 1Π, 3Π, 1Δ, and 3Δ remaining symmetries. Their permanent dipole moments are displayed in Figures 14−20. They are not negligible at all for these states and become more significant for the higher excited states. In particular, Figures 18 and 19 show the permanent dipole moment for the three 1Δ and 3Δ states. Consequently, we note a significant dipole moment in a particular region and then it tends to zero; for example, the 3 3Δ state has an important dipole moment equal to −5.21 au at a distance equal to 11.22 au and then it tends to zero. The region located at intermediate distance corresponds to the repulsive part of their deep potential wells. For a better understanding of the role of the avoided crossings, we have determined the transition dipole moment between the consecutive states of the symmetry 1Σ+. They are displayed in Figures 21−23. Indeed, Figure 21 corresponds to the transitions X−2 1Σ+, 2−3 1Σ+, and 3−4 1Σ+, Figure 22 includes the transitions 4−5 1Σ+, 5−6 1Σ+, 6−7 1Σ+, 7−8 1Σ+, and Figure 23 corresponds to the transitions 8−9 1Σ+ and 9− 10 1Σ+. We observe many peaks located at particular distances very close to the avoided crossings in adiabatic representation. Their positions were reported in Table 2.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge support of this work by King Abdul Aziz City for Science and Technology (KACST, Saudi Arabia) through the Long-Term Comprehensive National Plan for Science, Technology and Innovation program under Project No. 08-NAN148-7.

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REFERENCES

(1) Thorsheim, H. R.; Weiner, J.; Julienne, P. S. Phys. Rev. Lett. 1987, 58, 2420. (2) Fioretti, A.; Comparat, D.; Crubellier, A.; Dulieu, O.; MasnouSeeuwsand, F.; Pillet, P. Phys. Rev. Lett. 1998, 80, 4402. (3) Weiner, J.; Bagnato, V. C.; Zilio, S.; Julienne, P. S. Rev. Mod. Phys. 1999, 71, 1. (4) Masnou-Seeuws, F.; Pillet, P. Adv. At. Mol. Opt. Phys. 2001, 47, 53. (5) Bahns, J. T.; Stwalley, W. C.; Gould, P. L. Adv. At. Mol. Opt. Phys 2000, 42, 171. (6) Balakrishnan, N.; Dalgarno, A. Chem. Phys. Lett. 2001, 341, 652. (7) Bodo, E.; Gianturco, F. A.; Dalgarno, A. J. Chem. Phys. 2002, 116, 9222. (8) Rom, T.; Best, T.; Mandel, O.; Widera, A.; Greiner, M.; Hansch, T. W.; Bloch, I. Phys. Rev. Lett. 2004, 93, 073002. (9) Kajita, M. Eur. Phys. D. 2003, 23, 337. (10) Krems, R. V. Int. Rev. Phys. Chem 2005, 24, 99. (11) Avdeenkov, A. V.; Kajita, M.; Bohn, J. L. Phys. Rev. 2006, 73, 022707. (12) Krems, R. V. Phys. Rev. Lett. 2006, 96, 123202. (13) Demille, D. Phys. Rev. Lett. 2002, 88, 067901.

4. CONCLUSION The potential energy is calculated for 38 electronic states of LiRb molecule dissociating into Rb(5s, 5p, 4d, 6s, 6p, 5d, 7s, 6d) + Li(2s, 2p). The method used is based on nonempirical pseudopotentials, parametrized l-dependent polarization potentials, Gaussian basis sets and full valence CI calculation. We have determined the spectroscopic constants (Re, De, Te, ωe, ωexe, and Be) for all electronic states: 1−10 1,3Σ+, 1−6 1,3Π, and 1−3 1,3Δ. Our spectroscopic data for the ground state X 1Σ+ are compared with various theoretical works.32,40,41,23,24 The comparison has shown a good agreement with all these references. However, for the higher excited states and the other symmetries, our data were compared only with those of Korek et al. and for most of them the comparison is done by using the constants extracted from their adiabatic potential curves presented on their Web site. A general good agreement is 2959



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(54) Ghanmi, C.; Berriche, H.; Ben Ouada, H. Comput. Mater. Sci. 2007, 38, 494. (55) Berriche, H. J. Mol. Struct. 2003, 663, 101. (56) Müller, W.; Flesh, J.; Meyer, W. J. Chem. Phys. 1984, 80, 3297. (57) Foucrault, M.; Millie, P.; Daudey, J. P. J. Chem. Phys. 1992, 96, 1257. (58) Huron, B.; Rancurel, P.; Malrieu, J. P. Chem. Phys. 1973, 58, 5475. (59) Evangelisti, S.; Daudey, J. P.; Malrieu, J. P. Chem. Phys. 1983, 75, 91. (60) Schmidt-Mink, I.; Muller, W.; Meyer, M. Chem. Phys. 1985, 92, 263. (61) Bashkin, S. Stoner, J. O. Atomic Energy Levels Grot. Diagram; North Holland: Amsterdam, 1978; p 2. (62) Moore, C. E. Atomic Energy Levels; U.S. National Bureau of Standards Circular 467; US GPO: Washington, DC, 1958. (63) Boutalib, A.; Gadea, F. X. J. Chem. Phys. 1992, 97, 1144. (64) Gadéa, F. X.; Boutalib, A. J. Phys. B: At. Mol. Opt. Phys. 1993, 26, 61. (65) Khelifi, N.; Oujia, B.; Gadéa, F. X. J. Chem. Phys. 2001, 117, 879. (66) Khelifi, N.; Zrafi, W.; Oujia, B.; Gadéa, F. X. Phys. Rev. A 2002, 65, 042513. (67) Zrafi, W.; Oujia, B.; Berriche, H.; Gadéa, F. X. J. Mol. Struct. (THEOCHEM) 2006, 777, 87. (68) Zrafi, W.; Oujia, B.; Gadéa, F. X. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, 1. (69) Berriche, H. Ph.D. Thesis, Paul Sabatier University, 1995 (unpublished). (70) Gadea, F. X. Phys. Rev. A 1991, 43, 1160. (71) www-lasim.univ-lyon1.fr/spip.php?article274. (72) Ghanmi, C. Berriche, H. Unpublished. (73) Gadea, F. X. Thèse d’état, Université Paul Sabatier, Toulouse, 1987. (74) Gadea, F. X.; Pelissier, M. J. Chem. Phys. 1990, 93, 545. (75) Berriche, H.; Gadea, F. X. Chem. Phys. Lett. 1995, 247, 85. (76) Romero, T.; Agular, A.; Gadea, F. X. J. Chem. Phys. 1991, 110, 6219. (77) Dickinson, A. S.; Gadea, F. X. Phys. Rev. A 2002, 65, 052506. (78) Jendoubi, I.; Berriche, H.; Ben Ouada, H. Gadea, F. X. Progress in Theoretical Chemistry and Physics; Springer: Berlin, 2011 (in press) (79) Landolt-Börnstein, V. J. Molecular Constants; Springer-Verlag: Berlin, 1974; Vol. II/6. (80) Tarnovsky, V.; Bunimovicz, M.; Vusković, L.; Stumpf, B.; Bederson, B. J. Chem. Phys. 1993, 98, 3894.

(14) Yelin, S. F. ; Kirby, K. ; Côté, R. quant-ph. 0602030. (15) Kraemer, T.; Mark, M.; Walburger, P.; Danzl, J. G.; Chin, C.; Engesser, B.; Lange, A. D.; Pilch, K.; Jaakkola, A.; Nagerl, H. C.; Grimm, R. Nature 2006, 440, 315. (16) Kraft, S. D.; Staanum, P.; Lang, J.; Vogel, L.; Westerand, R.; Weidemuller, M. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, 5993. (17) Pashov, A.; Docenko, O.; Tamanis, K.; Ferber, R.; Knockel, H.; Tiemann, E. Phys. Rev. A 2005, 72, 062505. (18) Korek, M.; Allouche, A. R.; Fakhreddine, K.; Chaalan, A. Can. J. Phys. 2000, 78, 977. (19) Salami, H.; Ross, A. J.; Crozet, P. J. Chem. Phys. 2007, 126, 194313. (20) Fahs, H.; Allouche, A. R.; Korek, M.; Aubert-Frécon, M. J. Phys. B 2002, 35, 1501. (21) Hessel, M. Phys. Rev. Lett. 1971, 26, 215. (22) Muller, W.; Meyer, W. J. Chem. Phys . 1984, 80, 3311. (23) Aymar, M.; Dulieu, O. J. Chem. Phys. 2005, 122, 204302. (24) Aymar, M.; Dulieu, O. J. Chem. Phys. 2006, 125, 047101. (25) Mabrouk, N.; Berriche, H.; Gadea, F. X. AIP Conf. Proc. 2007, 963, 23. (26) Mabrouk, N.; Zrafi, W.; Berriche, H.; Ben Ouada, H. Lect. Ser. Comp. Comput. Sc. 2006, 7, 1451. (27) Mabrouk, N.; Berriche, H. J. Phys. B: At. Mol. Opt. Phys. 2008, 41, 155101. (28) Mabrouk, N.; Berriche, H. AIP Conf. Proc. 2009, in press. (29) Pavolini, D.; Gustavsson, T.; Spiegelman, F.; Daudey, J.-P. J. Phys. B: At. Mol. Opt. Phys. 1989, 22, 1721. (30) Allouche, A. R.; Korek, M.; Fakherddin, K.; Chalaan, A.; Dagher, M.; Taher, F.; Aubert-Fécon, M. J. Phys. B: At. Mol. Opt. Phys. 2000, 33, 2307. (31) Fahs, H.; Allouche, A. R.; Korek, M.; Aubert-Frécon, M. J. Phys. B 2002, 35, 1501. (32) Korek, M.; Allouche, A. R.; Kobeissi, M.; Chaalan, A.; Dagher, M.; Fakherddin, F.; Aubert-Frecon, M. Chem. Phys. 2000, 256, 1. (33) Korek, M.; Allouche, A. R.; Fakhreddine, K.; Chaalan, A. Can. J. Phys. 2000, 78, 977. (34) Rousseau, S.; Allouche, A. R.; Aubert-Frécon, M. J. M. Spectrosc. 2000, 203, 235. (35) Rousseau, S.; Allouche, A. R.; Aubert-Frécon, M.; Magnier, S.; Kowalczyk, P.; Jastrzebski, W. Chem. Phys. 1999, 247, 193. (36) Magnier, S.; Aubert-Frécon, M.; Millié, P. J. M. Spectrosc. 2000, 200, 96. (37) Magnier, S.; Aubert-Frécon, M.; Millié, P. Phys. Rev. A 1996, 54, 204. (38) Schmidt-Mink, I.; Müller, W.; Meyer, W. Chem. Phys. Lett. 1984, 112, 120. (39) SØrensen, L. K.; Fleig, T.; Olsen, J. J. Phys. B: At. Mol. Opt. Phys. 2009, 42, 165102. (40) Igel-Mann, G.; Wedig, U.; Fuentealba, P.; Stoll, H. J. Chem. Phys. 1986, 84, 5007. (41) Urban, M.; Sadlej, A. J. J. Chem. Phys. 1995, 103, 9692. (42) Smirov, A. D. J. Struct. Chem. 2007, 48, 21. (43) Korek, M.; Younes, G.; AL-Shawa, S. J. Mol. Struct. 2009, 899, 25. (44) Ivanova, M.; Stein, A.; Pashov, A.; Knöckel, H.; Tiemann, E. J. Chem. Phys. 2011, 134, 024321. (45) Dutta, S.; Altaf, A.; Elliott, D. S.; Yong, P. C. Chem. Phys. Lett. 2011, 511, 7. (46) Jendoubi, I.; Berriche, H.; Ben Ouada, H. AIP Conf. Proc. 2009, in press. (47) Durand, P.; Barthelat, J. Chem. Phys. Lett. 1974, 27, 191. (48) Durand, P.; Barthelat, J. Theor. Chim. Acta 1975, 38, 283. (49) Berriche, H.; Gadea, F. X. Chem. Phys. 1995, 191, 119. (50) Berriche, H.; Tlili, C. J. Mol. Struct. 2004, 682, 11. (51) Berriche, H. J. Mol. Struct. 2004, 682, 89. (52) Ghanmi, C.; Bouzouita, H.; Berriche, H.; Ben Ouada, H. J. Mol. Struct. 2006, 777, 81. (53) Berriche, H.; Ghanmi, C.; Ben Ouada, H. J. Mol. Spectrosc. 2005, 230, 161. 2960


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