ARTICLE

Adiabatic and quasi-diabatic investigation of the strontiumhydride cation SrH+: structure, spectroscopy, and dipolemomentsSana Belayouni, Chedli Ghanmi, and Hamid Berriche

Abstract: Ab initio investigation has been performed for the strontium hydride cation SrH + using a standard quantum chemistryapproach. It is based on the pseudopotentials for atomic core representations, Gaussian basis sets, as well as with full configurationinteraction calculations. A diabatisation procedure based on the effective hamiltonian theory and an effective metric is used toproduce the quasi-diabatic potential energy. Adiabatic and quasi-diabatic potential energy curves and their spectroscopicparameters for the ground and many excited electronic states of 1,3�+, 1,3�, and 1,3� symmetries have been determined. Theirpredicted accuracy is discussed by comparing our well depths and equilibrium positions with the available experimental andtheoretical results. Moreover, we localized and analyzed numerous avoided crossings between the electronic states of 1,3�+ and1,3� symmetries. The correction of the electron affinity of the H atom is also considered, for the 1–101�+ electronic states, toimprove the accuracy of the adiabatic potential energies of these states. In addition, we calculated the dipole moments, for awide range of internuclear distances in both diabatic and quasi-diabatic representations. The adiabatic permanent dipolemoments for the 101�+ electronic states revealed ionic characters related to electron transfer and yields both SrH(+) and Sr(+)Harrangements. The transition dipole moments between neighbor electronic states revealed many peaks around the avoidedcrossing positions.

Key words: pseudopotentials, configuration interaction, potential energy curves, spectroscopic parameters, dipole moments.

Résumé : Nous avons complété une étude ab initio du cation hydride du strontium SrH+ en utilisant une approche de chimiequantique standard. Elle est basée sur les pseudo-potentiels pour représenter le cœur atomique, sur des ensembles de base gaussienset des calculs avec interaction a pleine configuration. Nous utilisons une procédure de diabatisation basée sur la théorie du Hamil-tonien effectif et une métrique effective est utilisée pour produire l’énergie potentielle quasi-diabatique. Nous déterminons lescourbes d’énergie potentielle adiabatique et quasi-diabatique pour le fondamental et plusieurs états excités des symétries 1,3�+, 1,3� et1,3�. La précision de nos prédictions est analysée en comparant nos profondeurs de puits et nos positions d’équilibre avec les valeursthéoriques et expérimentales disponibles. De plus, nous localisons et analysons les nombreux croisements évités entre les étatsélectroniques des symétries 1,3�+ et 1,3�. Nous considérons aussi la correction de l’affinité électronique de l’atome H pour les états1–101�+, de façon a améliorer la précision des énergies potentielles adiabatiques de ces états. Les moments dipolaires sont calculés pourun large domaine des distances internucléaires, dans les deux représentations, diabatique et quasi-diabatique. Les moments dipolairesadiabatiques permanents pour les états électroniques 101�+ révèlent des caractères ioniques reliés au transfert d’électron et donnentles deux arrangements SrH(+) et Sr(+)H. Les moments dipolaires de transition entre états électroniques voisins révèlent plusieurs picsautour des positions des croisements évités. [Traduit par la Rédaction]

Mots-clés : pseudo-potentiel, interaction de configuration, courbes d’énergie potentielle, paramètres spectroscopiques, momentsdipolaires.

1. IntroductionIn the recent past, there has been considerable interest in the

creation of ultracold molecules at temperatures below 1 �K bymagneto-association [1, 2] or photo-association [3, 4], especially byusing heteronuclear molecules, such as alkali dimers, alkaline earthhydrides, and their corresponding ions. This opened an excit-ing prospect to test the variation of fundamental constants on boththe experimental and theoretical scales. The precise knowledge of

the long-range interactions between two different types of alkaliatoms is necessary for the understanding and realization of coldcollision processes and formation of cold and ultra-cold heteronu-clear molecules. These molecules can be produced through photoas-sociation of atoms or by a laser-cooled atomic vapor. For example,several cold and ultra-cold diatomic molecule have been formed,such as RbCs [5], KRb [6, 7], NaCs [8], and NaCs+ [9].

The literature reveals that the structural and spectroscopicproperties of the alkaline earth hydrides cations XH+ (X = Be, Mg,

Received 19 December 2015. Accepted 13 May 2016.

S. Belayouni. Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue del’Environnement, 5019 Monastir, Tunisia.C. Ghanmi. Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue del’Environnement, 5019 Monastir, Tunisia; Physics Department, Faculty of Sciences, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia.H. Berriche. Laboratoire des Interfaces et Matériaux Avancés, Département de Physique, Faculté des Sciences, Université de Monastir, Avenue del’Environnement, 5019 Monastir, Tunisia; Mathematics and Natural Sciences Department, School of Arts and Sciences, American University ofRas Al Khaimah, Ras Al Khaimah, UAE.Corresponding author: Hamid Berriche (email: hamid.berriche@fsm.rnu.tn).Copyright remains with the author(s) or their institution(s). Permission for reuse (free in most cases) can be obtained from RightsLink.

791

Can. J. Phys. 94: 791–802 (2016) dx.doi.org/10.1139/cjp-2015-0801 Published at www.nrcresearchpress.com/cjp on 9 June 2016.

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Ca, Sr, Ba, and Ra) have been developed rapidly on both the exper-imental and theoretical sides. In this context, the potential energycurves, the spectroscopic constants, and the permanent and transi-tion dipole moments of BeH + have been studied [10–12]. Moreover,many theoretical and experimental studies [13–17] have determinedthe structure and electronic properties of the magnesium hydridecation MgH+. Mølhave et al. [17] produced and cooled the molecularions MgH+ and MgD+ in a linear Paul trap. Other works studied themolecular properties of the CaH+ ionic system [18–23]. In addition,Allouche et al. [24] and Mejrissi et al. [25] performed a theoreticalstudy of the low-lying electronic states of the BaH+ ion. The stron-tium hydride cation SrH+ is also investigated extensively [26–32].Experimentally, only the well depth of the ground state has beenevaluated by Dalleska et al. [26]. Theoretically, the structure and spec-troscopic properties of the ground state of the strontium hydridecation SrH+ was studied for the first time by Fuentealba et al. [27].After that, Schilling et al. [28] successfully completed a theoreticalstudy for the strontium hydride cation SrH+ using a generalized va-lence bond plus configuration interaction calculations. They haveanalyzed the trends in bond energies, equilibrium geometries,vibrational frequencies, and metal orbital hybridizations. Re-cently, Abe et al. [29] realized an ab initio study on vibrationaldipole moments of XH+ where X = Mg, Ca, Zn, Sr, Cd, Ba, Yb, andHg. Also, in 2011 Katija et al. [30] evaluated a precise measurement ofthe �X1�nv�0,J�0,F�1/2,M�±1/2�¡ �X1�nv�1,J�0,F�1/2,M�±1/2� transition frequen-cies of the same molecular ion XH+. Aymar et al. [31] determined thepotential energy curves, permanent and transition dipole moments,and the static dipole polarizabilities of SrH+, CaH+, and BaH+ systems.They used a full configuration interaction performed in a two-valence electronic configuration space built from a large Gaussianbasis set. The structure and spectroscopic properties of the stron-tium hydride cation SrH+ have been calculated by Habli et al. [32].They used an ab initio approach based on large basis sets, nonem-pirical atomic pseudopotential for strontium core, correlation treat-ment for core valence through the effective core polarizationpotentials, and for the valence through full valence configurationinteraction.

Recently, the accurate data, such the potential energy curves,the spectroscopic parameters, and the transition and permanent di-pole moments, have been produced in our research group for severaldiatomic systems like LiH [33], LiNa [34], CsLi [35], CsNa [36] BeH+ [12],LiRb [37] and LiX (X = Na, K, Rb, Cs, and Fr) [38]. Our goal in this workis to extend our previous studies [12, 33–38] and determine the adia-batic and quasi-diabatic potential energy curves, their spectroscopicparameters, and the permanent as well as transition dipole mo-ments of the strontium hydride cation SrH+. We hope that this studymay help to explore further the photoassociation processes. In thefollowing section we briefly present the computational meth-ods. Section 3 is devoted to presenting our results, where we indicatethe adiabatic potential energy curves and their spectroscopic param-eters for the 1–101,3�+, 1–61,3�, and 1–61,3� electronic states, the quasi-diabatic potential energy curves, and finally the permanent andtransition dipole moments. Section 4 contains our conclusionsand a brief summary.

2. Methods of calculationWe used the non-empirical pseudopotential in its semi-local

form proposed by Barthelat and Durand [39], where the strontiumhydride cation SrH+ is modeled as a system with two valence elec-trons. In addition, we also considered the self-consistent field calcu-lation, which is followed by a full valence configuration interactioncalculation. For the simulation of the interaction between the polar-izable Sr2+ core with the valence electrons and the hydrogen nucleus,

a core polarization potential VCPP is used, according to the operatorformulation of Müller et al. [40]

VCPP � �1

2 �

f · f

where represents the dipole polarizability of the core and f

represent the electric field created by valence electrons and allother cores on the core .

f � �i

ri

ri3

F(ri, �) � �′≠

R′

R′

3Z

where riis a core–electron vector and R′ is a core–core vector.As reported by the formulation of Foucrault et al. [41], the cutoff

function F(ri, �) is taken to be a function of l to treat separatelythe interaction of valence electrons of different spatial symmetryand the core electrons. It has a physical meaning of excluding thevalence electrons from the core region for calculating the electricfield. In the Müller et al. [40] formalism, the cutoff function is uniquefor a given atom and is generally adjusted to reproduce the atomicenergy levels for the lowest states of each symmetry.

For the strontium and hydrogen atoms, we have used a (5s5p6d1f/5s5p3d1f) [42] and (9s5p3d/7s5p3d) [33] basis set of Gaussian-type or-bitals, where diffuse orbital exponents have been optimized toreproduce the atomic levels 1s, 2s, and 2p; 5s, 4d, 5p, 6s, 5d, 6p, 7s, 6d,and 7p; and 5s2, 5s5p, 5s4d, 5s4d, and 5s5p for H, Sr+, and Sr species,respectively. Following the formulation of Foucrault et al. [41], cutofffunctions with l-dependent adjustable parameters are fitted to repro-duce not only the first experimental ionization potential but also thelowest excited states of each l for H, Sr+, and Sr. In the present work,the core polarizability of Sr2+ is taken as Sr2� = 5.51 a0

3 [40] and theoptimized cutoff parameters for the lowest valence s, p, and d one-electron states of the Sr atom are 2.08205, 1.91905, and 1.64474 a.u.,respectively. To produce the energy levels of the neutral Sr atom, wehave performed a full configuration interaction. Table S1, summa-rize the data about the atomic energy levels of Sr+ (5s, 4d, 5p, 6s, 5d,6p, 7s, 6d, and 7p) and Sr(5s2, 5s5p, 5s4d, 5s4d, and 5s5p) and com-pares them with the available theoretical [42] and experimental [43]data. This table is given as supplementary material1.

For the quasi-diabatic study, our idea is to construct a unitarytransformation matrix that cancels the non-adiabatic coupling ele-ments. In this context, the quasi-diabatic wave function can be writ-ten as a linear combination of the adiabatic ones. The diabatisationmethod has been published previously in several works [44–52]. Wemention here only the most important features of the diabatisationmethod, which is based on effective Hamiltonian theory [53] and theeffective metric method [47]. The main purpose of this method is toevaluate the nonadiabatic coupling between the considered adia-batic states and to cancel it by an appropriate unitary transformationmatrix. This matrix gives us the quasi-diabatic energies and wavefunctions. Then the quasi-diabatic wave functions can be writtenas a linear combination of the adiabatic ones. This non adiabaticcoupling estimation is closely related to an overlap matrix between theadiabatic multiconfigurational states and the reference states corre-sponding to a fixed large distance equal to 105.00 a.u. The quasi-diabaticstates are deduced from the symmetrical orthonormalization of theprojection of the model space wave functions onto the selected adia-baticwavefunctions.Therecoverymatrixconstructedbytheprojectionis clearly a recovery matrix over nonorthogonal functions seeingthat the two sets are related to different interatomic distances. Thereference states corresponding to the adiabatic ones are taken at an

1Supplementary data are available with the article through the journal Web site at http://nrcresearchpress.com/doi/suppl/10.1139/cjp-2015-0801.

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infinite distance, taken to be equal to 105.00 a.u. At this distance alladiabatic states have reached their asymptotic limits, while adiabaticand quasi-diabatic states coincide. We take the origin on the stron-tium atom, which is the heavier atom. This diabatisation scheme isbased on the recovery matrix between the reference and the adiabaticstates, which correspond to a numerical estimation of the nonadiabaticcoupling, but do not involve the electric dipole matrix at all.

3. Results and discussion

3.1. Adiabatic potential energy curvesComputation of the potential energy curves is the first required

step to characterize any molecular systems. An accurate potentialenergy curve opens a window to investigate and interpret the behav-iour of the molecule at any experimental conditions. Using themethod of calculation reported in the previous sections, we haveinvestigated the adiabatic potential energy curves of 44 low-lyingelectronic states of 1,3�+, 1,3�, and 1,3� symmetries for the strontiumhydride cation SrH + dissociating into Sr+ (5s, 4d, 5p, 6s, 5d, 6p, 7s, 6d,and 7p) + H(1s, 2s, and 2p) and Sr(5s2, 5s5p, 5s4d, 5s4d, 5s5p) + H+ . Theadiabatic potential energy is performed for an interval of intermo-lecular distances ranging from 2.40 to 105.00 a.u. The 1�+ and 3�+

electronic states are depicted in Figs. 1 and 2, respectively, whereasthe 1,3�, 1,3� states are displayed in Figs. 3 and 4, respectively. In Fig. 1,we present the adiabatic potential energy curves of the 1–31�+ (Fig. 1a)

and 4–101�+ (Fig. 1b) electronic states without (solid lines) and with(dashed lines) the electron affinity for the strontium hydride cationSrH+. This Figure shows that the 1–51�+ electronic states are foundwith a unique well depth located respectively at 3.73, 4.64, 5.68, 8.32,and 5.82 a.u. In addition, we found that the ground state 11�+ disso-ciating into Sr+(5s) + H(1s) presents a well depth of 17 405 cm−1. Thisvalue is in excellent agreement with the available experimental re-sult (17 502 ± 480 cm−1) found by Dalleska et al. [26]. In a similar wayto the alkali dimers LiH [33], LiNa [34], LiCs [35], and CsNa [36], theimprint of a state behaving as (–2/R) can be clearly seen in Fig. 1,which is formed between the 4–101�+ electronic states and corre-sponding to the Sr2+H− structure. The same behavior has been ob-served previously for the CaH+ [23], SrH+ [32], and BaH+ [25] alkalineearth ions. In Fig. 2, we found that the 53�+ and 63�+ electronic statesdissociating, respectively, into Sr+(5d) + H(1s) and Sr+(6p) + H(1s) havea particular shape and present a high potential barrier of 636 and1550 cm−1 located at 8.49 and 9.05 a.u., respectively. This feature canbe explained by the interaction with the upper excited state.

In addition, Fig. 1 presents the correction due to the hydrogenelectron affinity. It is the difference between the electron affinitycalculated in our basis set and the known experimental value. Infact, the presence of the Sr base makes this correction dependenton R whereas it is constant for the atom. It is a question of calcu-lating the energy of the two systems H and the ion H− with a basisset on the ion Sr+, which makes this quantity dependent on theinteratomic distance. The calculation of this quantity is limited by

Fig. 1. Adiabatic potential energy curves without (solid lines) andwith (dashed lines) the electron affinity of the H atom for the 1–31�+

(a) and 4–101�+ (b) electronic states of the strontium hydride cationSrH+. [Colour online.]

Fig. 2. Adiabatic potential energy curves for the 10 lowest 3�+

electronic states of the strontium hydride cation SrH+. [Colour online.]

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the incomplete basis set we use. In the diabatic representation, thiscorrection concerns only the ionic state dissociating into Sr2+ + H−. Inthis representation, it induces a general shift of crossings betweenthe ionic curve and the neutral curve to larger internuclear dis-tances. The shift of the crossings is more important when the cross-ing takes place at long distance. It is about 0.1 a.u. for the first stateand increases quickly to be of an order of 10 a.u. for the higherexcited states. This can be explained by the behaviour in 2/R of theionic state. In the adiabatic representation, the diagonalisation of thediabatic matrix containing this correction in the ionic curve willdistribute it to all the adiabatic states that result from this diagonali-sation. This will imply thereafter: (i) a light change in the equilibriumdistance for the first electronic states, which will be increasinglyimportant for the excited states; and (ii) a change in the depth of thepotential wells, which can reach 100 cm−1 for some states. By takinginto account the electron affinity of the H atom, we observe thatthere is no change in the feature of the potential energy curves.However, they are shifted towards lower energy.

Our potential energy curves of the strontium hydride cation SrH+

present a similar shape to those obtained by Aymar et al. [31] andHabli et al. [32]. This is not surprising because we used the sameprocedure (exchange core-polarization and core polarization poten-tial), but a different cutoff radius. The differences between our re-sults and those of Aymar et al. [31] and Habli et al. [32] will bediscussed in detail in the section accorded to the spectroscopic pa-rameters. In addition, we observe that the potential energy curves

present many avoided crossings at short and large values of internu-clear distance between many excited states of 1,3�+ and 1,3� symme-tries located at intermediate and large values of internucleardistance. In several cases, these avoided crossings are responsible forthe presence of particular forms in the potential energy curves, suchas the appearance of the double wells and the barriers of potential.Here, we can say that the 6–101�+ electronic states presented doublewells and 8–103�+ electronic states have a barrier of potential. Mostof the avoided crossings can be explained by the interaction betweenthe electronic states of Sr(+)H and SrH(+) structures. We have localizedthe positions of the avoided crossings between the neighbor elec-tronic states of 1�+, 3�+, and 3� symmetries. These positions areshown in Table 1. �E represents the difference of energies at thepositions of the avoided crossing. For example, we quote the avoidedcrossings between 71�+ and 81�+ at 9.08 and 22.83 a.u., between 81�+

and 91�+ at 6.44 and 25.07 a.u., between 91�+ and 101�+ at 6.28 and29.00 a.u., and between 93�+ and 103�+ at 11.34 a.u.

3.2. Quasi-diabatic potential energy curvesIn addition to the adiabatic potential energy curves, we have

calculated the quasi-diabatic potential curve below the ionic limitSr2+H− related to all the electronic states of 1,3�+, 1,3�, and 1,3� sym-metries for the strontium hydride cation SrH+. In recent past, thequasi-diabatic method was applied successfully in our group onmany diatomic molecules like the mixed alkali diatomic moleculeand the quasi-diabatic potential energy curves for the LiH [33], LiCs

Fig. 3. Adiabatic potential energy curves for the 12 lowest 3� (solidline) and 1� (dashed line) of the strontium hydride cation SrH+.

Fig. 4. Adiabatic potential energy curves for the six lowest 1� (solidline) and 3� (dashed line) of the strontium hydride cation SrH+.

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[35], and CsNa [36] have been calculated. We extend here the samequasi-diabatic approach for the strontium hydride cation SrH+. Theorigin is taken at the strontium atom. We fixed the reference statesas the adiabatic ones in the larger internuclear distance equal105.00 a.u. Then we calculated the quasi-diabatic potential energycurves related to the 1,3�+, 1,3�, and 1,3� symmetries in the adiabaticrepresentation. In this section, we present only the quasi-diabaticpotential energy curves related to the 1�+ adiabatic representation.Figure 5 presents the quasi-diabatic potential energy curves relatedto the 1–101�+ adiabatic electronic states and named D1–10. We re-mark that the ionic quasi-diabatic curve noted as D1, dissociatinginto Sr2+H−, behaves as –2/R at intermediate and large internucleardistances. This ionic quasi-diabatic curve crosses the quasi-diabaticcurves D2–9 at different interatomic distances. The avoided crossingbetween the 1�+ electronic states discussed previously in the adia-batic representation are transformed in the quasi-diabatic one intoreal crossings. The lowest real crossings occur with the quasi-diabaticstate D2 dissociating into Sr+(4d) + H(1s) at 6.90 a.u., with the quasi-diabatic state D3 dissociating into Sr+(5p) + H(1s) at 5.96 a.u., with thequasi-diabatic state D4 dissociating into Sr+(5s) + H(1s) at 7.43 a.u., andwith the quasi-diabatic state D5 dissociating into Sr+(6s) + H(1s) at10.49 a.u. The ionic quasi-diabatic state D1 crosses the higher quasi-diabatic excited states D6–9 at much larger internuclear distances.For these crossings, we found, respectively, 10.71, 10.87, 10.42, and11.28 a.u. The second ionic quasi-diabatic curve noted in our work D8,dissociating into SrH+, is expected to cross the neutral excited statesD5–7 and D9–10 at 13.66, 15.35, 16.38, 23.13, and 26.63 a.u., respectively.Moreover, we show the presence of clear undulations in the shape ofthe quasi-diabatic potentials of D8–10 states. These undulations arerelated to the electron density and can be interpreted as resultingfrom the repulsive and attractive effects. The same undulations havebeen observed previously by Dickinson et al. [54, 55] in the quasi-diabtic study of several diatomic systems.

3.3. The spectroscopic parametersThe adiabatic potential energy curves have been used to extract

the spectroscopic parameters, such as the equilibrium distance(Re), the well depth (De), the electronic transition energy (Te), theharmonicity frequency ( e), the anharmonicity constant ( e�e), and

the rotational constant (Be). The spectroscopic parameters of theground and the low-lying electronic states of the different symme-tries 1,3�+, 1,3�, and 1,3� are collected in Table 2. These spectroscopicparameters are compared with the available experimental [26] andtheoretical [28, 29, 31, 32] results. To the best of our knowledge, theexperimental information on strontium hydride SrH+ are still lim-ited, except the ground (X1�+) state. As it seems from Table 2, theagreement between our well depth and the experimental valuefound by Dalleska et al. [26] is very good. In fact, we found a welldepth of 17 405 cm−1, while Dalleska et al. [26] present a well depth of17 502 ± 480 cm−1. The difference is about 98 cm−1. The experimentalresult presents a rather large uncertainty of ±480 cm−1.

There have been a few theoretical studies realized on the groundstate (X1�+). A general good agreement is observed between our spec-troscopic parameters and other theoretical [28, 29, 31, 32] values. Wefound the following spectroscopic parameters Re = 3.73 a.u.,De = 17 405 cm−1, e = 1387.75 cm−1, e�e = 19.88 cm−1, and Be =4.345 cm−1 for the ground state. These values are in very good agree-ment with the theoretical results of Habli et al. [32], who reported Re,De, e, e�e, and Be as 3.72 a.u., 17 588 cm−1, 1351.2 cm−1, 19.31 cm−1,and 4.356 cm−1, respectively. This is not surprising because we usedthe same method of calculation, but a different basis set of Gaussian-type orbital and different cutoff radius. Good agreement is observedbetween our spectroscopic parameters and those found by Aymaret al. [31] (Re = 3.73 a.u, De = 18 078 cm−1, and e = 1429 cm−1). We also

Table 1. Avoided crossing positions(in a.u.) between the neighbourelectronic states of the strontiumhydride cation SrH+.

State Rc (a.u.) �E (a.u.)

51�+/61�+ 15.92 0.00863With EA 15.93 0.0084171�+/81�+ 9.08 0.00332With EA 9.10 0.00302

22.83 0.00028With EA 22.87 0.0001881�+/91�+ 6.46 0.00401With EA 6.51 0.00403

25.07 0.00258With EA 25.10 0.0024891�+/101�+ 6.28 0.00323With EA 6.30 0.00324

29.00 0.0028829.03 0.00294

33�+/43�+ 4.05 0.0013683�+/93�+ 9.33 0.00045

14.36 0.0022093�+/103�+ 11.34 0.0008063�/73� 12.01 0.00199

Note: �E, difference of energies atthe positions of the avoided crossing;EA, H electron affinity.

Fig. 5. Quasi-diabatic potential energy curves D1–10 related to theten lowest 1�+ electronic states of the strontium hydride cation SrH+.[Colour online.]

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Table 2. Spectroscopic parameters of the ground and the low-lying electronic states of thedifferent symmetries 1,3�+, 1,3�, and 1,3� of the strontium hydride cation SrH+.

StateRe

(a.u.)De

(cm−1)Te

(cm−1) e

(cm−1) e�e

(cm−1)Be

(cm−1) References

X1�+ 3.73 17405 — 1384.75 19.88 4.345 This workWith EA 3.74 16948 — — 18.43 4.324 This work

17502±480 — — — — [26]3.96 15950.82 — 1346 — — [28]3.87 16374.53 — 1397.3 — 4.072 [29]3.73 18078 — 1429 — — [31]3.72 17588 — 1351.2 19.31 4.356 [32]

21�+ 4.64 9030 23071 726.21 5.44 2.811 This workWith EA 4.65 8681 22963 733.58 7.17 2.794 This work

4.64 8843 19845 728.8 5.248 2.806 [32]4.67 8830 — 712 — — [31]

31�+ 5.65 6714 34806 395.43 5.82 1.903 This workWith EA 5.67 6397 34667 393.44 5.43 1.883 This work

5.77 6462 29135 419.3 2.263 1.814 [32]5.81 6098 — 359 — — [31]

41�+ 8.32 9909 55281 513.48 4.19 0.872 This workWith EA 8.34 9668 55065 512.96 3.12 0.869 This work

8.29 10064 54497 505.1 4.509 0.879 [32]8.29 10140 — 510 — — [31]

51�+ 5.78 6133 64756 540.44 51.90 1.812 This workWith EA 5.79 5818 64613 532.77 62.83 1.802 This work

5.70 6135 58605 551.6 56.095 1.859 [32]5.72 6229 — 599 — — [31]

61�+ 4.65 2815 70430 699.18 47.40 2.785 This workWith EA 4.67 2471 70405 689.69 44.08 2.772 This work

4.64 2930 — 287.9 16.142 2.805 [32]2nd min 13.67 1556 65566 101.22 — — This workWith EA 13.80 1473 71403 108.46 — — This work

13.62 1725 71689 — — — [32]71�+ 5.42 7075 74394 555.38 16.210 2.058 This workWith EA 5.43 6754 — — — — This work

5.46 6804 68297 181.4 2.638 2.026 [32]2nd min 15.54 5751 75719 195.86 — — This workWith EA 15.56 5628 74970 197.43 — — This work

15.51 1725 — — — — [32]81�+ 5.24 2711 80402 399.91 11.49 2.158 This workWith EA 5.26 2267 — — — — This work

5.29 2409 74099 393.6 11.137 2.159 [32]2nd min 8.31 5419 77694 340.61 — — This workWith EA 8.34 5177 — — — — This work

8.25 5219 — — — — [32]91�+ 5.23 3937 81371 136.24 4.82 2.151 This workWith EA 5.24 3578 — — — — This work

5.21 3612 — 172.3 5.207 2.225 [32]2nd min 6.59 4306 81001 59.69 — — This workWith EA 6.61 4026 — — — — This work

6.54 4096 — — — — [32]3rd min 21.25 2579 82728 50.10 — — This workWith EA 21.34 2494 — — — — This work

21.95 2455 — — — — [32]101�+ 4.91 2677 84297 310.03 13.81 2.151 This workWith EA 4.93 2341 83762 — — — This work2nd min 4.89 2422 77130 312.3 10.675 2.526 [32]With EA 6.34 4221 82339 1090.52 — — This work3rd min 6.36 3923 — — — — This workWith EA 6.26 4007 — — — — [32]4th min 11.51 1723 84837 147.98 — — This workWith EA 11.57 1552 — — — — This work

11.51 1671 — — — — [32]26.84 1153 85821 43.75 — — This work26.98 1090 85012 51.02 — — This work

13�+ 8.00 170 17234 91.190 12.22 0.951 This work8.04 155 11493 88.3 13.600 0.934 [32]8.45 88 — 78 — — [31]

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Table 2 (continued).

StateRe

(a.u.)De

(cm−1)Te

(cm−1) e

(cm−1) e�e

(cm−1)Be

(cm−1) References

23�+ 8.03 311 31789 89.460 6.433 0.938 This work8.91 155 26201 62.4 10.260 0.834 [32]9.30 44 — 46 — — [31]

33�+ 11.77 36 41484 42.900 12.780 0.440 This work11.77 19 39630 32 12.006 0.436 [32]12.00 19 — 38 — — [31]

43�+ 4.44 2777 62412 683.23 42.020 3.066 This work4.63 2849 57656 6.83.7 31.480 2.818 [32]4.65 2838 — 876 — — [31]

Hump 9.50 107 — — — — This work53�+ 4.60 2717 68172 784.370 56.66 2.660 This work

4.59 2647 64259 752.0 18.485 2.867 [32]4.57 2801 — 584 — — [31]

Hump 8.49 754 — — — — This work63�+ 5.60 1785 71548 858.590 73.616 1.930 This work

5.41 1855 65693 456.8 –9.335 2.064 [32]Hump 9.05 1550 — — — — This work73�+ 5.22 3122 79576 725.100 42.100 1.985 This work

5.15 2994 73720 473.6 16.939 2.277 [32]83�+ 4.90 5053 80253 681.200 22.958 2.565 This work

4.90 4764 74958 541.7 15.886 2.516 [32]2nd min 10.63 1453 83853 235.360 9.531 0.535 This work

10.45 1211 — — — — [32]Hump 15.50 215 — — — — This work93�+ 4.83 3546 83013 577.580 23.519 2.589 This work

4.78 3296 77781 541.7 15.886 2.516 [32]2nd min 9.27 2229 84330 490.180 26.948 0.704 This work

8.91 1866 — — — — [32]3rd min 13.256 752 85808 158.700 8.372 0.344 This work

12.307 348 — — — — [32]103�+ 5.051 6202 84558 236.010 16.295 2.463 This work

5.019 6432 78821 393.1 12.395 2.313 [32]2nd min 10.84 4850 85914 310.380 4.965 0.514 This work

10.31 4517 — — — — [32]11� 5.85 3546 5158 337.57 52.170 0.322 This work

5.85 3522 — 207 — — [31]5.77 3476 5704 219.9 28.570 1.814 [32]

21� 5.19 1142 33981 369.030 37.230 2.413 This work5.14 1150 — 410 — — [31]5.13 1239 34330 404.8 35.277 2.295 [32]

31� 4.96 2332 62159 540.670 31.52 2.453 This work4.97 2350 — 559 — — [31]4.95 2383 62289 563.2 32.179 2.465 [32]

41� 5.08 2162 64774 409.29 25.00 2.432 This work5.07 2160 65125 484.3 26.910 2.350 [32]

51� 4.87 7549 71360 566.87 10.858 2.469 This work4.82 7384 71597 625.6 11.533 2.600 [32]

61� 4.99 3655 56507 427.70 25.420 1.848 This work5.00 3588 76513 516.7 29.264 2.416 [32]

13� 4.58 2277 29824 603.080 39.81 2.901 This work4.94 2125 24163 607.8 — — [31]4.52 2182 — 619 43.264 2.957 [32]

23� 4.93 2203 39316 553.340 31.99 2.492 This work4.95 2095 — 541 — — [31]4.92 2196 33405 552.3 34.543 2.496 [32]

33� 5.19 3051 67838 486.820 18.58 2.246 This work5.20 3087 — 495 — — [31]5.18 3143 61512 497.5 18.277 2.251 [32]

43� 5.02 2214 71118 502.530 30.93 2.401 This work5.00 2218 65074 496.2 29.624 2.416 [32]

53� 5.01 9034 76273 572.080 15.56 2.407 This work4.91 9052 69870 599.2 11.533 2.505 [32]

63� 5.04 3996 82.563 526.400 23.6 2.377 This work5.06 3837 76.215 529.6 26.3893 2.359 [32]

11� 5.35 850 31250 237.21 16.54 2.114 This work5.33 761 — 339.0 — — [31]5.32 856 25193 335.8 35.990 2.134 [32]

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observe that our spectroscopic parameters agree well with thevalues obtained by Abe et al. [28] using the complete active spacesecond-order perturbation theory method. However, we remarkthat the well depth of Schilling [28] is largely underestimated (De =15 951 cm−1) when compared with the experimental value of Dall-esta et al. [26] (De = 17 502 ± 480 cm−1) or the other theoreticalresults. For the first excited state, there is a very good agreement

between our equilibrium position as well as the well depth (Re =4.64 a.u. and De = 9030 cm−1) and those of Aymar et al. [31] (Re =4.67 a.u. and De = 8830 cm−1). However, our theoretical vibrationalconstant ( e = 726.21 cm−1) and the rotational constant (Be =2.811 cm−1) are in good agreement with the theoretical values ofHabli et al. [32] ( e = 728.8 cm−1 and Be = 2.806 cm−1). From thecomparison between our spectroscopic parameters, we remark

Table 2 (concluded).

StateRe

(a.u.)De

(cm−1)Te

(cm−1) e

(cm−1) e�e

(cm−1)Be

(cm−1) References

21� 5.15 1930 68958 382.99 19.00 2.282 This work5.07 1902 — 487 — — [31]5.05 1973 62678 508.7 34.395 2.368 [32]

31� 4.86 2369 79938 479.57 23.294 2.739 This work5.04 2235 76635 497.2 28.279 2.378 [32]

13� 5.34 892 31208 239.93 16.134 2.106 This work5.30 794 — 345 — — [31]5.29 885 25157 594.6 −203.474 2.158 [32]

23� 5.15 1935 68954 382.71 18.92 2.282 This work5.06 1908 — 486 — — [31]5.05 1978 62672 487.8 30.623 2.368 [32]

33� 4.69 2377 79930 1065.67 52.801 2.750 This work5.04 2238 76635 552.2 37.536 2.378 [32]

Note: EA, H electron affinity.

Fig. 6. Permanent dipole moments of the 1–31�+ (a) and 4–101�+ (b)electronic states of the strontium hydride cation SrH+. [Colour online.]

Fig. 7. Permanent dipole moments of the 1–43�+ (a) and 5–103�+ (b)electronic states of the strontium hydride cation SrH+. [Colour online.]

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first, that the correction of the electron affinity of the H atomincreases the equilibrium distances of all states 1�+. They are in-creased by 0.01–0.04 a.u. compared to the initial values withoutthe electron affinity of the H atom. Second, the potential welldepths are decreased by several tens of inverse centimetres.

The spectroscopic parameters of the higher excited states of1,3�+, 1,3�, and 1,3� symmetries are presented in Table 2 to com-pare with the available theoretical results of Aymar et al. [31] andHabli et al. [32]. In general, our spectroscopic parameters are closeto the values found by Aymar et al. [31] and those found by Habliet al. [32]. This is not surprising because we used the same proce-dure of calculation. In addition, several excited states exhibit dou-ble potential wells and sometimes triple wells, as in the case of the6–101�+ and 8–103�+ electronic states. The double or triple welldepths are due to the avoided crossing between many electronicstates. Their existences have generated substantial non-adiabaticcoupling, and have led to an undulating behaviour of the higherexcited states at large internuclear distances.

3.4. Permanent and transition dipole momentsThe knowledge of the dipole moment of a molecular system is

considered as a sensitive test for the precision of the calculatedelectronic wave functions and energies. In fact, the dipole momentsof the dipolar molecules have a great number of applications, such asthe control of ultracold chemical reactions [56], the creation of aplatform for quantum information processing [57, 58], and the ex-

amination of fundamental theories like the measurement of theelectron dipole moments [59, 60].

To understand the ionic behaviour of the excited electronic states,we have calculated the adiabatic permanent dipole moments forall the electronic states of 1,3�+, 1,3�, and 1,3� symmetries of thestrontium hydride cation SrH+. These adiabatic permanent dipolemoments are presented in Figs. 6–10. Figure 6 shows the adiabaticpermanent dipole moments of the 1–31�+ (Fig. 6a) and 4–101�+

(Fig. 6b) electronic states. In the Fig. 6a, we remark that, at shortdistances, the adiabatic permanent dipole moments of the 1–31�+

states exhibit positive and negative values with small amplitudes. Atlarge distances they approach zero and vanish. Figure 6b shows thatthe adiabatic permanent dipole moments of the 4–51�+ and 7–101�+

electronic states, one after another, behave as linear functions ofR, although there is some numerical noise around some abruptchanges. In addition, they drop to zero at particular distances corre-sponding to the avoided crossings between the neighbour electronicstates. The linear interpolation between the adiabatic permanentdipole moments of the 7–81�+ shows a positive linear variation be-cause of the ionic character of the SrH(+) structure. In the same way,we can observe that the linear interpolation between the adiabaticpermanent dipole moments of the 4–51�+ and 7–101�+ shows a neg-ative linear variation because of the ionic character of the Sr(2+)H(−)

structure. In addition, we remark that 7–81�+ states exhibit positiveand negative dipole moments. This significant change of sign in theadiabatic permanent dipole moments can be explained by thechange of the polarity in the SrH+ system, going from the Sr(2+)H(−)

Fig. 8. Permanent dipole moments of the first six 1� electronicstates of the strontium hydride cation SrH+. [Colour online.]

Fig. 9. Permanent dipole moments of the first six 3� electronicstates of the strontium hydride cation SrH+. [Colour online.]

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structure for the negative sign to the SrH(+) structure for the positivesign. This interpretation confirmed the sign convention as it is dem-onstrated in many previous studies [12, 34–36, 61–64].

The discontinuities between consecutive portions are due to theavoided crossings. Moreover, the positions of these abrupt changesare correlated to the avoided crossings between the adiabatic poten-tial energy curves. The abrupt changes and the avoided crossings areboth results of the manifestations of abrupt changes of the characterof the electronic wave functions. For example, we mention the posi-tions of the abrupt changes in the adiabatic permanent dipole mo-ments distribution between 71�+ and 81�+ at 22.94 a.u., between 81�+

and 91�+ at 25.55 a.u., and between 91�+ and 101�+ at 39.92 a.u. Thesepositions of real crossings between the adiabatic permanent dipolemoments are identical to the regions of internuclear distances exhib-iting avoided crossings between the adiabatic potential energycurves of 1�+ electronic states. It is clear that the positions of theirregularities in the R-dependence of the adiabatic permanent dipolemoments are correlated to the avoided crossings between the poten-tial energy curves, which are both manifestations of abrupt changesof the character of the electronic wave functions.

In Fig. 7, we depict the adiabatic permanent dipole moments ofthe 1–43�+ (Fig. 7a) and 4–103�+ (Fig. 7b) electronic states of thestrontium hydride cation SrH+. We remark that these dipole mo-ments pass through several maximums located at intermediate andlarge distances. In addition, there are many abrupt changes betweenthe adiabatic permanent dipole moments of the electronic states;33�+ and 43�+, 83�+ and 93�+, and 93�+ and 103�+ located at 4.06,9.28, and 11.24 a.u., respectively. We conclude that the positions ofthese abrupt changes coincide with the crossing positions previouslyseen in the adiabatic potential energy curves of the 3�+ electronicstates. The adiabatic permanent dipole moments have also been de-termined for the electronic states of the 1�, 3�, and 1,3� remainingsymmetries. Their adiabatic permanent dipole moments are dis-played in Figs. 8–10. As expected, these dipole moments are not

negligible and they become more significant for the higher excitedstates. The exception is the permanent dipole moments of 61,3�states, which tend to be a constants. The other permanent dipolemoments of 1–51,3� states vanish at large distance from zero.

The quasi-diabatic permanent dipole moments of the 1–61�+

and 7–101�+ electronic states of the strontium hydride cation SrH+

are depicted in Figs. 11a and 11b, respectively. These quasi-diabaticpermanent dipole moments are obtained using a simple rotation ofthe adiabatic permanent dipole moments matrix. From Fig. 11b, wecan see that the quasi-diabatic permanent dipole moments of the7–101�+ states present the same behaviour as in the adiabatic casewith abrupt variations. These abrupt variations are situated nearly atdifferent positions corresponding to the avoided crossings betweenthe neighbour electronic states mentioned in Table 1. For example,we can quote the crosses between the quasi-diabatic permanent di-pole moments of the 71�+ and 81�+ states localized at 21.14 and22.88 a.u. and accompanied by abrupt variations. At large distances,it is clearly observed that the quasi-diabatic dipole moments of the71�+ state are characterized by a linear divergence. Similar behav-iours are obtained between the quasi-diabatic permanent dipole mo-ments of 71�+ and 81�+ states located at 34.26 and 39.76 a.u.

In addition to the permanent dipole moments, we also calculatedthe transition dipole moments between neighbour electronic statesfor the electronic states of 1,3�+, 1,3�, and 1,3� symmetries. Here, wepresent only the transition dipole moments between the neighbour

Fig. 10. Permanent dipole moments of the first six 1� and 3� electronicstates of the strontium hydride cation SrH+. [Colour online.]

Fig. 11. Quasi-diabatic permanent dipole moments of the 1–61�+

and 7–101�+ electronic states of the strontium hydride cation SrH+.[Colour online.]

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electronic states of the 1�+ symmetry. They are displayed in Figs. 12aand 12b. The transition between the ground state (11�+) and the ex-cited state (21�+), dissociating into Sr+(5s) + H(1s) and Sr+(4d) + H(1s),respectively, is very large. It presents a maximum of 2.53 a.u. locatedat 7.44 a.u. We can conclude that around this distance there is animportant overlap between the corresponding molecular wave func-tions. At large distances, the 11�+–21�+ transition becomes a constantequal to 2.39 a.u., which is related to the atomic transition betweenSr+(5s) and Sr+(4d). Moreover, we remark that the 21�+–31�+ and 31�+–41�+ transitions, Sr+(4d) + H(1s) and Sr+(5p) + H(1s), and Sr+(5p) + H(1s)and Sr+(5d) + H(1s), present the same behavior. They decrease at smalldistances, then they pass by a minimums located at 7.67 and 8.13 a.u.,respectively, and finally they go to the absolute values of 1.63 and1.65 a.u., respectively. These values are in good agreement with thetransition dipole moments (1.69 and 1.84 a.u.) deduced from the the-oretical oscillator strength of Mitroy et al. [65]. The other transitionsbetween the high excited states present many peaks located at par-ticular distances very close to the avoided crossings in adiabaticrepresentation. We mention here the peaks observed in the 81�+–91�+ and 91�+–101�+ transitions located at 25.60 and 39.90 a.u.,respectively.

4. ConclusionThis work is focused on the structure and electronic properties

of the strontium hydride cation SrH+ dissociating into Sr+(5s, 4d,5p, 6s, 5d, 6p, 7s, 6d, and 7p) + H(1s, 2s, and 2p) and Sr(5s2, 5s5p,5s4d, 5s4d, 5s5p) + H+ It has been systematically investigated using

a standard quantum chemistry approach based on pseudopoentials,Gaussian basis sets, effective core polarization potentials, and fullconfiguration interaction calculations. The adiabatic potential en-ergy curves and their spectroscopic parameters for the ground andmany excited electronic states of 1,3�+, 1,3�, and 1,3� symmetries havebeen computed for a large and dense grid of internuclear distancesvarying from 2.40 to 105.00 a.u. The higher excited states have shownundulations related to avoided crossings or undulating orbitals ofthe atomic Rydberg states. They led to multiple potential barriersand wells in the potential energy curves. The agreement between thespectroscopic parameters obtained in our work and those ofthe previous studies [26, 28, 29, 31, 32] for the ground (X1�+) and thehigh excited states are shown to be satisfactory. The correction ofthe electron affinity of the H atom shows that there is no change inthe feature of the potential energy curves. Compared to the initialvalues without the electron affinity of the H atom, there are smallchanges in the equilibrium distances accompanied by an increase inthe potential well depths. In the quasi-diabatic representation, twoionic quasi-diabatic states are clearly observed. The first representsthat the Sr(2+)H(−) structure crosses the quasi-diabatic curves D2–9 atdifferent distances. The second ionic quasi-diabatic state dissociatinginto SrH(+), is expected to cross only the D5–7 and D9–10 states atlarge internuclear distances. Such avoided crossings became realcrossings in the quasi-diabatic representation.

For a better understanding of the ionic character of the electronicstates of the strontium hydride cation SrH+, we have calculated thepermanent and transition dipole moments. The permanent dipolemoments of the 1–101�+ electronic states have shown the presence ofthe ionic state, corresponding to the SrH+ structure, which is almosta linear feature function of R, especially for the higher excited statesand at intermediate and large distances. Moreover, the abruptchanges in the adiabatic permanent dipole moments are localized atparticular distances corresponding to the avoided crossings betweenthe neighbor electronic states.

References1. C.A. Regal, C. Ticknor, J.L. Bohn, and D.S. Jin. Nature, 424, 47 (2003). doi:10.

1038/nature01738.2. J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.C. Nägerl, and R. Grimm.

Science, 301, 1510 (2003). doi:10.1126/science.1088876. PMID:12934014.3. K. Winkler, F. Lang, G. Thalhammer, P.v.d. Straten, R. Grimm, and

J.H. Denschlag. Phys. Rev. Lett. 98, 043201 (2007). doi:10.1103/PhysRevLett.98.043201.

4. K.-K. Ni, S. Ospelkaus, M.H.G. de Miranda, A. Pe’er, B. Neyenhuis, J.J. Zirbel,S. Kotochigova, P.S. Julienne, D.S. Jin, and J. Ye. Science, 322, 231 (2008).doi:10.1126/science.1163861. PMID:18801969.

5. A.J. Kerman, J.M. Sage, S. Sainis, T. Bergeman, and D. DeMille. Phys. Rev. Lett.92, 153001 (2004). doi:10.1103/PhysRevLett.92.153001. PMID:15169280.

6. D. Wang, J. Qi, M.F. Stone, O. Nikolayeva, H. Wang, B. Hattaway,S.D. Gensemer, P.L. Gould, E.E. Eyler, and W.C. Stwalley. Phys. Rev. Lett. 93,243005 (2004). doi:10.1103/PhysRevLett.93.243005. PMID:15697805.

7. M.W. Mancini, G.D. Telles, A.R.L. Caires, V.S. Bagnato, and L.G. Marcassa.Phys. Rev. Lett. 92, 133203 (2004). doi:10.1103/PhysRevLett.92.133203. PMID:15089608.

8. C. Haimberger, J. Kleinert, M. Bhattacharya, and N.P. Bigelow. Phys. Rev. A,70, 021402 (2004). doi:10.1103/PhysRevA.70.021402.

9. J.P. Shaffer, W. Chalupczak, and N.P. Bigelow. Phys. Rev. Lett. 82, 1124 (1999).doi:10.1103/PhysRevLett.82.1124.

10. F.B.C. Machado and F.R. Ornellas. J. Chem. Phys. 94, 7237 (1991). doi:10.1063/1.460207.

11. L.F. Errea, B. Herrero, L. Mendez, I. Rabadan, and P. Sanchez. J. Phys. B: At.Mol. Opt. Phys. 27, L753 (1994). doi:10.1088/0953-4075/27/23/001.

12. M. Farjallah, C. Ghanmi, and H. Berriche. Eur. Phys. J. D, 67, 245 (2013).doi:10.1140/epjd/e2013-40141-3.

13. K. Højbjerre, A.K. Hansen, P.S. Skyt, P.F. Staanum, and M. Drewsen. New. J.Phys. 11, 055026 (2009). doi:10.1088/1367-2630/11/5/055026.

14. J.C. García-Madroñal, O. Mó, I.L. Cooper, and A.S. Dickinson. J. Mol. Struc.(Theochem.), 260, 63 (1992). doi:10.1016/0166-1280(92)87035-X.

15. C.M. Dutta, P. Nordlander, and M. Kimura. Chem. Phys. Lett. 264, 51 (1997).doi:10.1016/S0009-2614(96)01291-2.

16. M. Aymar, R. Guérout, M. Sahlaoui, and O. Dulieu. J. Phys. B: At. Mol. Opt.Phys. 42, 154025 (2009). doi:10.1088/0953-4075/42/15/154025.

17. K. Mølhave and M. Drewsen. Phys. Rev. A, 62, 011401 (2000). doi:10.1103/PhysRevA.62.011401.

Fig. 12. Transition dipole moments between neighbor electronicstates of 1�+ symmetry of the strontium hydride cation SrH+. [Colouronline.]

Belayouni et al. 801

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ww

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Uni

v of

Geo

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Lib

rari

es o

n 10

/05/

16Fo

r pe

rson

al u

se o

nly.

18. S. Canuto, M.A. Castro, and K. Sinha. Phys. Rev. A, 48, 2461 (1993). doi:10.1103/PhysRevA.48.2461. PMID:9909876.

19. J.B. Schilling, W.A. Goddard, III, and J.L. Beauchamp. J. Am. Chem. Soc. 108,582 (1986). doi:10.1021/ja00264a004.

20. J.B. Schilling, W.A. Goddard, III, and J.L. Beauchamp. J. Phys. Chem. 91, 5616(1987). doi:10.1021/j100306a024.

21. A. Boutalib, J.P. Daudey, and M. El Mouhtadi. Chem. Phys. 167, 111 (1992).doi:10.1016/0301-0104(92)80026-R.

22. H. Habli, H. Ghalla, B. Oujia, and F.X. Gadéa. Eur. Phys. J. D, 64, 5 (2011).doi:10.1140/epjd/e2011-10689-y.

23. H. Habli, R. Dardouri, B. Oujia, and F.X. Gadéa. J. Phys. Chem. A, 115, 14045(2011). doi:10.1021/jp204058y. PMID:22013968.

24. A.R. Allouche, F. Spiegelmann, and M. Aubert-Frécon. Chem. Phys. Lett. 204,343 (1993). doi:10.1016/0009-2614(93)90020-2.

25. L. Mejrissi, H. Habli, H. Ghalla, B. Oujia, and F.X. Gadéa. J. Phys. Chem. A, 117,5503 (2013). doi:10.1021/jp4025409. PMID:23701525.

26. N.F. Dalleska, K.C. Crellin, and P.B. Armentrout. J. Phys. Chem. 97, 3123(1993). doi:10.1021/j100115a010.

27. P. Fuentealba and O. Reyes. Mol. Phys. 62, 1291 (1987). doi:10.1080/00268978700102971.

28. J.B. Schilling, W.A. Goddard, III, and J.L. Beauchamp. J. Am. Chem. Soc. 109,5565 (1987). doi:10.1021/ja00253a001.

29. M. Abe, M. Kajita, M. Hada, and Y. Moriwaki. J. Phys. B: At. Mol. Opt. Phys. 43,245102 (2010). doi:10.1088/0953-4075/43/24/245102.

30. M. Kajita, M. Abe, M. Hada, and Y. Moriwaki. J. Phys. B: At. Mol. Opt. Phys. 44,025402 (2011). doi:10.1088/0953-4075/44/2/025402.

31. M. Aymar and O. Dulieu. J. Phys. B: At. Mol. Opt. Phys. 45, 215103 (2012).doi:10.1088/0953-4075/45/21/215103.

32. H. Habli, L. Mejrissi, N. Issaoui, S.J. Yaghmour, B. Oujia, and F.X. Gadéa. Int.J. Quant. Chem. 115, 172 (2015). doi:10.1002/qua.24813.

33. H. Berriche. PhD Thesis, Paul Sabatier University, France. 1995.34. N. Mabrouk and H. Berriche. H. J. Phys. B: At. Mol. Opt. Phys. 41, 155101 (2008).

doi:10.1088/0953-4075/41/15/155101.35. N. Mabrouk, H. Berriche, H.B. Ouada, and F.X. Gadéa. J. Phys. Chem. A, 114,

6657 (2010). doi:10.1021/jp101588v. PMID:20518475.36. N. Mabrouk and H. Berriche. J. Phys. Chem. A, 118, 8828 (2014). doi:10.1021/

jp5043427. PMID:25058020.37. I. Jendoubi, H. Berriche, H.B. Ouada, and F.X. Gadéa. J. Phys. Chem. A, 116,

2945 (2012). doi:10.1021/jp209106w. PMID:22360282.38. S. Bellayouni, I. Jendoubi, N. Mabrouk, and H. Berriche. Adv. Quant. Chem.

68, 203 (2014). doi:10.1016/B978-0-12-800536-1.00011-3.39. J.C. Barthelat and Ph. Durand. Theor. Chim. Acta, 38, 283 (1975). doi:10.1007/

BF00963468.40. W. Müller, J. Flesh, and W. Meyer. J. Chem. Phys. 80 3297 (1984).

41. M. Foucrault, Ph. Millie, and J.P. Daudey. J. Chem. Phys. 96, 1257 (1992).doi:10.1063/1.462162.

42. N. Boutasetta, A.R. Allouche, and M. Aulbert-Frécon. Phys. Rev. A, 53 3845(1996).

43. J.E. Sansonetti. J. Phys. Chem. Ref. Data, 41 013102 (2012). doi:10.1063/1.3659413.

44. F.X. Gadéa and A. Boutalib. J. Phys. B: At. Mol. Opt. Phys. 26, 61 (1993).doi:10.1088/0953-4075/26/1/006.

45. A. Boutalib and F.X. Gadéa. J. Chem. Phys. 97, 1144 (1992). doi:10.1063/1.463242.

46. F.X. Gadéa. PhD, Université Paul Sabatier, Toulouse, France. 1987.47. F.X. Gadéa and M. Pélissier. J. Chem. Phys. 93, 545 (1990). doi:10.1063/1.

459554.48. B. Sarkar and S. Adhikari. J. Chem. Phys. 124, 074101 (1990).49. M. Baer. Chem. Phys. Lett. 35, 112 (1975). doi:10.1016/0009-2614(75)85599-0.50. M. Baer. Phys. Rep. 358, 75 (2002). doi:10.1016/S0370-1573(01)00052-7.51. A.K. Paul, S. Sardar, B. Sarkar, and S. Adhikari. J. Chem. Phys. 131, 124312

(2009). doi:10.1063/1.3236839. PMID:19791886.52. S. Mukherjee, S. Bandyopadhyay, A.K. Paul, and S. Adhikari. J. Chem. Phys. A,

117 3475 (2013). doi:10.1021/jp311597c.53. F.X. Gadéa. Phys. Rev. A, 43 1160 (1991).54. A.S. Dickinson and F.X. Gadea. Phys. Rev. A, 65, 052506 (2002). doi:10.1103/

PhysRevA.65.052506.55. A.S. Dickinson and F.X. Gadea. J. Mol. Struct. (Theochem.), 621, 621 (2003).56. R.V. Krems. Phys. Chem. Chem. Phys. 10, 4079 (2008). doi:10.1039/b802322k.

PMID:18612510.57. J.J. Hudson, B.E. Sauer, M.R. Tarbutt, and E.A. Hinds. Phys. Rev. Lett. 89,

0230031 (2002). doi:10.1103/PhysRevLett.89.023003.58. M.G. Kozlov and D. DeMille. Phys. Rev. Lett. 89, 133001 (2002). doi:10.1103/

PhysRevLett.89.133001. PMID:12225020.59. D. DeMille. Phys. Rev. Lett. 88, 067901 (2002). doi:10.1103/PhysRevLett.88.

067901. PMID:11863853.60. T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau. Rep. Prog. Phys.

72, 126401 (2009). doi:10.1088/0034-4885/72/12/126401.61. M. Aymar, O. Dulieu, and F. Spiegelman. J. Phys. B: At. Mol. Opt. Phys. 39,

S905 (2006). doi:10.1088/0953-4075/39/19/S07.62. M. Aymar and O. Dulieu. J. Chem. Phys. 122, 204302 (2005). doi:10.1063/1.

1903944. PMID:15945719.63. M. Aymar and O. Dulieu. Mol. Phys. 105, 1733 (2007). doi:10.1080/

00268970701494016.64. I.S. Lim, W.C. Lee, Y.S. Lee, and G.H. Jeung. J. Chem. Phys. 124, 234307 (2006).

doi:10.1063/1.2204607. PMID:16821918.65. J. Mitroy and J.Y. Zhang. Phys. Rev. A, 77, 032512 (2008). doi:10.1103/PhysRevA.

77.032512.

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