Adiabatic and Diabatic Investigation of Numerous Electronic States for

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Adiabatic and Diabatic Investigation of Numerous Electronic States for the Alkali Dimer FrNa Soulef Jellali, Héla Habli, Leila Mejrissi, Brahim Oujia, and Florent Xavier Gadea J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b10739 • Publication Date (Web): 26 Dec 2018 Downloaded from http://pubs.acs.org on January 3, 2019

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The Journal of Physical Chemistry

Adiabatic and Diabatic Investigation of Numerous Electronic States for the Alkali Dimer FrNa Soulef Jellali *,a , Héla Habli a, Leila Mejrissi a , Brahim Oujia b, Florent Xavier Gadéa c a

Laboratoire de Physique Quantique et Statistique, Faculté des Sciences de Monastir, Université de Monastir, Avenue de l’Environnement 5019, Monastir, Tunisie,

b University c

of Jeddah, Faculty of Science, Physics Department , Jeddah , Kingdom of Saudi Arabia,

Laboratoire de Chimie et Physique Quantique, UMR5626 du CNRS, Université de Toulouse, UPS, 118 route de Narbonne, 31062, Toulouse Cedex 4, France. (*)

Corresponding author: E-mail: [email protected]

Abstract The current paper reports a global investigation of all excited states below the ionic limit Fr+ Na- of FrNa molecule following diabatic and adiabatic representations. The adiabatic and diabatic potential energy curves (PECs) for Σ+, Π and Δ symmetries have been calculated for a dense grid of internuclear distances. The transition and permanent dipole moments (TDM and PDM) have been reported for both representations. Regarding the pseudo-potential approach and Full valence Configuration Interaction (FCI), the ab initio computation has been performed. Furthermore, the diabatization method was achieved by the use of the variational effective Hamiltonian theory (VEH). This latter was served to assess the non-adiabatic coupling between the treated adiabatic states and to eliminate it employing a suitable unitary transformation matrix. The detailed computation of the PECs, PDM and TDM curves of the ground and excited states play a key role to coordinate experimental efforts in order to create cold and ultracold molecules in their ground stable rovibrational levels.

1. Introduction Several works put their efforts in the creation of the cold or ultracold molecules like NaCs, KRb and RbCs system1-7 by the use of photo-association of cold or ultracold alkali atoms. Recently, a new field can be investigated by developing trapping and laser cooling of radioactive atoms. Among them, we note Bose-Einstein condensation, cold atom-atom collisions, β decay and the electric dipole moment (EDM).8 To our knowledge, Lim et al.9 have achieved the first investigation of mixed species involving francium and rubidium or cesium atom. In ref 9, the spectroscopic properties for the neutral and 1 ACS Paragon Plus Environment

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cationic alkali dimers from K2 to Fr2 were computed. All these calculations have been performed using numerous approaches reposed on a relativistic coupled-cluster method or density functional theory. Furthermore, these relativistic ab initio methods were used by Derevianko et al.10 to determine the Van der Waals coefficients (VDW) for the heteronuclear alkali-metal dimers for Li, Na, K, Rb, Cs and Fr. Applying similar theory, Aymar et al.11 have carried out the first computation of the structural properties of francium diatomic compound FrCs, FrRb and Fr2, yielding to the PECs and electric dipole moments (PDM and TDM). In order to get an accurate knowledge of the long-range alkali-alkali interactions, a major interest has been put on studying the ground states and chiefly near the atomic asymptote. The exact given results are essential for the understanding and the accomplishment of diverse processes in cold collisions as the formation of ultracold molecules, as scattering length determination, and sympathetic cooling. The spectroscopic studies of vibrational and electronic states are needed to modeling and understanding the different steps in the phenomena of ultracold molecule formation. These latter are the photoassociation, the radiative stabilization into metastable vibrational levels and the stimulated emission pumping into the ground state. Therefore, numerous theoretical and experimental researchers were highly spanned in the investigations of structural and dipolar properties for alkali diatomic molecules like RbCs,12-14 NaRb and LiRb,15 NaCs, LiCs and LiNa,16, 17 KRb,18 LiK,19 NaK20, 21 and CsK.22 For all these systems, an interesting behavior is observed in the excited adiabatic potentials 1Σ+ manifested by the presence of numerous avoided crossings. As that is clear they showed significant nonadiabatic couplings between the electronic states, in which the majority of them are imprinted by the ionic charge transfer state. These couplings should be evaluated and taken into account in order to overtake the Born–Oppenheimer (BO) approximation. In the adiabatic representation, the coupling is founded by the kinetic energy nuclei’s operator, while the diabatic one offers only potential coupling. Despite the fact that the B.O. approximation enormously minimizes computational effort, the adiabatic representation has never really proved to be an appropriate basis for computing the elastic scattering or inelastic effect.23 Noteworthy, the direct calculation of the couplings between the adiabatic states is complicated because they may become nearly singular around the avoided crossings.24,25 The adiabatic representation encounters sharper couplings when the avoided crossings become increasingly narrow and the diabatic one is favored.26 Indeed, thanks to this chosen representation, both potential energy surfaces and couplings become smooth and the considered states maintain a physical meaning when the vibronic interaction should be included.

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The main purpose of this research is to treat the adiabatic and diabatic representations for FrNa molecular system. As no experimental or theoretical data are available, we discuss our results by comparison with the NaCs system.27,28 We have calculated the adiabatic and diabatic PECs for all symmetries Σ+, Π and Δ under the ionic limit Fr+ Na- for a dense grid with 370 distances. Also, we represent the dipolar properties such as transition and permanent dipole moments (TDM and PDM). In the present work, three main sections are attributed after the introduction. Section 2 sums up the different methods used for the adiabatic and the diabatic approaches. Section 3 reports the molecular results in adiabatic, diabatic representations together and the results of the electric dipole moments (PDM and TDM). Section 4 is allotted to concluding remarks. 2. Methods 2.1. Details of the adiabatic calculation Encouraged by the reliable results given by the pseudo-potential method in our previous works,28-38 we are basing on the same method to investigate the adiabatic properties for the FrNa molecule. In this way, we considered just the valence electron for both Fr and Na atoms and our own system is treated through two effective valence electrons system. The two polarizable cores of Na+ and Fr+ are replaced by a semi-empirical pseudo-potential proposed by Berthelat and Durand,39 supplemented by both formula CPP and ECP (respectively, Core Polarization Potentials and Effective Core Potentials). Consequently, at the self- consistent field (SCF) level, we computed the electronic energy taking into consideration the core-valence electron correlation approach developed by Muller et al.40 Then, we executed the FCI calculation involving the package codes,28-38 developed in LPQT (laboratory of Quantum Physics in Toulouse). The CPP expression is written as a function of both αc and 𝑓c (αc: dipole polarizability of the core (c); 𝑓c: electric field at center (c) created by the valence electron and all other centers’ cores) as shown in the following equations : 1

𝑉𝑐𝑝𝑝 = ― 2 ∑𝛼𝛼𝑐 𝑓𝑐 Such as:

2

(1)

𝑟𝑐𝑖

𝑅𝑐′𝑐

𝑐𝑖

𝑐′𝑐

𝑓𝑐 = ∑𝑖𝑟3 𝐹(𝑟𝑐𝑖, 𝜌𝓁𝑐) - ∑𝑐′ ≠ 𝑐𝑍𝑐 𝑅3

(2)

In expression (2), the term 𝑅𝑐′𝑐 represents a core-core vector and 𝑟𝑐𝑖 is the core-electron vector, while the Fl , written in expression (3), defines the l-dependent cut-off function explored by Foucrault group's.41

( )]

[

𝑟2𝑐𝑖

𝐹𝑙 = 𝑙 ― exp ― 𝜌2 𝑐


(3)


In this work, we optimized carefully two basis sets for the selected alkali atom Fr (7s, 7p, 6d, 8s, 8p, 7d, 9s, 9p) and Na (3s, 3p, 4s, 3d, 4p) to obtain an accurate calculation. We use large basis set (8s 6p 5d) and (6s 5p 4d) with (51, 41) functions for Fr and Na, respectively, contracted to (8s 4p 5d) and (6s 4p 2d). The number of these functions decreases to (45, 28). For each basis set, the cut-off radii parameters are reported in Table 1 and they are adjusted to reproduce the experimental energies42 for the lowest s, p and d levels of Fr and Na with fixed values of core polarizability αNa+=0.993 a.u, αFr+=20.38 a.u.11,43 We present, in Tables 2A and 2B, our theoretical atomic spectrum for the Fr and Na atoms as well as the corresponding energy differences (∆E). These values are compared firstly with the available experimental data,42 then, with theoretical data calculated with the polarizability αFr+= 20.38 a.u.11 Correspondingly, our atomic energies are in good agreement with both values. The difference between those works does not surpass 90 cm−1, which is found in Fr (7d) atomic level. This error is probably due to the effects of basis insufficiency and limitation of calculations. Usually, this good accordance obtained at the atomic level will be transmitted to the molecular energy yielding a very good approval of the molecular dissociation limits reported in Table 3. 2.2. Details of the diabatic calculation Our previous calculation relying on the BO approximation shows a shortfall results around the avoided crossing’s positions related to the strong non-adiabatic coupling between the investigated adiabatic states. These interactions are significant in various fields such as the nonradiative transition, inelastic collision and the pre-dissociation phenomena. To solve the problem, we adopt the diabatic approach, developed by F. X. Gadéa,44 which is more employed in many previous published paper in particular for the alkali hybrid series ( LiH, NaH, CsH and RbH)45-47 and alkali dimers (KLi, CsLi, RbLi and NaCs).28,34,48 In the current part, we will describe briefly the fundamental lines of this technique. To take into account the missed coupling in the adiabatic calculation, the diabatization method reposed on the variational effective Hamiltonian theory (VEH).49-51 It serves to assess the non- adiabatic coupling between the treated adiabatic states and to eliminate it employing a suitable unitary transformation matrix. In this method, the effective Hamiltonian used as a reference the model space spanned by n reference states (Φi) that represent the long rang adiabatic potential curves but in its alternative variational version. Very large interatomic distances are chosen here where all interatomic interaction dissipated. On the other hand, the target space is spanned by the n adiabatic states (Ψi) at interatomic distance R to be diabatised. Via VEH theory, the electronic Hamiltonian is calculated in the target space containing an effective basis set, instead of the 4 ACS Paragon Plus Environment

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computation of an effective Hamiltonian in the model space. It should be underlined that the variational properties are automatically kept. The VEH theory is developed firstly in nuclear physics by using two formalisms: one is “de Bloch”52 and the other is “des Cloizeaux”.53 From the “des Cloizeaux” formula, an overlap matrix A is considered as usual wave operator which is defined as follow: Aij= < Ψi ǀ Φj >. Using the unitary transformation (Uij= < Ψi ǀ Dj >, U+U= 𝟙), the n diabatic states (Di), corresponding to symmetrical orthogonalization of the projected model space vectors onto the target space, are orthonormal linear combinations of the n adiabatic ones. For appraising the overlap matrix, an effective metric is utilized for the overlap between the atomic basis sets at each R. In the case of the diatomic system, the lower half diagonal part is taken from one distance and the upper one from the other distance. Following the overlap matrix A, we performed a crude numerical estimation of the non-adiabatic coupling, as can be seen in ref 26. Thus, the electronic Hamiltonian H in the diabatic basis is calculated as a function of unitary transformation (U=A (A+A)-1/2) as shown in the following expression: < D ǀ H ǀ D > = U+ E U = (A+A)-1/2 A+ E A (A+A)-1/2 Notably, the overlap matrix depends on the heavier atom position. So, the common origin should be specified which is fixed here at the heavier francium atom. 3. Results and Discussions 3.1. Adiabatic and diabatic PECs and their spectroscopic constants First of all, reliable potential energy curves are needed for modeling and understanding cold collision phenomena. These behaviors offer divers physic meaning like non-radiative transition, inelastic collision, especially at the vicinity of the avoided crossings. Thus, we have investigated the FrNa diatomic molecule by both adiabatic and diabatic representations. We have calculated the adiabatic and diabatic PECs for all symmetries Σ+, Π and Δ under the ionic limit Fr+ Nawith very small interval steps down to 0.05 a.u, around to the ionic-neutral crossings. For the diabatization, the 10 lower adiabatic states below the ionic state for the 1Σ+ symmetry, and the 10, 16 and 6, lower adiabatic states, respectively, for symmetries, 3Σ+, 1,3Π and 1,3Δ are taken as references spanning the model space at the largest distances (R=200 a.u). Altogether, the diabatic theory has been performed for 42 states (20 Σ+, 16 Π and 6 Δ). In order to check the precision of ab-initio results, we have computed the spectroscopic constants related to each adiabatic state. Indeed, the vibrational levels are interpolated by the least-squares approach given as follow:



1 1 Ev  v( Re )  e (v  )  e  e (v  ) 2 2 2

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(4)

In this equation, Ev defines the energy of the vibrational level v. Re, ωe and ωeχe are the equilibrium distance, the vibrational frequency and the anharmonic constant, respectively. In Table 5, we report all these constants for different symmetries (Σ+, Π, ∆). For our best knowledge, these constants are calculated here for the first time. 3.1.1. 1,3Σ+ symmetry For best presentation, we report in Figure 1A a global graphic for adiabatic potential energy curves of all 1Σ+ electronic states, whereas in Figure 1B we give a zoom for the higher excited states (F, G, H, I, J, K, L and M)1Σ+. The curves of the ground state X1Σ+ and the first ones A1Σ+present a simple behavior. Both has a single potential well with depth equal to De=4506, 6365 (cm-1) at Re=7.12, 8.71 (a.u), respectively. In contrast, for the higher excited states, their shapes become more involved and more information can be extracted. Therefore, their curves are characterized by the widening of the well or the existence of double, triple and sometimes quadruple minimums. It can be the consequence of the avoided crossings between the neighbor states. For instance, the third state C1Σ+ presents a simple well (De=3498 cm-1 at Re=8.79 a.u), then this latter becomes very wide due to the avoided crossing with D1Σ+ at RAC= 12.6 a.u with smaller difference E= 164 cm-1. Also, the E1Σ+ state has double wells, one at 8.19 a.u with De=2573 cm-1 and the other at 18.21 a.u with De=2254 cm-1 when a special avoided crossing with the precedent state D1Σ+ is located at RAC= 19.2 a.u with small difference E= 579 cm-1. Beyond F1Σ+ state, as can be seen in Figure 1B, an interesting global behavior can be observed following multiple wells. Usually, it corresponds to the presence of undulating aspect. They aren’t due just to the avoided crossings but also to the undulation of atomic Rydberg orbitals. They are connected to a closed shell ion passing through antinodes in the Rydberg wave function of the other atom. Add to that we observe the existence of an underlying very attractive curve that yield series avoided crossings as reported in Table 4. There are two ionic characters in the higher states: the first corresponding to the ionic charge transfer state [Fr+ Na−] behaving as (−1/R) which is formed between the D-J1Σ+ adiabatic states, whereas the second corresponding to [Fr− Na+] is formed between the G-M1Σ+ adiabatic states. Noteworthy, in the entrance channel, starting asymptotically from state D1Σ+, the avoided crossing between the neighbor states at larger distances (from 20 to 80 a.u) will probably be passed diabatically caused by the weakness of the coupling; but then, much larger couplings will be manipulated experimentally at short distances, and charge transfer to Fr*Na can be foreseeable. 6 ACS Paragon Plus Environment

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After good examination of Figure 2, when drawn the adiabatic PECs of 3Σ+ states, we notice that the first electronic triplet state a3Σ+ present an almost repulsive character as observed for similar system (NaCs28, NaRb31). The adiabatic states (c, d) 3Σ+ have a simple potential depth about 2000 cm-1 in the vicinity of internuclear distance 8 a.u. Moreover, we also show numerous avoided crossings for the superior states (f, g, h, i, j, k, l, m and n) 3Σ+, which can be justified by the interaction with Fr+ Na- and Fr- Na+ ionic states. This underlying behavior contributes to the existence of potentials with multiple wells (double and even triple). Moving now to the diabatic representation, we display in Figure 3 and Figure 4 the diabatic potential energy curves for the 1Σ+ and 3Σ+ states and they are labeled with uppercase letter D1-10 and d1-10, respectively. It isn’t necessary to plot this representation with many distances since the variations of the potential energy and the electric dipole curves are definitely smoother. As can be seen from Figure 3, the ionic state D1 correlated to Fr+ Na- crosses all the curves and nicely behaves as -1/R as expected due to the electrostatic interaction between Fr+ and Na-. These real crossings located at the intermediate and large distances are corresponding to the avoided crossing between the 1Σ+ electronic states discussed previously in the adiabatic representation. The crossings occur with the lowest states D2-5 approximately at 9.82 a.u, 13.09 a.u, 14.16 a.u, and 14.51 a.u, respectively. The others are occurring at higher energies and much larger distances. Besides, the ionic state D1 presents a potential well depth at the interatomic distance R=6.9 a.u, while the lower states exhibit a repulsive character at shorter distances and various real crossings are localized. For instance, D2 crosses D3 at R approximately 7.27 a.u and D4 around 6.48 a.u; D3-D4 around 4.52 a.u, and D5-D6 near 6.32 a.u. Other real crossings can be observed between the higher diabatic states such as D6-D7 (17.26 a.u). Consequently, this comportment gives the reliability of the diabatization method. Moreover, we remark that the decreasing of the coupling’s magnitude between the corresponding diabatic states is related to the increasing of internuclear distance. Hence, the neutral-ionic crossings become weakly avoided since the ionic state D1 encounters highly excited states at larger distances. Figure 4 shows that the diabatic PECs for 3Σ+ states present a similar behavior found in the diabatic singlet ones once the charge transfer states have been removed; whereas the adiabatic ones have rather different shapes. This behavior is described by the presence of huge real crossings as well as various undulations with a strong repulsive character at shorter distances. This comportment is related to the nature of the Rydberg states that may extend to rather large distances for the highly excited states. Interestingly, the 3Σ+ diabatic crossings d1-d2 and d2-d3 are almost at the same positions than the corresponding 1Σ+ ones D2-D3, and D3-D4. This comportment is properly consistent with the Fermi model54, 55 where an atom is in a collision 7 ACS Paragon Plus Environment


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with a Rydberg electron, and the potential is, then, in proportionality to the Rydberg magnitude and to the scattering length. 3.1.2. 1,3Π and 1,3Δ symmetries Concerning 1,3Π and 1,3Δ symmetries, these adiabatic potential curves are depicted in Figure 5 and in the Supporting Information (Figure S1), respectively. They are computed for a dense and large grid of interatomic distances R from 4.5 to 200 a.u in order to exhibit the long-range avoided crossing. It should be mentioned that the behavior of adiabatic PECs of our FrNa alkali dimer is very similar to the behavior of CsLi alkali dimer.28 These findings are not surprising since we treated the interaction between heavy alkali atom and a light one using the same approach (pseudo-potential and CPP method). From Figure 5, we notice that the triplet states 3Π are slightly deeper than the singlet states 1Π. For instance, the 13Π state has a single potential well with De=6608 cm-1 at Re= 7.07 a.u deeper than 11Π state which posses a weak well with De=1152 cm-1 at Re= 8.15 a.u. The majority of these singlet and triplet curves exhibit a smoother behavior with a single potential well at different equilibrium distances. Furthermore, we note the presence of such avoided crossings between the higher excited states tied to the appearance of ionic state (as seen in Table 4). As shown in the Supporting Information (Figure S1), the adiabatic potential energy curves of 1Δ and 3Δ states correlated to the same dissociation limit almost degenerate. These states tend rapidly to their asymptotic limit at the distance 20 a.u. The 21Δ and 23Δ curves are quite closed having a potential well with depth value about 8000 cm-1 in the vicinity of internuclear distance 8 a.u. In the diabatic representation, the diabatic PECs for 1Π and 3Π states are displayed in Figure 6 and Figure 7. The

1,3Δ

states are drawn in the Supporting Information (Figure S2). From

these findings, we remark that the couplings become stronger at short distances, while at large distances the couplings decreases since the avoided crossing between the neighbor states are obviously passed diabatically to the neutral-ionic crossing at larger distances. 3.2. adiabatic and diabatic moment dipole In order to understand better the ionic character of

1,3Σ+, 1,3Π

and

1,3Δ

electronic states at the

avoided crossing positions, we have investigated the electric dipole moment for numerous excited states of FrNa molecule. An ab initio approach was used for both diabatic and adiabatic representations. The origin of our calculation is in the Francium atom. We can put in evidence the interplay between the ionic and the neutral states generating a direct illustration of the ionic character of the electronic wave-function. 8 ACS Paragon Plus Environment

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3.2.1. adiabatic and diabatic PDM In Figures (8-10), we display adiabatic PDM curves for all states of 1,3Σ+ and 1,3Π symmetries below the ionic limit Fr+ Na-, as a function of internuclear distances. The adiabatic PDM curves for 1,3Δ symmetry are presented in the Supporting Information (Figure S3). For 1Σ+ states, we remark a slow variation at short internuclear distances, while at large internuclear distances many crossings between F, G, H, I and J states are observed. These crossings are corresponding practically toward the numerous avoided crossings in the potentials. Thus, one after other, the dipole moment of these considered adiabatic states reaches the curve (−R) and then vanishes. However, the combination of these curves allowed clearly reproducing piecewise the (−R) function corresponding to the double charge transfer ionic state [Fr+ +Na-]. Concerning 3Σ+ states, the behavior of PDM function confirms the shape of potential’s curves. We notice that the crossing in the PDM function corresponds to the avoided crossings in PECs. As an example, the crossing between f3Σ+ and g3Σ+ situated at R=16 a.u can give potential consequences for the excitation or charge transfer efficiency. Interestingly, from the zoom of Figure 8 and Figure 9, we observe that the sign convention of the electric PDM for both states X1Σ+ and a3Σ+ of FrNa system is negative at small distances. This negative sign means that the centers’charge of the electrons relative to the center’s charge of the two nuclei is dislocated towards the Na atom. As usually, two contributions are engendered by the charge density distributed between Fr and Na species. With the effect of the electron charge transfer from Fr to Na atoms on the PDM variation, the induced polarization from the charge transfer can be also involved. The occurred polarization is induced from contributions of the excited unoccupied p and d orbitals, turning out that the contributions of charge transfer and induced polarization to the dipole moments have an opposite sign56. We find that the charge transfer is almost equal for the X1Σ+ and a3Σ+ states, but that of a3Σ+ state has a smaller dipole moment. In Figure 10, permanent dipole moment of

1,3Π

adiabatic states present an involved shape

described by discontinuities between the consecutive parts, due to the avoided crossings located in their potentials. Similarly, for 1Σ+, the combination of these parts formed piecewise the (−R) function. In contrast, the PDM for

1,3Δ

present some extrema then fall asymptotically toward

zero when the internuclear distance becomes larger. Here, charge-transfer ionic states are not detected. The significant minima appear for the third singlet and triplet state value nearly to (9 a.u) at R around 15 a.u (Supporting Information, Figure S3). The diabatic results of PDM function offer substantial information on the whole alkali dimers series (CsNa, NaLi...) thanks to the significant imprint of the ionic state clearly revealed for 9 ACS Paragon Plus Environment


these systems.28, 31, 34 Ionic states have potential effects on the excitation or charge transfer efficiency. The diabatic 1Σ+ PDM curves in a wide R interval are given in Figure 11. We get firstly a straight line as (-R) for the dipole of the ionic diabatic state Fr+ Na-. This line is formed by pieces from the numerous adiabatic states. Sharp or smooth transition variations can be observed at the crossings. On the other hand, the neutral diabatic state dipole rapidly falls to zero as R grows. This is fully consistent with a purely ionic diabatic state and proves the accuracy of our diabatization method on physical properties. Then, the diabatic PDM have been also determined for the electronic states of 3Σ+, 1Π and 3Π symmetries and displayed in the Supporting Information (Figure S4) and in Figures 12-13. From these graphs, we can see that these dipole moments are not negligible at short distances and they become more significant for the higher excited states. At large distance, they decrease progressively and trend to zero. Except for the permanent dipole moments of 3Π (Figure 13), we observed a negative linear variation behaves as -R. Thus, the adiabatic behavior PECs confirm the ionic character in the excited state 3Π from 83Π one. 3.2.2. adiabatic and diabatic TDM In order to design stimulated Raman steps that are optimized for a high molecular production rate to, for instance, the v=0 vibrational level of the ground state, experimentalists need an accurate knowledge of potentials and the TDM functions. We have depicted TDM functions of singlet and triplet Σ+ states in Figures (14-16). We notice that the X1Σ+→ A1Σ+ transition presents a huge maximum 1.34 a.u. localized at R= 8.01 a.u then decreases and vanishes to zero (Figure 14A). Therefore, an important overlap between the corresponding molecular wave functions is observed around this distance. The X1Σ+→ C1Σ+ transition presents an intense maximum which values about 4.32 a.u at R= 9.34 a.u. This latter decrease after this position and remains stable at 3.02 a.u, approximately. At large distances, the X1Σ+→ E1Σ+ transition becomes a constant equal to absolute value of 2.32 a.u., while the other transitions exhibit many extrema at smaller distances and fall to zero. Moreover, the appearance of these extrema in the TDM is in connection to the existence of the avoided crossings between the considered adiabatic states in the PECs. In Figure 15, we can remark that considerable changes are specifically indicated between the neighboring states. Until 20 a.u, a significant variation is spotted between A1Σ+→ C1Σ+ and A1Σ+→ D1Σ+ transitions, which is related to the strong configuration interactions. Add to that, they have an exchanged character


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at the position of avoided crossing between both states C1Σ+ and D1Σ+ nearly at 12.6 a.u, as shown in their PECs. From Figure 16, every one of these adiabatic 3Σ+ transitions is manifested by appearing of various extrema at different internuclear distances. It is worthy to mention that, the distances are corresponding to the avoided crossing positions. The sharp of these dipole moments is becoming more significant and it increases slowly as the crossings become faintly avoided. To complete this part we present the diabatic transition between the ionic state D1 and the diabatic excited states Di (i=2→5) as shown in Figure 17. We remark that these diabatic transitions have the same behavior that the adiabatic ones with a little difference in value. The analysis of the transition and the permanent dipole moment, also, enabled us to locate the positions of the avoided crossings identified in the adiabatic curves. 3.2. Vibrational Levels Spacing In regard to the importance of their vibrational properties, we have computed the energy levels as well their spacings (Gv - Gv-1) as a function of the number of vibrational levels v for few electronic states of

1,3Σ+

symmetry for our studied system. These results are obtained by the

resolution of the radial Schrodinger equation. Since the absence of theoretical or experimental references, we just present our results. The vibrational levels spacing of the ground and the first excited states (X, A, C, D) 1Σ+ reported in Figure 18 present an unusual variation which confirms the behavior in PECs. The spacings curve of the ground state X1Σ+ varies linearly and contains 76 vibrational levels since it presents very deep well (De= 4506 cm-1). The spacing between these levels are tight and not constant at the beginning corresponding to Morse-like anharmonic form in their potentials, then decrease slightly until vanishing at Fr (7s)+ Na (3s) dissociating limit. For the first excited state A1Σ+, there are numerous and very tight levels (v=154). We see the same comportment that characterized by the anharmonicity of the related potential well as their vibrational levels spacings are not constant. In contrast, a special behavior is detected for C1Σ+ and D1Σ+ which proves their shape in the PECs. In the beginning, the spacing curve of C1Σ+ state exhibit a smooth anharmonic shape up to v = 37; then, a sudden change reflecting a widening in the well, from v = 37, where the levels become more spaced then numerous and more tighter at dissociation limit. In Figure 19, the E1Σ+ state has 155 vibrational levels. Clearly, the spaces show a linear behavior up to v= 5 associated with an anharmonic potential. An abrupt change is registered several times at (v= 8, 12, 16, 19, 23, 27, 30, 34, 37, 41, 44, 48, 51, 54, 57 and 61), which is 11 ACS Paragon Plus Environment


corresponding to the accidental degeneracy and the appearance of a double minimum. At this juncture, the harmonic frequency in the outer well is less low than the inner well. However, as cited previously in ref 57, the vibronic coupling may shift the levels differently in the inner and the outer wells and this accidental degeneracy may be removed. For the higher excited states as (K, L, M) 1Σ+, drawn in Figure 20, we see clearly unusual potential curves with various wells results from these numerous avoided crossings which can trap vibrational levels in diverse R range. The behavior of the potential curve is well verified by the number and shape of vibrational level spacing. Figure 21 shows that a3Σ+ and e3Σ+ having a little vibrational level’s number (v=18, 11), which reflect the shallowness of the width of the well of potential (De=164, 630 cm-1); whereas we detect a numerous vibrational levels trap in c3Σ+ and d3Σ+. These latter shed lights on the importance of the width of the well of potential at the level of the dissociated limit.

4. Conclusion To conclude, a theoretical investigation of the FrNa molecule is carried out both in diabatic and adiabatic representation involving several states correlating up to rather higher energies. In the ab initio calculation, we apply the pseudo-potential approach to the alkali atoms (Fr, Na) to reduce the computation to a two effective electrons system where FCI can be easily achieved. Below Fr+ Naionic limit, almost states of various symmetries 1,3Σ+, 1,3Π and 1,3Δ have been investigated. The adiabatic and diabatic PECs, their related spectroscopic constants and the vibrational levels spacings have been derived. Irregular behavior is occurred by the presence of double or multiple wells due to the avoided crossings. These wells trap a large number of vibration levels. In order to understand better the ionic character of the electronic states of the FrNa molecule, the electric dipole moments, permanent or transition ones, have been carefully determined. The diabatic survey yields great insight into the strong interactions between the adiabatic electronic states. From these results, we clearly observed the ionic-neutral crossings at intermediate and large distances, especially in the 1Σ+ states. We get a straight line as (-R) for the dipole of the ionic diabatic state Fr+ Na-. This line is formed by pieces from the numerous adiabatic states and sharp or smooth transition variations which can be observed at the crossings. Therefore, the computation of dipolar properties presents the first step to obtain quantitative estimations of photo-absorption and molecular production rates in a gas of Fr and Na particles. As perspective, Frank-Condon factors between vibrational levels of the ground and various excited states will be evaluated. We predict to calculate the vibronic shifts like the inelastic collision cross-sections, the radiative or non-radiative lifetimes. 12 ACS Paragon Plus Environment

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REFERENCES (1) Wakim, A.; Zabawa, P.; Haruza, M.; Bigelow, N. P. Luminorefrigeration: Vibrational Cooling of NaCs. Opt. Expr. 2012, 20, 16083-16091. (2) Haimberger, C.; Kleinert, J.; Bhattacharya, M.; Bigelow, N. P. Formation and Detection of Ultracold Ground-State Polar Molecules. J. Phys. Rev. A, 2004, 70, 021402. (3) Wang, D.; Qi, J.; Stone, M. F.; Nikolayeva, O.; Wang, H.; Hattaway, B.; Gensemer, S. D.; Gould, P. L.; Eyler, E. F.; Stwalley, W. C. Photoassociative Production and Trapping of Ultracold KRb Molecules. J. Phys. Rev. Lett. 2004, 93, 243005. (4) Valtolina, G.; Covey, J.; De Marco, L.; Matsuda, K.; Tobias, W.; Ye, J. Implementation of In-Vacuum Electrodes for Manipulating Interactions Between Ultracold KRb Molecules. In APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts, (2017). (5) Kerman, A. J.; Sage, J. M.; Sainis, S.; Bergeman, T.; DeMille, D. Production and StateSelective Detection of Ultracold RbCs Molecules. J. Phys. Rev. Lett. 2004, 92, 153001. (6) Shimasaki, T.; Kim, J. T.; Zhu, Y.; DeMille, D. Continuous Production of Rovibronic +

Ground State RbCs Molecules via Short-Range Photoassociation to the b3Π1- C3𝛴1 - B1Π1 States. ArXiv e-prints (2018), arXiv:1802.01797 [physics.atom-ph] (7) Zhong-Hua, J.; Hong-Shan, Z.; Ji-Zhou, W.; Jin-Peng, Y.; Yan-Ting, Z.; Jie, M.; Li-Rong, W.; Lian-Tuan, X.; Suo-Tang, J. Photoassociative Production and Detection of Ultracold Polar RbCs Molecules. Chin. Phys. Lett. 2011, 28, 083701. (8) Sprouse, G. D.; Fliller, R. P.; Grossman, J. S.; Orozco, L. A.; Pearson, M. R. Traps for neutral radioactive atoms. Nucl. Phys. A. 2002, 701, 597-603. (9) Lim, I. S.; Schwerdtfeger, P.; Söhnel, T.; Stoll, H. Ground-State Properties and Static Dipole Polarizabilities of the Alkali Dimers from 𝐾𝑛2 to 𝐹𝑟𝑛2(n= 0,+ 1) from Scalar Relativistic Pseudopotential Coupled Cluster and Density Functional Studies. J. Chem. Phys. 2005, 122, 134307. (10) Derevianko, A.; Babb, J. F.; Dalgarno, A. High-Precision Calculations of Van der Waals Coefficients for Heteronuclear Alkali-Metal Dimers. J. Phys. Rev. A, 2001, 63, 052704. (11) Aymar, M.; Dulieu, O.; Spiegelman, F. Electronic Properties of Francium Diatomic Compounds and Prospects for Cold Molecule Formation. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, S905. (12) Pavolini, D.; Gustavsson, T.; Spigelmann, F.; Daudey, J. P. Theoretical Study of the Excited States of the Heavier Alkali Dimers. I. The RbCs molecule. J. Phys. B: At. Mol. Opt. Phys. 1989, 22, 1721. 13 ACS Paragon Plus Environment


(13) Allouche, A. R.; Korek, M.; Fakherddin, K.; Chalaan, A.; Dagher, M.; Taher, F.; AubertFrécon, M. Theoretical Electronic Structure of RbCs Revisited. J. Phys. B: At. Mol. Opt. Phys. 2000, 33, 2307. (14) Fahs, H.; Allouche, A. R.; Korek, M.; Aubert-Frécon, M. The Theoretical Spin-Orbit Structure of the RbCs Molecule. J. Phys. B: At. Mol. Opt. Phys. 2002, 35, 1501. (15) Korek, M.; Allouche, A. R.; Kobeissi, M.; Chaalan, A.; Dagher, M.; Fakherddin, K.; Aubert-Frécon, M. Theoretical Study of the Electronic Structure of the LiRb and NaRb Molecules. J. Chem. Phys. 2000, 256, 1. (16) Schmidt-Mink, I.; Müller, M.; Meyer, W. Potential Energy Curves for Ground and Excited States of NaLi from Ab Initio Calculations with Effective Core Polarization Potentials. J. Chem. Phys. Lett. 1984, 112, 120-128. (17) Petsalakis, I. D.; Tzeli, D.; Theodorakopoulos, G. Theoretical Study on the Electronic States of NaLi. J. Chem. Phys. 2008, 129, 054306. (18) Rousseau, S.; Allouche, A. R.; Aubert-Frécon, M. Theoretical Study of the Electronic Structure of the KRb Molecule. J. Mol. Spectrosc. 2000, 203, 235-243. (19) Rousseau, S.; Allouche, A. R.; Aubert-Frécon, M.; Magnier, S.; Kowalczyk, P.; Jastrzebski, W. Theoretical Study of the Electronic Structure of KLi and Comparison with Experiments. J. Chem. Phys. 1999, 247, 193-199. (20) Magnier, S.; Aubert-Frécon, M.; Millié, Ph. Potential Energies, Permanent and Transition Dipole Moments for Numerous Electronic Excited States of NaK. J. Mol. Spectrosc. 2000, 200, 96-103. (21) Magnier, S.; Millié, Ph. Potential Energies, Permanent and Transition Dipole Moments for Numerous Electronic Excited States of NaK. J. Phys. Rev. A. 1996, 54, 204. (22) Korek, M.; Allouche, A. R.; Fakhreddine, K.; Chaalan, A. Theoretical Study of the Electronic Structure of LiCs, NaCs, and KCs Molecules. Can. J. Phys. 2000, 78, 977-988. (23) Smith, F. T. Diabatic and Adiabatic Representations for Atomic Collision Problems. J. Phys. Rev. A. 1969, 179, 111. (24) Lichten, W. Resonant Charge Exchange in Atomic Collisions. J. Phys. Rev.A. 1963, 131, 229. (25) Nikitin, E. E. Theory of Nonadiabatic Reactions. In: Chemische Elementarprozesse, cd. H. Hartmann (Springer, Berlin, 1968). (26) Romero, T.; Aguilar, A.; Gadea, F. X. Towards the Ab Initio Determination of Strictly Diabatic States, Study for (NaRb)+. J. Chem. Phys. 1999, 110, 6219-6228.


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Page 15 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(27) Docenko, O.; Tamanis, M.; Ferber, R.; Pashov, A.; Knöckel, H.; Tiemann, E. Spectroscopic Studies of NaCs for the Ground State Asymptote of Na+ Cs Pairs. Eur. Phys. J. D, 2004, 31, 205-211. (28) Dardouri, R.; Issa, K.; Oujia, B.; Gadéa, F. X. Theoretical Study of the Electronic Structure of LiX and NaX (X= Rb, Cs) Molecules. Int. J. Quantum Chem. 2012, 112, 2724-2734. (29) Issa, K.; Issaoui, N.; Ghalla, H.; Yaghmour, S. J.; Mahros, A. M.; Oujia, B. Ab Initio Study of Ba+Arn (n = 1–4) Clusters: Spectroscopic Constants and Vibrational Energy Levels. J. Mol. Phys. 2016, 114, 118-127. (30) Khemiri, N.; Dardouri, R.; Oujia, B.; Gadéa, F. X. Ab Initio Investigation of Electronic Properties of the Magnesium Hydride Molecular Ion. J. Phys. Chem. A. 2013, 117, 8915–8924. (31) Chaieb, M.; Habli, H.; Mejrissi, L.; Oujia, B.; Gadéa, F. X. Ab Initio Spectroscopic Study for the NaRb Molecule in Ground and Excited States. Int. J. Quantum Chem. 2014, 114, 731747. (32) Habli, H.; Dardouri, R.; Oujia, B.; Gadéa, F. X. Ab Initio Adiabatic and Diabatic Energies and Dipole Moments of the CaH+ Molecular Ion. J. Phys. Chem. A. 2011, 115, 14045-14053. (33) Habli, H.; Mejrissi, L.; Issaoui, N.; Yaghmour, S. J.; Oujia, B.; Gadéa, F. X. Ab Initio Calculation of the Electronic Structure of the Strontium Hydride Ion (SrH+). Int. J. Quantum. Chem. 2014, 115, 172-186. (34) Dardouri, R.; Habli, H.; Oujia, B.; Gadéa, F. X. Ab Initio Diabatic Energies and Dipole Moments of the Electronic States of RbLi Molecule. J. Comput. Chem. 2013, 34, 2091-2099. (35) Khelifi, N.; Oujia, B.; Gadea, F. X. Dynamic Couplings, Radiative and Nonradiative Lifetimes of the A1Σ+ and C1Σ+ States of the KH Molecule. J. Phys. Chem. Ref. Data. 2007, 36, 191-202. (36) Mejrissi, L.; Habli, H.; Ghalla, H.; Oujia, B.; Gadéa, F. X. Adiabatic ab Initio Study of the BaH+ Ion Including High Energy Excited States. J. Phys. Chem. A. 2013, 117, 5503-5517. (37) Jellali, S.; Habli, H.; Mejrissi, L.; Mohery, M.; Oujia, B.; Gadéa, F. X. Theoretical Study of the SrLi+ Molecular Ion: Structural, Electronic and Dipolar Properties. J. Mol. Phys. 2016, 114, 2910-2923. (38) Habli, H.; Mejrissi, L.; Ghalla, H.; Yaghmour, S. J.; Oujia, B.; Gadéa, F. X. Ab Initio Investigation of the Electronic and Vibrational Properties for the (CaLi)+ Ionic Molecule. J. Mol. Phys. 2016, 114, 1568-1582. (39) Durand, P.; Barthelat, J. C. A Theoretical Method to Determine Atomic Pseudo-Potentials for Electronic Structure Calculations of Molecules and Solids. Theoret. Chim. Acta. 1975, 38, 283-302. 15 ACS Paragon Plus Environment


(40) Muller, W.; flesh, J.; Meyer, W.Treatment of Intershell Correlation Effects in Ab Initio Calculations by Use of Core Polarization Potentials. Method and Application to Alkali and alkaline Earth Atoms. J. Chem. Phys. 1984, 80, 3297-3310. (41) Foucrault, M.; Millié, Ph.; Daudey, J. P. Nonperturbative Method for Core–Valence Correlation in Pseudo-Potential Calculations: Application to the Rb2 and Cs2 Molecules. J. Chem. Phys. 1992, 96, 1257-1264. (42) NIST ASD Team, NIST Atomic Spectra, (The National Institute of Standards and Technology). http://physics.nist.gov/PhysRefData/ASD/levels_form.html. (43) Lim, I. S.; Laerdahl J. K.; Schwerdtfeger, P.Fully Relativistic Coupled-Cluster Static Dipole Polarizabilities of the Positively Charged Alkali Ions from Li+ to 119+. J. Chem. Phys. 2002, 116, 172-178. (44) Gadéa, F. X. Symmetry Between Model Space and Target Space in Effective-Hamiltonian Theory. J. Phys. Rev. A. 1987, 36, 2557. (45) Khelifi, N.; Oujia, B.; Gadéa, F. X. Ab Initio Adiabatic and Diabatic Energies and Dipole Moments of the KH Molecule. J. Chem. Phys. 2002, 116, 2879-2887. (46) Khelifi, N.; Zrafi, W.; Oujia, B.; Gadéa, F. X. Ab Initio Adiabatic and Diabatic Energies and Dipole Moments of the RbH Molecule. J. Phys. Rev. A. 2002, 65, 42513. (47) Zraﬁ, W.; Oujia, B.; Gadéa, F. X. Theoretical Study of the CsH Molecule: Adiabatic and Diabatic Potential Energy Curves and Dipole Moments. J. Phys. B: At. Mol. Opt. Phys., 2006, 39, 3815. (48) Dardouri, R.; Habli, H.; Oujia, B.; Gadéa, F. X. Theoretical Study of the Electronic Structure of KLi Molecule: Adiabatic and Diabatic Potential Energy Curves and Dipole Moments. J. Chem. phys. 2012, 399, 65-79. (49) Gadéa, F. X.; Pelissier, M. Approximately Diabatic States: a Relation Between Effective Hamiltonian Techniques and Explicit Cancellation of the Derivative Coupling. J. Chem. Phys. 1990, 93, 545-551. (50) Gadéa, F. X. Variational Effective Hamiltonians as a Generalization of the Rayleigh Quotient. J. Phys. Rev. A. 1991, 43, 1160. (51) Hoffmann, R. An Extended Hückel Theory. I. Hydrocarbons. J. Chem. Phys. 1963, 39, 1397-1412. (52) Block, C. Sur la Théorie des Perturbations des Etats Liés. Nuc. Phys. 1958, 20, 329. (53) Des Cloiseaux, J. Extension d'une Formule de Lagrange à des Problèmes de Valeurs Propres. Nuc. Phys. 1960, 20, 321-346.


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(54) Dickinson, A. S.; Gadéa, F. X. Undulations in Potential Curves Analyzed Using the Fermi Model: LiH, LiHe, LiNe, and H2 Examples. J. Phys. Rev. A. 2002, 65, 052506. (55) Dickinson, A. S.; Gadéa, F. X. The Fermi Model as a Key for Understanding Excited State Potential Energy Curves. J. Mol. Struct. (THEOCHEM). 2003, 621, 87-98. (56) Kotochigova, S.; Tiesinga, E. Ab Initio Relativistic Calculation of the RbCs Molecule. J. chem. Phys. 2005, 123, 174304. (57) Gemperle, F.; Gadéa, F. X. Breakdown of the Born-Oppenheimer Approach for a Diatomic Molecule: LiH in the D State. Europhys. Lett. 1999, 48, 513.

(A) 1 +

-0,20

L 1 +

M

1 +

1 +

K

J 1 +

I

1 +

Fr(8p)+Na(3s)

H

1 +

G

-0,24

Fr(8s)+Na(3s)

1 +

F

1 +

Energies(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


E

Fr(7s)+Na(3p) 1 +

D

Fr(6d)+Na(3s)

1 +

C

-0,28

Fr(7p)+Na(3s) 1 +

A

-0,32 Fr(7s)+Na(3s) 1 +

X -0,36

10

20

R(a.u)

30


40

50


(B)

Fr(7p)+Na(3p)

-0,20

1 +

M

Fr(7s)+Na(4p)

1 +

L

Fr(7s)+Na(3d) -0,21

1 +

K

Energies(a.u)

Fr(9p)+Na(3s) 1 +

Fr(7s)+Na(4s)

J

-0,22

1 +

I

Fr(9s)+Na(3s)

1 +

H

Fr(7d)+Na(3s)

-0,23 1 +

G

Fr(8p)+Na(3s)

-0,24 1 +

F 20

40

60

80

R(a.u)

100

120

Figure 1: (A) Global graphic for the adiabatic 1Σ+ potential energy curves dissociating below the ionic limit Na-Fr+. (B) Potential energy curves for higher excited states 1Σ+ of FrNa. Fr(7p)+Na(3p) Fr(7s)+Na(4p) Fr(7s)+Na(3d)Fr(9p)+Na(3s) Fr(7s)+Na(4s) Fr(9s)+Na(3s) Fr(7d)+Na(3s) Fr(8p)+Na(3s)

3 +

n

-0,21

3 +

m 3 + 3 + k j

3 +

l

3 +

i

3 +

h

3 +

g

-0,24

Energies(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 35

Fr(8s)+Na(3s)

3 +

Fr(7s)+Na(3p)

3 +

Fr(6d)+Na(3s)

f

3 +

e -0,27

d

Fr(7p)+Na(3s)

3 +

c -0,30

-0,33

3 +

a

10

Fr(7s)+Na(3s) 20

30

40

R(a.u)

50

60

70

Figure 2: Adiabatic 3Σ+ potential energy curves dissociating below the ionic limit Na-Fr+.


Page 19 of 35

-0,22

D10 D9 D7

-0,24

Energie (a.u)

D8

D6

D5

-0,26 D4

-0,28

D3

-0,30

-0,32 D1

D2

-0,34 20

40

60

80

100

120

140

R (a.u)

Figure 3: Diabatic 1Σ+ potential energy curves of FrNa

-0,22

d9

d10 d8 d6

-0,24 d5

Energie (a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-0,26

d7

d4

d3 d2

-0,28

-0,30

-0,32

d1

-0,34 10

20

30

40

50

R (a.u)

Figure 4: Diabatic 3Σ+ potential energy curves of FrNa 19 ACS Paragon Plus Environment

60


Fr(7p)+Na(3p) Fr(7s)+Na(4p)

1,3

8 

-0,20 1,3

1,3

7 

6 

Fr(7d)+Na(3s)

1,3

4 

Fr(8p)+Na(3s)

1,3

3 

-0,24

1,3

Fr(7s)+Na(3p)

2 

-0,26 1

1

Fr(7s)+Na(3d) Fr(9p)+Na(3s)

1,3

5 

-0,22

Energies(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 35

Fr(6d)+Na(3s)

3

1

-0,28

Fr(7p)+Na(3s) 1

B -0,30

3

b 10

20

30

40

50

60

R(a.u)

Figure 5: Adiabatic potential energy curves of 1Π states (solid lines) and 3Π states (dotted lines) dissociating below the ionic limit Na-Fr+.


Page 21 of 35

-0,18 1



8

-0,20

Energies(a.u)

7 6

-0,22

5 4

-0,24

3

-0,26

2 1 -0,28

10

20

R(a.u)

30

40

50

Figure 6: Diabatic 1Π potential energy curves of FrNa

-0,18 3



-0,20

Energies(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8 7

-0,22

6 5

-0,24

4

-0,26

2

1 -0,28

3

10

20

30

40

R(a.u) Figure 7: Diabatic 3Π potential energy curves of FrNa


50


M 0

-30

L

K

G

PDM(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-60

H I 5

-90

X

0

D -120

J

C

-5

A

-10

-15

E

-20

-150

F

-25

-30

5

10

15

20

20

40

25

30

60

80

100

120

140

160

R(a.u) Figure 8: Adiabatic Permanent Dipole Moment for the 1Σ+ states for the FrNa system with zoom for the PDM at short distance 10

0 i

-10

h k

-20

-30 d

5

-40

e a

0

l

n c

m

-5

-50

g

f -10

-60

5

10

10

15

20

20

30

R(a.u)

40

50

60

70

Figure 9: Adiabatic Permanent Dipole Moment for 3Σ+ states as a function of the internuclear distance (all in a.u.) for the FrNa system. 22 ACS Paragon Plus Environment

Page 23 of 35

10

0

1

8

-10

PDM(a.u)

1

5

-20

3

3

8

6

3

5 1

-30

4

1

1

1

6

3

B

3

5

4

1

-40

7 0

3

7

1

-50

b

2

-5

1

3

1

3

2 -60

-10

1

3

3

3

5

10

10

15

20

30

40

50

60

R(a.u) Figure 10: Adiabatic Permanent Dipole Moment for 1Π states (solid lines) and 3Π states (dotted lines) as a function of the internuclear distance (all in a.u.) for the FrNa system. 20 0 -20 -40

PDM(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-60 -80

15

10

-100 5

-120 -140

0

-5

-10

-160

5

10

20

15

40

20

60

80

100

120

140

160

R(a.u)

Figure 11: Permanent dipole moment for the 1Σ+ diabatic states of FrNa



5

5

6

4

3

0

1

2 7

PDM(a.u)

-5

-10

-15

-20

8 -25 8

16

24

32

40

48

R(a.u) Figure 12: Permanent dipole moment for the 1Π diabatic states of FrNa 10

0

-10

PDM(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-20

-30

-40

-50 8

16

24

R(a.u)

32

40

48

Figure 13: Permanent dipole moment for the 3Π diabatic states of FrNa


Page 25 of 35

(A)

5

4

X1 +C1 +

Adiabatic TDM(a.u)

3

2

X1 +E1 + X1 +A1 +

1

0

X1 +F1 +

X1 +G1 +

-1

X1 +D1 + -2

5

10

15

20

25

30

35

R(a.u) (B) 0,4 0,3

X1 +I1 +

0,2

Adiabatic TDM(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


X1 +J1 +

0,1 0,0 -0,1

X1 +L1 + X1 +K1 +

-0,2

X1 +H1 +

-0,3 -0,4

5

10

15

20

25

30

35

40

45

50

R(a.u)

Figure 14: Adiabatic transition dipole moments for: (A) X1Σ+→ (A-C-D-E-F-G) 1Σ+, (B) X1Σ+→ (H-J-K-L) 1Σ+


55


A1 + C1 +

6

A1 + D1 +

Adiabatic TDM(a.u)

4

C1 + E1 +

2

0

C1 + D1 +

D1 + F1 +

-2

D1 + E1 + -4

5

10

15

20

25

30

35

40

45

50

R(a.u)

Figure 15: Transition dipole moment between excited 1Σ+ adiabatic states. 6,0

3

3

e + f +

4,5

Adiabatic TDM(a.u)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 35

3,0

3

3

3

3

a + c +

c + f + 3

3

a + e +

1,5

3

3

a + f +

0,0

-1,5

3

3

a + d + 3

3

3

cd

-3,0 5

+

10

+

3

c + e +

15

20

25

30

35

R(a.u)

Figure 16: Transition dipole moment between selected 3Σ+ adiabatic states.


40

Page 27 of 35

3

D1D3

2

Diabatic TDM(a.u)

1

D1D4

0

D1D2

-1

D1D5

-2 -3 -4 -5

5

10

15

20

25

30

35

40

R(a.u)

Figure 17: Transition dipole moment between selected 1Σ+ diabatic states.

100 1

+

X 1 + A 1 + C 1 + D

80

60

-1

Gv-Gv-1(cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40

20

0 0

20

40

60

80

100

120

140

160

v

Figure 18: Vibrational level spacings for the ground and the first excited states (X, A, C, D) 1Σ+ of FrNa.



80 70

-0,250

60 -0,252

Energies(a.u)

-1

Gv-GV-1(cm )

50 40 1 +

E

30

-0,254

-0,256

20 -0,258

10 -0,260

0 0

20

40

60

v

80

100

120

140

8

160

12

16

R(a.u)

20

24

28

Figure 19: Vibrational spacing (left) and potential energy curves with vibrational levels (right) for E1Σ+ state of FrNa. 80

1

+

K 1 + L 1 + M

60 -1

Gv-Gv-1(cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 35

40

20

0 0

40

80

120

160

200

v

Figure 20: Vibrational level spacings for the higher excited states (K, L, M) 1Σ+ of FrNa.


Page 29 of 35

120 3

+

a 3 + c 3 + d 3 + e 3 + f

100

80 -1

Gv-Gv-1(cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60

40

20

0 0

15

30

45

60

75

v

Figure 21: Vibrational level spacings for the first 3Σ+ states of FrNa.


90


Page 30 of 35

Table 1: l-dependant cut-off radii for sodium and francium atoms (in a.u) l

Na

Fr

s

1.4187

3.1629

p

1.6484

3.027

d

0.7

3.1068

Table 2A: Theoretical energy level spectrum of francium atom compared with the experimental42 and theoretical energies11 (αFr+=20.38 a.u). IE (Fr) = 32848.87 cm-1 Atomic levels

Expt. (cm-1)

Theor. (cm-1)

This work (cm-1)

∆E1 (cm-1)

∆E2 (cm-1)

7s

-32848.87

-32848.87

-32848.87

0

0

7p

-19487.07

-19486.93

-19486.93

0.14

0

6d

-16499.14

-16470.92

-16499.22

0.08

28.3

8s

-13116.62

-13161.02

-13176.6

60

15.58

8p

-9372.35

-9448.16

-9425.78

53.43

22.38

7d

-8550.96

-8394.77

-8463.82

87.14

69.05

9s

-7177.85

-7161.96

-7192.18

14.33

30.22

9p

-5573.18

-5619.67

-5580.58

7.4

39.09

∆E1: Difference energy between Expt. and This work. ∆E2: Difference energy between Theor. and This work.

Table 2B: Theoretical energy level spectrum of sodium atom compared with the experimental.42 IE (Na) = 41449.451 cm-1 Atomic levels

Expt. (cm-1)

This work (cm-1)

∆E (cm-1)

3s

-41449.45

-41441.85

7.6

3p

-24481.81

-24475.41

6.4

4s

-15709.45

-15707.75

1.7

3d

-12276.59

-12269.99

6.6

4p

-11178.73

-11174.43

4.3

∆E: Difference between Expt. and This work.


Page 31 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3: Experimental42 and theoretical energies data for various molecular states of the FrNa below the ionic limit Fr+Na-. Asymptotic limit

Molecular states

Expt. (cm-1)

This work (cm-1)

∆E (cm-1)

Fr(7s) + Na(3s)

1,3Σ+

-74298.32

-74290.72

7.6

Fr(7p) + Na(3s)

1,3Σ+, 1,3Π

-60936.52

-60928.78

7.74

Fr(6d) + Na(3s)

1,3Σ+, 1,3Π, 1,3Δ

-57948.59

-57941.07

7.52

Fr(7s) + Na(3p)

1,3Σ+, 1,3Π

-57330.68

-57324.28

6.4

Fr(8s) + Na(3s)

1,3Σ+

-54566.07

-54618.45

52.38

Fr(8p) + Na(3s)

1,3Σ+, 1,3Π

-50821.80

-50867.63

45.83

Fr(7d) + Na(3s)

1,3Σ+, 1,3Π, 1,3Δ

-49996.41

-49905.67

90.74

Fr(9s)+Na(3s)

1,3Σ+

-48627.30

-48634.03

6.73

Fr(7s)+Na(4s)

1,3Σ+, 1,3Π

-48558.32

-48556.62

1.7

Fr(9p)+Na(3s)

1,3Σ+, 1,3Π

-47022.63

-47022.43

0.2

Fr(7s)+Na(3d)

1,3Σ+, 1,3Π, 1,3Δ

-45125.46

-45118.86

6.6

Fr(7s)+Na(4p)

1,3Σ+, 1,3Π

-44027.60

-44023.30

4.3

Fr(7p) + Na(3p)

1,3Σ+, 1,3Π

-43968.88

-43962.34

6.54

IL (Fr+Na-)= IE (Fr)-EA(Na)=28426.8003 (cm-1) ∆E: Difference between Expt. and This work. IE: Ionization Energy EA: Electron Affinity. IL: Ionic Limit.



Table 4: Avoided crossing positions RAC (in a.u) between the neighbor states and their difference energy E (in cm-1). States RAC E 1,3Σ+ A1Σ+/C1Σ+ 18 2739 C1Σ+/D1Σ+ 12.6 164 D1Σ+/E1Σ+ 19.2 579 1 + 1 + E Σ /F Σ 26.2 680 F1Σ+/G1Σ+ 10.8 880 44 85 G1Σ+/H1Σ+ 12 307 16.5 15 54 11 10.35 305 H1Σ+/I1Σ+ 20.5 33 78.8 13 I1Σ+/J1Σ+ 15.5 85 30 2 80 22 J1Σ+/K1Σ+ 9.85 924 K1Σ+/L1Σ+ 12.82 485 29.3 6 35 13 L1Σ+/M1Σ+ 7.8 125 10.65 419 27 504 e3Σ+/ f3Σ+ 10.75 125 f3Σ+/ g3Σ+ 9.55 26 g3Σ+/h3Σ+ 8.6 54 16.66 96 3 + 3 + hΣ /iΣ 8.5 612 i3Σ+/ j3Σ+ 7.8 274 23.25 19 7.5 76 j3Σ+/ k3Σ+ 15.08 52 k3Σ+/ l3Σ+ 13 190 30.5 72 3 + 3 + lΣ /mΣ 10.8 129 48 4 m3Σ+ /n3Σ+ 14.52 122 31 8 1,3Π 11Π/21Π 10.8 57 1 1 5 Π/6 Π 21.25 118 61Π/71Π 35 8 71Π/81Π 12.5 68 44.5 2 33Π/43Π 20.75 649 53Π/63Π 31.25 298 3 3 6 Π/7 Π 15 357 44.5 32 73Π/83Π 10.8 267 58 4


Page 32 of 35

Page 33 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 5: Spectroscopic constants of all electronic states of the different symmetries 1,3Σ+, 1,3Π and 1,3Δ of FrNa molecule. States Re (a.u) De (cm-1) Te (cm-1) ωe (cm-1) ωeχe (cm-1) Be (cm-1) (a) 1,3Σ+ states 1 XΣ 7.12 4506 0 96.92 0.481 0.0569 A1Σ 8.71 6365 12564 62.32 0.125 0.0380 C1Σ 8.79 3498 18328 56.18 0.231 0.0373 D1Σ 10.27 2150 20812 36.85 0.119 0.0273 E1Σ 8.19 2573 22.07 0.036 0.0430 22305 2nd well 18.21 2254 13.89 0.0068 0.0087 8.45 3617 11.69 0.0031 0.0404 F1Σ 25112 23.25 3481 9.12 0.0069 0.0053 G1Σ 8.85 3358 39.54 0.131 0.0368 2nd well 17.19 792 8.87 0.059 0.0097 26538 3d well 40.03 952 5.85 0.047 0.0018 H1Σ 8.51 3478 32.53 0.102 0.0398 nd 2 well 11.66 3248 27579 28.19 0.085 0.0212 3d well 46.6 1277 8.99 0.0603 0.0013 I1Σ 8.22 3219 48.34 0.17 0.0427 2nd well 20.42 601 27658 4.54 0.049 0.0069 3d well 77.23 80 1.57 0.0094 0.0004 J1Σ 8.32 3456 30.05 0.104 0.0417 nd 2 well 10.65 3161 29018 24.32 0.082 0.0254 3d well 30.23 1552 6.03 0.036 0.0031 K1Σ 9.82 3957 33.64 0.071 0.0299 31725 2nd well 14.40 2622 6.94 0.039 0.0139 L1Σ 9.22 3588 25.19 0.056 0.0339 32730 2nd well 11.70 3573 24.91 0.055 0.0211 1 MΣ 8.37 2789 25.62 0.059 0.0412 33252 2nd well 10.37 3094 31.60 0.084 0.0268 a3Σ 11.86 164 6305 15.58 0.366 0.0205 c3Σ 8.71 2181 16688 53.29 0.326 0.0381 d3Σ 8.08 2647 18629 61.86 0.364 0.0442 e3Σ 8.58 630 23380 57.71 0.337 0.0392 3 fΣ 8.39 73 13.99 0.374 0.0410 24851 2nd well 10.77 951 57.74 0.847 0.0249 g3Σ 9.56 3486 80.79 0.468 0.0316 26670 2nd well 22.14 182 43.89 0.511 0.0058 h3Σ 8.60 2923 58.98 0.297 0.0390 27181 2nd well 16.41 864 4.37 0.108 0.0107 i3Σ 8.55 3453 28485 75.07 0.425 0.0395 3 jΣ 8.38 2476 75.01 0.567 0.0411 29058 2nd well 17.13 173 41.61 0.663 0.0098 k3Σ 8.29 3369 43.04 0.144 0.0420 29791 2nd well 15.29 1576 1.10 0.059 0.0123 l3Σ 8.59 3116 22.13 0.056 0.0391 2nd well 12.90 2978 31485 19.53 0.045 0.0173 d 3 well 29.86 1829 2.14 0.039 0.0032 9.27 3517 27.29 0.073 0.0336 m3Σ 32093 43.85 1105 18.21 0.105 0.0015 8.33 2791 17.59 0.032 0.0416 n3Σ 33048 12.67 265 20.23 0.043 0.0179 33 ACS Paragon Plus Environment


B1Π 11Π 2nd well 21Π 31Π 41Π 51Π 61Π 71Π 2nd well 81Π b 3Π 13Π 23Π 33Π 43Π 53Π 63Π 73Π 2nd well 83Π 2nd well

8.15 8.02 11.58 10.74 8.19 8.34 8.39 8.30 8.47 34.57 8.86 7.07 8.29 21.66 8.48 8.78 8.26 8.29 8.52 41.22 10.20 47.11

1152 1734 323 933 2821 2897 4419 5095 4852 1110 3249 6608 2840 4.4 3483 3397 5346 5318 5895 1099 4949 64

11Δ 13Δ 21Δ 23Δ 31Δ 33Δ

7.91 8.09 8.35 8.28 8.06 8.16

1341 1110 3047 2981 6685 6084

(b) 1,3Π states 17110 48.23 72.98 19500 22.70 22783 56.42 25656 63.44 26624 61.21 28103 59.45 29234 48.35 51.61 30689 18.98 32539 42.48 11298 100.55 18479 53.85 24319 0.815 25452 39.42 26576 55.24 27085 51.00 29016 47.29 54.46 29730 36.02 52.37 32646 9.79 1,3 (c) Δ states 19765 60.08 20081 54.70 26532 69.45 26538 70.10 27480 69.52 28167 70.43


Page 34 of 35

0.504 0.766 0.585 0.746 0.354 0.323 0.197 0.131 0.158 0.118 0.138 0.361 0.235 0.014 0.094 0.232 0.133 0.137 0.149 0.205 0.138 0.022

0.0434 0.0449 0.0215 0.0250 0.0430 0.0415 0.0410 0.0419 0.0402 0.0024 0.0367 0.0577 0.0420 0.0061 0.0401 0.0374 0.0423 0.0420 0.0397 0.0017 0.0277 0.0013

0.671 0.674 0.379 0.400 0.161 0.176

0.0461 0.0441 0.0414 0.0421 0.0444 0.0433

Page 35 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic


Adiabatic and Diabatic Investigation of Numerous Electronic States for

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