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. 2023 Jun 14;127(25):5299–5311. doi: 10.1021/acs.jpca.3c00018

Spectroscopic Properties of the Alkali–Krypton Diatomic M–Kr (M = Rb, Cs, and Fr) van der Waals Systems Including the Spin–Orbit Coupling

Jamila Dhiflaoui †,*, Mohamed Bejaoui †, Hamid Berriche †,‡,*
PMCID: PMC10316391  PMID: 37313854

Abstract

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Using an ab initio approach based on pseudopotential technique, pair potential approach, core polarization potentials, and large Gaussian basis sets, we investigate interaction of heavy alkali–krypton diatomic M–Kr (M = Rb, Cs, and Fr) van der Waals dimers. In this context, the core–core interactions for M+–Kr (M = Rb, Cs, and Fr) are calculated at coupled-cluster single and double excitation (CCSD) level and included in the total potential energy. Therefore, the potential energy curves are performed for 14 electronic states: eight of 2Σ+ symmetry, four of 2Π symmetry, and two of 2Δ symmetry. Furthermore, for each M–Kr dimer, the spin–orbit coupling has been considered for the B2Σ+, A2Π, 32Σ+, 22Π, 52Σ+, 32Π, and 12Δ states. In addition, the transition dipole moment has been determined, including the spin–orbit effect using the rotational matrix issued from the spin–orbit potential energy calculations.

I. Introduction

Interactions of alkali metal atoms with rare gas atoms are of great importance in several areas of physics and chemistry. These interactions have been investigated experimentally1−15 and theoretically16−45 for many years and for many systems. In fact, they are considered as model systems for van der Waals molecules. To understand these weak interactions, it is crucial to test both theoretical methods and basis sets. The pair potentials themselves can be used by theoreticians to model systems using molecular dynamics and by spectroscopists to aid in the understanding and analysis of molecular spectra. Due to the closed shells of atoms and/or ions that form these systems, their ground state presents very weak interactions. In contrast, considerable bonding can exist in the excited states. The interactions here concern diatomic systems, one alkali or alkaline-earth atom or ion with a single rare gas atom. One might expect the behavior of these complexes to be quite uncomplicated, with bonding attributable to purely physical interactions and van der Waals forces.

In the earlier experimental studies,1−15 the interactions between alkali and rare gas atoms have been considered as model for investigating the collisional processes such as line broadening, quenching, and electronic energy transfer. In addition, alkali metal vapor lasers that are pumped by diode lasers have been the focus of several investigations.29−33 Readle et al. have demonstrated this approach in many published papers.29−31

Hedges et al.15 investigated the emission profiles of the cesium resonance lines broadened by collisions with inert gases, based to theoretical models without knowledge of the cesium density.

Theoretically, there are several calculations16−46 of the interactions between the alkali atoms with noble gas atoms. The M(2S) + Rg pairs (M = alkali; Rg = rare gas) were investigated by Baylis16 using pseudopotential method and spin–orbit coupling. They calculated the potential energy of the ground state (X2Σ+) and the first excited states (B2Σ+ and A2Π) and extracted their spectroscopic constants. The semiempirical potential model of Baylis16 was used with a few modifications by Pascale and Vandeplanque19 to study some excited states for M–Rg species. However, the determination of the potential energy curves of alkali atoms interacting with noble gas atoms was studied using the model potential (MP)16−28 where the total potential is considered as a system with a single electron.

Ehara and Nakatsuji35 calculated the Cs–Rg system (Rg = Xe, Kr, Ar, and Ne) at the SAC-CI level. Goll et al.36 studied the M–Rg (M = Li, Na, K, Rb, and Cs; Rg = Ar, Kr, and Xe) systems using the short-range gradient-corrected density functional method combined with long-range coupled-cluster methods (CCSD, CCSD(T)). More recently, the van der Waals systems involving alkali and alkaline-earth atoms interacting with rare gas atoms have been investigated by our group37−44 using a one-electron pseudopotential approach, large Gaussian basis sets, and full configuration interaction.

The precision of produced data for the ground and excited states for similar systems, such as Cs–Ar, Rb–Ar, Cs–Xe, Rb–Xe, and Rb–He complexes,37−39,42,45 has been demonstrated by the prediction of the B ← X absorption spectra of Cs–Ar, which was found to be in excellent agreement with the experimental result of Readle et al.29 Miller et al.46 used our data for RbHe42 to investigate the temperature-dependent rubidium–helium line shape. The prediction of the pressure effect on spectrum broadening by the Anderson–Talman unified pressure broadening theory and using our data has provided good agreement with experimental observations, which proves the high precision of our calculation and the used model.

In the present paper, we report a theoretical investigation of the M–Kr (M = Rb, Cs, and Fr) van der Waals dimers. The calculation is based on a pseudopotential technique reducing the M–Kr system to one electron, the valence electron. The all-electrons calculation is difficult especially for excited states. To reduce the M–Kr (M = Rb, Cs, and Fr) systems to a one-electron problem, we have treated the M+ and Kr as two closed-shell cores interacting with the alkali metal valence electron via a semilocal pseudopotential. This permits us to reduce the M–Kr systems to one electron–core and core–core interactions to be included using the CCSD(T) accurate potential.

In the present study, we aim to calculate accurate data for the ground and excited states of heavy alkali M(2S)–Kr pairs, which are crucial to evaluate the scaling possibilities for alkali metal–rare gas laser systems. In section II, we present the theoretical method of calculations. The results for Rb–Kr, Cs–Kr, and Fr–Kr are presented in section III. Finally we conclude in section IV.

II. Method of Calculation

The electronic structures of heavy alkali atoms Rb, Cs, and Fr interacting with the Kr atom are obtained by resolving the electronic Schrödinger equation. Therefore, the number of active electrons of M–Kr is reduced to only one electron using the l-dependent semilocal pseudopotential proposed initially by Durand and Barthelat.47 Furthermore, the core polarization pseudopotentials VCPP are incorporated using the l-dependent formulation of Müller et al.48 They account for the polarization of the alkali ionic cores as well as of the Kr atom considered as a whole. For each atom (λ = Rb, Fr, or Kr), the core polarization effects are described by the following effective potential:

II.

where λ is for either the M+ (M = Rb, Cs, and Fr) core or the krypton atom, α is the dipole polarizability, and f is the electric field created by valence electrons and cores of each center. The used polarizabilities of the alkali ions were taken as 9.245a03, 15.116a03, and 20.38a03 for Rb, Cs, and Fr, respectively,37−43 whereas that of the krypton atom was taken as 17.0a03.49

Cutoff radius has been adjusted for the three alkali atoms (Rb, Cs, and Fr) to reproduce the experimental energy level spectrum taken from ref (50). The cutoff parameters used here are 2.5213, 2.279, and 2.511 au for rubidium, 2.69, 1.58, and 2.810 au for cesium, and 3.16372, 3.045, and 3.1343 au for francium, respectively, for the lowest valence s, p, and d one-electron states.

To study the M–Kr systems, we have used the same uncontracted basis sets as in refs (37−43).

In turn, the one-electron SCF atomic calculations for the alkali atoms have been performed to test the quality of the basis sets. For the lowest valence s, p, and d one-electron states, the ionization potentials are deduced from the atomic data table50 and presented in previous publications.37−43

In addition, the electron–Kr effects have been treated through local pseudopotential. We have fitted the numerical pseudopotential of Yuan and Zhang51 using the analytical form according to

II.

For each symmetry (l = 0 (s) orbital, l = 1 (p) orbital, and l = 2 (d) orbital), the pseudopotential parameters α, Ci, and ni are extracted. They are given in Table 1.

Table 1. Values of Parameters in the Pseudopotential of e–Kr in Semilocal Potential Form Wl(r) = exp(αr2)∑i=0nCirni (All in Atomic Units).

l α Ci ni
0 0.9251 20.0 –1
1 0.282 2.08 1
2 0.508 21.21 1
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We have also used a basis set on the krypton atom. The use of a basis set on the Kr atom is essential to treat the steric distortion of the alkali valence orbitals resulting from their orthogonality to the Kr closed shells, which are represented by the pseudopotential. Since there are no active electrons on the Kr atom, the exponents were determined in order to provide correct overlap with the 3s and 2p orbitals of Kr and to extend toward the diffuse range.

In addition, to make them easy to use in a computing code based on the model potential, the core–core interaction of M+–Kr (M = Rb, Cs, and Fr) was achieved by using the numerical coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) potential of Hickling et al.52 The CCSD(T) numerical potential was interpolated with the analytical form of Tang and Toennies.53 It is considered as the sum of three terms: the first one is an exponential short-range repulsion (Aeff exp(−bR)), the second is the polarization contribution (D4=−1/2αKrR–4), and the third one is the long-range attractive term (D6R–6 – D8R–8 – D10R–10). The fitted parameters for Rb+–Kr, Cs+–Kr, and Fr+–Kr are presented in Table 2.

Table 2. Interpolation Parameters of the M+–Kr (M = Rb, Cs, and Fr) Core–Core Interaction (All in Atomic Units).

parameter Rb+–Kr Cs+–Kr Fr+–Kr
Aeff 269.41 316.586 279.684
b 1.64249 1.59915 1.56109
D4 6.30076 6.46908 7.26873
D6 594.698 610.39 639.961
D8 738.447 2629.35 738.741
D10 11857.1 12995.9 18301.8
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In fact, the precision of the M+–Kr (M = Rb, Cs, and Fr) interactions, VM+–Kr, is crucial to obtain correct potentials for the M–Kr (M = Rb, Cs, and Fr) Rydberg states.

The spin–orbit coupling is introduced for all M–Kr (M = Rb, Cs, and Fr) complexes using the semiempirical scheme of Cohen and Schneider.55 The spin–orbit coupling matrices are isomorphic to those given by Cohen and Schneider55 for the 2p5 and 2p53s configurations of Ne+ and Ne*, respectively. The matrices for the np and nd configurations are detailed in refs (37−43). The corresponding spin–orbit constants, ξ, are determined using the atomic spectra from the NIST database.50

III. Results and Discussion

III.1. Potential Energy Curves and Spectroscopic Constants of the M–Kr (M = Rb, Cs, and Fr) Dimers

a. M–Kr Ground States (X2Σ+)

The potential energy curves (PECs) for the ground states (X2Σ+) of the M–Kr systems have been determined for a large and dense grid of intermolecular distances. Figure 1 shows the repulsive character of the PECs of the X2Σ+ ground states correlating to Rb(5s) + Kr, Cs(6s) + Kr, and Fr(7s) + Kr, respectively. We remark that the equilibrium position for their wells increases as the mass of the alkali atom increases. However, the depth of these wells decreases as the mass increases. Similarly to the M–Rg (M = Rb, Cs, and Fr; Rg = He, Ar, and Xe) cases,37−43 these repulsive states exhibit weak van der Waals minima at large internuclear distances.

Figure 1.

Figure 1

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Potential energy curves of the X2Σ+ ground electronic states of the Rb–Kr, Cs–Kr, and Fr–Kr molecules.

The X2Σ+ states of all M–Kr van der Waals complexes are presented in Table 3 and compared with available theoretical16,35,36,45,56 and experimental57 works. They were determined for 86Rb–87Kr, 133Cs–87Kr, and 223Fr–87Kr isotopic species.

Table 3. Spectroscopic Constants of the X2Σ+ Ground States of M–Kr (M = Rb, Cs, and Fr) Systems.
system state Re (a0) De (cm–1) ωe (cm–1) ωexe (cm–1) Be (cm–1) refa
Rb–Kr X2Σ+ (5s) 9.91 80 9.39 0.69 0.014542 this work
    9.4 72.3       (16)
    9.52 78.7 9.6     (35)
    9.95 65.2 8.2     PBE/CCSD (36)
    9.84 71.3 8.5     PBE/CCSD(T) (36)
    9.89 69.3 8.4     LDA/CCSD(T)36
    10.47 48.1 6.6     CCSD(T)36
    9.84 71.3 8.5     (55)
    9.99 73       (57)
    10.21 60.74829       (58)
Cs–Kr X2Σ+ (6s) 10.17 76 6.50 0.11 0.011353 this work
    9.6 70.1       (16)
    9.63 91.4 8.9     PBE (36)
    10.14 68.1 7.0     PBE/CCSD36
    9.98 75.9 7.3     PBE/CCSD(T)36
    10.04 73.6 7.2     LDA/CCSD(T) (36)
    10.70 50.5 6.2     CCSD(T) (36)
    9.97 75.9 7.3     (55)
    10.28 74       (57)
    10.47 65.98271       (58)
Fr–Kr X2Σ+ (7s) 10.35 66 6.18 0.12 0.009237 this work
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a

Abbreviations: PBE, Perdew–Burke–Ernzerhof; PBE/CCSD, Perdew–Burke–Ernzerhof/coupled-cluster singles and doubles; PBE/CCSD(T), Perdew–Burke–Ernzerhof/coupled-cluster singles, doubles, and perturbative triples; LDA/CCSD, local spin density/coupled-cluster singles and doubles; CCSD(T): coupled-cluster singles, doubles, and perturbative triples.

Our values for the X2Σ+ states for Rb–Kr and Cs–Kr can be considered in excellent agreement with the experimental values.57 The calculated potential curves of the ground states of the Rb–Kr and Cs–Kr systems have a shallow minimum of 80 and 76 cm–1 situated at 9.91a0 and 10.17a0 respectively, whereas a well of 73 cm–1 is observed at 9.99a0 for Rb–Kr and 74 cm–1 at 10.28a0 for Cs–Kr dimer by the atomic scattering experiment of Buck and Pauly.57 Similarly, shallow wells were observed theoretically by Baylis16 with the semiempirical pseudopotential approach and by Goll et al.36 using several theoretical methods, but the most reliable results were obtained using the PBE/CCSD(T) combination of density functional and coupled-cluster theory.

For Fr–Kr, the theoretical spectroscopic information for the X2Σ+ ground state is still limited. It is important to note that the spectroscopic constants are determined here for the first time. The corresponding PECs are depicted in Figure 1, and the spectroscopic constants are presented in Table 3.

b. M–Kr Excited States (2Σ+, 2Π, and 2Δ)

The available data in the literature concern the two lowest states, A2Π and B2Σ+. The A2Π and B2Σ+ states are the first excited states of the M–Kr systems correlating to Rb(5p) + Kr, Cs(5p) + Kr, and Fr(7p) + Kr. Concerning the higher (3–7)2Σ+, (2–4)2Π, and (1–2)2Δ excited states, the literature is more scarce. These states make a relatively deep well with a short equilibrium distance. The calculated excited states for all M–Kr combinations are displayed in Figure 2 and grouped by molecular term symbol. The spectroscopic constants of states with 2Σ+, 2Π, and 2Δ symmetries are listed in Tables 4–6 and compared with theoretical15,16,35 works. Comparing the results of Rb–Kr and Cs–Kr dimers to the results of refs (15) and (16), we observe a general good agreement for the well depth De. The considered spectroscopic constants of the A2Π state are De = 428/406 cm–1 and Re = 6.97/7.36a0 for Rb–Kr/Cs–Kr. In fact, we note that for all alkali–krypton dimers, the A2Π state is seen to be the more attractive state and B2Σ+ state the more repulsive state.

Figure 2.

Figure 2

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Potential energy curves of the lowest and higher excited states of M–Kr dimers (M = Rb, Cs, and Fr).

Table 4. Spectroscopic Constants of the Excited States without and with Spin–Orbit Effect of Rb–Kr System.
state Re (a0) De (cm–1) Te (cm–1) ωe (cm–1) ωexe (cm–1) Be (cm–1) ref
A2Π (5p) 6.97 428 12389 37.10 1.38 0.02926 this work
A2Π1/2 (5p) 6.92 392 12266 38.61 1.37 0.029749 this work
  6.8 474         (16)
A2Π3/2 (5p) 6.97 428 12468 37.13 1.38 0.029262 this work
  6.8 552         (16)
B2Σ+ (5p) 13.47 53 12764 5.47 1.19 0.031932 this work
  13.7 17.3         (16)
B2Σ1/2+ (5p) 12.84 52 12844 5.65 1.18 0.031568 this work
32Σ+ (4d) 6.62 652 18783 58.92 1.11 0.032491 this work
  15.39 40 19395 3.91      
22Π (4d) 12.66 44 19390 4.77 0.19 0.008886 this work
12Δ (4d) 6.79 928 18506 51.83 1.02 0.03087 this work
42Σ+ (6s) 6.95 413 19768 34.57 2.04 0.029443 this work
52Σ+ (6p) 6.61 740 231395 9.31 1.16 0.032613 this work
  11.93 175 23704 6.69      
32Π (6p) 6.75 943 22936 49.80 1.00 0.031263 this work
62Σ+ (5d) 6.57 765 25024 61.30 1.10 0.032916 this work
  14.16 125 25664 3.08      
42Π (5d) 7.29 676 25113 35.62 0.52 0.026753 this work
22Δ (5d) 6.63 1155 24634 58.12 0.91 0.032353 this work
72Σ+ (7s) 6.72 821 25560 64 1.45 0.031473 this work
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Table 6. Spectroscopic Constants of the Excited States without and with Spin–Orbit Effect of Fr–Kr System.
state Re (a0) De (cm–1) Te (cm–1) ωe (cm–1) ωexe (cm–1) Be (cm–1) ref
A2Π (7p) 7.53 323 13140 34.13 1.25 0.025097 this work
A2Π1/2 (7p) 7.44 310 12706 37.30 1.13 0.025690 this work
A2Π3/2 (7p) 7.53 323 14380 34.12 1.28 0.025105 this work
B2Σ+ (7p) 14.03 50 13377 4.18 0.83 0.018554 this work
B2Σ1/2+ (7p) 13.46 46 14657 4.54 1.24 0.026249 this work
32Σ+ (6d) 14.81 37 16407 3.54 0.07 0.004507 this work
22Π (6d) 11.90 56 16423 5.41 0.15 0.010043 this work
12Δ (6d) 7.48 602 15877 40.71 0.94 0.025447 this work
42Σ+ (8s) 7.75 408 19346 22.84 0.67 0.016465 this work
52Σ1/2+ (8p) 7.28 440 23027 38.14 0.82 0.018667 this work
  13.43 148 23318 4.65      
32Π (8p) 7.41 617 22885 40.00 0.92 0.025882 this work
62Σ+ (7d) 7.47 210 24310 27.76 1.21 0.017710 this work
  11.32 150 24369 4.18      
42Π (7d) 7.73 562 23994 36.11 0.75 0.023790 this work
22Δ (7d) 7.28 748 23807 47.01 1.03 0.026863 this work
72Σ+ (9s) 7.41 621 25131 33.56 0.68 0.018000 this work
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In the X2Σ+ state, the alkali electron is mainly in a spherical sσ orbital; however, in the B2Σ+ state, the alkali electron is mainly in a pσ orbital which overlaps the krypton atom at a larger internuclear separation than the sσ orbital. In the A2Π state, the alkali electronic wave function has pπ character with a node along the internuclear axis, and the krypton atom can approach the alkali rather closely before the repulsive interactions dominate.

For all M–Kr (M = Rb, Cs, and Fr) complexes, the np (n = 5, 6, 7) 2Σ+ and nd (n = 4, 5, 6) 2Σ+ states are due to the excitation from the 6s nonbonding molecular orbital (MO) to the feebly antibonding npσ (n = 5, 6, 7) and ndσ (n = 4, 5, 6) MOs. Therefore, the potential energy curves are repulsive, and the systems become unstable. The repulsive interaction in the np (n = 5, 6, 7) 2Σ+ states seems to be smaller than that in the nd (n = 4, 5, 6) 2Σ+ states and starts from a smaller internuclear distance. For example, Rb–Kr shows that the B2Σ+ and 52Σ+ states have a shoulder humps located around 9.0a0, while the 32Σ+ and 62Σ+ states have a characteristic hump situated at 9.89a0 and 10.5a0, respectively. The humps of the 32Σ+ and 62Σ+ states are caused by an avoided crossing between the 62Σ+ and 72Σ+ states. The same effects are also observed for the both Cs–Kr and Fr–Kr, respectively, for np (n = 7, 8) 2Σ+ and nd (n = 6, 7) 2Σ+ states. Recently, these features were also observed in our group for the alkali and alkaline-earth atoms interacting with He, Ar, and Xe atoms.37−44 They were also noticed by Ehara and Nakatsuji35 in their theoretical calculation for the Cs–Rg systems.

For the higher excited states, the potential energy curves of 2Σ+, 2Π, and 2Δ symmetries have shallow minima. The Rydberg orbitals of these states of the alkali atoms are not directed toward the rare gas atom, but the ion core of alkali atoms polarizes the electron density of the rare gas atom, which is responsible for the attractive character.

For all M–Kr complexes, the spectroscopic constants presented in Tables 4–6 show that the well depths De of the A2Π and 12Δ states are found to be superior to those of the 22Π state and that the equilibrium position of the former is larger. The potential energy of the 72Σ+ state has a similar behavior to the ionic system, which is due to the avoided crossing between the 62Σ+ and 72Σ+ states. For Rb–Kr, Cs–Kr, and Fr–Kr, the dissociation energy and equilibrium distance of 12Δ states are calculated to be 928 cm–1 at 6.79a0, 700 cm–1 at 7.31a0, and 602 cm–1 at 7.48a0, respectively, which is explained by the polarizability and the radius of the alkali atom.

The difference potentials for the transitions from the ground electronic states ns 2Σ+ [n(Rb) = 5, n(Cs) = 6, and n(Fr) = 7] to the first and higher excited states ns 2Σ+ [n(Rb) = 6, 7, n(Cs) = 7, 8, and n(Fr) = 8, 9], np 2Σ+ [n(Rb) = 5, 6, n(Cs) = 6, 7, and n(Fr) = 7, 8], and nd 2Σ+ [n(Rb) = 4, 5, n(Cs) = 5, 6, and n(Fr) = 6, 7] of alkali-rare gas dimers M–Kr are investigated for M = Rb, Cs, and Fr. They are presented in Figure 3. The difference potentials for the transitions from ns 2Σ+ [n(Rb) = 5, n(Cs) = 6, and n(Fr) = 7] to nd 2Σ+ [n(Rb) = 4, 5, n(Cs) = 5, 6, and n(Fr) = 6, 7] have extrema at 5.89a0, 6.14a0, and 6.23a0 for Rb–Kr, Cs–Kr, and Fr–Kr, respectively. These values mirror the position of the hump in the nd 2Σ+ states provoked by the avoided crossing between the nd 2Σ+ state [n(Rb) = 4, n(Cs) = 5, and n(Fr) = 6] and the 7s 2Σ+ state. On the other hand, the excitation energies of the ns 2Σ+ state and the nd 2Π state [n(Rb) = 4, n(Cs) = 5, and n(Fr) = 6)] decrease with internuclear distance, and there exists a flat region at small distances.

Figure 3.

Figure 3

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Excitation energies from the ground state to the excited states of the M–Kr dimers (M = Rb, Cs, and Fr) as functions of the internuclear distance.

III.2. M–Kr (2Σ+, 2Π, and 2Δ) Excited States Introducing the Spin–Orbit Coupling

We have introduced the spin–orbit coupling for the first and higher excited states of the M–Kr systems correlated to the nΠ states for Rb, Cs, and Fr atoms, where n(Rb) = 5, 6, n(Cs) = 6, 7, and n(Fr) = 7, 8, by using the semiempirical scheme of Cohen and Schneider55 for the 2p53s and 2p5 configurations of Ne* and Ne+, respectively. The spin–orbit coupling constants ξ used in this calculation are as follows: ξ5p/6p(Rb) = 158.396/77.51 cm–1, ξ6p/7p(Cs) = 369.36/181.046 cm–1, and ξ7p/8p(Fr) = 1124.392/545.346 cm–1.

Figure 4 left shows the calculated PECs of M–Kr states dissociating into np + Kr and n2P1/2,3/2 + Kr, where n = 5, 6, and 7 for Rb, Cs, and Fr, respectively. The n2P1/2 and n2P3/2 states are obtained by diagonalization of the spin–orbit coupling matrix. The calculated spectroscopic constants for all M–Kr exciplexes are summarized in Tables 4–6.

Figure 4.

Figure 4

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M–Kr (M = Rb, Cs, and Fr) potential energy curves including spin–orbit coupling dissociating into n2P1/2,3/2 + Kr for (left) n = 5–7 and (right) n = 6–8. Red lines are for states including spin–orbit coupling; black lines are for states without spin–orbit coupling.

These new states that include spin–orbit effect demonstrate considerable quantitative variation and are qualitatively similar. The excitations (ΔTe) for n2P1/2 and n2P3/2 states are 202, 529, and 1574 cm–1 for Rb–Kr, Cs–Kr, and Fr–Kr, respectively. The larger splitting value for Fr–Kr states could be explained by the large spin–orbit constant, ξ7p(Fr) = 1124.392 cm–1, in comparison with ξ6p(Cs) = 369.36 cm–1 and ξ5p(Rb) = 158.396 cm–1.

For the Rb–Kr and Cs–Kr systems, including the spin–orbit effects, we show that our potential energy curves of the B2Σ1/2+, A2Π1/2, and A2Π3/2 states corresponding to the n2P1/2,3/2 + Kr (n = 5, 6) asymptote and those obtained experimentally by Hedges et al.15 are in good agreement. The same accord is observed also with theoretical results of Baylis16 and Ehara and Nakatsuji.35 For Cs–Kr, the dissociation energy and the equilibrium position are found to be De = 375/400 cm–1 and Re = 7.29/7.37a0 for the A2Π1/2 and A2Π3/2 states, respectively, to be compared with the experimental values15 of De = 300/350 cm–1, respectively.

By introducing the spin–orbit coupling, the first A2Π1/2 and A2Π3/2 excited states were studied by Baylis,16 who found two different values for De (112 and 439 cm–1 for the A2Π1/2 and A2Π3/2 states, respectively) and the same Re (7.18a0). Ehara and Nakatsuji35 found shallow potential wells of De = 163 and 283 cm–1, respectively. Our results for the A2Π3/2 state and those obtained by Baylis16 are in good agreement, while for the A2Π1/2 state, the well depth found by Baylis16 (112 cm–1) seems to be underestimated compared to the present work as well as the experimental work.15

Using the atomic spin–orbit constants ξ6p(Rb) = 77.51 cm–1, ξ7p(Cs) = 181.046 cm–1, and ξ8p (Fr) = 545.346 cm–1, the spin–orbit effect is also investigated for states correlating to Rb(6p), Cs(7p), and Fr(8p). Figure 4 right depicts the energies of the five states produced (52Σ+, 32Π, 52Σ1/2+, 32Π1/2, and 32Π3/2) without and with spin–orbit perturbation. We show that the PECs including the spin–orbit coupling are considerably similar to those of B2Σ+ and A2Π states in all of the M–Kr complexes from Rb to Fr.

We noticed that the molecular splitting energies at equilibrium position for the 32Π1/2 and 32Π3/2 states in M–Kr (M = Rb, Cs, and Fr) are smaller than the splittings found for the A2Π1/2 and A2Π3/2 states, which is due to the small spin–orbit constant for the np atomic limit, where n = 6, 7, and 8, for Rb, Cs, and Fr, respectively. The calculated data, before inclusion of the spin–orbit coupling, are presented in Tables 4–6. The present calculations illustrate a slight change for the equilibrium distance and the vibrational constant by introducing spin–orbit perturbation, but a larger change in vertical transition energy Te.

The literature about the higher excited states (52Σ1/2+, 32Π1/2, and 32Π3/2) with spin–orbit coupling is limited. To our best knowledge, no experimental or theoretical results are available for the M–Kr molecular systems.

For the 32Σ+, 22Π, and 12Δ excited states correlating to the Cs(5d) + Kr limit, the spin–orbit effect is introduced by using the following value: ξ5d(Cs) = 39.03356 cm–1. For Rb–Kr, it should be noticed that the spin–orbit coupling for the states dissociating into the Rb(4d) + Kr dissociation limit is not considered due to the small spin–orbit constant.

The new Cs–Kr molecular states, dissociating into 52D1/2 + Kr (32Σ1/2+ and 22Π1/2), n2D3/2 + Kr (22Π3/2 and 12Δ3/2), and n2D5/2 + Kr (12Δ5/2), are presented in Figure 5. The corresponding spectroscopic constants are presented here for the first time in Table 5, since there is no comparison of these values with other results.

Figure 5.

Figure 5

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Cs–Kr potential energy curves including spin–orbit coupling dissociating into 52D1/2,3/2,5/2 + Kr. Solid lines show states including spin–orbit coupling and dashed lines states without spin–orbit coupling.

Table 5. Spectroscopic Constants of the Excited States without and with Spin–Orbit Effect of Cs–Kr System.

state Re (a0) De (cm–1) Te (cm–1) ωe (cm–1) ωexe (cm–1) Be (cm–1) ref
A2Π (6p) 7.36 406 11219 32.92 1.06 0.021609 this work
A2Π1/2 (6p) 7.31 375 10881 34.80 1.07 0.021894 this work
    300         (15)
  7.46 163   35.1     (35)
  7.20 112         (16)
A2Π3/2 (6p) 7.38 406 11410 32.27 0.92 0.021494 this work
    350         (15)
  7.52 283   38.3     (35)
  7.20 439         (16)
B2Σ+ (6p) 13.45 56 11569 4.74 0.91 0.022881 this work
B2Σ1/2+ (6p) 12.84 52 12844 5.65 1.18 0.031568 this work
  16.2 17.1         (16)
32Σ+ (5d) 14.60 35 14592 3.91 0.11 0.005495 this work
32Σ1/2+ (5d) 13.65 41 14633 4.88 0.23 0.006282 this work
22Π (5d) 11.16 74   6.84 0.37 0.009402 this work
22Π1/2 (5d) 11.08 70 14534 7.21 0.83 0.009516 this work
    300         (15)
  7.46 163   35.1     (35)
22Π3/2 (5d) 11.14 74 14573 9.02 0.77 0.009437 this work
    350         (15)
12Δ (5d) 7.31 700 13927 39.02 0.79 0.021895 this work
    350         (15)
  7.37 441   46.2     (35)
12Δ3/2 (5d) 7.32 657 13887 39.02 0.81 0.021890 this work
12Δ5/2 (5d) 7.31 700 13966 39.01 0.81 0.021902 this work
42Σ+ (7s) 7.47 461 18154 27.98 0.91 0.020985 this work
  8.22 322   22.3     (35)
52Σ+ (7p) 7.11 599 21433 43.71 0.92 0.023160 this work
52Σ1/2+ (7p) 7.15 624 21499 42.44 0.83 0.022927 this work
32Π (7p) 7.24 741 21292 38.79 0.69 0.022331 this work
32Π1/2 (7p) 7.24 699 21153 39.02 0.73 0.022375 this work
32Π3/2 (7p) 7.26 734 21390 38.63 0.69 0.022231 this work
62Σ+ (6d) 7.41 220 22662 26.23 2.12 0.021341 this work
42Π (6d) 7.56 670 22212 37.02 0.68 0.020485 this work
22Δ (6d) 7.11 859 22023 44.78 0.81 0.023164 this work
72Σ+ (8s) 7.27 658 23775 36.34 0.84 0.022181 this work
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In addition, we note that the spin–orbit effect for the Cs n2P1/2,3/2 (n = 6, 7) states is much larger than that observed for the Cs(52D1/2,3/2,5/2) ones, which is due to the large spin–orbit constant for the np atomic limit.

III.3. Qualitative Prediction of the Broadening Spectrum of Alkali (Rb, Cs, and Fr) D2 Line

Using our potential energy curves for the lower (X2Σ+) and upper (B2Σ+ and A2Π) electronic states, the transition wavelengths were generated for all M–Kr systems using the following expression:43

III.3.

where V′(R) and V''(R) are the potential energy curves for the upper and lower electronic states respectively. In fact, the interaction potentials of the X2Σ+ state and the B2Σ+ and A2Π states and their differences are relevant to the broadening of the D2 and D1 lines45 of alkali atoms. The latter quantity provides information about contributions from dimers or collision pairs. Figure 6 shows the derived transition wavelengths λ(R) for all M–Kr (M = Rb, Cs, and Fr) dimers calculated from the difference potentials B2Σ+ – X2Σ+ and A2Π – X2Σ+ (Figure 6). As can be seen from Figure 6, two different internuclear separations could contribute to the same transition and its wavelength. From this figure we predict that the D2 broadening spectra are confined to the regions 770–865, 849–925, and 715–825 nm for Rb–Kr, Cs–Kr, and Fr–Kr complexes, respectively. Heaven and Stolyarov58 determined the maximum for the blue wing of the D2 line from Cs in Kr to be 835 nm using the QS approximation,45 which is in a general good agreement with the observed value of 841 nm. It is expected that our maximum will be at 849 nm, which is overestimated compared to the theoretical and experimental values.45 The same thing is observed for the Rb–Kr pair, as we found a value of 770 nm to be compared with the calculated and observed values58 of 753 and 759 nm.45 Once again, our wavelength is overestimated compared to the calculated and observed ones.45 This is due to the original used potential interactions for the X2Σ+ and B2Σ+ states. It seems that the potential used by Heaven and Stolyarov58 for the B2Σ+ state including spin–orbit coupling is more repulsive than ours, which makes their wavelength shorter. It is clear that the difference between our work and that of Heaven and Stolyarov58 is primarily related to the difference in the potential energy interaction calculation approaches. We used a simple model that reduced the M–Kr pair to only one valence electron interacting with the M+–Kr ionic system. However, Heaven and Stolyarov58 used more sophisticated ab initio approaches such as multireference configuration interaction (MRCI) and multireference averaged quadratic coupled cluster (MRAQCC) to produce the potential interactions used in the spectrum broadening simulations.

Figure 6.

Figure 6

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Transition wavelengths calculated from the difference potentials B2Σ+ – X2Σ+ and A2Π – X2Σ+ for the M–Kr dimers (M = Rb, Cs, and Fr).

III.4. Vibrational Analysis and Transition Dipole Moments of M–Kr (M = Rb, Cs, and Fr) Dimers Introducing the Spin–Orbit Coupling

By solving the nuclear radial Schrödinger equation, the vibrational level spacing of alkali–rare gas diatomic molecules M–Kr are investigated for the molecular states correlated asymptotically to the degenerate 2P (np) states, where n(Rb) =5, 6, n(Cs) = 6, 7, and n(Fr) = 7, 8, introducing the spin–orbit effect.

They were determined for 86Rb–87Kr, 133Cs–87Kr, and 223Fr–87Kr isotopic species using the Numerov algorithm with 30 000 points ranging from Rmin = 3.0a0 to Rmax = 200a0 and reduced masses μRb–Kr = 42.3123 a.u, μCs–Kr = 51.3939 a.u, and μFr–Kr = 60.9096 a.u.

The vibrational level spacings (Ev – Ev–1) in terms of the vibrational number v of the M–Kr dimers are depicted in Figures 7 and 8. As can be seen, our calculation for the 32Π and 52Σ+ higher excited states correlating asymptotically with 2P (np), where n(Rb) = 6, n(Cs) = 7, and n(Fr) = 8, indicates more vibrational levels than for the A2Π and B2Σ+ states correlating with 2P (np), where n(Rb) = 5, n(Cs) = 6, and n(Fr) = 7, which present, respectively, 40, 56, and 58 levels for Rb–Kr, Cs–Kr, and Fr–Kr dimers. This dissimilarity is obviously related to the difference in the well depths.

Figure 7.

Figure 7

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Vibrational level spacings of M–Kr (M = Rb, Cs, and Fr) diatomic molecules between the molecular states correlating asymptotically to the degenerate 2P (np) states, where n(Rb) = 5, n(Cs) = 6, and n(Fr) = 7.

Figure 8.

Figure 8

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Vibrational level spacing of M–Kr (M = Rb, Cs, and Fr) diatomic molecules between the molecular states correlating asymptotically to the degenerate 2P (np) states, where n(Rb) = 6, n(Cs) = 7, and n(Fr) = 8.

We have also determined vibrational levels for the B2Σ+ and A2Π states and 52Σ+ and 32Π states by introducing the spin–orbit coupling. For all systems where the spin–orbit coupling plays a more significant role, the variation of (Ev – Ev–1) with v is split into two curves. We are now interested in Rb interacting with the 87Kr isotope. We found 32 and 40 vibrational levels for Ω = 1/2 and Ω = 3/2, respectively. Moreover, when approaching the dissociation limit, the energy level spacing vanishes and the common behavior is a decrease as v rises. The same behavior has been observed also for all other dimers. The vibrational levels for heavy alkali–krypton diatomic molecules are more important than for the light alkali–rare gas molecules.59 To our knowledge, the vibrational levels for heavy alkali–krypton diatomic molecules have been calculated here for the first time.

The calculated transition dipole moment is an essential factor in determining the absorption spectrum. To investigate the nS → nP [n(Rb) = 5, n(Cs) = 6, n(Fr) = 7] transition dipole moments of the alkali atoms by the interaction with Kr atom, the spin–orbit coupling is included by using the energy matrix discussed previously issued from the diagonalization of the rotational matrix. The spin–orbit constants are estimated to be ξ5p(Rb) = 158.396 cm–1, ξ6p(Cs) = 369.36 cm–1, and ξ7p(Fr) = 1124.392 cm–1 from the atomic spectral data.50

The A ← X and B ← X transition dipole moments with and without spin–orbit effect are shown in Figures 9–11 as functions of the internuclear distance. These figures show that the X2Σ+ → B2Σ+ transition moments for Rb–Kr, Cs–Kr, and Fr–Kr have a maximum around 3.0a0. The X2Σ+ → A2Π transition dipole is split into two curves when the spin–orbit coupling is included. They are labeled as X2Σ+ → A2Π1/2 and X2Σ+ → A2Π3/2. The difference between the X2Σ+ → A2Π1/2 and X2Σ+ → A2Π3/2 transition dipole moments is significant at intermediate distances. The transition moments should converge to the value of the pure atomic 2P ← 2S transition for large internuclear separations.

Figure 9.

Figure 9

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Transition dipole moments for the X2Σ+ → A2Π, X2Σ+ → B2Σ+, X2Σ+ → A2Π1/2, X2Σ+ → A2Π3/2, and X2Σ+ → BX2Σ1/2+ transitions as functions of the internuclear distance for Rb–Kr dimer.

Figure 11.

Figure 11

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Transition dipole moments for the X2Σ+ → A2Π, X2Σ+ → B2Σ+, X2Σ+ → A2Π1/2, X2Σ+ → A2Π3/2, and X2Σ+ → B2Σ1/2+ transitions as functions of the internuclear distance for Fr–Kr dimer.

Figure 10.

Figure 10

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Transition dipole moments for the X2Σ+ → A2Π, X2Σ+ → B2Σ+, X2Σ+ → A2Π1/2, X2Σ+ → A2Π3/2, and X2Σ+ → B2Σ1/2+ transitions as functions of the internuclear distance for Cs–Kr dimer.

Reliable and accurate transition dipole moments at long-range internuclear distances in addition to precise potential energies are fundamental for prediction and design of the scaling characteristics of lasers driven by pumping of alkali–rare gas collision pairs.60 In fact, highly accurate potential interaction data are needed to predict the absorption spectra and broadening of alkali atomic lines. More precisely, the accurate transition dipole functions and interaction energies are used to simulate the absorption spectrum and broadening in the D2 line for Rb–Kr, Cs–Kr, and Fr–Kr dimers.

The long-range behavior of the permitted transition dipole moment was extensively explored in the papers of Chu and Dalgarno61 and Pazyuk et al.62 The latter demonstrated that the asymptotic behavior of the Rb2 and Cs262 transition dipole functions at R = ∞ tends to the limiting atomic value Bn/Rn with n = 3, 4, where the coefficients Bn can be calculated using the atomic wave functions and energies. A quantitative fitting of some transition dipole moments showed this behavior for Cs–Kr and Rb–Kr dimers. A detailed study of the transition dipole functions at long-range internuclear distances is in progress for a better understanding of the long-range behavior. This will permit the probing of their quality. This quality is related to that of the electronic wave functions, which were also used to evaluate the molecular energies.

IV. Conclusion

We have performed an accurate theoretical study of alkali metal (Rb, Cs, and Fr) atoms interacting with a rare gas (Kr) atom. The M+ (M = Rb, Cs, and Fr) cores and the electron–Kr interactions were replaced by a semilocal pseudopotential, and the M–Kr systems were reduced to one-electron SCF calculations. The potential energy curves and their corresponding spectroscopic constants have been determined. The spectroscopic parameters of the X2Σ+ ground states show a good agreement compared to the available theoretical15,16,35,36 and experimental57 works.

Moreover, the semiempirical scheme of Cohen and Schneider55 was included to study the spin–orbit effect. One can notice that the largest energy splitting is observed for the first excited states B2Σ+ and A2Π. Our spectroscopic constants and the earlier studies for the B2Σ1/2+, A2Π1/2, and A2Π3/2 excited states are compared with experimental57 and theoretical15,16,35,36 results. Good agreement was observed for the equilibrium distances and dissociation energies. Because of the spin–orbit effect, a splitting was also observed in the case of the vibrational level spacing and the transition dipole moment.

In addition, a qualitative prediction of the D2 broadening spectra was performed in this study. We found that broadening D2 spectra are confined to the regions 770–865, 849–925, and 715–825 nm for Rb–Kr, Cs–Kr, and Fr–Kr, respectively. Expected maxima for the blue-wing D2 lines are predicted to appear at 770, 849, and 715 for Rb–Kr, Cs–Kr, and Fr–Kr, respectively. Heaven and Stolyarov58 determined the maxima for the blue wing of the D2 line from Rb and Cs in Kr to be 753 and 835 nm using the QS approximation, in a good agreement with the observed values of 759 and 841 nm. Our predictions are 770 and 849 nm, which are overestimated compared to the theoretical and experimental values.48−58 This could be associated to the difference in the potential interactions for the X2Σ+ and B2Σ1/2+ states, especially at short and intermediate repulsive parts in both states’ potentials, making the difference in potential energy smaller and therefore giving longer wavelengths. It is clear that the difference between our work and that of Heaven and Stolyarov58 is primarily related to the difference in the potential energy interaction calculation approaches. The potential used by Heaven and Stolyarov58 for the B2Σ1/2 state including spin–orbit coupling is more repulsive than ours. Considering the simplicity of our used model, reducing the M–Kr pair to only one valence electron interacting with the M+–Kr ionic system, and the more sophisticated ab initio approaches used by Heaven and Stolyarov,58 our results are in general good agreement with theirs.

In addition, to study the structure and dynamics of large M+–Krn clusters, the one-electron pseudopotential approach can be used. Furthermore, to investigate the expansion effect of the alkali atom by collision with the Kr atom and also to evaluate transport coefficients for alkali atoms moving through a bath of Kr atoms, the accurate results of the M–Kr systems will be used.

The authors declare no competing financial interest.

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