Electronic structure of the polar molecules XF (X: Be, Mg, Ca) with rovibrational and dipole moment calculations

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

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Electronic structure of the polar molecules XF (X: Be, Mg, Ca) with rovibrational and dipole moment calculations Nayla El-Kork a, Nariman Abu el kher b, Farah Korjieh b, John Anwar Chtay b, Mahmoud Korek b,⁎ a b

Khalifa University, P.O. Box 57, Sharjah, United Arab Emirates Faculty of Science, Beirut Arab University, P.O. Box 11-5020, Riad El Solh, Beirut 1107 2809, Lebanon

a r t i c l e

i n f o

Article history: Received 14 April 2016 Received in revised form 16 December 2016 Accepted 16 January 2017 Available online 24 January 2017 Keywords: Ab initio calculation Electronic structure Spectroscopic constants Potential energy curves Dipole moments Rovibrational calculation

a b s t r a c t A theoretical investigation for the feasibility of laser-cooling is performed through the calculation of accurate potential energy curves, static dipole moments, spectroscopic constants and rovibrational calculations for 24, 26 and 27 highly excited electronic states for BeF, CaF and MgF molecules respectively. In order to understand the electronic structure of their lowest lying electronic states and to learn the characteristic behavior of their chemical bonding, a high level of calculation is realized by using the complete active space self-consistent field (CASSCF) with multi-reference configuration interaction MRCI method including single and double excitations with Davidson correction (+Q) for the three considered molecules. The comparison between the values of the present work and those available in the literature for several electronic states shows a good agreement. Fifty new excited electronic states have been investigated, in the present work, for the first time for the three studied molecules. © 2017 Elsevier B.V. All rights reserved.

1. Introduction A large amount of spectroscopic information and precise detailed of alkaline earth mono-fluoride molecules has been provided by the advances in high-resolution laser spectroscopic and molecularbeam techniques [1–19]. The bonding in these molecules is very ionic so their electronic structure can be pictured as a halogenoide anion X − and an alkaline earth cation M2 + . This ionic model can provide a powerful tool for the understanding of ionic compounds in terms of the properties of their constituent including diatomic, triatomic, clusters and molecular ions [20–39]. These molecules are amenable to direct laser cooling which may contribute significantly to various areas of research and wide range of applications [40]. They can be used as attractive candidates for quantum systems [41] and quantum computation related to the laser cooling of molecule [42]. Moreover, ultracold molecules are very useful for research in chemical dynamics [43–44], for controlling chemistry at low temperatures [45–46], high-resolution molecular spectroscopy [47], and nanolithography. Because of their permanent electric dipole moments, they lead to long-range and controllable anisotropic dipoledipole interactions [48]. However, laser cooling of molecules was thought to be a challenge because of their complex internal structure so while many atomic species can be cooled to temperature in

⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Korek).

http://dx.doi.org/10.1016/j.saa.2017.01.035 1386-1425/© 2017 Elsevier B.V. All rights reserved.

μK regimes using laser, it is not straightforward to apply cooling methods to molecules directly. Therefore, a theoretical calculation of the molecular electronic structure is required to explore all the possible decay pathways and identify an efficient cooling cycle. Calcium mono-fluoride (CaF) molecule is one of the members of the alkaline earth mono-halides. The potential energy curves (PECs) and the dissociation energies for the ground and first excited states of CaF and other alkaline earth mono-halides molecules were constructed by Rao et al. [5] in a small region around the equilibrium position. D'Incan et al. [6] and Verges et al. [7] observed some excited electronic states of CaF molecule while Kaledin et al. [8] accurately measured the spectroscopic constants of the ground and two excited states. The ground state and the valence excited A, B, and C states of some alkaline earth monohalides molecules were described by Rice et al. [9] using a ligand-field model and the dissociation energy D0; the ground state was computed by Partridge et al. [10] also. The spectroscopic constants for the ground state of CaF molecule have been calculated by using the self-consistent-field singles and doubles configuration-interaction (SCF-CSID) method by Langhoff et al. [11]. The ground and the valence excited A, B, and C states of some alkaline earth monohalides molecules were described by Törring et al. [12] using an electrostatic polarization model. The transition energy Te, the equilibrium position Re, the vibrational harmonic constant ωe , and the rotational constant B e of the ground state and five low lying excited states of CaF molecule have been calculated by Bundgen et al. [13], Yang et al. [14], Handy et al. [15]and C.E. Dykstra [16]. The transition energy T e and permanent and

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

transition dipole moment for the four low lying excited states of CaF and other alkaline earth monohalides molecules were calculated by Allouche et al. [17] using a ligand field approach method. Pelegrini et al. [18] calculated accurate potential energy curves, transition moments, spectroscopic constants, radiative transition probabilities and lifetimes for the ground state and the first excited state of CaF and other alkaline earth monohalides molecules using the CASSCF/ MRCI (complete active space self-consistent field, multi-reference configuration interaction) method. The analysis of the electronic structure of the magnesium monohalide species [19–49] has received a great deal of attention. It have been the subject of many experimental and theoretical studies, but few included their excited states. Rao and Lakshman [50] found the bands of MgF molecule in the disk and spot spectra. Based on the relatively high cosmic abundance of magnesium, MgF seems a likely candidate for a circumstellar molecule [51]. Several Studies were concentrated on the rotational and vibrational structure [50,52] and on its hyperfine structure [51,53,56]. Barrow and Beal [52] recorded and analyzed the high resolution of the 0–0 and 1–0 bands of the (A)2Π → X2Σ+

-113.75

(3) 2 Δ

E(Hartree) -113.8 -113.85

(2) 2 Σ−

(4) 2 Σ+

-113.95 -114

transition and also proposed that (A)2Π state was inverted. Walker and Richards [54] computed the spin-orbit coupling constants for MgF molecule and explained that the state (A)2Π of MgF was regular. In support with Walker and Richard conclusion Knight et al. [53] proved that the lowest (A)2Π excited state is regular. The spectra of MgF molecule have also been detected in the microwave and millimeter-wave regions as in the high-resolution infrared emission spectrum where Barber and co-workers [55] determined some of the Dunham coefficients for the ground X2Σ+ and the spin-rotation constants. Anderson et al. [51] used a direct absorption spectrometer in order to measure the pure rotational spectrum of MgF in its ground electronic state X2Σ+ where Peligrini et al. [18] calculated the spectroscopic constants, the radiative transition probabilities, the transition dipole moments (TDMs) and the lifetimes for the (A) 2 Π → X 2Σ + transitions. As early as 2015, Kang et al. [57] suggested that the lower mass and shorter spontaneous radiative lifetimes make the MgF molecule more suitable for laser cooling. They also investigated seven low-lying doublet electronic states and their spectroscopic constants using ab initio calculations.

(6) 2 Σ+

(5) 2 Σ+

-113.9

171

(2) 2 Δ

(1) 2 Σ−

(2) 2 Σ+

(1) 2 Δ

(3 )2 Σ+

-114.05 -114.1 -114.15

(1) 2 Σ+

-114.2 -114.25 0.9

1.4

1.9

2.4

2.9

3.4

R(Å)

3.9

3

μ(a.u)

(2) 2 Σ− (1) 2 Σ−

1

R(Å) 0.9

1.4

1.9

2.4

-1

2.9

3.4

(2) 2 Σ+ (3) 2 Σ+

-3

(1) 2 Δ

(1) 2 Σ+

(2) 2 Δ

-5

-7

(3) 2 Σ−

-9

Fig. 1. Potential energy and dipole moment curves of the 2Σ± and 2Δ electronic states of the molecule BeF.

3.9

172

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The beryllium monofluoride molecule (BeF) has been studied theoretically and experimentally in literature [12–21] but only for the lower 6 electronic states. Novikov and Gurvich [58] excited the BeF emission in a Schüer tube and reported the two electronic transitions B2Σ+ − X2Σ+ and C2Σ+ − X2Σ+. In theory, the spectroscopic parameters including the dissociation energy De have been widely studied in the past several decades [59–65] where these values showed wide variation [59–62]. Furthermore, some theoretical information [18,60,63,65] is available about the excited states of the BeF molecule. Moreover, vibrational manifolds such as vibrational levels, initial rotation and centrifugal distortion constants have been reported in the literature, which have important applications in the vibrational transition calculations. Since the alkaline earth mono-fluoride molecules are potential candidates for laser cooling with a lack in the quartet and highly excited electronic states, we perform in the present work theoretical calculations for the 3 alkaline earth monofluoride molecules BeF, MgF and CaF. Whereas, in previous works we investigated the electronic structure of the two molecules SrF and BaF [49,66]. The purposes of this investigation is to determine the accurate 27, 26, and

-113.72

27 potential energy curves of the highly excited electronic sates of the molecules MgF, CaF and BeF respectively along with the corresponding dipole moment curves and the spectroscopic constants. These investigated data are directly amenable to the calculation of the FCF and the radiative lifetime for the transition between two electronic states of the three considered molecules which are amenable to laser cooling. In part two, concerning the theoretical approach, the calculation has been performed by using the Complete Active Space Consistent Field (CASSCF) with Multireference Configuration Interaction (MRCI with Davidson correction) for the three molecules. Indeed, experiments on cooling and trapping of polar molecules have opened up exciting possibilities. In parts three and four, we present the calculated data with the corresponding discussion. In part five, a rovibrational study has been carried out by using the canonical function method [66–69]. The eigenvalue Ev, the rotational constant Bv, the centrifugal distortion constants Dv, and the abscissas of the turning points Rmin and Rmax have been calculated for the investigated electronic states up to the vibrational level v = 59. The comparison between the investigated data of the five alkaline earth monofluoride molecules is presented.

(5) 2 Π

E(Hartree)

(2) 4 Π

-113.77

-113.82

(4) 2 Π

-113.87

(3) 2 Π

(1) 4 Π

-113.92

(2) 2 Π

-113.97

-114.02

(1) 2 Π R(Å)

-114.07 0.9

1.4

1.9

2.4

2.9

3.4

3.9

4.4

4.9

2

μ(a.u)

(1) 2 Π

1

R(Å)

0 0.9

1.4

1.9

2.4

2.9

-1

-2

-3

-4

3.4

3.9

4.4

(2) 4 Π (1) 4 Π (2) 2 Π

(3) 2 Π

(4)2Π

-5 Fig. 2. Potential energy and dipole moment curves of the 2,4Π electronic states of the molecule BeF.

4.9

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(3) 4 Σ+

-113.75

(2) 4 Δ

173

(3) 4 Δ

(2) 4 Σ− (2) 4 Σ+

-113.8

-113.85

(1) 4 Σ− (1) 4 Δ

-113.9

(1) 4 Σ+ -113.95

1

1.5

2

2.5

3

3.5

4

4.5

R(Å)

5

1

μ(a.u)

(1) 4 Σ+ , (1) 4 Σ−

0.5

0 2.45 -0.5 -1

2.5

2.55

2.6

2.65

(2) 4 Δ

2.7

2.75

2.8

2.85

R(Å) 2.9

(2) 4 Σ−

-1.5 -2

(3) 4 Δ

-2.5 -3 -3.5

(1) 4 Δ

-4 Fig. 3. Potential energy and dipole moment curves of the 4Σ± and 4Δ electronic states of the molecule BeF.

2. Computational Approach An ab initio investigations of the low-lying electronic states of the alkaline earth monofluoride BeF, MgF and CaF molecules have been performed via Complete Active Space Self Consistent Field (CASSCF) using the Multireference Configuration Interaction (MRCI) method with Davidson correction (+ Q). By using the computational chemistry program MOLPRO [70] taking the advantage of the graphical user interface GABEDIT [71], the entire CASSCF configuration space was used as the reference in the MRCI calculations. This software is intended for highly accurate computations with extensive treatment of the electron correlation problem. The quality of the selected basis sets is checked by comparing our CI calculations for the ground and several excited electronic states of isolated Be, Mg, Ca and F atoms to the experimental data in NIST Atomic Spectra Database [72]. For the BeF molecule, the beryllium atom is treated in all electron schemes where the four electrons of this atom are considered using the VTZ [73] basis set for s, p, d and f functions. The nine electrons of the fluoride atom were treated using the Rydberg5 [74] and au-ccpcv5z basis sets for s, p and d functions. Among the 13 electrons explicitly considered for the BeF molecule, two techniques have been used. The first one is the CASSCF-MRCI method, where eight inner electrons are considered to be frozen in subsequent calculations so that 5 valence p electrons have been explicitly treated. The active

space contains 7σ(F: 2p 0, 3s, 3p0 , 3d 0, 4s; Be: 2p0 , 3s), 4π(F: 2p± 1, 3p± 1 , 3d± 1; Be: 2p 0 ) and 1δ(F: 3d ± 2) molecular orbital in the C 2v symmetry. This corresponds to 12 active molecular orbitals distributed into irreducible representation a1, b1, b2, and a2, in the following way: 7a1 , 4b1 , 4b2 , 1a 2 , noted [7,4,4,1]. The second one is the CASPT2-RS2 method where the 6 inner electrons are considered to be frozen in subsequent calculations so that seven valence electrons have been explicitly treated. The active space contains 9σ(F: 2s, 2p0, 3s, 3p0, 3d0, 4s; Be: 2s, 2p0, 3s), 4π(F: 2p± 1, 3p± 1, 3d± 1; Be: 2p0) and 1δ(F: 3d± 2) molecular orbital in the C2v symmetry. This corresponds to 14 active molecular orbitals distributed into the irreducible representation a 1 , b1 , b 2 , and a 2 , in the following way: 9a 1, 4b1 , 4b 2 , 1a2, noted [9,4,2,1]. The magnesium and the fluorine atoms are treated in all electrons schemes using the cc-pVDZ and the aug-cc-pCVDZ basis sets respectively. Among the 21 electrons considered for the magnesium monofluoride molecule, fourteen electrons were frozen in subsequent calculations, so that 7 valence electrons were explicitly treated. The active space 6σ(Mg: 3s, 3p0, 4s,3d0; F: 3s, 2p0), 3π(Mg: 3d±1, 3p±1; F: 2p±1), and 1δ(Mg: 3d±2) orbitals in the C2v symmetry are distributed into irreducible representation a1, b1, b2, and a2 in the following way 6a1, 3b1, 3b2, and 1a2 noted by [6,3,3,1]. In CaF molecule, the fluorine atom is treated in all electron schemes where the 9 electrons are considered using the Rydberg5 [74] basis set for the s, p and d functions. The calcium atom is treated

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-135.79

(5) 2Σ+

-135.84

(3) 2Σ+

(4) 2Σ+

-135.89 Energy (Hartree)

(2) 2Σ+ (1) 2Δ

-135.94

-135.99 X2Σ+ -136.04

μ(a.u)

-136.09 1

2

3

4

1

2

3

4

5

6

7

R (Å)

8

9

1

5

6

7

8

9

(5) 2 Σ+

-1

(2) 2 Σ+ (4) 2 Σ+

-3

X2 Σ+ -5

(1) 2 Δ (3) 2 Σ+

-7

-9

Fig. 4. Potential energy and dipole moment curves of the 2Σ± and 2Δ electronic states of the molecule CaF.

as a system with effective core potential, where 10 electrons are considered as inner electrons and the remaining 10 electrons are considered as valence electrons, using the ECP10MWB [75] basis set for the s, p, d and f functions. Among the 19 electrons, 12 electrons were frozen in subsequent calculations so that seven electrons were explicitly treated. The active space contains 8σ(Ca: 3s, 3d0, 4s, 4p, 5s;F: 1s, 2s, 3s), 2π(Ca: 4p± 1, 3d± 1), 1δ(Ca: 3d± 2) orbitals in the C2v symmetry; this corresponds to 8 active molecular orbitals distributed into irreducible representation a1, b1, b2, a2 in the following way: 8a1, 2b1, 2b2, 1a2 noted [9,2,2,1]. 3. Results and Discussion The representation of the potential energy curves (PECs) for a diatomic molecule, as a function of the internuclear distance, is a

characteristic of the molecular states that are based on the interatomic forces that bind the atoms together where these forces are either attractive or repulsive. These potential energy curves are of considerable importance in order to understand the dynamics, interpretation of the molecular spectra and other associated problems. By using the CASSCF-MRCI method for the three considered molecules, and in the representation 2s+1Λ±, the PECs for 24 electronic states have been calculated for the molecule BeF with five valence electrons and with an internuclear distance in the range 1 Å ≤ R ≤ 4.88 Å. Fourteen lowlying doublet electronic states (seven 2Σ+, two 2Σ−, two 2Δ and three 2Π) and 13 low-lying quartet states (four 4Σ+, three 4Σ−, two 4 Δ and four 4Π) have been investigated for the molecule MgF. For the molecule CaF the PECs are calculated for 370 internuclear distances in the range 1.55 Å ≤ R ≤ 8.95 Å for 10 doublet and 16 quartet electronic states.

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175

-135.63

(1) 4 Φ

E(Hartree)

(3) 4 Π

-135.68

(4) 4 Π -135.73

(3) 2 Π -135.78

(2) 4 Π -135.83

(1) 4 Π

-135.88

(2) 2 Π -135.93

(1) 2 Π R(Å)

-135.98 1.5

2

μ(a.u)

2

(1) 4 Π

2.5

3

3.5

4

4.5

(4) 4 Π

1

R(Å)

0 1.5

(2) 4 Π

2

(1) 4 Φ

2.5

3

3.5

4

4.5

5

(3) 2 Π

-1 -2

5

(3) 4 Π

(1) 2 Π

-3 -4

(2) 2 Π

-5 -6 Fig. 5. Potential energy and dipole moment curves of the 2,4Π electronic states of the molecule CaF.

The values of the dipole moment are among the most reliably predicted physical properties; they give useful information in discussing the bond nature, the ionicity, the polarity of the states and their interaction. Since the dipole moment at the Hartree–Fock level is usually large, and in order to obtain the best accuracy, Multireference Configuration Interaction (MRCI) wavefunctions were constructed using multiconfiguration self-consistent field (MCSCF) active space. The calculated values of the static dipole moments in terms of the internuclear distance R for BeF, MgF and CaF molecules are plotted with the PECs of these molecules in Figs. 1– 9, while the transition dipole moment curves are given in Fig. 10. The crossing between two potential energy curves corresponding to electronic states of different symmetries is allowed and their wavefunctions are adiabatic solutions of the Schrödinger equation. However, if these wavefunctions have the same symmetry, they

will mix with each other to give two adiabatic solutions, which no longer cross and the crossing becomes avoided. These adiabatic solutions of the Schrödinger equation are obtained by linear combinations of the diabatic ones where the variation method is used. In the range of considered R, some crossings and avoided crossings between different electronic states occur for the three considered molecules. The positions of these crossings and avoided crossings are given in Table 1 where Rc, Rac and ΔEac are respectively the internuclear distances at crossing, the avoided crossing and the energy difference at the avoided crossings between two corresponding states. The positions of the avoided crossings are of great importance for experimentalists because at the internuclear distance where an avoided crossing occurs the line of spectrum will disappear or will be shifted. The abrupt gradient change in the computed dipole moment curve (Figs. 1–9) for the 3 molecules is related to an avoided

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-135.68

(2) 4 Σ−

E(Hartree)

(2) 4 Δ

(3) 4 Δ

(4) 4 Σ+

Series1

-135.73

Series2 Series3

(1) 4 Σ−

Series4

(3) 4 Σ+

Series5

-135.78

Series6

(2) 4 Σ+

Series7

(1) 4 Δ

Series9

-135.83

Series8

(1) 4 Σ+ R(Å)

-135.88 1.7

2.2

2.7

3.2

3.7

4.2

4.7

5.2

5.7

6.2

2

μ(a.u) 1.5

(1) 4 Σ+

(1) 4 Σ−

1

(1) 4 Δ

(3) 4 Δ

4 0.5 (2) Δ

0

R(Å) 1.7

2.2

2.7

3.7

4.2

4.7

5.2

5.7

6.2

(3) 4 Σ−

-0.5 -1 -1.5

3.2

(2 )4 Σ+

(3) 4 Σ+

(2) 4 Σ−

-2 -2.5 -3 Fig. 6. Potential energy and dipole moment curves of the 4Σ± and 4Δ electronic states of the molecule CaF.

crossing between the PECs of two states of the same symmetry. The DMCs mirror this region as a sharp change in their slopes indicating a reversed polarity of the atoms or an ionic character transfer from one state to another. By plotting the PECs and the DMCs with vertical lines in the same scale and in the same figure, one can notice the agreement between the positions of the avoided crossing of two PECs and the crossing of their corresponding DMCs that confirm the calculation's validity of the studied excited electronic states. A compound can be considered ionic if its electric dipole moment curve, in terms of the internuclear distance R, shows a nearly linear trend close to the equilibrium position Re for his ground state [76] and curved if the compound has a dominating covalent character. This idea was quantified as the following: If , then ∂ μ/∂r = − q, the larger value of q, the more ionic charge (except in some case of polarization

effects) and ∂2μ/∂r2 is closer to zero as the ionicity increases. The negative linear slope of the dipole moment curves of the ground states of the molecules BeF, MgF and CaF (Figs. 1, 4, 7) confirm the presence of the ionic character of these molecules, and the curvature indicates the covalent nature of the same bond. This mixed nature can be confirmed experimentally by the evaluation of the hyperfine parameters. Each one of the investigated BeF, MgF and CaF molecules dissociates into its natural fragments at large internuclear distance, therefore for the most of investigated PECs for the different electronic states of the considered molecules each dipole moment curve is expected to approach zero smoothly. However, the significant increase in the absolute dipole moment values of (2)2Σ+ (Fig. 1), (2)4Π (Fig. 2) of the molecule BeF and the (2)2Σ+ (Fig. 7) of the molecule MgF points out a polarized state where the fluorine atom strongly attract the bonding electrons.

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177

-298.8

-298.9

(2) 4Σ(5) 2Σ+

-299

(3) 2Δ

(4) 2Σ+

Δ

(1) 2

(3) 2Σ+ -299.1

(2) 2Σ+ -299.2

(X)2Σ+ -299.3 1.2

1.7

2.2

2.7

3.2

3.7

4.2

1.2

1.7

2.2

2.7

3.2

3.7

4.2

1

0

-1

(4) 2Σ+ -2

(X)2Σ+

(3) 2Σ+ (5) 2Σ+

-3

(2) 2Σ+

-4

-5 Fig. 7. Potential energy and dipole moment curves of the 2Σ± and 2Δ electronic states of the molecule MgF.

4. Spectroscopic Constants The spectroscopic constants for the doublet and quartet electronic states of the considered diatomic molecules BeF, MgF and CaF are investigated by fitting the calculated potential energy values of the different investigated electronic states into a polynomial in terms of R around the internuclear distance at equilibrium R e. These constants are the harmonic vibrational frequencies ωe and ωexe, the relative energy separations Te, the equilibrium bond distances Re, and the rotational constants Be and αe. For the studied electronic states of the BeF molecule and from the Table 2 one can notice a large discrepancy between the

investigated values of the spectroscopic constants in literature either theoretically or experimentally particularly for the excited electronic states. In order to have more reliability of our work, the investigated data of this molecule has been done by using MRCI method with five and seven valence electrons and the RSPT2-RS2 technique using five valence electrons and larger basis set au-cc-pcv5z (Table 2). The comparison of our calculated values of Te, using five valence electrons MRCI method, with those given in literature either theoretical or experimental, for the electronic states (1)2Π, (2–5)2Σ+ and (1)2Δ showed an overall good agreement. An exception can be noticed by comparing with those calculated by using time-dependent density-functional theory with both

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-298.7 -298.75 -298.8 -298.85

(4) 4 Π

-298.9

(3) 4 Π (2) 4 Π

-298.95 -299

(3) 2 Π

(1) 4 Π

-299.05

(2) 2 Π

-299.1 -299.15

(1) 2 Π

-299.2

1.2

1.7

2.2

2.7

3.2

3.7

4.2

3.2

3.7

4.2

1.5

(1) 4 Π 1

(3) 2 Π

(2) 4 Π

0.5 0 1.2

-0.5

1.7

2.2

2.7

(4) 4 Π (3) 4 Π

-1 -1.5 -2

(1) 2 Π

(2) 2 Π

-2.5 -3 Fig. 8. Potential energy and dipole moment curves of the 2,4Π electronic states of the molecule MgF.

LSDxc/TDLSDxc and LB94xc/TDLSDxc functional [78] where the agreement deteriorates (Table 2). Our calculated value of Re are in very good agreement with those given in literature except that given in Ref. [79] for the second minimum of the electronic state (2) 2Σ+ where the relative difference is ΔRe/Re = 15.2. The very good agreement is obtained by comparing our calculated values of ωe for the ground and the first excited (1)2Π states with those given by Zun-Lue et al. [80] and Tai and Verma [81] respectively. This agreement is acceptable by comparing with the other investigated data in literature either theoretical or experimental. Similar results are obtained by comparing our calculated values of ωexe, B e and αe for the electronic states (1) 2 Π, (2, 5) 2 Σ+ and (1) 2 Δ with those given in literature. The good agreement is noticed

except for some values of Be, ωexe, and αe given in Refs. [79–81] for the second minimum of (2)2Σ+ and ground states respectively. The complete comparison of our calculated spectroscopic constants using MRCI + Q with five and seven valence electrons and RSPT2RS2 technique using five valence electrons with the investigated data in literature are summarized in supplementary material Table S1 [82]. By comparing our investigated data using the aucc-pcv5z basis set with those given in literature [97] we obtained, for the investigated electronic states, a very good agreement with the relative differences 0.6% ≤ ΔR e/Re ≤ 1.3 and 1.88% ≤ ΔTe / T e ≤ 4.84. For the second minimum of the (2)2 ∑ + electronic state the relative difference on the internuclear distance becomes larger where ΔRe/Re = 7.8%.

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179

-298.79

E(hartree)

(3)4Σ(2)4Δ

-298.84

(4)4Σ+

(3)4Σ+ -298.89

(2)4Σ-

-298.94

(2)4Σ+ -298.99

(1)4Δ (1)4Σ+

-299.04 1.4

1.9

(1)4Σ-

2.4

2.9

3.4

3.9

2.9

3.4

3.9

R(Å)

4.4

2

μ(a.u) 1.5

(1)4Δ (1)4Σ-

1

(1)4Σ+ (2)4Σ+

(2)4Σ0.5

(3)4Σ0 1.4 (4)4Σ+

-0.5

1.9

(2)4Δ

2.4

R(Å)

4.4

(3)4Σ+

-1 Fig. 9. Potential energy and dipole moment curves of the 4Σ± and 4Δ electronic states of the molecule MgF.

The calculated spectroscopic constants ωe, Re, Be, and Te of the MgF molecule are given in Table 4 along with the available data, either theoretical or experimental, in literature. One can notice the absence of these constants for some potential energy curves because of the crossing or avoid crossing near the minima of these curves. The comparison of our calculated values of Te with those given in literature for four electronic states shows a very good agreement with the relative difference 0.5%((3)2Σ+[57]) ≤ ΔTe/Te ≤ 1.6%((1)2Π[52, 53]). A larger relative differences (ΔTe/Te = 10.7%) is obtained by comparing our calculated value of T e, for the electronic state (3) 2Σ +, with that given by Barrow and Beale [52]. The PEC of (2)2Σ+ state exhibits two avoided crossings where the same behavior is observed by Kang et al. [57]; the second minimum occur with

(X)2Σ+ state nearly at Re = 3.669 Å and Te = 39.702.4 cm− 1 with Be = 0.123 cm− 1. These calculated values show a very good agreement with the theoretical data given by Kang et al. [57]. The first minimum of the (3) 2Σ + state is close to R e = 2.139 Å and T e = 38.425.2 cm− 1. The comparison of our calculated spectroscopic constants Te, Re, ωe, Be and αe, for this minimum, with the experimental data [52] demonstrate a good agreement with the relative differences 5.1%, 2.1%, 7.2%, 5.1%, and 8.9% respectively. While the comparison of our calculated values of these constants with the theoretical data [57,77] shows a very good agreement except the value of ωe where the relative differences are 15.8% and 17.9%. (Table 3). The comparison of our calculated values of Re for the electronic states X2 Σ+ , (1) 2Π, and (3) 2 Σ+ with the theoretical data

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X2 Σ+ −(2)2 Σ+ (CaF)

2

(1) 2 Π−(2) 2 Π (BeF) 1

R(Å)

0

μ(a.u)

0.9

1.4

1.9

2.4

2.9

3.4

-1

X 2Σ+-(2) 2Σ+ (MgF)

-2

(1) 2 Π−(2)2 Π (CaF)

-3

-4

-5

Fig. 10. Relative position of the lowest-lying molecular states for the BeF, CaF, MgF, SrF, and BaF molecules.

given in literature [18,77,57], shows an excellent agreement with the relative difference 1.0%(X2Σ+,[57]) ≤ ΔRe/Re ≤ 1.9%(X2Σ+[18]). Similar agreement is obtained by comparing our calculated values with the experimental data [52,53,55] for the two states X2 Σ +, (1)2Π where the relative difference ΔRe/Re ≤ 1.8% while this agreement becomes larger for the electronic state (3)2Σ+ with ΔRe/Re = 20.5%. From Table 3, one can notice the discrepancies between the values of ω e given in literature for the three electronic states X2Σ+, (1)2Π, and (3)2Σ+. By comparing the theoretical data in literature with our calculated value, one can find 3.0%(X2Σ+[77]) ≤ ΔRe/ Re ≤ 20.7%(X2Σ+[57]), while the comparison with the experimental data gives 0.3%((3)2Σ+[52]) ≤ ΔRe/Re ≤ 10.5%(X2Σ+[55]). Our calculated values of ωexe are in good agreement with those given in literature either theoretical or experimental. No comparison for our calculated values of α e and the spectroscopic constants for the other electronic states of Table 3 since they are given here for the first time. The ground state for CaF molecule obtained in the present work is a sigma Σ+ state which is confirmed as compared to theoretical [11, 13–14,18,39], and experimental [8,10,79,83–84,86] results. The presently predicted relative order of the five lowest excited electronic states (1) 2 Π, (2) 2 Σ+, (1)2 Δ, (3)2 Σ + and (2)2 Π is in agreement

with the theoretical [9,13,17–18,39] and experimental [6–8,10,12, 79,83–85] data. This agreement deteriorates for the two electronic states (3) 2Σ + and (2) 2Π [12,84]. The presently MRCI calculated values of Te is in excellent agreement with the experimental data given in literature for the first excited electronic state (1)2Π with a relative difference 0.56%[86] ≤ (ΔTe/Te) ≤ 0.78%[84], while this relative difference becomes 1.43%[18] ≤ (ΔTe/Te) ≤ 8.57%[17] by comparing with the theoretical data in literature. For the second excited state (2)2Σ+ our calculated values of Te are in good agreement with either the theoretical or the experimental values given in literature with the relative difference 1.28%[13] ≤ (ΔTe/Te) ≤ 7.93%[7]. For the electronic states (1)2Δ and (3)2Σ+ one can notice (Table 4) a large discrepancy between the theoretical investigated data in literature. By comparing these values to our calculated Te, we can find a very good agreement with the theoretical values given by Bundgen et al. [13] and Yang et al. [39] with relative difference 1.0%[39] ≤ (ΔTe / Te) ≤ 1.43%[13]. This relative difference becomes larger by comparing with the values of Yang et al. [39] calculated by other different techniques. Therefore, the different theoretical methods used for the calculation of T e have large influence on the investigated values for these 2 electronic states. Similar results have been obtained by comparing our calculated values of Te for the electronic state (2)2Π with

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196 Table 1 Positions of the crossings Rc, the avoided crossings Rac, and the energy difference ΔE between the two corresponding electronic states of the molecules BeF, MgF, CaF. Crossing

Avoided crossings Rc (Å)

State 1/State 2

Rac (Å)

ΔE × 103 (hartree)

(1)2 Δ/(2)2 Δ A2Π/(2)2Π (2)2Π/(3)2Π

(2)2 Δ/(5)2 ∑+

1.73 1.63 1.49 1.59 1.46

(2)2 Δ/(1)2 ∑− (3)2 Δ/(2)2 ∑−

1.53 1.74

(4)2Π/(5)2Π (2)2 Δ/(3)2 Δ

1.52 1.74 1.35 2.28 1.73 2.78 1.78 1.73

3.20 19.91 3.51 5.34 1.31 4.44 27.11 3.49

2.64 4.54 7.30 2.07 2.50 3.33

(3)2 ∑+/(4)2 ∑+ (4)2 ∑+/(5)2 ∑+

State 1/State 2 BeF (1)2 Δ/(2)2 ∑+ (1)2 Δ/(3)2 ∑+ (2)2 Δ/(4)2 ∑+

CaF (1)2Δ/(3)2 ∑+ (1)2Δ/(4)2 ∑+ (3)4Δ/(5)4 ∑+ (1)4Φ/(3)4Π (3)4Δ/(2)4 ∑− MgF (3)2Σ+/(1)2Δ (4)2Σ+/(1)2Δ

(3)2Π/(4)2Π

2.18 2.00 2.76

(5)2 ∑+/(6)2 ∑+ (4)4 ∑+/(5)4 ∑+

2.15 2.45

(X)2Σ+/(2)2Σ+ (2)2Σ+/(3)2Σ+ (3)2Σ+/(4)2Σ+

3.67 2.14 1.78 2.74 2.08 2.92 2.32

2.17 4.82

5.02 2.05

(4)2Σ+/(5)2Σ+ (1)2Π/(2)2Π

4.84 2.53 1.58 2.82 1.85 9.53 1.48

those given in literature. Our MRCI calculated values of Re are in very good agreement either with the theoretical or the experimental data given in literature (Table 4). The value of the harmonic frequency ωe is an indicator for the geometrical shape of the potential energy curves. Again, here we notice the influence of the theoretical technique of calculation on the investigated value of ωe. The relative difference between our calculated values of ωe and those given by Yang et al. [14] varies between 0.04% (BD(T) method with BS2 basis set) and 14.2% (B3LYP method with BS2 basis set) for the ground state and the excited states (1)2Π, (2)2Σ+, (1)2Δ, and (3)2Σ+. This relative difference becomes Δωe/ωe = 19.4% for the (2)2Σ+ state calculated by Yang et al. [39] using MRCI method with cc-pvqz basis set. By comparing our calculated values of ωe with those obtained experimentally, one can find a relative difference varies between 1.08% [84] for the state (2) 2 Π and 12.52% [6] for the state (1) 2 Δ. Our MRCI calculated values of Be are in good agreement with the most calculated values in literature using the different techniques for the electronic states X2 Σ +, (1) 2 Π, (2)2 Σ +, (1) 2 Δ, and (3)2 Σ+ . The comparison with the experimental data of Be for these states shows an overall good agreement with the relative difference 4.01% [6] ≤ (ΔTe/Te) ≤ 11.11% [12]. In Tables 2, 3, and 4 one can find that the investigation of 7, 10, and 17 new electronic states for the molecules BeF, MgF and CaF respectively. Since our potential energy curves are adiabatic within the Born-Oppenheimer approximation, we can notice for these electronic states that, for the 2Σ±, 2Π and 2Δ there are many avoided crossings for the three studied molecules. While for the new quartet investigated electronic states the bounding between the alkaline earth fluoride molecules are weak where most of the potential energy curves are either unbound or shallow. By using the au-cc-pcv5z basis set with f atomic orbital, we noticed that, the new calculated values of Te for the different electronic states of the molecule CaF are shifted down with respect to those

181

calculated by using s, p, d basis set. By comparing the new calculated values of Te and ωe, for the different electronic states, with those obtained experimentally (Refs. [6,7,95]) we notice an improvement in our results in order to become in very good agreement with the experimental data. Since the transition dipole moment (TDMs) is very useful in designing the photoassociation experiments and to predict transitions, we present in Fig. 10 (Table 8) the transition dipole moment curves in terms of the internuclear distance R between the electronic states 2Σ+ − 2Σ+ and 2Π − 2Π for the three considered molecules. The TDMs curves show a sudden changes and jumps at different internuclear distances, which can be explained by the radiationless electronic transitions (no change in spin). These transitions are most probable when two potential energy curves cross or come very close to one another, and they happen without an appreciable alteration of energy and position of the nuclei. Then, electronic motion suddenly switches from one electronic state to another, and the electronic wavefunctions of these states are mixed. Due to the spin forbidden transitions between two atomic orbitals at asymptotic limits the TDMs curve vanish when R ~∞. The transition energies T e of the different excited electronic states of the five alkaline-earth mono-fluoride are plotted, as spotlight, with respect to the ground state in Fig. 11. The ground state is X2Σ+ and the first excited electronics state is (1)2Π for each of the considered five molecules. Starting from the second excited electronic state an inversion of states is noticed between these different molecules while other states maintain the same ordering. It is also noticed from this figure that the transition energy of some excited states with respect to the ground state gets lower from BeF to BaF. Such inversion of states and decreasing trend in energy gap as we go down group II, are direct consequence of the impact of atomic electronegativity (χ(Be) N χ(Mg) N χ(Cs) N χ(Sr) N χ(Ba)) on the molecular orbitals which they form. In fact, it is known that atomic orbitals of atoms that have a higher electronegativity are lower in energy with respect to those of lower electronegativity [87]. The transition from one electronic state to another being originated from the transfer of an electron from one molecular orbital to another. Thus, the transition energy Te between 2 electronic states of a molecule of lower electronegativity alkaline earth is less than that of the transition energy Te between the same electronic states of a molecule with higher electronegativity alkaline earth. By comparing the energy levels of the considered molecules, one can find the absence of some electronic states from the energy level diagram which is due to the avoided crossing at the of the internuclear distance Re or the presence of undulations in the PECs of higher excited states of the five molecules. In order to examine the theoretical trend of the spectroscopic constants of the five molecules XF (X: alkaline-earth molecules) we give in the supplementary material82 the different values of these constants for the investigated electronic states. According to their definitions and mathematical expressions [88], most of the values of the spectroscopic constant follow the right trend (as given in the Table S2). One can notice the exception of some cases as: the values of ωe for the ground and (1)2Π states of BaF molecule [66] (using the VDZ Valence Double Zeta 3-21G basis set through the CASSCF/MRCI method with five valence electrons) and the shallow either doublet or quartet electronic states. We remark also that most of the quartet electronic states of the CaF molecule are shallower than those the BaF molecule. 5. The Vibration Rotation Calculation The eigenvalue Ev, the rotational constant Bv, the centrifugal distortion constant Dv, and the abscissas of the turning point Rmin and Rmax have been calculated for the alkaline-earth molecules By using the

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Table 2 Spectroscopic constant of the lowest electronic states of the molecule BeF. States

Te (cm−1)

ΔTe/Te %

Re (Å)

× 2 ∑+

0.00

0.00

1.348a1 1.374a2 1.342a3 1.359a4 1.372b1 1.3614b2 1.361b3 1.35b4 1.369b5 1.3637b6 1.3711b7 1.369b8 1.37b9 1.3531b10 1.352b11 1.363b12 1.371c1 1.360c2 1.386d1 1.370d2

1.78 0.99 0.96 0.14 1.55 1.16 1.71 1.55 1.63 0.37 0.29 1.11 1.70 0.89 2.81 1.63

1.366f1 1.371g1

1.33 1.70

k

(1)2Π

1st min

(2)2 ∑ +

2nd min

(3)2Σ+

34,579.1a1 32,922.1a2 33,021.6a4

33,233.7c5

3.89

34,814.0h3 35,214.2j1 36,359.5j2 34,423.8j3 34,359.2j4 34,270.5j5 32,504.1 j6 32,343.9L1 33,542.8k 53,591.0a1 52,739.6a2 51,983.4a3 51,120.3a4 49,552.58e1 49,570.5h2 50,844.5h3 38,472.67j1 41,263.3 j2 51,264.6j3 51,208.1j4 51,175.9j5 42,916.8j6 50,182.8k 64,569.42a1 51,556 h3 54,160.2a4 52,117.8k 54,986.9a1 54,592.2a2 53,265.6a3 53,120.2a4 50,364.9e1 50,364.0h2 53,595.0h3 39,755.1j1 48,909.5j2 53,232.6j3 53,232.64j4

0.67 1.83 5.14 4.49 0.63 0.89 6.00 6.46 1.88

7.53 7.50 5.12 28.21 23.00 4.34 4.44 4.50 19.91 4.84 20.15h3

8.40 8.40 2.53 27.70 11.05 3.19 3.19

ΔRe/Re %

1.364 1.366a1 1.414a2 1.401a4 1.395c3 1.394c4 1.393c5 1.437d1 1.432d2

0.72

1.387h3

1.53

1.397L1 1.399k 1.313a1 1.352a2 1.304a3 1.340a4

2.26 1.0

1.335h2 1.321h3

1.67 0.60

1.334k 3.011a1 2.552 h3 2.890a4 2.775k 1.297a1 1.330a2 1.294a3 1.310a4

1.3

1.325h2 1.317h3

2.23 1.62

ωe (cm−1)

Δωε/ωe %

Be (cm−1)

1357.2a1 1237.0a2 1381.3a3 1340.3a4 1236.12b1 1265.6b2 1247.36b3

8.92 6.74 8.09

1.516a1 1.459a2 1.531a3 1.4601a4 1.465b1 1.488b2 1.489b3

3.35 1.86 1.78

9.110b1 9.120b2 9.120b3

3.17 3.07 3.07

1258.0b5 1250.0b6 1265.7b7 1272.5b8 1240.0b9 1339.3b10 1280.0b11 1250.0b12 1272.8c1 1265.9c2 1245.0d1 1255.0d2 1265.54e1 1239.0f1

7.30 7.89 6.74 6.24 8.63 1.31 5.68 7.89 6.21 6.72 8.26 7.53 6.75 8.70

1.472b5

2.90

8.800b5

6.47

1.469b7 1.472b8

3.10 2.90

9.260b7 9.520b8

1.58 1.17

8.340b10

11.36

1267.0h1

6.64

1329.3a1 1165.1a2 2.12 2.05 2.01 5.19 4.83

1175.4c3 1172.6c4 1154.7c5 1116.0d1 1324.0d2 1171.0h1 1183.0h3

11.57 11.78 13.13 16.04 0.39 11.90 11.00

1174.2L1

11.66

1469.8a1 1266.6a2 1494.7a3

15.24

ΔBe/Be %

ωexe (cm−1)

Δωexe/ωexe %

9.409a1

αe (cm−1)

Δαe/αe %

0.01680a1 0.02000a2 0.01740a3

9.981a3

0.01750b1 0.01685b2 0.01760b3

4.16 0.29 4.76

0.01690b7 0.01695b8

0.59 0.89

1.469c1 1.491c2

3.10 1.64

8.820c1 9.350c2

6.25

0.01692c1 0.01700c2

0.71 1.19

1.489e1

1.75

9.422e1

0.13

0.01864e1

10.95

1.479i1 1.489i2

2.42 1.78

0.01685i1 0.01760i2

0.29 4.76

1.476a1 1.379a2 1.4120a4 1.412 c3 1.419 c4 1.420 c5

4.33 3.86 3.79

8.80 c3 8.78 c4 8.40 c5

1.433h3

2.91

13.50h3

1.413L1

4.26

8.78L1

0.01500a2

1.599a1 1.508a2 1.621a3

0.01713 c3 0.01610 c4 0.01750 c5

0.0170L1

12.892a1 11.887a3

1362.07e1 1351.0h2 150.0h3

7.32 8.08 2.25

1.5641e1 1.547h2 1.580h3

2.81 3.25 1.18

12.99e1 12.6h2 13.1h3

367.7a1 410.0h3

11.50

0.305a1 0.424h3

39.01

1.8h3

0.76 2.26 1.61

0.02106e1

7.8 1579.9a1 1391.3a2 1612.9a3 1419.72e1 1420.0h2 1402.0h3

1.639a1 1.557a2 1.646a3 9.58 9.57 10.71

1.5788e1 1.570h2 1.584h3

10.537a1

0.0163a1 0.0192a2 0.0164a3

10.572a3 3.67 4.20 3.35

11.08e1 9.9h2 11.5h3

5.15 6.04 9.13

0.01837e1

12.69

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

183

Table 2 (continued) Te (cm−1)

States

j5

(1)2Δ

(4)2 ∑ +

2

(5) ∑

+

(6)2 ∑ + (1)4Π

(2)4 ∑ + (2)4Δ (2)4 ∑− (3)4Π (4)4 ∑+

53,127.7 43,118.4j6 53295k 62,360.4a1 61,973.9a2 59,309.2a3 47,998.1j1 56,418.5j2 64,781.8a1 64,252.1a2 61,277.1a3 44,957.3j1 56,773.4j2 69,255.4a1 68,955.4a2 48,498.1j1 66,008.4j2 77,930.6a1 77,393.4a3 96,270.6a1 101,367.3a2 85,208.5a3 98,243.5a1 98,788.4a1 103,928.6a2 99,311.1 a1 104,422.1a2 105,159.9a2 88,383.6a3 115,055.6a3

ΔTe/Te % 3.38 21.58 2.37

Re (Å)

ΔRe/Re %

1.324k 1.292a1 1.327a2 1.290a3

0.6

ωe (cm−1)

Δωε/ωe %

Be (cm−1)

1612.a1 1412.8a2 1621.4a3

1.650a1 1.565a2 1.656a3

1.281a1 1.320a2 1.282a3

1665.9a1 1431.2a2 1666.5a3

1.678a1 1.582a2 1.677a3

1.314a1 1.350a2

1509.8a1 1335.4a2

1.595a1 1.512a2

1.524a1 1.468a3 1.925a1 1.941a2 1.959a3 1.712a1 1.705a1 1.757a2 1.700a1 1.769a2 1.750a2 1.722a3 1.708a3

2187.1a1 1836.a3 435.7a1 460.1a2 445.4a3 475.2a1 491.7a1 433.1a2 504.0a1 482.8a2 445.5a2 488.1a3 582.9a3

1.187a1 1.278a3 0.743a1 0.730a2 0.717a3 0.937a1 0.944a1 0.898a2 0.948a1 0.886a2 0.899a2 0.918a3 0.936a3

ΔBe/Be %

ωexe (cm−1)

Δωexe/ωexe %

αe (cm−1)

Δαe/αe %

23.03 9.52

30.6 12.36

29.97 4.68

For present work using MRCI calculation with 5 valence electrons, a2For present work using MRCI calculation with 7 valence electrons, a3For present work using CASPT2 calculation with 5 valence electrons. a4For present work using MRCI calculation with 7 valence electrons and the au-cc-pcv5z basis set. b1Ref. [80], b2(exp.)Ref. [80], b3(exp.)Ref. [80], b4Ref. [80], b5Ref. [80], b6 Ref. [80], b7Ref. [80], b8Ref. [80], b9Ref. [80], b10Ref. [80], b11(SCF)Ref. [80], b12(CI(SD))Ref. [80]. L1Ref. [92], c1Ref. [63]. c2(exp.)Ref. [63], c3Ref. [63], c4(exp.)Ref. [63], c5(exp.)Ref. [63]. d1(CI)Ref. [65], d2(variational calculations)Ref. [65]. e1(exp.)Ref. [81], f1Ref. [61], g1Ref. [93], h1(exp.)Ref. [79], h2(exp.)Ref. [79], h3Ref. [79], h4Ref. [79], i1(exp.)Ref. [62], i2(exp.)Ref. [62], j1(LSDxc/TDLSDxc)Ref. [78], j2(LB94xc/TDLSDxc) Ref. [78], j3(UCIS)Ref. [78], j4(ROCIS)Ref. [78], j5(XCIS)Ref. [78], j6(TDDFT)Ref. [78]. kRef. [97]. a1

Table 3 Spectroscopic constants for the lowest electronic states of MgF molecule. State 2 +

X Σ

2

(1) Π

(2)2Σ+ (ext) 2 +

(2) Σ (int)

(1)4Σ+ (1)4Δ (1)2Δ (3)2Σ+

(1)4Σ− (1)2Σ− (2)2Δ (2)2Σ− (3)4Π (3)4Σ+ (4)4Π a

Ref.

Te (cm−1)

MRCI + Q [a] Theo [b] Theo [c] Theo [d] Exp [e] Exp [f] MRCI + Q [a] Theo [b] Theo [d] Exp [e, g] MRCI + Q [a] Theo [d] MRCI + Q [a] Theo [c] Theo [d] Exp [e]

0 0

MRCI + Q [a] Theo [d] Exp [e]

0 0 0 28,254.3 27,674.0 27,834.1 27,816.1 38,425.2 38,430.5 39,702.4 37,679.7 56,921.2 57,210.8 57,263.5 47,710.2 47,968.9 42,589.6 57,320.7 57,337.7 72,976.7 73,082.7 88,866.1 90,881.7 91,174.3

ΔΤe/Te %

2.1 1.5 1.6 0.0

5.1

0.50 10.7

Re (Å) 1.778 1.745 1.75 1.761 1.75 1.746 1.767 1.735 1.747 1.747 3.482 3.585 1.755 1.698 1.737 1.719 2.92 3.35 3.717 2.138 1.699

ΔRe/Re % 1.9 1.6 1.0 1.6 1.8 1.8 1.1 1.1 3 3.2 1.0 2.1

20.5

3.939 4.196 3.469 3.726 2.527 2.98 2.299

Present work, bRef. [18], cRef. [90], dRef. [91], eRef. [52], fRef. [55], gRef. [53].

ωe (cm−1) 700.4 774.6 721.6 555 721.6 774.6 720.5 790.4 653.1 746 172.8 172.8 710.9 823.2 837.9 762.1 60.28 35.98 27.97 820.6 898.5 823.2 18.02 16.58 31.47 36.08 169.1 88.03 206.4

Δωe/ωe %

Be (cm−1)

ΔBe/Be %

0.499 10.6 3.0 20.7 2.9 10.5 9.7 9.3 3.5

15.8 17.9 7.2

9.4 0.3

0.519 0.508 0.519 0.519 0.506

4.0 0.9 4.0 4.0

0.524 0.521 0.130 0.123 0.513 0.551 0.532 0.539 0.186 0.140 0.114 0.345 0.335 0.551 0.1 0.089 0.132 0.113 0.247 0.177 0.299

3.6 3.0

αe × 103 (cm−1)

ωexe (cm−1)

Δωexe/ωex e %

4.7 4.7

4.37 3.81 4.97

4.37 3.81 4.97

4.7

4.94

4.94

4.8

4.6 4

4.6 4

4.4

4.4

5.6 4.4

4.9 5.04

4.9 5.04

5.1

5.6 2.17 4.34 1.47 5.88 5.04

5.6

4.54 0.52 1.94 1.52 1.29 10.3 5.46

4.54 0.52 1.94 1.52 1.29 10.3 5.46

5.4 7.4 3.7 5.1

2.9 59.7

1.2 4.9 13.8 2.2 3.5 4.1 11.0 9.0

5.88 5.04

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Table 4 Spectroscopic constants for the lowest electronic states of CaF molecule. State

Te (cm−1)

X2 Σ +

0.0

ΔTe/Te %

Re (a.u) 3.808a1 3.808a2 3.812b1 b2

(1)2Π

(2)2Σ+ (1st min)

16,655.2a1 16,573.5a2 16,543.5a3 16,421.0c 15,701.4d1 15,607.5d2 15,607.5d3 15,626.6d4 17,712.0e1 18,217.0f 17,978.0g 16,526.0k 16,530.0m 16,526.8o 16,526.8p 16,526.8q 16,529.1r 16,562.3t 20,326.5a1 21,015.9a2 19,012.9a3 19,646.1d1 19,564.6d2 19,536.3d3 19,512.2d4 20,069.0e1 21,486.0f 21,463.0g

1.43 6.07 6.71 6.71 6.58 5.97 8.57 7.36 0.78 0.76 0.78 0.78 0.78 0.76 0.56

ΔRe/Re %

0.10

3.734 3.740b3 3.748b4 3.693b5 3.696b6 3.834b7 3.810b8 3.873b9 3.738b10 3.730b11 3.793b12 3.742b13 3.735b14 3.797b15 3.763b16 3.751b17 3.767b18 3.703b19 3.699b20 3.738b21 3.706b22 3.682b23 3.712b24 3.725c 3.784d1 3.768d2 3.772d3 3.780d4 3.733e1 3.843e2 3.740h1 3.713h2 3.716h3 3.740i1 3.713i2 3.689j 3.717k 3.688o 3.688p 3.688q 3.717t 3.782a1 3.794a2

1.98 1.82 1.60 3.11 3.03 0.68 0.05 1.68 1.87 2.09 0.40 1.76 1.95 0.29 1.20 1.52 1.09 2.84 2.95 1.87 2.75 3.42 2.59 2.23 0.63 1.06 0.95 0.74 2.01 0.91 1.82 2.56 2.48 1.82 2.56 3.23 2.45 3.25 3.25 3.25 2.45

3.692c 3.744d1 3.733d2 3.737d3 3.744d4 3.699e1

524.3a1 518.6a2 568.8a3 567.6b1 563.4b2 571.2b3 582.7b4 577.3b5 583.8b6 547.8b7 569.1b8 543.3b9 560.1b10 571.4b11 532.3b12 571.3b13 581.5b14 531.7b15 580.2b16 574.8b17 602.9b18 581.3b19 573.1b20 610.8b21 586.5b22 579.2b23 615.1b24 612.5c 586.8d1 580.0d2 576.4d3 572.4d4 581.2e1 559.0e2 591.0h1 587.0h2 583.0h3 591.0i1 587.0i2 588.6j 581.1k 588.6o

Δωe/ωe %

Be (cm−1)

ΔBe/Be %

0.322a1 0.322a2 7.63 6.94 8.21 10.02 9.18 10.19 4.29 7.87 3.50 6.39 8.24 1.50 8.23 9.84 1.39 9.63 8.79 13.03 9.81 8.52 14.16 10.60 9.48 14.76 14.40 10.65 9.60 9.04 8.40 9.79 6.21 11.29 10.68 10.07 11.29 10.68 10.92 9.77 10.92

0.318b7 0.322b8 0.311b9 0.334b10 0.336b11 0.325b12 0.334b13 0.335b14 0.324b15 0.330b16 0.332b17 0.329b18 0.341b19 0.341b20 0.334b21 0.340b22 0.345b23 0.340b24

1.26 0.00 3.54 3.59 4.17 0.92 3.59 3.88 0.62 2.42 3.01 2.13 5.57

5.57 3.59 5.29 6.67 5.29

0.327d1 0.329d2 0.329d3 0.327d4 0.335e1

1.53 2.13 2.13 1.53 3.88

0.344j 0.344k 0.344o

6.39 6.39 6.39

0.344q 0.338t

6.39

10.92 9.77

2.44 1.01 1.31 1.20 1.01 2.24

588.6q 581.1t 531.9a1 511.4a2 563.4a3 624.0c 586.7d1 587.9d2 584.2d3 578.6d4 579.9e1

14.76 9.34 9.53 8.95 8.07 8.28

0.334d1 0.335d2 0.335d3

2.40 2.69 2.69

0.334d4 0.342e1

2.40 4.68

3.689k

2.52

586.8k

9.36

0.343k

4.96

3.661o 3.661p 3.661q

3.31 3.31 3.31

594.5o

0.349o

594.5q

10.53 13.98 10.53

0.342q

6.59 7.16 4.68

3.689t

2.52

586.8t

9.36

0.343t

4.96

0.74 2.41 1.12 0.88 1.63

522.7a1 472.5a2 512.6a3 586.2d1 577.8d2 574.5d3 571.4d4 551.5e1

3.797a1 3.860a2 3.46 3.89 4.04 4.17 1.28 5.40 5.30

ωe (cm−1)

3.769d1 3.752d2 3.755d3 3.764d4 3.736e1

4.73 0.326a1 0.324a2

0.324a1 0.313a2 10.83 19.39 9.02 8.52 5.22

0.329d1 0.332d2 0.331d3 0.330d4 0.335e1

1.52 2.41 2.11 1.82 3.28

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

185

Table 4 (continued) State

Te (cm−1) j

(2nd min)

(1)2Δ

18,841.0 18,850.0m 18,840.2o 18,840.2q 18,833.1r 18,844.5t 30,018.5a1 27,662.1a2 25,207.3a1 25,337.3a2 22,112.6a3 21,767.7d1 21,189.2d2 20,940.1d3 20,696.9d4 24,851.0e1 22,552.0f 21,530.8n 21,573.9r 21,530.8s

ΔTe/Te % 7.88 7.83 7.89 7.89 7.93 7.86

Re (a.u)

ΔRe/Re %

Be (cm−1)

ΔBe/Be %

8.68

0.343j

5.54

2.76 2.76

572.4o 572.4q

8.68 8.68

0.342o 0.349q

5.26 7.16

7.67

3.831d1 3.818d2 3.822d3 3.830d4 3.776e1

1.12 1.47 1.36 1.15 2.60

566.1t 185.9a1 184.6a2 480.4a1 462.4a2 498.2a3 523.2d1 519.5d2 513.8d3 506.1d4 558.9e1

8.18 7.53 6.50 5.08 14.05

0.319d1 0.321d2 0.320d3 0.319d4 0.328e1

2.51 3.12 2.81 2.51 5.18

3.766r

2.87 3.93

528.6r

9.12 12.52

0.329r 0.324s

5.47 4.01

3.695o 3.695q

j

Δωe/ωe %

572.4

6.307a1 6.154a2 3.874a1 3.914a2 15.80 18.96 20.38 21.79 1.43 11.77 17.08 16.84 17.08

ωe (cm−1)

2.76

3.695

j

0.118a1 0.123a2 0.311a1 0.305a2

2 +

(3) Σ (1st min)

(2)2Π

34,277.7a1 33,868.0a2 31,850.0a3 33,937.1d1 31,105.9d2 30,349.4d3 29,837.6d4 32,741.0e1 30,772.0k 30,158.6r 30,771.9t 35,035.1a1 36,085.7a2 32,060.0a2 34,580.6d1 33,841.0d2 33,310.5d3 32,693.6d4 32,765.0e1 32,138.0f 32,644.0g 30,216.0l 30,270.0m 30,216.0r 30,255.1t 30,284.4t

(4)2Σ+ (2nd min) (3rd min) (1)4Σ+ (1)4Π (5)2Σ+ (1st min) (2nd min) (3)2Π

(6)2Σ+ (1st min) (2nd min) (1)4Δ (2)4Π (2)2Δ (2)4Σ+ (3)4Σ+

a1

3.685a1 3.707a2 1.00 10.20 12.94 14.88 4.69 11.39 13.66 11.39

3.717d1 3.673d2 3.672d3 3.679d4 3.652e1

0.86 0.33 0.35 0.16 0.90

4.024a1 4.131a2 1.31 3.53 5.18 7.16 6.93 9.01 7.32

3.907d1 3.879d2 3.869d3 3.860d4 3.804e1

2.99 3.74 4.01 4.25 5.78

15.95 15.74 15.95 15.80 15.69

3.800l

5.89

573.4a1 599.5a2 589.2a3 634.5d1 635.9d2 636.7d3 638.2d4 595.6e1 650.7k

0.344a1 0.340a2 9.63 9.83 9.94 10.15 3.73 11.88

650.7t 486.9a1 403.9a2

11.88

491.4d1 487.1d2 483.2d3 477.8d4 518.5e1

0.92 0.04 0.77 1.90 6.09

481.7k

1.08

481.7t

0.338d1 0.347d2 0.347d3 0.346d4 0.350e1

1.78 0.59 2.02 0.58 1.71

0.288a1 0.273a2 0.305d1 0.310d2 0.312d3 0.313d4 0.323e1

5.57 7.10 7.69 7.99 10.84

0.324l

11.11

1.08

48,535.1 45,695.9a2 50,476.6a1 47,018.5a2 40,380.4a1 36,420.8a2 40,557.3a1

a1

6.339 6.099a2 11.405a1 11.296a2 11.618a1 11.913a2 13.200a1

a1

131.9 118.0a2 30.7a1 31.0a2 15.6a1 15.1a2 3.6a1

0.117a1 0.125a2 0.036a1 0.037a2 0.035a1 0.032a2 0.027a1

43,547.1a2 51,339.9a1 47,026.6a1 44,180.1a2 42,116.2a3

3.903a2 7.913a1 5.085a1 4.837a2

464.2a2 106.0a1 217.3a1 245.2a2

0.306a2 0.075a1 0.181a1 0.199a2

49,468.3a2 49,470.4a2 51,239.5a1 50,512.6a3 51,406.7a1 51,575.7a1 50,112.7a3 51,647.1a1 47,168.7a2 60,370.2a1 57,881.1a2

4.495a2 10.763a2 11.171a1

454.9a2 69.5a2 10.1a1

0.231a2 0.040a2 0.036a1

10.436a1 9.751a1

20.4a1 19.7a1

0.042a1 0.048a1

11.020a1 12.324a2 7.681a1 7.354a2

17.9a1 17.5a2 40.5a1 67.7a2

0.038a1 0.032a2 0.079a1 0.086a2 (continued on next page)

186

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

Table 4 (continued) State 4

(3) Π (1)4Σ− (2)4Δ (4)4Π (4)4Σ+ (3)4Δ (2)4Σ− (3)4Σ−

Te (cm−1) a1

64,383.5 64,606.8a1 65,557.4a1 63,778.8a3 65,703.6a1 65,765.0a1 69,686.3a1 69,929.1a1 78,128.5a1

ΔTe/Te %

Re (a.u)

ΔRe/Re %

ωe (cm−1)

Δωe/ωe %

Be (cm−1)

a1

11.407 12.708a1 12.247a1

a1

15.7 5.2a1 13.7a1

0.035a1 0.029a1 0.032a1

11.994a1 11.850a1 13.007a1 11.232a1 7.944a1

15.8a1 13.4a1 13.5a1 10.2a1 33.9a1

0.032a1 0.032a1 0.027a1 0.035a1 0.074a1

ΔBe/Be %

Present values calculated using MRCI method. a2 Present values calculated using RSPT2 method. a3For present work using MRCI calculation with 7 valence electrons, f orbital and the aucc-pcv5z basis set. b1Ref. [14] (ab initio values calculated using the BD(T) method with BS1 basis set). b2Ref. [14] (ab initio values calculated using the BD(T) method with BS2 basis set). b3 Ref. [14] (ab initio values calculated using the BD(T) method with BS3 basis set). b4Ref. [14] (ab initio values calculated using the B3LYP method with BS1 basis set). b5Ref. [14] (ab initio values calculated using the B3LYP method with BS2 basis set). b6Ref. [14] (ab initio values calculated using the B3LYP method with BS3 basis set). b7Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the BD(T) method with BS1 basis set). b8Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the BD(T) method with BS1 basis set). b9Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the BD(T) method with BS1 basis set). b10Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the BD(T) method with BS2 basis set). b11Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the BD(T) method with BS2 basis set). b12Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the BD(T) method with BS2 basis set). b13Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the BD(T) method with BS3 basis set). b14 Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the BD(T) method with BS3 basis set). b15Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the BD(T) method with BS3 basis set). b16Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the B3LYP method with BS1 basis set). b17Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the B3LYP method with BS1 basis set). b18Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the B3LYP method with BS1 basis set). b19Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the B3LYP method with BS2 basis set). b20Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the B3LYP method with BS2 basis set). b21Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the B3LYP method with BS2 basis set). b22Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the B3LYP method with BS3 basis set). b23Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the B3LYP method with BS3 basis set). b24Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the B3LYP method with BS3 basis set). cRef. [18] (ab initio values calculated using the CASSCF/MRCI method). d1Ref. [39] (ab initio values calculated using the MRCI method with cc-pvtz basis set). d2Ref. [39] (ab initio values calculated using the MRCI method with cc-pvqz basis set). d3Ref. [39] (ab initio values calculated using the MRCI method with cc-pv5z basis set). d4Ref. [39] (ab initio values calculated using the MRCI method with complete basis set (CBS)). e1Ref. [13] (ab initio values calculated using the MRD-CI method). e2Ref. [13] (ab initio values calculated using the SCF method). fRef. [17] (Theoretical values calculated using a ligand field approach without polarization effects). gRef. [9] (RMF values without polarization effects). h1Ref. [94] (Theoretical values calculated using the SCF method). h2 Ref. [94] (Theoretical values calculated using SDCI method). h3Ref. [94] (Theoretical values calculated using the CPF method). i1Ref. [11] (Theoretical values calculated using the SCF method). i2Ref. [11] (Theoretical values calculated using the CISD method). jRef. [83] (Experimental values). kRef. [84] (Experimental values). lRef. [12] (Experimental values). mRef. [85] (Experimental values). nRef. [95] (Experimental values). oRef. [8] (Experimental values). pRef. [79] (Experimental values). qRef. [10] (Experimental values). rRef. [7] (Experimental values). sRef. [6] (Experimental values). tRef. [86] (Experimental values).

a1

canonical functions approach [67–69] and the cubic spline interpolation between each two consecutive points of the potential energy curves obtained from the ab initio calculations. The calculations have been done for a certain number of states for each molecule as example (the calculation for other electronic states can be done upon request from the authors). For the molecule BeF the vibrational level has been obtained up to v = 59 for the ground state (Table 5). The comparison of our calculated values, for the ground state, of the rotational constant Be with those given in literature[81,88] shows a very good agreement with relative differences 1.8% for v = 0 and 2% for v = 1 and 2; this relative difference becomes lager for the value of centrifugal distortion constant D e. While the comparison, for these constants, with the theoretical data in literature [80] shows relative differences 2.78% ≤ ΔBe /B e ≤ 4.97% and 0.42% ≤ ΔD e/ De ≤ 28.6%. For the turning points, Rmin and Rmax and the eigenvalue Ev there are only theoretical data in literature. The comparison of our calculated values of Rmin and Rmax with those given in literature shows good agreements with the relative differences 1.61% ≤ ΔRmin/Rmin ≤ 2.10% and 1.76% ≤ ΔRmax/Rmax ≤ 8.5%. While the comparison of our calculated values of Ev with those given in literature [80] shows an increasing relative difference from 6.4% for v = 0 until 20.0% for v = 59 which is explained by the use of different theoretical techniques of calculation. The comparison of our rovibrational calculated data, for the molecule MgF, with the few available data in literature [52,90] for the ground state up to v = 7 shows very good agreements with the relative differences ΔEv/Ev ≤ 2.7%, ΔBv /B v ≤ 3.7%, ΔDv /D v ≤ 6.0%, ΔR min /

Rmin ≤ 1.8% and ΔRmax/Rmax ≤ 1.7%. By comparing our calculated value of the vibrational constants for the molecule CaF with those given experimentally in literature [39], one can find that an overall acceptable agreement for the values of Ev with the relative difference 6.0% ≤ ΔEv/Ev ≤ 10.1% for the ground and the first excited states. The agreement becomes good by comparing our calculated values with those obtained experimentally in references [13,17, 18] with the relative differences ΔB v/B v ≈ 6.2%, ΔDv/Dv ≈ 4.3%, ΔRmin/Rmin ≤ 2.54% and ΔRmax/Rmax ≤ 2.52%. The comparison with the theoretical rovibrational data in literature is related to the used theoretical technique of calculation, [39] by comparing our calculated values with this data one can find 1.7% ≤ ΔEv / Ev ≤ 15.0%, 0.05% ≤ ΔRmin /Rmin ≤ 2.31% and 2.63% ≤ ΔRmax / Rmax ≤ 7.9% (Tables 6 and 7). 6. Conclusion In the present work, ab-initio calculations have been performed to investigate the PECs, the rovibrational calculation, and DMCs of the doublet and quartet low lying electronic states of the XF (X: Be, Mg, Ca) molecules. The calculations have been done by using the RSPT2-RS2 method and the CASSCF-MRCI calculations with Davidson correction (+ Q). Twelve PECs and twenty three DMCs are represented for the first time. The spectroscopic constants R e, ωe , T e, Be and μ are calculated for 27, 26 and 27 highly excited electronic states for BeF, CaF and MgF molecules respectively. The comparison between our calculated constants and those available in literature, either theoretical or

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

187

E(cma ) 99000

(1) 4 Π

(2) 4 Δ (2) 4 Σ+ (2) 4 Σ− (3) 4 Σ+

(4) 4 Π

(3) 4 Π

79000

(3) 4 Σ−

(4) 2 Σ+ (2) 2 Δ (2) 2 Σ− (5) 2 Σ+ (ΑC) (3) 2 Π

59000

(2) 4 Δ

(4)

(3) 2 Π

(1) 2 Δ (3) 2 Σ+

(3) 4 Δ 4 Σ+

(3−4) 4 Π (1) 4 Σ−

(AC) (5) 2 Σ+

(1) 2 Δ

(2) 4 Σ−

(1) 4 Σ+

(1) 4 Δ (2) 4 Σ+ (1) 4 Π (2) 2 Δ

(3) 4 Δ

(4) 4 Π (5) 4 Π (6) 4 Π (5) 4 Δ (1) 4 Γ (2) 4 Φ

(2) 2 Σ+ AC (2)2 Π

(4) 2 Σ+ (3) 2 Π

(3) 2 Σ+ (5) 2 Σ+ (2) 2 Σ+

39000

(1) 4 Σ+

AC (2) 2 Π

(AC) (4) 2 Σ+

(3) 2 Π (1) 4 Σ+

(2) 2 Π (3) 2 Σ+

(1) 2 Π

(3) 2 Σ+

(AC) (3) 2 Π (3) 2 Σ+

(2) 2 Π (1) 2 Π

(1) 2 Δ

(2) 2 Π (1) 2 Δ

(2) 2 Σ+ 19000

(6) 2 Π (5) 2 Σ+ (1) 4 Σ+ (2) 2 Δ (4) 2 Δ (1) 4 Π (2) 4 Π (3) 4 Π (2) 4 Σ+ (3) 4 Σ+ (4) 4 Σ+ (1) 4 Δ (2) 4 Δ (4) 4 Δ (1) 2 Σ− (2) 2 Σ− (3) 2 Σ− (1) 4 Σ− (2) 4 Σ− (3) 4 Σ− (1) 4 Φ (1) 2 Φ (2) 2 Φ

(2) 2 Σ+ (1) 2 Π

(1)2 Π

(2) 2 Σ+

(1) 2 Δ

(1) 2 Π

X2Σ+ -1000

0

BeF

X2Σ+ 5

MgF

X2 Σ +

X2Σ+ 10

CaF

15

SrF

X2Σ+ 20

BaF

25

AC: Avoided crossing Fig. 11. The relative energy separation Te for the five molecules BeF,MgF, CaF, SrF, and BaF.

experimental, shows an overall very good accordance. The double nature, ionic/covalent, of the ground state of alkaline-earth mono-fluoride molecules has been verified from the dipole moment curves. Upon comparison between the relative energy separations T e of the excited electronic states with respect to the ground state of the five XF (X: Be, Mg, Ca, Sr, Ba) molecules, similar behaviors are noticed for the first excited state and some relative shifts of the transition energies take place between higher and lower close lying states. By studying each of the spectroscopic constants for the five alkaline-earth monofluoride molecules, one can find that they are following the regular trend except those of some shallow PECs. Moreover, for the first time, fifty new excited electronic states have been investigated for the first time for the three studied molecules.

Acknowledgments The authors would like to acknowledge the use of the Ankabut High Power Computer Facility. Cluster and the Khalifa University High Power Computer shared facility, Abu-Dhabi UAE. In addition, the authors would like to thank the Lebanese National Council for Scientific Research for the financial support of the present work by the grant program for scientific research (no. 1-1-13). Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.saa.2017.01.035.

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Table 5 Values of the eigenvalues Ev, the abscissas of the turning points Rmin and Rmax the constants BV and Dv for the different vibrational levels of X2Σ+, (2)4Π, (2)4Σ+, and (2)4Δ states of the BeF molecule. v X2 Σ + 0

1

2

3

4

5

6

7

8

9

10

11 12 13 14 15 16 17 18 19 20

Ev (cm−1)

ΔEv/Ev %

677.68a1 634.075b 617.15m

6.43 8.93

2017.21a1 1888.092b1 1852.26m

6.40 8.17

3339.81a1 3124.45b1 3077.67m

6.44 7.85

4646.27a1 4343.333b1

Rmin (Å)

ΔRmin/Rmin %

Rmax (Å)

ΔRmax/Rmax %

1.2887a1 1.3102b1

1.668

1.4166a1 1.4423b1

1.814

1.2493a1 1.2696b1

1.624

1.4720a1 1.4998b1

1.888

1.2241a1 1.2438b1

1.609

1.5131a1 1.5427b1

1.956

6.52

1.2048a1 1.2240b1

1.593

1.5484a1 1.5798b1

2.027

5936.29a1 5544.919b1

6.59

1.1887a1 1.2077b1

1.598

1.5805a1 1.6135b1

2.087

7210.92a1 6729.378b1

6.67

1.1751a1 1.1937b1

1.582

1.6103a1 1.6450b1

2.154

8471.32a1 7896.876b1

6.78

1.1630a1 1.1815b1

1.590

1.6384a1 1.6751b1

2.239

9718.91a1 9047.568b1

6.90

1.1522a1 1.1705b1

1.588

1.6648a1 1.7039b1

2.348

10,953.45a1 10,181.605b1

7.04

1.1424a1 1.1606b1

1.593

1.6911a1 1.7319b1

2.412

12,172.14a1 11,299.129b1

7.17

1.1335a1 1.1516b1

1.596

1.7166a1 1.7592b1

2.481

13,375.42a1 12,400.279b1

7.29

1.1252a1 1.1432b1

1.599

1.7415a1 1.7860b1

2.555

14,571.40a1 13,485.183b1 15,767.68a1 14,553.965b1 16,965.78a1 15,606.742b1 18,154.73a1 16,643.623b1 19,307.75a1 17,664.713b1 20,426.31a1 18,670.109b1 21,545.11a1 19,659.902b1 22,651.45a1 20,634.177b1 23,729.60a1 21,593.013b1 24,804.17a1 22,536.484b1

7.45 7.69 8.01 8.32 8.50 8.59 8.75 8.90 9.00 9.14

1.1175a1 1.1355b1 1.1102a1 1.1283b1 1.1033a1 1.1216b1 1.0968a1 1.1153b1 1.0908a1 1.1094b1 1.0852a1 1.1037b1 1.0798a1 1.0984b1 1.0747a1 1.0934b1 1.0699a1 1.0886b1 1.0652a1 1.0839b1

1.610 1.630 1.658 1.686 1.705 1.704 1.722 1.740 1.747 1.755

1.7652a1 1.8123b1 1.7822a1 1.8383b1 1.8049a1 1.8641b1 1.8292a1 1.8896b1 1.8528a1 1.9150b1 1.8757a1 1.940b1 1.8986a1 1.9655b1 1.9215a1 1.9907b1 1.9441a1 2.0158b1 1.9668a1 2.0411b1

Bv (cm−1)

ΔBv/Bv %

Dv × 106 (cm−1)

ΔDv/Dv %

7.543a1 7.755b1 7.865b7 7.367b8 8.13e1 8.28k1

2.81 4.26 2.33 7.71 9.70

7.488a1 7.710b1 7.888b7 7.630b8 7.90e1 8.28k1

2.96 5.34 1.89 5.55 10.6

7.413a1 7.667b1 7.827b7 7.647b8 7.33e1

3.42 5.58 3.15 1.23

2.668

1.508a1 1.466b1 1.464b7 1.463b8 1.48006e1 1.48011k1 1.4658m 1.492a1 1.440b1 1.4471b7 1.444b8 1.461491 1.462471 1.4481m 1.476a1 1.423b1 1.4297b7 1.427b8 1.44278e1 1.4312m 1.460a1 1.407b1 1.4132b7 1.411b8 1.444a1 1.390b1 1.3971b7 1.394b8 1.429a1 1.375b1 1.3808b7 1.377b8 1.415a1 1.359b1 1.3641b7 1.361b8 1.401a1 1.343b1 1.3475b7 1.345b8 1.386a1 1.327b1 1.331b7 1.329b8 1.371a1 1.311b1 1.3146b7 1.313b8 1.357a1 1.296b1 1.2984b7 1.297b8 1.347a1 1.280b1

1.766

1.340a1

5.386a1

3.285

1.332a1

6.077a1

3.301

1.315a1 1.289a1 1.219b1

8.585a1 8.791a1 7.184b1

3.357 3.428

2.78 2.91 2.9 1.85 1.84 2.80 3.48 3.00 3.21 2.04 1.97 2.94 3.59 3.13 3.31 2.25 3.03 3.63 3.20 3.35 3.63 3.73 3.24 3.46 3.77 3.37 3.63 3.95 3.59 3.81 4.13 3.81 3.99 4.25 3.96 4.11 4.37 4.11 4.23 4.49 4.31 4.42 4.97

5.43

1.275a1 1.269a1

7.372a1 7.623b1 7.820b7 7.419b8 7.280a1 7.581b1 7.817b7 7.366b8 7.173a1 7.540b1 7.728b7 6.406b8 7.031a1 7.498b1 7.669b7 7.506b8 6.951a1 7.459b1 7.695b7 6.988b8 7.070a1 7.420b1 7.630b7 7.366b8 7.033a1 7.383b1 7.605b7 7.688b8 6.379a1 7.346b1 7.555b7 6.406b8 5.686a1 7.310b1

3.40 6.07 0.63 4.13 7.37 1.18 5.11 7.73 10.69 6.64 9.07 6.75 7.30 10.70 0.53 4.95 7.92 4.18 4.97 8.13 9.31 15.15 18.43 0.42 28.56

18.28

4.830a1

1.248a1

6.334a1 8.463a1

1.234a1 1.224a1 1.145b1

4.821a1 7.732a1 7.064b1

3.523 3.601 3.688 3.777

6.45

8.63

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

189

Table 5 (continued) v 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

Ev (cm−1)

ΔEv/Ev %

a1

25,856.39 23,464.657b1 26,892.18a1 24,377.591b1 27,914.13a1 25,275.345b1 28,917.26a1 26,157.965b1 29,914.08a1 27,025.498b1 30,893.43a1 27,877.980b1 31,860.62a1 28,715.446b1 29,537.922b1 33,752.00a1 30,345.429b1 34,676.44a1 31,137.985b1 35,589.33a1 31,915.599b1 36,490.45a1 32,678.277b1 37,375.06a1 33,426.018b1 38,249.32a1 34,158.817b1 39,106.84a1 34,876.662b1 39,954.70a1 35,579.535b1 40,788.63a1 36,267.414b1 41,609.47a1 36,940.270b1 42,419.48a1 37,598.068b1 43,217.50a1 38,240.767b1 44,001.64a1 38,868.312b1 44,774.62a1 39,480.674b1 45,534.20a1 40,077.768b 46,283.25a1 40,659.536b1 47,019.91a1 41,225.903b1 47,742.76a1 41,776.789b1 48,454.23a1 42,312.104b1 49,153.63a1 42,831.750b1 51,830.35a1 43,335.622b1 52,468.64a1 43,823.604 b1 53,094.32a1 44,295.572b1 53,707.10a1 44,751.390b1 55,468.03a1 45,190.911b1 56,026.71a1 45,613.978b1 56,571.55a1 46,020.417b1 57,101.06a1 46,410.044b1 57,614.46a1 46,782.655b1

Rmin (Å)

ΔRmin/Rmin %

a1

9.25 9.35 9.45 9.54 9.65 9.76 9.87 9.97 10.09 10.20 10.32 10.44 10.56 10.69 10.81 10.95 11.08 11.22 11.36 11.51 11.66 11.82 11.98 12.15 12.32 12.49 12.67 12.86 16.38 16.47 16.57 16.67 18.52 18.58 18.65 18.72 18.80

1.0605 1.0796b1 1.0566a1 1.0754b1 1.0526a1 1.0715b1 1.0487a1 1.0677b1 1.0450a1 1.0639b1 1.0414a1 1.0605b1 1.0380a1 1.0571b1 1.0347a1 1.0539b1 1.0315a1 1.0508b1 1.0283a1 1.0478b1 1.0253a1 1.0449b1 1.926 1.0421b1 1.0197a1 1.0394b1 1.0171a1 1.0368b1 1.0146a1 1.0344b1 1.0121a1 1.0319b1 1.0093a1 1.0297b1 1.0065a1 1.0274b1 1.0040a1 1.0253b1 1.0019a1 1.0232b1 1.0003a1 1.0212b1 0.9989a1 1.0193b1 0.9976a1 1.0175b1 0.9965a1 1.0157b1 0.9955a1 1.0139b1 0.9945a1 1.0123b1 0.9936a1 1.0107b1 0.9927a1 1.0092b1 0.9894a1 1.0077b1 0.9886a1 1.0063b1 0.9878a1 1.0049b1 0.9871a1 1.0037b1 0.9850a1 1.0024b1 0.9843a1 1.0020b1 0.9836a1 1.0010b1 0.9830a1 0.999b1 0.9824a1 0.9980b1

Rmax (Å)

ΔRmax/Rmax %

Bv (cm−1)

3.860

1.203a1

6.687a1

3.883 3.978

1.195a1 1.175a1

6.257a1 7.394a1

1.167a1

4.894a1

ΔBv/Bv %

Dv × 106 (cm−1)

ΔDv/Dv %

a1

1.801 1.779 1.795 1.811 1.808 1.834 1.840 1.855 1.871 1.896 1.911 1.926 1.931 1.936 1.951 1.956 2.021 2.076 2.121 2.125 2.089 2.042 1.994 1.926 1.848 1.789 1.721 1.662 1.849 1.790 1.731 1.681 1.76 1.798 1.769 1.627 1.587

1.9895 2.0663b1 2.0134a1 2.0916b1 2.0361a1 2.1171b1 2.0586a1 2.1426b1 2.0814a1 2.1683b1 2.1041a1 2.1943b1 2.1270a1 2.2204b1 2.1498a1 2.2467b1 2.1728a1 2.2732b1 2.1956a1 2.3001b1 2.2186a1 2.3272b1 2.2420a1 2.3546b1 2.2654a1 2.3824b1 2.2890a1 2.4106b1 2.3127a1 2.4391b1 2.3366a1 2.4681b1 2.3607a1 2.4975b1 2.3848a1 2.5274b1 2.4088a1 2.5580b1 2.4339a1 2.5890b1 2.4586a1 2.6207b1 2.4836a1 2.6530b1 2.5088a1 2.6861b1 2.5345a1 2.7199b1 2.5605a1 2.7545b1 2.5868a1 2.7899b1 2.6134a1 2.8265b1 2.6403a1 2.8639 b1 2.7516a1 2.9026b1 2.7805a1 2.9425b1 2.8098a1 2.9837b1 2.8397a1 3.0263b1 2.9343a1 3.0706b1 2.9673a1 3.1166b1 3.0025a1 3.1646b1 3.0371a1 3.2147b1 3.0741a1 3.2673b1

4.080 4.175 4.286

1.151a1 1.071b1 1.140a1

4.391

1.123a1

7.113a1

4.507

1.114a1

5.135a1

4.620

1.098a1

6.935a1

4.759

5.022

1.087a1 0.998b1 1.073a1 0.984b1 1.061a1

5.164

1.047a1

4.513a1

5.312

1.032a1

7.022a1

5.465

1.021a1

4.401a1

5.627

1.006a1

5.549a1

5.794

0.994a1

5.531a1

5.979

0.981a1

4.195a1

6.193

0.967a1

4.622a1

6.372

0.954a1

5.219a1

6.593

0.940a1

3.396a1

6.820

0.925a1

4.736a1

7.067

0.914a1

3.621a1

7.315

0.899a1

3.322a1

7.576

0.885a1

4.240a1

7.851

0.872a1

3.383a1

8.154

0.858a1

3.075a1

8.468

0.843a1

3.328a1

5.487

0.789 a1

3.320a1

5.826

0.777a1

3.593a1

6.189

0.764a1

3.298a1

6.571

0.750a1

2.297a1

4.645

0.708a1

3.884a1

5.031

0.694a1

3.349a1

5.398

0.679a1

4.143a1

5.847

0.664a1

4.788a1

6.284

0.650a1

3.957a1

4.894

6.95

8.18 8.29

7.253a1 6.996b1 5.679a1

5.691a1 6.991b1 5.462a1 6.997b1 7.002a1

3.54

22.84 28.10

(continued on next page)

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Table 5 (continued) v 58 59 (2)4Π 0 1 2 3 4 5 6 7 8 9 10

Ev (cm−1) a1

58,112.39 47,138.033b1 58,591.14a1 47,475.938b1

ΔEv/Ev % 18.88 18.97

214.9 631.0 1026.9 1399.3 1746.4 2069.7 2370.9 2651.6 2911.8 3150.7 3365.5

ΔRmin/Rmin %

Rmin (Å) a1

0.9819 0.9971b1 0.9813a1 0.9961b1

1.548 1.508

Bv (cm−1)

ΔRmax/Rmax %

Rmax (Å) a1

3.1131 3.3226b1 3.1543a1 3.3809b1

ΔBv/Bv %

Dv × 106 (cm−1)

6.729

a1

0.633

5.753a1

7.183

0.616a1

6.138a1

1.822 1.763 1.721 1.689 1.663 1.641 1.623 1.606 1.592 1.579 1.568

2.050 2.160 2.250 2.334 2.414 2.494 2.577 2.663 2.753 2.851 2.960

7.350 7.160 6.963 6.772 6.590 6.398 6.177 5.934 5.689 5.415 5.132

0.8864 0.9306 1.0338 1.1985 1.3400 1.4115 1.4190 1.5033 1.6726 2.0491 2.1906

(2)4 ∑+ 0 239.9 1 702.3

1.622 1.564

1.837 1.943

9.217 8.928

1.4040 1.3993

(2)4Δ 0 1

1.625 1.559

1.823 1.931

9.257 9.054

1.1879 1.2650

279.4 789.5

ΔDv/Dv %

a1

Present work with 5 valence electrons MRCI calculation, b1, b7, b8Ref. [80], e1(exp.)Ref. [81], k1(exp.)Ref. [89], mRef. [97].

Table 6 Vibrational energies Ev, the abscissas of the turning points Rmin and Rmax, the rotational constants Bv and centrifugal distortion constants Dv, for the first six vibrational levels of the electronic states (X)2Σ+ and (1)2 Π of CaF molecule. v X2 Σ + 0

1

2

3

4

5

EV (cm−1)

260.91a 289.86d 283.80b1 281.70b2 285.60b3 291.30b4 288.60b5 291.90b6 273.30b7 283.80b8 271.20b9 279.40b10 285.00b11 265.70b12 285.00b13 290.00b14 265.40b15 289.50b16 286.70b17 300.80b18 290.00b19 285.90b20 304.80b21 292.60b22 288.90b23 306.90b24 777.84a 865.48d 1289.91a 1435.62d

ΔEV/EV %

ΔRmin/Rmin %

1.948a 9.99 8.07 7.38 8.64 10.43 9.59 10.62 4.53 8.07 3.79 6.62 8.45 1.80 8.45 10.03 1.69 9.88 8.99 13.26 10.03 8.74 14.40 10.83 9.69 14.98 10.13

10.15

a

1797.18 2000.28d

10.15

2299.63a 2559.46d

10.15

2797.38a

Rmin (Å)

1.903b1 1.930b2

1.903a 1.860b1 1.904b 1.874a 1.833b1 1.888b2 1.852a 1.812b1 1.875b2 1.833a 1.795b1 1.864b2 1.817a

Rmax (Å)

ΔRmax/Rmax %

2.089a 2.36 0.93

2.31 0.05 2.24 0.74 2.21 1.23 2.12 1.66

2.037b1 2.007b2

2.149a 2.094b1 2.039b2 2.194a 2.136b1 2.062b2 2.231a 2.172b1 2.081b2 2.265a 2.205b1 2.099b2 2.297a

BV ∗ 101 (cm−1)

ΔBV/BV %

DV ∗ 107 (cm−1)

ΔDV/DV %

3.21a 3.42c

6.14

4.900a 4.688c

4.52

3.19a 3.40c

6.18

4.890a 4.689c

4.29

3.16a 3.37c

6.23

4.870a 4.689c

3.86

2.55 4.09

2.63 5.39 2.72 6.40

3.14

a

4.850

a

2.72 7.21 3.11a

4.830a

3.09a

4.830a

2.72 7.91

N. El-Kork et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 177 (2017) 170–196

191

Table 6 (continued) v

EV (cm−1)

ΔEV/EV %

Rmin (Å)

ΔRmin/Rmin %

Rmax (Å)

ΔRmax/Rmax %

3113.16d

10.14

1.780b1 1.855b2

2.08 2.05

2.235b1 2.114b2

2.77 8.66

(1)2Π 0 274.86a 292.54d

6.04

800.17a 872.48d

8.29

1

2

3

4

6

a

1313.25 1445.58d

9.15

1833.13a 2011.81d

8.88

2344.37a 2571.20d

8.82

2840.75a 3123.73d

9.06

1.936a 1.888e 1.889f 1.886a 1.848e 1.846f 1.855a 1.820e 1.819f 1.832a 1.799e1 1.798f 1.814a 1.782e 1.780f 1.798a 1.767e 1.765f

2.073a 2.022e 2.023f 2.130a 2.081e 2.079f 2.173a 2.122e 2.122f 2.210a 2.158e 2.158f 2.244a 2.191e 2.190f 2.274a 2.222e1 2.221f

2.54 2.49 2.06 2.17 1.92 1.98 1.83 1.89 1.80 1.91 1.75 1.87

BV ∗ 101 (cm−1)

2.52 2.47 2.35 2.45

ΔBV/BV %

ΔDV/DV %

3.26a 3.47c

6.05

5.020a 4.810c

4.37

3.26a 3.45c

5.51

5.350a 4.814c

11.13

a

a

3.23 3.42c

2.40 2.40

DV ∗ 107 (cm−1)

5.55

4.420 4.818c

3.20a

5.260a

3.19a

6.200a

3.17a

3.990a

8.26

2.41 2.41 2.42 2.47 2.34 2.39

a Present work, b1Ref. [14] (ab initio values calculated using the BD(T) method with BS1 basis set). b2Ref. [14] (ab initio values calculated using the BD(T) method with BS2 basis set). b3Ref. [14] (ab initio values calculated using the BD(T) method with BS3 basis set). b4Ref. [14] (ab initio values calculated using the B3LYP method with BS1 basis set). b5Ref. [14] (ab initio values calculated using the B3LYP method with BS2 basis set). b6Ref. [14] (ab initio values calculated using the B3LYP method with BS3 basis set). b7Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the BD(T) method with BS1 basis set). b8Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the BD(T) method with BS1 basis set). b9Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the BD(T) method with BS1 basis set). b10Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the BD(T) method with BS2 basis set). b11Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the BD(T) method with BS2 basis set). b12Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the BD(T) method with BS2 basis set). b13Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the BD(T) method with BS3 basis set). b14 Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the BD(T) method with BS3 basis set). b15Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the BD(T) method with BS3 basis set). b16Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the B3LYP method with BS1 basis set). b17Ref. [5] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the B3LYP method with BS1 basis set). b18Ref. [5] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the B3LYP method with BS1 basis set). b19Ref. [5] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the B3LYP method with BS2 basis set). b20Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the B3LYP method with BS2 basis set). b21Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the B3LYP method with BS2 basis set). b22Ref. [14] (Theoretical values calculated using the MS model of potential energy function for the ab initio energies found from the B3LYP method with BS3 basis set). b23Ref. [14] (Theoretical values calculated using the HP model of potential energy function for the ab initio energies found from the B3LYP method with BS3 basis set). b24Ref. [14] (Theoretical values calculated using the TT model of potential energy function for the ab initio energies found from the B3LYP method with BS3 basis set). cRef. [18] (Experimental values). d Ref. [96] (Experimental values).eRef.[96] (Experimental values calculated using Lakshman and Rao's method). fRef [96] (Experimental values calculated using Morse method).

Table 7 Values of the eigenvalues Ev, the rotational constants Bv, the centrifugal distortion constants Dv, and the abscissas of the turning points for the different vibrational levels of X2Σ+ and (1)4Σ+ states of MgF molecule. v

Ev (cm−1)

0

350.2 359.6c

1

1041.3 1067.4c

ΔEv/Ev %

4 5 6 7 8 9

Dv × 106 (cm−1)

ΔDv/Dv %

1.0238

2.7 5.16e 4.937

3.6

1.0820e 1.0143

e

3.7

e

1.0750 1.0129 1.0167

4.799

0.9893

4.754

1.0079

4.712

1.0007

4.667

0.9741

4.624 4.582

0.9993 0.9868

ΔRmax/Rmax %

1.6

1.850 1.820c

1.6

1.673 1.645c

1.7

1.908 1.877c

1.1

1.646 1.618c 1.626 1.597c 1.58c

2.4 2.4

Rmax (Å)

6.0

2.4 4.845

ΔRmin/Rmin %

1.715 1.687c

Rmin (Å)

5.7

2.6 5.122 4.889

1725.1 1766.3c 2400.6 2457.3c 3067.5 3140.7c 3727.6 3815.3c 4379.6 4481c 5023.7 5140c 5660.9 6290.2

ΔBv/Bv %

4.983

2

3

Bv × 10 (cm−1)

1.6

2.4 2.3 2.3

1.7 1.8 1.8

1.594 1.567c 1.581 1.554c 1.570 1.543c 1.560 1.550

1.7 1.7 1.7

1.951 1.920c 1.988 1.957c 2.021 1.987c 2.052 2.021c 2.082 2.05c 2.110 2.078c 2.137 2.164

1.6 1.6 1.7 1.5 1.5 1.5

(continued on next page)

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Table 7 (continued) v

Ev (cm−1)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (1)4Σ+ 0 1

6912.2 7527.3 8134.9 8734.8 9326.7 9910.4 10,487.6 11,060.7 11,628.4 12,188.1 12,742.4 13,292.5 13,837.3 14,378 14,913.3 15,440.6 15,960 16,472.7 16,980.3 17,481.9 17,977.2

4.539 4.497 4.456 4.413 4.368 4.326 4.291 4.257 4.216 4.176 4.144 4.109 4.076 4.045 4.005 3.962 3.922 3.887 3.853 3.815 3.779

0.9637 0.9851 0.9895 0.9850 1.0049 0.9632 0.8665 0.9057 1.0050 0.9054 0.8438 0.9016 0.8320 0.8887 0.9893 0.9892 0.9490 0.8589 0.9049 0.9201 0.8980

1.541 1.533 1.525 1.518 1.511 1.505 1.500 1.493 1.488 1.482 1.478 1.473 1.468 1.463 1.460 1.455 1.451 1.447 1.443 1.440 1.436

2.19 2.215 2.24 2.265 2.29 2.313 2.337 2.36 2.383 2.406 2.429 2.451 2.472 2.494 2.516 2.539 2.562 2.584 2.607 2.629 2.651

29.5 85.0

1.805 1.715

7.1491 7.0473

2.720 2.610

3.189 3.440

ΔEv/Ev %

Bv × 10 (cm−1)

ΔBv/Bv %

Dv × 106 (cm−1)

ΔDv/Dv %

Rmin (Å)

ΔRmin/Rmin %

ΔRmax/Rmax %

Rmax (Å)

First entry is for the present work, cRef. [90], eRef. [52].

Table 8 Values of the transition dipole moment in terms of the internuclear distance R for the transition X2Σ+ − (2)2Σ+ and (1)2Π_(2)2Π of the molecules BeF, MgF and CaF. MgF molecule 2 +

2 +

R (Å)

X Σ

− (2) Σ a.u

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 1.78 1.8 1.82

−0.27379777 −0.88795581 −1.28215264 −1.4179875 −1.46747095 −1.48917957 −1.49629436 −1.50033183 −1.50249354 −1.5033409 −1.50333981 −1.50256416 −1.50189555 −1.50116674 −1.49959893 −1.49883045 −1.49810089 −1.49748432 −1.49687046 −1.49633177 −1.49587635 −1.49551031 −1.49865862 −1.49840028 −1.49824217 −1.49818472 −1.49822718 −1.49831895 −1.49853765 −1.49884279 −1.4992207 −1.49965461 −1.50012527 −1.50060747 −1.50107045 −1.50147627 −1.50177811 −1.50191814 −1.50182421 −1.50140532 −1.5005608 −1.4991075

CaF molecule 2 +

X Σ

2 +

− (2) Σ a.u

2.60178486 2.54982623 2.49632944 2.44341003 2.39459251 2.35302972 2.31931291 2.29386054 2.27569307 2.25802085 2.23414063 2.22320084 2.21668203 2.20970267 2.20109293 2.19093547 2.17933278 2.16055172 2.14971941 2.13871675 2.1274886 2.11621988 2.10457523 2.09211442 2.07924629 2.06613122 2.05287524 2.03958822 2.02640599 2.01346784 1.99511909 1.98153256 1.96819796 1.95502175 1.94203744 1.92922986 1.91662566 1.90423608 1.89207049 1.88014081 1.86847684 1.85701513

BeF molecule 2

2

(1) Π_(2) Π a.u

(1)2Π_(2)2Π a.u

−2.33888202 −2.37322983 −2.4057651 −2.432299 −2.45104296 −2.46047454 −2.45965629 −2.44998938 −2.42849178 −2.39593881 −2.3146773 −2.25594428 −2.19314703 −2.12790213 −2.08356293 −2.03864455 −2.01061949 −1.98391398 −1.9449206 −1.88951793 −1.83908239 −1.7940187 −1.74623695 −1.69911886 −1.65529436 −1.61428503 −1.57559326 −1.53903647 −1.50483659 −1.46963873 −1.44817369 −1.41237363 −1.37144859 −1.32917606 −1.28464802 −1.23936814 −1.1934828 −1.1481411 −1.10390952 −1.0608277 −1.02004001 −0.9812138

0.91977271 0.88649388 0.85443593 0.82227272 0.79074775 0.75816769 0.72697369 0.69725889 0.66504087 0.62831241 0.59705422 0.56330551 0.52702889 0.49834729 0.47276287 0.43728977 0.40254633 0.36836894 0.34629472 0.30413533 0.29465634 0.2499109 0.22468106 0.20450416 0.19591217 0.2039963 −1.57452205 −1.57469469 −1.59585846 −1.63439074 0.41389162 0.4514825 0.49680721 0.54988284 0.62215086 0.70633666 0.87714993 0.95606156 −1.60718675 −1.71897081 −1.68881909 −1.64090723

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193

Table 8 (continued) MgF molecule

CaF molecule

R (Å)

X2Σ+ − (2)2Σ+ a.u

X2Σ+ − (2)2Σ+ a.u

(1)2Π_(2)2Π a.u

(1)2Π_(2)2Π a.u

BeF molecule

1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18 2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.36 2.38 2.4 2.42 2.44 2.46 2.48 2.5 2.52 2.54 2.56 2.58 2.6 2.62 2.64 2.66 2.68 2.7 2.72 2.74 2.76 2.78 2.8 2.82 2.84 2.86 2.88 2.94 2.96 2.98 3 3.02 3.04 3.06 3.08 3.1 3.12 3.14 3.16 3.18 3.2 3.22 3.24 3.26 3.28 3.3 3.32

−1.49686406 −1.49356521 −1.48885031 −1.48222226 −1.47299154 −1.46019943 −1.44252125 −1.41811177 −1.38398349 −1.33438183 −1.25931978 −1.1490224 −0.99952 −0.81778614 −0.6208732 −0.43238185 −0.26983841 −0.13615603 −0.02879413 0.05844775 0.12984162 0.18989233 0.24099248 0.2855711 0.32518223 0.36094197 0.39368072 0.42405014 0.44392152 0.4713203 0.4975129 0.52272974 0.54716851 0.57099323 0.59433952 0.61767061 0.64048214 0.65631712 0.67927376 0.70219718 0.72515301 0.74817545 0.77130091 0.79454387 0.81790871 0.8414143 0.86508185 0.88893049 0.90222209 0.91901115 0.94087237 0.9525787 0.96356453 1.01556988 1.02037882 1.03833887 1.0557357 1.07108616 1.07911467 1.09139248 1.10426129 1.11040337 1.11667623 1.12159373 1.12002893 1.12915921 1.13008468 1.12767545 1.1178693 −3.50344359 −3.77395035 −3.90586774 −3.95972015

1.84583383 1.83496011 1.82438057 1.81415567 1.80428056 1.79475047 1.78553908 1.7765874 1.76787939 1.75920531 1.75091629 1.74276504 1.73477092 1.72692806 1.71920047 1.71159837 1.70410797 1.69671875 1.68922957 1.68215926 1.67515097 1.66817886 1.66121459 1.6543855 1.64732062 1.64013875 1.63277518 1.62514846 1.61716596 1.60869926 1.59958792 1.5896211 1.57851823 1.56590032 1.55123438 1.53378548 1.51247504 1.48224521 1.4461136 1.39724682 1.32945426 1.23326823 1.10165323 0.92700188 0.71989678 0.50274153 0.29960076 0.12379861 −0.02256179 −0.14307879 −0.24308956 −0.32739191 −0.39982919 −0.57154659 −0.61910919 −0.66362095 −0.70566647 −0.74600086 −0.78502598 −0.82327343 −0.85847275 −0.8941034 −0.92910401 −0.96357168 −0.99755731 −1.03110551 −1.06419691 −1.09679338 −1.12859406 −1.16002975 −1.19080675 −1.22085644 −1.25000427

−0.94414842 −0.90937285 −0.87695162 −0.84607793 −0.81721832 −0.78999829 −0.76427499 −0.74001401 −0.71691635 −0.69495479 −0.67361779 −0.65381956 −0.63476171 −0.61628903 −0.59865173 −0.58064569 −0.56452215 −0.54806536 −0.53651625 −0.52066299 −0.50512597 −0.48980871 −0.47319568 −0.45773407 −0.44207693 −0.4262172 −0.41013763 −0.39373007 −0.37696938 −0.35975619 −0.34201259 −0.32367067 −0.30463014 −0.28481619 −0.26414999 −0.24256461 −0.22002248 −0.19220317 −0.16681601 −0.14028615 −0.11294197 −0.08492553 −0.05632713 −0.0248256 0.00502988 0.03772135 0.0701611 0.10263586 0.13624527 0.1705814 0.20606996 0.24311117 0.27559956 0.37898407 0.41264232 0.44542675 0.47721381 0.50762378 0.53663255 0.56218255 0.5934845 0.61200454 0.63764119 0.65346616 0.67050949 0.68562808 0.70044638 0.71043301 0.71845854 0.72150621 0.72265473 0.72665511 0.73385169

−1.56859182 −1.46766174 −1.33334689 −1.17894233 −1.01875301 −0.86636523 −0.73244677 −0.611049 −0.51557172 −0.43287622 −0.36074592 −0.25662825 −0.16305903 −0.05390005 0.09901295 0.08287051 0.053422735 0.04800958 0.04762011 0.04754283 0.04696551 0.04621843 0.04546643 0.04472945 0.04306509 0.04352856 0.04367671 0.0437387 0.04410442 0.04455437 0.04613848 0.04151577 0.04208248 0.04352138 0.04346672 0.03506814 0.03496721 −0.00055369 −0.0171309 −0.03322713 −0.03029549 −0.01778423 −0.00880844 −0.00285906 0.00089321 0.00338742 0.00505868 0.00620898 0.00702121 0.00764841 0.00805023 0.00833165 0.00852283 0.0087397 0.00873385 0.00869903 0.00860385 0.00852306 0.00843258 0.00832982 0.00821704 0.00809608 0.00796849 0.00783602 0.00769841 0.00755825 0.00741583 0.00727029 0.00712786 0.00698465 0.00684106 0.00669522 0.00655151 (continued on next page)

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Table 8 (continued) MgF molecule

CaF molecule

R (Å)

X2Σ+ − (2)2Σ+ a.u

X2Σ+ − (2)2Σ+ a.u

(1)2Π_(2)2Π a.u

(1)2Π_(2)2Π a.u

BeF molecule

3.34 3.36 3.38 3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.58 3.6 3.62 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78 3.8 3.82 3.84 3.86 3.88 3.9 3.92 3.94 3.96 3.98 4 4.02 4.04 4.06 4.08 4.1 4.12 4.14 4.16 4.18 4.2 4.22 4.24 4.26 4.28 4.3 4.32 4.34 4.36 4.38 4.4 4.42 4.44 4.46 4.48 4.5 4.52 4.54 4.56 4.58

−3.98697387 −3.98052676 −3.98396582 −3.96819808 −3.93520489 −3.90856848 −3.83666214 −3.75368546 −3.69494442 −3.6571084 −3.59632055 −3.53097384 −3.46160918 −3.38838692 −3.31185952 −3.23260574 −3.15104168 −3.06779708 −2.98328385 −2.89813499 −2.81188758 −2.7267652 −2.64234665 −2.55893194 −2.47686931 −2.39647091 −2.31790034 −2.24135276 −2.16704646 −2.09499968 −2.02527688 −1.95797672 −1.89306293 −1.83075499 −1.77046676 −1.71241071 −1.65667419 −1.60283313 −1.55094785 −1.50090182 −1.45258352 −1.40569013 −1.36048088 −1.31659308 −1.2737712 −1.23221429 −1.19163143 −1.15168114 −1.11264712 −1.07417024 −1.03584585 −0.99804867 −0.96009124 −0.92230549 −0.88399546 −0.84551858 −0.80618092 −0.7664612 −0.72564512 −0.71392455 −0.71543466

−1.2802588 −1.29888978 −1.33185629 −1.35687997 −1.38092028 −1.40865825 −1.42724825 −1.44493021 −1.46919551 −1.48163185 −1.4737323 −1.44384045 −1.19468761 1.25764194 1.31006932 1.38740874 1.49751043 1.55697794 1.58965235 1.61183394 1.62922006 1.6332579 1.63375288 1.63345965 1.63420457 1.62325461 1.61198829 1.59741158 1.58170012 1.56286167 1.54182614 1.51894907 1.49461346 1.46901129 1.43530628 1.41530785 1.3877147 1.35960537 1.33199205 1.3036387 1.27548931 1.2475227 1.21976209 1.1945835 1.16513133 1.13320275 1.11245822 1.08645779 1.06103284 1.03617692 1.01160184 0.98747719 0.96370971 0.94046153 0.91765431 0.89525687 0.87326315 0.8516524 0.83041269 0.80953526 0.78901113 0.76883131 0.74911093 0.72932824 0.7100872 0.69107701 0.67215498 0.65329615 0.63426708 0.61521831 0.59705655 0.57941285 0.56263121 0.54635675

0.74156715 0.74998139 0.75817079 0.76309013 0.76671018 0.76792209 0.7662551 0.76202522 0.75654816 0.74644264 0.73027009 0.70220085 0.68230746 0.65744504 0.62892837 0.59909448 0.5640575 0.52692777 0.48831148 0.44865238 0.40647201 0.36306633 0.31865302 0.26417307 0.23680807 0.18863732 0.13258996 0.08885835 0.04545297 0.00313842 −0.039397 −0.08114055 −0.12152877 −0.16125245 −0.19995481 −0.23765658 −0.27416177 −0.29957324 −0.3477144 −0.37718924 −0.40898913 −0.43957891 −0.46891897 −0.49699851 −0.52382061 −0.549377 −0.57367257 −0.59670838 −0.61848806 −0.63901496 −0.65829096 −0.6763167 −0.6930896 −0.7087248 −0.72297216 −0.73594137 −0.74761615 −0.75796869 −0.7669657 −0.77456242 −0.78069896 −0.78530367 −0.78823371 −0.78821115 −0.78840661 −0.7850601 −0.77868091 −0.76755923 −0.7500596 −0.72624195 −0.70220289 −0.67723245 −0.6530445 −0.62838814

0.00640837 0.00626444 0.00612317

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195

Table 8 (continued)

R (Å)

MgF molecule

CaF molecule

X2Σ+ − (2)2Σ+ a.u

X2Σ+ − (2)2Σ+ a.u

(1)2Π_(2)2Π a.u

0.53054075 0.5151407 0.50012311 0.48545964 0.47111851 0.45711293 0.44342328 0.43007289 0.41699911 −0.37584072 −0.34972669 −0.32791669 −0.30717921 −0.287564 −0.26908728 −0.25172512 −0.23545034 −0.22022003 −0.20598084 −0.192688 −0.18028917 −0.1687286 −0.15795139 −0.15037879 −0.1442176 −0.14084753 −0.13422295 −0.13029651 −0.12502258 −0.1203593 −0.11626599 −0.11170274 −0.10663535 −0.10002637 −0.0938449 −0.08806381 −0.08265561 −0.07760609 −0.0728575 −0.06841166 −0.06424457 −0.06034268 −0.05666791 −0.05322608 −0.04999462 −0.0469591 −0.04410749 −0.04142725 −0.03890693

−0.60318588 −0.57718778 −0.55128589 −0.52496852 −0.49852626 −0.47229862 −0.44631413 −0.42085628 −0.39627355 −0.37584072 −0.34972669 −0.32791669 −0.30717921 −0.287564 −0.26908728 −0.25172512 −0.23545034 −0.22022003 −0.20598084 −0.192688 −0.18428917 −0.1757286 −0.17095139 −0.16537879 −0.1602176 −0.15584753 −0.15022295 −0.14429651 −0.13802258 −0.1293593 −0.12126599 −0.11370274 −0.10663535 −0.10002637 −0.0938449 −0.08806381 −0.08265561 −0.07760609 −0.0728575 −0.06841166 −0.06424457 −0.06034268 −0.05666791 −0.05322608 −0.04999462 −0.0469591 −0.04410749 −0.04142725 −0.03890693

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BeF molecule

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

(1)2Π_(2)2Π a.u

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