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Electronic structure with dipole moment calculations of the high-lying electronic states of BeH, MgH and SrH molecules To cite this article: Nayla El-Kork et al 2018 J. Phys. Commun. 2 055030
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J. Phys. Commun. 2 (2018) 055030
https://doi.org/10.1088/2399-6528/aac3f8
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Electronic structure with dipole moment calculations of the highlying electronic states of BeH, MgH and SrH molecules
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25 November 2017 REVISED
11 April 2018 ACCEPTED FOR PUBLICATION
11 May 2018 PUBLISHED
25 May 2018
Nayla El-Kork1, Israa zeid2, Hadeel Al Razzouk2, Sara Atwani2, Racha Abou arkoub2 and Mahmoud Korek2 1 2
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
E-mail:
[email protected] and
[email protected] Keywords: ab initio calculation, electronic structure, spectroscopic constants, potential energy curves, dipole moments Supplementary material for this article is available online
Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Abstract By using the complete active space self consistent field (CASSCF) with multi-reference configuration interaction MRCI+Q method including single and double excitations with Davidson correction, the 26, 27 and 25 low-lying doublet and quartet electronic states in the representation 2s+1Λ(+/−) (without spin orbit interaction) of the molecules BeH, MgH and SrH have been investigated. The potential energy curves, the internuclear distance Re, the harmonic frequency ωe, the permanent dipole moment μ, the rotational constant Be and the electronic transition energy with respect to the ground state Te are calculated. Using the canonical approach the eigenvalue Ev, the rotational constant Bv and the abscissas of the turning points Rmin and Rmax have been calculated for the investigated electronic states. The comparison between the values of the present work and those available in the literature for several electronic states shows very good agreement.
1. Introduction The mono-hydrides of the alkaline-earth metals are expected to be present in sunspots, stars, nebulae, and the interstellar medium. These species, have received considerable attention from both the experimentalists and the theoreticians, because the molecules SrH, MgH and CaH are relatively easy to synthesize in the gas-phase, have interesting ground states, due to astrophysical significance [1–5] and the apparent simplicity of their electronic structure. Since the excited electronic states of the considered three molecules showed complex spectra their electronic states have been the subject of much interest from both experimental spectroscopists and quantum chemists. Since the collision process between cold atoms allowing high precision measurements with hydrogen isotopes, precise determination of molecular potentials and atomic lifetimes the alkaline-earth-metal hydrides have been the subject of extensive research in the past years. These molecules are ideal candidates for the production of the polar molecules and direct laser cooling [6–16] as well as for the ultracold fragmentation. Moreover, these Hydrides containing alkaline-earth metals offer a wide variety of interesting applications. For example, they are prototypes for hydrogen storage materials. They have been predicted to be stable at pressures that can be achieved in a diamond anvil cell and they are predicted to display high-temperature superconductivity. With small amounts of hydrogen (0.1%–1%) the Hydrides containing alkaline-earth metals are investigated as viable magnetic, thermoelectric or semiconducting materials. The BeH molecule is less popular for experimentalists because of the toxicity of the Be-containing molecules, while it is extensively studied theoretically by using ab initio method. The early work on the electronic emission spectra of BeH molecule has been done by Watson and Fredrickson [17]. Recently, there is more demanding of this molecule due to the existence of near-degeneracy effects and low-lying states [18] and it is being studied in the surfaces of fusion reactors ITER (International Thermonuclear Experimental Reactor) through the Joint European Torus (JET) of the European Community Fusion Program. Since the visible emission of MgH was observed in the Sun and then in many stars and because of the relative abundances of the magnesium isotopes in stellar atmospheres, this molecule attracted the attention of the © 2018 The Author(s). Published by IOP Publishing Ltd
Dissociation of atomic levels Be+H
Dissociation energy limit of BeH levels (cm−1)
Be (1s22s2, 1S)+H (1s, 2S) Be (1s22s2p, 3P0)+H (1s, 2S) Be (1s22s2p, 1P0)+H (1s, 2S) Be (1s22s3s, 1S)+H (1s, 2S) Be (1s22s3p, 3P0)+H (1s, 2S) Be (1s22p2, 1D)+H (1s, 2S) Dissociation limit of atomic levels Mg+H
2
6
2 1
0a 22 115a 43 861a 56 412a 60 167a 60 888a Dissociation energy limit of MgH levels (cm−1)
2
a
Mg(2p 3s , S)+H(1s, S) Mg(3s3p, 3P0)+H(1s, 2S) Mg (3s3p, 1P0)+H (1s, 2S) Mg (3s4s, 3S)+H (1s, 2S) Mg (3s4s, 1S)+H (1s, 2S) Mg (3s4p, 3P0)+H (1s, 2S) Mg(3s3d, 1D)+H(1s, 2S) Dissociation of atomic levels Sr+H
0 20 972a 34 814a 43 677a 48 003a 50 870a 51 810a Dissociation energy limit of SrH levels (cm−1)
Sr(5s2, 1S)+H (1s, 2S) Sr(5s5p, 3P0)+H (1s, 2S) Sr(5s4d, 3D)+H (1s, 2S)
0a 12 892a 20 645a
Sr(5s5p, 1P0)+H (1s, 2S) Sr(5s4d, 1D)+H (1s, 2S)
21 118a 21 385a
a b
Molecular states of BeH X2Σ+ (2)2Σ+, (1)2Π, (1)4Σ+, (1)4Π (3)2Σ+, (2)2Π (4)2Σ+ (1)2Σ−, (3)2Π, (1)4Σ−, (2)4Π (5)2Σ+, (1)2Δ, (4)2Π Molecular states of MgH +
XΣ (2)2Σ+, (1)2Π, (1)4Σ+, (1)4Π (3)2Σ+, (2)2Π (4)2Σ+, (5)2Σ+, (2)4Σ+ (6)2Σ+ (7)2Σ+, (3)2Π, (3)4Σ+, (2)4Π (8)2Σ+, (1)2Δ, (4)2Π 2
Molecular states of SrH X2Σ+ (2)2Σ+, (1)2Π, (1)4Σ+, (1)4Π (3)2Σ+, (1)2Δ, (2)2Π (2)4Σ+, (1)4Δ, (2)4Π (3)2Π, (4)2Σ+ (5)2Σ+, (2)2Δ, (4)2Π
Total dissociation energy limit of Be+H atoms (cm−1)
Relative error (%)
0b 21 978b 42 565b 54 677b 58 907b 56 882b
0.0 6.4 2.9 3.1 2.1 6.6
Total dissociation energy limit of Mg+H atoms (cm−1)
Relative error (%)
b
0 21 850b 35 051b 41 197b 43 503b 47 844b 46 403b
0.0 4 0.7 5.7 9.4 5.9 10.4
Total dissociation energy limit of Sr+H atoms (cm−1)
Relative error (%)
0b 14 573b 18 232b
0.0 11.5 11.7
21 698b 20 149b
2.7 5.8
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Table 1. The lowest dissociation limits of BeH, MgH and SrH molecules.
Present work. Experimental values from the NIST atomic spectra Data base. N El-Kork et al
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Figure 1. Potential energy and dipole moments curves of 2Σ± and 2Δ states of BeH molecule.
astrophysical community very early [19–21]. High resolution infrared emission spectra have been measured by Seto et al [22] for MgH molecule with the record of vibration–rotation emission spectrum. The electronic transition of the strontium hydride molecule SrH has been investigated since the thirties [23–26]. Recently, experimental [27] and theoretical [28–30] studies have been occurred to investigate the low lying electronic sates and to calculate the potential energy curves with the molecular parameters. By using diode laser spectrometer Magg et al [31] and Birk et al [32] recorded three vibration–rotation bands. In the present work, we employed the ab-initio method to investigate the potential energy curves (PECs) and the static dipole moment curves (DMCs) of the low-lying 26, 27 and 25 doublet and quartet electronic states of the BeH, MgH and SrH molecules respectively. The spectroscopic constants as the equilibrium internuclear distance Re, the harmonic frequency ωe, the vibrational constant Be, the transition energy with respect to the ground state minimum Te, and the permanent dipole moment μ are calculated for the bound states of these molecules. For the study of the rovibrational problem there are many important theories and techniques in literature with computer programs as LEVEL [33–36] or the Duo program [37]. In order to obtain high order precision there is a need for large order centrifugal distortion constant as Dv, Hv, K QvK [38–44]. By using the canonical function approach one can obtain these constants with the high values of vibrational levels even near dissociation by one single and simple routine. This method provides strong evidence for our assumption that the higher-order of Dv, Hv, Lv K. and the higher vibrationel v and rotational level J for any electronic state and any type of potential energy curves (either experimental, empirical or theoretical) are as accurate as the low-order values. In this technique we use the compact form en=〈Φ0RΦn−1〉 (R=1/r) of a CDC (e1=Bv, e2=Dv, e3=Hv K..) of any order n where Φn is the solution of the differential equation Φ″n+f(r)Φn=s(r) where f(r) and s(r) are given, as well as their initial values at an arbitrary origin. This is done by using a simple method for the computation of the functions Φn, as successive solutions of the ‘rotational Schrödinger equations’ by taking Φn orthogonal to Φ0 (the vibrational wavefunction), and by deriving exact values of Φn(ro) and Φn′(ro) which are the initial values of Φn at an arbitrary point ro [38, 45–47].
2. Computational approach 2.1. Ab initio calculation In the present work an ab initio calculation of the lowest-lying electronic states of BeH, MgH and SrH has been performed via CASSCF and MRCI (single and double excitation with Davidson correction) calculations. Multireference CI calculations (MRCI) were performed to determine the correlation effects. The potential energy calculations for the states L() of the molecules have been carried by using CAS-SCF method. The calculations have been performed via the computational chemistry program MOLPRO [48] taking the advantage of the graphical user interface GABEDIT [49]. This software is intended for high level accuracy correlated ab initio
3
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Figure 2. Potential energy and dipole moments curves of 2Π states of BeH molecule.
Figure 3. Potential energy and dipole moments curves of 4Σ± and 4Δ states of BeH molecule.
calculations. MOLPRO has been run on a PC- computer with UNIX-type operating systems. For the three studied molecules BeH, MgH and SrH, the one electron hydrogen atom is treated using for s, p, d, and f functions the correlation-consistent polarized Quadruple-Zeta basis set, augmented with sets of diffuse functions aug-ccpVQZ [50]. 4
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Figure 4. Potential energy and dipole moments curves of 4Π states of BeH molecule.
In the BeH molecule, the beryllium atom is treated in all electron schemes, the 4 electrons are considered using for s, p, d, and f functions of the triple zeta basis set VTZ [51]. The quality of chosen basis sets for the H, Be, Mg, and Sr isolated atoms is checked by comparing our calculation of the lowest energy values of the asymptotic energy at each dissociative asymptote with those obtained experimentally by NIST Atomic Spectra Database [52] (table 1). This comparison shows a relative difference ranges between 0.7%ΔE/E11.7%. For some of the highest electronic states the dissociation limits are not obtained because of the undulation of potential energy curves of these states. For these undulations the short-range electronic interactions are significant at certain energy points. They are related to the singularities in the electronic Hamiltonian operator and giving rise to the Coulomb cusp in the electronic wave function and appearance of cusps in the exact wave function. Also, these undulations are explained by the breakdown of the Born-Oppenheimer (B.O) approximation since the interactions between electronic states are significant at certain energy points, and the responsible term for the socalled ‘non-adiabatic effects’ can be very important and it cannot be neglected. However, the overall good relative error given in table 1 can ensure the accuracy of our calculated data. An heteronuclear diatomic molecule belong to C∞v group. Since MOLPRO can handle only Abelian pointgroups the linear molecules are treated in C2v instead of C∞v. The states Σ+, Πz, Δ0, and Δx2−y2 belong to the irreducible representation number 1, the states Πx, Δxy belong to the irreducible representation 2, the states Πy, Δyz belong to the irreducible representation 3, and Δzx belong to the irreducible representation 4. Among the 5 electrons explicitly considered for BeH (4 electrons for beryllium and 1 electron for hydrogen) two inner electrons were frozen in subsequent calculations so that 3 valence electrons were explicitly treated. The active space contains 7 s (Be: 2s , 2p0 , 3s , 3p0 , 4s , 3d 0; H : 1s ), 3p (Be: 2p1 , 3p1 , 3d1) and 1d (Be: 3d2) orbital which corresponds to 15 active molecular orbitals distributed into irreducible representation a1, b1, b2, and a2, in the following way: 8a1, 3b1, 3b2, 1a2, noted [1, 3, 8]. For the molecule MgH, the 12 electrons of the magnesium atom are considered using for s, p, and d functions the cc-pVQZ basis set. Among the 13 electrons of the considered molecule, ten electrons were frozen in subsequent calculations, so that 3 valence electrons were explicitly treated. The active space contains 7σ (Mg: 3s, 3p0, 3d0, 4s; H: 1s, 2s, 2p0), 3π (Mg: 3p±1, 3d±1,; H: 2p±1), and 1δ (Mg: 3d±2) orbitals in the C2v symmetry; this corresponds to 15 active molecular orbitals distributed into irreducible representation a1, b1, b2, a2 in the following way: 8a1, 3b1, 3b2, 1a2, noted [1, 3, 8]. In the molecule SrH, the strontium species is considered using for s, p, d and f functions the Effective Core Potential ECP28MWB basis set, where 28 electrons are considered as inner electrons and the remaining 10 electrons are considered as valence electrons. Among the 11 electrons, 8 electrons were frozen in subsequent calculations, so that 3 electrons were explicitly treated. The active space contains 2σ (Sr: 4d0; H: 1s), 1π(Sr: 4d±1),
5
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Table 2. Positions of the crossings and avoiding crossing between the different electronic states of molecules BeH, MgH, and SrH. Crossing between (n1) state1/(n2) state2
Molecule
State 1
State 2
BeH
12Δ 12Δ 14Δ
62Σ+ 12Σ− 24Σ−
1/6 1/1 1/2
24Δ
24Σ−
2/2
24Δ
34Σ−
2/3
Rc(Å) 1.343 2.543 2.383 3.643 1.223 1.983 3.023 3.963
Avoiding crossing State 1 22Σ+ 32Σ+
32Σ+ 42Σ+
42Σ+ 52Σ+ 52Σ+
52Σ+ 12Δ 62Σ+
62Σ+ 22Π 32Π 42Π 14Σ+ 14Δ 24Π
72Σ+ 32Π 42Π 52Π 24Σ+ 24Δ 34Π
State 1
MgH
Δ Σ+
ΔE*10−3
Re(Å)
4.254 83 30.028 33 13.755 58 12.952 38 15.724 43 18.320 91 3.694 17 1.358 54 3.459 63 6.187 56 8.525 29 24.930 41 0.225 74 11.356 53
1.323 1.883 4.963 2.463 2.063 2.023 3.183 2.143 1.683 1.283 2.183 1.563 1.623 1.483
Crossing between (n1)state1/(n2)state2
Rc(Å)
State 2
State 2
1/7 1/8 1/8 1/8 2/1 5/1 5/1 6/2
Σ− Δ
2
2
4
4
1.959 682 3.759 682 4.179 682 7.119 682 1.939 682 3.799 682 6.079 682 2.539 682
Avoiding crossing State
State n+1/State n
RAC (Å)
ΔEAC (cm−1)
Σ+
Π Σ+
4/3 5/4 6/5 7/6 7/6 8/7 8/7 4/3 2/1
6.179 682 2.799 682 2.579 682 2.199 682 4.699 682 3.799 682 7.259 682 4.119 682 1.779 682
1877.87 2486.88 398.72 1412.36 2068.04 278.27 452.84 176.06 1552.91
State 1
State 2
Crossing between (n1)state1/(n2)state2
2
2 4
SrH
Δ Δ
Σ+ Σ+
2
2
4
4
Rc(Å)
1/3 2/3 Avoiding crossing
State 2 + Σ 2 + Σ 2 + Σ 2 + Σ 4 + Σ 4 Π
State n+1/State n 1/2 2/3 4/5 5/6 2/3 2/4
6
RAC (Å)
ΔEAC (cm−1)
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Figure 5. Potential energy and dipole moments curves of 2Σ± and 2Δ states of MgH molecule.
and 1δ (Sr: 4d±2) orbitals in the C2v symmetry; this corresponds to 6 active molecular orbitals distributed into irreducible representation a1, b1, b2, a2 in the following way: 3a1, 1b1, 1b2, 1a2, noted [1, 3].
3. Results 3.1. Potential energy curves For the BeH molecule, the calculations have been performed for 442 internuclear distances in the range 0.67 ÅR7.70 Å for 26 electronic states in the representation 2s+1Λ(±). The number of electronic states obtained in the present work is 14 doublet electronic states (8 in symmetry 1, 5 in symmetry 2 and1 in symmetry 4), and 12 quartet electronic states (6 in symmetry 1, 3 in symmetry 2 and 3 in symmetry 4). The potential energy with the dipole moment curves of these electronic states are given in figures 1–4 while the complete tables with the figures of these potential energy curves are given in the supplementary materials. In the considered range of R, some crossings and avoided crossings occur between some potential energy curves of at different values of internuclear distances for the doublet and quartet electronic states. The positions of these crossings and avoided crossings are given in table 2. The calculations for MgH molecule have been performed for 488 internuclear distances in the range 0.80 ÅR8.9 Å in the representation 2s+1Λ(±). The number of electronic states obtained in the present work is 17 doublet states (9 in symmetry one, 6 in symmetry two, and 2 in symmetry four) and 13 quartet electronic states (7 in symmetry one, 4 in symmetry two, and 2 in symmetry four). The potential energy with the dipole moment curves of these electronic states are given in figures 5–8 while the complete tables with the figures of these potential energy curves are given in the supplementary materials. In the considered range R, crossings and avoided crossings occur between some of the investigated potential energy curves. The positions of these crossings and avoided crossings are given in table 2. 7
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Figure 6. Potential energy and dipole moments curves of 2Π states of MgH molecule.
Figure 7. Potential energy and dipole moments curves of 4Σ± and 4Δ states of MgH molecule.
The calculations of the SrH molecule have been performed for 649 internuclear distances in the range 1.2 ǺR7.3 Ǻ in the representation 2s+1Λ(±). The number of electronic states obtained in the present work is 10 doublet electronic states (6 in symmetry one and 4 in symmetry two), and 11 quartet states (7 in symmetry one and 4 in symmetry two). The potential energy with the dipole moment curves of these electronic states are 8
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Figure 8. Potential energy and dipole moments curves of 4Π states of MgH molecule.
Figure 9. Potential energy and dipole moments curves of 2Σ± and 2Δ states of SrH molecule.
given in figures 9–12 while the complete tables with the figures of these potential energy curves are given in the supplementary materials. In the considered range R, crossings and avoided crossings happen in the doublet and quartet electronic states where their positions are listed in table 2. 9
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Figure 10. Potential energy and dipole moments curves of 2Π states of SrH molecule.
Figure 11. Potential energy and dipole moments curves of 4Σ± and 4Δ states of SrH molecule.
3.2. Dipole moment 3.2.1. BeH molecule The dipole moment operator is among the most reliably predicted physical properties. The expectation value of this operator is sensitive to the nature of the least energetic and most chemically relevant valence electrons. The HF dipole moment is usually large, as the HF wave function over estimates the ionic contribution. To obtain the best accuracy of this operator, multireference configuration interaction (MRCI) wave function were constructed using multi configuration Self-consistent field (MCSCF) active space. All the calculation were performed with 10
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Figure 12. Potential energy and dipole moments curves of 4Π and 4Φ states of SrH molecule.
the MOLPRO [48] program. The variation of the static dipole moment curves in term of the internuclear R are plotted with potential energy curves in the same figures in order to show the agreement between the positions of the avoided crossings of the potential energy curves and the crossing of the dipole moment curves. This agreement, which is represented by vertical lines, is a criteria of the validity and the accuracy of the present work for the titled molecules. The static dipole moments curves with the potential energy curves for the 3 considered molecules BeH, MgH, and SrH are given as function of the internuclear distance R in figures 1–12 while the complete tables with the figures of these static dipole moments curves are given in the supplementary materilas. The dipole moment curves of the electronic states (4)2Σ+, (5)2Π, (4)4Σ+ of BeH, (4)2Σ+, (4)4Σ+ of MgH, and (6)2Σ+ of SrH dissociate at infinity as ionic character, while these dipole moment curves of the other electronic states tends to zero at infinity.
4. Comparison and discussion 4.1. BeH molecule The calculations of the spectroscopic constants such as the vibrational harmonic constant ωe, the internuclear distance at equilibrium re, the rotation constant Be, the centrifugal distortion constant De, and the electronic transition energy with respect to the ground state Te have been done by fitting the energy values around the equilibrium position to a polynomial in terms of the internuclear distance, the degrees of these polynomials are determined from the evaluation of the statistical error for the coefficients. These values are displayed in table 3 together with the available data in literature either theoretical or experimental. Due to the avoided crossing at their minima, we have not carried out the calculations for the spectroscopic constants for the states 32∑+, 22∏, 32∏, 24∑+, 14Δ, and 24Δ. 4.1.1. Comparison with the experimental values As shown in table 3 the comparison of the present values of Te for the electronic states A2∏, C2∑+, and 42∏ with the experimental data given by Colin and DeGreef, and Colin et al [54, 64, 60], showed a very good agreement with relative differences δTe/Te=1.31%, 0.52%, 10.36%, respectively. Similar very good agreement is obtained by comparing our value of Te for the state A2∏ with that given by O’Neil and Schaefer [60, 65, 66] with relative 11
state X2 ∑+
A2∏
12 C2 ∑+
14∏ 14 ∑+ 32 ∑+(2nd min) 42 ∑+(1st min)
(2nd min)
0.00
20 267.09a 20 033j 18 270i 20 647.78b 20 002.54c 20 019.7647k 22 480.97h 22 496.12g 30 732.62a 22 367i 44 441.13b 30 407.09b 30 891.022c 39 551.13a 39 578.96a 56 187.91a 56 261.48a 78 292i 49 361.11b 48 877.17e 62 843a 63 684.76a 89 406i 54 119.78b 71 000.08a
ΔTe/Te%
Re(Å)
0.00
1.347a 1.343f 1.3457b 1.3426d 1.3424l 1.341 04m 1.341 07m 1.340 99m 1.341 44m 1.342 4n 1.338a
1.15 9.85 1.87 1.31 1.22 10.92 11
1.3377b 1.3335d 1.333f
ΔRe/Re%
2.3066b 2.301d
Δωe/ωe%
2043.93a 0.3 0.1 0.33 0.34 0.44 0.44 0.44 0.412 0.34 0.03 0.34 0.37
2.3a 27.22 44.6 1.06 0.52
ωe(cm−1)
2049b 2060.78d 2061.2352l 1987.68m 1987.51m 1987.48m 1986.55m 2061.416n 2072.68a
0.25 0.82 0.84 2.67 2.76 2.76 2.8 0.856
35.8 46.15k 37.326 66l 37.433 n
Be(cm−1)
2079b 2088.58d 2089.9530k 2088.38l
0.83 0.75
ΔBe/Be%
10.255a 10.266b 10.3164d 10.319 92l 10.1914m 10.1910m 10.1922m 10.1853m 10.319 n 10.38a
0.3 0.8
1008.22a 0.28 0.043
ωexe(cm−1)
0.1 0.6 0.63 0.62 0.624 0.612 0.68 0.624 0.1 0.74
10.39b 10.4567d 10.466 305k 10.467 15l
0.83 0.83
3.501a
3.907a 5.84a 3.3a 1.34a
1053b 1061.12d 1036.59l 2642.8a 17.44a 455.63a 1993.23a
4.34a 1.46a
315.93a 1668.15a
0.99a 8.71a
2.41a
1301.80a
3.22a
4.44 5.25 2.7
3.496b 3.5141d 3.511l 4.36a 0.55a 1.75a 10.33a
0.14 0.4 0.28
39.16 12.3 13.12
40.38 15.02
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(2nd min) 52∑+(1st min)
Te(cm−1)
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Table 3. Spectroscopic constants for the lowest electronic states of the molecule BeH.
state
Te(cm−1)
42∏
65 504.48a 78 393i 58 636.48b 58 717.139c 66 139.68a 77 741.85a 71 586.74a 54 603.71b 59 624i 74 423.80a 75 191.15a 75 590.11a 83 590.39a 77 618.63a 79 666.22a 85 242.77a 86 085.35a 88 393.12a 94 088.65a 93 337.84a 119 950.46a 120 207.15a
14∑− 24∑+ (2nd min) 12Δ
13
52∏ 12∑72∑+(1st min) (3rd min) 24∏ 62∑+(3rd min) 34∑+(1st min) (2nd min) 34∏ 44∑+ (1st min) (2nd min) 54∑−(1st min) (2nd min)
ΔTe/Te%
Re(Å)
ΔRe/Re%
ωe(cm−1)
Δωe/ωe%
ωexe(cm−1)
Be(cm−1)
1.34a
2939.75a
10.33a
1.51a 4.77a 1.58a
1520.22a 126.45a 1294.02a
8.15a 0.84a 7.49a
1.36a 1.76a 1.54a 5.31a 5.42a 5.85a 3.2a 6.2a 2.41a 2.02a 5.05a 1.62a 2.2a
2027.59a 851.35a 2071.53a 134.23a 20.87a 141.82a 314.98a 198.46a 513.90a 359.15a 479.55a 1106.51a 923.21a
10.05a 6.01a 7.86a 0.75a 0.72a 0.54a 1.82a 0.49a 3.2a 4.71a 0.73a 7.1a 3.84a
19.67 10.48 10.36
ΔBe/Be%
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Table 3. (Continued.)
23.72 16.71
a
The values given in bold are for the present work, Ref. [53], c Ref. [54], d Ref. [55], e Ref. [56], f Ref. [57], g Ref. [53], h Ref. [58], i Ref. [59], j Ref. [60], k Ref. [61], l Ref. [62] m Ref. [62] n Ref. [63]. b
N El-Kork et al
Te (cm−1)
State
ΔTe/Te
1.744 23a 1.72b 1.729 68c 1.743d 1.726 19e
0.00 (1) Σ 2
+
(1)2Π
14
19 130a 19 397b 19 278c 19 170d 14 704e 19 216.8f
Re(Ao)
1.40% 0.77% 0.21% 23.14% 0.45%
1.724g 1.713h 1.730i 1.69a 1.67b 1.679c 1.699d 1.6612e g
(2) Σ 2
+
(1)4Π
(3)2Σ+
0.55% 1.06% 0.63% 16.95%
2.35% 1.93% 0.62%
1.39% 0.83% 0.07% 1.03% 1.16% 1.79% 0.81% 1.18% 0.65% 0.53% 1.70%
1.668 1.673h 1.680i 2.63a 2.59b 2.596c 2.622d 2.617e 3.96a 1.71a
1.30% 1.00% 0.61%
1.67b 1.67c 4.1a
2.34% 2.34% 1.22%
1.52% 1.29% 0.30% 0.49%
4.05b 1.72a 5.80%
1.73b
ωe(cm−1) 1486.61a 1453.66b 1495.26c 1484.4d 1494.68e 1495.25f 1598g 1597h 1497i 1570.66a 1551.33b 1598.4c 1561.8d 1572.02e 1599.5f 1742g 1692h 1598i 812.35a 831.47b 828.4c 818.6d 789.64e 69.35a 1483.93a 1498.43b 1445.4c 392.53a
Δωe/ωe
2.22% 0.58% 0.15% 0.54% 0.58% 6.97% 7.43% 0.70% 1.23% 1.77% 0.56% 0.09% 1.84% 11% 7.72% 1.36%
1722.12b
30.57a 29.43b 31.64c 30.989d 34.25e 31.637f 21.5g 25.7h 32.4i 32.56a 32.15b 31.9c 33.17d 28.046e 32.536f 22.4g 30.0h 31.9i
Δωexe/ωexe
Be
ΔBe/ Be
5.72a 3.73% 3.50% 1.37% 12.04% 3.50% 29.67% 15.93% 5.98% 1.26% 2.03% 1.87% 13.86% 0.07% 31.20% 7.86% 2.03%
5.6443f 5.86g 5.94h 5.818i 6.03a
1.32% 2.45% 3.32% 1.71%
6.1913f 6.266g 6.231h 6.178i 2.51a
2.67% 3.75% 3.33% 2.45%
2.35% 1.97% 0.77% 2.79% 8.27a 46.38a 0.97% 2.60% 5.34%
371.56b 1763.178a 0.58%
ωexe(cm−1)
2.33%
50.33b
1.1a 5.93a 8.52%
1.72a
1.03a
51.44a
5.86a
47.98b
6.73%
N El-Kork et al
22 174a 22 051b 22 410c 22 034d 18 414e 32 409a 1st min: 36 251a 35 401b 35 551c 2nd min: 41 325a 41 069b 1st min: 40 454a 38 106b
ΔRe/Re
J. Phys. Commun. 2 (2018) 055030
Table 4. Spectroscopic constants for the lowest electronic states of the molecule MgH.
Te (cm−1)
(4)2Σ+
38 485c 2nd min: 47 211a 46 833b 1smin: 45 679a 41 550b 41 240c 2nd min: 44 883a 1st min: 46 561a 43 138b 2nd min: 53 795a 48 973a 42 861b 42 573c 49 403a 1st min: 57 950a 2nd min: 61 749a 58 211a 42 790b 2nd min: 59 187a 59 309a 1st min: 60 623a 2nd min: 61 936a 1st min: 60 672a 2nd min: 62 642a
(2) Π 2
15
(5)2Σ+
(3) Π 2
(4)2Π (7)2Σ+
(1)2Δ (6)2Σ+ (5)2Π (3)4Σ+
(8)2Σ+
ΔTe/Te 4.87%
0.80% 9.04% 9.72%
Re(Ao)
ΔRe/Re
1.79c 5.5a
4.07%
5.35b 1.69a 1.64b 1.682c 2.49a
2.72%
12.48% 13.07%
1.69b 2.77a 1.92a 1.67b 1.68c 1.89a 2.34a
2.96% 0.47%
1.75a 1.64b 6.7a 1.8a 2.83a 8.12a
ωexe(cm−1)
218.52b 1297.302a 1528.52b 1740c 461.78a
3.98%
13.02% 12.50%
1631.23b 1134.19a 1171.26a 1466.61b
32.56a
1360.3a 491.82a
5.95a
ΔBe/ Be
17.82% 34.12% 571.9a
2.78a 5.62a
11.11% 1.9a
2.25a 4.66a
25.22% 14.54a 335.27a
259.82a 1260.5a 1633b 7.58a
Be
6.60%
1768.37a 752.16a
6.28%
Δωexe/ωexe
0.57a
1468.05a
4.7a
26.50%
Δωe/ωe
205a
1.76a 7.35%
ωe(cm−1)
0.015a
4.85a 3.13a 0.77a 5.65a
29.55% 1188.62a
0.35a
49.38a 13.47a
5.35a 2.17a
35 011.7a
0.217a
2.13a
1403.69a
232.54a
3.82a
3.77
774.04a
5.16a
1.22a
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State
J. Phys. Commun. 2 (2018) 055030
Table 4. (Continued.)
16
State
Te (cm−1)
(1)4Σ− (2)4Π (4)4Σ+ (6)2Π
62 049a 62 107a 65 913a 1st min: 66 156a 2nd min: 65 621a 66 162a 1st min: 67 353a 2nd min: 67 754a 73 653a 76 894a 89 469a 91 352a
(9)2Σ− (3)4Π
(5)4Σ+ (1)4Δ (2)4Δ (6)4Σ+
ΔTe/Te
Re(Ao)
ΔRe/Re
ωe(cm−1)
Δωe/ωe
ωexe(cm−1)
1.95a 3.7a 3.5a 2.4a
1010.97a 119.39a 282.73a 560.22a
59.33a 46.9a 6.49a 352.85a
3.65a
268.46a
4.27a
2.19a 2.42a
585.68a 589.82a
129.56a 7.48a
6.18a 4.85a 4.05a 1.95a 2.05a
Δωexe/ωexe
Be 4.56a 1.25a 1.42a 2.96a
ΔBe/ Be
J. Phys. Commun. 2 (2018) 055030
Table 4. (Continued.)
1.3a 3.6a 2.97a 0.25a
508.75a 68.02a 1042.05a 738.4a
1200.23a 42.27a 56.08a 118.66a
0.69a 1.03a 4.55a 4.1a
a
Present work, Ref. [39], c Ref. [43], d Ref. [40], e Ref. [44], f Ref. [41], g Ref. [42], h Ref. [42]. b
N El-Kork et al
J. Phys. Commun. 2 (2018) 055030
N El-Kork et al
Table 5. Spectroscopic constants of the low-lying electronic states of SrH molecule State X2∑+
(1)2∏
(2)2∑+
(1)2Δ (2)2∏ (1)4∏ (1)4∑+ (3)2∑+
(3)2∏ (2)4∏ (1)4Δ (4)2∏ (2)2Δ (3)4∏ (1)4Φ (2)4Δ
Re(A) 2.211a 2.210b 2.160c 2.146d,e 2.172a 2.160b 2.121d 2.160a(1) 3.454a(2) 2.200b(1) 2.117d(1) 2.217a 2.170b 2.178 5.081 8.089 2.122(1) 3.874(2) 6.633(3) 2.816 4.57 6.299 2.784 4.876 2.531 2.851 2.465
% ΔRe/Re
0 2.3 3
0.5 2.4
1.8 2 2.1
ωe (cm−1)
%Δωe/ωe
1166.2a 1167b 1206.2c 1207d 1205.6e 1214.5a 1175b 1253.9d 1192a(1) 463a(2) 1288b(1) 1234.3d(1) 1131a 1043b 1150 103.6 14.17 1227(1) 290.5(2) 82.4(3) 483 41.7 106.9 342.7 78.5 587.4 339.1 641.6
0 3.2 3.3 3.2 3.3 3.1
7.4 3.4 8.4
Be (cm−1) 3.45a 3.43b 3.63c 3.67d,e 3.58a 3.57b 3.72d 3.62a(1) 1.41a(2) 3.47b(1) 3.73d(1) 3.46a 3.44b 3.56 0.65 0.26 3.75(1) 1.12(2) 0.38(3) 2.13 0.81 0.42 2.14 0.71 0.26 0.2 0.27
% ΔBe/Be
Te (cm−1)
% ΔTe/Te
0 0.6 4.7 5.8
0.2 3.7
4.3 2.9 0.6
13 112a 13 804b 13 500.6d 14 734a(1) 18 413a(2) 14 952b(1) 14 312.7d(1) 19 174a 16 851b 21 107 27 042 27 054 31 178(1) 34 418(2) 35 043(3) 33 222 34 774 34 804.7 34 929 35 497 45 919 47 062 49 809
5 2.8
1.4 2.9 13.7
a
Present work, Ref. [45], c Ref. [42], d Ref. [38], e Ref. [46]. b
difference δTe/Te=1.15%. For the 42∑+ state, the comparison between our calculated value of Te with that given by Lefebvre-Brion and Colin, and Pitarch-Ruiz et al [56, 67], showed a less agreement with a relative error of δTe/Te=13.12%. The comparison between our calculated values for Re, ωe, and Be for the electronic states states X2∑+, A2∏, and C2∑+ with those obtained experimentally given by Colin and DeGreef [64], showed a very good agreement with relative differences δRe/Re=0.33%, δωe/ωe=0.82%, and δBe/Be=0.6% for X2∑+, δRe/Re=0.34%, δωe/ωe=0.8%, and δBe/Be=0.74% for A2∏, andδRe/Re=0.043%, δωe/ωe=5.25%, and δBe/Be=0.4%. 4.1.2. Comparison with the theoretical values The comparison between our calculated values of Te with the calculated theoretical values given in literature by Matchado et al and Petsalakis et al [55, 68] shows a very good agreements for the electronic states A2∏ and C2∑+ with relative differences 1.87% and 1.06% respectively, and less agreements for the states 42∑+, 42∏, and 52∑+ with the relative differences 12.3%, 15.02% 10.48% respectively. A similar less agreement is obtained by comparing our values of Te with those of Cade and Huo [53], Leif et al [58], and Frank et al [59] with relative differences 10.9%, 11.0% and 9.8% respectively for the electronic state A2∏. The agreement deteriorate by comparing our calculated values of Te with those given in literature for the electronic states 12Δ [59, 68], C2∑+, 42∑+, 52∑+, 42∏ [59]. By comparing our calculated values for Re, ωe, and Be for the electronic states states X2∑+, A2∏, and C2∑+ with those obtained theoretically by Petsalakis et al [68], one can find an excellent agreement with relative differences δRe/Re=0.33%, δωe/ωe=0.25%, and δBe/Be=0.1% for X2∑+, δRe/Re=0.03%, δωe/ωe=0.3%, and δBe/Be=0.1% for A2∏, and δRe/Re=0.28%, δωe/ωe=4.44%, and δBe/Be=0.14% for C2∑+. Similar excellent agreement is obtained by comaparing our values of these constants with those of Focsa et al [61] with the relative differences 0.3% and 0.37% for X2∑+ and A2∏ states respectively. The 17
J. Phys. Commun. 2 (2018) 055030
N El-Kork et al
Figure 13. Transition dipole moment curves between the ground state X2Σ+ and the (2, 3)2Σ+ and (1, 2)2Π states of the molecule BeH.
comparison between the present values for Re and those given by Cade and Huo, and Herzberg [57, 69] showed an excellent agreement with the relative differences 0.3% and 0.37% for X2∑+ and A2∏ states respectively. 4.2. MgH molecule In the present work, the calculated spectroscopic constants of the molecule MgH are given in table 4 along with those found in the literature either experimentally or theoretically. The comparison of our calculated values of the vibrational harmonic frequency constant ωe for the ground state X2∑+ with those given in literature shows a very good agreement with the relative difference Δωe/ωe equal 2.22% [70], 0.15% [71], 0.54% [72, 73], 0.58% [74]. A less accuracy is obtained for ωe by comparing our values with those given in [73] by using MC and HF methods where Δωe/ωe are respectively 6.97% and 7.43%. For the values of Re, there is a very good agreement between our result and those given in [70–75] where 0.07%ΔRe/Re1.79%. As for the values of Be, there is also a very good agreement between our result and the ones given in [72, 73, 75] where 1.32%ΔBe/Be 3.32%. For the first excited state (1)2∏ the comparison of the 5 values of Te, the 7 values of Re, the 8 values of ωe, and the 4 values of Be found in literature [72, 73, 75] with those of the present work showed a very good agreement 0.21%δTe/Te1.4%, 0.53%ΔRe/Re1.7%, 0.09%Δωe/ωe1.84%, 2.45% ΔBe/Be3.75%. A less agreement is noticed by comparing our results with those calcultated by using MC and HF methods [73]. The comparison between our results and those found in literature for the excited state (2)2∑+ shows also the very good agreement with the relative differences 0.55%ΔTe/Te1.06% (except for the result given in [75] where the relative error increases to 16.95%), 0.3%ΔRe/Re1.52%, and 0.77%Δωe/ωe2.79%. No comparison for the value of Be since it is given here for the first time. The comparison of our results with those given by J-M Mestdagh et al [70] for the excited state (3)2Σ+ shows a very good agreement for the calculated spectroscopic constants where ΔTe/Te=2.35%, ΔRe/Re=2.34%, and Δωe/ωe=0.97% for the first minimum. For the second minimum of the same state, we got ΔTe/Te=0.62%, ΔRe/Re=1.22%, and Δωe/ωe=5.34%. By comparing our results with the experimental ones [74], we get very good results for the first minimum of the same state where ΔTe/Te=1.93%, ΔRe/Re=2.34%, and Δωe/ωe= 2.6%. No comparison for the value of Be since it is given here for the first time also. The same table 4 shows the very close agreement between our work and that obtained by the experimental work [74]and Mestdagh et al [70] for the state (4)4Σ+ where, for the first minimum, 4.87%ΔTe/Te5.8%, 0.58% ΔRe/Re4.07%, and Δωe/ωe=2.33%. For the second minimum, ΔTe/Te=0.8%, ΔRe/Re=2.72%, and Δωe/ωe=6.6%
18
J. Phys. Commun. 2 (2018) 055030
N El-Kork et al
Figure 14. Transition dipole moment curves between the ground state X2Σ+ and the (2, 3)2Σ+ and (1, 2)2Π states of the molecule MgH.
Figure 15. Transition dipole moment curves between the ground state X2Σ+ and the (2, 3)2Σ+ and (1, 2)2Π states of the molecule SrH.
compared to [70]. The results of Be are also found for the first time for the considered state. The values obtained for the internuclear distance at equilibrium Re and the electronic transition energy with respect to the ground states Te for first minima of both states (2)2∏ and (5)2∑+ shows a good agreement compared to [70], but the accuracy becomes less for the values of the vibrational harmonic constant ωe compared to the same reference. The accuracy deteriorates by comparing our results with those given by Mestdagh et al [70] for the states (3)2∏ and (1)2Δ. It was not possible to obtain such constants for some states that made avoided crossing with a neighbor one, these states are given in table 2.
19
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4.3. SrH molecule The comparison of our calculated values in the present work with those given in literature (table 5) for the constants ωe, and Be for the ground state X2Σ+ shows very good agreemenst with the relative differences Δωe/ωe equal 0.0% [76], 3.2% [32], 3.3% [27] and ΔBe/Be=0.6% [45], 5.8% [32], 4.7% [27]. The comparison of our results for the state (1)2∏ with those given theoretically by Leininger and Jeung [64] shows a very good agreement for Te, ωe and Be with the relative difference ΔTe/Te=5%, Δωe/ωe=3.3%, and ΔBe/Be=0.2%. While the comparison of these constants with those obtained experimentally [27] shows also very good agreement with the relative differences ΔTe/Te=2.8%, Δωe/ωe=3.1% ΔBe/Be=3.7%. The comparison between our results and those obtained theoretically in literature for the excited state (2)2Σ+ shows also the very good agreement with the relative difference: ΔTe/Te=1.4%, Δωe/ωe=7.4% and ΔBe/Be=4.3% with the results of Leininger and Jeung [76]. Similar results are obtained by comparing our data with those obtained experimentally [27] with the relative difference ΔTe/Te=2.9%, Δωe/ωe=3.4% and ΔBe/Be=2.9%. The spectroscopic constants of the state (1)2Δ are compared only with the theoretical data given by Leininger and Jeung [76] since there is no experimental data is available yet. It shows a less agreement for Te with a relative difference ΔTe/Te=13.7% which is may be due to an upward displacement in the potential energy. However, the relative error for ωe shows somehow a good agreement with a relative difference Δωe/ωe=8.4% and an excellent agreement for Be with a relative difference ΔBe/Be=0.6%. The three published values for Re for the state X2Σ+ shows a very good agreement with our calculated value with relative differences 0.0% [22, 76]
