Journal of Quantitative Spectroscopy & Radiative Transfer 201 (2017) 64–74
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Theoretical electronic structure with dipole moment and rovibrational calculation of the low-lying electronic states of the HgF molecule Soumaya Elmoussaoui, Wael Chmaisani, Mahmoud Korek∗ Beirut Arab University, Faculty of Science, 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 25 March 2017 Revised 20 May 2017 Accepted 30 June 2017 Available online 4 July 2017 Keywords: Ab initio calculation Electronic structure Static and transition dipole moment Ionic fraction Einstein spontaneous emission constants Vibrational levels Franck–Condon factor
a b s t r a c t With the lack of evitable data about the electronic structure of the low-lying states of the HgF molecule unlike its counter parts of mercury monohalides, spectroscopic study of eighteen low-lying doublet and quartet electronic states in the representation 2s+1 (+\-) of this molecule is conducted. Adiabatic potential energy curves of those states are investigated using the complete active space self consistent field (CASSCF) calculation with multi-reference configuration interaction (MRCI) method including single and double excitations with Davidson correction (+Q). The spectroscopic constants as Re , ωe , ωe xe , Be , α e , Te and μe are calculated for the bound states. The dissociation energy D00 and the percentage ionic character fionic are also computed. The transition dipole moment μTDM between some doublet low-lying states is studied and some emission coefficients as the Einstein spontaneous coefficients A21 , the spontaneous radiative lifetime τ spon , and the oscillator strength f21 are thus determined. A rovibrational study has been done to investigate the vibrational levels of the low-lying bound states and the vibrational constants Ev , Bv , Dv , Rmin and Rmax are reported. The Franck–Condon factors have also been calculated for the most probable transitions between the excited states and the ground state. The comparison between the values of the present work and those short information available in the literature shows good accordance. © 2017 Published by Elsevier Ltd.
1. Introduction Ever since the development of spectroscopic studies, there has been an interest in studying various diatomic molecular systems in their gas phase. By the end of the 1970s, the search for efficient, high-power diatomic electronic transition lasers in the visible and ultraviolet region has drawn special attention to one of the more promising candidates, the mercury monohalides, in particular. These coherent radiation sources are technologically important as they were shown to be the most powerful gas lasers in the visible spectral range. In literature, the ground state and the B−X transition of the HgX (X: F, Cl, Br, I) diatomic molecules have been the subject of many experimental [1–8] and theoretical studies [9–17], but few have included their excited states other than the B-state [18–23]. Liao et al. [9] studied, theoretically, the free molecules of HgX (X: F, Cl, Br, I) in the solid state using the local density-functional method to determine the ground state vibrational frequency and bond length. Those two ground state spectroscopic constants are also calculated by Khalizov et al. [10] from electronic structure calculations performed using DFT and highlevel ab initio methods. High-level relativistic ab initio calculations
∗
Corresponding author. E-mail addresses:
[email protected],
[email protected] (M. Korek).
http://dx.doi.org/10.1016/j.jqsrt.2017.06.040 0022-4073/© 2017 Published by Elsevier Ltd.
as the CCSD(T) and the normalized elimination of the small component (NESC) methods are employed by Cremer et al. [11] to test the possible bonding situations that may occur between Hg and other atoms or small radicals. Spin-orbit density functional study of HgXn (X=F, Cl, Br, and I; n = 1, 2, and 4) in the gas phase is conducted by Kim et al. [12] using Relativistic Effective Core Potentials (RECP) to determine the ground state molecular geometry and vibrational frequency. Motivated by the scarcity of information about the HgF molecule, Howell [18] studied its emission spectrum as developed in a high frequency discharge. He reported the vibrational analysis up to v = 9 of both states corresponding to the C2 −X2 + transition. The ground state rotational constant Be was calculated by Dmitriev et al. [19] while calculating the spin rotational Hamiltonian for HgF. By summarizing the information about the electronic structure of the mercury monohalide molecules available in literature, it is noticed that fewer studies have considered the HgF [9–12] molecule than its counterparts HgCl [9–16,18,20–22], HgBr [9–18,21,22] and HgI [9,11–15,18,21–23]. The lack of interest concerning HgF may be referred to the experimental difficulties arising from fluorine which is the most oxidizing and most electronegative element and has a small radius. On the other hand, convenient basis sets and active space are crucial for ab-initio calculations considering excited states, therefore, available studies have mostly considered the ground state. For this reason and consid-
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ering the lack of evitable data on the excited states other than the two lowest ones, we were motivated to conduct a spectroscopic study to investigate the low-lying electronic states of the mercury monofluoride (HgF) molecule. Ab-initio calculations are carried out to plot the potential energy curves (PECs) and the static dipole moment curves (DMCs) of 18 low-lying doublet and quartet electronic states of HgF correlating with the three lowest dissociation asymptotes by considering 9 and 11 valence electrons consecutively while treating different number of core electrons for the mercury atom. The spectroscopic constants as the equilibrium internuclear distance Re , the harmonic frequency ωe , the anharmonicity correction ωe xe , the rotational constant Be , the rotational-vibrational constant α e , the transition energy with respect to the ground state minimum Te , and the permanent dipole moment μe are calculated for the bound states. By using these constants, the dissociation energy D00 and the ionic factor fionic are also computed. The comparison between the values of the present work and those available in the literature shows very good agreement and coherency among the other mercury monohalide molecules [3,7–16,18–22]. The behavior of sixteen excited electronic states are presented by this work for the first time. The transition dipole moment between some doublet low-lying states is studied and employed to determine some emission coefficients as the Einstein spontaneous coefficients A21 , the spontaneous radiative lifetime τ spon , and the oscillator strength f21 . Motivated by the insufficient knowledge about the vibrational levels and the corresponding transitions that can take place among the three lowest bound states of HgF molecule, vibrational levels with J = 0 are investigated and several corresponding vibrational constants are calculated. Further calculations are done to investigate the Franck–Condon factors that mirror the diagonal transition probability between the low-lying excited states and the ground state. 2. Computational method With the complete active space self consistent field (CASSCF) wave functions being taken as reference, calculations are performed using the multireference configuration interaction (MRCI) method with Davidson correction (+Q), single and double excitations, as applied in the computational chemistry program MOLPRO [24] and taking the advantage of the graphical user interface GABEDIT [25]. The calculations are performed with four different sets of core-valence electronic selections. For the 80 electrons of the neutral mercury atom, the energy-consistent quasi-relativistic effective core potential ECP78MWB [26] basis set is used to treat 78 electrons within the core and 2 electrons as valence. On the other hand, the nine electrons of the neutral fluorine atom are described by the all electron augmented correlation-consistent quadruplezeta (aug-cc-pCVQZ) basis set [27] that includes polarized and diffuse functions. Among the 89 electrons considered for the mercury monohalide molecule, the wavefunctions of 11 electrons are to be determined. While considering nine valence electrons for the HgF molecule, the 1 s orbital of the F atom is kept doubly occupied and frozen in the subsequent calculations. The active space thus contains 6σ (Hg: 6 s, 6p0 , 7 s; F: 2 s, 2p0 , 3 s) and 2π (Hg: 6p±1 ; F: 2p±1 ) orbitals in the C2v symmetry group distributed into irreducible representations as 6a1 , 2b1 , 2b2 , 0a2 noted by [6, 2, 2, 0]. For the calculations with 11 valence electrons, the two electrons occupying the 1 s orbital are considered as valence adding a σ molecular orbital to the active space represented by [7, 2, 2, 0] in irreducible representations. In order to study the influence of the core electrons of mercury on the electronic structure, calculations with similar active spaces and irreducible representations are carried out with the use of ECP60MWB basis set [28,29] for Hg which treats 60 electrons within the core and 20 electrons as valence. While for the fluorine atom the same basis set (aug-cc-pCVQZ for
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all the 9 electrons) is used. In the subsequent calculations, the electrons occupying the 5s2 ,5p6 and 5d10 orbitals of Hg are frozen such that 2 electrons of Hg are left to be explicitly treated as valence electrons. The two electrons in the 1s orbital of F atom, are initially frozen then freed summing up to 9 then 11 valence electrons, respectively. In the range of the internuclear distance R around the equilibrium position of its ground state, the HgF molecule is assumed mainly ionic. As function of R, the potential energy curves and static dipole moment curves of the doublet and quartet states in the representation of 2s+1 ± are calculated for at least 114 internuclear distances in the range 1.41 A˚ ≤ R ≤ 4.77 A˚ 3. Results and discussion By employing the four preceding combinations of basis sets, eighteen low-lying states correlating with the three lowest dissociation asymptotes are investigated; 12 doublet states (four 2 + , four 2 , two 2 and two 2 − ) and 6 quartets (two 4 + , two 4 , one 4 and one 4 − ). The atomic fragments of HgF at the molecular dissociation asymptotes of the correlating molecular states and the calculated asymptotic energy are provided in Table 1. The (2)2 + state is polarized at the dissociation limit as it dissociates into Hg+ (2 S) and F− (1 S) ionic fragments. The comparison with the experimental energy values of the natural dissociative asymptotes provided by the NIST atomic spectra database [30] shows acceptable accordance for all the selected basis sets where the relative difference ranges between 2.53% ≤ E/Eexp ≤ 8.52%. The PECs and DMCs of the electronic states of interest are plotted in terms of the internuclear distance R and given in Figs. 1 and 2. These curves which are obtained using (Hg: ECP78MWB, F: aug-cc-pCVQZ with nine valence electrons) describe nearly identical behavior to those obtained using the remaining sets of calculation. Among the investigated states, seven are found to be bounded. The energy data of these states around the equilibrium position is fitted into a polynomial in terms of the internuclear distance R in order to calculate their spectroscopic constants Re , Te , ωe, ωe xe , Be , α e , and μe . The calculated constants with the experimental and theoretical values available in the literature are given in Table 2. This table also gives the values of the dissociation energy D00 as estimated in terms of the calculated molecular spectroscopic constants ωe and ωe xe [31]. In the absence of any experimentally determined ground state X2 + equilibrium distance, good agreement is noticed when comparing our calculated bond distances obtained using (Hg: ECP60MWB) to those available in literature where the relative difference ranges between 0.0% ≤ Re /Re ≤ 6.6%. Nevertheless, the harmonic frequency values ωe obtained using (Hg: ECP78MWB) are in very good agreement with those reported from both experimental and theoretical methods such that the relative difference range is 3.5% ≤ ωe /ωe ≤ 5.9%. The ground state dissociation energy in literature varies significantly upon varying the method of determination, e.g., method of calculation, selected basis sets, chosen active space … etc. The theoretical values and the corresponding limitations are weighted upon comparison with experimental data that is unfortunately rare. It is noticed that the number of core electrons have affected mainly the ground state dissociation energy, the transition energies of the (2)2 + state and the highly excited electronic states. Although the detection of the B2 + −X2 + transition of mercury monohalide molecules (HgX, X: Cl, Br, I) was a milestone in laser technology and has appealed the attention of several experimental and theoretical researchers, this transition is peculiarly neglected when it comes to the HgF molecule. Nothing is reported about the (2)2 + state of HgF although papers had been published including this state of the monohalides of cadmium and zinc, all of which have exhibited sort of similarity of electronic constitution and spectra. In order to judge between the transition energy values of the (2)2 +
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S. Elmoussaoui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 201 (2017) 64–74 Table 1 Atomic fragments at the natural molecular dissociation asymptotes of HgF and the correlating molecular states.
First asymptote Hg (1 Sg ) + F (2 Pu ) Second asymptote Hg (3 Pu ) + F (2 Pu )
Third asymptote Hg (1 Pu ) + F (2 Pu )
Correlating states
Eexp † (cm−1 )
Ecal (cm−1 )
%|Eexp − Ecal |/Eexp
X2 + , (1)2
0
0
–
(3)2 + , (4)2 + , (2)2 , (3)2 , (1)2 , (1)2 − ,(1)4 + , (2)4 + , (1)4 , (2)4 , (1)4 , (1)4 −
37,645 (J = 0) 39,412 (J = 1) 44,043 (J = 2) 40,367∗ 54,069
35,955(a1) 35,248(a2) 35,973(a3) 36,691(a4) 50,835(a1) 50,717(a2) 49,462(a3) 50,586(a4)
10.93(a1) 12.68(a2) 10.89(a3) 9.11(a4) 5.98(a1) 6.20(a2) 8.52(a3) 6.44(a4)
(4)2 , (2)2 , (2)2 −
This work: MRCI+Q calculations. F: aug-cc-pCVQZ basis set with: (a1) 9 valence electrons, Hg: ECP78MWB, (a2) 11 valence electrons, Hg: ECP78MWB, (a3) 9 valence electrons, Hg: ECP60MWB, (a4) 11 valence electrons, Hg: ECP60MWB † Ref. [30] ∗ Averaged over the fine structure levels (J = 0, 1, 2) and theoretical values are compared with. (a)
Fig. 1. Potential energy curves (PEC) of the doublet and quartet electronic states of the HgF molecule.
state that are obtained by the two different ECPs, we had to predict the behavior of this state by deductive reasoning. Since both zinc and mercury are group IIB elements and share many unique characteristics due to the weak metallic bonding of their outer most 4s2 /6s2 electrons, respectively, and upon comparing the transition energy values of this state among the other mercury monohalides [1–5,7,8,14–16,21,22], it is concluded that a behavior similar to that
of zinc monohalides [32–35] can be expected. The detailed analysis related to the electronegativity of the associated halogen is provided in a previous work [32] concerning the comparison among the excited states of the four zinc monohalides supported the prediction that the spectroscopic constants of the (2)2 + state obtained by using the ECP78MWB basis set for mercury are more reasonable. This conclusion has been verified by the constants cal-
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Fig. 2. Static dipole moment curves (DMC) of the doublet and quartet electronic states of the HgF molecule.
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S. Elmoussaoui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 201 (2017) 64–74 Table 2 Spectroscopic constants of the bound molecular states of HgF; the equilibrium bond distance (Re ), the transition energy with respect to the ground state minimum (Te ), the vibrational frequency (ωe ), the anharmonicity correction (ωe xe ), the rotational constant (Be ), the vibrational-rotational constant (α e ), the permanent dipole moment (μe ), the dissociation energy (D00 ) and the ionic character (fionic ). State
˚ Re (A)
Te (cm−1 )
ωe (cm−1 )
X2 +
1.913(a1) 1.911(a2) 2.136(a3) 2.110(a4)
0
461.8(a1) 464.7(a2) 434.5(a3) 427.6(a4) 490.8(b) 439.0(c) 486.2(d)
2.100(c) 2.048(d) 2.080(e) 2.003(f) 2.110(g)
(2)2 +
(2)2
(3)2 + (1)4 +
(1)4
(4)2 +
2.878(a1) 2.841(a2) 2.751(a3) 2.714(a4) 1.885(a1) 1.882(a2) 2.090(a3) 2.092(a4) 2.230(g) 1.903(a1) 1.901(a2) 2.931(a1) 2.951(a2) 3.377(a3) 3.397(a4) 3.398(a1) 3.447(a2) 3.649(a3) 3.767(a4) 2.316(a1) 2.295(a2)
ωe xe (cm−1 ) 3.79(a1) 3.56(a2) 4.20(a3) 3.20(a4) 4.05(b)
446.0(g)
25,699(a1) 26,051(a2) 19,230(a3) 20,097(a4) 38,935(a1) 39,141(a2) 38,152(a3) 38,462(a4) 39,060(b) 35,910(g) 52,114(a1) 51,954(a2) 52,921(a1) 52,733(a2) 43,988(a3) 45,025(a4) 53,208(a1) 53,008(a2) 44,151(a3) 45,118(a4) 58,051(a1) 59,162(a2)
231.8(a1) 238.2(a2) 271.1(a3) 269.0(a4) 521.2(a1) 538.7(a2) 480.5(a3) 468.6(a4) 506.0(b) 333.0(g) 463.9(a1) 457.1(a2) 55.7(a1) 54.5(a2) 42.2(a3) 36.2(a4) 25.8(a1) 23.5(a2) 22.6(a3) 23.0(a4) 339.5(a1) 331.7(a2)
Be (cm−1 )
α e (cm−1 )
μe (a.u.)
D00 ∗ (cm−1 )
fionic †
0.2654(a1) 0.2660(a2) 0.2130(a3) 0.2181(a4)
0.0025(a1) 0.0024(a2) 0.0022(a3) 0.0019(a4)
1.57(a1) 1.56(a2) 1.60(a3) 1.49(a4)
13,831(a1) 14,933(a2) 11,021(a3) 14,062(a4) 14,625(b)
0.43(a1) 0.43(a2) 0.40(a3) 0.37(a4)
1.30(e)
11,746(e) 14,357(f)
0.33(e)
0.2220(g)
0.26(a1) 0.29(a2) 0.34(a3) 0.53(a4) 10.51(a1) 10.10(a2) 5.38(a3) 10.33(a4) 10.05(b)
0.1173(a1) 0.1203(a2) 0.1283(a3) 0.1319(a4) 0.2733(a1) 0.2744(a2) 0.2224(a3) 0.2223(a4)
0.0 0 018(a1) 0.0 0 020(a2) 0.0 0 023(a3) 0.0 0 039(a4) 0.0045(a1) 0.0043(a2) 0.0040(a3) 0.0037(a4)
2.38(a1) 2.21(a2) 2.12(a3) 1.96(a4) 1.53(a1) 1.48(a2) 1.31(a3) 1.30(a4)
50,605(a1) 49,046(a2) 54,595(a3) 33,982(a4) 6205(a1) 6914(a2) 9702(a3) 5084(a4)
0.41(g) 0.49(h)†† 0.36(i)†† 0.49(j)†† 0.44(a1) 0.41(a2) 0.41(a3) 0.38(a4) 0.43(a1) 0.42(a2) 0.33(a3) 0.33(a4)
2.74(a1) 3.47(a2) 1.88(a1) 1.90(a2) 1.37(a3) 1.71(a4) 1.53(a1) 1.72(a2) 1.79(a3) 1.56(a4) 3.15(a1) 3.25(a2)
0.2681(a1) 0.2687(a2) 0.1131(a1) 0.1115(a2) 0.0851(a3) 0.0839(a4) 0.0839(a1) 0.0814(a2) 0.0711(a3) 0.0682(a4) 0.1811(a1) 0.1846(a2)
0.0020(a1) 0.0025(a2) 0.0066(a1) 0.0070(a2) 0.0063(a3) 0.0064(a4) 0.0054(a1) 0.0061(a2) 0.0056(a3) 0.0046(a4) 0.0018(a1) 0.0020(a2)
1.08(g) 1.19(a1) 1.19(a2) 0.16(a1) 0.16(a1) 0.06(a1) 0.05(a1) 0.07(a1) 0.07(a2) 0.04(a3) 0.04(a4) 0.37(a1) 0.28(a2)
19,435(a1) 14,843(a2) 179(a1) 153(a2) 80(a3) 78(a4) 95(a1) 73(a2) 60(a3) 76(a4) 8969(a1) 8298(a2)
0.26(g) 0.33(a1) 0.33(a2) 0.03(a1) 0.03(a2) 0.01(a3) 0.01(a4) 0.01(a1) 0.01(a2) 0.01(a3) 0.01(a4) 0.08(a1) 0.06(a2)
This work: MRCI+Q calculations. F: aug-cc-pCVQZ basis set with: 9 valence electrons, Hg: ECP78MWB, (a2) 11 valence electrons, Hg: ECP78MWB, (a3) 9 valence electrons, Hg: ECP60MWB, (a4) 11 valence electrons, Hg: ECP60MWB (b) Ref. [18]Exp , (c) Ref. [9]DFT , (d) Ref. [10]QCISD/L2&G6 , (e) Ref. [11]NESC , (f) Ref. [12]CCSD(T) ,
1.63(g)
(a)
(a1)
ω2
(g)
Ref. [19]SCF with ARECP basis set .
Ref. [31], D00 = 4 ωee xe − ω2e + ωe4xe † Ref. [36], fionic = μ/eRe †† From electronegativity calculations using Pauling electronegativity χ F = 4, χ Hg = 1.9; (h) Ref. [37], fionic = 0.16 (χ F − χ Hg ) + 0.035 (χ F − χ Hg )2 (i) Ref. [37], fionic = (χ F − χ Hg )/(χ F + χ Hg ) (j) Ref. [38], fionic = 1 − Exp ( − [(χ F − χ Hg )3/2 /(χ F .χ Hg )3/4 ]) ∗
culated for the (2)2 state when compared to the experimental data [18] and confirmed by the vibrational calculations corresponding to the (2)2 −X2 + transition that will be shown in the discussion to come. Hence, we here report for the first time that the (2)2 + state should lie above 25,0 0 0 cm−1 looking forward to experimental comparisons. At internuclear distances greater than 3.75 A˚ , the undulation of the PEC of the (2)2 + state is noticed. It can be justified by either the breakdown of the Born-Oppenheimer approximation at this region or due to significant short-range electronic interactions at certain energy points. These interactions result in singularities in the electronic Hamiltonian operator thus giving rise to the Coulomb cusp in the electronic wavefunction and to the cusps appearance in the exact wavefunction. For the (3)2 + and (4)2 + states, it is noted that investigating their bounding nature depends on the selected basis set for calculation. With the (Hg: ECP78MWB) basis set, these states are bound but they are dissociative when (Hg: ECP60MWB) basis set is employed. This may be attributed to the size of valence basis, which implies a smaller number of core electrons to be represented by the quasi-
relativistic effective core potential thus larger valence wavefunctions. Another factor that happens to have similar effect on the certain studied states is the choice of valence number of electrons. With nine valence electrons considering either ECPs (Hg: ECP78MWB or ECP60MWB), the (2)2 and (2)2 − states are very shallow states hence nearly dissociative. However, upon increasing the number of valence electrons to 11, these states are strongly dissociative. It is also noted that when an ECP with a smaller number of core electrons is employed, the equilibrium distance of the naturally dissociated bound state is slightly greater than that calculated when more electrons are described by the core functions. This can be related to the attractive force exerted by the core electrons on the outermost ones due to increased inner potential. For the quartet states, the double excitations and the occupancy of antibonding molecular orbitals lead to dissociative nature as indicated from the small values of D00 and ωe and the relatively large equilibrium distance. Six sextet states have also been investigated to rule out their influence if any. They are all bound but lying at very high energy; (1)6 + at 151,169 cm−1 , (2)6 + at 157,211 cm−1 , (1)6 at
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Fig. 3. Dipole moment curves (DMC) of the X2 + and (2)2 + states of HgF with their first and second derivatives.
151,606 cm−1 , (2)6 at 157,633 cm−1 , (1)6 at 151,615 cm−1 , and (1)6 − at 151,839 cm−1 . The static dipole moment of the considered states is calculated and plotted as function of the internuclear distance R and given in Fig. 2. Note that this figure and all the other figures in this work are plotted as obtained using (Hg: ECP78MWB, F: augcc-pCVQZ) basis sets with nine valence electrons. With the mercury atom at the origin, the sign convention is defined such as the negative value corresponds to the Hgδ + Fδ − polarity. The DMCs of the X2 + and (2)2 + states will be mainly discussed here to give an example on how these curves can help in indicating the bond nature. The ground state DMC presents negatively increased values with the largest magnitude |μ| = 2.944 a.u. near R = 2.92 A˚ and giving |μe | = 1.57 a.u. at the equilibrium distance Re = 1.913 A˚
Whereas the negative values of the (2)2 + state DMC sharply increase as the internuclear distance increases due to the state’s polarized character at the dissociation limit. One can understand the behavior of a bond from such curves by noticing the behavior of μ, ∂ μ/∂ R, ∂ 2 μ/∂ R2 as function of R (Fig. 3). At large internuclear separations, the X2 + state is purely covalent (μ → 0 and ∂ μ/∂ R → 0) which is an expected behavior at the natural dissociation limit Hg(1 Sg )+F(2 Pu ). Meanwhile, the (2)2 + state is primarily ionic as μ processes large negative values and ∂ μ/∂ R ∼ -2.2 before dissociating into ionic fragments Hg+ (2 S)+F− (1 S). As the internuclear distance decreases, ionic character begins to mix into the ground state and covalent character enters into the (2)2 + state. Near the ˚ the flipequilibrium separation of the excited state Re = 2.878 A, ping and crossing of the ∂ 2 μ/∂ R2 curves indicate that both states are nearly 50-50 mixtures of covalent and ionic character. At this region, the transition dipole moment curve TDMC (Fig. 4) between these two states describes a vertical jump. Since the ionicity increases as ∂ 2 μ/∂ R2 is closer to zero with μ = 0, one can point out that close to the equilibrium separation of the ground state, the ground state bond becomes mostly ionic (∂ 2 μ/∂ R2 ∼ 0.9) and that
of the excited states becomes mostly covalent (∂ 2 μ/∂ R2 ∼ -7.3). The mixed nature of these two states can also be confirmed by calculating the percentage ionic character fionic =μ/eRe [36] which is another parameter that helps to quantify the nature of a molecular bond where 0 ≤ fionic ≤ 1, with f = 1 being the ionic extreme. Fig. 5 shows the different values of ionic character of the states of interest as function of R. The purely ionic or covalent nature is easily illustrated at large internuclear distances where fionic bears large or nearly zero values, respectively. The mixed (50-50) nature (ionic/covalent) of the bond takes place in the region near the equilibrium position of the (2)2 + state where the ionic fraction becomes equal and intersects. Near the equilibrium position of the ground state, it is obvious that the ground state bond is more ionic than the excited state bond where |fionic (X2 + )| > |fionic ((2)2 + )|. By applying the same equation considering the dipole moment at the equilibrium position, the ionic character of the remaining states is calculated and given in Table 2. Moreover, for the ground state, further calculations concerning fionic are done using the Pauling electronegativity values of Hg and F and by applying different equations available in literature [37,38]. The electronic density at each internuclear distance can also be mirrored from the dipole moment curves of the different states as the quantity of charge is calculated using q = μ/R. The abrupt gradient change of the DMCs can be related to the occurrence of an avoided crossing between the PECs of two states of the same symmetry. At this region, the wavefunctions of these states will mix with each other to give two adiabatic solutions such that the corresponding PECs do not cross. DMCs mirror this interaction as a sharp change in their slopes indicating a reversed polarity of the atoms. For the provided PECs, the avoided crossing positions are apparent at 1.77 A˚ between (1)2 -(2)2 with energy separation E = 1756 cm−1 and at 2.31 A˚ between (3)2 + -(4)2 + with energy separation E = 3404 cm−1 . Two crossings take place between (2)2 + and (2)2 at 1.53 A˚ and 1.95 A˚ when the energy is 63,930 cm−1
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Fig. 4. Transition dipole moment curves (TDMC) between the three lowest-lying doublet bound states of HgF.
Fig. 5. The ionic character of the ground and first excited state of HgF as function of the internuclear distance R.
and 39,177 cm−1 , respectively. Such behavior leads to difficulty in accurate descriptions near the vicinity of these positions and to perturbations in the vibrational levels of (2)2 , as well as predissociation of the upper 2 state. Moreover, the TDMCs in Fig. 4 also show multiple 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 as for (2)2 -(2)2 + , and they happen without an appreciable alteration of energy and position of the nuclei due to interfering of the vibrational levels. Then, the electronic motion may suddenly switches from one electronic state to another, and the electronic wavefunctions of these states might mix leading to lack of reliability of theoretical predictions at those positions. Confirmations are thus sought from experimental results as will be
shown later when the Franck–Condon factors aided the comparison of our theoretically calculated vibrational levels corresponding to the transitions between the (2)2 and X2 + states. Furthermore, the quasi-zero value of the TDM between two states at large internuclear distances R as for the (2)2 −X2 + transition can be referred to the spin forbidden transition from Hg (1 Sg ) to Hg (3 Pu ). The molecular orbitals (MOs) of the HgF molecule constructed from the atomic orbitals of Hg and F other than those represented by the ECP78MWB are (1σ ) 2σ 3σ 1π 4σ 2π 5σ 6σ 7σ in which the 1σ MO is fully occupied in the configurations to be discussed below and thus the active space contains 6σ and 2π as detailed previously where 9 valence electrons are to be distributed. Though the covalent configuration of the ground state X2 + (1σ 2 ) 2σ 2 3σ 1 1π 4 4σ 2 is dominant at long distances, the ionic configuration (1σ 2 ) 2σ 2 3σ 2 1π 4 4σ 1 is dominant for shorter internuclear
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71
Table 3 Einstein spontaneous coefficient (A21 ), spontaneous radiative lifetime (τ ), the classical radiativedecay rate of the single-electron oscillator at the emission frequency (γ cl ), and the emission oscillator strength (f21 ) as calculated from the transition dipole moment between the low-lying states of HgF employing MRCI+Q method with ECP78MWB for Hg and aug-cc-pCVQZ for F. Transition
|μ21 | (a.u.)
|μ21 | × 1030 (C.m)
ω21 × 10 − 15 (rad.s−1 )
A21 × 10 − 8 (s−1 )
τ spon (ns)
γ cl × 107 (s−1 )
|f21 |
X2 + -(2)2 +
1.988(a1) 1.862(a2) 1.142(a1) 1.090(a2) 1.224(a1) 1.079(a2) 0.059(a1) 0.076(a2) 0.167(a1) 0.236(a2)
16.857(a1) 15.786(a2) 9.684(a1) 9.241(a2) 10.379(a1) 9.146(a2) 0.497(a1) 0.642(a2) 1.416(a1) 2.003(a2)
4.840(a1) 4.907(a2) 7.333(a1) 7.372(a2) 9.816(a1) 9.786(a2) 2.493(a1) 2.466(a2) 2.482(a1) 2.413(a2)
1.3592(a1) 1.2416(a2) 1.5598(a1) 1.4430(a2) 4.2966(a1) 3.3059(a2) 0.0 0 02(a1) 0.0 0 03(a2) 0.0013(a1) 0.0024(a2)
7.36(a1) 8.05(a2) 6.41(a1) 6.93(a2) 2.33(a1) 3.02(a2) >103 (a1) >103 (a2) >103 (a1) >103 (a2)
14.68(a1) 15.09(a2) 33.70(a1) 34.06(a2) 60.38(a1) 60.01(a2) 3.89(a1) 3.81(a2) 3.86(a1) 3.65(a2)
0.309(a1) 0.274(a2) 0.154(a1) 0.141(a2) 0.237(a1) 0.184(a2) <10−3 (a1) <10−3 (a2) 0.001(a1) 0.002(a2)
X2 + -(2)2 X2 + -(3)2 + (2)2 + -(2)2 (2)2 -(3)2 +
(a1) 9 valence electrons (a2) 11 valence electrons
distances between 1.4 A˚ and 2.8 A˚ corresponding to a charge transfer from Hg to F. As a result of the avoided crossing between X2 + ˚ the ionic/covalent natures are exand (2)2 + states close to 2.8 A, changed. Therefore, for the (2)2 + state, (1σ 2 ) 2σ 2 3σ 1 1π 4 4σ 2 is dominant at short distances and (1σ 2 ) 2σ 2 3σ 2 1π 4 4σ 1 at long distances before ionic fragmentation. For the (1)2 state, the main contributing configuration at long distances is (1σ 2 ) 2σ 2 3σ 2 1π 3 4σ 2 whereas (1σ 2 ) 2σ 2 3σ 2 1π 4 4σ 2π 1 is dominant at distances shorter than 1.77 A˚ where an avoided crossing with the (2)2 state takes place resulting in an exchange of the main configurations between these two states before and after this region. For higher excited states where double excitations take place, e.g., (3)2 state, repulsive potentials can be justified by the occupancy of antibonding MOs. Based on the shape and behavior of the PECs, DMCs and TDMCs of the investigated states, and upon comparing with the other mercury monohalides, the low-lying bound states are represented as potential candidates for molecular transitions. Therefore, we were motivated to calculate the Einstein coefficient of spontaneous emission A21 . From this constant the spontaneous radiative lifetime τ spon , the classical radiative decay rate of the single-electron oscillator at the emission frequency γ cl and the emission oscillator strength f21 are determined by using the transition dipole moment value between two states μ21 at its equilibrium position of the upper state applying the following formulas [39]:
ω21 = 2 π ν21 A21 =
2
τspon =
3 ω21 μ221 3 ε0 hc 3
1 A21
γcl =
2 e2 ω21 6 π ε0 me c 3
f21 =
− A21 3 γcl
where ν 21 is the transition frequency between the two states, h is Plank’s constant, ε 0 is the vacuum permittivity, me is the mass of electron, e is the electronic charge, and c is the velocity of light. The calculated constants are given in Table 3. It is known that strong electronic transitions have A21 values of the order of 108 – 109 s−1 and life times typically 1–10 ns. By referring to these two constants as well as the magnitude of the oscillator strength f21 , one can deduce that certain transitions worth further investigations as for the (2)2 + −X2 + , (2)2 −X2 + , and (3)2 + −X2 + .
Certainly, the radiative lifetime calculated here for (2)2 is expected to be longer than the real (shorter) lifetime due to both avoided crossings that may lead to predissociation and to crossings with another state as discussed above. Attempting to clarify the image of the molecular transitions among the lowest lying states of HgF, vibrational levels of the concerned states with the Franck– Condon factors of these transitions are calculated. A rovibrational calculation have been done for the X2 + , (2)2 + , (3)2 + and (2)2 states obtained from the ab-initio calculation with the basis sets (Hg: ECP78MWB, F: aug-cc-pCVQZ) and nine valence electrons. The eigenvalue Ev , the rotational constant Bv , the centrifugal distortion constant Dv , and the abscissas of the turning points Rmin and Rmax are calculated by using the canonical functions approach [40–42] and considering the reduced mass of 202 Hg and 19 F atoms. Table 4 provides these constants for the X2 + and (2)2 states and the vibrational data of the other states is available in the supplementary material (Table SM.1). At high vibrational levels, the separations between successive levels is expected to approach zero corresponding to the continuous spectrum that indicates the dissociation of the molecule. Errors affecting the reliability of high vibrational levels may truncate due to several factors, e.g., the errors related to the dissociation limit arising from the selected basis sets and active space, the crossing and avoided crossings with higher and lower states … etc. The values of the FCFs describe the overlap of the vibrational wavefunctions for the spontaneous radiative transitions. By using the LEVEL8.2 program [43], the FCFs values are calculated for transitions occurring between the lowest excited states and the ground state. The (1)2 unbound state is assumed not to participate in transitions due to its dissociative nature. The equilibrium position of the PEC of the (2)2 + state is considerably shifted from that of the X2 + state, therefore, higher vibrational levels of the ground state participate in the (2)2 + −X2 + transition. This is confirmed by the highest values of the calculated Franck–Condon factors corresponding to this transition and given in Table SM.2 in the supplementary material. Whereas the equilibrium positions of the (2)2 and (3)2 + states are almost vertical above the ground state so transitions take place between low vibrational levels of both states. Table 5 represents the calculated Franck–Condon factors corresponding to the (2)2 −X2 + transition whose detailed transition energy values among the vibrational levels are the only values available experimentally in literature [18]. Table 5 also gives these values as calculated in this work and as reported experimentally along with the relative difference between them. Excellent agreement is shown. The FCFs of the (3)2 + −X2 + are calculated as well and given in Table SM.3. In addition, calculating FCFs of certain transitions is of another special importance as they shed light on the possibility of direct laser cooling for a
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Table 4 Vibrational levels of the X2 + and (2)2 states of HgF. State
v
Ev (cm − 1 )
Bv × 10 (cm − 1 )
Dv × 107 (cm − 1 )
˚ Rmin (A)
˚ Rmax (A)
X2 +
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
228.57 685.41 1139.68 1589.96 2036.99 2480.38 2920.09 3356.29 3788.73 4217.50 4642.55 5063.81 5481.27 5894.91 6304.69 6710.55 7112.47 7510.40 7904.29 8294.08 8679.70 9061.09 9438.17 9810.84 10,179.02 10,542.62 10,901.49 11,255.51 11,604.51 11,948.35 12,286.84 12,619.77 12,946.90 13,267.98 13,582.72 13,890.79 14,191.82 14,485.41 265.06 792.68 1320.41 1846.25 2369.46 2888.21 3401.00 3907.13 4405.43 4895.25 5376.35 5848.39 6311.01 6764.06 7207.45 7641.04 8064.73 8478.45 8882.11 9275.49 9658.23 10,029.97 10,390.35 10,738.93 11,075.29 11,399.04 11,709.65 12,006.61 12,289.38 12,557.30
2.6434 2.6244 2.6055 2.5871 2.5682 2.5502 2.5318 2.5137 2.4957 2.4776 2.4597 2.4418 2.4239 2.4060 2.3882 2.3703 2.3524 2.3344 2.3164 2.2983 2.2800 2.2617 2.2431 2.2244 2.2055 2.1863 2.1667 2.1468 2.1265 2.1058 2.0845 2.0625 2.0399 2.0164 1.9920 1.9666 1.9399 1.9119 2.7108 2.6779 2.6441 2.6103 2.5769 2.5452 2.5145 2.4852 2.4573 2.4303 2.4037 2.3778 2.3524 2.3269 2.3014 2.2757 2.2496 2.2230 2.1958 2.1678 2.1390 2.1090 2.0777 2.0449 2.0106 1.9746 1.9366 1.8963 1.8535 1.8079
3.5081 3.4474 3.4559 3.3957 3.3913 3.3768 3.3382 3.3468 3.3206 3.3100 3.3068 3.2964 3.2895 3.2888 3.2877 3.2869 3.2899 3.2943 3.3052 3.3174 3.3321 3.3512 3.3775 3.4029 3.4314 3.4753 3.5226 3.5783 3.6369 3.7095 3.7978 3.8957 4.0068 4.1406 4.2952 4.4759 4.6825 4.9193 2.8090 2.5983 2.4481 2.3148 2.2753 2.2842 2.2958 2.3805 2.4655 2.5283 2.6191 2.7300 2.8278 2.9151 3.0139 3.1131 3.2018 3.2963 3.4227 3.5841 3.7592 3.9440 4.1554 4.3919 4.6493 4.9601 5.3061 5.7028 6.1885 6.7521
1.852 1.811 1.784 1.764 1.747 1.732 1.719 1.707 1.697 1.687 1.678 1.670 1.662 1.654 1.648 1.641 1.629 1.625 1.623 1.618 1.612 1.608 1.603 1.598 1.593 1.590 1.586 1.582 1.578 1.574 1.571 1.567 1.564 1.561 1.558 1.555 1.552 1.550 1.831 1.796 1.776 1.761 1.749 1.739 1.730 1.723 1.715 1.709 1.702 1.696 1.690 1.685 1.680 1.674 1.670 1.665 1.661 1.656 1.652 1.648 1.645 1.641 1.638 1.635 1.632 1.629 1.626 1.624
1.982 2.037 2.077 2.111 2.142 2.170 2.197 2.223 2.247 2.271 2.294 2.317 2.339 2.361 2.383 2.404 2.425 2.446 2.467 2.488 2.508 2.529 2.550 2.570 2.591 2.612 2.633 2.654 2.676 2.698 2.720 2.742 2.765 2.789 2.813 2.838 2.863 2.890 1.953 2.007 2.047 2.082 2.114 2.144 2.173 2.200 2.227 2.253 2.279 2.304 2.330 2.355 2.380 2.405 2.430 2.455 2.481 2.507 2.534 2.561 2.589 2.619 2.649 2.681 2.715 2.750 2.787 2.826
(2)2
S. Elmoussaoui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 201 (2017) 64–74
73
Table 5 Franck-Condon factors for the (2)2 (v ) ↔ X2 + (v ) system transition of HgF using the PECs obtained from MRCI+Q calculations employing Hg: ECP78MWB, F: aug-cc-pCVQZ basis sets with 9 valence electrons. v \ v
0
1
2
3
4
5
6
7
8
9
0
6.29[−2]
3.48[−3]
3.42[−7]
7.88[−6]
2.83[−6]
1.09[−6]
4.65[−7]
1.99[−7]
8.46[−8]
1
9.34[−1] 38,990(a) 39,053(b) 0.16%(c) 6.51[−2]
9.84[−2]
8.89[−3]
8.24[−6]
4.75[−5]
2.07[−5]
8.71[−6]
3.94[−6]
1.77[−6]
2
9.52[−4]
8.27[−1] 39,060(a) 39,064(b) 0.01%(c) 1.07[−1] 38,606(a) 38,585(b)
7.63[−1]
1.14[−1]
1.52[−2]
1.08[−4]
1.68[−4]
8.32[−5]
3.78[−5]
1.79[−5]
2.19[−2]
4.82[−4]
4.44[−4]
2.42[−4]
1.17[−4]
2.85[−2]
1.38[−3]
9.70[−4]
5.68[−4]
9.45[−2]
3.45[−2]
3.03[−3]
1.80[−3]
8.02[−2]
3.94[−2]
5.59[−3]
7.50[−1] 39,504(a) 39,143(b) 0.92%(c) 7.89[−2]
6.58[−2]
4.27[−2]
5.25[−2]
1.22[−2]
7.69[−1] 39,570(a) 39,161(b) 1.04%(c) 5.28[−2]
3
1.17[−4]
0.05%(c) 1.49[−3]
4
1.11[−4]
6.34[−4]
39,134(a) 39,078(b) 0.14%(c) 1.30[−1] 38,683(a) 38,602(b) 0.21%(c) 1.18[−3]
5
3.10[−5]
5.13[−4]
1.96[−3]
39,660(a) 39,573(b) 0.22%(c) 7.30[−1] 39,209(a) 39,093(b) 0.30%(c) 1.39[−1] 38,762(a) 38,621(b) 0.37%(c) 3.82[−4]
6
5.32[−6]
1.39[−4]
1.36[−3]
4.51[−3]
1.15[−1] 39,733(a) 39,587(b) 0.37%(c) 7.20[−1] 39,286(a) 39,107(b) 0.46%(c) 1.36[−1] 38,842(a) 38,638(b) 0.53%(c) 1.73[−5]
7
4.83[−7]
2.09[−5]
3.41[−4]
2.70[−3]
8.56[−3]
1.07[−1] 39,804(a) 39,598(b) 0.52%(c) 7.25[−1] 39,361(a) 39,116(b) 0.63%(c) 1.24[−1] 38,921(a) 38,648(b) 0.71%(c) 1.23[−3]
8
1.14[−10]
1.03[−6]
4.06[−5]
5.97[−4]
4.40[−3]
1.41[−2]
7.38[−1] 39,434(a) 39,133(b) 0.77%(c) 1.04[−1] 38,998(a) 38,661(b) 0.87%(c) 5.07[−3]
9
4.86[−8]
1.64[−7]
3.68[−7]
4.88[−5]
8.23[−4]
6.22[−3]
2.08[−2]
7.79[−1] 39,631(a) 39,175(b) 1.16%(c)
(a)Ecal : Calculated transition energy (b)Eexp : Experimental transition energy [18] (c)Relative difference %|Eexp − Ecal |/Eexp
molecule. In this endeavor, we shall refer to the main criterion for direct laser cooling process which is that FCFs should be highly diagonal indicating a limited number of lasers required to keep that molecule in a closed-loop cycle. For the HgF molecule, the (2)2 + −X2 + transition has small FCF values where the highest among them are obviously non-diagonal (Table SM.2).Whereas the (2)2 −X2 + and (3)2 + −X2 + transitions do have large diagonal FCF values, but they also have non-negligible off-diagonal values (Tables 5 and SM.3), i.e., the off-diagonal values are greater than 2%. For this reason along with the intervening electronic states, e.g., (1)2 and (2)2 + , to which the upper state could radiate and terminate these cycling transitions, the direct laser cooling for HgF compound is considered practically difficult. 4. Conclusion In this work, the electronic structure of 18 low-lying doublet and quartet electronic states of the mercury monofluoride HgF molecule has been analyzed from ab-initio calculations as implemented in the MRCI+Q method with 9 and 11 valence electrons and using different number of core electrons for mercury. The obtained PECs and DMCs are plotted from which the spectroscopic constants Re , ωe , ωe xe , Be , α e , Te , μe and D00 are then calculated for
the bound states. In general, good accordance have been shown upon comparison with the data available in literature. As the excited state (2)2 + is peculiarly neglected in literature, a deductive reasoning based on comparisons with the diatomic monohalides of other group IIB elements led us to predict that this state should lie above 25,0 0 0 cm−1 . The vibrational study and Franck–Condon factors calculation carried out in this work have supported this conclusion looking forward to experimental confirmation. Moreover, an explicit study has been conducted to analyze the bond nature of the two lowest bound states, X2 + and (2)2 + , during which the electric dipole moment μ, its first and second derivatives, ∂ μ/∂ R and ∂ 2 μ/∂ R2 , the transition dipole moment μTDM and the percentage ionic character fionic are related. Those two states have been confirmed to show a mixed nature (ionic/covalent) as function of the internuclear distance and thus predictions are made concerning their relative molecular system transition. Further investigations have been motivated concerning possible strong transitions among the low-lying states of the molecule. Therefore, Einstein spontaneous coefficients A21 and the spontaneous radiative lifetime τ spon are determined from the transition dipole moment value between two states μ21 at its equilibrium position of the upper state. For those states, the vibrational constants Ev , Bv , Dv , Rmin and Rmax are reported. Realizing the importance of the Franck–Condon fac-
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